Uni-directional transport in billiard chains

University of Ljubljana Faculty of mathematics and physics Physics department

Martin Horvat

Uni-directional transport in billiard chains Doctoral thesis

ADVISER: Assoc. Prof. Dr. Tomaˇz Prosen

Ljubljana, 2006

Univerza v Ljubljani Fakulteta za matematiko in fiziko Oddelek za fiziko

Martin Horvat

Enosmerni transport v biljardnih verigah Doktorska disertacija

MENTOR: izr. prof. dr. Tomaˇz Prosen

Ljubljana, 2006

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Abstract A spatially extended two-dimensional billiard system named serpent billiard is studied in quantum and classical picture. It is composed of bends and straight wave-guide segments forming a channel (billiard chain) with smooth and parallel walls. The billiard possesses a property called the uni-directional transport by which the classical particles traversing in one direction do not change its direction of travel. The classical phase-space separates into two disjointed components corresponding to the left and right unidirectional motion. The classical billiard is discussed in terms of jump models, where dynamics over a basic cell is decomposed into a Poincaré map and a time function measuring the travelling time across a basic cell. The phase space of the Poincaré map is mixed, but for suitable parameters the phase space is almost chaotic. The dynamical properties of the periodic serpent billiard are numerically examined by calculating time correlation functions, Lyapunov exponents and diffusion of particles along the chain. As the result of the singularity of the time function the transport is marginally normal after subtracting the average drift, which is supported by analytical argument. The quantum serpent chain and more general linear chain of scatterers are discussed in the scattering matrix formalism. The uni-directionality is lost in the quantum billiard, due to tunnelling between classical invariant components resulting in small reflection. The transport properties of the chains are studied in the process of lengthening, called dynamical approach, where the effect of (partial) localisation i.e. decay of transmission along the chain, is studied in more detail. It was found that in general (partial) localisation in chains is quite common, but not in serpent billiards. Consequently, in the random serpent billiard the reflection increases up to some length linearly. The behaviour of transmission with increasing length was studied analytically in the general one-dimensional case, whereas in other cases only numerically. The transmission seems in average to decrease faster in chains with higher dimensional scatterers. The classical uni-directionality has an deep impact on the quantum-mechanical properties of serpent billiard. Many of this properties are inherited from the bends, which are accounted to break integrability of the serpent billiards. One of this properties is the similar quantum reflection, which is studied in bend as the function of the energy and geometrical parameters. It was found to be usually several orders smaller then transmission, but at specific wave-numbers strongly increases up to the order of the transmission. These events called reflection resonances are examined and explained using a one-dimensional scattering model. In the same spirit also numerical results for the Wigner-Smith time are explained. The statistics of the scattering matrix indicates that longer periodic serpent billiards are similar to random scatterer in which reflection is much smaller than transmission. The energy levels of a closed (compactified) serpent billiard are repulsed but paired together due to the classical uni-directionality. The pairs obey the Wigner level statistics. The spacing between the levels in the pair is connected to the reflection and the Wigner-Smith time in the billiard. In addition, the correspondence between the classical and quantum scattering is examined on the classical surface-of-section (SOS) as the chain is increased. The correspondence is quantified by the overlap between the classical probability density and the corresponding Wigner function, called the quantum-classical fidelity (QCF). In strongly chaotic cases the QCF decays exponentially as the chain is lengthened, with the rate equal to the maximal Lyapunov exponent.

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PACS numbers: 3.65.Sq Semi-classical theories and applications 03.65.Yz Decoherence; open systems; quantum statistical methods 05.40.Fb Random walks and Levy flights 05.45.-a Nonlinear dynamics and nonlinear dynamical systems 05.45.Mt Quantum chaos; semi-classical methods 05.45.Pq Numerical simulations of chaotic systems 05.60.Cd Classical transport 05.60.Gg Quantum transport 46.40.Cd Mechanical wave propagation (including diffraction, scattering, and dispersion) 72.10.-d Theory of electronic transport; scattering mechanisms

Keywords: quantum chaos, billiard chains, extended systems, wave-guide, unidirectional transport, two-dimensional channels, quantum-classical correspondence, level statistics, transport, scattering

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Povzetek V tem delu smo ˇstudirali odprt in prostorsko neomejen biljardni sistem imenovan serpentinast biljard v kvantni in klasiˇcni sliki. To je biljardna veriga oz. valovod z gladkimi in vzporednimi stenami sestavljen iz zavojev in ravnih delov. Klasiˇcni delec, ki potuje v neko smer vzdolˇz biljarda, ne more spremeniti smeri potovanja. To lastnost imenujemo enosmerni transport. Klasiˇcni fazni prostor tako razpade na dve disjunktni invariantni komponenti, od katerih pripada ena gibanju delca v levo, druga pa v desno stran verige. Dinamika vdolˇz kanala je razstavljena na preslikavo skakanja (Poincaréjeva preslikava) med dvema koncema osnovnih celic in funkcijo ˇcasa, ki meri ˇcas potovanja ˇcez osnovno celico v odvisnosti od vstopne toˇcke. Preslikava skakanja ima meˇsan fazni prostor z velikostjo kaotiˇcne komponente odvisno od ˇsirine kanala. Za doloˇcene ˇsirine je preslikava skoraj popolnoma kaotiˇcna. Numeriˇcno smo ˇstudirali tudi eksponente Ljapunova, autokorelacijske funkcije in difuzijo delcev vzdolˇz verige. Zaradi singularnosti v ˇcasovni funkciji ugotovimo, da je difuzija pri odˇsteti povpreˇcni translaciji marginalno normalna. Slednje potrjujemo tudi z analitiˇcnimi argumenti. Kvantne serpentinaste biljarde in bolj sploˇsne verige obravnavamo v okviru sipalnega formalizma. V kvantni sliki serpetinasti biljardi zgubijo enosmernost transporta zaradi tuneliranja med invariantnimi komponentami, ki ima za posledico neniˇcelno refleksijo. Transportne lastnosti skozi sistem analiziramo v procesu daljˇsanja, kar imenujemo dinamiˇcni pristop, pri ˇcemer se posvetimo pojavu (delne) lokalizacije t.j. eksponentno upadanje transmisije z dolˇzino verige. Ugotovimo, da je lokalizacija v sploˇsnem precej pogosta, vendar ne v naˇsih serpentinastih biljardih. Poslediˇcno, v nakljuˇcnih serpentinastih biljardih refleksija naraˇsˇca poˇcasi – linearno – z dolˇzino verige. V sploˇsnem je upadanje transmisije hitrejˇse v verigah sipalcev veˇcjih dimenzij. Klasiˇcni enosmerni transport ima globok vpliv na kvantne lastnosti serpentinastih biljadov. Veliko teh lastnosti je podedovano od zavojev, ki so zasluˇzni za zlom integrabilnosti v serpetinastih biljardih. Ena od teh lastnostih je podobnost v refleksiji, ki je podrobno ˇstudirana v zavoju kot funkcija valovne dolˇzine in geometriˇcnih parametrov. Izkaˇze se, da je reflekcija obiˇcajno nekaj redov manjˇsa kot transmisija, razen pri specifiˇcnih valovnih dolˇzinah, kjer moˇcno naraste, in jih imenujemo rekleksijske resonance. Slednje razloˇzimo s pomoˇcjo enodimenzionalnega sipalca. S podobnim pristopom tudi razloˇcimo singularno obnaˇsanje ˇcasovnega zamika Wignerja in Smitha. Z uporabo statistike sipalne matrike ugotovimo, da imajo daljˇsi serpentinasti biljardi doloˇceno podobnost z nakljuˇcnimi sipalci, ki imajo skoraj zanemerljivo refleksijo glede na transmisijo. Prav tako ˇstudiramo lastnosti spektrov zaprtih serpentinastih biljardov. Energijski nivoji se med seboj odbijajo a so grupirani v pare, kot posledica klasiˇcnega enosmernega transporta. Razdalje med pari so podrejene Wignerjevi statistiki, ki jo tipiˇcno najdemo v kaotiˇcnih sistemih. Razdalja med nivoji v paru je v relaciji z refleksijo in ˇcasovnim zamikom Wignerja in Smitha. Dodatno smo si ogledali korespondenco med klasiˇcnim in kvantnim sipanjem na klasiˇcni preseˇcni ploskvi v procesu podaljˇsevanja periodiˇcne verige. Korespondenco kvantitativno ocenimo s prekrivalnim integralom med klasiˇcno verjetnostno porazdelitvijo in pripadajoˇco Wignerjevo funkcijo kvantnega sistema, ki ga imenujemo kvantno-klasiˇcna zvestoba. V moˇcno kaotiˇcnih primerih zvestoba eksponentno upada z dolˇzino verige, pri ˇcemer je hitrost upadanja podana z maksimalnim koeficientom Ljapunova.

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PACS ˇ stevila: 3.65.Sq Semiklasiˇcna teorija in aplikacije 03.65.Yz Dekoherenca; odprti sistemi; kvantne statistiˇcne metode 05.40.Fb Nakljuˇcna hoja in Lévyjevi poleti 05.45.-a Nelinearna dinamika in nelinearni dinamiˇcni sistemi 05.45.Mt Kvantni kaos; semi-klasiˇcne metode 05.45.Pq Numeriˇcne simulacije kaotiˇcnih sistemov 05.60.Cd Klasiˇcni transport 05.60.Gg Kvantni transport 46.40.Cd Propagacija valov vkljuˇcno z uklonom, sipanjem in disperzijo 72.10.-d Teorija elektronskega transporta; sipalni formalizem

Kljuˇ cne besede: kvantni kaos, verige biljardov, neomejeni sistemi, valovodi, enosmerni transport, dvodimenzionalni sistemi, kvantno-klasiˇcna korespondenca, statistika energijskih nivojev, transport, sipanje

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Acknowledgement I would like to thank Tomaˇz Prosen for his father-like support and guidance during the years I worked in his group. I could not wish for a better intuitive, smart and supportive adviser. These were fascinating years, full of new ideas and discoveries that lead to the presented thesis. I would also like to thank Mirko Degli Esposti for his warm hospitality during the stay in Bologna (Italy) and interesting discussions. In particular, I would like to thank for the careful and critical reading of the manuscript and pointing out the misleading statements. The last, but not the least, I would like to thank may colˇ leges Marko Znidariˇ c, Gregor Veble and Iztok Piˇzorn for all the help and making a fruitful scientific environment. In particular Gregor has greatly contributed to the quality of the dissertation.

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Zahvala Najlepˇse se zahvaljujem Tomaˇzu Prosenu za njegovo skoraj oˇcetovsko podporo in vodstvo med leti dela v njegovi skupini. Teˇzko si bi ˇzelel boljˇsega mentorja. To so bila zanimiva leta, polna novih idej in odkritij, ki so vodila v predstavljeno disertacijo. Zahvalil se bi tudi Mirku Degli Espostiju za prijetno bivanje ˇ posebno se bi rad v Boloniji (Italija) in zanimive diskusije. Se zahvalil za pozorno branje besedila disertacije in opozorilom na doloˇcene okorno postavljene trditve. Na koncu, vendar niˇc manj pomembno, bi se rad zahvalil svoˇ jim kolegom Marku Znidariˇ cu, Gregorju Vebletu in Iztoku Piˇzornu za vso pomoˇc in prijetno delovno ozraˇcje. Posebno Gregor je veliko pripomogel h kvaliteti disertacije.

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Dedication I dedicate this dissertation to my mother Milena and late father Ladislav Horvat.

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Posvetilo Disertacijo posveˇcam svoji mami Mileni in pokojnemu oˇcetu Ladislavu Horvatu.

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Contents 1 Introduction 1.1 From classical mechanics to quantum chaos . . . . . . . . . . . . . . . . . . . . . 1.2 Billiards and extended systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 4

2 Classical serpent billiard 2.1 Dynamics in the serpent billiard . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The transport properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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3 General linear chains of quantum scatterers 3.1 Chain of one-dimensional scatterers . . . . . . 3.1.1 Static generating scattering matrix . . 3.1.2 Noisy generating matrix . . . . . . . . 3.2 Chain of d-dimensional scatterers . . . . . . .

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4 Scattering across a single bend 4.1 The scattering matrix of a bend . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Numerically stable scheme for scattering matrix calculation . . . . . . . . . . . . 4.3 Quantum transport across the bend . . . . . . . . . . . . . . . . . . . . . . . . .

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5 Quantum serpent billiards 5.1 Transport properties . . . . . . . . . . . . . . . . . . . 5.1.1 The periodic serpent billiards . . . . . . . . . . 5.1.2 The random serpent billiards . . . . . . . . . . 5.2 Statistical properties of scattering . . . . . . . . . . . . 5.3 Spectrum of the compactified serpent billiard . . . . . 5.4 Quantum classical correspondence on the classical SOS

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6 Conclusions

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Bibliography

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Appendices

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A The jump models of elements in the serpent billiards

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B Concatenating scattering matrices

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C The classical probability scattering matrix

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xiv D The D.1 D.2 D.3 D.4

cross-product of Bessel functions The properties of mode numbers . . . . . . . . . . . . . . . . . . Numerical evaluation of mode functions in a bend . . . . . . . . The overlap of mode-functions in different geometries . . . . . . . The number of modes in the straight and the bended wave-guide

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E Perturbative calculation of the bend’s scattering matrix

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F Wave-functions in the coordinate space

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G The Gaussian packet and the Wigner function in an ∞-well

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Daljˇ si i ii iii

slovenski povzetek 137 Klasiˇcni serpentinast biljard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Sploˇsne kvantne verige sipalcev . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 Serpentinast biljard v kvantni sliki . . . . . . . . . . . . . . . . . . . . . . . . . . 143

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Chapter 1

Introduction 1.1

From classical mechanics to quantum chaos

The modern understanding of deterministic dynamics that describes isolated systems in the nature have two basic threads of development. These are classical mechanics and quantum mechanics. The first was initiated by Newton, I. (1643 - 1727) and generalised by Lagrange, J. L (1736 - 1813) and Hamilton W. R. (1805 - 1865). This development paved the way for the birth of the Hamiltonian chaos introduced by Poincaré, J.H. (1854 - 1912) in a specific stellar three body problem. Soon the chaotic dynamics characterised by sensitivity on initial condition was discovered in many systems and the theory evolved under contributions of people such as Birkhoff, G.D. (1911 - 1996), Kolmogorov, A.N. (1903 - 1987), Cartwright, M.L. (1900 - 1998), Littlewood, J.E. (1885 - 1977), Smale, S. (1930 -) and Arnold V.I. (1937-) just to mention a few. Excellent references of there achievements are e.g. (Abraham and Marsden, 1978; Arnold and Avez, 1989; Goldstein, 1980). Their work led to the establishment of the Hamiltonian theory of chaos (dynamical systems), where we examine dynamical properties of Hamiltonian (conservative) systems and commonly classify systems in into chaotic, mixed and regular systems (Ott, 1993; Katok and Hasselblatt, 1997). Roughly speaking, in the chaotic systems all the neighbouring trajectories exponentially diverge and therefore the dynamics is practically unpredictable for long times, in regular systems we have an opposite situation in which trajectories do not diverge and usually we have some integrals of motions. The mixed systems posses both regular and chaotic dynamics in the phase space. They are the most common ones and therefore most practically interesting, but also difficult to analyse and to obtain some concrete general results. The term chaos often means hyperbolicity which guarantees the decomposition of the tangent space at each phase space point into expansion and contraction subspaces. A systematic study of hyperbolic systems from the geometric theory perspective was initiated by Smale (Smale, 1967). In passing years from the initial discovery interest in chaotic systems only grew as we found that processes that we encounter in nature are at least partially chaotic and bring deterministic stohasticity on the long time scale. The other thread, quantum mechanics (Messiah, 2000; Griffiths, 2004; Sakurai, 1994) has its birth in the early 20th century with the discussion of photons by Planck, M. (1858 -1947) and Einstein, A. (1879 - 1955) and basic quantisation principles by Bohr, N. ( 1885 - 1962) followed by a more general description by Schrödinger, A. (1887 - 1961) and Heisenberg, W. K. (1901 - 1976). Each of them gave its own formulation but with the same content, which Dirac, P. (1902 - 1984) unified and generalized. This development resulted in the Copenhagen interpretation of Quantum mechanics (Audi, 1973; Jammer, 1966) in 1927 that is commonly accepted in the physics community today. The latter explains how one should formulate the quantum mechanical description and understand its results, in particular the process of measurement

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Chapter 1. Introduction

(Wheeler and Zurek, 1983). The mathematical description of Quantum mechanics based on operator theory over Hilbert spaces used today was developed by Dirac, P. and von Neumann, J. (1903-1957). The quantum mechanics has given many satisfactory answers and raised many more questions. Some of them are still an object of discussion and in particular interesting here is the question of classical-quantum correspondence basically asking : How does the classical world come out of the quantum mechanical description. From the Ehrenfest theorem (Messiah, 2000) and correspondence of observables given in the Copenhagen interpretation the quantum mechanics yields in the limit of high energies (or small effective Planck constant) the laws of classical mechanics. But the question remains how does it converge and how do classical properties influence the quantum mechanical description. The latter question raised research in many areas of physics and mathematics connected to the quantum physics. The study of this question in single and many-particle classically chaotic systems is particularly interesting and is today a well established research area called quantum chaos (chaology) (Casati et al., 1995). In the latter we search for connections between quantum and classical behaviour in complex dynamics (usually chaotic) and different signatures of classical chaos in quantum systems (Haake, 2001; Reichl, 1992). The most famous signature is the influence of the classical dynamics onto the quantum spectral statistics, but there are many other interesting discoveries, see in e.g. (Reichl, 2004). The quantum mechanics of classically chaotic systems in the matrix formulation shows many similarities with purely random processes, connecting it to the random matrix theory (RMT) (Mehta, 1991), which is not perfectly understood by now. A classical system is described in phase space and can be represented by the probability density function. By using the phase-space representation (Lee, 1995) of the corresponding quantum system, we can compare the quantum and the classical system more directly in terms of phase-space functions. There exist many phasespace representations, but the most commonly used are the Wigner and the Hussimi functions (Takahashi, 1989). These representations are important tools to establish connections between the classical and corresponding quantum system and have been used in many useful results e.g concerning semi-classical eigen-states of regular and chaotic systems (Berry, 1977b; Voros, 1976, 1977) and consequently about the spectral statistics in mixed systems (Berry and Robnik, 1984). The research in the field of quantum chaos is still continuing, bringing into light new connections between the two different descriptions of a particular system, and the presented thesis is a part of it.

1.2

Billiards and extended systems

The billiard systems in two and three-dimensions have been used as a study case of dynamics for a very long time. They can be remarkably simple in geometrical and computational sense both in the classical and quantum descriptions, but can produce a broad band of possible types of dynamics from regular, mixed to purely chaotic regime of motion. There are few very famous cases which have been proved to be fully chaotic (uniformly hyperbolic) and ergodic i.e. the Sinai billiard (Sinai, 1970) and the Bunimovich stadium billiard (Bunimovich, 1974). Until the introduction of the latter by Bunimovich L., chaotic billiards were thought to need a non-convex boundaries e.g. disk in the Sinai billiard (Lorentz gas), to produce the exponential divergence of orbits. These were called ”dispersion“ billiards, where neighbouring parallel orbits diverge when they collide with dispersing components. Bunimovich showed that, by considering the orbits beyond the focusing point of a convex border, it was possible to obtain exponential divergence of trajectories. In these chaotic focusing billiards, neighbouring parallel orbits converge at first, but divergence prevails over convergence on average. If the billiard is integrable (e.g. circle, ellipse) the divergence and the convergence of trajectories are balanced. More recently, Bunimovich merged the two approaches to chaos in billiard systems and unified the theory in (Bunimovich, 2003). By making a hole into a billiard through which trajectories can escape into infinity we

1.2. Billiards and extended systems

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obtain an extended (open) system. The paradigmatic case of open systems are billiards connected to different number of infinite wave-guides. In the quantum discussion these objects are sometimes called quantum dots or micro-cavities. These objects are not compact and therefore have to be considered as scatterers (Newton, 2002). It has been shown theoretically and experimentally that the transport properties through the chaotic cavity posses some universal properties known as Ericson fluctuations (Stöckmann, 1999). There are also many others universal features that can be modelled using RMT (Beenakker, 1997; Reichl, 2004). The cavities with two open ports connected together forming a structure of linear topology represent another interesting object referred to as billiard chains or chains of cavities (quantum dots) or more general as chains of scatterers. The general chains of scatterers represents a useful model for macroscopic structures like multi-layered structures, real-life wires, nano-tubes etc. This is the reason that chains attracted a great scientific attention in the past and still do. The work on this subject stretches from the early 80s to the present. The past studies mainly focused on the chain of randomly chosen scatterers and its average properties. Very good reviews on this work are (Erdös and Herndon, 1982; Beenakker and van Houten, 1991; Nakamura, 1997) and (Kramer and MacKinnon, 1993), where the main approach of analysis is the transfer matrix formalism (Newton, 2002). Other interesting articles discussing the random or at least noisy linear chain of scatterers using the same approach are (Cahay and Datta, 1988; Abrahams and Stephen, 1980; Andereck and Abrahams, 1980; Kirkman and Pendry, 1984), where we would like to expose the more detailed work (Langley, 1996). There were also some attempts to understand the chains using purely scattering matrices, but unfortunately they have only partially used the advantages of this approach. The most interesting articles on this subject are (Anderson et al., 1980; Anderson, 1982) and (Ko and Inkson, 1988). There are again focusing on average properties of random chains. More recently there were some interesting studies using scattering formalism. The most relevant here is the stability analysis of scattering matrix merging procedures (Mayer and Vigneron, 1999), and the random walk in chains by (Cwilich, 2002). The billiard chains are more specific spatially extended systems than the general chain of scatterers. The periodic or disorder case of the latter is a promising field of research with a variety of direct applications, e.g. in nano-physics, fiber optics, electromagnetic cavities, etc. It is fair to say that studies of these have been under-represented as compared to a vast amount of work which has been dedicated to closed billiards. Nevertheless, one has to mention several basic results in these types of systems. First, one can study the escape rates from finite portions of an infinite billiard chain, like in the Lorentz channel (Gaspard, 1998). Second, one can study classical transport properties, such as diffusion and transport of heat along the billiard chains in order to understand the dynamical (microscopic) origin of the macroscopic transport laws (Alonso et al., 1999, 2002; Li et al., 2003). Third, one can study the relation between deterministic diffusion in a classical billiard chain, Anderson-like dynamical localisation in the corresponding quantum chain, and the nature of its spectral fluctuations (Dittrich and Doron, 1994; Dittrich et al., 1997, 1998). The diffusion in macroscopic systems can be separated into three classes: normal, anomalous, ballistic and marginal normal (Metzler and Klafter, 200). And fourth, there is an interesting effect of localisation transition in the presence of correlated disorder, which has been studied in the case of billiard chains both theoretically (Izrailev, 2003) and experimentally (Kuhl et al., 2000). Generally we expect normal diffusion in chaotic channels and ballistic in regular channels. In polygonal channels, we can have normal or anomalous diffusion, as reported in (Alonso et al., 2002; Sanders and Larralde, 2006). The detection of marginal normal diffusion is difficult and therefore is rarely reported. In this thesis we are interested in a billiard chain composed of bends and straight wave-guide segments with parallel walls named the serpent billiard. This type of billiard possesses an unique property of unidirectional transport, meaning that the particles traversing along the chain can-

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not change their direction of travel. The walls of the billiard are smooth. We can think of the serpent billiard as a straight wave-guide with additional bends that break the integrability of the straight wave-guide. Therefore it is essential to study the bends in order to understand the behaviour of the serpent billiard. The bends themselves are interesting quantum objects. The wave propagation in bended wave-guides has a long and rich history of research that dates back to Lord Rayleigh (Strutt, 1897) and continues to the present day. Initially bends were investigated from the view point of electromagnetic wave theory, but more recently also quantum mechanical aspects attracted a lot of attention. A large number of contributions in this field of research is mainly due to fact that the bends are typical elements incorporated in the designs of the wave-guides and acquiring their properties to sufficiently high precision seems to be problematic in the regimes of high energies and high curvatures even today. Contributions to the research of quantum particles in bends can be separated into two branches. These are the studies of bound states, their existence (Jensen and Koppe, 1971; Exner and Seba, 1989; Exner, 1992) and spectra (Lin and Jaffe, 1996), and the scattering properties, both reviewed in (Londergan and Carini, 1999). In order to describe the quantum particles over our open billiard several approaches have been used in the past: the Green function approach (Spivack et al., 2002), the finite difference mesh calculations (Lent, 1990) and the mode-matching techniques (MMT) using natural modes, eigen-functions of the Laplace operator, for the bend (Cocran and Picina, 1966; Accation and Bertin, 1990; Sols and Macucci, 1990; Sprung and Wu, 1992; Lin and Jaffe, 1996; Rashid and Kodama, 2002) and other bases (Amari and J., 2000). The work (Lin and Jaffe, 1996) is especially interesting as it raises the question how to stabilise the calculations and gives a MMT method that is stable, but unfortunately a bit ambiguous. The MMT based on natural modes is also called modal approach and is used in our discussion. The modal approach looks the most promising to deal with bends, because of its simplicity, power of interpretation and good precision of results, but it hides some problems that are examined in the thesis. Using the analogy between the quantum theory and the EM theory we can connect our work on the serpent billiard in the quantum picture with the EM wave propagation of longitudinal magnetic waves (Cocran and Picina, 1966). We must view this doctoral thesis as an introduction to quantum and classical analysis of a certain type of spatially extended systems, as to clarify all the properties in detail one would need a lifetime.

1.3

Outline

In Chapter 2 we introduce the geometrical structure of the serpent billiard and discuss its classical properties in the framework of jump models. We concentrate on the periodic serpent billiard. In order to understand the dynamics along the billiard we analyse first the dynamical properties of a single basic cell. The conclusions are then used in the study of transport properties, where we give some heuristic calculations to explain numerical results. The material in this section was published in (Horvat and Prosen, 2004). The general chains of quantum scatterers are discussed in the Chapter 3 and presented in (Horvat and Prosen, 2005). We focus mainly on the transport properties of the chain in the process of lengthening. This gives us a basic idea of what to expect in our quantum serpent billiard. In the Chapter 4 we introduce the quantum scattering formalism and apply it to the analysis of a single bend on the straight wave-guide as reported in (Horvat and Prosen, 2006). We introduce a stable method for calculation of the scattering matrix of a bend in the modal approach. The latter was possible by using the extensive knowledge about the modal structure of the Laplace operator, given in the Appendix. In the bend we are interested in the transport properties – reflection and Wigner-Smith delay times as a function of wave-number and curvature. In Chapter 5 we continue the discussion with the study of the quantum serpent billiard in the scattering formalism. The formalism is introduced in a bit more general form to simplify our work. The study can be separated into

1.3. Outline

5

three parts. In the first we concentrate on the transport properties of the billiard chain as a function of the wave-number and its length. The second is devoted to the understanding of the energy spectrum of the closed (compactified) serpent billiard and searching for the signatures of the classical uni-directionality. In the last, third, part we examine the quantum-classical correspondence (QCC) over the classical SOS in the on-shell scattering. We compare the evolution of the classical density and the corresponding Wigner function on the classical SOS as the chain is lengthened. These results correspond to those published in (Horvat et al., 2006), where we discuss QCC in classically chaotic systems over compact phase-space in the Weyl-Wigner formalism. The behaviour of the Wigner function in chaotic system with compact phase-space is separately discussed in (Horvat and Prosen, 2003a,b). The conclusions of the thesis follow in the Chapter 6.

6


7

Chapter 2

Classical serpent billiard We are discussing a class of classical billiard channels with an unusual and distinct dynamical property that a particle traversing in one direction of the billiard can not change its direction of motion, called uni-directionality. Specifically, we focus on billiard chains with parallel walls composed of straight and bended sections of specific curvature named serpent billiards. The general serpent billiard is composed of two types of basic elements: bends of inner radius r = q and outer radius r = 1 and straight wave-guide segment of length d and width a = 1 − q. The walls are parallel to each other and the cross-sections perpendicular to walls are of constant width a. We construct the serpent billiard so that the bend is connected to a straight wave-guide and this is connected to another bend, but with inverted direction of bending from the previous one. This procedure is then repeated to form arbitrarily long chains. In figure 2.1 we show two examples of such billiards: periodic and custom-built. The periodic serpent billiard are billiards with the smallest possible period of two basic cells and the basic cell is of a fixed form. One basic cell is composed of a bend of an angle β and of a straight wave-guide of a length d. We refer to parameters q, β and d as configuration of the basic cell or of the periodic serpent billiard. The custom-built serpent billiards are chains with form parameters of basic cells chosen arbitrarily e.g. at random or to form a chain with a longer period as the basic chain. The developed theory is applicable to custom-built as well to periodic serpent billiards, but the numerical results are shown for simplicity sake only for the periodic case.

n=3

n=4 d

n=1

r=1 q r=

β β

d

n=2

(a)

(b)

Figure 2.1: An example of a periodic billiard with the configuration β = 2π/3, d = 1/2 (a) and a custom-built serpent billiard (b) at inner radius q = 0.6.

We note that the property of uni-directionality can be proved for a more general billiard

8

Chapter 2. Classical serpent billiard

channel which is bounded by an arbitrary pair of parallel smooth curves. Namely, it is easy to prove the following observation. Let the billiard motion in R2 be bounded by two smooth C 1 curves Cj , j = 1, 2, with natural parametrisation s → ~rj (s). The curves C1 and C2 should never intersect and they should be parallel in the following sense: for any s ∈ R, a line L(s) intersecting C1 perpendicularly at ~r1 (s) should also intersect C2 perpendicularly, say at a point ~r2 (τ ) defining a map τ = σ(s). The function σ : R → R should be a monotonously increasing invertible function, i.e. σ 0 (s) > 0 for all s, or in other words, the lines L(s) should not intersect each other inside the billiard region. Then the billiard motion is uni-directional, i.e. the sign of the tangential velocity component ~v · (d/ds)~rj (s) stays constant for all collision points of an arbitrary trajectory. To prove this observation, it is sufficient to consider two subsequent collisions of a segment of trajectory with a velocity of unit length |~v | = 1. We may assume the first collision to take place at ~r1 (s) ∈ C1 and write sin α := ~v · (d/ds)~r1 (s) > 0. Then we consider two possible cases: (a) The next collision happens on the curve C2 , say at the point ~r2 (τ ). Then the angle of incidence can be again written as sin β := ~v ·(d/dτ )~r2 (τ ). The angles between the segment of the trajectory and lines L(s) and L(σ −1 (τ )) are denoted by α and β, respectively. Since the latter two do not cross inside the billiard region, it follows that the sign of α and β should be the same (positive). We may have β = 0 only if α = 0, i.e. when the motion takes place along L(s) which is a periodic orbit. See fig. 2.2a. (b) Another possibility is that the next collision happens on the same curve, i.e. at ~r1 (s0 ). Writing sin β = ~v · (d/ds)~r1 (s0 ) we again observe that the sign of the angles α and β should be the same (positive), considering that the other ends of the lines L(s) and L(s0 ) at points ~r2 (σ(s)) and ~r2 (σ(s0 )) should lie on the same side of the trajectory segment since the latter should not cross C2 . See fig. 2.2b. Thus we have proved that L(s) is a family of marginally stable periodic orbits, of vanishing overall measure, which separates the phase space of the billiard into two halves of uni-directional motions. C2

C2

r2(τ) β β L (s)

αα

r2(σ(s)) L (σ (τ))

r2(σ(s’))

−1

C1

L (s’)

L (s)

β β r1 (s’)

α α r1 (s)

r1(s)

(a)

C1

(b)

Figure 2.2: The schematic diagram shows two possible cases (a,b) of subsequent collisions needed for the proof of uni-directionality.

It is perhaps worth stressing that the conditions of parallelism as expressed in the statement imply also that the width of the channel should be constant, |~r1 (s) − ~r2 (σ(s))| = const. In the following sections we shall discuss dynamical properties of the serpent billiards in terms of the so-called jump models. First we analyze the jump model of the billiard chain and give its characteristic properties that determine the basic properties of the model. Then, we study the correlations and transport properties of the model analytically and numerically showing results for some configurations of the serpent billiard. Examples of the periodic serpent billiards most frequently used in analysis are shown in figure 2.3, where the basic cell of the billiard is determined by parameters: inner radius q, angle of the bend β and length of the straight section d. We should note that a detailed understanding of the dynamics of such a class of billiards may

2.1. Dynamics in the serpent billiard

9

have useful applications, in particular in fiber optics, electromagnetic waveguide propagation, etc. β=π

β = π/2

β = π/4

(a)

(b)

Figure 2.3: Examples of serpent billiards at q = 0.6 with different angles β as indicated in the figure and lengths of the straight section d = 0, 1 (a,b), with a few typical trajectories.

2.1

Dynamics in the serpent billiard

Let us consider the Hamiltonian dynamics of a particle inside the serpent billiard channel which is composed of bends with inner radius q and straight sections of width a = 1 − q. Due to unidirectionality of the motion, as shown above, we choose to consider only the propagation in one direction along the channel as it is shown in the figure 2.3. The forward dynamics of the billiard can be written in terms of dynamics within a given element and a transition to an adjacent basic element. In order to fully describe the dynamics we only need to know a Poincaré-like mapping,

10


which maps coordinates of an entry into a cell to the coordinates of an exit, and the time spent between entry and exit (the entry into an adjacent element). Thus we formulate the dynamics of our billiard chain in terms of a jump map model (Zumofen and Klafter, 1993). We shall define the jump model more precisely below. The trajectory of a particle with fixed unit velocity |v| = 1 is calculated only at the instances, when passing through the borders of elements. The cross-sections (CS) of the channel taken perpendicular to the walls are of constant width a = 1 − q. Let coordinate y ∈ [0, a] represent the position along the CS. The trajectory intersects the border of an element at some position y ∈ [0, a] and with the projection of the velocity on y-axis denoted by vy ∈ [−1, 1]. To simplify the notation, we introduce the phase space of possible intersection-points defined as S = {(y, vy ) : y ∈ [0, a], vy ∈ [−1, 1]} .

(2.1)

The particle travelling from left to right along the chain passes an element by entering through the left border at x = (y, vy ) and leaving thought the the right border at x0 = (y 0 , vy0 ). Let us introduce the mapping of entry points into exit ones referred to as the jump or Poincaré map denoted by P : S → S x0 = P(x) , x0 , x ∈ S, (2.2) and the time function T : S → R+ , where T (x) measures the time needed by a particle to pass an element depending on its entry point x. The pair (P, T ) represents a jump model of the element and encompass all its dynamical properties. Note that a bend and a straight wave-guide are integrable systems, but the serpent billiard is not. The integrability is broken by connecting elements, which have incompatible integrals of motion. To construct the trajectory along the billiard in the jump model approach we need the jump models of individual elements and a procedure to combine them to form an uninterrupted flow of trajectories. Let us denote by (Pb,β , Tb,β ) and (Ps,d , Ts,d ) the jump models for the bend of angle β and for the straight wave-guide of length d, respectively, with their explicit forms given in Appendix A. The jump maps are given using the parametrisation of the CS as shown in figure 2.4a and 2.4b for the straight wave-guide segment and the bend, respectively. In addition we introduce the inversion of the phase space I : S → S written as I(y, vy ) = (a − y, −vy ) .

(2.3)

By following the construction plans for the serpent billiard given in the beginning of this section and making necessary inversions of the phase space as indicated in figure 2.4 we can write a jump model of a whole serpent billiard represented by the pair (Pn , Tn ). The phase-space inversions are needed in order to match the parametrisation of CS in the billiard with that assumed by the jump models. The number of bends in the billiard shall be used as a measure of length. The jump map of a periodic chain of length n composed of bends of angle β and straight sections of length d reads Pn = (I ◦ Ps,d ◦ Pb,β ) ◦ Pn−1 = Pn ,

P = I ◦ Ps,d ◦ Pb,β ,

Tn (x) = Tn−1 (x) + Ts,d (Pb,β ◦ Pn−1 (x)) + Tb,β (Pn−1 (x)) =

n−1 X

(2.4) T (Pn (x)) ,

(2.5)

m=0

whereas for a custom-built billiard with custom form parameters, the set of angles of the bend and lengths of straight sections {(βi , di ) : i = 1, . . . , n}, the jump model is written as Pn = I ◦ Ps,dn ◦ Pb,βn ◦ Pn−1 ,

Tn (x) = Tn−1 (x) + Ts,dn (Pb,βn ◦ Pn−1 (x)) + Tb,βn (Pn−1 (x)) .

(2.6) (2.7)


11

x

x

φ

y out

n=3 y x

r

dir e

cti

d

x

on of t

rav el

y

r’

β

n=2

φ’ y

y

x

φ x

0

y

y in

φ r

a

(a)

n=1 y

r

(b)

(c)

Figure 2.4: Schematic figures of a straight wave-guide segment (a) and of a bent (b) with coordinate systems used in the jump model description and the serpent billiard with the cross-section parametrisation, which is changing orientation along the chain (c).

The custom-built serpent billiards represent a large class of channels and therefore we expect a large diversity in dynamics. Its general properties are explained by considering the periodic case. In the periodic serpent billiard it is useful to define the jump map of the basic cell i.e. Poincaré map P and the time function of the basic cell T (x). Notice that the jump map and the time function are given in the form of a recursion with initial condition P0 (x) = x and T0 (x) = 0. Let us consider for a moment a more general case of dynamics in the serpent billiard, when particles traverse to the left and to the right at the same time with the momentum k or energy E = 12 k 2 . The CSs on both sides of the scatterer are the same and parametrised as an interval I = [0, a]. In order to distinguish between them we introduce additional set of labels {L, R} and add them to the definition of the left and right CS denoted by CL = (L, I) and

CR = (R, I),

(2.8)

respectively. We would then need a more general structure to describe the dynamics than the jump model (Pserp , Tserp ) of the serpent billiard. In this case the serpent billiard is discussed as a classical scatterer. Using an auxiliary map κk (y, vy ) = (y, kvy ) we reintroduce the scaled notation of dynamics that includes the total momentum k. First we define the scaled jump model e serp ) of the scaled time function Ten : S˜ → R and the scaled jump map given by the pair (Teserp , P e serp : S˜ →: S˜ defined as P e serp = κk ◦ Pserp ◦ κ−1 , P k

Teserp = kTserp ◦ κ−1 k

(2.9)

Se = κk S = {(y, p = kvy ) : y ∈ I , vy ∈ [−1, 1]} .

(2.10)

SSOS = SL ∪ SR

(2.11)

˜ corresponding to the trajectories crossing a CS, is given by where the scaled phase-space S,

with p denoting the projection of the momentum vector onto CS. We introduce then the canonical ˜ and SR = phase-spaces of trajectories crossing the left and right CS denoted by S L = (L, S) ˜ respectively. With these phase-spaces we define the SOS of the scatterer (R, S),

12


The mapping of ingoing trajectory xin from the left or from the right side into outgoing trajectory ˜ serp : SSOS → SSOS of the scatterer: xout is then given by a Poincaré map P e serp xin = xout . P

(2.12)

We denote by Pα→β , where α, β ∈ {L, R}, the different possible mappings between the phasee serp in a matrix block form as spaces SL and SR . Using this notation we can write the map P # " −1 e 0 P P P L→L L→R serp e serp = P . (2.13) = e PR→L PR→R Pserp 0

The diagonal blocks correspond to the classical reflection and are zero due to uni-directionality in our serpent billiard. This generalised Poincaré map is a classical analog of a quantum scattering matrix. In the same way as presented we could manage scatterers with many leads. In order to illustrate the richness of dynamics in the serpent billiard we look at the propagation of a set of rays along a periodic chain. The rays are prepared with a Gaussian distribution g : S → R+ in the phase space of the most left cell of the chain, forming a classical beam. This is depicted for an area of few basic cells in figure 2.5. The initial beam is unfocused by hitting 1 vy

0.5 0 -0.5 -1

-1

0

1

2

3

4

5

6

7

8

Y

Figure 2.5: Ensemble of 20 trajectories with a Gaussian distribution in the phase space on the left border of the most left cell. The serpent billiard configuration is β = π, q = 0.6 and d = 0.

the circular walls and the initial distribution spreads across the phase space of the CS as the number of collisions increases. The effect of beam spreading is described by the standard geometrical optics and is the source of chaos (Bunimovich, 2003) as explained in the introduction. The trajectory dynamics along the billiard is more transparently presented in the phase-space S positioned at the borders of the basic cells, where the evolution of a Gaussian shaped beam is given by g n (x) = g(P−1 x∈S. (2.14) n (x)), An example of such an evolution is shown in figure 2.6 for a specific case, where the phasespace is expected to be fully chaotic, that corresponds to the results in figure 2.5. By comparing these two results, note that the parametrisation of the CS is inverted by the Poincare map after trajectory passes the borders of the basic cell. In the figure 2.6 we can clearly see unstable and stable manifolds of the mapping. With increasing n, the initial Gaussian distribution spreads along the unstable manifold uniformly covering the chaotic component, which in this case is the whole phase space. The non-continuous flow of trajectory-intersections in the phase space indicates that the Poincaré map Pn is non-smooth. Another illustrative information about the gross dynamics in the serpent billiard is obtained from phase-space portraits of the Poincare map Pn roughly representing gn (x) in the limit n → ∞. We calculate the portraits for different configurations of periodic serpent billiards shown for β = π, d = 0 in figure 2.8, for β = π, d = 1 in figure 2.9, for β = π/2, d = 0 in figure 2.10 and for β = π/4, d = 0 in figure 2.11. Note that the phase portraits are symmetric in y around the mean radius (1 + q)/2. We observe that the phase space of the serpent billiard is


13

Figure 2.6: The evolution of a Gaussian shaped beam of rays in the phase-space S along the serpent billiard as defined in eq. (2.14) for n = 0(1)11 increasing from the left to the right and downwards, as shown in fig. 2.5 in the coordinate space.

in general of mixed type with chaotic and regular regions coexisting in the phase space S. By inserting straight segments between the bends in the chain of some length d > 0 the phase space structure changes drastically, making the serpent billiard more chaotic. This is nicely depicted in figures 2.8 and 2.9 for the β = π. A similar scenario occurs at other angles β = π/2, π/4, just little less drastic, but the figures with results are omitted. By decreasing the angle β the trajectories can pass the basic cell with less scattering from circular walls and consequently the regular regions increase in the size. This can be seen by comparing the portraits 2.8, 2.10 and 2.11 for angles β = π, π/2 and π/4, respectively, at d = 0. It is believed that for long enough straight wave-guide segments at fixed q and β the serpent billiard could be even ergodic (Bunimovich, 2005). It seems apparent that the smaller the angle the longer should the straight segments be in order to achieve ergodicity. In presented examples the chaotic component is dominant in size over the regular components. In certain cases the regular components are practically negligible, so that the jump map appears to become (almost) fully chaotic and ergodic. For example, we were unable to locate a single regular island at parameter values q = 0.6, β = π and d = 0. The latter configuration represents the paradigmatic periodic serpent billiard. Roughly speaking, the relative area of regular island p decreases with increasing q and with increasing d. Numerically we examine the relative area p for our paradigmatic example and results are shown in figure 2.7. The holes in the plot correspond to situations where we were unable to find any stable islands. We quantify the exponential instability of trajectories inside the chaotic component of the phase-space S for a periodic serpent billiard by calculating the average Lyapunov exponent λ as described in e.g. (Reichl, 1992). The λ as a function of q is calculated for several classes of periodic serpent billiards and the results are shown in figure 2.12. We observe that the chaoticity as measured by λ, being equal to the dynamical Kolmogorov-Sinai entropy, is always positive and is increasing with q for all observed configurations. This is generally expected as the number of collisions within the jump increases with q. Namely, a bigger number of collisions between entry and exit sections implies less correlation between the angular momenta of the adjacent basic cells meaning stronger integrability breaking. In serpent billiards with small angles of bends we do not expect a monotonic increase of λ with q as at certain values of q there exist strong islands of regular motion.

14


1 0.1

1-p

0.01 0.001 1e-04 1e-05 1e-06

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 q

1

Figure 2.7: Numerical estimation of the total relative area of a chaotic SOS components p in the phase space S as a function of parameter q in the periodic serpent billiard as β = π and d = 0. The ratio is obtained by sampling of 1000 random trajectories starting inside the chaotic component over a phase space grid of a size 1000 × 1000. The length of trajectories, measured in the number of jumps, was 10 6 .

Figure 2.8: Phase-space portraits for β = π and d = 0 at q = 0.1, 0.3, 0.5 and q = 0.6, 0.7, 0.9 (bottom)

Important fingerprints of dynamics, in many ways complementary to the Lyapunov exponents, are the time-correlation functions. These reflect the mixing property of the system which implies a decay of an arbitrary initial phase space measure to equilibrium. Time correlation functions are also directly related to transport, which is studied in the next section, through a linear response formalism. We discuss the discrete time correlation function of the jump map between two observables φ(x) and ψ(x), which is defined as Pt−1 n=0 φ(Pn (x))ψ(Pn+τ (x)) . (2.15) Cφ,ψ (τ, x) = lim P t−1 t→∞ n=0 φ(Pn (x))ψ(Pn (x))

The correlation functions are normalised such that always, Cφ,ψ (0, x) ≡ 1, even for observables which are not in L2 (SL ). We consider auto-correlation functions of very regular observables such as the phase space coordinates, namely Cy,y and Cvy ,vy , and the auto-correlation function of the time function CT,T which is even more interesting for two reasons: (i) CT,T is directly related to particle transport as described in the next section, and (ii) T is not in L2 (SL ) as discussed below. Different configurations of serpent billiards do not differ strongly in the behaviour of the


15

Figure 2.9: Phase-space portraits for β = π and d = 1 at q = 0.1, 0.3, 0.5 and q = 0.6, 0.7, 0.9 (bottom)

Figure 2.10: Phase-space portraits for β = π/2 and d = 0 at q = 0.1, 0.3, 0.5 and q = 0.6, 0.7, 0.9 (bottom)

correlation function over the chaotic component. Therefore we discuss the numerical data only for our paradigmatic case given by parameters β = π, d = 0 and q = 0.6, which are presented in figure 2.13. The data strongly suggest that the correlation function CT,T typically exhibits exponential decay ∼ exp(−constτ ) for most of values of q, except that for the small parameter values q < 0.3 the initial exponential-like decay seems to turn into an asymptotic algebraic decay ∼ t−const . On the other hand the correlation decay of non-singular observables, like C x,x and Cvy ,vy seems to behave as an algebraic function for all values of q. It is interesting to observe that the qualitative nature of correlation decay seems to be quite different for different classes of observables such as y compared to T . However, in all the cases the time correlation functions decay strongly, which is a firm indication of the mixing property of the serpent billiard inside the chaotic component. The complementary part to the Poincaré map Pn (x) in the jump model of the serpent billiard is the time function Tn (x). The parallel walls in the billiard enable bouncing ball orbits of a particle in the CS direction that do not propagate along the chain. These orbits cause the square-root singularity in the time function for vy = ±1. It is straight-forward to derive the

16


Figure 2.11: Phase-space portraits for β = π/4and d = 0 at q = 0.1, 0.3, 0.5 (top) and q = 0.6, 0.7, 0.9 (bottom) 3.5

3

d=0 d=1

3 2.5

2.2

d=0 d=1

2.5

1.6

2

1.5

1.4

1.5

λ

λ

λ

2

0.5

0.5

0

0

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 q

1

1.2 1

1

1

d=0 d=1

2 1.8

0.8 0.6 0.4 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 q

(a)

1

0.2

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 q

(b)

(c)

Figure 2.12: The average Lyapunov exponent λ as a function of q, calculated from a single trajectory of length 106 for periodic serpent billiards with form parameters of basic cell β = π, π/2, π/4 (a,b,c) and d as written in the figure.

leading order of the time function in the limit |vy | → 1 for the straight wave-guide segment: Ts,d (y, vy ) =

1 d (1 − |vy |)− 2 , 2

(2.16)

and of the bend: Tb,β (y, vy ) ≈

1 βa (1 − |vy |)− 2 , 2r

Γ=r

q

1 − vy2 q .

(2.17)

The singular property of time function in basic elements is inherited by the chain. In order to understand the transport properties along the periodic serpent billiard we need to discuss the distribution of time-of-flight across a single basic cell, denoted by p(t), for a very long chaotic trajectory. It is defined over a chaotic component Ω ∈ S with the jump model of the basic cell (P, T ) as X px (t) = δ[t − T (Pn (x))] , x∈Ω, (2.18) n∈Z

where Ω ∈ S is the chaotic component. We can see that pP(x) = px and by assuming ergodicity in the chaotic component we can rewrite px (t) as Z d2 x δ(t − T (x)) , (2.19) p(t) = Ω

1

2.1. Dynamics in the serpent billiard 1

17 1

q=0

10-2

10-2

10-3 -4

10-3 -4

10

10-5

q=0.3

-1

10 C(τ)

C(τ)

-1

10

10 1

102

10

10-5

103

1

τ 1

q=0.6

-1

q=0.9

-1

10

10

10-2

10-2

C(τ)

C(τ)

103

τ

1

10-3 -4

10-3 -4

10

10-5

102

10

10 1

102

10

103

10-5

1

τ

10

102 τ

103

Figure

2.13: The average auto-correlation functions: hCT,T (x, τ )iΩ (full curve), hCy,y (x, τ )iΩ (dashed), and Cvy ,vy (x, τ ) Ω (dots) as functions of the number of jumps τ . The calculation is performed using trajectories of length 5 · 105 and averaged over 104 initial conditions x in the chaotic component. In the plot for q = 0.6 and q = 0.9 we insert the functions exp(−0.0508193 τ ) and exp(−0.00234154 τ ), respectively, in order ti guide the eye. The noisy plateaus indicate the level of statistical fluctuation.

valid for almost all x ∈ Ω. The square-root singularity of the time function T (x) with onedimensional support implies the asymptotic form of the distribution p(t) ∼ t−3 ,

t→∞.

(2.20)

This asymptotic property does not essentially depend on the full ergodicity of the map, as we find the same asymptotic behaviour in numerical simulations of p(t) for different configurations of the basic cells shown in figure 2.14. The same asymptotics is expected by considering the timeof-flight across sections of a custom-built serpent billiard. It is obvious that the only important condition for the universal decay of p(t) is that the chaotic component should extend to the lines of singularity vy = ±1. In the periodic serpent billiard we have calculated the average time hT i and time minimal time Tmin of flight for the basic cell on a chaotic component: Z 1 d2 x T (x) , Tmin = lim inf T (x) , (2.21) hT i = x∈Ω vol(Ω) Ω and the results shown in figure 2.15. Please note that the maximal time on a chaotic component is infinite due to the singularity. It is interesting that the average time hT i to pass the cell increases approximately linearly with increasing q, within a statistical error. In cases β = π/2 and β = π/4, where the direct transition across the bent is possible, the minimal time increases monotonically with q from Tmin = d at q = 0. Situation Tmin > 0 means that by travelling along the chain the particles gain a non-zero time delay in each basic cell. Consequently, if we measure travelled length in number of basic cells, the speed of particles is bounded from below. The fine structure of the time function reflects the geometry of the billiard. The time function

18

Chapter 2. Classical serpent billiard β = π, d = 0

β = π, d = 1

10

0.001 1e-04

0.1 0.01 0.001 1e-04

1e-05

1e-05

1e-06

1e-06

1e-07

1e-07

1e-08

10

100

q=0.1 q=0.2 q=0.3 q=0.4 q=0.5 q=0.6 q=0.7 q=0.8 q=0.9 10 t-3

1

P(t)

0.1 0.01 P(t)

10

q=0.1 q=0.2 q=0.3 q=0.4 q=0.5 q=0.6 q=0.7 q=0.8 q=0.9 10 t-3

1

1e-08

1000

10

100

t

β = π/2, d = 0 10

0.001 1e-04

0.1 0.01 0.001 1e-04

1e-05

1e-05

1e-06

1e-06

1e-07

1e-07 1

10

100

q=0.1 q=0.2 q=0.3 q=0.4 q=0.5 q=0.6 q=0.7 q=0.8 q=0.9 10 t-3

1

P(t)

0.1 0.01 P(t)

β = π/4, d = 0 10

q=0.1 q=0.2 q=0.3 q=0.4 q=0.5 q=0.6 q=0.7 q=0.8 q=0.9 10 t-3

1

1e-08

1000

t

1000

1e-08

0.1

1

10

t

100

1000

t

Figure 2.14: Distribution of the time-of-flight p(t) across a basic cell in periodic serpent billiards with different form parameters q, β and d as wriiten in the figure. 6

, d=0 Tmin , d=0 , d=1 Tmin , d=1 f1(t) f2(t)

7 6 5

4

2

3

1 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 q

(a)

1

0


4 3.5 3 2.5

3

4

2

4.5


5

2 1.5 1 0.5 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 q

(b)

1

0

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 q

(c)

Figure 2.15: The average time hT i and the minimal time Tmin of flight across a basic cell on the chaotic component for β = π, π/2, π/4 (a,b,c). Inserted fitting functions are f1 (q) = 2.45558 + 2.49604 q, f2 (x) = 4.0383 + 2.436 q, f3 (x) = 1.2184 + 1.304 q, f4 (x) = 2.8037 + 1.213 q, f5 (x) = 0.51047 + 0.8607 q and f6 (x) = 2.1866 + 0.60872 q

of the basic cell has a profile with discrete steps nested as onion leaves, where leaves could be labelled by the number of collisions with walls. This is also visible in density plots of the time functions in figure 2.16 calculated for the most common basic cells. By approaching the line of singularity, the nesting of leaves becomes denser; making the time function “continuous” in the limit |vy | → 1. The dynamics across the basic cell does not posses any strict symmetry in the phase space and therefore the time function is non-symmetric. In the case β = π the time function is almost symmetric around vy = 0. If we would observe the time function of a longer periodic chain Tn (x) the detailed structure is mixed by the chaotic dynamics so that the function becomes on average symmetric around vy = 0 with lines of singularity at |vy | = 1.

1

2.2. The transport properties 1

0.5

0.5

0.5

0

0

0

vy

vy

-0.5

-0.5

0

0.1

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y

0.5

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0.7

β = π, d = 1

-1 1

vy

1

β = π, d = 0

1

19

-1 1 0.9 0

0.8

-0.5

0.05

0.1

0.15

0.2 y

0.25

0.3

-1 1 0.4 0

0.35

0

0

0

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-0.5

0

0.1

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β = π/2, d = 0

-1 1

vy

0.5

vy

0.5

vy

0.5

y

0.5

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0.7

-1 1 0.9 0

0.8

0.05

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0.25

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0

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vy

0

vy

0.5

vy

0.5

-0.5

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0.1

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β = π/4, d = 0

-1 1

y

0.5

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0.7

-1 1 0.9 0

0.8

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0.1

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0.2 y

0.25

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0

0

vy

0

vy

0.5

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-1 0

0.1

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0.3

0.4

y

0.5

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0

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0.8

1

-1 0.9 0

0.02

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0.04

0.02

0.04

y

0.06

0.08

0.1

y

0.06

0.08

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0.08

0.1

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0.1

-0.5

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0.04

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0.5

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0.02

y

-0.5

0.05

0.1

0.15

2

(a)

0.2 y

(b)

0.25

3

0.3

0.35

-1 0.4 0

4

5

y

(c)

Figure 2.16: Density plots of the time function T (y, vy ) over the phase space S for various values of β and d as indicated in the figure and for q = 0.1, 0.6, 0.9 (a,b,c)

2.2

The transport properties

We are interested in the transport properties of classical particles traversing along the serpent billiard in the context of the jump model. We discuss only periodic serpent billiards as there exists at least an approximate relation between the number of traversed basic cells n and the actually travelled physical length L. The latter is for the basic cell with parameters q, β and d written as L ≈ |(1 + q) sin(β/2) + d cos(β/2)|n . (2.22) Although we discuss transport properties for the simplest periodic billiard, where the period is composed of two equal basic cells, the presented theory can also be applied to a billiard of a larger period by finding the jump model of that period. The period then represents a unit of physical length. The uni-directionality property dictates that the particles move just in one direction along the chain and so we focus only on these which travel from the left to the right. The basic cells visited by the particles are labelled with non-negative integer n ∈ N ∪ {0}. The transport length is measured as the number of traversed basic cells. This means that our length is an

20


integer connected to an actual physical distance by equation (2.22). Let us consider the serpent billiard with the jump model of the basic cell given by the Poincaré map P : S → S and by the time function T : S → R+ . We are interested in the question, how does an ensemble of particles, initially in the first basic cell, spread over the chain when we subtract the average propagation (drift). The phase space of the Poincaré map can be generally separated into chaotic and regular regions. For particles prepared in a regular region (island) Λ ⊆ S we can see particles spreading ballistically, where the width of the distribution of particles over the cells σ(t) increases linearly with time and is given by σ(t) = t[min{T (x)}−1 − max{T (x)}−1 ] . x∈Λ

x∈Λ

(2.23)

Thus we concentrate on the more non-trivial and interesting case of transport on the chaotic component. The particles are prepared with some density function ρ(x) over the chaotic component Ω ∈ S of the first (most left) basic cell in the chain. We describe the transport of particles in terms of their distribution over the basic cells denoted by Pn (t) which is defined as Z t Pn (t) = dτ [pn (τ ) − pn+1 (τ )] , n ∈ N ∪ {0} , (2.24) 0

Here we introduce the distribution pn (τ ) of the time for particles to pass n cells Tn (x) (2.5) on the chaotic component x ∈ Ω: Z d2 x ρ(x)δ[t − T (Pn (x))] , p0 (t) = δ(t) . (2.25) pn (t) = Ω

The chaotic dynamics is assumed ergodic on the chaotic component Ω and so without losing generality we express ρ(x) with the characteristic function of a chaotic component χ Ω as ρ(x) = χΩ (x)/ vol(Ω). The main characteristics of transport along the chain are the first two central moments of Pn (t) expressed as Z t ∞ X X dτ pn (τ ) , (2.26) n Pn (t) = hni = σn2

=

n=1 ∞ X

n=1

0

2

n∈N

(n − hni) Pn (t) =

Z

t

dτ 0

X

n∈N

n pn (τ ) − hni (hni + 1) .

(2.27)

The time Tn (x) is a sum of n times that are chosen by a chaotic process given with fast decaying correlations. By neglecting remaining correlations in the map P between sampled times we express pn (t) as the multiple convolution pn (t) = (p ∗ . . . ∗ p)(t) , | {z }

(2.28)

n

where the distribution of time to pass one cell p(t) is given by eq. (2.19). We assume that pn (τ ) with increasing n converges to some limiting distribution p∗n (τ ) (Gnedenko and Kolmogorov, 1954). In the time-asymptotics t 1 we expect that most particles travel a long way along the chain having n 1. In this limit we can approximate pn (t) with p∗n (t) and treat n as a continuous variable and so we write Z t ∞ X 1 ∂ m p∗n Pn (t) − (τ ) . (2.29) dτ m! ∂nm −∞ m=1

To compute the transport properties we use the leading order of this approximation that reads Z t ∂ dτ p∗n (τ ) . (2.30) Pn (t) − ∂n −∞

2.2. The transport properties

21

In figure 2.17 we give an illustration of the distribution Pn (t) evolving in time for different configurations of the serpent billiards. For a given t, the Pn (t) has a finite range in n, because the minimal time of flight Tmin > 0, and is given by nmax = bt/tmin c. At large times Pn (t) has a strong peak around the position hni(t) = t/hT i resembling a Gaussian function. In the usual case, the chaotic sea extends to the line of singularity in the time function that enables extremely slow jumps yielding long tails in Pn (t) from the peak up to the origin n = 1. The distribution Pn (t) is strongly asymmetric due to uni-directionality.

0.001

0.01

1e-04

1

t=5 t=10 t=50 t=102 t=103 t=104

0.1

Pnt

t

0.01 Pn

1

t=5 t=10 t=50 t=102 t=103 t=104

0.001

0.01

1e-04

0.001 1e-04

1e-05

1e-05

1e-05

1e-06

1e-06

1e-06

1e-07

1e-07

1

10

100 n

(a)

1000

10000

1

10

100 n

1000

(b)

t=5 t=10 t=50 t=102 t=103 t=104

0.1

Pnt

1 0.1

10000

1e-07

1

10

100

1000

10000 100000

n

(c)

Figure 2.17: Evolution of the distribution of particles over cells Pn (t) in a periodic serpent billiard with q = 0.6, d = 0 and β = π, π/2, π/4 (a,b,c). The distribution is calculated with N = 10 7 points chosen uniformly over the chaotic component in the first cell.

We distinguish two fundamentally different cases of transport depending on the distance between the chaotic region and support of time singularity denoted by D = lim sup (y,vy )∈Ω ||vy |− 1|. When D > 0, the chaotic sea is well separated from the line of singularity and p(τ ) is a bounded distribution implying the existence of all its moments and so the Fourier transform of p(τ ) is an analytic function around the origin. Currently this is known to happen only in the serpent billiards with q = 0, where the entire area of singularity is covered by the regular region. This case is related to the research of truncated Lévy flights discussed in (Koponen, 1994; Mantegna and Stanley, 1994). Then following the central limit theorem (Feller, 1968), p. 258 the limiting distribution p∗n is a Gaussian distribution 1 τ − nhT i ∗ √ pn (τ ) = √ G , (2.31) nσ nσ R R where we use two central moments hT i = dt t p(t) and σ 2 = dt (t − hT i)2 p(t) of times needed to travel over a single cell and the standard Gaussian distribution x2 1 G(x) = √ e− 2 . 2π

(2.32)

The convergence of pn towards the expected limiting distribution p∗n in terms of probability is guarantied by the Berry-Eséen theorem (Feller, 1968), p. 542 written as Z x r3 dτ [pn (τ ) − p∗ (τ )]| ≤ C 2 , ∀ n , | (2.33) nσ −∞ as well as in terms of distributions by the Gram-Chalier expansion (Chebyshev, 1890), that states i h 3 r3 (2.34) pn (τ ) = p∗ (τ ) 1 + 3 + O(n− 2 ) , 6σ n

where r3 = (t − hT i)3 and C ≈ 0.7655 is an absolute constant found in (Shiganov, 1986). By plugging the limiting distribution p∗ (x) in the formula for the leading order of the asymptotic

22


form of the distribution of particles over cells we get t + hT in τ − nhT i √ Pn (t) √ , G 2 nσ nσ

(2.35)

We are only interested in the main leading contributions to the probability distribution P n (t) and this is where t ≈ nhT i. By making a local expansion in n around t/hT i and taking only the first order we obtain for Pn (t) a Gaussian distribution   n − t/hT i  1 . (2.36) Pn (t) q G q σ t/hT i3 σ t/hT i3 From the formula above we read the leading order approximation of the first two central moments: t σ2 hni , σn2 t. (2.37) hT i hT i3 This implies that in the discussed case (D > 0) we have a normal diffusion of particles along the chain after subtracting the average drift. The normal diffusion is usually expected in chaotic channels as examples studied in e.g. (Dittrich and Doron, 1994; Gaspard, 1993) show. (ii)In a more common case, when D = 0 the distribution of times has a algebraic tail p(t) ∼ t−3 as τ → ∞. This yields the Fourier transform of p(τ ) of the form 1 log pˆ(ω) = i ωhT i − ω 2 σ 2 (α − log(ω)) + O(ω 3 ) , 2

(2.38)

with that the distribution pn (t) can be expressed as pn (t) = F −1 [exp(n log pˆ)](t) ,

(2.39)

where σ and α are constants depending on details of the distribution p(t). The Fourier transform F and its inverse F −1 are Z Z 1 iωt −1 ˆ ˆ f (ω) = F[f ](ω) = dtf (t)e , f (t) = F [f ](t) = dω fˆ(ω)e−iωt . (2.40) 2π By choosing an appropriate scaling of the time variable around the average position we obtain the following limiting distribution 1 t − hT i n ∗ √ pn (t) = √ . (2.41) G σ n log n σ n log n The “fat” algebraic tails persist in pn (t) with increasing n and causing strong deviations of pn (t) from p∗n (t) in the sense of probability. It is therefore meaningful to show convergence only in the sense of distributions. By properly expanding pn (t) around its limiting form we obtain h 1 i 3 pn (t) = p∗n (t) 1 + O n− 2 (log n)− 2 . (2.42) We see that convergence is much slower than in the previous case with D > 0. The distribution has a Gaussian form with the standard deviation increasing by the log-normal law. The leading order of the distribution of particles over the cells with local expansion in n around hni then reads t 1 n − hni0 t σ2t 02 log Pn (t) 0 G , hni = , σn = . (2.43) σn σn0 hT i hT i hT i3

2.2. The transport properties

23

where hni0 and σn0 are the leading orders of the first two central moments hni and σn , respectively. We see that leading term of Pn (t) is again a Gaussian distribution with the width indicating that the particles exhibit marginally normal diffusion, when we subtract the average drift. Because of the slow convergence of pn towards p∗n , we expect a strong deviation of Pn (t) from the expected form (2.43) for finite times. We have tested our analytical results by performing extensive numerical simulations. An example of hni(t) and σn2 (t) for the configuration q = 0.6, β = π and d = 0 is shown in figure 2.18. We stress that our numerical data are indeed consistent with the marginally normal diffusion. 108 7

9

10

measurement σ2n ~ t log(t)

108

106

107

10

4

105

10

3

104

2

10

5

σn

1010

106

100 2

10

(b)

measurement vt

σn /t

(a)

3

10

102 2

10

10

3

10

4

5

6

10

10 t

10

7

8

10

9

10

102 10 102

50 0

103

104

4

105

5 106

6 7 8 log10(t)

9

107

109

108

t

Figure 2.18: The average position hni (a) and the spread of particles σn2 (b) along the billiard with −1 q = 0.6, β = π and d = 0 as a function of time t. The average transport velocity v = hT i , where hT i = 3.94538 is obtained from the data of figure 2.15. The error bars represent the standard deviation of the measurement. The statistics is performed over an ensemble of 25000 initial points. The fitted line in the inset is σn2 /t = −0.044374 + 5.2636 log 10 t.

24


25

Chapter 3

General linear chains of quantum scatterers The term linear chain of quantum scatterers is used here to denote a sequence of connected quantum scatterers building a structure of linear topology. We examine chains of scatterers on an infinite straight wave-guide referred to as two-way (channel, port) scatterers. Similarly we could also treat any types of connected quantum scatterers in a linear sequence. The scatterers on the chain are conservative quantum systems of arbitrary complexity of wave-dynamics with two identical waveguides attached on the left and the right side. The latter serve as attachmentports to the neighbouring scatterers on the chain, as shown in figure 3.1. We try to leave from discussion all geometric parameters of the wave-guides and work purely algebraically to make results as general as possible and applicable to any type of scatterers. We are interested in

L(left) aL bL

waveguide

scatterer

R(right) bR aR waveguide

Figure 3.1: Schematic picture of a single scatterer representing a basic cell of the chain in discussion.

general on-shell1 transport properties of these chains, in the process of lengthening, that we call the dynamical approach. We give here only a sketch of the general theory. For details the reader can see (Londergan and Carini, 1999) or the introductory part of the Chapter 4, where a bend plays the role of the scatterer. In the on-shell scattering (Newton, 2002) we try to solve the quantum eigen-equation over the whole region composed of the scatterer and both infinite wave-guides symbolically written as ˆ H|ψi = E|ψi ˆ represents the Hamiltonian of the system. The wave functions in the left |ψL i and in where H the right waveguide |ψR i are expanded in basis of functions {|e± n i}n∈N having unit probability current d X + L,R − |ψL,R i = aL,R (3.1) n |en i + bn |en i , n=1

where superscript + and − corresponds to the phase propagation from left-to-right and rightto-left, respectively. The number of basis functions involved in the expansion, denoted by d, is 1

The term on-shell means working at sharply defined energy.

26

Chapter 3. General linear chains of quantum scatterers

called the dimension of scatterers. Usually, the full functional basis is infinite (d = ∞), but in practice, we set d to some finite value depending on the problem and the desired precision of calculation. The smoothness of the wave function across the borders between the wave-guides and scatterer is a condition that connects the wave functions on both sides of the scatterer. The connection between the two sides expressed in vectors of expansion coefficients a L,R = {aL,R n }n∈N and bL,R = {bL,R } is given in the form of a scattering matrix S or a transfer matrix T: n n∈N L L b a , (3.2) , ψout = Sψin = ψout , ψin = aR bR L R a a T ψL = ψ R , ψL = , ψ = , (3.3) R bL bR with superscripts L and R corresponding to the left and the right side of the scatterer. The vectors of expansion coefficients ψin,out and ψL,R are called SOS (surface of section) states and CS (cross section) states, respectively. In the operator treatment of the problem, as in e.g. (Prosen, 1995), these states are defined in the Hilbert space over the CS of the straight waveguide. By following the definitions (3.2) and (3.3) we express the scattering matrix S and the transfer matrix T as block matrices L R r t x1 x2 S= , T = , (3.4) tL r R x3 x4 where r L,R represent the reflection matrices and tL,R the transmission matrices of the left and right side of the scatterer. The matrices of this form are sometimes called two-way (channel, port) scattering and transfer matrices, because they connect two distant openings of the scatterer. Purely from definitions (3.2) and (3.3) we obtain the relations between scattering and transfer matrices by transformations # " −1 −1 −x−1 x3 x−1 tL − r R tR r L r R tR 4 4 , S[T ] = (3.5) T [S] = −1 −1 −1 x1 − x2 x−1 tR −tR rL 4 x3 x2 x4 Basis functions |e± n i, which alter their amplitude along the wave-guide are called travelling waves or open modes and other are called evanescent waves or closed modes. We are interested in properties of asymptotically long chains. It is a general opinion that in this case we may neglect evanescent waves and work with the basis of travelling waves. In this case the matrices S and T fulfil the following relations id 0 S † S = SS † = id , T † KT = T KT † = K, K = , (3.6) 0 −id implying that the scattering matrix S is unitary, S ∈ U (2d), and the transfer matrix T is hyperbolic, T ∈ U (d, d). Then by introducing transport probability matrices corresponding to the reflection Πx and transmission Σx Πx = rx† rx ,

Σx = tx† tx ,

Πx + Σx = id ,

x = L, R .

(3.7)

we define the two basic measures of transport of the scatterer : (i) the average transmission (reflection) probability T (R) as R = hΠx i,

T = hΣx i = 1 − R ,

x = L, R .

(3.8)

2 ) as and (ii) the standard deviation of the transmission (reflection) probability σ T2 (σR 2 σR =

1 2 1 2 2 = Πx − R 2 = σ T Σx − T 2 , d+1 d+1

x = L, R ,

(3.9)

27 where we have used the symbol h•i = d1 tr{•}. We see that only two quantities can be independent and so we use either (R, σR ) or (T , σT ) as a measure of transport. We build the chain of scatterers recursively by connecting additional scatterers to one end of the chain. This procedure is illustrated in figure 3.2. The chain’s length is measured in units of the number of scatterers composing the chain. An existing chain of length n with the scattering matrix Sn is extended with an additional elementary scatterer described by a generating scattering matrix S, forming a chain of length n + 1, and with the scattering matrix S n+1 : L R L R rn tn r t . Sn = . (3.10) S= L R tLn rnR t r The recursion for the scattering matrices of the chain then reads Sn+1 = Sn S .

(3.11)

where we introduced a binary operation representing concatenation of scattering matrices. It is explicitly written as L L −1 L rn+1 = rnL + tR n r L tn , R rn+1

=

rnR

+

−1 tLn rR L0 tR n

tLn+1 = tL L−1 tLn , ,

tR n+1

=

−1 t R L 0 tR n

(3.12) ,

(3.13)

with expressions L = 1 − rnR rL and L0 = 1 − r L rnR . The operation is derived and explained in Appendix B. We think of the recursion (3.11) as a discrete dynamical system defined over the space of scattering matrices. We study two types of chain-generation: either we take fixed (static) generating scattering matrix S, or S is taken randomly during the growth of the chain. Notice that the iteration (3.11) can be very elementary written in the transfer matrix formalism as Tn+1 = T Tn , (3.14) where T = T [S] is the generating transfer matrix and Tn = T [Sn ] is the transfer matrix of the chain of length n. But as can be read from the relations (3.6), matrix elements of Tn are not bounded in size. This implies that the map (3.14) is generally unstable and so numerically of limited use.

1

2

3

n−1

n

n+1

S1 S

S

S

S S

Sn Figure 3.2: Schematic picture of a discussed linear chain of scatterer. It is build by recursive lengthening of the initial chain S1 with scatterers S. Scatterer S can be taken as fixed or n-dependent.

We work in the frame of the scattering matrix formalism to remove any divergence present in e.g. transfer matrix formalism. The presented dynamical approach enables intuitively clear insight into the finite and infinite systems of composed scatterers. In the following sections, besides the dynamical properties of the system (3.11) we also present the transport properties of the chain in the absence and presence of small and strong noise. Using the dynamical approach we bring forward of the chain in the nice self explanatory way and we additionally reproduce some of the well known results. We focus mainly on chains of one-dimensional scatterers (d = 1) in which we study the transport along the chain numerically and analytically. In addition, we discuss the chains of general d-dimensional scatterers, where we give only numerical results. The latter results are partially explained by comparing to the one-dimensional case.

28

3.1


Chain of one-dimensional scatterers

Here we discuss chains of linearly connected one-dimensional scatterers. We describe the chain and individual scatterers on it by 2 × 2 unitary scattering matrices S parametrised as " # L R p Aeiα Beiβ L L R S(A, α , β , β ) = (3.15) , A = 1 − B2 , L L +β R −αL ) iβ i(β Be −Ae where A and B are reflection and transmission intensity, respectively, α L is the reflection phase, β R and β L are the transmission phases. Note that T [S] = B 2 , R[S] = A2 and σR = σT = 0. In introduced parametrisation (3.15) the scattering matrix Sn of the chain with some length n and the generating scattering matrix S read Sn = S(An , αnL , βnL , βnR ) ,

S = S(A, αL , β L , β R ) .

(3.16)

The recursion relation/map for the chain generation (3.11) can now be written explicitly. We discuss two cases of chain generation. In the first case the chain is periodic with fixed S and in the second case S is chosen randomly at each iteration step creating a random (custom build) chain. The appropriate explicit form of the map differs a bit for the two cases.

3.1.1

Static generating scattering matrix

The initial chain is described by the scattering matrix S1 and the generating matrix S is constant along the chain. Then the explicit form of the recursion relation for chain generation (3.11) by using a new variable χn = βnL + βnR − αnL + α reads s A2n + A2 + 2An A cos χn An+1 = , (3.17) 1 + 2An A cos χn + (An A)2 χn+1 = χn + 2λ + arg{(1 + An Ae−iχn )(An + Ae−iχn )} ,

(3.18)

L βn+1 R βn+1

(3.19)

= =

βnL + β L + arg{(1 + An Ae−iχn )} , βnR + β R + arg{(1 + An Ae−iχn )} .

(3.20)

where we introduced a constant λ = (β L + β R )/2 of the generating scattering matrix. The relevant parts of the recursion are expressions (3.17) and (3.18) representing an independent twodimensional dynamical system. Depending on parameters of the generating matrix S, namely A and λ, we distinguish two topologically different types of dynamics: (i) quasi-periodic or ballistic with an elliptic fixed-point and (ii) attraction to a single fixed-point, called localisation. The phase-space portraits for both types of dynamics are plotted in figure 3.3. The study of our two-dimensional dynamical system shows that the type of dynamics depens on the sign of the following expression D = A2 − (sin λ)2 , (3.21) yielding for D < 0 the ballistic and for D > 0 the localised dynamics. The expression D is a discriminant in the eigenvalue problem of the transfer matrix T [S] (det(T [S] − κ id) = 0): L

R

β −β √ ei 2 cos λ ± D κ± = √ 1 − A2

(3.22)

and it is easy to see that D < 0 ⇒ κ1 = κ∗2 and |κ1,2 | = 1 ,

D > 0 ⇒ κ 1 < 1 < κ2 .

(3.23)

3.1. Chain of one-dimensional scatterers

29 1 0.9 0.8 0.7 0.6 An

0.5 0.4 0.3 0.2 0.1 0

-3

-2

-1

(a)

0 χn

1

2

3

(b)

Figure 3.3: The phase space portraits of the dynamical system (3.17) and (3.18) generated at parameters A = 0.5, λ = 0.628319 (a) and A = 0.5, λ = 0.314159 (b). In (b) we insert arrows to sketch the directions of the flow in the vicinity of the fixed point, which are v1 = (1, 0) and v2 = (−0.945261, 0.326315).

We can conclude that a growing chain in the transfer matrix formalism (3.14) for D < 0 will be numerically stable and for D > 0 some matrix elements of the transfer matrix T n [Sn ] will diverge. The phase-space portrait of ballistic dynamics indicates an existence of an additional integral of motion of dynamics (3.17) (3.18), with an elliptic fixed point (Ae , χe ) given by Ae = u −

p

u2 − 1 ,

χe = λ +

π + mπ , 2

u=

sin λ , A

(3.24)

where m ∈ Z is such that Xe ∈ / (−π/2, π/2). By examining the dynamics of the system and using the correspondence between the scattering and the transfer matrices given by (3.5) we derive the integral of motion I(An , χn ) =

sin λ + An A sin(λ − χn ) . 1 − A2n

(3.25)

By considering the last expression we conclude that the scattering matrix of the chain evolves along its length on a one-dimensional manifold. In the case of localised dynamics the whole phase-space converges to a single fixed point attractor (Aa , χa ): Aa = 1 ,

χa = λ + arg{iu +

p 1 − u2 } ,

u=

sin λ . A

(3.26)

Hence, the chain converges with increasing length to a state of zero transmission (perfect reflection). The convergence is exponential. This √ can be best seen by locally expanding the dynamics around the fixed-point in variables B = 1 − A2 and χn = χa + δχn and so we obtain Bn , δχn ∼ |κ+ |−n .

(3.27)

Notice that the norms of all matrix elements converge to limiting values, but the phases do not, with the exception of rL . Similar behaviour was encountered in the higher-dimensional chains.

3.1.2

Noisy generating matrix

Here we discuss chains that are composed of randomly chosen scatterers. Such chains of scatterers are known as random chains or random wires. The scattering matrix S n of a chain of length n is constructed from its initial state S1 by merging with generating matrices S that are randomly

30


chosen on each step of lengthening. To simplify the discussion we define three disjoint sets of generating scattering matrices: Mb = {S ∈ U (2) : D < 0} ,

(3.28)

Mm = {S ∈ U (2) : D = 0} .

(3.30)

Ml = {S ∈ U (2) : D > 0} ,

(3.29)

We call Mb the set of ballistic matrices, Ml , the set of localised matrices and Mm the set of marginal matrices. In order to measure the volume of introduced sets we use a uniquely defined invariant measure over unitary matrices µH , called the Haar measure (Conway, 1990; Reichl, 2004), and normalised so that µH (U (2d)) = 1. The Haar measure of the marginal matrices µH (Mm ) is obviously zero thereby making this set uninteresting for our general discussion. The measures of the other two sets are the following: µH (Mb ) =

2 π2

Z

1

dx 0

Z

π

dθ 0

µH (Mb ) = 1 − µH (Ml ) =

Z

π 0

dδ H[(sin θ)2 (1 − x2 ) − (sin δ)2 ] ,

1 , 2

(3.31) (3.32)

where H[x] = {1 : x > 0; 0 : x < 0} is the Heaviside step-function. It is important to think about the role of these sets when used in construction of chains. Let us construct a chain of length n in which we use m(n) localised generating matrices from Ml . It is evident that in the case the ratio m(n)/n in the limit n → ∞ is finite then the transmission of the chain converges exponentially toward the zero. In the opposite case, when the chain is constructed mostly of ballistic generating matrices from Mb , the transmission can in the worst case decrease linearly with the length of the chain. Now we consider chains of one-dimensional scatterers in which the generating scattering matrix can vary with the length of the chain. In the parametrisation (3.16), the dynamics of the chain’s scattering matrix in the process of lengthening is determined by the recursion Bn+1 =

Bn B p , 1 + 2An A cos(φn + αL ) + (An A)2 L

(3.33) L

φn+1 = φn + 2λ + arg{(1 + An Ae−i(φn +α ) )(An + Ae−i(φn +α ) )} , L βn+1 = βnL + β L + arg{(1 + An Ae−i(φn R βn+1

=

βnR

R

+ β + arg{(1 + An Ae

+αL )

)} ,

−i(φn +αL )

)} ,

(3.34) (3.35) (3.36)

where we introduce additional variable φn = βnL + βnR − αnL = χn − αL . This four dimensional dynamical system can be reduced to a two dimensional one. It is described by the transmission intensity Bn , phase φn and evolution equations (3.33) and (3.34). The iteration step is controlled by three parameters A, λ and αL that can change along the chain. This is one more parameter in comparison to the static case of the generating matrix. Just for illustration, we show in figure 3.4 an example of the chain generated by a noisy scattering matrix S chosen deep enough in the ballistic set Mb . We see that the presence of a small noise does not strongly deform the trajectory. It just induces a small variation around unperturbed trajectory given by the integral of motion (3.25) for the average generating scattering matrix hSi. Let us now discuss a bit more general case of random chains. These chains are built using generating matrices from some the set A intersecting the set of localised matrices M l so that µH (A ∩ Ml ) > 0. We are mainly interested in the transmission properties of asymptotically long chains. Intuitively we expect that the transmission will converge exponentially towards zero exhibiting exponential localisation. In the limit of long chains the transmission is very small and

3.1. Chain of one-dimensional scatterers

31

Figure 3.4: The phase space portrait in the ballistic regime for parameters A0 = 0.5 and λ0 = 0.628319 without the noise (small points) and with sample trajectories obtained in the presence of noise in the generation matrix S (large points). The latter is defined by A = A0 ζ, λ = λ0 + ζ, where = 0.001 and ζ ∈ [−1, 1] is uniformly distributed stochastic variable.

we can replace the exact evolution with the following approximation: Bn+1 = Bn f (B(n), φn + α(n)) ,

p φn+1 = φn + 2λ(n) + 2 arg{1 + 1 − B(n)2 e−i(φn +α(n)) } , x f (x, y) = q , √ 1 + 2 1 − x2 cos(y) + (1 − x2 )

(3.37) (3.38) (3.39)

where B(n), α(n) and λ(n) are parameters of the generation scattering matrix S ∈ A in n-th iteration step. The matrix S is picked randomly at each iteration step and so the recursion equations (3.37) and (3.38) represent a stochastic dynamical system. By taking into account the equation (3.37) the transmission through the chain of length n can be written as a product Bn = B 1

n Y

f (B(k), φk + α(k)) .

(3.40)

k=1

In order to understand the scaling of the transmission scales with the length n we introduce the transmission decay rate In = n1 log(Bn ), which reads n

1X log[f (B(k), φk + α(k))] , In = n k=1

n1.

(3.41)

We would like to express the distribution of In over an ensemble of realisations of the chain in the limit n 1, defined as Pn (I) = hδ(I − In )istoch. , (3.42) where h. . .istoch. denotes the average over S ∈ A. To obtain this, we need to know the dynamics of the variable φn determined by (3.38). The average distribution of the position of the dynamical system φn starting at point φ1 is defined by ρn (φ; φ1 ) = hδ(φ − φn )istoch. .

(3.43)

It is meaningful to assume that ρn has a limiting distribution independent of the initial position φ1 that is written as ρ(φ) = lim ρn (φ; φ1 ) . (3.44) n→∞

32


In some simple stochastic processes we can analytically express ρ(φ), but generally this is not the case. If ρ(φ) exists then by using the central limit theorem (Feller, 1970) it is straightforward to show that the limiting distribution of In is a Gaussian distribution r (I − I)2 n exp −n Pn (I) = , n1. (3.45) 2πσI2 2σI2 with the first I and the second moment σ 2 given by

2 I = hlog[f (B, φ + α]i , σI2 = log[f (B, φ + α]2 − I ,

(3.46)

where h. . .i is an average over the stochastic variables B and α, and over the limiting distribution ρ(φ) (3.44). The result (3.45) implies that the transmission decays exponentially with the rate distributed by a Gaussian function (3.45). A similar result was already reported in (Anderson et al., 1980) and (Abrahams and Stephen, 1980) using scaling techniques and transfer matrix formalism, respectively, that are technically complicated compared to our derivation. The theoretical predictions are supported by numerical studies of which two examples are shown in figure 3.5. There is a good agreement between the measured distribution of In and the one predicted theoretically. 3.5 3

4

measurments theory

3.5 3 2.5

2

Pn(I)

Pn(I)

2.5

1.5

2 1.5

1

1

0.5 0 -1.8

measurments theory

0.5 -1.6

-1.4

-1.2

-1

-0.8

-0.6

-0.4

0 -2.2

I

(a)

-2

-1.8

-1.6

-1.4

-1.2

-1

I

(b)

Figure 3.5: Theoretical and measured distribution of the average decay rate In at n = 100 for two examples of random chains, where A = 0.1, B ∈ [0, 1] uniformly distributed. Variable a was uniform over [0, 2π] in case (a) and over [0.5, 0.7] in case (b). The theoretical curves are given with I = −1, σ 2 = 1.4674 in the case (a) and with I = −1.56325, σ 2 = 1.19556 in the case (b).

3.2

Chain of d-dimensional scatterers

We continue our discussion with the general linear chains of d-dimensional scatterers. The large parametric space of scattering matrices2 makes the analytical discussion very limited and so the results are mostly numerical. The scattering matrices of the chain are created using the general recursion relation (3.11) in the scattering matrix formalism. We limit ourselves to periodic chains with fixed generating scattering matrices. Here our discussion relies on the transfer matrices in the larger extent than in the onedimensional case. Let us review some algebraic properties of the transfer matrices. The symmetry (3.6) yields the following relations between eigenvalues and eigenvectors of the transfer matrix T 1 T v = κv ⇐⇒ (Kv)† T = ∗ (Kv)† , (3.47) κ 2

The set of unitary matrices U (2d) is Lie group of dimension (2d)2 (Elliott and Dawber, 1979).

3.2. Chain of d-dimensional scatterers

33

and for two eigenvalues κ1,2 with corresponding left eigenvectors v1,2 the symmetry (3.6) gives an additional equation (κ1 κ∗2 − 1)v2† Kv1 = 0 . (3.48) Meaning that every eigenvalue κ, with the left eigenvector v, has a corresponding an eigenvalue 1/κ∗ , with the right eigenvector Kv. An eigenvalue on the unit circle κ = eiω , ω ∈ R without degeneracies with assigned left-hand eigenvector v implies the right eigenvector Kv and the property v † Kv 6= 0. For the left eigenvector v corresponding to the eigenvalue living outside the unit circle we get the identity v † Kv = 0. The dynamical system of scattering matrices Sn defined by the recursion (3.11) has all Lyapunov exponents (Reichl, 1992) equal to zero and so it is not chaotic. This becomes evident from the following discussion. Let us assume that we have a generating scattering matrix S and corresponding transfer matrix T [S] with its eigen spectrum {κ1 , . . . , κ2d } and corresponding eigenvectors vi in the columns of the matrix P = [vi ]2d i=1 , Then we can write the dynamics of the chain in the process of lengthening in the transfer matrix formalism as −1 Tn = P diag{eκi (n−1) }2d T1 . i=1 P

(3.49)

We distinguish two types of dynamics of the scattering matrix description of the chain depending on the spectral properties of the transfer matrix: (i) If all eigenvalues are on the unit circle, |κi | = 1, ∀i the dynamics of Sn is quasi-periodic. This type of dynamics we call ballistic motion and the corresponding generating scattering matrices forms a set of ballistic unitary matrices Mb . (ii) If there is an eigenvalue outside the unit circle then the transmission T [Sn ] of the chain decays exponentially towards some plateau value around which it oscillates as the chain is lengthened. Let us explain this behaviour in a purely periodic chain, where Tn = T [S]n . Following the definition of the transfer matrix (3.4) and the relation to the scattering matrix (3.5) the average transmission probability can be expressed by the bottom diagonal block of the matrix T n as Tn =

1 −1 tr{x−† 4,n x4,n } . d

(3.50)

The right eigenvectors of T [S] are the rows in the inverse transition matrix P −1 = [uTi ]2d i=1 . We write the left eigenvector as viT = [αi , βi ] ∈ C2d and the right eigenvector as uTi = [ζi , ηi ] ∈ C2d by introducing the upper halves vi , ζi ∈ Cd and the lower halves βi , ηi ∈ Cd of eigenvectors. Then according to (3.49) the dynamics of the block x4 in Tn reads x4,n =

2d X

eκi n βi ηiT .

(3.51)

i=1

We denote by K = {i : |κi | > 1} the set of indices corresponding to eigenvalues outside the unit circle. The vectors ui , vi with i ∈ K satisfy the identity vi† Kvi = u†i Kui , which implies that upper and lower halves of these vectors are non-trivial: kαi k2 = kβi k2 6= 0 and kζi k2 = kηi k2 6= 0. We introduce projection matrices PuL and PuR onto the set of vectors {βi }i∈K and {ηi }i∈K , respectively. Then the transmission can be expressed as a sum of non-decaying and decaying terms: T [Sn ] = T0 [Sn ] + O(exp(−βIn)) ,

|κ|max = max{|κm |} , β ∈ {1, 2} m

(3.52)

with non-decaying term written as T0 [Sn ] = tr{[(id − PuL )x4,n (id − PuR )]−† [(id − PuL )x4,n (id − PuR )]−1 } ,

(3.53)

34


Haar measure(ballistic matrices)

where we introduce decay rate I analogous to that in the one-dimensional case. If the off-diagonal blocks of the matrix x4 expressed in terms of introduced projectors are zero, (id − PuL )x4 PuR = PuL x4 (id−PuR ) = 0, then the coefficient β = 1, otherwise β = 1. The latter is the statistically more common situation. In case the block x4,n is approximately a random matrix then T0 [Sn ] ∼ #K/d. The presented dynamics of transmission is called localised motion and the set of corresponding generating scattering matrices are calledlocalised matrices denoted by M l . In case that the projection Pu is a trivial, when #K = d, then the transmission decays to zero and we are talking about total localisation, otherwise we only have partial localisation. The set of generating (unitary) scattering matrices is split into set of ballistic M b and localised matrices Ml . The separation is made on the ground of eigenvalues of the corresponding transfer matrices and it is interesting to know the Haar measures of these two sets. The measures are obtained numerically by generating unitary matrices uniformly with respect to the Haar measure (Stewart, 1988) and checking the spectrum of the corresponding transfer matrix for eigenvalues outside of unit circle. The result is plotted in figure 3.6, where we see that the measure of ballistic matrices decreases very fast with dimension d approximately as exp(−O(d 2 )). We conclude that finding chains of non-exponential decay is infinitesimally small in the semi-classical limit d → ∞. 1

measurments fit, exp(-f(x))

0.1 0.01 0.001 1e-04 1e-05 1e-06

0

5

10

15 d(d+1)

20

25

30

Figure 3.6: The Haar measure of the set of ballistic unitary matrices for different dimensions of scatterers d. The fitted function is f (x) = 0.465792x − 0.238663, where x = d(d + 1).

It is instructive to study the eigenvalues of transfer matrices corresponding to localised scattering matrices Ml as they give information about the length-scales of transmission decay T [Sn ]. We define the set of eigenvalues of the transfer matrix T [S] Σ[S] = {κ : det(T [S] − κid) = 0} ,

(3.54)

and denote by du (S) the number of eigenvalues in Σ[S] outside the unit circle. We investigate the distribution of the maximal modulus of values in Σ[S] Z dµH (S)δ(t − max |Σ[S]|) , t ≤ 1 , (3.55) Pmax (t; d) = U (2d)

and distribution of the relative number of eigenvalues outside circle denoted by P max Z dµH (S)δ(t − du (S)/d) , t ∈ [0, 1] , Pu (t; d) =

(3.56)

U (2d)

over the set of scattering matrices S ∈ U (2d) taken uniformly with respect to the Haar measure µH . From the distributions Pmax and Pu we can learn about the decay rates and the percentage of unstable eigenvalues involved in the decay of transmission by lengthening the chain, respectively.


35 25

1 0.1 0.01 0.001 1e-04 1e-05 1e-06 1e-07

0.35

Pmax (t)

0.3 0.25 0.2 0.15 0.1

d=2 d=3 d=4 d=5 d=6 d=7 d=8 d=9 -3 t 1

10

100

1000

d=2 d=5 d=10 d=20 d=50 d=100

20

Pu(x)

0.4

15 10 5

0.05 0

2

4

6

8

10

12

14

16

0

0

0.2

t

(a)

0.4 0.6 x = du/d

0.8

1

(b)

Figure 3.7: The distributions of maximal eigenvalues by absolute values of the transfer matrices P max (a) and the distribution of the relative number of eigenvalues outside the unit circle P u (b).

Both of distributions are numerically calculated for several dimensions d and shown in figure 3.7. In figure 3.7.a we see that Pmax has algebraic asymptotics, Pmax (t) ∼ t−3 as t → ∞, and is zero on the unit circle t = 1. By closer analysis one can find that the distribution P max (t) also has a limiting semi-classical form 2 lim √ Pmax d→∞ d

! √ d t; d = P (t) , 2

(3.57)

with P (t) having asymptotic algebraic tails dependence, when t → ∞, given by P (t) = at−3 ,

. a = 16.0 ± 0.05 .

(3.58)

This √ scaling of Pmax (t) implies that the average decay factor increases with the dimension d as t ∼ d and consequently the decay rate increases as I ∼ 12 log(d). The scaling law does not work for the one-dimensional case (d = 1), but this is not very surprising. The number d u (S) corresponds to the dimension of space that is exponentially stretched from left to the right side when the chain lengthened. From results for the distribution Pu shown in 3.7a we see that the dimension du on average increases with increasing dimension d and converges towards d. We conclude that on average the transmission at higher dimensions d decays faster to a lower asymptotic plateau given by hT [Sn ]0 in (3.53), where h•in is the average over the chain length n. In addition we are interested in the transport measures R and σR in the introduced sets of scattering matrices and in the correlation between the two measures. We calculate points mR (S) = (R, σR )(S) ∈ [0, 1] × [0, 21 ] for S uniformly sampled over sets Mb , Ml and U (2d) with respect to the Haar measure. The results are plotted in figure 3.8. We see that the points mR (Mb ) occupy the available space more intensely at smaller values of the average reflection R. This means that it is more probable that a ballistic matrix has smaller average reflection. The points mR (S) in localised scattering matrices are scattered over all available space given by the set mR (U (2d)). Let us look at the set of points mR (U (2d)) in more detail. We find that for d = 2 the points mR are to some extent correlated in the region given by the inequality 1 1 1 (3.59) − R− . σR ≤ √ 2 3 2 The correlations between the measures decrease with increasing dimension d making them statistically independent. Therefore the distribution d2 P/dσR dR of points mR at high dimensions

36


(a)

(b)

(c)

Figure 3.8: The measures of reflection (r, σr ) for the ballistic (a), localized (b) and general (c) set of 1 unitary scattering matrices at different dimension d, where f (x) = 3− 2 ( 12 − |x − 21 |).

can be expressed as a product of distributions dP/dR and dP/dσR of R and σR , respectively. We write this as d2 P dP dP (R, σR ) (R) (σR ) , d1. (3.60) dσR dR dR dσR Both of distributions dP/dR and dP/dσR converge with increasing dimension to a Gaussian as shown in figure 3.9. With increasing dimension the distributions of R and σR become more narrow. The first stays centred around R = 21 and the second moves towards zero with increasing d. Their first two central moments are given in figure 3.10, where we see that dP/dR is centred 25

200

d=10 d=15

180

20

d=10 d=15

160 140

dP/dR

dP/dσR

15

10

120 100 80 60

5

40 20

0 0.35

0.4

0.45

0.5 R

0.55

0.6

0.65

0 0.075 0.08 0.085 0.09 0.095 0.1 0.105 0.11 0.115 0.12 0.125 σR

Figure 3.9: Distribution of the average reflection probability dP/dR, standard deviation of reflection probability dP/dσR for general unitary matrices U (2d) uniformly distributed with respect to Haar measure.

around R = 12 with dispersion decreasing as O(d−1 ), whereas the average and dispersion of σR decreases with dimension as O(d−p ) and O(d−p−1 ), respectively, where p ≈ 0.48. The difference of one order in the dependence of average value and standard deviation on the dimension d is typical for distributions of statistical averages, as are our measures of reflection. The ballistic scattering matrices are rare in the set of unitary matrices. By using generating matrices from the ballistic set the resulting scattering matrix of the chain is also contained in that set. The reflection measures R and σR behave quasi-periodically, with increasing length and we examine them in three examples of the chain in which the generating scattering matrix is fixed and just chosen randomly. In figure 3.11 we show the evolution of the reflection measures mR (Sn ) along the length n of the chain in the (R, σR ) plane. We see that points {mR (Sn )}n∈N do not cover all available space and their distribution in general strongly differs from the distribution of mR (U (2d)). It seems that points mR (Sn ) are bounded to some open region determined by the


37

1 0.1 0.1 0.01

0.01

0.001 ( − 2)½

( − 2)½ 1/2

0.3262*d−0.4812

−1.008

0.001

−1.4844

0.2554*d 1

10 d

100

1e−04

(a)

0.1167*d 1

10 d

100

(b)

Figure 3.10: The first two central moments of distributions of the average reflection probability dP/dR (a) and of the standard deviation of refle ction probability dP/dσR (b).

generating matrix S and the initial scattering matrix S1 . In addition, we examine the Fourier

example 1

(a)

example 2

(b)

Figure 3.11: The reflection measures mR (Sn ) as the length of the chain n increases for n = 1 to 65356 in three examples of two (a) and three (b) dimensional ballistic chains.

power spectrum of the average reflection of the chain R(Sn ) as this is increased in length n. In figure 3.12 we plot the power spectral density (PSD) of the set {R(Sn )}N n=1 (N = 1000) for chains discussed in the figure 3.11. We know that in the one-dimensional case the dynamics is quasi-periodic with a simple Fourier spectrum. In chains of higher dimensional scatterers the Fourier spectrum becomes complicated in general. The peaks are distributed on the whole range of frequencies making a spectrum rather dense, but still far away from continuous. We can therefore conclude that the dynamics of the scattering matrix in the ballistic chains for finite dimensions is still quasi-periodic.

38


0.01

0.85

0.001

0.8

1e-04

0.75

1e-05

PSD(R)n

Rn

0.9

0.7 0.65

1e-07

0.6

1e-08

0.55

1e-09

0.9 0.85 0.8 0.75 0.7 0.65 0.6 0.55 0.5 0.45 0.4 0.35

0

50

100 n

150

1e-10

200

0

100

200

300

400

500

300

400

500

n 0.001 1e-04 1e-05 PSD(R)n

0.5

Rn

1e-06

1e-06 1e-07 1e-08 1e-09

0

50

100 n

150

1e-10

0

100

200 n

example 1

(a)

200

example 2

(b)

Figure 3.12: The evolution of the average reflection Rn with increasing length n for n ∈ [0, 1000], where we show only n ∈ [0, 200], (a) and the one-sided power Fourier spectrum density PSD(R) n of this evolution (b) in two examples of the one and two dimensional ballistic chains (top row, bottom row).

39

Chapter 4

Scattering across a single bend The design of the serpent billiard incorporates straight and bent segments. We think of the serpent billiards as linear chains of quantum scatterers, where the bends have the role of the latter. The bends are the only non-trivial parts of the serpent billiard that have to be understood before attacking the main problem – the serpent billiard. The bend is discussed as a two-way scatterer in the scattering matrix formalism introduced in Chapter 3, with the difference that here the scatterer is specific and work is performed on the full functional basis of modes. We are interested in the scattering of a non-relativistic particle across our two dimensional open billiard composed of a bend and infinite straight wave-guides attached to its ends, which we call leads or asymptotic regions. Schematic picture of the billiard is depicted in figure 4.1. Notice that the walls are connected so that they continue smoothly. The bend has the inner (outer) radius r = q (1) and the angle equal to β and the straight wave-guides are of width a = 1 − r. The scattering is managed in the mode matching techniques (MMT), referred to as the modal approach, and presents an effective method to master such on-shell scattering. The

φ

y

B β

A r

x

r=1

r=q

x

y

C

Figure 4.1: A schematic picture of a finite bend with the inner radius r = q and outer radius r = 1 on a straight wave-guide of width a = 1 − q.

area of the billiard, denoted by Ω, is separated into three sections: A – left lead, B – bend and C – right lead. We search for the wave function ψ(r ∈ Ω), which is the solution of the stationary

40

Chapter 4. Scattering across a single bend

Schrödinger (Helmholtz) equation on the domain Ω with Dirichlet boundary conditions: −∆ψ(r) = k 2 ψ(r) ,

ψ|r∈∂Ω = 0 ,

(4.1)

where E = 21 k 2 is the energy and k the corresponding wave-number. In the leads A and C, by using the Cartesian coordinate system, r = (x, y), the equation (4.1) reads 2 ∂2 ∂ + ψ = k 2 ψ , ψ|y=0,1−q = 0 , (4.2) ∂x2 ∂y 2 and in the bend, described in polar coordinate system, r = (r, φ), the equation (4.1) is of the form 2 1 ∂2 1∂ ∂ + + ψ = k 2 ψ , ψ|r=q,1 = 0 , (4.3) ∂r2 r ∂ r2 ∂φ2 The modal approach suggests that we first solve the equation (4.1) on each section of the open billiard separately. These partial solutions are called mode functions. In the case of the infinite straight wave-guide (4.2) and the open bend (4.3) the mode functions are given by ansätze ψ ∝ u(y) exp(igx) and ψ ∝ U (r) exp(iνφ), respectively, so the equations (4.2) and (4.3) are rewritten: d2 u + (k 2 − g 2 )u = 0 , dy 2 d2 U 1 dU ν2 2 + + k − U = 0, dr2 r dr r2

u|0,a = 0 ,

y ∈ [0, a] ,

(4.4)

U |q,1 = 0 ,

r ∈ [q, 1] ,

(4.5)

with the channel width a = 1−q. The constants g, µ ∈ C are called mode numbers. The equations (4.4) and (4.5) give a discrete set of possible solutions for the modes (the mode functions and the corresponding mode numbers) denoted by pairs (gn , un (y)) and (νp , Up (r)), where p, n ∈ N. The modes in the straight wave guide are given by expressions r πn 2 π 2 2 2 k = gn + , un (x) = sin nx . (4.6) a a a whereas the expression for the modes in the bend are more complicated and its discussion follows. Using introduced mode pairs we can formulate modes in the straight part of the wave guide as e± n (r) = un (y)

exp(±ign x) , √ gn

(4.7)

exp(±iνn φ) √ νn

(4.8)

and in the bent part fp± (r) = Up (r)

where the sign ± distinguishes two directions of phase (probability flux) propagation and the √ square-root of a complex number z = |z| exp(iφ), φ ∈ [0, 2π) is defined as z = |z|1/2 exp(iφ/2). The mode functions in the bend Up (r) are proportional to the well known cross-products of Bessel functions (Cochran, 1964) written as Zν (k, r) = Jν (kr)Yν (k) − Yν (kr)Jν (k) ,

(4.9)

with the Bessel functions of the first Jν and the second kind Yν (Olver, 1972). The allowed values of ν called the mode numbers are determined from the Dirichlet boundary conditions 2 . The mode Zν,k (q) = 0 and are only real or purely imaginary and ordered so that νp2 > νp+1 numbers and the corresponding mode functions are the building stones of the modal approach

4.1. The scattering matrix of a bend

41

for scattering across a bend. They are discussed in Appendix D in much detail. For convenience we introduce two scalar products on the cross-section of our open billiard defined as Z 1 Z a dr (a, b) = a(r) b(r) , ha, bi = dy a(y) b(y) . (4.10) q r 0 and define mode functions Up (r) as Zνp ,k (r) , Up (r) = q (Zνp ,k , Zνp ,k )

νp ∈ Mk,q,+ ,

p∈N,

(4.11)

The sets of mode functions {Up (r)}p∈N and {un (y)}n∈N are complete orthogonal basis of Hilbert spaces ((•, •), L2 [q, 1]) and (h•, •i, L2 [0, a]), respectively. An essential ingredient in the scattering calculation are the overlaps of mode functions Anp = hun , Up i ,

Bnp = (un , Up ) ,

(4.12)

that represent transition matrices between the sets of mode functions in different regions, X X un (x) = Bnp Up (r) , Up (r) = Anp un (x) , (4.13) p

n

yielding an important identity for our numerical analysis AB T = AT B = id .

(4.14)

Mode functions corresponding to real and imaginary mode numbers are called open modes or travelling waves and closed modes or evanescent (decaying) waves, respectively. The solution ψ(r) in the whole open billiard region (4.1) is expressed as X + − − a+ r ∈ ΩA , (4.15) ψ(r) = n en (r) + an en (r) , n

ψ(r) =

X

+ − − λ+ p fp (r) + λp fp (r) ,

p

ψ(r) =

X

+ − − b+ n en (r) + bn en (r) ,

n

r ∈ ΩB ,

r ∈ ΩC ,

(4.16)

(4.17)

where ΩA,B,C are domains corresponding to sections A, B and C, respectively. The expansion ± ± coefficients a± r) is smooth n , λp and bn are determined by the condition that the wave function ψ(~ across the whole domain. In the first section we will present the scattering matrix formalism (Newton, 2002) using the modal approach. In the quest for a controllable accuracy and stability of our calculation, we present in the second section a numerically stable MMT method for the calculation of scattering matrix of the single bend. In the method we have incorporated the knowledge about the mode structure of the Laplacian in the bend, which is presented in the Appendix D. In the third section, we continue with the discussion of numerical results and some analytical estimates of the quantum transport properties obtained by the developed method.

4.1

The scattering matrix of a bend

The scattering matrix S (Newton, 2002) presents a formal connection between the incoming and outgoing “waves” with respect to the scatterer, which is in our case a region of the bend.

42


± = {b± } We organise vectors of expansion coefficients a± = {a± n }n∈N and b n n∈N of waves in asymptotic regions into incoming and outgoing SOS states introduced in Chapter 3 and denoted by vin = (a+ , b− ) and vout = (a− , b+ ), respectively. Then the scattering matrix S maps the incoming into the outgoing SOS states

Svin = vout .

(4.18)

which has in our case a simple symmetric block form R T , S= T R

(4.19)

with R and T being the reflection and the transmission matrix, respectively. By reordering rows and columns in S so that matrix elements concerning open (subscript o) and closed (subscript c) waves are separated and grouped together Soo Soc , (4.20) S→S= Sco Scc we can find that the matrix S (4.20) is generalised unitary (Prosen, 1995). This property is defined by the following relations † † Soo Soo = Soo Soo = id ,

† i Soo Sco † i Soc Soo † i Sco Sco

(4.21)

= Soc ,

= Sco ,

(4.23)

† i Soc Soc

=

(4.22)

= Scc −

† Scc

.

(4.24)

The symmetry relations (4.21 - 4.24) result from the probability flow conservation and are in our case equivalent to the condition AB T = id (4.14). Due to the time-reversal symmetry of the physical problem the scattering matrix is symmetric ST = S ,

ST = S .

(4.25)

The existing block symmetry (4.19) simplifies the calculation of the scattering matrix to a discussion of a particular incoming wave e+ n from left side represented by the following wave function ansatz X ψ(r) = e+ e− r ∈ ΩA , (4.26) n (r) + m (r) Rmn , ψ(r) =

X

m

fp+ (r)

− − Λ+ pn + fp (r) Λpn ,

p

ψ(r) =

X m

r ∈ ΩB ,

e+ m (r) Tmn . r ∈ ΩC ,

(4.27) (4.28)

The matrix elements Rmn , Tmn and Λ± pn are determined by the condition that the wave function ψ(r) is smooth across the entire region. The continuity of ψ(r) and its normal derivative on connecting cross-sections between regions ΩA,B,C give the following systems of matrix equations id + R = M (Λ+ + Λ− ) , +

−

id − R = N (Λ − Λ ) , with

1

1

M = G 2 AV − 2 ,

T = M (FΛ+ + F −1 Λ− ) , +

T = N (FΛ − F 1

N = G− 2 BV

1 2

,

−1

−

Λ ),

(4.29) (4.30) (4.31)

4.2. Numerically stable scheme for scattering matrix calculation

43

which we write using the diagonal matrices V = diag{νn }n∈N , G = diag{gn }n∈N and F = exp(iV ), and the transition matrices A and B (4.12). The elimination of matrices H ± from equations (4.29) and (4.30) yields the blocks of the scattering matrix S reading T = (C − FDC −1 FD)−1 F(C − DC −1 D) ,

R = (C − FDC

−1

FD)

−1

(FDC

−1

FC − D) ,

(4.32) (4.33)

that we express by using the following additional matrices: 1

1

1

1

C = M −1 + N −1 = V 2 B T G− 2 + V − 2 AT G 2 ,

(4.34)

1 2

(4.35)

D=M

−1

−N

−1

T

=V B G

− 21

−V

− 21

T

1 2

A G ,

where we take into account the relation AB T = id. We choose the present form of the transmission matrix T (4.32) and the reflection matrix R (4.33) in order to increase its numerical stability by minimizing the use of inverses and avoiding the use of F −1 .

4.2

Numerically stable scheme for scattering matrix calculation

The bend on a straight wave-guide is almost a paradigmatic example for testing numerical schemes and ideas on how to accurately calculate the scattering matrix. In particular, the high curvature case q → 0 turns to be highly non-trivial. Here we give a simple and stable procedure to obtain the scattering matrix of a bend with a clear precision control for practically all curvatures. The scattering across a bend of angle β having the inner radius q at the wave-number k is described by the scattering matrix S(β) (4.19) composed of the reflection matrix R(β) (4.33) and the transmission matrix T (β) (4.32). In practice, we work with finitely dimensional matrix approximations, denoted by RN (β), TN (β) ∈ CN ×N ,

SN (β) ∈ C2N ×2N ,

AN , BN ∈ RN ×N ,

(4.36)

where N is the number of modes used in the asymptotic regions. The main objective is to construct these finitely dimensional matrices RN (β) and TN (β) so that: 1. calculations are numerically stable and precisem 2. SN (β) satisfy the time reversal symmetry (4.25) and are generalized unitary (4.21 - 4.23)m 3. the sub-blocks of matrices RN (β) and TN (β) of dimension N 0 ∈ (No , N ] are calculated with controllable accuracy, where No = bka/πc is the number of open modes in the asymptotic region. The recipe to achieve these may be separated into two parts. In the first part we cure the numerical instability caused by the maximal element r(β) = exp(βνN ) in FN (β), which is exponentially diverging with increasing N . This is achieved by separating the bend into 2 n identical subsections of angle β 0 = 2−n β so that εnum r(β 0 ) ≈ 1, where εnum is the numerical precision (in double precision εnum = 2−52 ). The number n is calculated as | log εn | n = max 0, 1 + log2 . (4.37) βνN The calculation of the scattering matrix of the bend subsection SN (β 0 ) is stable to high precision. By joining these elementary subsections and corresponding scattering matrices together we obtain the full matrix of the bend SN (β) using the recursion n times SN (2−m+1 β) = SN (2−m β) SN (2−m β) .

(4.38)

44


The symbol represens the concatenation of scattering matrices associated to scatterers on the wave-guide explained in B. In the second part we focus on the numerical accuracy of the scattering matrices S N (β 0 ), where our main concern goes to the problem that S(β 0 )N diverges with increasing N . This can be traced-back to the transition matrices AN and BN and their violation of the identity AB T = id (4.14) valid in the infinite dimensional case. We eliminate the problem by deforming T = id and make them non-singular. We AN and BN so that they fulfill the condition AN BN make the SVD decomposition of the truncated matrix AN = UN ΣN VNT , modify their singular ˜ values ΣN = diag{σi }N σ i }N i=1 to ΣN = diag{˜ i=1 so that they fit in bounds obtained for the infinite dimensional case derived in Appendix D and given by (D.62) 1 σi : σi ∈ [q 2 , 1] σ ˜i = (4.39) σ ∗ : otherwise and again generate both matrices ˜ N VNT , AÑ = UN Σ

˜ N = UN Σ ˜ −1 VNT , B N

(4.40)

or the alternative using the matrix BN as a base for generation of both matrices. The value of σ ∗ > 0 can be chosen almost arbitrarily and the most elegant choice is σ ∗ = 1. Presented adaptations of matrices AN and BN , (4.40) make the scattering matrix SN (β 0 ) and consequently SN (β) generalised unitary with a well behaved limit N → ∞, at least in the sub-space of N 0 modes. In the following we give a numerical verification of the latter in two steps. In the first step we check the precision of transition between modes in the bend and in the straight wave-guide in the sub-space of dimension N 0 < N by using the deformed transition maÑ . We discuss the following measures of the transition error, when transitioning trices AÑ and B from the modes in the asymptotic region (infinite wave-guide) to the ones in the bend   N N X X ˜mp − δnm | , (4.41) Añp B εs→b (N, N 0 ) = max 0 |hun |  |Up )(Up | − id |um i| = max 0 | n,m≤N

p=1

n,m≤N

p=1

and when in the opposite situation # "N N X X 0 ñp − δpr | . Añr B εb→s (N, N ) = max 0 |(Up | |un ihun | − id |Ur )| = max 0 | p,r≤N

n=1

p,r≤N

(4.42)

n=1

We can see that the introduced transition errors (4.41) and (4.42) measure the violation of the identity AB T = id in the subspace of dimension N 0 , when working with finite number of modes N . It is expected, but not obvious, that the errors vanish in the limit N → ∞ for a fixed N 0 , inner radius q and wave-number k. This conjecture is strongly supported by numerical studies. In figure 4.2 we show the dependence of transition errors on Nc = N − No at some fixed number of closed modes on sub-block Nc 0 = N 0 − No = 10 and two values of No = 10, 100. The errors decrease with increasing Nc down to a certain plateau given by the precision of mode functions. We find that the main error is contained in the first few real modes of the bend. A conservative estimate for the plateau is of order 10−8 . The transition errors (4.41) and (4.42) increase with decreasing q indicating that we need more modes to achieve equally small error as for higher q. From the results shown it seems that the transition from modes in the bend to the ones in the asymptotic region is less accurate than vice versa. We did not find any analytic approximation for transition errors and so we only numerically estimate the dimension of the whole functional space Nt = Nc + No necessary for the transition errors to be are smaller than some ε for some wave-number k and inner radius q Nt (N 0 , ε) = min{N : εs→b (N, N 0 ) < ε ∧ εb→s (N, N 0 ) < ε} ,

(4.43)

4.2. Numerically stable scheme for scattering matrix calculation

45

1

1

0.1

0.1

0.01

0.01

0.001

0.001

1e-04

1e-04

εs -> b

εs -> b

where N 0 = Nc 0 + No is the dimension of the observed sub-space. In our numerical analysis ε ≈ 10−7 .

1e-05

1e-06

1e-07

1e-07

1e-08

1e-08

1e-09

1e-09

1

10

100

1

Nc

0.1

q = 0.1 q = 0.2 q = 0.3

10

100 Nc

0.1

q = 0.4 q = 0.5 q = 0.6

0.01

0.01

q = 0.7

0.001

0.001

q = 0.8

1e-04

1e-04

q = 0.9

εb -> s

εb -> s

1e-05

1e-06

1e-05

1e-05

1e-06

1e-06

1e-07

1e-07

1e-08

1e-08

1e-09

10

100

1e-09

10

100 Nc

Nc

Figure 4.2: The transition errors εs→b (top row) and εb→s (bottom row) as a function of the total number of used closed modes Nc in the sub-space of dimension N 0 = Nc + No where Nc0 = 10 and No = 10, 100 (left, right).

In the second step we discuss the convergence of RN (β) and TN (β) on some fixed sub-space of dimension N 0 < N with increasing N . Both matrices are calculated with presented algorithms to improve precision and enable stability. The convergence is monitored through the relative difference of matrices at subsequent changes of dimension N called measure of convergence : εR (N, N 0 ) =

kRN +1 (β) − RN (β)kN 0 , kRN (β)kN 0

kTN +1 (β) − TN (β)kN 0 , kTN (β)kN 0

εT (N, N 0 ) =

(4.44)

where we introduce a norm kAkM = maxi,j≤M |Aij | in the sub-space of dimension M . The measures (4.44) give a simple upper bound for the deviation of matrices from their asymptotic limit kR∞ (β) − RN (β)kN 0 ≤ C1 (N 0 )

∞ X

M =N +1 ∞ X

kT∞ (β) − TN (β)kN 0 ≤ C2 (N 0 )

εR (M, N 0 ) ,

(4.45)

εT (M, N 0 ) ,

(4.46)

M =N +1

with expressions C1 (N 0 ) = maxN ≥N 0 kRN (β)kN 0 and C2 (N 0 ) = maxN ≥N 0 kTN (β)kN 0 being of order 1. The rhs of eqs. (4.45) and (4.46) can be thought of as estimates of accuracy of the matrices RN (β) and TN (β). We numerically study the dependence of measures εR,T on the number of all considered closed modes Nc for two cases No = 10, 100. The results are shown in figures 4.3 and 4.4 for sub-space dimensions Nc 0 = 0, and Nc 0 = 10, respectively. In the case Nc 0 = 0 we talk about the convergence of the open-open block of the scattering matrix also referred to as the semi-quantal approximation (Bogomolny, 1992). With increasing number of the considered closed modes Nc the measures of convergence decrease as εR,T ∼ Nc −4 . In the

46


0.01

0.01

1e-04

1e-04

1e-06

1e-06 εR

εR

case Nc 0 = 0, where the sub-block includes only open modes, we see two rates of decrease with increasing number of closed modes Nc . This is similar to the behaviour of the transition errors, discussed earlier, and therefore it is understood. In both cases we can see that ε R,T decrease down to some plateau around 10−12 almost equal to machine precision and this is surprisingly better than the transition error discussed above.

1e-08

1e-08

1e-10

1e-10

1e-12

1e-12

0.01

1

10

100

0.01

Nc

1e-04

1e-04

1e-06

1e-06

q=0.1 q=0.2 q=0.3 q=0.4 1

10

100 Nc

q=0.5 q=0.6 q=0.7

εT

εT

q=0.8

1e-08

1e-10

1e-12

1e-12 10

100

~ t-4

1e-08

1e-10

1

q=0.9

1

Nc

10

100 Nc

Figure 4.3: The measures of convergence εR,T (Nc + Nc , No ) (top, bottom) depending on the total number of closed modes Nc for two numbers of open modes No = 10, 100 (left, right) and various inner radii q as indicated in the figure. The bend is of angle β = π.

The presented method for calculations of the scattering matrix and its accuracy control works well for all q > 0. Nevertheless, the high curvatures q → 0 are difficult to discuss as for their precise treatment we need to consider a large number of modes. To conclude, by insisting that the scattering matrix is accurately calculated at least on some subspace we have to increase the number of all considered modes in the bend so that we reach sufficient precision and this can be done by our method in a stable and controlled way.

4.3

Quantum transport across the bend

The scattering matrix of our open billiard describes the stationary quantum transport of the particle over the bend. We discuss here the most important and obvious measures of the transport. These are the reflection and transmission probabilities and Wigner-Smith delay times (Wigner, 1955; Smith, 1960). Since the classical particle cannot scatter back as discussed in (Horvat and Prosen, 2004) we are particularly interested in the reflection probability. The latter can serve as a measure of quantum tunnelling between the two classically invariant components of the phase space corresponding to right and left going particles. The scattering of a straight incoming ray across the bend of an angle β = π is illustrated in figure 4.5. The ray follows the classical trajectories, but as it is of finite width its parts are scattered differently when hitting the curved wall. The parts of the ray travel different lengths yielding an interference, which increases with the travelled length. We are discussing the scattering across the bend at some wave number k, inner radius q

4.3. Quantum transport across the bend 1

0.01

0.01

1e-04 1e-06

1e-06

εR

εR

1e-04

1e-08

q=0.1 q=0.2 q=0.3

1e-12

1e-12 10

100

1e-04

Nc

1e-05

q=0.4 10

100 Nc

1e-05

q=0.5 q=0.6

1e-06

1e-06

q=0.7

1e-07

1e-07

q=0.8

1e-08

1e-08

q=0.9

εT

εT

1e-08 1e-10

1e-10

1e-04

47

1e-09

1e-09

1e-10

1e-10

1e-11

1e-11

1e-12

1e-12

1e-13

10

100

1e-13

10

100 Nc

Nc

Figure 4.4: The measures of convergence εR,T (Nc + Nc , No + 10) (top, bottom) depending on the total number of closed modes Nc for two numbers of open modes No = 10, 100 (left, right). See also caption of the figure 4.3. q = 0.2

0

0.02

q = 0.6

0.04

0.06

0.08

0.1

0.12

0.14

Figure 4.5: The wave functions |ψ(x, y)|2 obtained in the scattering of an incoming ray of a Gaussian shape, with a unit probability flow, across bends with inner radii q = 0.2, 0.6 (left, right). The calculations are performed at wave-number k with 100 open modes in the straight wave-guide.

and using N modes describing the wave function over the asymptotic region, where N ≥ N o = bka/πc. The scattering is described by the transmission matrix T (4.32) and the reflection matrix R (4.33). Let us consider an incoming wave described by the vector of expansion coefficients a = {an ∈ C}N n=1 coming from the left asymptotic region into the bend

ψin (r) =

N X

n=1

a n e+ n (r) .

(4.47)

48


Then the transmitted jT and reflected probability flux jR are written as jR = a† Πa ,

jT = a† Σa ,

j 0 = a† a = j R + jT ,

(4.48)

The matrices Π and Σ are obtained from the open-open mode block of R and T denoted by subscript oo † † Π = Roo Roo , Σ = Too Too , where Π + Σ = id . (4.49) We describe the transport properties of the bend by the first two central moments of the probability current jR,T in the statistics over incoming states a. The states are uniformly distributed on a 2N-dimensional sphere of radius j0 (Newton, 2002; Prosen et al., 2002). The average probability current is then given by R=

hjR iα 1 = tr{Π} , j0 No

T =1−R,

and the standard deviation of probability current is written as

(jR − hjR iα )2 α 1 tr{Π2 } 2 2 2 = − R = σT . σR = No + 1 No j02

(4.50)

(4.51)

2 , which measures In the following we only discuss the average reflection R and the dispersions σ R the spread of the reflected scattered wave. The basic form of the transmission matrix T can be determined from the semi-quantal calculations, whereas for the reflection matrix R it can not, as it is a purely quantum phenomenon and so there is no trivial explanation for reflection. The gross structure of matrices R and T , similarly as for the matrices A and B, do not change significantly with increasing wave-number k. In figure 4.6 we show the density plot of R and T at No = 100 open modes in the leads. The high probabilities in the matrix T can be explained classically by calculating the classical scattering matrix Tclass introduced in (Méndez-Bermudéz, 2002) and defined in Appendix C. The classical and quantum transmission matrix display similar patterns, but due to the quantum interference we can not establish a clear correspondence. In the matrix T we have a large area of high values so we can expect that the transmission probability of each individual mode should be very high. In the reflection matrix it is important to notice that the area of strongest reflection is concentrated around the open modes with indices close to the last open mode N o . At the so called resonant wave numbers km = πm/a (m ∈ N) a new open mode appears in the asymptotic region causing an intense increase in the reflection matrix elements corresponding to open modes at high indices. To visualise the changes we show in figure 4.7 a zoom-in of the scattering matrix calculated for k = k100 , q = 0.6 around the index of the highest open mode No . The changes to R and T are small but significant and centred around the index N o . In order to clarify the contributions to the total reflection we plot in figure 4.8 the reflection probability of individual modes Πnn for wave-numbers near and far from the resonance. We learn that the highest open mode is the one with the strongest reflection and at resonant wavenumbers k ≈ kNo the reflection probability for “all” modes increases. In particular the highest open mode has almost a perfect reflection ΠNo ,No ≈ 1. In the vicinity of the resonance we could effectively approximate the average reflection as R ≈ ΠNo ,No /No . The latter can be seen in figure 4.9b. The resonant wave-numbers kNo are important markers on the wave-number-axis standing for an anomalously strong reflection. This is illustrated in figure 4.9a, where we plot the average reflection R as a function of the wave-number around k = k100 . We see the average reflection R has a strong sharp maximum at resonant wave numbers and decreases in an oscillating manner with increasing wave number until crossing the next resonant wave number. The frequency of

4.3. Quantum transport across the bend

49

RN

TN

140

140

120

120

100

100

80

80

60

60

40

40

20

20

Tclass

100 90 80 70 60 50 40 30

1

1

20

-9

40

-8

60

80

-7

100

1 140 1

120

-6

20

-5

20

-4

40

60

-3

80

100

-2

120

-1

140

0

10 1

1 10 20 30 40 50 60 70 80 90 100

-5

-4.5

-4

-3.5

-3

Figure 4.6: The density plot of the scattering matrices log 10 |(RN )nm | and log10 |(TN )nm | and of the classical analog log10 |(Tclass )nm | calculated at inner radius q = 0.6, wave-number k = 100.5π/a and total number of modes N = 150. before

130

RN :

120

120

110

110

110

100

100

100

90

90

90

80

80

80

80

90

100

110

120

70 130 130 70

80

90

100

110

120

70 130 130 70

120

120

120

110

110

110

100

100

100

90

90

90

80

80

80

70

70

80

-9

90

100

-8

110

120

-7

70 130 70

-6

80

90

100

-5

after

130

120

70 130 70

TN :

middle

130

110

-4

120

70 130 70

-3

80

90

100

110

120

130

80

90

100

110

120

130

-2

-1

0

Figure 4.7: Zoom in the density plot of the reflection log 10 |(RN )nm | and transmission matrix log10 |(TN )nm | around n, m = bka/πc (black solid line) for wave numbers k = (100 − 10−3 )a/π (be. fore), k = 100a/π (middle) and k = (100 + 10−3 )a/π (after).

oscillations increases with increasing wave number. From the numerical results we see that R decreases with increasing q. In narrow channels at large wave-numbers we find analytically using a perturbative approach (see Appendix E) and numerically that R ∼ a2 /No .

(4.52)

The resonant behaviour of the reflection around the resonant wave-number can be partially explained by neglecting all open modes expect the one with the higher index No and treat it as an independent 1D scatterer. For 1D scatterers (d = 1) connecting highest open modes in the

50


0.01 1e-04 1e-06 1e-08 1e-10 1e-12

0.01

Πnn

1e-04 1e-06 1e-08

r=100.001 r=100.7

0 0.2 0.4 0.6 0.8 1

1e-10

r=250.001 r=250.8

1e-12

0

0.2

0.4

0.6

0.8

1

n/No

Figure 4.8: The diagonal matrix elements Πnn at wave-numbers k = rπ/a, as indicated in the figure, and fixed inner radius q = 0.6. Note that No = brc.

bend and in the straight wave-guide the reflection and transmission matrix elements read as i sin(νNo β)(K − K −1 ) , 2 cos(νNo β) − i(K + K −1 ) sin(νNo β) 2 , 2 cos(νNo β) − i(K + K −1 ) sin(νNo β)

R1D = −

(4.53)

T1D =

(4.54)

where we introduce a coefficient that incorporates most of dynamical properties of the modes K=

gNo ANo ,No ∈ [0, 1]. νp BNo ,No

(4.55)

The interval of the coefficient was found empirically. By introducing an additional phase shift µ = 2 arctanh K the reflection and transmission coefficients can be rewritten in a more compact form sin(βνNo ) sin(iµ) R1D = − , T1D = . (4.56) sin(βνNo + iµ) sin(βνNo + iµ) and then the whole scattering matrix reads 1 sin(iµ) − sin(βνNo ) S1D = . sin(iµ) − sin(βνNo ) sin(βνNo + iµ)

(4.57)

At k = kNo , the mode number gNo ∼ µNo and the phase shift µNo become zero yielding a perfect reflection in our 1 scatterer R1D = −1 and T1D = 0. By assuming a phase shift φ = βνp approximately independent of the wave-number k we can make a local expansion of the reflection around the resonant wave-numbers writing 1

0 ≤ δk 1 , (4.58) 1 + O(δk) , 1 + 2iL cot(φ)δk 2 p where we have introduced a constant L = 2/kn K. The reflection probability in the same order of approximation then reads R1D (kn + δk) = −

kR1D k2 = 1 − 4L cot(φ)2 |δk| + O(δk 2 ) .

(4.59)

This means that we have (at least in the 1d model approximation) a linear descent with k from the resonant value of reflection. The resonances are nicely explained by the 1D model. This treatment is meaningful, because the matrices A and B are approximately diagonal at

4.3. Quantum transport across the bend

2 No /(1-q)2

1

q=0.2 q=0.6 q=0.9

1 0.1

51

r rσR ΠN ,N

1 0.8

o

0.6

0.1

0.4 0.2

0.01 0.01

-1 -0.5 0 0.5 1 1.5 2 4

10 (r-r0)

0.001 0.001

1e-04 1e-05

o

2

|R1d|

0

0.5

1

1.5

2

k(1-q)/π - 100

1e-04

-1

-0.5

0

0.5

1

100(r-r0), r = k(1-q)/π

(a)

(b)

Figure 4.9: The scaled average reflection No R/(2a2 ) for a bend of an angle β = π and several values of q as indicated in the figure around the resonant wave numbers k = 101π/a (a) and the detailed view at the area around the resonance wave-number using different measures of reflection (b).

(No , No ) with algebraic decay explained in the previous sections. If the modes would be strictly independent we would have the identity ΠNo ,No = kR1D k2 , but the algebraic tails make the 1D model hold only up to a very crude order-of-magnitude around the resonance, as we can see in figure 4.9b. The measures of reflection R and σR on a larger range of wave numbers k are depicted in figure 4.10. From the figure we conclude that R strongly oscillates with peaks at resonant wave number kn and its upper bound decreases proportionally to k −1 , as predicted. The numerical results in figure 4.10.b indicate that σR < R, and R ∼ σR as k goes to infinity. To get rid of oscillations and obtain an overall average behaviour of R and σ R we calculate 1

1

〈R〉 k-1

0.1

0.9 0.8

0.01

0.7

0.001

0.6

1e-04

0.5

1e-05

0.4 0.3

1e-06

0.2

1e-07 1e-08

σR/ 〈R〉 0.7

0.1 10

100 k

1000

0

100

1000 k

Figure 4.10: The average reflection R and the relative deviation of reflection σR on a larger interval of wave numbers k at q = 0.6.

their integrals across the wave-number axis. The results are shown in figure 4.11 and yield the following dependence Z k Z k σR dκ = O(log(k)) . (4.60) R dκ ∝ k0

k0

This indicates together with previous conclusions that the reflection measures have on a large wave-number scale the following asymptotic behaviour R ∼ σR = O(k −1 ) .

(4.61)

52


It seems that this relation (4.61) is valid for an arbitrary inner radius q. This is a very useful information in the study of wave-guides and general billiards that include bends. 0.1

∫k〈R〉(κ) dκ k 0.08 ∫ σR(κ) dκ 0.06

f1(k) f2 (k)

0.04 0.02 0

10

100 k

1000

Figure 4.11: The integral of the average reflection R and the standard deviation of reflection σ R across a larger interval of wave numbers k for a bend with an inner radius q = 0.6 and an angle β = π. The inserted lines are f1 (k) = 0.0110301 + 0.0102197 log k and f2 (k) = −0.0155328 + 0.00692377 log k.

A bit deeper insight into the scattering formalism gives the Wigner-Smith delay time τ ws (Wigner, 1955; Smith, 1960), which measures the quantum analog of geometric length traveled by the wave. In the semi-classical limit, where we could apply geometric optics, τ ws is equal to the average geometric length of classical trajectories across the open billiard. By using the hermitian variant of the lifetime matrix † Q = Soo

dSoo , dik

(4.62)

the Wigner-Smith delay time is defined as τws

1 1 1 † dRoo † dToo + , = tr{Q} = tr Roo tr Too 2No No dik No dik

(4.63)

where we used the block symmetries of our matrix S (4.19). The τws can be thought of as the average of delay times corresponding to individual modes defined as h i o 1 nh † 0 i n † 0 = Roo Roo τws = + Too Too , (4.64) No nn nn with derivative defined as (•)0 = d/dk. We refer to the first and the second term in above expression as the reflection and the transmission delay time, respectively. Examples of times n }No calculated for q = 0.6 and wave-numbers k near and far from resonances are plotted in {τws n=0 figure 4.12. The reflection in modes at low lying indices is negligible and almost constant with respect to the wave-number k making the transmission delay time the dominant contribution to n . A rather opposite situation occurs at high lying indices, where reflection is stronger and the τws n . Only very near to the resonance cat it become reflection delay time becomes noticeable in τws dominant. Because the mode numbers decrease with increasing index the (transmission) delay time increases with increasing index. Notice that the highest open mode carries the leading contribution to the delay time and is strongly increased in the vicinity of the resonant wavenumber. Numerical results shown in figure 4.13 point to a similar dependence of Wigner-Smith delay time τws on the variable k as the average reflection R, just the oscillations are smoothed out. The time τws strongly increases near the resonance wave-numbers k = kNo due to intense changes in the scattering matrices R and T in the area around the index of the newly open mode.

4.3. Quantum transport across the bend 1000

r=100.001

53

1000

r=100.7 r=250.001

r=100.001 r=100.7 r=250.001

100

r=250.7

r=250.7

100

n

τws

10 1 0.95

10

1

0

0.1

0.2

0.3

0.4

0.96

0.5 n/No

0.97

0.6

0.98

0.7

0.99

0.8

1

0.9

1

n Figure 4.12: Wigner-Smith times for individual modes τws for different wave-numbers k = πr/a near and far from the reflection resonances at q = 0.6.

12

12

q=0.2 q=0.6 q=0.9

10

10 9 8

6 T0.9 T0.6 T0.2

4 2

τws

τws

8

7 6 5

T0.9 T0.6 T0.2

4

0 -2

q=0.2 q=0.6 q=0.9

11

3 -1

-0.5

0

0.5

1

k(1-q)/π - r

2

-1

-0.5

0

0.5

1

k(1-q)/π - r

Figure 4.13: The Wigner-Smith delay time τws for the bend of angle β = π around the reflection resonance at r = 11, 101 (left,right). The solid horizontal lines with the labels T0.2,0.6,0.9 represent the classical delay times at q = 0.2, 0.6, 0.9, which are approximately given with the formula T q = 2.45863 + 2.48696q.

Qualitatively we can explain the singular behaviour by treating the highest open modes in regions as part of an independent 1D scattering model in which the delay time is given by 1D ∗ 0 ∗ 0 τws = ={T1D T1D + R1D R1D }=

β sinh(2µ)νp0 − sin(2βνp )µ0 . cosh(2µm ) − cos(2βνp )

(4.65)

The first term in the nominator of equation (4.65) is the contribution of transmission and the second term i sthe contribution of reflection. By slowly increasing the wave-number across the region of the resonance we can notice three different regimes: before, in the vicinity and after the resonance wave-number. Before the resonance k < kNo , a new real mode appears in the bend, which makes the propagation across the bend very slow. From the formula (4.65) we learn 1D . In the that this results in a large transmission time and consequently in a large time delay τ ws instance, when crossing the reflection resonance a new mode appears in the straight wave-guide, 1D ∼ (k − k )− 12 for k > k which causes a square-root singularity τws No No and its sign is determined by 2βνNo . This reflection term has a short-scale influence to the behaviour of the time delay and can enhance or reduce its size. Obviously, this is a very non-classical situation. By going further away from the resonance wave-numbers the reflection contribution to the time delay is levelled by an increasing transmission term due to a very slow propagation of the mode in the asymptotic region, which again increases the transition time. So we can experience one or two peaks of the time delay in the vicinity of the reflection resonance. Away from the reflection resonance the time delay drops even below the classical time. The latter we assume is due to reflection phenomena

54


which reduces the classically expected phase shift. The presented 1D scattering model has only an instructive purpose and does not represent any useful approximation, similarly as was this the case in the discussion of the reflection. It just explains qualitatively the behaviour of the delay time in the serpent billiard in the vicinity of the reflection resonance, which is shown in figure 4.14. 2000 1800 1600 1400

500

1000

0

800 600

2000

o-1 τN ws

500

o τN ws No-1 τws

400 300

1500

200 100

-500 -0.3 -0.2 -0.1

1000 0

0.1 0.2

100(r-11)

400

rτws

600

o τN ws

1000

1200

2500

rτws

1500

0 -100 -0.3 -0.2 -0.1

0

0.1 0.2 0.3

100(r-11)

500

200 0 -0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0 -0.5

-0.4

-0.3

100(r-101)

(a)

-0.2

-0.1

0

0.1

0.2

0.3

100(r-101)

(b)

Figure 4.14: The Wigner-Smith delay time as the function of wave-numbers k = πr/(1 − q) in the vicinity of the reflection resonances at r = 11, 101 in a bend with the inner radius q = 0.2, 0.6 (a,b) and the angle β = π.

55

Chapter 5

Quantum serpent billiards The serpent billiard is an open system composed of bent and straight segments of a waveguide. We attach to both ends of the billiard infinite straight wave-guides referred to as leads or asymptotic region. Consequently the whole serpent billiard can be treated as a single scatterer on an infinite wave-guide. For the details on construction see Chapter 2. The wave-phenomena across the serpent billiard area are described in the scattering matrix formalism using the modal approach. We are interested in the characteristics of the on-shell scattering in the energy range below which some semi-quantal approach can be accurately applied. Therefore it is difficult to make any analytic predictions and so our discussion relies mainly on numerical results. We consider the serpent billiard of width a with a sequence of bends and straight segments, where the bends have the inner radius q = 1−a. The borders of the billiard and leads, sometimes called connection ports, are the cross-sections at which we describe the scattering across the billiard in the classical and quantum picture. The cross-section (CS) of the billiard is parametrised as an interval I = [0, a]. By considering the trajectories coming from one side and leaving on the other side the classical dynamics is described in terms of the jump model (Tserp , Pserp ), which is energy independent. The latter is composed of the Poincaré map Pserp : S → S – a mapping of entry points into exit points on the opposite sides of the billiard, and of the time function Tserp : S → R measuring the lengths of trajectories across the billiard in dependence of the entry point, see Chapter 2 for details. By considering incoming trajectories from both sides of the billiard at some energy E = 21 k 2 , the dynamics across the scatterer is described by the Poincaré e serp : SSOS → SSOS (2.13) defined on the SOS of the billiard SSOS = SL ∪ SR (2.11). map P The SOS of the scatterer SSOS is composed of the canonical phase-spaces corresponding to the left and right CS denoted by SL and SR , respectively, and is energy dependent. In the quantum picture the scattering is described by a scattering matrix. This maps the incoming waves into outgoing ones with respect to the CS of the billiard, where the waves are given in terms of eigen-modes of the Laplacian in the asymptotic region. We are searching for the solution of the Helmholtz equation on the whole area of the open billiard, the serpent billiard and the leads, denoted by Ω: −∆ψ(r) = k 2 ψ(r) ,

ψ|r∈∂Ω ,

(5.1)

with the Dirichlet condition at a specific wave-number k. In the scattering approach we are interested in the wave-function ψ(r) in the asymptotic region. The wave-function in the left and right asymptotic region of the open billiard is denoted by ψL andψR , respectively, and given by the ansatz X + L,R − ψL,R = aL,R aL,R ∈C, (5.2) n en (r) + bn en (r) , n n∈N

where e± n (r) (4.7) are the eigen-functions of the Laplace operator on the infinite straight wave-

56

Chapter 5. Quantum serpent billiards

guide: e± n (r)

exp(±ign x) = un (y) , √ gn

gn2

2

=k −

πn 2 a

,

un (y) =

r

π 2 sin ny . y ∈ I a a

(5.3)

The superscript in e± n (r) labels the direction of the phase propagation: + for left-to-right (rightgoing) and − for right-to-left (left-going). We introduce vectors of expansion coefficients a L,R = {aLn }n∈N and bL,R = {bLn }n∈N corresponding to the left-going and the right-going parts of the wave-function ψL,R , respectively. The wave-function on the whole billiard should be smooth implying that the vectors aL,R and bL,R are linearly dependent. This dependence is expressed by a two-way scattering matrix S written as a block matrix L R r t , (5.4) S= tL r R which connects the vectors of expansion coefficients in the following way L L b a . , ψout = Sψin = ψout , ψin = aR bR

(5.5)

The scattering matrix S represents the solution of the Helmholtz equation in the asymptotic eserp of the serpent billiard. It is generalised region and is a quantum analog of the Poincaré map P unitary as defined by equations (4.21), (4.22), (4.23) and (4.24) and symmetric S T = S due to the time-reversal symmetry of dynamics. In equation (5.5) we define vectors of coefficients ψin and ψout that correspond to parts of the wave-functions ψin (x, y) and ψout (x, y) entering and leaving the scatterer, respectively. We look at bends and straight segments as scatterers on an infinite straight wave-guide. More precisely, we identify three types of scatterers as is pointed out in Chapter 2. These are a bend, a straight segment of the wave-guide and an inversion of the CS given by the map I : y 7→ a − y. We treat the quantum serpent billiard as a chain of quantum scatterers. Such chains of scatterers with unitary scattering matrices were discussed in Chapter 3. We isolate an individual scatterer from the chain, attach to its ends straight wave-guides and calculate its scattering matrix. We denote by Sb,β , Ss,d and SI the scattering matrices of the bend with an angle β, of the straight segment of length d and of the inversion of the CS, respectively. The scattering matrix S b,β has a symmetric block form (4.19) with reflection and transmission block given by equation (4.33) and (4.32), respectively, in Chapter 4. The other two scattering matrices Sd,d and SI are obtainable in a straight-forward calculation and are written as 0 eiGd , G = diag{gn }n∈N , (5.6) Ss,d = eiGd 0 and SI =

(−1)M 0

0 (−1)M

,

M = diag{n + 1}n∈N .

(5.7)

We introduce in Appendix B the binary operation of concatenation , with which we calculate the scattering matrix of a “larger” scatterer SAB from the scattering matrices SA and SB of its parts A and B, respectively, as SAB = SA SB . The scattering matrix Sn of a periodic billiard of n cells, each composed of a bend of an angle β and a straight segment of a length d, is then written as Sn,period = Scell,β,d . . . Scell,β,d , | {z } n

Scell,β,d = Sb,β Ss,d SI .

(5.8)

57 For convenience we introduced the scattering matrix of the basic cell S cell,β,d . By writing the length of the chain in binary code n = (bm . . . b1 b0 )2 , with bi ∈ {0, 1}, we can calculate the scattering matrix Sn,period fast by the following recipe Sn,period = Sb0 m 2m . . . Sb0 1 2 Sb0 0 ,

(5.9)

where we assume S00 S = S S00 = S and matrices S20 i for i ∈ [0, m] are obtained from the recursion S10 = Scell,β,d . (5.10) S20 m+1 = S20 m S20 m , In case of a custom-built serpent billiard of n cells with form parameters {β i , di }i∈[1,n] the scattering matrix is given by Sn,custom = Scell,β1 ,d1 . . . Scell,βn ,dn ,

(5.11)

where n is called the length of the chain. Because the serpent billiard is of constant width and all bends have identical curvature the presented calculation of the chain’s scattering matrix is in our opinion optimal from the numerical viewpoint and the computing time consumption. In figure 5.1 we give an illustration of a Gaussian ray entering in differently composed periodic billiards from the left. The construction of a Gaussian ray is given later. The description of the wave-function calculated across the billiard area includes evanescent modes, which can diverge along the billiard if they are not considered correctly. The recipe for stable evaluation of the wave-function along the billiard in the coordinate space is described in Appendix F. The presented scattering matrices describe the scattering of waves in the asymptotic region across the scatterer in terms of modes of the Laplace operator (Hamiltonian) on the asymptotic region. This modal approach can be generalised making it independent of mode definition. Thereby we rewrite the scattering matrix in an operator form (Prosen, 1995) which can be very useful, because it enables discussing scattering in more classical terms. In order to do that we introduce the functional space corresponding to the configuration space C = C L ∪ CR (2.8) of the billiard, composed of the left and right CS at the border, on which these operators are defined. This is done in few steps. First we introduce a Hilbert space H(I) of L2 functions on a CS I with R the standard scalar product ha|bi = I dy a(y)∗ b(y) and orthonormal basis {un (y) = hy|un i}n∈N (5.3). Next we introduce a two-level Hilbert space Hside spanned by two abstract orthonormal vectors {|Li, |Ri} labelling the left and right CS of the scatterer. Then the Hilbert space H(C) of functions on configuration space C of the billiard can be written as H(C) = H(I) × Hside ,

(5.12)

B = {|un i|si = |n, si : n ∈ N, s ∈ {L, R}} .

(5.13)

with the orthonormal basis reading

Then we introduce the reflection r L,R and transmission tL,R operators corresponding to the scattering matrix S 5.4) written as X X L rˆL,R = |nirn,m hm| , tˆL,R = |nitLn,m hm| , (5.14) n,m∈N

n,m∈N

which act one the Hilbert space H(C). The SOS state vectors ψin and ψout represent expansion coefficients of the incoming and outgoing waves in the asymptotic region, respectively. We rewrite these SOS vectors into states in H(C) as X X L |ψin i = aLn |n, Li + bR |ψout i = aR (5.15) n |n, Ri , n |n, Ri + bn |n, Li , n∈N

n∈N

58

Chapter 5. Quantum serpent billiards β=π

β = π/2

β = π/4

0

0.005

0.01

(a)

0.015

0.02

0.025

0.03

0.035

0.04

(b)

Figure 5.1: The density plot of wave-functions |ψ(x, y)|2 , which are the solutions of the on-shell scattering of a Gaussian ray across differently composed serpent billiards with q = 0.6 and d = 0, 1 (a,b) and at k = 396.626 (No = 50). The Gaussian ray is formed on √ the left SOS of the left most cell of the billiard chain at the position y = 0.2, ky = 0 and σ 2 = (1 − q)/( 2πNo ) = 0.0031915.

in order to associate them more directly to the classical SOS. Consequently the scattering matrix of the serpent billiard S (5.4) can be rewritten in the form of an operator Sb : H(C) → H(C) Sb = rˆL ⊗ |LihL| + rˆR ⊗ |RihR| + tˆL ⊗ |RihL| + tˆR ⊗ |LihR| .

(5.16)

Let us consider the position basis {|yi}y∈I on the CS. Then we can introduce the orthonormal base of the Hilbert space H(C) written as B 0 = {|y, si = |yi|si : y ∈ I, s ∈ {L, R}}. In the base

59 B 0 the scattering matrix can be written point-wise on CSs at the border of the scatterer in the form of a matrix 0 L hy |ˆ r |yi hy 0 |tˆR |yi 0 0 b [hy , s |S|y, si]s,s0 ∈{L,R} = , (5.17) hy 0 |tˆL |yi hy 0 |ˆ rR |yi

which has precisely the form of the scattering matrix in the modal description. Similarly, we rewrite the scattering matrices of the bend Sb,β , of the straight segment Ss,d and of the inversion SI into the operator form as Sbb,β = rˆb,β ⊗ (|LihL| + |RihR|) + tˆb,β ⊗ (|RihL| + |LihR|) , Sbs,d = tˆd ⊗ (|RihL| + |LihR|) , SbI = Iˆ ⊗ (|RihL| + |LihR|) ,

(5.18) (5.19)

by using operators of CS inversion Iˆ and of propagation in a straight segment tˆd defined by X X tˆd = eign d |nihn| , Iˆ = (−1)n |nihn| . (5.20) n∈N

n∈N

In the point-wise representation of the scattering matrix (5.17) we can write elegantly its semiclassical (WKB) approximation (Merzbacher, 1997). Then following the recipe given in references (Bogomolny, 1992; Prosen, 1995), the standard leading order semi-classical approximation is written as b ≈ hy 0 |(tˆR )sc |yi ⊗ |LihR| + hy 0 |(tˆL )sc |yi ⊗ |RihL| , hy 0 |S|yi (5.21)

with the transmission blocks expressed as a sum over classical trajectories connecting points y and y 0 on different CSs at the border of the serpent billiard labelled by j,

hy 0 |(tˆL )sc |yi =

1 X √ 2πi j

hy 0 |(tˆR )sc |yi =

X

1 √ 2πi

j

s

∂ 2 Sj (y 0 , y) iSj (y0 ,y)−iπνj (y0 ,y) e , ∂y 0 ∂y

(5.22)

s

∂ 2 Sj (y, y 0 ) iSj (y,y0 )−iπνj (y,y0 ) e . ∂y 0 ∂y

(5.23)

Along the j-th trajectory connecting the phase-space points (y, vy ) and (y 0 , vy0 ) = Pserp (y, vy ) the classical action Sj and the Maslov phase νj read as Sj (y 0 , y) = kTserp (y, vy ) ,

1 ν j = n j − zj , 2

(5.24)

where nj and zj are integers determining the number of hits to the wall and the number of zeroes of ∂ 2 Sj (y, y 0 )/∂y 0 ∂y along the trajectory, respectively. The semi-classical scattering matrix does not include reflection contributions as expected due to the classical uni-directionality. In the similar way as for a whole serpent billiard, we can write the semi-classical approximation of scattering matrix for a bend as hy 0 |Sbb,β |yi ≈ hy 0 |(tˆb,β )sc |yi ⊗ (|LihR| + |RihL|) .

(5.25)

ˆn , (tˆL )sc = [(tˆb,β )sc tˆd I]

(5.26)

Taking this into account we see that calculation of the scattering matrix of the serpent billiard in the semi-classical approximation reduces to the simple multiplication of transmission matrices (blocks) of all elements in the chain. The transmission blocks in the scattering matrix of the periodic billiard (5.8) are approximated by (tˆR )sc = [Iˆ tˆd (tˆb,β )sc ]n

60


and in the scattering matrix of the custom-built billiard (5.11) as (tˆL )sc = (tˆb,β1 )sc tˆd1 Iˆ . . . (tˆb,βn )sc tˆdn Iˆ ,

(tˆR )sc = Iˆ tˆdn (tˆb,βn )sc . . . Iˆ tˆd1 (tˆb,β1 )sc .

(5.27)

We see that in the leading order of the semi-classical approximation the calculation of the scattering matrix is significantly simplified. But as it leaves the reflection out of the discussion this approach is not very useful in studying quantum phenomena. In the first section we discuss the transport properties of the periodic and custom-built serpent billiards. In addition, we examine the statistical properties of the scattering matrix of the chain as it is increased in length. The serpent billiard can be closed forming a compact billiard. We investigate spectral properties of such billiards in the second section, where we focus on the influence of classical uni-directionality on the energy spectrum. In the last section we study the quantum-classical correspondence (QCC) in the classical phase-space using the Wigner function representation of states. As a quantitative measure of the correspondence we introduce the overlap of the Wigner function and the classical density on the SOS of the scatterer, called the quantum-classical fidelity (QCF), and investigate its behaviour as the chain is lengthened.

5.1

Transport properties

The transport properties of the serpent billiard are investigated on a certain energy shell. This means that we study only the stationary transport, because for the time-dependent transport we have to consider some set of energy shells. We are particularly interested in how the transport properties evolve with the length of the serpent billiard and change with the energy. The length is defined as the number of bends in the billiard. We are studying the transport through a simple periodic serpent billiard, as defined in Chapter 2, and totally random serpent billiards i.e. custom-built billiards with the form parameters of bends and straight segments chosen randomly. The quality of the stationary transport in the serpent billiard is measured similarly as for the bend, which was studied in Chapter 4. Let us now consider the serpent billiard of length n and width a at the energy E = 21 k 2 . The scattering across the chain into the asymptotic region is described by the infinite dimensional scattering matrix S (5.4) at some wave-number k. The waves in the asymptotic regions are described by the open and closed modes. The number of the first is No = k(1 − q)/π. We are interested in the waves in the asymptotic limit to which only open modes contribute. Then the transport properties are determined by the sub-block of S corresponding to open modes, which are written in the form of a finite block matrix Soo

R L r t = L oo R oo ∈ C2No ×2No , t oo r oo

(5.28)

o where the symbol [A]oo = {Aij }N i,j=1 denotes the sub-block of matrix A corresponding to open modes. The general unitarity of S implies that Soo (5.28) is an unitary matrix from the group U (2No ). We introduce matrices of reflection probabilities corresponding to S as

† ΠL,R = rL,R oo rL,R oo .

(5.29)

The transport properties of the billiard are then described in terms of the average reflection R and the standard deviation (uncertainty) σR of the reflection defined as R = hΠL,R i ,

2 σR =

2 1 ΠL,R − R2 No + 1

(5.30)

5.1. Transport properties

61

where we use h•i = No −1 tr{•}. The introduced measures are statistical quantities. But if we consider a single arbitrary SOS state of an incoming wave ψ = [a, b], where a and b represent left and right incoming waves, respectively, then its reflection probability reads Rψ =

1 † † a Π a + b Π b . L R a† a + b † b

(5.31)

The reflection in our serpent billiards is purely a quantum phenomenon and is usually several orders smaller than the transmission R T . That makes the reflection the most interesting transport quantity to investigate.

5.1.1

The periodic serpent billiards

The periodic serpent billiard are geometrically the most simple case of the serpent billiard. They are useful in determining of basic transport properties of the general class of serpent billiards. We discuss a periodic serpent billiard of length n with the scattering matrix S n (5.8) composed of n basic cells. The basic cell is constructed of a bend and a straight segment and to it we associate the scattering matrix Scell . The width of the billiard is a = 1 − q, the angle of the bend is β and the length of the straight segment is d, where q is the inner radius of the bend. First we study the dynamics of the scattering matrix Sn with increasing length using the average reflection R. Depending on the parameters of the basic cell the chain exhibits ballistic or (partially) localised scattering matrix dynamics as defined in Chapter 3. We show an examples of each type of dynamics in figure 5.2a. In the ballistic case the reflection just oscillates with increasing n, whereas in the case of localised dynamics it saturates exponentially to some plateau. We can understand this behaviour by writing the lengthening of the billiard chain in the transfer matrix formalism using the semi-quantal approximation (Bogomolny, 1992) in which only open modes are considered relevant. The transfer matrix formalism was introduced in Chapter 3 as an unstable counterpart to the scattering formalism. There we show that the chain of length n is equivalently described by the scattering matrix Sn or the transfer matrix Tn . The lengthening of the chain from the right side by a single cell described by the scattering matrix S cell or transfer matrix Tcell is then written as Sn+1 = S Sn ,

Tn+1 = T Tn .

(5.32)

The two formalisms are connected by a transformation of scattering matrices into transfer matrices T [•] and its inverse transformation S[•] which are defined by (3.5). These transformations are generally unstable as they assume that some sub-matrices of arguments are invertible. By restricting to open modes the serpent billiard is described by the unitary scattering matrix S oo (5.28) and transfer matrix Too = T [Soo ]. The transfer matrix Too has the following symmetry relations id 0 † † Too KToo = Too KToo = K , K = , (5.33) −id 0 id T T Too JToo = Too JToo = J , J = , (5.34) −id 0 which show that T is a hyperbolic and symplectic matrix, T ∈ U (No , No ) ∩ Sp(2No ). The symmetry relations (5.33) and (5.34) imply that an eigenvalue λ of the matrix Too induces three eigenvalues λ∗ , λ∗ −1 , λ−1 with following properties Too v = λv ,

Too (KJv) = λ∗ (KJv) ,

(Jv)T Too =

1 (Jv)T , λ

(Kv)† Too =

1 (Kv)† . (5.35) λ∗

62

Chapter 5. Quantum serpent billiards 5 4.5

1

4

0.5

3

ℑ{λ}

103

3.5 2.5 2 1.5

-0.5

1 0.5 0

0

-1 0

500 1000 1500 2000 2500 3000 3500 4000 n

(i)

-1

-0.5

0

0.5

1

0.5

1

ℜ{λ}

0.045 0.04

1

0.035

0.5

0.025

ℑ{λ}

0.03 0.02 0.015

-0.5

0.01 0.005 0

(ii)

0

-1 0

10 20 30 40 50 60 70 80 90 100 n

(a)

-1

-0.5

0 ℜ{λ}

(b)

Figure 5.2: The average reflection R as the function of chain length n (a) and spectrum of the corresponding transfer matrix Too in the ballistic case at k = 50.5π/(1 − q) (i) and the partially localised case at k = 50.01π/(1 − q) (ii) in periodic serpent billiard with q = 0.6, β = π and d = 0.

Notice that the number of eigenvalues outside the unit-circle is always even. In the semi-quantal approach the transfer matrices of periodic chain of length n and of the basic cell are given by Too,n = T [Soo,n ] and Too,cell = T [Soo,cell ], respectively. The approximate relation between the two transfer matrices is obtained from (5.32) and is given by Too,n ≈ (Too,cell )n .

(5.36)

The approximation in the expression (5.36) is assumed to decrease with increasing wave-numbers (energy). Therefore we can treat serpent billiards on reduced space approximately as chains of unitary scatterers. Then from the results of Chapter 3 we conclude the following. If all eigenvalues of Too,cell lie on the unit circle we obtain the ballistic scattering dynamics and if there are eigenvalues outside the unit-circle we get the (partially) localised scattering dynamics. The two cases of dynamics with corresponding spectra of Too,cell are shown in figure 5.2. In the case of the localised scattering dynamics shown in the figure the transfer matrix has two eigenvalues outside the unit circle. We find empirically that localised scattering dynamics is more common at wave-numbers k just above the resonant ones π/aN. Some explicit conditions for parameters to achieve localisation where, however, not found. In our studies we did not find more than two eigenvalues outside the unit circle which implies that T∞ ∼ 2/No . We suspect that the highest open modes, one in each lead, are responsible for this partial localisation. It is probable that there can be more than two eigenvalues out-side the unit circle. In particular, the limit of high curvatures q 1 was not investigated very thoroughly.


63

In order to calculate the scattering matrix Sn of the chain as precisely as possible it is also necessary to include evanescent modes in the wave-description. The measures of reflection e.g. average reflection Rn in discussion here converges with increasing the number of evanescent modes Nc to its exact value. An example of this convergence is shown in figure 5.3 for a periodic serpent billiard of different lengths and at two different energies. We can see that at both energies

0.01

n=100

0.001

n=1000

1e-04

10-4 e-0.06 Nc

1e-05 1e-06 1e-07 1e-08 1e-09

0.1

n=10

0

20

40

60

80 100 120 140 160 Nc

|(Nc) / (200) - 1|

|(Nc) / (180) - 1|

0.1

n=10

0.01

n=100 n=1000

0.001

10-3 e-0.05 Nc

1e-04 1e-05 1e-06 1e-07 1e-08

0

20

40

60

80 100 120 140 160 180 Nc

Figure 5.3: The convergence of reflection with increasing number of evanescent modes included in the calculation in the serpent billiard of different length n at wave-numbers k = π/ar, r = 10.5, 50.5 (a,b) with q = 0.6, β = π and d = 0.

and all lengths of the billiard the convergence towards assumed exact value is approximately exponential. In longer serpent billiards we experience slightly slower convergence. The reflection Rn can have rather complicated behaviour along the length of chain and this depends on parameters of the chain endued by x = {k, q, β, d}. Small perturbations of parameters can lead to generally different dependence of reflection along the chain. We are considering the case of ballistic dynamics of the scattering matrix Sn along the chain. Then according to the theory of error propagation (Stoer, 2005) the reflections in two chains with similar parameters diverges linearly with increasing length n: Rn (x + δx) − Rn (x) ∼ O(Rn )kδxkn .

(5.37)

The expected dependence (5.37) is supported by numerical results shown in figure (5.4) for perturbations of parameters in wave-number direction. The basically same dependence can be obtained for perturbations of parameters in other directions. This means that small errors and perturbations to the scattering matrix of the basic cell will only linearly grow with the length of the chain, but the properties of the resulting quasi-infinite chains can be nevertheless very different. The dependence of the average reflection Rn (k) on the wave-number k is a complicated function and cannot be simply explained. There are several properties of the serpent billiard which affect the form of the function Rn (k) at certain length n. These are (i) the reflection resonances in scattering across the bend that produce a sharp increase of the reflection at k ∈ π/aN, see figure 5.5a, (ii) possibility of (partially) localised dynamics of the scattering matrix that exponentially increase the reflection with the length n and (iii) the weak tunnelling between the two classically invariant phase-space components corresponding to the left and right traversing particle along the serpent billiard. We present the function R n (k) at fixed n from different aspects to point out its basic features. In figure 5.5 we show Rn (k) on a small interval with fine sampling of k and on a larger range of wave-numbers k for serpent billiards of different length. We see that the reflection Rn (k) as a function of k on average increases with increasing length n. But, at some fixed n the reflection decreases with increasing k. The numerical results indicate that the envelope of the function Rn (k) scales with the wave-number as k −α , where

64

Chapter 5. Quantum serpent billiards 1e+08

simulation 4n

simulation 10n

1e+07

10000

Rn 〈|δRn|/|δk|〉loc

1000

-1

Rn-1〈|δRn|/|δk|〉loc

100000

100

1e+06 100000 10000 1000

10

10

100

1000

100

10000

10

100

1000

n

10000 100000 1e+06 n

(a)

(b)

Figure 5.4: The difference in the reflection |δR| at small perturbations in the wave-number δk as a function of length n in periodic serpent billiards with q = 0.6, β = π, d = 0 and k = 10.5π/(1 − q), 50.5π/(1 − q) (a,b). The difference |δR| is locally averaged over the chain length. 0.1

0.001

0.01

1 0.01

1e-04 1e-05

1e-04 1e-05

1e-06 1e-07 99.5

0.001

1e-06 100

100.5

101

101.5

102

102.5

1e-07

10

100

k(1-q)/π n=1

n=10

(a)

200

r = ka/π n=1000

n=10

2 r-1

(b)

Figure 5.5: The average reflection Rn as a function of wave-number k around the 101st reflection resonance of the bend (a) and on larger interval calculated at k ∈ 0.1π/aN (b) in chains with parameters q = 0.6, β = π and d = 0 and different lengths n.

. α = 1 ± 0.2. From the figure it is obvious the reflection resonances of the bend introduced in Chapter 4 are inherited by serpent billiards. Another illustrative aspect of Rn (k) is its value statistics across the wave-number axis. In order to make different energy regimes more compatible to each other we discuss the rescaled average reflection ρ(k) = N o Rn (k) = tr{[rnL ]†oo [rnL ]oo }. We calculate the distribution of ρ(k) and logarithm log ρ(k) across the k-axis denoted by P ρ (t) and Plog ρ (t) and defined as Z Z Pρ (t) = dk δ(t − ρ(k)) , Plog ρ (t) = dk δ(t − log ρ(k)) . (5.38) The distribution Plog ρ is a more convenient object to discuss, because the values of the rescaled reflection ρ stretch over a few orders of magnitude. The numerically obtained distribution Plog ρ (t) is presented for chains of different length in figure 5.6. We see that Plog ρ is zero below some value of log ρ. This is not necessarily true as there is a limit to the numerical precision. The distribution Plog ρ for a bend (n = 1) has a narrow peak at reflections about two orders of magnitude above the numerical precision. In longer chains we obtain in addition to the peak at low reflections also a smaller peak at high reflections near to ρ = 1. The latter peak is mainly formed of reflections ρ at wave-numbers corresponding to localised scattering matrix dynamics.


65

1

0.6

0.9

0.5

0.8

0.4

0.6

dP/dx

dP/dx

0.7 0.5 0.4

0.3 0.2

0.3 0.2

0.1

0.1 0

n=10 n=100 n=1000

-6

-5

-4

-3 -2 x=log10(r)

-1

0

0

-5

(a)

-4

-3

-2 -1 0 x=log10 (r )

1

2

(b)

Figure 5.6: The distribution of cka/πcRn (k) across the wave-numbers in a chain with q = 0.6, β = 0, d = 0 and different length n = 1(a), 10, 100, 1000(b).

The lower peak in the distribution Plog ρ spreads with increasing length n. From the connection between the distributions Pρ (t) = t−1 Plog ρ (log t) we conclude that the rescaled reflection ρ has approximately (practically) a singular distribution Pρ (t) ∼ t−1 around the origin. In order to get the dependence of Rn on the wave-number k, which is independent of the chain length n, we calculate its average over the chain length N 1 X Rn , R = lim N →∞ N

(5.39)

n=1

that depends solely on the wave-number k and parameters of the basic cell q, β and d. The R as a function of k is for one example of the serpent billiard presented in figure 5.7. We can 100 10 1 rR

0.1 0.01 0.001 1e-04 1e-05 1e-06 0 20 40 60 80 100 120 140 160 180 200 r = ka/π Figure 5.7: The average reflection probability averaged over the chain length R in the periodic serpent billiard with q = 0.5, β = π and d = 0 as a function the wave number k = π/ar (a = 1 − q), which is sampled in a coarse-grained sense.

clearly see that the envelope of the function R scales as k −1 and so we can with some certainty conclude that the average reflection R in (periodic) serpent billiard scales with the wave-number k approximately as R ∼ k −1 .

66


5.1.2

The random serpent billiards

A random serpent billiard is a custom-built billiard at some fixed width a = 1 − q composed of basic cells of random form. The scattering matrix of a random serpent billiard S n of length n is obtained by the following iteration Sn = Sn−1 Scell,βn ,dn ,

(5.40)

where the form parameters βn , dn in the scattering matrix of the basic cell Scell,β,d are chosen randomly at each iteration step. The parameters β, d are sampled uniformly over a certain interval. We consider three cases of intervals: (i) β ∈ [0, π/2] or (ii) [0, π] or (iii) [0, 2π], with d ∈ [0, 1] in all cases. Examples of billiards for each individual case of sampling intervals are shown in figure 5.8. We study the behaviour of the average reflection Rn of the random billiard as

(a)

(b)

(c)

Figure 5.8: Three random billiards used in calculations at k = 10.5π/a = 82.4668 and q = 0.6 = 1 − a with uniformly sampled d ∈ [0, 1] and β uniformly sampled over intervals [0, π/2] (a), [0, π] (b) and [0, 2π] (c). Only 2000 cells in each billiard are shown.

the function of length n. In all considered cases of random billiards we obtain similar results. This is expected, because by using random cells in the construction, the scattering matrix of the chain Sn becomes quickly randomised with increasing length and yields a statistically similar matrix in all considered cases. Therefore we discuss in more detail only one case of chain generation, where the angle β ∈ [0, π] and d ∈ [0, 1] The reflection as a function of n for a few realisations of the random serpent billiard at different k and q is shown in figure 5.9. We see from numerical results that the rescaled reflection ρn (k) = No Rn evolves with n similarly for different k and at fixed q. We find that as long as the average reflection Rn is small compared to the uncertainty of the reflection σR,n the rescaled reflection ρn increases linearly with n, ρn ∼ αn for

hRn i ζn = √ .1. No σR,n

(5.41)

In the figure we see that the length of the linear trend increases with decreasing q. From the equation above we can conclude that that reflection of the chain scales with the wave-number k as hRn i ∼ k −1 . We found √ in Chapter 3 for general unitary scattering matrices of dimension 2d that on average hRi ∼ dσR . By considering this together with equation (5.41) we see that at ζn ∼ 1 the Sn on open modes is similar to a random unitary matrix. Beyond the length at which ζ n ∼ 1 the reflection enters into a slower than linear increase. In the limit of very long chains, which is numerically not achievable, we expect that the reflection ρn exponentially relaxes to some plateau denoted by ρ∞ . This means that random serpent billiards exhibit (partial) localised scattering dynamics. In the case of narrow channels (small curvatures), e.g. q = 0.6, 0.9, the plateau seems to be ρ∞ ≈ 1, but in wide channels the plateau is usually much higher. Therefore to minimise


67 q = 0.2

100

3.5 3 ζn

0.1

2.5 2

0.01

1.5 1

0.001 1e-04

r=50.5

4

10-3 n

1

r=10.5

4.5

r=50.5

10

No

5

r=10.5

0.5 1

10

100

1000

0

10000 65536

1

10

100

n

1000

10000 65536

1000

10000 65536

1000

10000 65536

n

q = 0.6 10

8.5 10-5n

0.1

r=10.5

1.4

r=50.5

1

r=50.5

1.2 1

0.01

ζn

No

1.6

r=10.5

0.8 0.6

0.001

0.4 1e-04 1e-05

0.2 1

10

100

1000

0

10000 65536

1

10

100

n

n

q = 0.9 1.4

r=10.5

0.1

r=50.5

1.2

0.01

3 10-6n

1

0.001

ζn

No

1

1e-04

0.8 0.6

1e-05

0.4

1e-06

0.2

1e-07

1

10

100

1000

10000 65536

r=10.5 r=50.5

0

1

10

n

100 n

(a)

(b)

Figure 5.9: The average reflection Rn in random serpent billiards with increasing chain length n (a) and 1 the ratio between the average reflection and the standard deviation of the reflection ζ n = No − 2 Rn /σR,n (b) for different inner radii and wave-numbers k = πr/(1 − q) as indicated in the plots At each wavenumber we show ζn for three realisations of the chain with β ∈ [0, π] and d ∈ [0, 1].

the reflection through a random serpent billiard we should use bends of high curvature (q → 0). In the spirit of the presented theory in Chapter 3 we could say that ρ∞ is an effective number of eigenvalues of the transfer matrix corresponding to basic cells T [Scell,oo,βm ,dm ] outside the unit circle. This indicates that in wide channels the number of unstable eigenvalues could be in fact more than 2, which was empirically found, or the scattering matrix is not that random as assumed. In additionally, the long linear dependence of the reflection ρn with n could mean that the ratio between the number of localised and ballistic scattering matrices used to create

68


the chain is small or the localisation in used basic cells is generally small.

5.2

Statistical properties of scattering

The open billiard here in consideration can be thought of as a cavity connected to infinite leads. There exist many wonderful results in the case when the cavity is classically fully chaotic (Ott, 1993) done in the spirit of random matrix theory (RMT) (Brody et al., 1981; Guhr et al., 1998) with mathematical details given in (Mehta, 1991). The short reviews of these results are found in e.g. (Beenakker, 1997; Reichl, 2004; Stöckmann, 1999). The serpent billiard i¡s a very specific type of a system and cannot be treated as a generic chaotic cavity. This is due to the classical uni-directionality and strong coupling of the cavity with leads, which make the serpent billiard an interesting model to study. Here we try to present some aspects of the serpent billiard that display statistical nature. In the following sub-sections we only discuss periodic serpent billiards. It is instructive to take a look at the propagation of waves across serpent billiards of different lengths. We are considering a incoming Gaussian ray, defined in Appendix G, from one side of the billiard and observe how it is scattered as a function of length n. The Gaussian ray intersects the CS of the billiard at the position y0 with the average momentum in CS direction p0 and has the width equal to σ. We are observing the waves in the asymptotic region described in the basis of eigen-functions of the Laplacian. Therefore the Gaussian ray is given in the form of a vector of expansion coefficients g = {GI,n,α }n∈N (G.9) in the incoming SOS state for the left side of the serpent billiard g , (5.42) ψin = 0 where I = [0, a] is the interval of the CS and α = (y0 , p0 , σ) is the vector with all the parameters of the Gaussian ray. We then observe the evolution of the outgoing wave as we increase the length n of the billiard chain L rn g ψout,n = Sn ψin = . (5.43) tLn g An example of such evolution is shown in figure 5.10, where we consider a periodic serpent billiard with form parameters q = 0.6, d = 0 and β = π at the wave-number k = 1574.72. The number of open modes in this example is No = 200. 1

1e-08

0.01

1e-10 |gr,m|2

|gt,m|2

1e-04 1e-06 1e-08

1e-14 1e-16

n=1 n=2 n=5

1e-10 1e-12

1e-12

1

n=1 n=2 n=5

1e-18 50

100

150

200

m

250

1e-20

1

50

100

150

200

250

m

(a)

(b)

Figure 5.10: Transmitted (a) and reflected (b) parts of a Gaussian packet scattered over periodic billiards of different lengths n with d = 0, q = 0.6 and β = π at k = π200.5/(1 − q) = 1574.72. The solid vertical line indicates the last open mode.

5.2. Statistical properties of scattering

69

The expansion coefficients of the transmitted wave gt,m = [tLn g]m spread fast across all open modes with increasing length n and decay fast in the region of evanescent modes at indices n > No . The open modes near to the last one are exceptions an behave statistically different at lengthening of the chains. This indicates that a transmitted wave across longer chains is a random wave (Berry, 1977a). The coefficients gt,m behave approximately as complex Gaussian variables cm = xm + iym ∈ C, with xm and ym being real and Gaussian distributed over R:

hcm i = 0 , |cm |2 = No −1 . (5.44)

The number of open modes increases with increasing k. Obtaining the open modes in the bends becomes extremely difficult with increasing k, as explained in Chapter 4. Therefore unfortunately, we cannot numerically test the random-wave hypothesis to sufficient precision. In addition, we can see that the coefficients of the reflected wave gr,m = [rnL g]m for different lengths of the chain strongly are enhanced at indices near to the highest open mode. This property is inherited from the bends of which the serpent billiard is composed, see Chapter 4 for details on bends. In figure 5.11 we present the coefficients of the transmitted gt,m and the reflected part gr,m using a larger number of closed modes. We see that the coefficients corresponding to closed modes decrease with increasing index as |gt,m |2 , |gr,m |2 ∼ m−5 ,

m→∞.

(5.45)

The algebraic decay is the consequence of the power type coupling between the modes in the bend and in the straight segment determined by the transition matrices A and B (4.12). This is the reason why the closed modes have to be included in order to obtain an accurate description of the wave-functions. 1e-06

0.01

1e-08

1e-04

1e-10 1e-12 |gr,m|2

|gt,m|2

1e-06 1e-08 1e-10

1e-16 1e-18

1e-12

1e-20

1e-14 1e-16

1e-14

1e-22 1

10

100

800

1

10

100

m

800

m

(a)

(b)

Figure 5.11: Transmitted (a) and reflected part (b) of a Gaussian packets scattered over a periodic serpent billiard of length n = 4 with d = 0, q = 0.6 and β = π at k = π50.5/(1 − q) = 396.626. The inserted line is ∝ n−5 .

We now simplify discussion by considering only symmetric serpent billiards so that on the classical SOS the chain looks the same from the left and from the right end. The scattering matrix Ssymm,n of a billiard composed of n bends is built by the following procedure Ssymm,n = Scell,β,d . . . Scell,β,d Sb,β , | {z }

(5.46)

n

where is the operation of concatenation of scattering matrices and S cell,β,d (5.11) in the scattering matrix of the basic cell. The scattering matrix Sn has then the following block symmetric form rn tn Ssymm,n = . (5.47) tn rn

70


We are interested in the evolution of the reflection rn and the transmission tn with increasing length n of the chain. An example of such development is shown in figure 5.12. We are interested n=1

n=2

n=5

-10

-8

-6

(a)

-4

-2

0

-3

(b)

-2

-1

(c)

0

1

2

3

(d)

Figure 5.12: The density-plot of the amplitude and phase of reflection matrix elements, log10 |ri,j |, arg{ri,j } (a,c), and of the amplitude and phase of transmission matrix elements, log10 |ti,j |, arg{ti,j } (b,d). These are calculated for different chain lengths n and at fixed wave number k = 200.5π/(1 − q) = 1574.72. The solid horizontal and vertical line indicate the last open mode.

only in the open-open block of transmission and reflection matrices. With increasing length n the initial area of high intensities in transmission matrix elements [tn ]ij (i, j ∈ [1, No ]) spreads across the whole open-open block. The matrix elements become random in the norm as well as in the phase. In the reflection block we cannot detect any stronger deformation of the initial matrix element as we lengthen the chain. There is only a slight increase of the intensity around the highest open mode. In addition, we calculate the distribution of the real and imaginary parts of transmission and reflection matrix elements. The results for different lengths of the chain n are presented in figure (5.13). To enable better comparison between results obtained at different values of the average reflection we discuss distributions of rescaled matrix elements 2No tij 1−R

and

2No rij . R

(5.48)

5.3. Spectrum of the compactified serpent billiard

71

We see that the distribution of transmission elements converge with increasing length of the chain to a Gaussian shape, whereas reflection matrix elements preserve a very sharp, possibly exponential type, of distribution around zero. This indicates that the open-open block of trans10

10

n=1 n=2 n=5 n=10 n=50 G(x,0,1)

1

1

dP/dx

0.1

dP/dx

0.1

0.01

0.01

0.001

0.001

1e-04

-8

-6

-4

-2 0 2 x = ℑ[tn]ij 2 No/(1-)

10

4

6

1e-04

8

-6

-4

-2 0 2 x = ℑ[tn]ij 2 No/(1-)

4

6

8

n=1 n=2 n=5 n=10 n=50 n=100

1

0.1 dP/dy

0.1

0.01

0.001

0.01

0.001

1e-04 -15

-8

10

n=1 n=2 n=5 n=10 n=50 n=100

1

dP/dy

n=1 n=2 n=5 n=10 n=50 G(x,0,1)

1e-04 -10

-5 0 5 y = ℑ[rn]ij 2 No/

(a)

10

15

-15

-10

-5 0 5 y = ℑ[rn]ij 2 No/

10

15

(b)

L No Figure 5.13: The distributions of real and imaginary parts (a,b) of the reflection matrix elements {r ij }i,j=1 L No (bottom) and of real and imaginary parts (a,b) of transmission matrix elements {t ij }i,j=1 (top) for different length of the chain. The calculations are performed at the wave-number k = 200.5π/(1 − q) = 1574.72 and G(x, x, σ 2 ) is a Gaussian distribution.

mission matrix has some similarity with random unitary matrices. We assume that all matrix elements [tn ]ij = ψij + iξij (ψij , ξij ∈ R) for i, j ∈ [1, No ] are distributed in the same way and independent of each other. Then a meaningful solution satisfying R + T = 1 and R T is that the variables ψij and ξij are approximately Gaussian, x2 dP dP 1 e− 2σ2 + O(R) , (x) = (x) = √ dψij dξij 2πσ 2

(5.49)

with the dispersion σ 2 = (1−R)/(2No ). This result agrees perfectly with data presented in figure 5.13. In the presented studies we assume to be far away from reflection resonance discussed for a single bend in Chapter 4. Namely these could cause certain non-generic behaviour in the transmission part of the scattering matrix of the chain and additionally R T is not satisfied.

5.3

Spectrum of the compactified serpent billiard

The serpent billiard is an open system and possesses only a small number of bound states that are discussed in e.g. (Exner, 1992; Lin and Jaffe, 1996). We are interested in the energy spectrum of a closed system composed of serpent billiards and how the classical uni-directionality affects

72


the spectrum. We obtain the spectrum of the closed system in the scattering matrix formalism using the eigenvalue condition developed in (Bogomolny, 1992) and refined in (Prosen, 1996). There are three obvious ways how to close (compactify) the serpent billiard: (i) by closing both ends separately by one way scatterers or (ii) together with a two-way scatterer and (iii) connecting both ends together. All three ways are depicted in figure 5.14. Let us define these compactification methods in more detail. (i) S1

S

S2 (iii) S

(ii) S S’ Figure 5.14: There ways to compactify the serpent billiard represented by the two-way scattering matrix S:(i) by connecting to each end some scatterer with scattering matrices S 1,2 , (ii) by connecting both ends to a two-way scatterer with the scattering matrix S 0 and (iii) by joining the ends of the serpent billiard together.

The serpent billiard is described by a two way-scattering matrix S (5.4) and represents a map (5.5) of SOS state vectors – incoming into outgoing ones. In the case (i), we have two oneway scatterers with scattering matrices S1,2 connected to the left and right side of the serpent billiard. These ends intercept the outgoing parts of the SOS state of the serpent billiard and map them back into incoming waves by their scattering matrices as S1 [bL ] = [aL ] ,

S2 [aR ] = [bR ] .

(5.50)

Taking this into account and the mapping defined by S (5.5) we can write the condition for the eigen-state of the closed system, referred to as eigen-condition, using the outgoing SOS state S1 0 S(k) ψout = ψout . (5.51) 0 S2 If the ends are closed by flat walls (perfect mirrors) the scattering matrices are S 1,2 = −id. In the case (ii), we assume having an additional two-way scatterer with the scattering matrix S 0 . It is connected to both ends of the serpent billiard and maps its outgoing SOS state back into the incoming SOS state: S 0 ψout = ψin . (5.52) This yields the eigen-condition for the incoming SOS state S 0 S(k)ψin = ψin .

(5.53)

In the last case (iii), we close the serpent billiard on itself by connecting its ends together thereby forming a compact billiard. This imposes the following condition onto the CS (SOS) states 0 e−iξ id ψL = eiξ ψR ⇒ J(ξ)ψout = ψin , J(ξ) = , (5.54) eiξ id 0


73

where we introduce the Bloch phase ξ in order to generalise our discussion to periodic chains of serpent billiards with the spectrum given by the Bloch theorem (Ashcroft and Mermin, 1976). By combining the conditions for SOS states with the definition of the scattering matrix S (5.5) we get the following eigen-condition −iξ e tL e−iξ rR . (5.55) J(ξ)S(k)ψout = ψout , J(ξ)S(k) = eiξ rL eiξ tR This compactification is in some sense more pure as it does not incorporate additional scatterers. In all presented cases of compactification the obtained eigen-condition has a form of a matrix equation C11 C12 CS (k)ψx = ψx , CS (k) = , x = in, out , (5.56) C21 C22 where CS (k) is called the compactified scattering matrix and depends on the wave-number k. The equation (5.56) has a nontrivial solution at eigen-wave-numbers k – eigen-energies E = 12 k 2 , when det(CS (k) − id) = 0 . (5.57) The basis of the null-space ker{CS (k) − id} at some eigen-wavenumber k represents a set of eigen-SOS states corresponding to k. The set of eigen-wavenumbers (eigen-energies) is called the spectrum of the compactified serpent billiard. The matrix CS (k) has two blocks proportional to the transmission and other two blocks proportional to the reflection of the serpent billiard. We found that in general case reflection is many orders smaller in magnitude than the transmission, T R. This implies that two blocks in CS (k) have much smaller norm than the other ones. This has a deep impact on the spectrum of the problem, especially in the limit of high energies. Here we focus on the spectral properties of the case (iii). We are only interested in billiards that do not possess an additional discrete symmetry, besides uni-directionality present in the semi-classical limit. In the spectrum of these objects we find pairs of eigen-energies very near to each other, much closer than the predicted mean level spacing given by the Weyl formula. By this formula the average energy level density in 2D billiards reads dn A s = k− + O(k −1 ), dk 2π 4π

dn A s √ + O(E −1 ) , = −√ dE 2π 28π E

(5.58)

where we use E = 21 k 2 and A and s are the area and circumference of the billiard, respectively dn in dependence of E is a constant, (Stöckmann, 1999). In 2D billiards the leading term of dE yielding the mean energy spacing approximately given by 2π/A. The pairing can be explained in the semi-classical limit k → ∞, where we can neglect the influence of evanescent modes in the compactified scattering matrix CS (k). The reflection becomes negligible (δC = 0) making CS (k) a diagonal matrix with blocks e−iξ tL and eiξ tR , conjugate to each other lim tL t†R = id ,

k→∞

(5.59)

on the sub-block of travelling waves (open modes) due to the time-reversal symmetry. The left and right going waves do not interact with each other. Then by solving the eigen-condition for a single direction, i.e. finding a vector a such that tL a = a, we obtain the solutions of the eigen-condition of the serpent billiard (5.56) in a form of a linear superposition of vectors 0 a . (5.60) and a 0 This means that the solutions of the eigen-condition are two-fold degenerate in the semi-classical limit. At finite wave-numbers k the degeneracy is broken in the presence of a small reflection and

74


the degenerate eigen-energies are split into pairs that we find in the spectrum. Let us assume that we find a pair of solutions (ki , ψi )i=1,2 CS (ki )ψi = ψi ,

(5.61)

in close vicinity of wave-number k0 = (k1 + k2 )/2. We have assumed that CS (k) is analytic around k0 . By inserting the first order expansion of the compactified scattering matrix around k0 dC CS (k0 + δk) = CS (k0 ) + δ + O(δk 2 ) , (5.62) dk into the first equation we find that CS (k0 ) gives two approximate solutions to the eigen-condition CS (k0 )(ψ1 ± ψ2 ) = ψ1 ± ψ2 + O(δk) .

(5.63)

which means that CS (k) at k0 is already approximately degenerate in explained way. To find something more about the splitting we study how the reflection is compensated by the shift in the wave-number δk from the position k0 . We start by rewriting the first order expansion in k of compactified scattering matrix in the following form CS (k0 + δk) = C0 + δC + O(δk 2 ) + O(δkkrL,R k) , where we introduce matrices C0 and δC −iξ e tL (k0 ) 0 , C0 = 0 eiξ tR (k0 )

δC =

eiξ t0L (k0 ) e−iξ rR (k0 ) eiξ rL (k0 e−iξ t0R (k0 )

(5.64)

.

(5.65)

We refer to C0 as the leading (classical) part and to δC as the small (quantum) corrections. First we find the best approximate solutions for ψ in the eigen-condition of the leading part C0 ψ = ψ. The approximate solution is two-fold degenerate and is formed of the left-going and right-going waves denoted by u and v, respectively, 0 a . (5.66) , v= u= b 0 Vectors a and b are the closest approximations to solutions of the eigen-condition of blocks of C0 : eiξ tL (k0 )a = a + O(krL ak) , e−iξ tR (k0 )b = b + O(krR bk) , (5.67) and should be very similar to each other as was observed in the semi-classical limit. In equation √ (5.67) we have taken into account that krL,R ak ∼ R. We are interested in the leading order approximation of the eigen-condition solution. Then the eigen-SOS state ψ are expressed as a linear combination of vectors u and v, ψ = αu + βv ,

(5.68)

where α, β ∈ C are expansion coefficients. By plugging this ansatz in the eigen-condition (5.56) using the approximation of CS (k) (5.64) we obtain an approximate eigen-condition valid only in the vicinity of k0 √ (5.69) δC(αu + βv) = αu + βv + O(δk 2 ) + O(δk R) + O(R) . By multiplying the above equation from the left with u† and v † it is transformed into the matrix equation for the expansion coefficients reading −iξ ie τL δk −e−iξ ρR α =0, (5.70) β eiξ ρL ieiξ τR δk


75

with matrix element τL,R and ρL,R defined as τL = −ia† t0L (k0 )a , †

ρR = −a rR (k0 )b ,

ρL = b† rL (k0 )a , τR =

ib† t0R (k0 )b

.

(5.71) (5.72)

where we use (•)0 = d/dk. The matrix elements depend on δk and we obtain non-trivial solutions for coefficients only for values of δk that make the matrix in equation (5.70) singular. These values are r ρR ρL δk± = ± . (5.73) τL τR

They are not strictly real numbers due to approximations made or more precisely, they are real only up to order of approximations. By inserting the solution (5.73) back into equation (5.70) we obtain eigen-SOS states corresponding to eigen wave-numbers k± = k0 + δk± reading 1 ∓p1 a . (5.74) ψ± = p 2 p2 b p1 + p22

Notice that the shift δk± (5.73) in the wave-number and eigen-SOS vectors (5.74) are independent of the Bloch phase in this order of approximation. For example, in an one-dimensional scatterer with the scattering matrix reading √ √ iα iβR L Re 1 − Re √ , (5.75) S1D = √ 1 − ReiβL − Rei(βL +βR −αL ) where phases β L,R and αL are functions of k and the compactified scattering matrix is written as √ √ 1 − Rei(βL +ξ) − √Rei(βL +βR −αL +ξ) √ CS1D = , (5.76) Rei(αL −ξ) 1 − Rei(βR −ξ) the solutions for the wave-number shift δk± are given by s R δk1D,± = ± 0 ∈R. (1 − R)βL0 βR

(5.77)

The restrictions made to the model are physically meaningful, because amplitudes usually vary only slowly with the wave-number k in contrast to phases that are roughly proportional to k. The vectors of coefficients corresponding to values of δk are then written as 1 ∓p1 √ √ α =p 2 (5.78) , p1 = τ R ρR , p2 = τ L ρL . 2 β 1D,± p 2 p1 + p2

We expect p1 and p2 to be of the same order, because of the approximate symmetry between the left and right going waves. By returning to the general case, we obtain from displacements δk± (5.73) the estimate for the wave-number (energy) spacing in the observed pair r ρR ρL ∆ksplit = φi (k) mod 2π}i=1 , ui (k) = bi (k) where M is approximately twice the number of open modes in sections of the serpent billiard 1 . The phases φi (k) from Msp monotonically increase with increasing energy E = 21 k 2 as we can see in figures 5.17. Due to the classical unidirectional transport these phases are paired. An intersection of the phase φi (k) with the abscissa φ = 0 represents a solution of the eigencondition. This can be obtained by closely following the curve of the individual phase φ i (k) in the variable k. The intersection of a close pair of phases with abscissa represents solutions 1

It is interesting to notice that the number of open modes in different sections can differ only by one.

Chapter 5. Quantum serpent billiards π

π

π/2

π/2

0

0

φ

φ

78

-π/2

-π

-π/2

31

31.5

32

32.5

-π 500

33

500.5

501

E

(a)

501.5 E

502

502.5

503

(b)

Figure 5.17: The phases φi (k) of eigen-pairs in Msp (k) in two different energy regimes (a) and (b). We plot phases with red and green depending on the sign of the phase of the overlap arg s(ψ i ).

corresponding to an energy splitting. Due to the vicinity of intersections it is difficult to correctly resolve them. In the figure we can see avoided crossing between pairs of phases, when these come close to each other. The number of avoided crossing increases with increasing wave-number. We find that the left and right going waves in eigen-solutions are usually of the same order of magnitude kai k ≈ kbi k, which is also supported by our previous theoretical discussions of the splitting. In order to distinguish different intersections with abscissa of nearby evolving phases in variable k we calculate the overlaps of the upper and lower components of eigenvectors in Msp (k) si (k) = s(ui ) = ai (k)† bi (k) . (5.88)

π

π

π/2

π/2

0

0

γ

γ

The absolute value of the si (k) is not fixed and due to fine mixing of right and left going waves we expect si (k) to be practically non-zero in all cases. We are interested only in the phase of overlaps arg(si (k)), which are shown in figure 5.18 as functions of the energy for our serpent billiard. We

-π/2

-π

-π/2

31

31.5

32 E

(a)

32.5

33

-π 500

500.5

501

501.5 E

502

502.5

503

(b)

Figure 5.18: The phase of overlaps arg(si (k)) of eigen-pairs in Msp (k) for our compactified serpent billiard in two different energy regimes (a) and (b).

see that their dependence on the energy is highly complicated, but we are not interested in their behaviour over the energy axis, but just in the vicinity of solutions of the eigen-condition. The theory predicts that the overlap of eigen-SOS states ψ± in some split classical degeneracy (aka. splitting) is approximately r p1 p2 † s± = ± a b∈C. (5.89) 2 p1 + p22


79

From this expression we conclude that eigenvectors in Msp (k), which are paired in the energy and denoted by ψi and ψi+1 have an overlap with an approximate π phase difference s(ψi ) = −s(ψi+1 ) ,

arg s(ψi ) = π + arg s(ψi+1 ) .

(5.90)

In figure 5.19 we show the phases of overlaps, denoted by arg si (k), which corresponding to eigenvalues in Msp (k) with phases φi near to zero. We see that the difference between arg(si (k)) agrees with the theoretical predictions. The overlap is therefore expected to be in the opposite quadrants of the complex plane and this gives us the possibility to distinguish and resolve both solutions in the splitting. Nevertheless it is problematic automatically resolve any splitting π

π

π/2

π/2

0

0

-π/2

-π/2

-π

31

31.5

32 E

(a)

32.5

33

-π 500

500.5

501

501.5

502

502.5

503

E

(b)

Figure 5.19: The phases of overlaps arg(si (k)) (green) for eigenvalues in Msp (k) with phases |φi (k)| ≤ 0.1 and corresponding phases ψi (k) (red) calculated in two different energy regimes (a) and (b).

correctly. Therefore we do not expect to obtain all eigen-energies of the billiard in the numerically accessible range of energies. Let us say that a compactified serpent billiard occupies a domain Γ ∈ R2 in the coordinate space in which we consider a free particle. The classical phase space the compactified serpent billiards X = {(q, p) : q ∈ Γ, p ∈ R2 } can be separated into three invariant disjoint phasespace components i.e. X→ and X← of left and right traversing particles, respectively, and X· of marginally stable periodic motion perpendicular to the walls. The phase spaces X → and X← are of the same finite volume just with inverted momentum coordinate. The phase space X · is a 3D manifold separating the two phase-spaces X→ and X← . The volume of X· is therefore zero and can be neglected in following discussion. The dynamics in phase spaces X → and X← is in our examples (almost fully) chaotic. By naively considering the Berry-Robnik conjecture (Berry and Robnik, 1984) and the principle of uniform semi-classical condensation (Robnik, 1988, 1998) one would expected that a general eigen-solution can be assigned to specific invariant phase-space on which its Wigner function condenses and the spectrum of the system would therefore be strictly two-fold degenerate. But this is not the case. The phase-spaces X →,← communicate with each other through tunnelling across X· . This causes the splitting of energy levels and mixing of states “living” on separate phase-spaces. By living, we mean that the Wigner function condenses on that domain. The energy-level statistics is an important fingerprint of quantum dynamics and reflects the properties of the classical dynamics. We study this statistics in our compactified serpent billiard. Let us denote by {Ei }i∈N the energy spectrum of our billiard. In figure 5.20 we plot the level spacing between the consequent energy levels ∆Ei = Ei+1 − Ei as the function of energy Ei . We can see in the figure that the splittings in our billiard can be organised into the bottom and top band. The bottom band of the nearest neighbouring level spacings is the collection

80

Chapter 5. Quantum serpent billiards 100 1

∆ Ei

0.01 1e-04 1e-06 1e-08 1e-10

0

1000 2000 3000 4000 5000 6000 7000 8000 9000 10000 Ei

Figure 5.20: The spacings between nearest neighbouring levels ∆Ei = Ei+1 − Ei at energies Ei in the discussed serpent billiard. Using ≈ 10000 levels in energy interval [61,10000].

of discussed energy splittings and is well separated from the origin ∆E = 0. The upper band represents spacings between the levels in the consequent pairs. At reflection resonances k ∈ π/aN the envelope of the lower band strongly increases. This is expected from known behaviour of the reflection around the resonances and its connection to the splitting is given by equation (5.80). In order to compare the level distributions between different examples we perform the unfolding procedure (Haake, 2001) of the spectrum and thereby obtaining a set of rescaled ˜i }i∈N with an uniform local average level density. The set of level spacings is denoted energies {E ˜ ˜i }i∈N . We plot in figure 5.21 the level density distribution P (S) of the unfolded by {Si = Ei+1 − E spectra defined as N X P (S) = lim δ(S − Si ) . (5.91) N →∞

i=1

Let us introduce the distributions of nearest neighbouring level spacings – splittings P s (S) and the distribution of spacings between the pairs Pp (S). Then we can write P (S) as a sum of both distributions P (S) = (1 − f )Ps (S) + f Pp (S) , (5.92) with f = 1/2 in theory, but in practice, where we cannot resolve all splittings, it can be slightly bigger than one-half. Because the splittings are many orders of magnitude smaller than the average level spacing, both contributions are clearly visible in figure 5.21. The presented calculations in our serpent billiard are done using f = 0.5. We see that the distribution P s (S) has a sharp peak near to the origin S = 0. We see that there is some level repulsion between the splittings as well between the pairs of energy level. According to the Bohigas-Giannoni-Schmidt conjecture (Bohigas et al., 1984) the evolution operators of classically chaotic systems with the time-inversion symmetry are effectively GOE. The corresponding level spacing statistics is given with the Wigner distribution PGOE (S; ρ) =

π 2 − π (ρS)2 , ρ Se 4 2

(5.93)

where ρ is the average level density. We found that levels spacing of pairs in our billiard obey the GOE statistics Pp (S; ρ) = PGOE (S) with the average level density ρ = (2 − µ)−1 , where µ is the average splitting of unfolded levels Z µ = dS S Ps (S) . (5.94)

5.3. Spectrum of the compactified serpent billiard 10

0.16

P(S)

10

0.14

1

0.1

PGOE

0.18

P(S)

P(S)

0.2

100

1

81

0 1 2 3 4 5 6 7 8 9 10 2

10 S

0.12 0.1 0.08 0.06

0.01

0.04 0.02

0.001

0

1

2

3

4 S

5

6

7

0

8

0

1

2

3

4 S

(a)

5

6

7

8

(b)

Figure 5.21: The level spacing distribution P (S) on the whole range of spacings S with the inset showing the zoom around the origin (a) and in the region above the typical nearest neighbouring level spacing (b). See also caption of figure 5.20.

Then for spacings much larger than the typical splitting, the cumulative level spacing distribution W (S) reads Z S π 2 ds P (s) = 1 − f e− 4 (ρS) . WGOE (S) = (5.95) 0

From figure 5.22 we see that the predicted cumulative distribution holds nicely in our billiard on the whole range on which the model function is valid. There exist also eigen-solutions called bouncing-ball modes (Bäcker et al., 1997; Burq and Zworski, 2005), which are not paired. But they are in our case rare events on the energy axis, with their total number N bb increasing with 1 the energy as Nbb ∼ E 2 (Veble, 2006). Therefore they are not investigated here. This modes correspond to the classical dynamics of perpendicular bouncing between the walls. 1

WGOE(S)

0.9

0.1

0.8 1-W(S)

0.7 W(S)

0.6 0.5 W(S)

0.4 0.3 0.2 0.1 0

0

0.6 0.5 0.4 0.3 0.2 0.1 0

1

0.01 0.001 1e-04

0

3

4

5 S

6

7

0.1 0.2 0.3 0.4 0.5 S 2

3

4

5

6

7

8

S

Figure 5.22: The cumulative distribution of level spacings W (S) for both examples of considered compactified serpent billiards. See also caption under figure 5.20.

The obtained statistics in the serpent billiards is not unexpected. Namely, by considering the Berry-Robnik conjecture we would expect the two-fold degenerated spectrum with the Wigner statistics. But because we can not neglect the tunnelling between invariant components the degeneracy is raised and the results pairs inherit the Wigner statistics. In the article (Veble et al., 2006) authors discuss a billiard called “Monza”, which is another example of a compactified serpent billiard. They observe the spectral statistics that coincides with the one in our disordered serpent billiard.

82

5.4


Quantum classical correspondence on the classical SOS

It is commonly excepted that in the limit of high energies or small effective Planck constants a quantum system behaves similarly to its classical counterpart. The basic statement in this direction is given by the Ehrenfest theorem (Messiah, 2000). The quantum-classical correspondence (QCC), described in detail e.g. in (Bolivar, 2004), can be studied in different ways depending on the system. The most common way is to use an observable and compare its classical and corresponding quantum expectation value. In open systems most frequently used quantities are the transport properties expressed by reflection and transmission coefficients and the WignerSmith delay time in different forms. There is a large number of interesting discussions of QCC for closed and open systems of which the review can be found in e.g. (Reichl, 2004; Haake, 2001) and reference therein. In chaotic systems results are usually compared to predictions of RMT. In a recent publication (Schomerus and Jacquod, 2005) the authors shortly reviewed past research in the area of QCC in open systems. They analyze QCC in a two way scattering system called open kicked rotator and the deviations from RMT predictions. We study the QCC in the classical phase-space using quasi-distributions – the Wigner functions (Lee, 1995) in a quantitative way. This approach was recently introduced for discrete chaotic maps on a compact phase space (Horvat et al., 2006). The serpent billiard is a two-way scatterer on an infinite straight wave-guide. We are discussing correspondence between the classical and quantum scattering across the serpent billiard at some fixed energy E = 12 k 2 in the spatial domain. In our analysis we use a classical Gaussian ray i.e. classical bundle of trajectories forming a Gaussian type of distribution on the classical phase-space at one CS and corresponding quantum Gaussian ray. The later is given as a SOS state with a Gaussian packet defined at the same CS. The Gaussian ray can be understood for example as a laser-ray used in optics. We basically study QCC by comparing the scattering of a Gaussian ray coming from one side into the billiard, both in classical and quantum picture. Just for an illustration, we show in figure (G.5) a Gaussian ray coming from the left into the bend and scattered across the billiard area in the coordinate space. In this example the scattering is described in the quantum case by a wave-function ψ(x) across the billiard area and in the classical case by a ensemble of trajectories passing through the billiard. The quantum Gaussian ray

(a)

(b)

(c)

Figure 5.23: The amplitude |ψ(x)|2 (a) and the phase arg{ψ(x} (b) of a wave function ψ(x) and the classical trajectories of a Gaussian packet prepared in the SOS and scattered across a bend with parameters q = 0.6, β = π and √ d = 0 and k = 789.325 (No = 100). The initial packet is positioned at y = a/2 = 0.2, . p0 = 0 with σ = a/ 2πNo = 0.0159.

hits the outer walls and thereby creates inferences. These decrease the correspondence with the

5.4. Quantum classical correspondence on the classical SOS

83

classical Gaussian ray. The ray refocuses, when leaving the wall. At the focus point the phase arg{ψ(x)} exhibits a saddle type spatial dependence. The caustic of phase clearly shows the direction of the probability flow. This becomes evident by considering the WKB approximation, where ψ(x) is expressed by a real valued function S as ψ(x) ∝ exp(ikS). The calculation of the local probability flow j then gives j = ={ψ ∗ ∇ψ} ∝ ∇S .

(5.96)

In this example, we see quite clear qualitative correspondence between the quantum and classical picture. In general the study of QCC in the coordinate space of the serpent billiard is computationally too demanding. Therefore we discuss the correspondence of waves and trajectories as they pass the cross-sections (CSs) of the scatterer. This is possible in the quantum and classical scattering formalism. The on-shell scattering through the CS and across the scatterer is determined in the quantummechanical picture by the scattering matrix operator Sbserp (5.16) mapping the incoming SOS state |ψi into the outgoing SOS state |ψ 0 i as |ψi0 = Sbserp |ψi ,

|ψi, |ψi0 ∈ H(C) .

(5.97)

The operator Sb is not unitary, but instead is generalised unitary. In the classical picture the e serp (2.13) scattering is defined by the Poincaré map P e serp (x) , x0 = P

x, x0 ∈ SSOS ,

(5.98)

bserp (Arnold and Avez, 1989), or here more appropriately by the Perron-Frobenius operator P which maps the distribution of incoming trajectories ρ into the distribution of outgoing trajectories ρ0 bserp ρ(x) = ρ(Pe −1 (x)) , ρ0 (x) = P ρ0 , ρ ∈ C ∞ (SSOS ) . (5.99) serp

The classical phase space SSOS (2.11) and Hilbert space H(C) (5.12) are composed of two parts assigned to the left and right CS. According to Bogomolny (Bogomolny, 1992) the scattering e serp . There is matrix operator Sbserp is a quantum analog (quantisation) of the Poincaré map P no general recipe for quantisation of maps, but some general understanding can be obtained from (Berry et al., 1979; Hannay, 1980; Degli Esposti and Graffi, 2003). The classical map is area-preserving and defined on a compact phase space. Maybe it seems contradictory that we can have a compact classical phase space and an infinite dimensional Hilbert space H(C L,R ). But in each asymptotic region (lead) only open modes correspond to non-decaying waves and therefore we have at some energy E = 21 k 2 only No = ak/π basis vectors with a clear classical correspondence. Consequently the effective dimension of the Hilbert space H(C) is 2N o , with the factor 2 representing the number of independent CSs. The configuration space C on which we define the SOS states H(C) is composed of two CS represented by intervals I. Each of them can be understood as an infinite well. Then a general SOS state |ψi = |ψL i ⊗ |Li + |ψR i ⊗ |Ri ∈ H(C) (5.100) can be thought of as a linear superposition of states |ψL i and |ψR i in left and right infinite well, respectively. The state in the infinite well hy|ψi with y ∈ I can be represented in the WeylWigner formalism on the Euclidean phase-space (Lee, 1995; de Groot and Suttorp, 1972) by a real valued quasi-probability density called the Wigner function. It is defined on the phase space Sqm = {(y, p) : y ∈ I, p ∈ R}

(5.101)

84


and given with the formula fψ (y, p) = 1 W π

Z

min(y,a−y) − min(y,a−y)

dv ei2pv hψ|y + vihy − v|ψi .

(5.102)

R fψ (x) is normalised so that f The Wigner function W Sqm dx Wψ (x) = 1. More details on the Wigner function of a quantum system in the infinite well are presented in Appendix G. The classical phase-space Se of trajectories passing the CS from one side at a given momentum k is the subset of Sqm : Se ⊂ Sqm . From experience we know that the effective part of the e The quantum analog of the classical phase-space Sqm,SOS is Sqm is approximately equal to S. composed of phase-spaces Sqm.L = (L, Sqm ) and Sqm,R = (R, Sqm ) corresponding to the left and right infinite well, respectively, and reads Sqm,SOS = Sqm,L ∪ Sqm,R .

(5.103)

The phase-space Sqm,SOS is the definition domain of the Wigner functions Wψ of SOS states |ψi (5.100), which is explicitly given by fψ (x) . Wψ (s, x) = W s

(5.104)

tr{ˆ ω (s0 , x0 ) ω ˆ (s0 , x0 )} = δs0 ,s δ(x0 − x) .

(5.106)

In the general Weyl-Wigner formalism we operate with point operators {ˆ ω (s, x)} (s,x)∈Sqm,SOS that form a complete base of operators on the Hilbert space H(C) and are defined as r Z min(y,a−y) 2 0 0 ω ˆ (s, x) = ω ˆ (x) ⊗ |sihs| , ω ˆ (q, p) = dv eipv |q + vihq − v| , (5.105) π − min(y,a−y) with the orthogonality property

By using the point operators the Wigner function can be elegantly expressed by 1

ω (s, x)|ψi , Wψ (s, x) = (2π)− 2 hψ|ˆ

(5.107)

and has the role of a rescaled expansion coefficient of the density operator |ψihψ| in the basis of point operators. This implies we can write the density operator of the pure state as Z 1 d2 ζ Wψ (ζ)ˆ ω (ζ) . (5.108) |ψihψ| = (2π) 2 Sqm,SOS

We consider a SOS state |ψi scattered into a state |ψ 0 i = Sbserp |ψi with the corresponding Wigner function W = Wψ and W 0 = Wψ0 , respectively. The point operators enable us to write the whole scattering process in terms of Wigner functions as W 0 = Sbw,serp W 0 ,

(5.109)

where we introduce the Weyl-Wigner scattering operator Sbw,serp acting explicitly through an integral kernel Z † b ω ˆ (ξ)Sbserp ω ˆ (ζ)}g(ζ) . (5.110) d2 ζ tr{Sbserp Sw,serp g(ξ) = Sqm,SOS

The presented description of the quantum scattering using Wigner functions in the SOS is analogous to the classical description of the scattering using the probability densities (5.99). The


85

Weyl-Wigner scattering operator Sbw,serp is not strictly unitary or more precisely, it is approximately unitary on the subset of functions defined on the classical-phase space S SOS . We refer to this form of the quantum and classical scattering description, defined by equations (5.110) and (5.99), as the SOS method or the SOS approach to scattering. The operator Sbserp is the quantum bserp and therefore we expect that Sbw,serp converges to analog of the Perron-Frobenius operator P 1 2 b Pserp with increasing energy E = 2 k in the set of smooth functions C ∞ (SSOS ) defined on the classical phase-space SSOS : bserp . (5.111) lim Sbw,serp (k) = P k→∞

Such a limiting procedure can be explicitly performed in some simple systems, but this problem has attracted less attention in the past. A similar problem is more strongly represented. This is the convergence of the quantum dynamics in the Weyl-Wigner representation towards its classical analog by decreasing the effective Planck constant. For a review of this topic see e.g. in (Hillery et al., 1984; de M. Rios and Ozorio de Almeida, 2002; Ozorio de Almeida and Brodier, 2005). Recently in (Dittrich et al., 2006), a semi-classical Wigner time propagator is presented in a compact form. We are discussing the scattering of a Gaussian ray across periodic serpent billiards of different lengths n in the SOS approach. The Gaussian ray enters the billiard area from the side s ∈ {L, R}. In the classical scattering picture the ray is described by a probability density function of e incoming trajectories ρ(s, x) = φ(x) of a Gaussian shape defined on the phase-space S L = {s, S}. We call it the classical Gaussian packet to which we assign some phase-space position (y 0 , p0 ) and width σ and is expressed using the function gI,α (G.11) defined in Appendix G as φ(x) = gI,α (x) ,

x∈S,

(5.112)

where α = (y0 , p0 , σ) is the vector of parameters. The corresponding quantum Gaussian ray is defined in the scattering formalism as a SOS state |ψi = |φi|si ∈ H(C) and called the quantum Gaussian packet. It is expressed on the CS interval using the function GI,α (y) (G.5) as hy|φi = GI,α (y) , y∈I. (5.113) R The state is normalised so that kφk22 = I dy hφ|yihy|φi = 1. In order to contain the leading part of the quantum Gaussian packet in the basis of open modes for all No or energies E = 12 k 2 we 1 have to work with p0 p and the width σ = O(No − 2 ). In this case the norm kφk2o of the state φ contained in open modes scales as 2

kφk2o = 1 − O(e−[σ(|p0 |−k)] ) ,

k →∞.

(5.114)

√ We choose σ = a/ 2πNo so that the Gaussian packet at p0 = 0 has symmetric uncertainties in the phase-space. The leading contribution to the reflected rˆL |φi and transmitted part tˆL |φi of a Gaussian ray |φi is confined in open modes making the operator Sbserp effectively unitary. The Wigner function of the incoming Gaussian packet is denoted by W (s, x). We consider only Gaussian packets (rays) positioned well inside the classical phase-space for which the Wigner function W and the classical density ρ are very similar: W (s, x) = ρ(s, x) + e−|O(k)| .

(5.115)

The error estimate on r.h.s. in equation (5.115) was rigorously derived for a similar problem of Gaussian packets in the toroidal geometry in reference (Degli Esposti et al., 2006). Notice that the quantum W (s, x) and classical ρ(s, x) phase-space functions describing the incoming Gaussian ray are either identical, or converge to each other in the semi-classical limit k → ∞. In figure 5.24 we show an example of such a scattering described in the SOS approach, where we

86

Chapter 5. Quantum serpent billiards 1

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Figure 5.24: The classical (top row) and the corresponding quantum scattering (bottom row) of a Gaussian packet on classical SOS Scl = [0, a] × [−1, 1] across a single cell of the serpent billiard with parameters q = 0.6, β = π and d = 0√and k = 789.325 (No = 100). The initial packet (a) is positioned at y = a/2 = . 0.2, p0 = 0 with σ = a/ 2πNo = 0.0159. The classical scattering is represented by a density of points (b), and the quantum by the WF of the transmitted part tLserp |φi/ktLserp |φik (b) and of the reflected part . L L L rˆserp |φi/kˆ rserp |φik (kˆ rserp |φik = 1.8 · 10−5 ) (c) of the initial packet φ.

consider a Gaussian ray coming from the left into a single bend. This example was previously depicted in figure 5.23 in the coordinate space in terms of wave-functions and trajectories. The transmitted part of the classical and the quantum Gaussian packet on SOS are very similar. The high positive values of the Wigner function coincide with high values of the classical density. The classical packet is folded and split into two parts that are close to each other. The proximity of the parts creates interference fringes in the Wigner function of the transmitted part on the scale of the effective Planck constant O(2π/No ) (Zurek, 2001). The Wigner interference fringes are purely quantum phenomenon and consequently diminish the QCC. In the dynamics of a system the occurrence of fringes are the first signs of decoherence. The statistical nature of fringes in chaotic systems on a compact phase space is discussed in (Horvat and Prosen, 2003b). The reflected part of the quantum Gaussian packet does not have a classical correspondence. We can see that the main contributions to the reflection are near to the borders of classical phase space in the momentum axis |p| = k. e serp,n the Poincaré map for scattering Let us denote by Sbserp,n the scattering operator and by P across a periodic serpent billiard of length n. The scattering at given n in the SOS approach is described in the classical picture by the probability density function ρ(n) and in the quantum picture by the Wigner function W (n) over the classical phase space SSOS and its quantum analog Sqm,SOS , respectively. Then the evolution of the classical system along the length of chain n is given by e −1 (ξ)) = P bserp,n ρ(ξ) , ξ ∈ SSOS , (5.116) ρ(n) (ξ) = ρ(P serp,n and of the quantum counterpart written as

W (n) (ξ) = WSbserp,n |ψi (ξ) = Sbw,serp,n W|ψi (ξ) ,

ξ ∈ Sqm,SOS .

(5.117)


87

Our aim is to understand the behaviour of the L2 norm of the difference between the Wigner function W (n) and the probability density ρ(n) as the serpent billiard is increased in length n, defined as Z (n) (n) 2 (n) 2 (n) 2 dξ ρ(n) (ξ)W (n) (ξ) . (5.118) kρ − W k2 = kρ k2 + kW k2 − 2 SSOS

where we already used the fact that the classical density function is zero outside the classical SOS SSOS . The Perron-Frobenoius (Weyl-Wigner scattering) operator is (approximately) unitary and so the L2 norms of both functions are (approximately) conserved : kρ(n) k22 = kρk22 and kW (n) k22 ≈ kW k22 . Practically the only quantity in the expression (5.118) that changes with the length n is the overlap of the Wigner function W (n) and the classical density ρ(n) named the quantum-classical fidelity (QCF) and is defined by Z Z (n) (n) e serp,n (ξ)) , F (n) = dξ ρ (ξ)W (ξ) = dξ ρ(ξ)W (n) (P (5.119) SSOS

SSOS

where we have used the fact that P serp,n is symplectic. The QCF (5.119) is a measure of the quality of the QCC. Notice that the QCF can be expressed as an overlap of the initial classical density and the object called the classically echoed Wigner function written as e serp,n (ξ)) . H (n) (ξ) = W (n) (P

(5.120)

This is a useful object to study, because its deformation from the initial form is directly connected to the change of QCC. In our case, where Gaussian rays are coming from a specific side s the expression for the QCF can be significantly simplified and reads Z Z 2 −1 e e f fˆs F (n) = d x φ(Pserp,n (x))Wtˆsserp,n |φi (x) = d2 x φ(x)W (5.121) tserp,n |φi (Pserp,n (x)) , Se

Se

e serp,n : S˜ → S˜ The expression tˆsserp,n : H(Cs ) → H(Cs ) is the transmission matrix operator and P is the Poincare (jump) map from one to the other side of the serpent billiard. The QCF in our case just the overlap of phase-space functions corresponding to transmitted parts in quantum fφ and the classical density and classical system. By using the similarity of the Wigner function W φ at high enough wave-numbers (5.115) we can approximate the QCF as Z e fφ (x)W fˆs dx W F (n) = F 0 (n) + exp(−|O(k)|) , (5.122) F 0 (n) = tserp,n |φi (Pserp,n (x)) , Se

which is a more practical formulation of QCF, because then there is no need for classical densities in the formulation. Notice that in our scattering experiment F 0 (0) = (2π)−1 + exp(−|O(k)|). If the effective part of quantum phase space Sqm corresponding to one side of the billiard is equal to the classical phase space Se then we can find for the approximated QCF an upper bound F 0 (n) ≤ ktsserp,n φk2 F 0 (0) .

(5.123)

Note that ktsserp,n φk2 ∼ T . The property (5.123) can be very useful in numerical studies. We conclude that the quality of QCC is best described by the relative QCF defined as G(n) =

F 0 (n) . F 0 (0)

(5.124)

The relative QCF G(n) is our measure of QCC. The decrease of G(n) from the value 1 is understood as notable violation of the QCC. The classical Gaussian ray enters on the side s and exits on the opposite side s 0 . We present in figures 5.25 and 5.26 the development of the classical density ρ(n) , the Wigner function W (n)


n=3

n=2

n=1

88

-0.15

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0

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(c)

Figure 5.25: The Wigner functions Wψn (x) of the renormalized transmitted part tˆLserp,n |φi/ktˆLs,serp,n |φik e −1 (x)) of the Gaussian packet of points on of a Gaussian packet (a), the classical distribution φ(P serp,n e serp,n (x)) (c) for different lengths n of a the SOS φ(x) (b) and classically echoed Wigner function Wψ (P periodic serpent billiard as indicated in the figure at k = 789.325. The basic-cell configuration is q = 0.6, β = π and d = 0.

and the classically echoed Wigner function H (n) on Ss0 with increasing length n in two periodic serpent billiards with the basic-cell configuration β = π, d = 0 and β = π/4, d = 0, respectively. The calculations were performed at q = 0.6, k = 789.325. The classical phase-space of the first billiard is (almost) fully chaotic. The classical packet ρ(n) is with increasing n undergoing the process of stretching and folding back onto the phase space as in a kind of a Horseshoe map (Ott, 1993). The quantum packet W (n) follows the classical counterpart for a short time before quantum interference between parts of the packet breaks the QCC. We already established that with increasing length n the SOS state Sbn |ψi has similar statistical properties as a random wave in open modes of a straight channel defined by equation (5.44). Then according to the analysis in (Horvat and Prosen, 2003b) the Wigner function W n converges with increasing length n to a random scalar field defined on the classical phase space S˜ with with Gaussian distribution of values given by 1 1 (5.125) P (w) = p exp − 2 (w − W )2 , 2 2σw 2πσw where the first two (central) moments are approximated as 1 1 . 1 2 . 1 , σw = − W = , V V 2π V

(5.126)

using the volume of the classical phase space V = 2ak. It is instructive to define the relative standard deviation in units of the average Wigner function κ = σw /W . At large k we can


89

n=3

n=2

n=1

write κ2 ≈ ak/π = No and see that the distribution (5.125) diverges in the semi-classical limit k → ∞. The classical unstable directions are clearly visible from the dynamics of ρ (n) . The classically echoed Wigner function H (n) measuring the QCC deforms with increasing n and eventually spreads across the whole phase-space. The latter indicates the breakdown of QCC. The directions in which the classically echoed Wigner function stretches are similar to the contours of the time function. Following the arguments in (Horvat et al., 2006) we could say the e −1 . In direction of deformation coincides with the stable manifold of the inverse Poincare map P

-0.15

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Figure 5.26: The basic-cell configuration is q = 0.6, β = π/4 and d = 0. For description of figures see the caption under the figure 5.25.

the second billiard the phase space is of the mixed type with a large regular component. The Gaussian packet is initially positioned in the vicinity of the regular island. The classical ρ (n) and quantum W (n) packet with increasing length n follow each other on the stable manifold without a stronger accumulation of interference. Nevertheless there is some decay of QCC, but it is much slower than in the first billiard chain. We are trying to understand the evolution of QCF with increasing length n in our experiment with a Gaussian ray coming form s-side through the use of the classically echoed Wigner function on the classical SOS, which corresponds to the opposite s0 -side e e (n) (x) = W fˆs H tserp,n |φi (Pserp,n (x)) ,

x ∈ S˜ .

(5.127)

e serp,n = P e n of Poincaré e serp,n (2.4) can be written as a product P The classical Poincaré map P e =P e serp,1 . In the semi-quantal approximation the transmission maps of the single basic-cell P s operator (tˆserp,n )sc (5.26) can be given as a product (tˆsserp,n )sc = tˆn of transmission operators tˆ = (tˆsserp,1 )sc corresponding to the basic cell. The operator tˆ is semi-quantal quantisation of the map

90


e In the limit of high wave-numbers the classically echoed Wigner function is approximately P. e n (x)) . e n (x) ≈ W fˆn (P H t |φi

(5.128)

This approximation is meaningful exclusively in the usual cases of classically uni-directional serpent billiards, where the reflection is several orders of magnitude smaller than the transmission. Consequently, at high wave-numbers the evolution of the transmitted waves with the length n in the SOS approach is approximately equivalent to the time evolution of a discrete e and corresponding quantum evolution operator dynamical system given by classical map P e The effective dimension of the Hilbert ˆ t : H(I) → H(I) with the compact classical phase space S. space H(I) is equal to the number of open modes in the straight wave-guide N o = bka/πc. The time evolution of the QCF in classically chaotic discrete maps on the compact phase-space is discussed in article (Horvat et al., 2006). They use several well known toy models of discrete dynamical systems, e.g Perturbed cat map (Garcia-Mata and Saraceno, 2005) and Sawtooth map (De Bièvre and Degli Esposti, 1998) in the toroidal geometry. The simplicity of the models enables computation of QCF up to much higher dimensions of effective Hilbert space and precision than we are able to do in our serpent billiards. With the results of analysis in the article we can predict the behaviour of the QCF in our scattering. In order to simplify discussion the length of the serpent billiard is denoted by t (t ≡ n) and referred to as “time”. In strongly chaotic serpent billiard we expect that the QCF begins to strongly decay at some length (time) T 1 , when the packet starts to interfere with itself. The QCF then decays exponentially with the rate given by e as the maximal Lyapunov exponent λmax of the Poincaré map P 1

G(t) ∼ (λmax ~) 3 e−λmax (t−T1 ) ,

(5.129)

where the ~ is effective Planck constant in our case equal to 2π/No . The duration time of the QCF decay is approximately the famous Ehrenfest time scale TE = | log ~|/λmax on which the Wigner function spreads across the whole available phase-space. The decay ends approximately at the length (time) T2 ≈ T1 + TE after which the wave function saturates toward a state statistically equal to a random wave (Berry, 1977a). Using the random wave approximation we find that G(t) saturates to a plateau given by Gplateau = No −1 . In the figure 5.27 we show some examples of the QCF as a function of length n in some typical periodic billiards at q = 0.6. The calculations are performed with the Gaussian packet positioned in the middle of the phase space and various wave-number values k. In the strongly chaotic cases β = π in which the phase space is (almost) fully chaotic, the relative QCF G(n) decays exponentially down to the plateau 1/No in a way predicted by the formula (5.129). The case β = π/2 is still chaotic but already contains some small regular islands and therefore the QCF decays (exponentially) fast, but does not follow predicted dependence that well. In these two cases the relaxation of the QCF down to the plateau happens on the Ehrenfest time scale TE . The case β = π/4 represents rather opposite situation from the previous one, because there are large regular islands in the phase space on which the leading part of the initial Gaussian packet lays. This example corresponds to the dynamics of the classical and quantum phase-space functions in figure 5.26. The classical and quantum packet with increasing length n just rotates on the regular component. Therefore the QCF decays slowly and the decay is visibly slower for larger wave-numbers. As explained in the article (Horvat et al., 2006) the decay occurs on the time scale proportional to the well known Heisenberg time scale tH = No . The value of the wave-number k plays an important role in the functional dependence of the QCF with length n. We generally expect that in the semi-classical limit k → ∞ the QCC is perfectly represented by G = 1. But we wonder how does the QCC improve with increasing wavenumber k. Let us shortly discuss the QCF in dependence of k at fixed n, which could be in theory obtained by combining the semi-classical approximation and the Weyl-Wigner formalism. But

5.4. Quantum classical correspondence on the classical SOS 1

91

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Figure 5.27: The evolution of the relative QCF G with the chain length n in three examples of periodic serpent billiards with cell configuration q = 0.6, d = 0 and β = π, π/2, π/4 (a,b,c) for various wavenumbers k = πr/(1 − q) as indicated in the figure. The solid lines are theoretically predicted decays determined up to a constant factor. 1/50

1/200 this is too difficult task, because of1/100 the complexity of the system and the difficulty to distinguish between the semi-classical errors and exact results. Therefore we analyse the QCF as a function of k only numerically. The results for serpent billiards of lengths n = 1, 2, 4 with q = 0.6, d = 0 and β = π, π/2, π/4 are presented in figure (5.28) in which the Gaussian packet is placed in the middle of the phase space. We should recall that the phase space in the case β = π is practically fully chaotic, whereas in the cases β = π/2, π/4 it is of mixed type. In the last case β = π/4 the regular component is very large. It is interesting to notice that for small wave-numbers the QCF is in all cases rather large. This is because only few open modes describe the Gaussian packet on SOS at small k, which cannot strongly interfere on small length scales as the accumulated phase in all modes is practically the same. The QCF in a single bend is much lower in the case β = π than in the cases β = π/2, π/4 due to additional hits to the wall, which enhance the interference effect between parts of a Gaussian ray. In the chaotic periodic serpent billiard the ergodic length ˜ is of the order n = 4. Therefore the phase space properties are scale nerg of Poincaré map P unimportant for understanding the QCF in very short serpent billiards e.g. with n = 1, 2, as shown in figures 5.28a and 5.28b. In these cases only the collisions to the walls can be blamed for the decrease of the QCF. In chains with a chaotic phase-space and with the length n ∼ n erg we see that the QCF decreases with increasing k as O(k −1 ), which means that G is at its plateau value. Beyond some wave-number kbreak ∝ eλmax n , which is proportional to the stretching factor of the phase-space, the QCF is again increasing, as expected. In the case β = π/4 the Gaussian packet is positioned near to the regular component and so the QCC is basically just improving with increasing k. At the current state of computations it is unfortunately impossible to go to much higher wave number than presented.

92


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Figure 5.28: The relative QCF G as a function of the wave-number k = rπ/(1 − q) in the scattering of a Gaussian SOS state, positioned at y = (1 − q)/2, py = 0, across a single cell n = 1 (a) and a periodic chain of length n = 2 (b) and n = 4 (c). The configuration of basic cell is d = 0, q = 0.6 and β as written in the figure.

93

Chapter 6

Conclusions In the presented thesis we discuss a specific class of billiard chains in two-dimensions called the serpent billiards with a special dynamical property of classical uni-directional motion. We analyse the dynamical properties of such billiards in the quantum and classical picture. In the classical picture we have proved unidirectional motion for a more general class of billiard channels with parallel walls. The dynamics along the serpent billiard channel is described in terms of a variant of a jump model (Zumofen and Klafter, 1993), namely the jump-Poincaré map between the surfaces of section of two adjacent basic cells of the billiard, and the time function i.e. the time needed to traverse the basic cell as a function of the position on the surface of section. We have shown that the jump map is chaotic with generally mixed phase space. The relative size of the largest chaotic component is increasing with decreasing width of the channel. The latter dependence is not strictly monotonic, because of bifurcations of regular components, but for narrow channels the chaotic component is typically largely dominant. It is easy to find parameter values for which all possible islands of stability are undetectably small for numerical (experimental) purposes. The numerically measured maximal Lyapunov exponent shows that the chaoticity in a chaotic component is monotonically increasing with narrowing the channel. The transport of particles in the typical case along the channel measured in the number of traversed basic cells exhibits a marginally-normal diffusion σn2 ∼ t log t (when the drift term is subtracted) due to square-root singularity of the time function. This singularity is a consequence of parallel walls, or saying in dynamical terms, it is due to a family of marginally stable bouncing ball trajectories bouncing perpendicularly between the walls. In order to understand the quantum analog of the classical serpent billiards we investigate the general linear chains of unitary scatterers using the so-called dynamical approach. In the latter approach we study properties of the chains of scatterers in the scattering matrix formalism as they growth in length. The growth is realised by concatenation of additional scatterers at the end of the existing chain. In this numerically stable approach we discussed the chains of one-dimensional and d-dimensional scatterers. We were able to reduce the one-dimensional case to a simple two-dimensional abstract dynamical system in which we obtained asymptotic results analytically. There we also learnt, that depending on the scattering matrices used in creation of the chain, we distinguish two types of dynamics, namely ballistic and localised. The first corresponds to some quasi-periodic motion of the chain’s scattering matrix and the second to the motion in which the transmission decays exponentially to zero with increasing chainlength. We also give some analytic results for both types of motion and reproduce well known results about the transmission distributions in the case where we choose scatterers in the chain randomly. In the chains composed of d-dimensional scatterers we only give numerical results. We can again separate the motion into ballistic and localised according to the building scattering matrices, by mapping them into the transfer matrix formalism and examining their spectra. In the localised dynamics the transmission decays exponentially towards some plateau, where

94

Chapter 6. Conclusions

the largest eigenvalue of the transfer matrix outside the unit circle determines the speed of the decay. The plateau of the transmission generally decreases with increasing number of eigenvalues outside the unit-circle. The Haar measure of localised scattering matrices in U (2d) is increasing rapidly with the dimension of scatterers d, namely as 1−exp(−O(d2 )). By examining eigenvalues of transfer matrices corresponding to scattering matrices S ∈ U (2d) taken uniformly w.r.t. to the Haar measure we find that the average decay rate of the transmission I increases with the dimension as I ∼ 21 log(d). Our studies show that the percentage of eigenvalues of transfer matrices outside the unit-circle increases with increasing dimension d. This means that in the semi-classical limit long chains will have on average very low transmission. The serpent billiard is composed of bends and straight segments. The bends break the classical integrability on a straight wave-guide in a “soft” manner so that they preserve unidirectionality. We examine the effect of the latter on the wave-phenomena by studying the scattering over a single bend connected on each open side to straight leads. We derive mathematical properties and numerical methods for computing the mode functions and the corresponding mode numbers of the Laplace operator defined over the bend. We take special care of closed (evanescent) modes in the bend. The obtained modal structure and its properties are used in developing a stable numerical scheme for the calculation of the scattering matrix with a controllable precision. The numerical scheme is applied to obtain general stationary transport properties. In particular we focus on the average reflection R, which measures quantum violation of classical uni-directionality. In a bend of width a we find that the reflection at wave-numbers k ∈ π/aN strongly increases up to R ∼ No −1 , where No is the number of open modes in the straight waveguide. These so-called reflection resonances are explained using a one-dimensional model based on the highest open modes in the bend and in the straight wave-guide. The average reflection is decreasing slowly with increasing wave-number k, as R ∼ k −1 . Additionally, we present results on the Wigner-Smith delay in the bend, where we examine in more detail the behaviour around the reflection resonances. This is again explained using a one-dimensional model. The quantum description of the particle in a serpent billiard is the main topic of this thesis. We discuss the serpent billiard as a chain of two types of quantum scatterers – bends and straight wave-guide segments. These scatterers are very specific due to the underlying classical uni-directionality. Therefore the transport properties are not satisfactorily explained by results obtained in the general chains. In the study of the transport properties we discuss periodic and random serpent billiards. We find that the reflection is usually several orders of magnitude smaller than the transmission. But there are a few exceptions. At wave-numbers corresponding to reflection resonances of the bend k ∈ π/aN the reflection of the serpent billiard strongly increases. In addition, we find that periodic chains constructed from basic cells of a certain configuration exhibit (partial) localisation as found in the general chains. The localisation is more common at wave-numbers near to the reflection resonances. In the random serpent billiards the reflection is found to increase linearly, R ∼ n, up to certain length of the chain. By lengthening the serpent billiard we find that the transmission blocks in the scattering matrix become statistically similar to random unitary matrices. We take special interest in the spectrum of closed nonsymmetrical serpent billiards obtained in the scattering matrix formalism, where we concentrate on signatures of classical uni-directionality. The spectrum is composed of pairs of energy levels that are very near to each other. The pairs represent a split semi-classical degeneracy of left and right going waves due to the presence of reflection. The reflection is here directly connected to the tunnelling between classically invariant phase-spaces of left and right traversing particles. We found that the splitting between √ levels in the pair ∆k scales with the average reflection and the Wigner-Smith time as ∆k ∼ R/τws . In not to long random billiard this implies that ∆k ∼ 1 n− 2 with n. As expected, the pairs have the Wigner type of level statistics typical for chaotic systems. In addition, we investigate the classical-quantum correspondence (QCC) of scattering in the classical SOS of periodic serpent billiards as these are increased in length n. The QCC is

95 quantified by an overlap between classical density of trajectories and the corresponding Wigner functions named the quantum-classical fidelity (QCF) F (n). We find that QCF in strongly chaotic billiards decays exponentially as F ∼ e−λmax n with increasing length n to a plateau equal to No −1 , where λmax is the maximal Lyapunov exponent. We connect the presented results with those obtained in chaotic discrete maps (Horvat et al., 2006). The serpent billiards are interesting dynamical objects, with rather specific properties. They can be used as models in real world problems of transport, such as optical fibers or waveguides. The small reflection in long chains makes the serpent billiard useful to the field of massive transport of microwave energy. We believe that the presented methods of analysing the dynamics along the chain can be useful in discussing other unidirectional billiard channels or even more general billiard chains. But the discussion of the serpent billiards still leaves some loose ends. Namely, we still do not fully understand the tunnelling between classically invariant phase-space components, the resulting strength of the reflection and the condition to obtain (partial) localisation in the serpent billiard chain. Additionally we show a need for a more general formulation of the tunnelling, which can be more directly incorporated into the properties of the energy spectrum.

96

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105

Appendix A

The jump models of elements in the serpent billiards A serpent billiard is constructed from two types of elements i.e. bends and straight segments of a wave-guide. In these elements particles enter from one side and leave through the other side thereby obeying the property called the unidirectional motion. We describe the exit (entry) point of the trajectory with a position y and a projection of the direction v y on the entry (exit) cross-section, which together form a point (y, vy ) on the entry (exit) phase space S. The dynamics over an element is described by a jump model (T, P), which is composed of a jump (Poincaré) map P : S → S and a time function T : S → R, introduced in Chapter 2. The Poincaré map P associates to an every entry point (y, vy ) ∈ S on one side of the element an unique exit point (y 0 , vy0 ) = P(y, vy ) on the other side of the element, and the time function T measures the geometric length of that trajectory, equal to T (y, vy ). The jump models of the straight segment (Ts , Ps ) (a) and the bend (Tb , Pb ) (b) are presented in the following. (a) The straight wave-guide of width a and length d is schematically depicted in figure 2.4a. The exit point (y 0 , vy0 ) of the trajectory entering at the point (y, vy ) is given by the Poincaré map Ps (y, vy ) =

(˜ y − na, vy ) : n = even , (a − y˜ + na, −vy ) : n = odd

d vy y˜ = y + q , 1 − vy2

n = b˜ y /ac ,

(A.1)

and the time spend to cross the element reads d Ts (y, vy ) = q . 1 − vy2

(A.2)

(b) The bend with an angle β and an inner radius q is shown in figure 2.4b. It is more convenient to describe trajectories over the bend in polar geometry. Therefore we transform the points in the phase space S into the radial coordinates using the formula (r, φ) = (y + q, − arccos v y ), as radius r and angle φ. The particle’s entry (y, vy ) and exit (y 0 , vy0 ) point are translated into (r, φ) and (r 0 , φ0 ), respectively. Over the bend the angular momentum Γ is conserved representing the identity connecting the entry and exit point Γ = r sin φ = r 0 sin φ0 .

(A.3)

In order to describe the trajectory across the bend we give just an explicit formula for the map g : φ 7→ φ0 and t(φ) depending on the entry point (r, φ). The trajectory over the bend has few different scenarios that are discussed each individually in the following. Let us first define some

106

Appendix A. The jump models of elements in the serpent billiards

auxiliary variables α = arcsin Γ , δ 00

γ = π − arcsin(Γ/q) , β − δ0 0 = β − δ − nδ , n = , δ

(A.4) (A.5)

where bxc is the largest integer not larger than x, δ is the angle between the consequent hits to the walls and δ 0 is the angle from the entry surface to the first hit to the wall (δ 0 < δ), which is measured from the centre of the bent. In the case β < δ 0 , where the trajectory avoids hitting walls we have φ0 = φ + β , p Tb = r2 + r02 − 2rr 0 cos β .

(A.6)

φ0 = φ + β − (n + 1)δ , p p Tb = 1 + r 2 − 2r cos δ 0 + 1 + r 02 − 2r 0 cos δ 00 + 2n cos α ,

(A.8)

(A.7)

The angles δ and δ 0 are determined for each case separately below, depending on the value of the angular momentum. If Γ ≤ q, the particle is hitting the outer wall of the bend. Then in the case β > δ 0 , writing δ 0 = π − α − φ and δ = π − 2α, we have (A.9)

otherwise, in the case β ≤ δ 0 , the particle does not hit any walls and so trajectory is described by equations (A.6) and (A.7). p If Γ < q, we write δ = γ − α and ∆t = 1 + q 2 − 2q cos δ and we have to discuss two cases: (i) when the particle first hits the inner wall, φ < π/2 φ0 = φ + β − (n + 1)δ + δnodd (γ + β) , (A.10) √ p 02 0 00 1 + r − 2r cos δ : n odd q 2 + r2 − 2rq cos δ 0 + n∆t + p 2 Tb = , (A.11) 02 0 00 q + r − 2qr cos δ : n even

and (ii) when particle first hits the outer wall, φ > π/2

φ0 = φ + β − (n + 1)δ − δneven (γ − α) , p p q 2 + r02 − 2qr 0 cos δ 00 : n odd 2 0 1 + r − 2r cos δ + n∆t + √ Tb = , 1 + r 02 − 2r 0 cos δ 00 : n even

(A.12) (A.13)

where we use δnodd = (1 : n odd; 0 otherwise) and δneven = (1 : n even; 0 otherwise). The δ 0 is equal to π − γ − φ in case (i) and π − α − φ in case (ii). The presented map is valid only when β > δ 0 otherwise we have to consider the case where the trajectory avoids hitting the wall. We can see that in the direction of the bouncing ball orbits |vy | = 1 the jump model approach fails and therefore has to be excluded from the discussion. This does not present any loss of generality in the study of dynamics along the serpent billiard, because the particles in bouncing ball orbits do not traverse.

107

Appendix B

Concatenating scattering matrices We present here a method to concatenate scattering matrices (Mayer and Vigneron, 1999) associated to two consequent scatterers on a straight wave-guide. We refer to the straight wave-guide outside the scatterer as a asymptotic region. In addition, we give a procedure to calculate a cross-section (CS) state living on the border between the scatterers. The scattering across a two-way scatterer into the asymptotic region is described by a scattering matrix S. The latter represent a formal connection between incoming ψ in and outgoing ψout surface-of-section (SOS) states (vectors), with respect to the scatterer, that read aL bL Sψin = ψout , ψin = , ψout = , (B.1) bR aR where a and b denote vectors corresponding to left-to-right (right) and right-to-left (left) going waves in the asymptotic region, respectively. For details see Chapters 3 and 4. Let us now consider two scatterers marked by letters A and B with corresponding scattering matrices S A and SB of block form written as L rA,B tR A,B . (B.2) SA,B = R tLA,B rA,B We concatenate both scatterers A and B in the order AB so that they connect at border and build a “larger” scatterer with the scattering matrix S written as L R r t S= . (B.3) L t rR This is depicted in figure B.1, with CS states (vectors) living on the left border ψLT = [aL , bL ] T = [a , b ] of the scatterer and between both scatterers ψ T and on the right border ψR R R mid = [a, b]. In order to describe scattering over a composed scatterer and obtain the scattering matrix S L(left)

aL bL

A S

S

S b B a

R(right)

aR

bR

Figure B.1: Schematic figure of concatenation of two scatterers on an infinite straight wave-guide.

we discuss scattering for the case of an incoming wave from the left (a L 6= 0, bR = 0) and right (aL = 0, bR 6= 0) independently shown in figures B.2a and B.2b, respectively. An incoming wave

108

Appendix B. Concatenating scattering matrices

L a (a = r R b ) and partially from the left (right) side aL (bR ) is partially reflected in b1L = rA L R B R transmitted in a1 = tLA aL (b1 = tR b ) on first scatterer from its side. The transmitted part is R B R b ) passing over then partially transmitted over the next scatterer in a1R = tLB tLA aL ( bL = tR t A B R L tL a ( a1 = r R tR b ). The latter the composed scatterer and partially reflected back as b1 = rB A L A B R L L reflected part is again partially transmitted out of the composed scatterer in b 2L = tR A rB t A a L 2 L R R 2 R L L 2 L R R aR = tB rA tB bR ) and partially reflected back as a = rA rB tA aL (b = rB rA tB bR ). In this manner the separation of incoming waves continues into infinity and thereby producing contributions to the outgoing waves (SOS states) and the CS state in the middle of the composed scatterer as depicted in figure B.2.

S

SA aL 1

bL

1

b 2L 3

bL 4

bL

bL

b

2

b

3

b

4

5 bL

b6L

b

5

b

6

b7L

SB

S

SA

SB bR

1

a1 a2 a3 a3

bL

aR2

2 bL

aR3

3 bL

2

aR

a4R

a4 6

a

b

b

1

a1R

3

b

4

bL

+

b

b

4 bL

5

b

5

aR5

bL

aR6

6 bL

6

b

1

a1 2

a

3

a

4

a

2

aR

aR a 3R a4R aR5

aR

5

a

6

a

a6R a7R

Figure B.2: Schematic figure of scattering of left incoming wave over the two consequent scatterers.

P n By summing together P nall the contributions to the outgoing waves on the left, b L = n bL , and right side, aR = n aL for both cases independently we can write the scattering matrix of the composed scatterer S in the form L L −1 L r L = rA + tR A rB L tA ,

r

R

=

R rA

+

R 0 −1 R tLA rB L tA

tL = tLB L−1 tLA , ,

R

t =

0 −1 R tR tA BL

(B.4) ,

(B.5)

R r L and L0 = 1 − r L r R . The equations (B.4) and (B.5) are also where we define L = 1 − rA B A B obtainable from the expression S = S[T [SB ]T [SA ]], where S[•] and T [•] are transformations between the scattering and transfer matrices defined by equations (3.5). But this procedure gives us less physical insight into the problem. If the reflected waves an and bn between inner boundaries of both scatterers decay strongly we could neglect higher indices n and so obtain some useful approximate expressions of (B.4) and (B.5). The presented concatenation of two scattering matrices SA,B into S is a binary operation on the set of scattering matrices that we represent with a symbol . By using this notation the equations (B.4) and (B.5) can be rewritten as

S = S A SB .

(B.6)

Notice that unitary scattering matrices U (2d) form a group (Scott, 1987) with the operation . In addition, we sum up all contributions the left-to-right P n and right-to-left going waves P to n between both scatterers denoted by a = n a and b = n b , respectively. The result yields

109 the CS state ψmid at the inner border of both scatterers expressed by a linear mapping P acting on the incoming SOS state hai P11 P12 ψmid = Pcs ψin , ψmid = , Pcs = , (B.7) P21 P22 b where the matrices Pij (i, j = 1, 2) are P11 = L−1 tLA ,

R 0 P12 = rA L

−1 R tB

,

L −1 L P21 = rB L tA ,

P22 = L0

−1 R tB

.

(B.8)

The presented calculations of the scattering matrix and CS state are performed in the framework of the scattering formalism and therefore are numerically stable. The latter is in practice extremely important.

110

Appendix B. Concatenating scattering matrices

111

Appendix C

The classical probability scattering matrix We discuss here a two-dimensional system connected to a two-dimensional flat straight waveguides referred to also as leads or asymptotic regions. The classical probability scattering matrix Scl (Méndez-Bermudéz, 2002) is a classical object describing scattering in this systems inspired by its quantum analog – the quantum on-shell scattering matrix Sqm . In order to understand the first, we have to look at the formulation of the latter. Without loss of generality, we consider here only a one-way scattering matrix associated to a scatterer, which is connected to an infinite straight wave-guide (asymptotic region) of width a, as shown in the figure C.1. y

scatterer

infinite wave−guide χ out a

χ in x

Figure C.1: Scheme of a scatterer ending an infinite straight wave-guide.

In the quantum one-shell scattering we wish to obtain the solution of the eigen-problem of the whole system i.e the scatterer and the asymptotic region ˆ E (x, y) = EψE (x, y) , Hψ

(C.1)

ˆ in the asymptotic just on the asymptotic region ψ(x, y) (x < 0). The Hamilton operator H region {(x, y) : x ∈ (−∞, 0], y ∈ [0, a]} is simply a Laplace operator ˆ = − 1 ∆ = − 1 [∂x2 + ∂y2 ] , H 2 2

(C.2)

and across the area of the scatterer may not be specified. We refer to pˆx,y = −i∂x,y as operators of the momentum in the x and y direction, respectively. Wave-functions in a straight wave-guide at certain energy E = 21 k 2 are described by an ansatz X + − − ψ(x, y) = a+ (C.3) n en (x, y) + an en (x, y) , n∈N

112

Appendix C. The classical probability scattering matrix

where a± n ∈ C are the expansion coefficients in eigen-functions of the Laplace operator defined as r π exp(±ik x) 2 ± √ n , (C.4) en (x, y) = sin ny a a kn p with traverse momenta kn = k 2 − (πn/a)2 . The superscript ± denotes the direction of the phase flow along the straight wave-guide. The square-root of a complex number z = |z| exp(iφ), √ φ[0, 2π], is defined as z = |z|1/2 exp(iφ/2). Notice that functions e± n (x, y) have a probability flux of a unit size and traverse momenta kn can be either real or imaginary. The functions e± n with real (imaginary) kn are called open (closed) modes. The open modes have a clear classical correspondence, whereas the close modes are purely quantum. The number of open modes is ˆ = (ˆ px , pˆy ), No = bka/πc. The function e± n is not eigen-functions of momentum operator p but can be written as a superposition of such with momentum vectors p = (±k n , µn) and p = (±kn , −µn): iµny±ikn x e± − e−iµny±ikn x . (C.5) n (x, y) ∝ e We can say that a e± n (x, y) represents contributions of two probability flows of well defined direction. Notice that the phase flow remains finite in the asymptotic region only for open modes. By imposing smoothness of ψE across the border between the billiard and straight wave− guide the vectors ψin = {a+ n }n∈N and ψout = {an }n∈N become linearly depended. They are expressed by a quantum scattering matrix Sqm as Sqm ψin = ψout .

(C.6)

Through the use of the scattering matrix we describe the quantum on-shell scattering of waves from the scatterer back into the asymptotic region. In the classical one-shell scattering we study how a point particle enters and escapes from the area of the scatterer. The Hamilton function in the asymptotic region is H = 21 p2 , where p is classical momentum vector. As the energy is conserved, we introduce dimensionless speed v = p/|p| to simplify the discussion. The trajectory intersects the border of the scatterer (x = 0) at the point y at some angle θ towards the normal to the border as depicted in figure C.1. The intersections forms a phase-space S, which we call SOS of the scatterer S = {(y, vy = sin θ) : y ∈ [0, a], vy ∈ [−1, 1]} ,

(C.7)

where we introduces a canonical variable vy – the projection of the speed on the border instead of the angle θ. The particle enters the scatterer at the phase-space coordinates x in ∈ S and exits at xout ∈ S, with some time delay. The scattering is elegantly described by a mapping of entry points into exit points refereed to as the Poincaré map P : S → S P(xin ) = xout .

(C.8)

+ In introduced notation, an open mode e− n (en ), n ∈ [1, No ], contributes to the probability flow in the direction vy = ±n/No . We therefore introduce phase-space regions corresponding to directions associated to individual open modes e± n denoted by En , reading

En = {(y, vy ) ∈ S : y ∈ [0, a], No |vy | ∈ [n − 1, n]} . The classical probability scattering matrix is then written as Z Z dy δ(x − P (y)) , n, m ∈ [1, No ]. dx [Scl ]nm = En

(C.9)

(C.10)

Em

The results presented in (Méndez-Bermudéz, 2002) show that the classical probability scattering matrix (C.10) allows to predict global structures of its quantum analog. The matrix S cl can be useful to recognise signatures of classical dynamics in quantum scattering matrix S qm and can serve as an additional check to usually tedious computation of Sqm .

113

Appendix D

The cross-product of Bessel functions Here we present an analysis of the mode structure of a Laplacian on the domain of a open bend, see figure D.1, which are used in the description of the wave phenomena across the bend at given wavenumber k. The bend has the inner radius r = q and the outer radius r = 1. This is closely related to the work of Cochran (Cochran, 1964, 1966a,b), with the difference that our work is directed toward the application of the mode structure in scattering calculations across the bend. Additionally, we give explicit expressions of mode functions in the bend enabling fast and accurate numerical computation, where we take great care of closed (evanescent) modes. y

r=1 r=q r

φ x

Figure D.1: Schematic picture of an open bend of an inner radius q and outer radius 1 parametrised in the polar coordinates, as radius r and angle φ.

The mode functions in the bend Up (r) are proportional to well known cross-products of Bessel functions (Cochran, 1964) of the first Jν and of the second kind Yν (Olver, 1972) in the form Zν (k, r) = Jν (kr)Yν (k) − Yν (kr)Jν (k) , (D.1) or Zν,k (r) =

J−ν (kr)Jν (k) − Jν (kr)J−ν (k) , sin(νπ)

ν∈ / Z,

(D.2)

where the allowed values of mode numbers ν are determined by Dirichlet boundary conditions Zν,k (q) = 0. In equation (D.2) we have used the relation Yν (z) = (Jν (z) cos(νπ) − J−ν (z))/ sin(νπ) valid for orders ν ∈ / Z.

114

D.1

Appendix D. The cross-product of Bessel functions

The properties of mode numbers

The set of mode numbers at the given wave-number k ∈ R, k > 0 and inner radius q ∈ (0, 1) is denoted by Mk,q = {ν ∈ C : Zν,k (q) = 0}. The functions Zν,k (r) are even Z−ν,k (r) = Zν,k (r) and analytic in the order parameter ν (Cochran, 1966a) yielding the following symmetry of the set of mode numbers: Mk,q = −Mk,q , M∗k,q = Mk,q . (D.3) We additionally conclude that the mode numbers are taking real or imaginary values, Mk,q ⊂ R ∪ iR ,

(D.4)

and there is only a finite number of real modes and an infinite number of imaginary modes for a finite wave-number k. We prove the symmetry (D.4), by changing the variable r = e−x and transforming the Bessel equation (4.3) and the corresponding boundary condition into the equation d2 Z + k 2 e−2x − ν 2 Z = 0 , Z|x=0,log q = 0 , (D.5) 2 dx

which can be interpreted as a one dimensional stationary Schödinger equation with the Hamiltonian H and potential V : 2 2 −2x : x ∈ [0, − log q] ˆ = −ν 2 Z , H ˆ = − d + V 0 (x) , V 0 (x) = −k e . (D.6) HZ 2 ∞ : elsewhere dx

8

8

ˆ is a Hermitian operator and the eigenvalues are real numbers −ν 2 ∈ Because the Hamiltonian H R we find that ν ∈ R ∪ iR. From the form of the potential V 0 (x), depicted in figure D.2, we see that there are only of finite number of real mode numbers ν 2 > 0 and infinite number of imaginary mode numbers ν 2 < 0. The independent solution of the equation (D.6) are orthogonal with respect to the measure dx = r −1 dr, which implies that w(r) = r −1 is the weight function for modes Up (r) in the bend.

V ’(x)

0

−log (q)

x

2

−k

Figure D.2: The analog of the quantum potential in the stationary Schrödinger equation (D.6) for the mode functions in the bend.

The properties (D.3) and (D.4) enable a decomposition of Mk,q into two disjoint subsets of mode numbers laying on the positive Mk,q,+ and the negative Mk,q,− real and imaginary axes: Mk,q,+ = {ν ∈ Mk,q : k we can make use of the Debye approximation of the Bessel functions for imaginary orders (Temme, 2003; Erdélyi, 1955) and write the formula o n y y ∗ (β, ζ)ei[ξ−ζ−y (arcsinh k −arcsinh kr )] = G(α, ξ)G 2 √ , (D.42) Ziy (k, r) = π (1 − e−2πy ) ξζ by using the substitutions ξ=

p

y2 + k2 ,

ζ=

p y 2 + (kr)2 ,

α−1 = 1 +

k2 , y2

β −1 = 1 +

(kr)2 . y2

(D.43)

The expression G(x, y) is given in the form of an asymptotic series G(x, y) =

∞ X (−i)m vm (x) , ym

(D.44)

m=0

where polynomials vm (t) are generated by the following recursion formula Z k−1 1 k+1 t 1 (1 − 5τ )τ 2 vk (τ )dτ . vk+1 (t) = (1 − t) k vk (t) + 2t v 0 k (t) + t− 2 2 16 0

(D.45)

The first few vm (t) read as v0 (t) = 1,

v1 (t) =

1 5 − t, 8 24

v2 (t) =

3 77 385 2 − t+ t , 128 576 3456

,... .

(D.46)

The formulae for the evaluation of cross products of Bessel functions in different regimes of parameters (D.30, D.36, D.39, D.42) enable a stable high precision calculation of the modal functions for imaginary mode numbers.

D.3

The overlap of mode-functions in different geometries

We are considering scattering across a bend on a straight wave-guide (open-billiard) shown in figure 4.1. The main ingredient in the modal description of the scattering are the overlap integrals of mode functions of the Laplacian in the straight waveguide and in the bend. These overlap integrals “tell” about the compatibility of both scattering regions and are discussed in the following.

D.3. The overlap of mode-functions in different geometries

121

The set of cross products of Bessel functions Zν,k = {Zν,k (r ∈ [q, 1]) : ν ∈ Mk,q,+ } at a given wave number k and inner radius q is complete and orthogonal w.r.t. the weighting function w(r) = r −1 found in Section D.1. The orthogonality relation for functions Zν,k ∈ Zν,k then reads (Luke, 1962) Z 1 ∂Zα,k ∂Zβ,k k dr w(r)Zα,k (r)Zβ,k (r) = δα,β Zα+1,k (1) (1) − qZβ+1,k (q) (q) . (D.47) 2ν ∂ν ∂ν q For convenience we define over the border connecting the bend and straight wave-guide, referred to as the cross-section of our open billiard, two different scalar products denoted by (·, ·) and h·, ·i and written as Z a Z 1 dr a(r) b(r) , ha, bi = dy a(y) b(y) . (D.48) (a, b) = 0 q r Let us now introduce the mode functions and corresponding mode numbers at some fixed wave number k and inner radius q for different regions of the open billiard. In the bend the mode number νp and mode function Up (r) read as νp ∈ Mk,q,+ ,

Zνp ,k (r) Up (r) = q , (Zνp ,k , Zνp ,k )

p∈N,

(D.49)

2 where we order the mode numbers so that νp2 > νp+1 , and in the straight lead connected to the bend we have the mode number gn and mode function un (x) defined by r r πn 2 π 2 , un (y) = gn = k 2 − sin ny , n ∈ N . (D.50) a a a

We see that both regions of the open billiard possess real and imaginary mode numbers that we also refer to as open modes or travelling waves and closed modes or evanescent waves, respectively. The overlap integrals of mode functions are Anp = hun , Up i ,

Bnp = (un , Up ) ,

(D.51)

where we use the relation between the coordinates r = q +y. In figure D.5 we show a density plot of matrix elements log |Anp | and log |Bnp | at two values of inner radii q = 0.2, 0.6 calculated with No = 100 open modes in both geometries. We see that matrices Anp and Bnp have a similar form for all q and k, namely starting at small indices with a wide area of high values that squeezes to almost a single intensified point at n, p ≈ No again spreading in a triangular shape with increasing indices. The parameter q has a strong influence on the shape of area with high values in matrices A and B. In the case of small values of q in contrast to larger q, the area of high values in A and B covers almost the whole open-open block of indices and with crossing of the narrowing at n, p ≈ No spreads faster with increasing indices. The shape of matrices A and B is similar so we show numerical results only for the matrix Anp . We found numerically that the area of high values in matrices A and B scales with the number of open modes No as Anp , Bnp ∼ F (n/No , p/No ), where F is some well behaved function. This is demonstrated by plotting the matrix elements |Anp | in relative indices n/No and p/No for different number of open modes No as we show in figure D.6. Additionally, numerical evidence is found that highest and tail (smallest) values of matrix elements Anp and Bnp scale differently with No : −1

highest values : |Anp |, |Bnp | . No 2 Fmax (n/No , p/No ) , tail values : |Anp |, |Bnp | .

No−2 Ftail (n/No , p/No )

.

(D.52) (D.53)

122

Appendix D. The cross-product of Bessel functions q = 0.2

log10 Anp

300 250

250

200

200

150

150

100

100

50

50

log10 Bnp

1 300 1

50

100

150

200

250

1 300 1 300

50

100

150

200

250

300

1 300 1

50

100

150

200

250

300

-5

-4

250

250

200

200

150

150

100

100

50

50

1

1

50

-10

100

-9

150

-8

q = 0.9

300

200

-7

250

-6

-3

-2

-1

0

Figure D.5: Density plot of the matrix elements log 10 |Anp | (top-row) and log10 |Bnp | (bottom-row) for q = 0.2 (left) and 0.9 (right) as indicated in the figure. The number of open modes is N o = 100 and the number of all considered modes is N = 300.

No = 100

2

0.5

1.5 p/No

1

1

0.5

0

0.5

1 n/No -10

-9

-8

1

0.5

0 2 0

1.5

No = 400

2

1.5 p/No

p/No

1.5

0

No = 200

2

-7

0.5

1 n/No

-6

-5

0 2 0

1.5

-4

-3

0.5

-2

1 n/No -1

1.5

2

0

Figure D.6: Density plot of matrix elements log 10 |Anp | for different values open modes No = 100, 200, and 400 open modes as indicated in the figure at inner radius q = 0.6.

where Fmax and Ftail are some well behaved functions. Knowing this enables a better precision control of scattering calculations. From figure D.7 we see that Fmax is an envelope function to 1

the maximal values of No2 |Anp | and that Fmin can be chosen to fit the tails of No2 |Anp | almost to an equality. We prove the scaling (D.53) by using an asymptotic approximation of the mode

D.3. The overlap of mode-functions in different geometries

0.1 0.01

1000 o

0.001 1e-04 1e-05 1e-06 1e-07 1e-09

0

100 10 1 0.1 0.01 0.001

(a)

1e-08

No = 50 No = 100 No = 200 No = 400

10000

No2 |An, N y|

o

No1/2 |An, N y|

100000

No = 50 No = 100 No = 200 No = 400

1

(b)

1e-04 0.5

1 n/No

1.5

123

2

1e-05

0

0.5

1 n/No

1.5

2

Figure D.7: The cuts in the index space of matrix Anp at fixed p/No = 0.5 and different number of open modes No = 100, 200, and 400 at inner radius q = 0.6.

function in the bend Up =

s

2 sin | log q|

πp log r log q

,

p1,

(D.54)

which yields the following asymptotic behaviour of the matrix elements 5

Anp

6 | log q| 2 q 2 (−1)p + (−1)n+1 np−3 + O(p−5 ) ≈ 2 3 π (1 − q) 2

n = fixed, p → ∞,

(D.55)

5

Anp

2 (1 − q) 2 −2 ≈ 2 q (−1)p+1 + (−1)n pn−3 + O(n−5 ) p = fixed 1, n → ∞. (D.56) 3 π | log q| 2

The off-diagonal diagonal elements Anp and Bnp have a algebraic decay with a pre-factor decreasing with increasing q and singular at q = 0. Meaning that for large enough q it is approximately justified that we express open modes in the bend by the open modes in the straight wave-guide and vice versa, which is also indicated by relation (D.52). Taking into account the definition of the matrix elements Anp and Bnp (D.51) and the completeness of the mode functions at given k and q we discovers that matrices A and B are transition matrices between the sets of mode functions in different regions, X X un (x) = Bnp Up (r) , Up (r) = Anp un (x) , (D.57) p

n

yielding the relation AB T = AT B = id .

(D.58)

In practice we work with finite sets of modes, where the identity (D.58) does not hold exactly. This we can see from figure D.8, where we plot AB T for different inner radii q at No = 100 and at the number of all considered modes N = 300. The mismatch from the identity (D.58) at the same dimension increases with decreasing inner radius q meaning that scattering numerical computation at smaller values of q should be less accurate. The discrepancy between AB T and the identity on (truncated) finite dimensional spaces is strongly non-uniform in indices. Before going into practical aspects of this problem, we examine the convergence of matrix elements (AB T )nm to δnm with increasing the number of all considered modes N = No + Nc at fixed No with Nc being the number of closed modes. An example of such convergence is illustrated in figure D.9. The convergence proceeds by the standard scenario, where the agreement between [AB T ]np and δnp propagates block-wise from low to higher indices with increasing N . The propagation is slow due to the triangular shaped area of high intensities

124

Appendix D. The cross-product of Bessel functions q = 0.2

q = 0.6

-10

-9

-8

-7

-6

q = 0.9

-5

-4

-3

-2

-1

0

Figure D.8: Density plot of matrix elements log 10 |(AB T )nm − δnm | for q = 0.2, 0.6 and 0.9 (from left to right). On the abscissa and ordinate we plot indices n and m, respectively. The number of open modes is No = 100 and the number of all modes is 300. Nc = 1

Nc = 100

-10

-9

-8

-7

-6

Nc = 500

-5

-4

-3

-2

-1

0

Figure D.9: Density plot of the matrix elements log 10 |(AB T )nm − δnm | for different numbers of closed modes Nc = 1, 100 and 500, as indicated in the figure, at inner radius q = 0.2 and No = 100 open modes. Labels on the abscissa and the ordinate are indices n and m, respectively.

in the closed-closed modes block of A and B. The speed of propagation to higher mode indices increases with increasing inner radius q. The SVD decompositions (Demmel, 1997) of matrices A and B is useful for improving and stabilising the scattering calculations. From the definition of transition matrices (D.51) and completeness of mode functions we get (AAT )nn0 T

(BB )nn0

= hun , r un0 i ,

= un , r−1 un0 ,

(AT A)pp0 (B T B)

= (Up , r Up0 ) ,

pp0

= (Up , r

−1

Up 0 ) .

(D.59) (D.60)

which we use to determine the boundaries of images 1 kAak2 ∈ [q 2 , 1] , kak2

kBak2 h − 1 i ∈ 1, q 2 kak2

for kak 6= 0 .

(D.61)

These results together with AB T = id can be used to determine the form of the SVD decomposition √ (D.62) A = U ΣV T , B = U Σ−1 V T , Σ = diag {σi ∈ [ q, 1]}i∈N ,

D.4. The number of modes in the straight and the bended wave-guide

125

where U and V are orthogonal matrices. We give here an example of the SVD decomposition of the finite dimensional matrices A and B at q = 0.2 and No = 100. In figure D.10 we plot singular values, and in figure D.11 the corresponding matrices U and V as density plots, where vectors are ordered by decreasing associated singular value. We see that the relative dimension of the 1 2

0.8

1.5 σi

σi

0.6 0.4

Ni=100 Ni=200 Ni=300 Ni=400 Ni=500 1/2 q

0.2 0

0

1

Ni=100 Ni=200 Ni=300 Ni=400 Ni=500 -1/2 q

0.5

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

0

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

i/N

1

i/N

Figure D.10: The singular values of the matrices A (left) and B (right) for different number of closed modes Nc = 100, 200, 300, 400 and 500 at q =0.2 and No = 100. The horizontal lines represent the 1 1 theoretical bounds of singular values [q 2 , 1] (a) and [1, q − 2 ] (b).

space violating the given boundaries for the singular spectra converges with increasing space dimension N = No + Nc , where Nc is number of imaginary modes. These relative dimension at q = 0.2 is around 20%. It is important to see that vectors in matrices V and U , which correspond to singular values violating the boundaries, have non-zero components only at closed modes. We conclude that due to finite dimensional representation of matrices A and B we have deviations from the infinitely dimensional case only at high laying closed modes that span a space of almost fixed relative dimension for some value of q.

D.4

The number of modes in the straight and the bended waveguide

Here we discuss properties of the number of open modes in the bend No,bend (k, q) = card{±∞

(F.3)

Let us introduce the scattering and transfer matrix formalism for an individual element on the chain. We imagine that an element is taken out of the chain and connected on each side to an infinite straight wave-guide, referred to as leads or asymptotic regions, so that the walls are smooth. The direction of the ξ and x coordinate axes are parallel at the border and connected to each other by the relation dx = g(y)dξ. The Laplace eigen-basis in the straight wave-guide are r r π πn 2 2 exp(ign x) ± , un (y) = sin ny , gn = k 2 − en = un (y) √ , (F.4) gn a a a

where we use the same terminology about the nature of functions e± n as introduced for the element. The wave function in the asymptotic regions ψL (x, y) and ψR (x, y) in the left and in right wave-guide, respectively, are then written as X L,R − ψL,R (x, y) = anL,R e+ (F.5) n (x, y) + bn en (x, y) . n

We refer to functions un (y) and zp (y) as modes of the Laplacian or short modes in the straight wave-guide and element, respectively. Each individual element has a right and left border referred to as cross-sections. To simplify description we define two types of states “living” on the cross-section area: (i) the surface-of-section (SOS) states and (ii) the cross-section (CS) states. (i) The SOS states ψin (y) and ψout (y) corresponding to incoming and outgoing waves, respectively, are defined as L L X X bn a , (F.6) un (y) ψin (y) = un (y) Rn , ψout (y) = aR bn n n

n

and (i) the CS states ψL (y) and ψR (y) corresponding to left and right side of the element, respectively, we write as " # X aL,R n ψL,R (y) = un (y) L,R , (F.7) bn n

where the top row represents the component on the left side and bottom on the right side of the element. The two-way scattering Se and transfer Te matrix of the element in the chain are defined as block matrices t11 t12 rL tR , (F.8) , Te = Se = t21 t22 tL rR and connect the vectors of wave-function coefficients in asymptotic region a L,R and bL,R in the n n following way L R L L a a a b = (F.9) , Te Se R = L R b bR a b

The matrices Se and Te are obtained from the condition that the wave-function is smooth across the border of element and asymptotic regions. Notice that Se maps the incoming into outgoing SOS state , whereas Te maps the left into right CS state. Due to exponential divergence of evanescent waves transfer matrices are practically unobtainable and therefore we work using only scattering matrices. We assume to have a CS state expressed in modes in the straight waveguide at the border of the element. In order to express the wave function across that border from the asymptotic

131 region into the element we need to have transition matrices between the modes in the element and in the asymptotic region: Z Z Anp = dy un (y)zn (y) , Bnp = dy w(y)un (y)zn (y) , (F.10) Z Z enp = dy w(y)un (y)zn (y) , enp = dy g(y)w(y)un (y)zn (y). A B (F.11)

which enable to change the basis of CS states from modes in the asymptotic region to the modes in the element, writing them as " # X αnL,R ψL,R (y) = zn (y) , (F.12) βnL,R n The vectors of expansion coefficients in the element αL,R and β L,R are connected to ones in the straight wave-guide aL,R and bL,R by transition matrices P and P˜ as n a α a −1 α . (F.13) =P , =P β b β b The transition matrices have block form written as 1 P = 2

P+ P− P− P+

,

P

−1

1 = 2

"

P˜ + Pe− Pe− Pe+

#

,

(F.14)

where we introduce diagonal matrices g = diag{gn }n and µ = diag{µp }p and the following matrices 1 1 1 1 1 e − 21 ± µ− 12 Bg e 12 . P ± = g 2 Aµ− 2 ± g − 2 Bµ 2 , Pe± = µ 2 Ag (F.15)

epn = Bnp , B epn = Anp and If the element is a bent, then g(y)w(y) = 1, which implies that A ± T ± e (P ) = ±P . These identities significantly simplify the computation. The wave function across the area of the element is written as a sum X ψ(ξ, y) = αn zn+ (ξ, y) + βn zn− (ξ, y) , (F.16) n

where the expansion coefficient α are obtained from the conditions imposed on the borders. We assume now having the left [aL , bL ]T and right CS state [aR , bR ]T expressed in modes of the straight wave-guide. Using the CS transition matrix P −1 we obtain corresponding expansion coefficients for the left [αL , βL ]T and right [αR , βR ]T CS state in modes of the Laplacian on the area of the element. Then the stable formula for the wave function is X ψ(ξ, y) = αnR zn+ (L − x, y) + βnL zn− (x, y) . (F.17) n

Notice that, we only use that parts of the left and right CS states, which are stable across the area of the element. In order to obtain the wave function across the element of the chain we need the CS states at both borders of elements. The CS/SOS states are for convenience expressed by expansion coefficients in basis modes in the straight wave-guide. Let us now consider a scattering experiment in a chain of n element with the incoming SOS state vector given by aL ψin = . (F.18) bR

132

Appendix F. Wave-functions in the coordinate space

The scattering matrix of the chain S is calculated by combining scattering matrices of its parts with the operation : (F.19) S = Se . . . Se . {z } | n

We cut the chain into two parts (e.g. with of m and n − m elements) and treat them as separate scatterers with scattering matrices S1 and S2 (S = S1 S2 ): S1,2 =

rL1,2 t1,2 R 1,2 rR t1,2 L

.

(F.20)

The CS state at the inner border of both scatterers denoted by ψcs can be expressed by a linear map P = P (S1 , S2 ) (B.7) given in Appendix B acting on the incoming wave ψcs = P ψin .

(F.21)

In this way we can find all CS/border between elements along the chain. Note that the calculations of CS states does not depend on the structure of the chain so it can be easily applied to our serpent billiard. By combing calculations of CS states along the chain together with the way to calculate the wave function across a single element we can obtained the wave-function in the coordinate space across the whole chain.

133

Appendix G

The Gaussian packet and the Wigner function in an ∞-well The understanding of quantum mechanical systems depends on their representation in terms known to the classical mechanics. Here we introduce a Gaussian packet and a Wigner function (Wigner, 1932; Lee, 1995) that bring the quantum and classical description of a system closer together. The system of discussion is a particle confined in an infinite well on an interval I = [0, a]. The coherent state represent a classical particle in the quantum picture and the Gaussian packet its usual approximation (Klauder and Skagerstam, 1985; Perelomov, 1986). We consider particles of unit mass and variables scaled so that ~ = 1. The classical particle of the free Hamiltonian Hfree = 21 p2 is at the given time determined by its position y ∈ I and momentum p ∈ R in the phase-space X: X = {(y, p) : y ∈ I; p ∈ R} = I × R . (G.1)

The quantum particle in the well is described by a wave-function ψ(y) (y ∈ I) with the boundary ˆ free = values ψ(0) = ψ(a) = 0. The Hamilton operator of the free quantum particle is given by H 1 2 ˆ , where pˆ = −i∂y is the momentum operator. The eigen-functions of the free Hamiltonian 2p are r π 2 un (y) = sin ny , (G.2) a a which form the basis of the Hilbert space L2 (I) of functions over I. We use Gaussian packets as a quantum representation of a (classical) particles positioned somewhere in the classical phase space X. We define a Gaussian packet GI (y) as a state defined over the interval I approximately of a Gaussian shape and obeying the boundary conditions GI (0) = GI (a) = 0. It is derived by deforming a coherent state defined over R, with the position (y0 , p0 ) ∈ R2 in the phase-space and width σ, written as Gα (y) =

1

− 1 e

(πσ 2 ) 4

(y−y0 )2 2σ 2

eip0 (y−y0) ,

(G.3)

where we introduced a vectors of form parameters α = (y0 , p0 , σ). The deformation is most elegantly performed by using an operator PbI , which performs an appropriate periodisation of functions on interval I, defined as X (PbI f )(y) = f (y − 2na) − f (−y − 2na) . (G.4) n∈Z

The image of the operator PbI has the property (PbI f )(0) = (PbI f )(a) = 0. Then the Gaussian packet over the interval I is given by the formula GI,α (y) = Aα (PÎ Gα )(y) ,

(G.5)

Appendix G. The Gaussian packet and the Wigner function in an ∞-well

134

where A is a normalisation constant. The latter can be written in a clearer form by introducing auxiliary dimensionless variables σ ˜ = σπ/a, p˜ = p0 a/πa and y˜ = y0 /a: − 1 2 σ ˜ −(˜σp˜)2 √ e h(˜ σ , y˜) , Aα = h(π/˜ σ , p˜) − π

X

2

cos(2πyn)e−(nx) .

(G.6)

We frequently use the Gaussian packet GI (y) expanded in the basis {un (y)}n∈N as X GI,α (y) = Gn,α un (y) ,

(G.7)

h(x, y) =

n∈Z

n∈N

where the expansion coefficients GI,n,α are given by Z Z GI,n,α = dy GI,α (y)un (y) = dy Gα un (y) , I R r i πσ 2 1 h i π nx0 − 1 σ2 (p0 + π n) −i π nx0 − 21 σ 2 (p0 − π n) a 2 a a a . e e −e e = Aα a i

(G.8) (G.9)

In figure G.1 we show three examples of Gaussian packet GI,α (y) and corresponding expansion coefficients GI,α,n . From the figure we can clearly see the result of moving the Gaussian packet 2.5

0.9

α=(0.1,0,0.1) α=(0.9,50,0.1) α=(0.5,0,0.1)

2

0.7 0.6 0.5

|GI,α,n|

1.5

|GI,α(y)|

α=(0.1,0,0.1) α=(0.9,50,0.1) α=(0.5,0,0.1)

0.8

1

0.4 0.3 0.2

0.5

0.1 0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

0

1

(b)

y

0

2

4

6

40

20

20

20

0

0

0

p

40

-20

-20

-20

-40

-40

-40

(c)

0

0.2

0.4

-0.05

y

0.6

0.8

0

10

0.05

0.2

0.4

0.1

y

8

10

12

14

16

n

40

p

p

(a)

0

0.6

0.15

0.8

10

0.2

0.2

0.25

0.4

y

0.6

0.8

0.3

Figure G.1: A Gaussian packet on the interval I = [0, 1] for different parameters α represented in coordinates (a) and by expansion coefficients (b).

higher in the momentum and nearer to the walls. The introduced Gaussian packets G I,α could be also used in a Hussimi-like distribution (Takahashi, 1989) defined as 2 Z HI,σ (y, p) = dx GI,(y,p,σ) (x)ψ(x) , I

(G.10)

1

135 which is a continuous representation of a quantum state ψ(y) over the classical phase-space. The classical analog of the quantum Gaussian packet GI,α (y) is a probability density distribution gI,α over the phase-space X written as 1 −2 2 e−σ (x−x0 ) , (G.11) gx0 ,σ (x) = √ 2 πσ R with an appropriate normalisation constant B so that X d2 x gI,α (x) = 1. The Wigner-Weyl formalism in Euclidean phase space R2n is well known and presented in (de Groot and Suttorp, 1972; Lee, 1995). The Wigner function Wψ (y, p) of a state ψ(x) defined over the real line x ∈ R is given by Z 1 Wψ (y, p) = dv eipv ψ ∗ (y + v/2)ψ(y − v/2) , (G.12) 2π R gI,α (y, p) = B gp0 ,1/σ (p)PÎ gy0 ,σ (y) ,

where we have used that ~ = 1. The wave-function ψ(y) in our case is non-zero only for y ∈ I and is given as series expansion in the basis {un (x)}n∈N : X ψ(x) = an un (x) , an ∈ C . (G.13) n∈N

By plugging this series into formula (G.12) we obtain the Wigner function expressed in terms of expansion coefficients an Wψ (y, p) =

∞ X

n,m=1

an a∗m Fnm (y/a, pa) ,

(y, p) ∈ X .

(G.14)

We introduce symbols Fnm (x, y) representing Wigner functions of basis function un (x) defined as Z 2 Fnm (x, y) = dη sin(πn(x + η)) sin(πn(x − η))e−2iyη , (G.15) π |η|≤max{x,1−x} and explicitly written as 1 Fnm (x, y) = eiπ(n−m) S((π(n + m) − 2y)z) + e−iπ(n−m) S((π(n + m) + 2y)z) (G.16) z − eiπ(n+m) S((π(n − m) − 2y)z) − e−iπ(n+m) S((π(n − m) + 2y)z) , (G.17) where we use z = min{x, 1 − x} and S(x) = sin(x)/x. Wigner functions on our phase-space X has the following normalisation properties: Z Z 1 2 dx Wψ (x) = kψk2 , dx Wψ (x)2 = kψk42 , (G.18) 2π X X with k • k22 denoting the L2 norm over interval I. In figure G.2 we show Wigner functions of basis elements un (y). We see that the Wigner function of a particular un (y) condenses around the classical momentum |pn | = πa n. This property is used in the construction of the classical probability scattering matrix presented in Appendix C.

Appendix G. The Gaussian packet and the Wigner function in an ∞-well

136

200

200

150

150

100

100

50

50 0

p

p

0 -50

-50

-100

-100

-150

-150

-200 n=10

0

0.2

0.4

y

0.6

0.8

1

-200 n=20

200

200

150

150

100

100

50

50

-50

-50

-100

-100

-150

-150

-200 n=30

0

0.2

-0.3

0.4

-0.2

y

0.6

0.8

-0.1

1

-200 n=40

0

0.2

0.4

0

0.2

0.4

y

0.6

0.8

1

0.6

0.8

1

0

p

p

0

0

0.1

y

0.2

0.3

Figure G.2: The density plots of Wigner functions corresponding to basis functions u n (y) for n indicated in the figure.

i. Klasiˇ cni serpentinast biljard

137

Daljˇ si slovenski povzetek Na podroˇcju dinamiˇcnih sistemov med ˇstudijskimi sistemi ˇze dalj ˇcasa prevladujejo biljardi. To so sistemi lokalno prostega delca na kompaktnem obmoˇcju obdanem s steno razliˇcnih lastnosti. Zaradi enostavnosti bomo privzeli trde stene. Biljardi oz. bolj natanˇcno (klasiˇcni ali kvantni) delci v biljardih so raˇcunsko enostavni sistemi in posedujejo vso znano bogastvo dinamike konzervativnih kompleksnih sistemov, glej npr. (Ott, 1993). Popolnoma izolirani sistemi v naravi ne obstajajo, saj so na nek naˇcin vedno sklopljeni z okolico. V biljardih to realiziramo tako, da ga odpremo preko lukenj v steni skozi katere lahko delec oz. del valovanja uide. Takˇsen sistem imenujemo odprt (razˇsirjen) sistem oz. v kvantni sliki kvantna pika (ang. quantum dots) ali mikrovalovna votlina (ang. micro-wave cavity) in ga praviloma obravnavamo kot sipalec. Analiza teh odprtih biljardov je veliko doprinesla k razumevanju transporta skozi strukture velikostnega reda de Broglijeve valovne dolˇzine (Reichl, 2004). Odprte biljarde z dvema luknjama pa lahko poveˇzemo v linearno verigo biljardov, ki so poseben primer sploˇsih verig sipalcev. V primeru, da je veriga neskoˇcna govorimo o prostorsko neomejenem sistemu. Sploˇsne verige sipalcev so razmeroma dobro preuˇcene, vendar je tehnika analize veˇckrat nekoliko nepregledna. Zato v disertaciji predstavimo t.i. dinamiˇcni prostop, kjer verigo v sipalnem formalizmu opazujemo v procesu podaljˇsevanja kot abstrakten dinamiˇcni sistem. Izkaˇze se, da je to edini stabilen pristop k analizi transportnih lastnosti konˇcnih in neskoˇcnih verig. Biljadne verige so bile v primerjavi s sploˇsnimi verigami po ˇstevilu ˇstudij nekoliko zapostavljene. Kljub temu je pomembno omeniti doloˇcene zanimive ˇstudije v teh sistemih. Prva je raziskava ˇcasov pobega iz doloˇcenega omejenega obmoˇcja neskoˇcnega biljarda, npr. v Lorentzovem kanalu (Gaspard, 1998). Druga je ˇstudija klasiˇcnih transportnih zakonov in prenosa toplote (Alonso et al., 1999, 2002; Li et al., 2003) in tretja je preuˇcevanje relacij med diffuzijo in Andersonovo dinamiˇcno lokalizacijo in spektralnimi lastnostmi (Dittrich and Doron, 1994; Dittrich et al., 1997, 1998). V doktorski disertaciji preuˇcujemo specifiˇcen tip verige biljardov imenovan serpentinast biljard. To je kanal sestavljen iz zavojev in ravnih odsekov, tako da so stene gladke in vzporedne, in ima konstanten pravokotni presek. Zunanji in notranji polmer zavojev je konstanten vzdolˇz verige in njuno razmerje doloˇca krivinski polmer zavoja. Dva primera biljardov sta prikazana na sliki i. Klasiˇcni delec v serpentinastem biljardu ne more spremeniti smeri potovanja, kar imenujemo enosmerni transport. Zaradi te lastnosti fazni prostor razpade na dve invariantni komponenti, od katerih pripada ena gibanju delca v levo, druga pa v desno stran verige. V kvantni sliki je enosmernost transporta krˇsena s tuneliranjem med invariantnimi komponentami. Rezultat tuneliranja je majhna, vendar zaznavna, refleksija dobljena v sipalnem opisu biljarda.

i

Klasiˇ cni serpentinast biljard

Sploˇsne dinamiˇcne lastnosti serpentinastih biljardov pridobimo z obravnavo periodiˇcnih primerov v katerih definiramo osnovno celico sestavljeno iz enega zavoja in enega ravnega odseka. Zavoj ima kot β, notranji polmer q in zunanji polmer 1, ravni odsek pa ima dolˇzino d. Dinamiko klasiˇcnih delcev vzdolˇz biljarda zastavimo v okviru modelov skakanja (ang. jump model) (Zumofen and Klafter, 1993), kjer pot delca opiˇsemo preko sekaliˇsˇc trajektorije z mejami osnov-

138

Daljˇ si slovenski povzetek

n=3

n=4 d

n=1

r=1 q r=

β β

d

n=2

(a)

(b)

Slika i: Primer periodiˇcnega biljarda s konfiguracijo β = 2π/3, d = 1/2 (a) in poljubno zgrajen serpentinast biljard (b) pri notranjem polmeru zavoja q = 0.6.

nih celic. Sekaliˇsˇce je doloˇceno s pozicijo na preseku y ∈ [0, a = 1 − q], z relativno projekcijo hitrosti na presek vy ∈ [−1, 1] in s ˇcasom dogodka. Izhodne in vhodne toˇcke tvorijo fazni prostor S = {(y, vy ) : y ∈ [0, a], vy ∈ [−1, 1]}. Zaradi enosmernega transporta se lahko pri analizi omejimo le na eno invariantno komponento, pripadajoˇco desno potujoˇcim delcem. Dinamiˇ cne lastnosti serpentinastega biljarda Dinamiko skozi osnovno celico opiˇsemo s preslikavo skakanja (Poincaréjevo preslikavo) P : S → S med vhodnimi in izhodnimi toˇckami in funkcijo ˇcasa T : S → R, ki meri ˇcas preleta. Fazni prostor Poincaréjeve preslikave je v sploˇsnem meˇsanega tipa, kot je to vidno iz slike ii. Pri doloˇcenih parametrih doseˇzemo skoraj popolni kaos. Velikost kaotiˇcne komponente se spreminja s parametri sistema in pribliˇzno velja, da se z veˇcanjem q in d poveˇcuje relativna velikost kaotiˇcne komponente. Kaotiˇcnost dinamike vdolˇz periodiˇcne biljardne verige okarakteriziramo s povpreˇcnim

Slika ii: Fazni portreti pri vrednosti parametrov β = π in d = 0 z notranjim polmerom q = 0.1, 0.3, 0.5 (zgoraj) in q = 0.6, 0.7, 0.9 (spodaj).

eksponentom Ljapunova (Reichl, 1992) preslikave skakanja preko ene osnovne celice. Eksponent Lyapunova priˇcakovano raste skoraj monotono z veˇcanjem parametra q in d, saj se z oˇzanjem

i. Klasiˇ cni serpentinast biljard

139

biljarda poveˇca ˇstevilo odbojev in tako ojaˇca meˇsanje trajektorij. Z veˇcanjem dolˇzine ravnega odseka d pa podaljˇsujemo prepotovano pot in postopoma uniˇcujemo otoke regularne dinamike. Delci, ki pridejo preko preseka v zavoj oz. ravni segment s smerjo hitrosti skoraj pravokotno na 3.5

d=0 d=1

3 2.5 λ

2 1.5 1 0.5 0

0

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9

1

q

Slika iii: Povpreˇcni eksponent Ljapunova λ izraˇcunan na trajektoriji dolˇzine 106 v periodiˇcnem serpentinastem biljardu kot funkcija z oblikoˇcnimi parametri β = π in d = 0 kot funkcija notranjega polmera q.

ˇ steno, potujejo preko elementa zelo dolgo. Casovna funkcija osnovne celice ima zaradi paralelnosti sten korensko singularnost oblike 1

T (y, vy ) ∼ (1 − vy )− 2 ,

(i)

v smeri pravokotno na steno |vy | = 1. Poslediˇcno ima distribucija ˇcasov preleta ene celice p(t) potenˇcno asimptotiˇcno odvisnost, kot je prikazano na sliki iv(a). Funkcijska odvisnost ˇcasovne 10

1

0.1

0.5

0.01 P(t)

vy

q=0.1 q=0.2 q=0.3 q=0.4 q=0.5 q=0.6 q=0.7 q=0.8 q=0.9 10 t-3

1

0

0.001 1e-04 1e-05

-0.5

1e-06 1e-07 1e-08

-1 0 0

0.05

1

0.1

0.15 2

0.2 y

0.25

3

0.3

4

0.35

0.4

10

(a)

100

1000

t

5

(b)

Slika iv: Diagram intenzitet ˇcasovne funkcije T (y, vy ) preko obmoˇcja faznega prostora S (a) in porazdelitev ˇcasov preletov preko osnovne ˇcelice p(t) izraˇcunani pri β = π in d = 0, za razliˇcne q.

funkcije je nezvezna in po strukturi podobna ˇcebulni lupini, kjer lahko posameznemu listu v lupini pripiˇsemo doloˇceno ˇstevilo trkov trajektorije s steno, glej sliko iv(b). Transport vzdolˇ z serpentinastega biljarda Transport vzdolˇz verige je opisan s pomoˇcjo evolucije porazdelitve delcev po celicah P n (t). Pri raziskovanju transportnih lastnosti postavimo ansambel delcev v fazni prostor prve celice in spremljamo njegovo ˇcasovno evolucijo z uporabo porazdelitve Pn . Tipiˇcna evolucija porazdelitve

140


Pn je prikazana sliki v, kjer je jasno vidna propagacija glavnine porazdelitve oblikovane v paket vzdolˇz verige in njeno poˇcasno ˇsirjenje. Porazdelitev delcev po celicah lahko izrazimo kot Z t dτ [pn (τ ) − pn+1 (τ )] , n ∈ N ∪ {0} , (ii) Pn (t) = 0

kjer je pn porazdelitev akumuliranega ˇcasa potovanja ansambla preko n celic. Zaradi singularnosti ˇcasovne funkcije doloˇcena koliˇcina delcev moˇcno zaostaja za njeno glavnino. Slednje je tudi razlog za marginalno normalen transport, kot bomo videli v nadaljevanju. Predpostavimo, da je 1

t=5 t=10 t=50 t=102 3 t=104 t=10

0.1

Pnt

0.01 0.001 1e-04 1e-05 1e-06 1e-07

1

10

100

1000

10000

n

ˇ Slika v: Casovni razvoj porazdelitve delcev preko celic Pn (t) v periodiˇcnem serpentinastem biljardu pri q = 0.6, d = 0 and β = π. Izraˇcuni so izvedeni z N = 107 toˇckami enakomerno porazdeljenimi preko kaotiˇcne komponente v zaˇcetni celici.

dinamika ergodiˇcna na kaotiˇcni komponenti. Zato lahko porazdelitev pn (t) aproksimiramo kot n-kratno konvolucijo porazdelitve ˇcasov preleta preko ene celice p(t) (Feller, 1970) Z −1 ˆ dtf (t) exp(iωt) , (iii) pn (t) = (p ∗ . . . ∗ p)(t) = F [exp(n log pˆ)](t) , f (ω) = Ff (ω) = | {z } R n

ki jo izrazimo s Fourierovo transformiranko porazdelitve preletov pˆ

1 log pˆ(ω) = i ωhT i − ω 2 σ 2 (α − log(ω)) + O(ω 3 ) . 2

(iv)

Porazdelitev pn z veˇcanjem ˇstevila n limitira k p∗n (t)

1 = √ G σ n log n

t − hT i n √ σ n log n

,

(v)

kjer je G standardna Gaussova porazdelitev. Nas zanima predvsem ˇcasovna asimptotika v kateri je preteˇzni del ansambla daleˇc vzdolˇz verige, torej v celicah z n 1. V tej limiti lahko razliko porazdelitev pod integralom v (ii) nadomestimo s Taylorjevim razvojem v spremenljivki b in porazdelitev Pn aproksimiramo s pribliˇzkom Z t ∂ Pn (t) − dτ p∗n (τ ) , (vi) ∂n −∞ iz ˇcesar sledi 1 Pn (t) 0 G σn

n − hni0 σn0

,

t hni = , hT i 0

2 σn0

σ2t log = hT i3

t hT i

,

(vii)

ii. Sploˇ sne kvantne verige sipalcev

141

kjer sta hni0 in σn0 asimptotiˇcna pribliˇzka prvih dveh centralnih momentov porazdelitve Pn . Iz tega zakljuˇcimo, da je difuzija delcev v eno smer marginalno normalna ob odˇstetem povpreˇcnem drsenju. Na sliki vi je prikazana ˇcasovna evolucija prvih dveh centralnih momentov porazdelitve Pn v periodiˇcnem serpentinastem biljardu pri q = 0.6, β = π and d = 0, ki podpira teoretiˇcne ugotovitve. 108 7

9

10

measurement σ2n ~ t log(t)

108

106

107

10

4

105

10

3

104

2

10

5

σn

1010

106

100 2

10

(b)

measurement vt

σn /t

(a)

3

10

102 2

10

10

3

10

4

5

6

10

10

7

10

8

10

9

10

t

102 10 102

50 0

103

104

4

105

5 106

6 7 8 log10(t)

9

107

109

108

t

Slika vi: Teˇziˇsˇce hni (a) in disperzija σn2 (b) delcev po celicah biljarda kot funkcija ˇcasa t pri q = 0.6, β = π and d = 0. Rezultati so statistiˇcno povpreˇcje preko ansambla 25000 trajektorij. Vrisana ˇcrta je enaka σn2 /t = −0.044374 + 5.2636 log 10 t.

ii

Sploˇ sne kvantne verige sipalcev

Sploˇsne kvantne verige sipalcev razumemo kot sipalce nameˇsˇcene na ravnem valovodu neke ˇsirine. Celotno verigo in posamezne dele razumemo kot sipalce na neskonˇcnem valovodu in jih ekvivalentno opiˇsemo v formalizmu sipalnih matrik in prenosnih matrik (ang. transfer matrix) (Newton, 2002). Sipalne matrike, oznaˇcene s S, in prenosne matrike, oznaˇcene s T , imajo naslednjo bloˇcno obliko L R x1 x2 r t . (viii) , T = S= x3 x4 tL r R kjer r L,R predstavlja refleksijo z leve oz. desne strani in tL,R transmisijo iz leve proti desni oz. obratno. Sipalna matrika predstavlja formalno povezavo med vhodnimi in izhodnimi deli valovanja v valovodu glede na sipalec, prenosna matrika pa povezavo med valovanjem na levem in desnem robu sipalca. Pri razpravi se omejimo na opis valovanja v potujoˇcih valovnih naˇcinih (ang. traveling modes) in zanemarimo razpadajoˇce naˇcine (ang. decaying or evanescent modes). Valovanje v ravnem vodniku opiˇsemo z d potujoˇcimi valovnimi naˇcini, kjer d imenujemo dimenzija sipalca. K sipalcem pripadajoˇca sipalna matrika je potem unitarna matrika iz grupe U (2d). Stacionarne transportne lastnosti sipalca ovrednotimo s povpreˇcno refleksijo hRi in variacijo 2 , ki sta definirani kot refleksije σR E D R = rx† rx ,

2 σR =

i 1 hD x† x 2 E (r r ) − R2 , d+1

x = L, R .

(ix)

pri ˇcemer je h•i = d−1 tr{•}. Povpreˇcna transmisija (prepustnost) je enaka T = 1 − R. Nas zanimajo stacionarne transportne lastnosti skozi verigo v procesu njenega podaljˇsevanja. Stanje verige opiˇsemo s sipalno matriko, ki je numeriˇcno stabilen opis sipalca. Podaljˇsevanje verige razumemo kot dinamiˇcni sistem nad stanji verige – sipalnimi matrikami. Veriga eno-dimenzionalnih sipalcev

142


Eno-dimenzionalni sipalci na ravnem valovodu so opisani s sipalnimi matrikami S ∈ U (2), ki jih parametriziramo kot # " L R p Aeiα Beiβ L L R , A = 1 − B2 , S(A, α , β , β ) = (x) L L +β R −αL ) iβ i(β Be −Ae pri ˇcemer je R = A2 refleksija sipalca in T = B 2 = 1 − A2 transmisija sipalca. V primeru, da prvotno verigo podaljˇsujemo z identiˇcnimi sipalci S(A, αL , β L , β R ) je dinamika verige S(An , αnL , βnL , βnR ) opisana z rekurzivno zvezo An+1 =

s

A2n + A2 + 2An A cos χn , 1 + 2An A cos χn + (An A)2

(xi)

χn+1 = χn + 2λ + arg{(1 + An Ae−iχn )(An + Ae−iχn )} , L βn+1 R βn+1

= =

(xii)

βnL + β L + arg{(1 + An Ae−iχn )} , βnR + β R + arg{(1 + An Ae−iχn )} .

(xiii) (xiv)

kjer uvedemo konstanto λ = (β L + β R )/2. Kot opazimo, sta relevantni le enaˇcbi (xi) in (xii) in predstavljata dvo-dimenzionalen dinamiˇcni sistem. Glede na predznak izraza D = A 2 − (sin λ)2 loˇcimo dva tipa dinamike sipalne matrike in sicer: za D < 0 imamo balistiˇcno dinamiko pri kateri refleksija le oscilira vzdolˇz verige, za D > 0 pa dobimo lokalizirano dinamiko, kjer transmisija eksponentno upada z dolˇzino verige. V prvem primeru lastne vrednosti pripadajoˇce transmisijski matriki leˇzijo na enotskem krogu, v drugem pa je ena lastna vrednost zunaj enotskega kroga in druga znotraj. Slednji sta si po absolutni vrednosti obratno sorazmerni in njun logaritem je enak hitrosti padanja transmisije. Primera obeh tipov dinamike sta prikazana na sliki vii. Sipalce 1 0.9 0.8 0.7 An

0.6 0.5 0.4 0.3 0.2 0.1 0

(a)

-3

-2

-1

0 χn

1

2

3

(b)

Slika vii: Fazni portet dimeniˇcnega sistema – verige enodimenzionalnih sipalcev – podan z enaˇcbama (xi) in (xii) pri parametrih A = 0.5, λ = 0.628319 (a) in A = 0.5, λ = 0.314159 (b). Na sliki (b) s puˇsˇcicami skiciramo lastne smeri toka okoli fiksne toˇcke enake: v1 = (1, 0) in v2 = (−0.945261, 0.326315).

loˇcimo na dva tipa, balistiˇcne in lokalizirane, glede na to kakˇsen tip dinamike sipalnih matrik generira ko z njimi podaljˇsujemo verigo. Opazimo, da imamo v primeru balistiˇcne dinamike en integral gibanja. V primeru uporabe sipalca z dodatkom ˇsuma pri gradnji verige, trajektorija verige z doˇzino le oscilira okoli integrala gibanja izraˇcunanega za povpreˇcni sipalec. Nekoliko podrobneje si ogledamo dinamiko sipalnih matrik, kjer uporabljamo nek deleˇz nakljuˇcno izbranih lokaliziranih sipalcev. Transmisija v tem primeru eksponentno pada kot Tn ∼ exp(−In) , pri ˇcemer je hitrost padanja I porazdeljena po Gaussovi porazdelitvi. Veriga veˇ c-dimenzionalnih sipalcev

iii. Serpentinast biljard v kvantni sliki

143

Veˇc-dimenzionalne verige preuˇcujemo le numeriˇcno, pri ˇcemer se omejimo na verige, kjer zaˇcetno verigo podaljˇsujemo z identiˇcnimi sipalci. Glede na uporabljen sipalec loˇcimo dva tipa dinamike verige v procesu podalˇsevanja: balistiˇcno in lokalizirano dinamiko. V primeru balistiˇcne dinamike je gibanje sipalne matrike verige kvazi-periodiˇcno, medtem ko v lokalizirani dinamiki transmisija ˇ je plato razliˇcen od niˇc govorimo eksponentno pada do nekega platoja okoli katerega oscilira. Ce o delni lokalizaciji, sicer o popolni. Za lokalizacijo so ponovno krive lastne vrednosti prenosnih matrik sipalca zunaj enotskega kroga. Z veˇcjim ˇstevilom lastnih vrednosti zunaj enotskega kroga se plato transmisije tipiˇcno niˇza. Hitrost eksponentnega padanja z dolˇzino verige doloˇcuje lastna vrednost najveˇcje velikosti, oznaˇcena z λmax , na naslednji naˇcin Tn ∼ konst. + O(exp(−In)) ,

Haar measure(ballistic matrices)

kjer I = log |λmax | imenujemo hitrost naraˇscanja refleksije oz. razpada transmisije. Nas predvsem zanima razumevanje statistiˇcnih lastnosti verig ob veˇcanju dimenzije sipalcev d in kakˇsno vlogo imata oba tipa sipalcev poimenovana glede na dinamiko verige, ki jo generirata, to so balistiˇcni in lokalizirani sipalci. Na sliki viii je prikazana Haarova mera (Reichl, 2004) balistiˇcnih sipalcev v mnoˇzici vseh sipalcev U (2d). Vidimo, da se Haarova mera balistivˇcnih sipalce z dimenzijo d pada kot exp(−O(d2 )). Absolutna velikost maksimalne lastne vrednosti λmax in ˇstevilo lastnih 1

measurments fit, exp(-f(x))

0.1 0.01 0.001 1e-04 1e-05 1e-06

0

5

10

15 d(d+1)

20

25

30

Slika viii: Haarova mera mnoˇzice balistiˇcnih unitarnih matrik za razliˇcne dimenzije sipalcev d. Vrisana krivulja je podana s formulo f (x) = 0.465792x − 0.238663, kjer je x = d(d + 1).

vrednosti zunaj enotskega kroga du so karakteristiˇcne vrednosti dinamike refleksije vzdolˇz verige. Ogledamo si porazdelitev vrednosti λmax , oznaˇcene s Pmax (λ), in relativnega ˇstevila nestabilnih lastnih vrednosti du /d, oznaˇcenega s Pu (u), za sipalce v U (2d). Porazdelitvi sta za razliˇcne dimenzije d prikazani na sliki ix. Iz rezultatov ugotovimo, da maksimalna lastna vrednost λmax in dimenzija du v povpreˇcju rasteta z dimenzijo d. Analiza pa prav tako pokaˇze, da ima P max (λ) doloˇcene skalirne lastnosti iz katerih sledi, da v povpreˇcju hitrost razpada transmisije I skalira z dimenzijo d kot I ∼ d1 log(d). Tako lahko zakljuˇcimo, da z naraˇsˇcajoˇco dimenzijo d postane balistiˇcna sipalna dinamika zelo redek pojav. Tipiˇcna veriga je zato vsaj delno lokalizirana, pri ˇcemer konˇcna refleksija (plato) in hitrost konvergence rasteta z naraˇsˇcajoˇco dimenzijo d.

iii

Serpentinast biljard v kvantni sliki

Serpentinast biljard v kvantni sliki razumemo kot verigo sipalcev s posebnimi lastnosti. Vlogo sipalcev igrajo zavoji, ki vzdolˇz verige alternirajo svojo orientacijo in zlomijo klasiˇcno integrabilnost sistema. Za razumevanje lastnosti serpentinastih biljardov je zato bistveno poznavanje kvantne narave enojnega zavoja. Zanima nas vpliv klasiˇcnega enosmernega transporta na kvantno naravo serpentinastih biljadov, npr. na transportne lastnosti pri doloˇceni energiji, na lastnosti spektra zaprtih biljardov in na korespondenco med klasiˇcno in kvantno dinamiko. Sipanje

144


1 0.1 0.01 0.001 1e-04 1e-05 1e-06 1e-07

Pmax (t)

0.3 0.25 0.2 0.15 0.1

d=2 d=3 d=4 d=5 d=6 d=7 d=8 d=9 -3 t 1

10

100

25

15 10

1000 5

0.05 0

d=2 d=5 d=10 d=20 d=50 d=100

20

Pu(x)

0.4 0.35

2

4

6

8

10

12

14

0

16

0

0.2

t

(a)

0.4 0.6 x = du/d

0.8

1

(b)

Slika ix: Porazdelitev maksimalnih lastnih vrednosti prenosne matrike po absolutni vrednosti P max (a) in porazdelitev relativnega ˇstevila lastnih vrednosti zunaj enotskega kroga P u (b) za sipalce s sipalnimi matrikami v U (2d).

iz asimptotiˇcnega obmoˇcja nazaj v le-tega skozi serpentinast biljard in njegove dele opiˇsemo s formalizmom sipalnih matrik na energijski lupini E = 21 k 2 , kjer k imenujemo valovno ˇstevilo. Valovanje pri neki energiji v asimptotskem obmoˇcju razvijemo po valovnih naˇcinih. Slednje delimo na potujoˇce (ang. traveling) in razpadajoˇce (ang. decaying ali evanescent) valovne naˇcine. Vseh naˇcinov je neskoˇcno, a potujoˇcih je konˇcno in enako No = ka/π, pri ˇcemer je a ˇsirina kanala. Razpadajoˇci naˇcini so vir numeriˇcnih problemov. Sipanje preko enojnega zavoja na ravnem valovodu Zavoj razumemo kot samostojni sipalec na ravnem valovodu, ki ga obravnavamo pri fiksni energiji E = 21 k 2 . Zavoj oklepa nek kot β in ima zunanji polmer r = 1 ter notranji polmer r = q, ki ga ˇ lahko razumemo kot ukrivljenost zavoja. Sirina kanala je potem a = 1−q. Kvantno sipanje preko zavoja opiˇsemo z neskoˇcno dimenzionalno sipalno matriko S zapisano v bazi valovnih naˇcinov v asimptotiˇcnem obmoˇcju r t , (xv) S= t r ki je simetriˇcna zaradi simetrije sipalca. Reflekcijsko r in transmisijsko t matriko izraˇcunamo natanˇcno na konˇcnem prostoru dimenzije N 0 , ki je nekoliko veˇcja od No . Za izraˇcun uporabimo posebaj v ta namen razvito metodo, ki upoˇsteva lastnosti Laplacevega operatorja, in se tako izognemo numeriˇcnim nestabilnostim povezanim z razpadajoˇcimi naˇcini. V kvantni sliki se ne ohranja klasiˇcna enosmernost transporta, saj dobimo pri sipanju tipiˇcno neko od niˇc razliˇcno refleksijo. Za ovrednotenje krˇsitve klasiˇcne enosmernosti uporabimo povpreˇcno refleksijo R in disperzijo refleksije σR , kjer uporabimo del sipalne matrike S pripadajoˇce odprtim naˇcinom. Kot priˇcakovano je povpreˇcna refleksija R nekaj redov manjˇsa od transmisije T = 1 − R, razen pri t.i. resonaˇcnih valovnih dolˇzinah k ∈ π/aN, ko se refleksija moˇcno dvigne. To se lepo vidi na sliki x, kjer si ogledamo povpreˇcno refleksijo preko intervala treh resonanˇcnih valovnih dolˇzin. Pri resonanˇcnih valovnih ˇstevilih se v ravnem valovoda pojavi nov potujiˇc valovni naˇcin, ki se zelo intenzivno sipa nazaj. Omenjen valovni naˇcin tudi stran od resonanˇcnih valovnih ˇstevil prispeva glavnino h celotni refleksiji. Iz slike vidimo, da se z manjˇsanjem parametra q refleksija naraˇsˇca pri fiksnem ˇstevilu k(1 − q)/π. Majhna refleksija priˇca o tem, da je tuneliranje med klasiˇcnimi invariantnimi komponentami zelo ˇsibko, in njena jakost upada z energijo kot R ∼ k −1 .

(xvi)

Slednje potrjuje dejstvo, da ovojnica grafa refleksije kot funkcije z valovnim ˇstevilom R k k skalira −1 0 0 kot O(k ) in integral povpreˇcne refleksije po valovnih ˇstevilih R(k )dk skalira kot O(log(k)),


145

q=0.2 q=0.6 q=0.9

2 No /(1-q)2

1 0.1 0.01 0.001 1e-04 1e-05

0

0.5

1

1.5

2

k(1-q)/π - 100

Slika x: Povpreˇcna refleksija No R/(2a2 ) v zavoju s kotom β = π in pri razliˇcnih vrednostih q kot funkcija valovne dolˇzine k okoli resonanˇcnega valovnega ˇstevila k = 101π/a. 1

〈R〉 k-1

0.1

0.1

∫k〈R〉(κ) dκ k 0.08 ∫ σR(κ) dκ

0.01 0.001

f1(k) f2 (k)

0.06

1e-04 1e-05

0.04

1e-06 0.02

1e-07 1e-08

10

100

1000

0

10

100 k

k

(a)

1000

(b) R

R Slika xi: Povpreˇcna refleksija R (a) in integrala mer refleksije Rdk in σR dk (b) preko daljˇsega intervala valovnih ˇstevil k pri q = 0.6. Vrisani krivulji v delu (b) sta f1 (k) = 0.0110301 + 0.0102197 log k in f2 (k) = −0.0155328 + 0.00692377 log k.

kar je vidno na sliki xi. Naslednja zanimiva koliˇcina je ˇcasovni zamik Wignerja in Smitha τws , ki predstavlja kvantno analogijo klasiˇcne geometriˇcne poti pri prehodu sipalca, in je definiran kot 1 † droo † dtoo tr roo τws = + tr too , (xvii) No dik dik kjer [•]oo predstavlja blok matrike pripadajoˇc potujoˇcim valovnim naˇcinom. Pri preuˇcevanju tega zamika vzdolˇz osi valovnega ˇstevila k opazimo, da v bliˇzini resonanˇcnih valovnih ˇstevil ˇcasovni zamik moˇcno naraste in z viˇsanjem valovnega ˇstevila pade pribliˇzno na klasiˇcno priˇcakovano vrednost, glej sliko xii. Resonanˇcno obnaˇsanje lahko kvantitativno pravilno razloˇzimo z enodimenzionalnim sipalcem, ki povezuje najviˇsje potujoˇce valovne naˇcine v asimptotiˇcnem obmoˇcju s pripadajoˇcim valovnim naˇcinom v zavoju. Stacionarni transport Transport skozi periodiˇcne in nakljuˇcne serpentinaste biljarde preuˇcujemo na neki energijski 2 , ki sta izraˇ lupini. Transport merimo s povpreˇcno refleksijo Rn in disperzijo refleksije σR cunani na enak naˇcin kot v primeru enojnega zavoja. Zanima nas predvsem odvisnost transporta od dolˇzine verige.

146

Daljˇ si slovenski povzetek 12

q=0.2 q=0.6 q=0.9

11 10 9 τws

8 7 6 5

T0.9 T0.6 T0.2

4 3 2

-1

-0.5

0

0.5

1

k(1-q)/π - r

ˇ Slika xii: Casovni zamik Wignerja in Smitha τws preko nekega intervala okoli 100. refleksijske resonance v zavoju s kotom β = π. Polne vodoravne ˇcrte z oznakami T0.2,0.6,0.9 predstavljajo klasiˇcni ˇcas preleta pri q = 0.2, 0.6, 0.9 pribliˇzno podanem s formulo Tq = 2.45863 + 2.48696q.

V periodiˇcnih serpentinastih biljardih ugotovimo, da imamo moˇznost balistiˇcne ali lokalizirane dinamike sipalnih matrik, podobno kot v sploˇsnih verigah. Lokalizirano dinamiko lahko ponovno pripiˇsemo lastnim vrednostim prenosne matrike pripadajoˇce osnovne celice, ki so zunaj enotskega kroga. Primer dinamike povpreˇcne refleksije in pripadajoˇc spekter prenosne matrike je prikazan na sliki xiii. 5 4.5

1

4

0.5

3

ℑ{λ}

103

3.5 2.5 2 1.5

-0.5

1 0.5 0

0

-1 0

500 1000 1500 2000 2500 3000 3500 4000 n

(i)

-1

-0.5

0

0.5

1

0.5

1

ℜ{λ}

0.045 0.04

1

0.035

0.5

0.025

ℑ{λ}

0.03 0.02 0.015

-0.5

0.01 0.005 0

(ii)

0

-1 0

10 20 30 40 50 60 70 80 90 100 n

(a)

-1

-0.5

0 ℜ{λ}

(b)

Slika xiii: Povpreˇcna refleksija R v odvisnosti od dolˇzine verige n (a) in spekter pripadajoˇce prenosne matrike Too v primeru balistiˇcnega transporta pri k = 50.5π/(1 − q) (i) in delne lokalizacije pri k = 50.01π/(1 − q) (ii) v periodiˇcnem serpentinastem biljardu s konfiguracijo: q = 0.6, β = π, d = 0.


147

Zaradi visoke transmisije osnovne celice in intenzivnega meˇsanja transmisijskih delov sipalne matrike z daljˇsanjem verige je za lokalizirano dinamiko tipiˇcno, da refleksija konvergira k platoju pribliˇzno kot du /No , kjer je du ˇstevilo lastnih vrednosti zunaj enotskega kroga. Lokalizirana dinamika v parametriˇcnem prostoru je bolj pogosta v bliˇzini resonanˇcnih valovnih ˇstevil k ∈ π/aN, a je precej bolj redka kot v sploˇsnih verigah unitarnih sipalcev. Veliko transportnih lastnosti serpentinastih biljardov je podedovano od lastnosti pozameznih zavojev, ki ga tvorijo. Povpreˇcna refleksija R je preko parametriˇcnega prostora sploˇsno majhna v primerjavi s transmisijo T = 1 − R, razen pri resonanˇcnih valovnih ˇstevilih π/aN, kjer refleksija moˇcno naraste. To je vidno na sliki xiv(a), kjer prikazujemo R v odvisnosti od valovnega ˇstevila za razliˇcno dolge biljarde. Z viˇsanjem energije refleksija poˇcasi upada in to je neodvisno od dolˇzine verige, 1 0.01

0.1 0.01

0.001 1e-04 1e-05

1e-04 1e-05

1e-06 1e-07 99.5

0.001

1e-06 100

100.5

101

101.5

102

1e-07

102.5

10

100

k(1-q)/π n=10

n=1

200

r = ka/π n=1000

n=10

(a)

-1

2r

(b)

Slika xiv: Povpreˇcna refleksija Rn kot funkcija valovnega ˇstevila k okoli k = 101. refleksijske resonance v zavoju (a) in preko veˇcjega intervala v toˇckah k ∈ 0.1π/aN (b) v verigi s parametri q = 0.6, β = π, d = 0 in razliˇcnimi dolˇzinami n.

glej sliko xiv(b). Z dolˇzino verige se refleksija zaradi toˇck z lokalizirano sipalno dinamiko tipiˇcno nekoliko poveˇca. Ta aspekt ˇse ni detajlno raziskan. V naˇsih numeriˇcnih raziskavah, v katerih smo omejeni na neko konˇcno energijsko skalo, ugotovimo, da povpreˇcna refleksija skalira z valovnim ˇstevilom kot R ∼ k −1±0.2 . Transport skozi nakljuˇcne biljarde neke dolˇzine n merimo z reskalirano refleksijo ρ n (k) = No Rn in primer rezultatov je prikazan na sliki xv. Iz numeriˇcnih ˇstudij ugotovimo, da reskalirana 10

r=10.5 r=50.5

1

8.5 10-5n

No

0.1 0.01 0.001 1e-04 1e-05

1

10

100

1000

10000 65536

n

Slika xv: Povpreˇcna refleksija Rn v nakljuˇcnih biljardih kot funkcija dolˇzine n pri q = 0.6 in razliˇcnih valovnih ˇstevilih k = πr/(1 − q) kot je prikazano na sliki. Pri vsakem valovnem ˇstevilu prikaˇzemo tri realizacije nakljuˇcne verige z β ∈ [0, π] and d ∈ [0, 1].

refleksija ρn v nakljuˇcnih biljardih podobno linearno raste z n za razliˇcne vrednosti k in q do

148


√ neke dolˇzine ko je R ≈ No σR , kar je tipiˇcno za nakljuˇcne sipalce. Ta dolˇzina se podaljˇsuje z veˇcanjem q oz. tanjˇsanjem kanala. Z nadaljnim daljˇsanjem verige refleksija preide iz linearnega naraˇsˇcanja v bolj poˇcasnega in priˇcakujemo, da konvergira k nekem platoju, niˇzjem od N o . Statistika sipalnih matrik V klasiˇcni sliki se prvotno koncentriran snop klasiˇcnih trajektorij se pri potovanju vzdolˇz verige razmaˇze po preseˇcni ploskvi in podobno se zgodi v kvanti sliki, kjer z daljˇsanjem verige postane sipanje precej podobno nakljuˇcnemu. Ta nakljuˇcnost se odraˇza tudi v sipalni matriki S, kjer se omejimo na podprostor pripadajoˇc potujoˇcim valovnim naˇcinom. Na sliki xvi prikaˇzemo porazdelitev reskaliranih refleksijskih in transmisijskih elementov sipalne matrike: 2No tij , 1−R

2No rij . R

(xviii)

Opazimo, da z naraˇsˇcajoˇco dolˇzino porazdelitev transmisijskih elementov [t n ]ij = ψij + iξij 10

10

n=1 n=2 n=5 n=10 n=50 G(x,0,1)

1

1

dP/dx

0.1

dP/dx

0.1

0.01

0.01

0.001

0.001

1e-04

-8

-6

-4

-2 0 2 x = ℑ[tn]ij 2 No/(1-)

10

4

6

1e-04

8

-6

-4

-2 0 2 x = ℑ[tn]ij 2 No/(1-)

4

6

8

n=1 n=2 n=5 n=10 n=50 n=100

1

0.1 dP/dy

0.1

0.01

0.001

0.01

0.001

1e-04 -15

-8

10

n=1 n=2 n=5 n=10 n=50 n=100

1

dP/dy

n=1 n=2 n=5 n=10 n=50 G(x,0,1)

1e-04 -10

-5 0 5 y = ℑ[rn]ij 2 No/

10

15

-15

-10

-5 0 5 y = ℑ[rn]ij 2 No/

10

15

Slika xvi: Porazdelitev realnih in imaginarnih matriˇcnih elementov refleksijske r in transmisijske matrike t za razliˇcne dolˇzine verig n izraˇcunane pri valovnem ˇstevilu k = 200.5π/(1 − q) = 1574.72, kjer je G(x, x, σ 2 ) Gaussova porazdelitev.

(ψij , ξij ∈ R) hitro konvergira h Gaussovi porazdelitvi oblike x2 dP dP 1 (x) = (x) = √ e− 2σ2 + O(R) , dψij dξij 2πσ 2

(xix)

z disperzijo σ 2 = (1 − R)/(2No ). Medtem, ko se porazdelitev refleksijskih matriˇcnih elementov z dolˇzino skoraj ne spreminja in ima pribliˇzno eksponentno obliko. Spekter zaprtega serpentinastega biljarda


149

Energijski spektri biljardov in signature klasiˇcne dinamike v le-teh so ˇze dalj ˇcasa predmet raziskav. S povezavo zaˇcetka in konca serpentinastega biljarda dobimo zaprt biljard s klasiˇcnim enosmernim transportom vzdolˇz obsega. Zanimajo nas njegove spektralne lastnosti in vpliv klasiˇcnega enosmernega transporta na energijski spekter. Za testni objekt uporabimo serpentinast biljard brez geometriˇcnih simetrij, ki je prikazan na sliki xvii. Energijski spekter raˇcunamo

Slika xvii: Primer serpentinastega biljarda uporabljenega v izraˇcunih energijskega spektra.

v sipalnem formalizmu po postopku opisan v (Prosen, 1996). V semi-klasiˇcni limiti so nivoji dvojno degenerirani, pri ˇcemer lahko vsakega razstavimo na levo in desno potujoˇce valovanje. Pri konˇcni energiji pa se degeneracija razcepi, zaradi tuneliranja med klasiˇcno invariantnimi komponentami faznega prostora, v par lastnih energij z lastnimi stanji v katerih so enako moˇcno zastopana v levo in v desno potujoˇca valovanja. Spekter je razen redkih izjem sestavljen le iz bliˇznjih parov nivojev. Razdalja med nivoji v razcepu degeneracije skalira z refleksijo R in ˇcasom Wignerja in Smith τws odprtega biljarda √ R . (xx) ∆krazcep ∼ τws Razcepi so tipiˇcno kakˇsen red ali veˇc manjˇsi kot povpreˇcne razdalje med nivoji. V nakljuˇcnih biljardih, kjer refleksija linearno naraˇsˇca z dolˇzino, ugotovimo, da razcep skalira z dolˇzino verige n kot 1 ∆krazcep ∼ n− 2 . (xxi) Zaradi bliˇzine nivojev v razcepu je v praksi teˇzko vedno razloˇciti oba nivoja v paru. V nadaljevanju dela spekter renormaliziramo tako, da je gostota nivojev konstanta. Ena od tipiˇcnih spektralnih karakteristik je porazdelitev razmikov med zaporednimi reskaliranimi nivoji P (S). V serpentinastih biljardih se izkaˇze, da lahko P (S) zapiˇsemo kot vsoto porazdelitve razmikov med zaporednimi pari Pp (S) in porazdelitve razmikov med nivoji v razcepu Ps (S) P (S) = (1 − f )Ps (S) + f Pp (S) .

(xxii)

Po teoriji je f = 1/2. V praksi, kjer ne razloˇcimo vseh nivojev v parih, je f lahko tudi veˇcji od polovice. Loˇcitev porazdelitve na dva prispevka je moˇzen, saj so razcepi degeneracij tipiˇcno precej manjˇsi kot razdalja med pari. Za izbran biljard je funkcija P (S) prikazana na sliki xviii, kjer sta obe porazdelitvi dobro vidni. Ob upoˇstevanju tuneliranja med invariantnima komponentama in domneve Berryja in Robnika (Berry and Robnik, 1984), po priˇcakovanjih ugotovimo, da so razmiki med pari porazdeljeni po Wignerjevi porazdelitvi Pp (S) =

π 2 − π (ρS)2 , ρ Se 4 2

(xxiii)

ki je tipiˇcna za spekter kaotiˇcnih sistemov (Bohigas et al., 1984). Izraz ρ = (2−µ)−1 je povpreˇcen razmik med pari in µ ∼ k δkrazcep je povpreˇcen razmik med racepljenimi nivoji.

150

Daljˇ si slovenski povzetek 10

0.16 0.14 P(S)

1

0.1

PGOE

0.18

10

P(S)

1 P(S)

0.2

100

0 1 2 3 4 5 6 7 8 9 10 102 S

0.12 0.1 0.08 0.06

0.01

0.04 0.02

0.001

0

1

2

3

4

5

6

7

0

8

0

1

2

3

4

S

5

6

7

8

S

(a)

(b)

Slika xviii: Porazdelitev razmikov med energijskimi nivoji P (S) na celotnem intervalu razmikov S (a) in poveˇcava obmoˇcja izven razcepa klasiˇcne degeneracije (b), pri ˇcemer smo vzeli f = 0.5. V levi sliki vloˇzen diagram prikazuje poradelitev P (S) v bliˇzini izhodiˇsˇca S = 0.

Kvantno-klasiˇ cna korespondenca nad faznim prostorom Klasiˇcna in kvantna dinamika vzdolˇz serpentinastega biljarda sta si zelo podobni in podobnost se poveˇcuje z viˇsanjem energije. To podobnost raziˇsˇcemo na primeru korespondence med klasiˇcnim in kvantnim sipanjem ˇzarka vzdolˇz periodiˇcnega serpentinastega biljarda. Sipanje ˇzarka primerjamo na klasiˇcni preseˇcni ploskvi Se (ang. surface-of-sections) na kateri klasiˇcni sistem opiˇsemo z verjetnostno porazdelitvijo ρ : Se → R+ in kvantni sistem z Wignerjevo funkcijo W : Se → R. Kvantno-klasiˇcno korespondenco ovrednotimo s prekrivalnim integralom klasiˇcne porazdelitve in Wignerjeve funkcije imenovana kvantno-klasiˇcna zvestoba F , Z dx ρ(x)W (x) . (xxiv) F = Se

V primeru popolne korespondence se F pri sipanju ne spremeni, kar seveda ni moˇzno pri konˇcni energiji. Zanimiva je predvsem kvantno-klasiˇcna zvestoba F sipanega ˇzarka, ki je na vstopu opisan z Gaussovo porazdelitvijo, kot funkcija dolˇzine verige n. Za primer moˇcno kaotiˇcnega biljarda je zvestoba F (n) prikazana na sliki xix. Z vleˇcenjem paralel med dalˇsanjem verige 1

r=50.5 r=100.5 r=200.5 3 e- 1.8923n

G

0.1 1/50 0.01 1/100 1/200

(a)

0

2

4

6

8

10

n

ˇ Slika xix: Casovni razvoj relativne kvantno-klasiˇcne zvestobe G(n) = F (n)/F (0) z dolˇzino verige n v periodiˇcnem serpentinastem biljardu s konfiguracijo osnovnih celic q = 0.6, d = 0 in β = π pri razliˇcnih valovnih ˇstevilih k = πr/(1 − q) ko je nakazano v sliki. Polne linije so teoretiˇcne napovedi doloˇcene do konstantnega faktorja natanˇcno.

in diskretnimi preslikavami ugotovimo, da relativna zvestoba F (n)/F (0) eksponentno upada z


151

dolˇzino n do platoja 1/No z eksponentom, ki je enak maksimalnemu eksponentu Ljapunova λmax klasiˇcne Poincaréjeve preslikave skoka, kot F (n) ∼ exp(−λmax n) ,

(xxv)

in kjer je No ˇstevilo potujoˇcih valovnih naˇcinov v ravnem valovodu. Teoretiˇcno napoved podpirajo numeriˇcni rezultati.

152


Izjava o avtorstvu

Izjavljam, da sem v predloˇzeni doktorski disertaciji uporabljal rezultate lastnega raziskovalnega dela.

V Ljubljani, dne 25. september 2006.

Martin Horvat