Classical Fields Approximation for the dynamics of Bose-Einstein ...

` DEGLI STUDI DI TRENTO UNIVERSITA dipartimento di fisica

corso di laurea magistrale in fisica

Classical Fields Approximation for the dynamics of Bose-Einstein Condensates at finite temperature

Relatore: Prof. Franco Dalfovo Correlatori: Dr. Iacopo Carusotto Dr. Marek Tylutki

Candidato: Fabrizio Larcher

ANNO ACCADEMICO 2013/2014

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Contents Introduction

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1 Bose-Einstein condensation, Gross-Pitaevskii equation and solitons 1.1 Brief history of Bose-Einstein condensation . . . . . . . . . . . . . . . . 1.2 The Gross-Pitaevskii equation . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Solitonic solution for the GPE . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 What is a soliton? . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Russell, the Wave of Translation and the KdV equation . . . . . 1.3.3 Solitonic solution of the Gross-Pitaevskii equation . . . . . . . . 1.3.4 Soliton dynamics in an inhomogeneous condensate . . . . . . . .

8 8 10 12 12 13 14 18

2 Numerical solution of the GPE 2.1 Preliminaries . . . . . . . . . . . . . . . 2.1.1 The Split-Step Fourier method . 2.1.2 The Discrete Fourier Transform . 2.1.3 Discretisation . . . . . . . . . . . 2.2 Dimensionless Gross-Pitaevskii equation 2.3 The numerical method for the GPE . . 2.3.1 Kinetic evolution . . . . . . . . . 2.3.2 Potential evolution . . . . . . . . 2.3.3 The algorithm . . . . . . . . . . 2.3.4 Imaginary-time evolution . . . . 2.3.5 Test: 1D Harmonic Oscillator . . 2.3.6 Real time evolution of the soliton

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3 Classical Fields Approximation 3.1 The model . . . . . . . . . . . . . . . . . . . . . . . 3.2 The method . . . . . . . . . . . . . . . . . . . . . . 3.3 Non-interacting Bose gas . . . . . . . . . . . . . . . 3.3.1 Box confinement . . . . . . . . . . . . . . . 3.3.2 Harmonic confinement . . . . . . . . . . . . 3.3.3 Schrödinger evolution . . . . . . . . . . . . 3.4 Interacting Bose gas: spontaneous soliton creation 3.4.1 Momentum cut-off and condensate fraction 3.4.2 Energy functional and results . . . . . . . . 3

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4

Contents

4 Evaporative cooling 55 4.1 Cooling atoms: not a simple work . . . . . . . . . . . . . . . . . . . . . 55 4.2 Numerical simulation of evaporative cooling . . . . . . . . . . . . . . . . 57 5 Superfluidity and critical velocity in a BEC 5.1 Numerical simulations . . . . . . . . . . . . . 5.1.1 Zero temperature . . . . . . . . . . . . 5.1.2 Finite temperature . . . . . . . . . . . 5.1.3 Sound speed determination . . . . . .

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Conclusions A Second quantisation formalism A.1 Introduction to first quantisation . . A.2 Fock Space and second quantisation A.3 Representation of operators . . . . . A.3.1 One-body operators . . . . . A.3.2 Two-body operators . . . . .

63 65 66 71 73 75

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B Landau criterion for superfluidity

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Acknowledgements

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Introduction Physics. It was a humid, rainy and pretty cold day of the middle of September. The year was 2008. In the wide and dusty room about one hundred and twenty young faces were listening to the tall man. With calm and controlled voice he was describing them the great unknowns wavering upon their future: “You really have to understand why you decided to enrol at Physics and Mathematics. Are you really resolute, or was it just because outside is cold?”. Deafening silence. Less than one year after that speech, about one third of the people in that room had realised that maybe outside it was not so cold. The tall man was Professor Franco Dalfovo. Five years later, I had the great honour to be assisted by him, Dr. Iacopo Carusotto and Dr. Marek Tylutki in my Master’s Thesis project, but at the time, I could barely imagine it. He was right, one should really understand the reasons behind such a daring choice. Physics can be tough, requires a lot of dedication and passion, and a certain amount of abnegation. It has the very peculiar property to make you feel even more ignorant as you study it, and forces you ceaselessly to rearrange your beliefs: it is a great school for humility. So why should one face such a challenge? For sure, I am not yet able to answer entirely, but I have some clues. At first, the wonder. The wonder in front of the crazy order one is able to perceive (just to perceive) when studying Nature. Newton described himself as a child playing on a beach. I would rather put that child in a huge toy shop1 . The boy has a bandage on his eyes, and cannot fully understand where he is, or which kind of objects surround him. Still, he is seduced by their charm, and does whatever he can to grab them. It is the old, obscure passion of alchemists: to unveil the deepest of the mysteries and to be able to dominate the Universe by knowing its secret Rules. Funnily, a mission doomed to failure by those same rules. Then, the people. The common representation of physicists, and of scientists in general, is that of a strange, dazed person with crazy hair, bustling about disturbing machinery or a blackboard full of unintelligible computations. With all due exceptions, 1

One could argue that it would have been pretty difficult for Newton to find a toy shop.

5

6

Introduction

it is quite an accurate description. From students to researchers, up to professors, this eccentricity is often a marker of curiosity and intellect, and listening to a conversation around the Department of Physics is always an enriching experience. Finally, believe it or not, physics is fun. It is indeed fun to ask questions about everything, to open discussions about things that common people would simply assume as given or irrelevant, and not to resign until the question is answered (or it is determined an answer to be impossible, which is more or less the same). It might be a little less fun for parents, partners and friends, forced to repeatedly fight against the polemic frame of mind that physics shapes. It is fun to work in obscure laboratories or on chaotic desks, and to be able to recreate conditions not existing spontaneously in any other place in the universe, to observe things that should not exist. Not an insignificant achievement, for some clever monkeys on a little rock. In conclusion, in spite of the struggle, delusion and quarrel that may be caused by it, it is worthwhile to live physics, entirely and passionately. This work. The present thesis should represent the crowning achievement of a five years path. Among the great variety of topics encountered in my studies, I was conquered by the behaviour of ultracold atoms. The first time I was stricken by it, was during a legendary public lecture held in September 2010 at the University of Trento by the Nobel Laureate Prof. Wolfgang Ketterle. From that moment I tried my best to prepare myself to work, one day, in this field. I then realised I had a huge luck to be enrolled at the University of Trento, where the INO-CNR BEC Centre is located. The main activity of the centre is the study of ultracold atoms, both from a theoretical and from an experimental point of view. One of the recent achievement of the group was the publication ([1]) on Nature Physics of the first direct observation of solitons inside a Bose-Einstein Condensate (BEC), created via the Kibble-Zurek Mechanism (KZM). One of the authors was Prof. Dalfovo, who proposed me to study the formation and the dynamics of solitons inside a confined 1D bosonic condensate at nonzero T , which thus represents the leading topic of this work. In that paper, measurements are taken at finite temperature, while the great majority of theoretical and experimental investigations in the literature about solitons regards zero temperature. The group is hence interested in the development of theoretical models for treating finite temperature systems, and this thesis work consists in acquiring some familiarity with the techniques involved in such problem. Among these, we studied the so-called Classical Fields Approximation (CFA), developed by Brewczyk et al.2 . We first started with the study of the solution of the time dependent Gross-Pitaevskii equation (GPE), then we reproduced interesting computations already performed by Karpiuk et al. ([3]) and by Witkowska et al. ([4]) on the spontaneous generation of solitons at finite temperature. Hence we tested the method on different situations, taking the zero temperature case as a springboard for the extension to finite temperature. The work is ongoing, and the results are far from complete, but the final goal is to apply the method to a systematic study of the generation and of the dynamics of solitons at finite temperature, including the relevant 3D case. In the first chapter, we briefly review the main historical and theoretical aspects regarding Bose-Einstein Condensation, including the key Gross-Pitaevskii equation; we give also a look to the particular class of solitonic solutions for the GPE, and analyse 2

See [2], [3] and the references therein.

7

some of their properties. In Chapter 2 we expose a simple method to find a numerical solution for the GPE, in order to get the time evolution of a determined initial state, named the Split-Step Fourier method. We apply the method to the relevant case of an harmonic confinement, and perform some test simulations. The problem of simulating numerically an atomic cloud in thermal equilibrium at finite temperature is overcome with the help of the Classical Fields Approximation. This is introduced in Chapter 3, together with its application to both a non-interacting and an interacting system. Some differences between these two cases are explained, and the spontaneous creation of solitons only due to thermal fluctuations is deduced from the results. Having built the machinery needed to produce a thermal state, it would be a pity not to use it to reproduce some funny experiments. At this purpose, from the end of Chapter 3, the work is split in two directions. In Chapter 4 we review and simulate the clever mechanism of evaporative cooling, while in Chapter 5 we study the conditions under which superfluidity is possible in a 1D trapped gas. Both chapters are provided with comparisons with other experimental and theoretical realisations. Finally, an appendix is added in which some of the building blocks of the theoretical formalism behind BEC are reported.

Chapter 1

Bose-Einstein condensation, Gross-Pitaevskii equation and solitons The study of Bose-Einstein condensation has been one of the most exciting and persistent scientific efforts of modern physics. Its connections with superfluidity and the difficulty in the experimental realisation made it a central research field in physics for several decades. Introduced in 1924, about one year before the formulation of Quantum Mechanics by Heisenberg and Schödinger1 , this concept has seen contribution of many of the most brilliant and famous minds of the last century. The list of their names is impressive: starting from the first two contributors, S. N. Bose and the universally known A. Einstein, it protracts and includes sacred monsters as F. London and L. Tisza, L. D. Landau and N. N. Bogoliubov, O. Penrose and L. Onsager, R. P. Feynman and many others2 . In this chapter we briefly review the history and development of Bose-Einstein condensation and introduce the main equation governing a condensate in the limit of zero temperature, in weakly interacting Bose gases: the Gross-Pitaevskii equation. We then study in particular the solitonic solution, and its properties, which will be of great use in the rest of the work. Sections 1.2 and 1.3 are reported mainly from [6] and [7].

1.1

Brief history of Bose-Einstein condensation

At first, Satyendra Bose. The brilliant Indian physicist, being not able to convince any European journal to publish his work on the quantum statistics of photons, sent the English manuscript to his idol, Albert Einstein. He remained so impressed by that work to translate it from English to German and to submit for Bose to the “Zeitschrift f¨ ur Physik”, at the time the most important journal for physics. He then extended Bose’s 1

E. Schr¨ odinger first heard about the de Broglie hypothesis by reading Einstein’s paper on BEC. We might daringly say it was Bose-Einstein condensation to push the concept of wavefunction, and not the opposite! 2 A short review of the history of Bose-Einstein condensation can be found in [5].

8

1.1. Brief history of Bose-Einstein condensation

9

ideas for photons so to include also the concepts just developed by a young de Broglie in its PhD thesis: particles may behave as waves, and so they could be governed by the same statistics. One of the consequences is that indistinguishable particles obeying Bose statistics, which would later been called bosons, may share the same quantum state. Einstein suggested that a very low temperature could cause them to “condense” into the lowest accessible quantum state, and to realise a new form of matter. It was a revolutionary idea, but difficult to digest. The paper was criticised and almost forgotten until 1938, when Fritz London brought new light in the field, by looking at the possibility to explain the new fascinating phenomenon of superfluidity. He proposed the idea that somehow BEC was involved in the phase transition showed by superfluid 4 He, and that Einstein’s relation for T BEC was a good estimate of the observed transition temperature. Lazlo Tisza, after a long chat with London, in one night developed the famous two-fluid model for superfluidity, stating that since a BEC acts like a new collective degree of freedom, it could move Bose, Einstein and the condensation. without friction hence causing superfluidity. This model was further developed by Lev Landau and Isaak Khalatnikov, who also introduced the idea that superfluid liquid could be described in terms of a gas of weakly interacting quasiparticles. Nikolay Bogoliubov later developed a formal explanation of Landau’s conjectures, and lead off the golden period of the rush for the theory of interacting Bose gas. This period culminated in 1961 with the independent elaboration by Eugene Gross and Lev Pitaevskii of the Gross-Pitaevskii equation (GPE), the key equation for Bose-Einstein Condensation. On the experimental point of view, many years and the development of smart techniques3 were necessary for the first experimental realisation of a BEC. For more than 70 years, what prevented experimentalists to get condensation was mainly the difficulty to reach the very low temperature involved in the process. The twenty-year work of S. Chu, C. Cohen-Tannoudji and W. Phillips on laser cooling, which eventually earned them a Nobel Prize in 1997, was a fundamental step in the survey. The final outburst arrived in 1995, when E. Cornell and C. Wieman were lastly able to produce the first Bose-Einstein condensate at JILA lab. Their discovery was rewarded in 2001 with another Nobel Prize, together with W. Ketterle. From that moment an enormous variety of experiments and hypotheses has been made, and new amazing properties of this intriguing new phase of matter have been discovered and proven4 . That path still continues nowadays, with an increasing number of groups working on condensates and on their possible use in connection with other fields of physics. 3

The most common of these techniques are explained in Chapter 4. Among these, the possibility to realise atom lasers, exploiting the coherent nature of matter inside a BEC. 4

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Chapter 1. BEC, GPE and solitons

1.2

The Gross-Pitaevskii equation

The Gross-Pitaevskii equation is a fascinating tool to describe a dilute quantum system of bosonic particles at low temperature. It has been independently developed by Eugene P. Gross5 and Lev P. Pitaevskii6 in 1961, and is a masterpiece of simplicity and elegance. The treatment reported here is brief and incomplete, and a much better derivation can be found in [10]. We make use of some of the concepts developed in appendix A. Let us consider a gas of N bosons interacting via a two-body isotropic poGross and Pitaevskii tential V2 . Being isotropic it only depends on the distance among the particles V (r1 − r2 ) = V (|r1 − r2 |). In terms of the matrix elements of the two-particle operator in the coordinate space, we require ⟨r1 r2 | V2 |r′ 1 r′ 2 ⟩ = δ(r1 − r′ 1 )δ(r2 − r′ 2 )V (|r1 − r2 |) .

(1.1)

The representation in the Fock space of the interaction potential operator is then V =

1 2

∫

∫

dr

ˆ † (r, t)Ψ ˆ † (r′ , t)V (|r − r′ |)Ψ(r ˆ ′ , t)Ψ(r, ˆ t) dr′ Ψ

(1.2)

Then the time-dependent Hamiltonian in coordinate representation is written in the second quantisation formalism as ˆ H(t) = 1 + 2

∫

∫

)

(

2 2 ˆ t) ˆ † (r, t) ℏ ∇ + V (r, t) Ψ(r, drΨ 2m

∫

dr

(1.3)

ˆ † (r, t)Ψ ˆ † (r′ , t)V (|r − r′ |)Ψ(r ˆ ′ , t)Ψ(r, ˆ t) dr′ Ψ

If the gas is enough dilute, i. e. when the diluteness condition n|a3 | ≪ 1

(1.4)

holds (here n = N/V is the gas density, and a is the s-wave scattering length), atomic interactions are dominated by low energy two-body s-wave collisions. These are essentially elastic hard-sphere collisions between two atoms, and can be modelled in terms of the contact potential V (|r − r′ |) = g δ(r − r′ ) , (1.5) where g is the coupling constant which may be expressed in terms of the s-wave scat5 6

Structure of a quantised vortex in boson systems, Il Nuovo Cimento, 1961 [8]. Vortex lines in an imperfect Bose gas, Soviet Physics JETP-USSR, 1961 [9].

11

1.2. The Gross-Pitaevskii equation

tering length a as g=

4πℏ2 a . m

(1.6)

Then we integrate in r′ and get ˆ H(t) =

∫

ˆ † (r, t)H ˆ t) + g ˆ 0 Ψ(r, drΨ 2

ˆ0 = where we defined H

(

ℏ2 ∇2 2m

∫

ˆ † (r, t)Ψ ˆ † (r, t)Ψ(r, ˆ t)Ψ(r, ˆ t), drΨ

(1.7)

)

+ V (r, t) .

ˆ t) is determined by the Heisenberg’s equaThe time evolution of the operator Ψ(r, tion ] ˆ t) [ ∂ Ψ(r, ˆ t), H(t) ˆ iℏ = Ψ(r, . (1.8) ∂t Equation (1.8) can be solved by using the commutation relations (A.33): ∫

ˆ t) ∂ Ψ(r, ˆ − dr′ Ψ ˆ t)H(t) ˆ † (r′ , t)H ˆ ′ , t)Ψ(r, ˆ t) ˆ 0 Ψ(r = Ψ(r, iℏ ∂t ∫ g ˆ † (r′ , t)Ψ ˆ † (r′ , t)Ψ(r ˆ ′ , t)Ψ(r ˆ ′ , t)Ψ(r, ˆ t) − dr′ Ψ 2 ∫ ( ) ˆ − dr′ Ψ(r, ˆ t)H(t) ˆ t)Ψ ˆ † (r′ , t) − δ(r − r′ ) H ˆ ′ , t) ˆ 0 Ψ(r = Ψ(r, − (

=

g 2

∫

(

)

ˆ t)Ψ ˆ † (r′ , t)Ψ ˆ † (r′ , t)Ψ(r ˆ ′ , t)Ψ(r ˆ ′ , t) − 2δ(r − r′ ) × dr′ Ψ(r, ˆ † (r′ , t)Ψ ˆ † (r′ , t)Ψ ˆ † (r′ , t) ×Ψ )

ℏ2 ∇2 ˆ t) + g Ψ ˆ † (r, t)Ψ(r, ˆ t)Ψ(r, ˆ t). + V (r, t) Ψ(r, 2m

(1.9)

The great peculiarity of the Bose-Einstein condensation is the macroscopic occupation of a single-particle state of the system. It is then appropriate to decompose the Bose ˆ t)⟩ and a field operator in terms of a highly populated mean field term ψ(r, t) = ⟨Ψ(r, ˆ t): fluctuation term δ Ψ(r, ˆ t) = ψ(r, t) + δ Ψ(r, ˆ t) Ψ(r, (1.10) At T = 0 the lowest energy state is macroscopically occupied, and we can consider fluctuations to be small. Equation (1.9) becomes: ∂ iℏ ψ(r, t) = ∂t

(

)

ℏ2 ∇2 − + V (r, t) + g|ψ(r, t)|2 ψ(r, t). 2m

(1.11)

which is finally the Gross-Pitaevskii equation (GPE). The corresponding energy at time t can be computed applying (1.10) to the Hamiltonian (1.7), and is ∫

Eψ (t) =

(

ℏ2 g dr − |∇ψ(r, t)|2 + V (r, t)|ψ(r, t)|2 + |ψ(r, t)|4 2m 2

)

.

(1.12)

12


In case of time-independent potentials V (r, t) = V (r), GPE admits stationary solutions, hence wave functions for which |ψ(r, t)|2 = const. The condensate wave function evolves thus in time according to µ

ψ(r, t) = ψ0 (r)e−i ℏ t ,

(1.13)

and the GPE reduces to (

1.3 1.3.1

)

ℏ2 ∇2 − + V (r) − µ + g|ψ0 (r)|2 ψ0 (r) = 0. 2m

(1.14)

Solitonic solution for the GPE What is a soliton?

Up to now we have filled the pond of our theoretical background with a good amount of notions, and we are able to splash around with confidence. Nevertheless, one crucial topic is still missing: the soliton. In physics, solitons are self-reinforcing solitary waves, which maintain their shape while travelling at constant speed. The three properties usually ascribed to them are: • their form is constant; • they are spatially localised; • when they collide with other solitons, they emerge substantially unchanged, except for a phase shift. In particular, the shape-conserving property of solitons may be ascribed to the delicate balance between the “linear” dispersing and the “nonlinear” pumping effect in the medium. They appear in different fields, as in fluid mechanics, optics, biophysics and atomic physics7 . For example, optical solitons may exist in an optical device where the diffraction is balanced by the Kerr effect8 , or can be found by studying the lowfrequency collective motion in proteins and DNA. As we will see they also exist in atomic physics: in particular we will analyse the solitonic solution for the 1D GrossPitaevskii equation, and some of its intriguing properties that will have an important use in the following.

7

A noticeable series of examples, and a general discussion about solitons, could be found in [11] Kerr effect: the refractive index of a material changes in response to an applied electric field, and in particular it depends on the square modulus (intensity) of the field. 8

1.3. Solitonic solution for the GPE

1.3.2

13

Russell, the Wave of Translation and the KdV equation

In 1834 a young Scottish civil engineer called John Scott Russell was hired for a summer job to experiment the most efficient design for canal boats, particularly for the Union Canal near Edinburgh. He passed some months observing the barges, which were usually pulled by horses or mules from the shore. One August day the rope broke and the barge suddenly stopped. In its own words ([13]): I was observing the motion of a boat which was rapidly drawn along a narrow channel by a pair of horses, when the boat suddenly stopped - not so the mass of water in the channel which it had put in motion; it accumulated round the prow of the vessel in a state of violent agitation, then suddenly leaving it behind, rolled forward with great velocity, assuming the form of a large solitary elevation, a rounded, smooth and well-defined heap of water, which continued its course along the channel apparently without change of form or diminution of speed. I followed it on horseback, and overtook it still rolling on at a rate of some eight or nine miles Russell an hour [14 km/h], preserving its original figure some thirty feet [9 m] long and a foot to a foot and a half [300-450 mm] in height. Its height gradually diminished, and after a chase of one or two miles [2-3 km] I lost it in the windings of the channel. Such, in the month of August 1834, was my first chance interview with that singular and beautiful phenomenon which I have called the Wave of Translation. Russell then reproduced this phenomenon in several controlled laboratory experiments using a wave tank, and was able to catch the main features of solitons ([12]). In 1895 the Dutch physicist Diederick Korteweg and his student Gustav de Vries derived and equation which could describe Russell’s experiment, the so-called KortewegDe Vries (KdV) equation ∂t ϕ + ∂x3 ϕ + 6ϕ∂x ϕ = 0 . (1.15) Equation (1.15) shows that the evolution in time of the wave’s height is driven by the sum of two terms: one is dispersing (∂x3 ϕ), while the other is reinforcing (ϕ∂x ϕ). The authors find a periodic solution and a solitary wave solution sharing the same properties of Russel’s wave of translation. Their work was somehow forgotten until 1965, when Norman Zabusky and Martin Kruskal realised a numerical solution of the KdV equation. This awakened a lot of successive studies, including the 1973 demonstration of the integrability and the existence of solitonic solution for the nonlinear Schrödinger equation, performed by Vladimir Zakharov and Alexei B. Shabat.

14


Figure 1.1: Russell’s original drawing about the wave of translation, from [13].

1.3.3

Solitonic solution of the Gross-Pitaevskii equation

As already explained, the solitonic solution corresponds to a localised modulation of the density profile, moving in the medium at constant velocity. The particularity of such a solution is that its shape is preserved in time. Since the perturbation in density is actually a (in general not complete) depletion, they are usually referred to as grey solitons. Their existence is directly related to the nonlinearity of the GPE, which compensates the dispersive quantum pressure term arising from the kinetic energy. Let us focus on the 1D homogeneous case (V (x) = 0), and consider a travelling solution with constant speed v. At a distance far from the perturbation one is allowed to suppose the gas density to approach its unperturbed uniform value n: |ψ0 | → n. This gives to the patient reader a suggestion: it is useful to introduce a dimensionless function f (x, ξ) such that: √ ψ0 (x) = nf (x, ξ), (1.16) ℏ being ξ = √2mng the so-called condensate healing length. Because of the previous observation, f has the two constraints

lim f (x) = 1

x→±∞

and

lim ∂x f (x) = 0.

x→±∞

(1.17)

15


Hence, given the fact that our solution moves at constant speed, say v, we can rewrite ψ(x, t) =

√ µ nf (ζ)e−i ℏ t ,

(1.18)

in which we defined the dimensionless ζ = x−vt ξ . Then, we can substitute (1.18) into (1.11) and get 2iU

∂f ∂2f + (1 − |f |2 )f, = ∂ζ ∂ζ 2

(1.19)

where we used dt = − vξ dζ and imposed µ = ng and U≡ being c ≡ density n.

√

ng m

mvξ v =√ ℏ 2

√

m v =√ , ng 2c

(1.20)

the sound speed predicted by GP theory for a uniform condensate of

And now some boring calculations. Multiply (1.19) by f ∗ and subtract its complex conjugate. You should get (

2iU

∂f ∗ ∂f ∗ f + f ∂ζ ∂ζ

)

−

∂2f ∗ ∂2f ∗ f + f = 0. ∂ζ 2 ∂ζ 2

(1.21)

Now, remembering the Leibniz rule for the derivative of a product, a lot of terms cancel out, and one remains with ∂ ∂ζ

(

2iU |f |2 −

∂f ∗ ∂f ∗ f + f ∂ζ ∂ζ

)

= 0,

(1.22)

which means the parenthesis to be constant. By imposing this constant to be 2iU one finally writes ∂f ∗ ∂f ∗ 2iU (1 − |f |2 ) + f − f = 0. (1.23) ∂ζ ∂ζ Let us put this equation apart for a moment, and search for another equation for f . Since f is in general a complex function we can divide it f = fR + ifI . By plugging it into equation (1.19), and by taking just the imaginary part of the result, we obtain: 2U

∂fR ∂ 2 fI = + (1 − fR2 − fI2 )fI . ∂ζ ∂ζ 2

(1.24)

Equation (1.24) can be simplified by considering solutions for which fI is constant, hence ∂fR = (1 − fR2 − fI2 )fI . (1.25) 2U ∂ζ

16


Now, equations (1.25) and (1.23) coincide if we impose fI = Equation (1.25) becomes

√ 2U = v/c.

(1.26)

√ ∂fR v2 2 = 1 − fR2 − 2 . ∂ζ c

(1.27)

By integrating this last equation one gets the relation for the real part of f : 

√

fR (ζ) =



√

v2 ζ 1 − 2 tanh  √ c 2

v2 1 − 2 . c

(1.28)

We are finally able to write the solitonic solution 

√ v ψS (x − vt) = n i + c

√



v2 x − vt 1 − 2 tanh  √ c 2ξ



√

µ v2 1 − 2  e−i ℏ t . c

(1.29)

Let us now put a sense in this effort and comment some aspects of our new friend. Note that the density profile n(x − vt) = |ψ0 (x − vt)|2 has a minimum in the centre of the soliton corresponding to n(0) = nv 2 /c2 . This value drops to zero just in the case of a soliton with zero velocity, hence not moving at all. This kind of solitons, having a complete density depletion in the centre, are evocatively called dark solitons. The width of the soliton is determined by the healing length ξ, but is amplified by a factor (1 − v 2 /c2 )−1/2 , which is increasingly large as v → c. Phase and energy of the soliton. It is worth to analyse the phase jump of a soliton on the two sides of the density depletion9 . Since the real part of ψ0 (ζ) is proportional to tanh (ζ), then Re(z) ≥ 0 ⇔ ζ ≥ 0. The phase jump is ∆θ(ψ0 ) = lim θ(ψ0 ) − lim θ(ψ0 ) ζ→−∞

(

= π − 2 arctan

ζ→+∞

)

v/c √ . 1 − v 2 /c2

(1.30)

Using the properties of the trigonometric functions tan(arcsin(x)) = 9

x 1 − x2

and

arcsin(x) = π/2 − arccos(x),

The phase of a complex number z = Re(z) + iIm(z) is defined as

( θ(z) = arctan

Im(z) Re(z)

(

= π + arctan

)

Im(z) Re(z)

if Re(z) ≥ 0,

) if Re(z) < 0.

The addition of the factor π if Re(z) < 0 is necessary since the function arctan (x) lies between (−π/2, π/2). This would cause the lost of information about the semi-plane in which the complex number lies.

17


1.2 1 0.8 0.6 0.4 t = −14

0.2 0 -10

-8

-6

t = −7

-4

t=0

-2

t=7

0

2

4

6

x Figure 1.2: Time evolution of the square modulus of a grey soliton with v = 0.3c.

1.2 1 0.8 v = 0.8 v = 0.6c v = 0.4c v = 0.2c v=0

0.6 0.4 0.2 0 -6

-4

-2

0

2

4

6

x Figure 1.3: Soliton turning from grey to dark with decreasing v.

18


we get:

( )

v . c

∆θ(ψ0 ) = 2 arccos

(1.31)

Then, for a dark soliton, the wave function √ ψDS (x) = n tanh

(

x √ 2ξ

)

(1.32)

is real and odd, and the phase jump is π. Another interesting aspect to investigate is the soliton energy. In [6], the energy ϵ of the soliton was computed to be the difference between the grand canonical energies in the presence and in the absence of the soliton ∫

ϵ=

∞

−∞

(

)

ℏ2 dψ0 2 g + (|ψ0 |2 − n2 ) 2m dx 2

and the result is

(

4 v2 ϵ = ℏcn 1 − 2 3 c

dx,

(1.33)

)3/2

.

(1.34)

Equation (1.34) is very intriguing. One may note, for instance, that taking a small value for the velocity v leads to: 1 ℏn 2 4 v . ϵ = ℏcn − 3 2 c

(1.35)

The second term of equation (1.35) tells us that the soliton behaves as a particle of negative mass10 ms = − ℏn c . This may appear surprising, but it is clear when one observes the density perturbation to be a hole rather than a particle. Moreover, being a (quasi)particle with negative kinetic energy causes its energy to decrease when its velocity increase. Any dissipative effect, like collisions with thermal excitations, will then result in an acceleration of the soliton, leading eventually to its disappearance for v → c.

1.3.4

Soliton dynamics in an inhomogeneous condensate

In the previous subsection we treated the theory of the solitonic solution in a homogeneous condensate, hence in case V (x) = 0. In our numerical simulation, however, we want to work with condensates being somehow trapped, therefore we have to deal with the inhomogeneous case as well. We will see that by using some approximations, we can use again some of the relations already computed. Let us suppose at first that the condensate length, which we call 2L, is much larger than the width of the soliton: L ≫ l0 = 10

√ ℏ 2ξ = √ . mng

This relation will be also derived later in equation (1.46).

(1.36)

19


For such a large condensate one could work in the frame of the semiclassical Landau dynamics for the superfluidity, where the quantity (1.35) plays the role of the quasiparticle’s Hamiltonian. The requirement (1.36) also allows to consider the motion of the condensate as a whole, induced by the soliton oscillations, as much slower than the soliton itself, hence the condensate as being at rest. Given (1.36), we can use a local density approximation: we assume equation (1.34), which was computed for the homogeneous case, to be also valid for the inhomogeneous case V (x) ̸= 0, provided the substitution of c by its local value c(X), being X the position of the centre of the soliton. In the first approximation, then, the soliton wave function remains the same, with the only precaution to consider X and v as functions of time and connected by dX(t) dt = v(t). The soliton dynamics is consequently governed by the energy conservation relation (1.34), which can be rewritten as ϵ=

4 ℏm (c(X)2 − v 2 )3/2 = const, 3 g

(1.37)

where µ = gn = mc2 . The last equation also implies (

u2 ≡ c(X)2 − v 2 = c(X)2 −

dX dt

)2

to be constant.

(1.38)

In order to find the relation for c(X) we can also approximate the density by assuming that the kinetic energy term (quantum pressure) in the GP energy functional (1.12), proportional to |∇Ψ|2 , is negligible compared to the mean field energy density. This corresponds to neglecting the first term in the GPE (1.14) so that the density becomes: nTF (x) =

µ − V (x) , g

(1.39)

This is known as Thomas-Fermi approximation. Substituting (1.39) into the definition for c(X) we find: µ − V (X) V (X) gnTF c(X)2 = = = c20 − , (1.40) m m m √

where c0 = µ/m is the sound speed at X = 0. By substituting (1.40) into (1.38) one finally finds (

m

dX dt

)2

+ V (X) = m(c20 − u2 ).

(1.41)

Equation (1.41) describes the classical motion of a particle having mass11 2m and energy m(c20 − u2 ) in the potential V (X). Let us now finally consider the interesting case of an harmonic trapping of the 2 x2 . The distribution of the sound velocity is c2 (X) = c2 − condensate V (x) = 21 mωho 0 11 Given the fact that the soliton energy decreases while its velocity increases, it would be more correct to say that it behaves like a particle of mass −2m moving in a potential −V (X). Note, moreover, that the “effective mass” −2m is not the “physical mass”.

20

1 2 2 2 ωho X ,


and the equation of motion (1.41) takes the form (

dX m dt

)2

=

ω2 µ − u2 − ho X 2 . m 2

(1.42)

Equation (1.42) describes a harmonic oscillation with frequency of ωho ωsol = √ . 2

(1.43)

The same results can be found with a different approach in [14], and represents a very important feature of soliton motion, due to its peculiarity. Some final considerations on the solitons may be done. At first, one could ask himself what the amplitude of the oscillation is. It can be very easily computed by dX searching for the turning point coordinate XT , hence the one for which dt = 0: XT

XT =

2 2 ωho

(

)

µ 2u2 − u2 = L2 − 2 , m ωho

(1.44)

2µ where L = mω 2 is the Thomas Fermi radius. The soliton then oscillates between ±XT , ho where its velocity and central density drop to zero. The grey soliton becomes dark, and due to energy conservation (1.37) it bounces, becoming grey again. Another characteristic of solitons is the propagation without change in the density profile. Indeed, by looking at the density perturbation near to the soliton:

δn(x) = |ψS (x)|2 − n (

=n 1−

v2 c2

   √ ) 2 x−X v tanh2  1 −  − 1 (

=

c2

l0

(1.45)

)

mu2 (x − X)u cosh−2 . g l0 c

This quantity does not depend on time for a given x − X, because thanks to equation (1.38), u is an integral of motion. It is also useful to compute the solitonic “physical mass”. As already said, a soliton is a region of decreased density, and so we may suppose it to contain a negative “number of atoms” ∫

NS (ψ) =

+∞ −∞

δn(x) dx = −

2ℏ u. g

(1.46)

Thus the atom depletion (1.46) only depends on u, being itself an integral of motion. Hence a soliton is a quasiparticle of constant negative mass mNS , which depends on the energy of the system and is not equivalent to the “effective mass” 2m.

Chapter 2

Numerical solution of the GPE The Gross-Pitaevskii equation (1.11) is a particular nonlinear partial differential equation. One easy and effective numerical way to find the solution of such an equation, and in general of nonlinear Schrödinger equations (NLSE), is the so called Split-step Fourier method, which will here be presented. The name is explicative, since the method consist in splitting the ”kinetic” and the ”potential” term (which includes the non linear one) of the time evolution, and to propagate the system separately. In particular it results useful to work in the Fourier space during the kinetic evolution. The Fourier transform is achieved by using the known FFT algorithm, allowing this method to be of noticeable efficiency. The most complete source about it is [15]. A good overview of such a method, including a parallel implementation, can be found in [16], while an accurate evaluation of the convergence is reported in [17].

2.1

Preliminaries

Let us recall the 1D GPE (1.11): ∂ iℏ ψ(x, t) = ∂t

(

)

ℏ2 ∇2 − + Vex (x, t) + g|ψ(x, t)|2 ψ(x, t) 2m

(2.1)

This equation is a particular case of the so called nonlinear Scrödinger equation, which assumes general form: iut = −Juxx + V u + q|u|2 u (2.2) where u is a complex function and q is a real number.

2.1.1

The Split-Step Fourier method

Consider the general evolution equation of the form: ut = (T + V)u

(2.3)

u(x, 0) = u0 (x)

(2.4)

21

22

Chapter 2. Numerical solution of the GPE

In the case of equation (2.2) one can write: T u = iJuxx

Vu = −i(V + q|u|2 )u

(2.5)

The solution of the equation (2.3) may be propagated from one time level to the next by means of u(x, t + ϵ) = exp {ϵ(T + V)}u(x, t) (2.6) where ϵ denotes the time step. In general equation (2.6) is first-order accurate, but it is exact in the particular case in which the operators T and V are time independent (as explained in [15]). The time-splitting procedure now consists in replacing the right-hand side of equation (2.6) by an appropriate combination of products of the exponential operators exp (ϵT ) and exp (ϵV). For this purpose is very useful to consider the Baker-CampbellHausdorff (BCH) formula for two operators A and B, id est: (

exp (λA) exp (λB) = exp

∞ ∑

) n

λ Zn

(2.7)

n=1

where Z1 = A + B and the remaining operators Zn are commutators of A and B, commutators of A and [A, B], etc. They could be very complicated, for example: 1 Z2 = [A, B] , 2 where [A, B] = AB − BA is the commutator of A and B, and Z3 =

1 ([A, [A, B]] + [[A, B], B]) . 12

Using equation (2.7), one get the first-order approximation of the exponential operator in equation (2.6) as follows: exp {ϵ(T + V)} = exp (ϵT ) exp (ϵV) + O(ϵ2 )

(2.8)

Note, of course, that this equation is exact whenever T and V commute. At this point, the split step method consist in separating the advancement in time into two steps: first by solving u′t = T u (2.9) and then advancing the solution by solving ut = Vu′ considering as initial condition of the latter the solution of the former.

(2.10)

23

2.1. Preliminaries

Note that in equation (2.8) the operators T and V may be interchanged without interchanging the order of the method.

2.1.2

The Discrete Fourier Transform

Consider a sequence {fs } of length M , obtained by taking samples of a continuous function f at equal intervals. Then the discrete Fourier transform (DFT) is the sequence f˜k given by f˜k =

M −1 ∑

2π

fs ei M ks ,

0≤k 1). 2

See [19] for a rigorous treatment of the optimal cut-off determination.

40

Chapter 3. Classical Fields Approximation

This leads to a double consideration. On the one hand, the occupation numbers may be considered continuous, leading ultimately to the equipartition of energy in every single-particle mode: ⟨nj ⟩ϵj = KB T , where ⟨nj ⟩ is the mean occupation of the singleparticle state of energy ϵj . When assuming an infinite number of accessible modes, then, this leads to an ultraviolet divergence analogous to the one in the famous blackbody problem. On the other hand, one can assume the (finite number of) most relevant modes to carry all the information about the thermal excitation, and so consider the population of the higher energy state to be zero. These two considerations suggest one to introduce an artificial cut-off in the theory, namely Kmax in relation (3.7). The finiteness of the expansion (3.7) is a highly desirable achievement when treating the problem numerically, given the very finite typical lifetime of the researchers. This aspect, however, requires a lot of attention and discussion, since the cut-off choice may affect the statistical properties of the simulated system. In [19] one can find an extended rigorous treatment of the Kmax determination for the cases faced in this work. The associate energy functional is then determined by equation (1.12) to be: ∫

Eψ =

[

]

pˆ2 g dxψ (x) + V (x) ψ(x) + 2m 2 ∗

∫

dx|ψ(x)|4 .

(3.8)

Considering this approximation, and working in the canonical ensemble, the classical probability distribution3 of finding the system in a given configuration of the mode amplitudes αk is: 1 P (αk ) = exp [−βEψ ] , (3.9) Z ∑

where Z = {αk } exp [−βEψ ] is the canonical partition function and β = 1/kB T . Note that the condition of a fixed number of atoms N transforms in both the cases treated in this work and presented later in a constraint on the amplitudes: ∑

|αk |2 = N.

(3.10)

|k|≤Kmax

The crucial introduction of the finite temperature T is then accomplished in two ways: in the Boltzmann factor e−βEψ inside the probability distribution and in the cut-off Kmax .

3.2

The method

In order to generate the canonical ensemble we must then sample the phase space of the classical amplitudes αk . The first guess could be to sample the entire space uniformly, but this turns out to be extremely inefficient. The question is then how to sample only the relevant portions of the phase space. A very efficient answer is to use the Monte Carlo method to generate a Markovian process of a random walk inside the phase space. All states (i.e. all configurations αk ) “visited” during this walk become members of the statistical ensemble, and therefore are used in the ensemble averages. 3

In a strict sense, this is only exact for non-interacting particles. In the following we will postulate this to be valid also for a weakly interacting Bose gas, given the correct choice of the cutoff Kmax .

41

3.2. The method

The Metropolis algorithm is a very simple way to realise this purpose. For each observable O one can define the statistical average as: N 1 ∑ ⟨ψs |O|ψs ⟩. ⟨O⟩ = N j=1

(3.11)

One is then required to generate N samples of the classical field ψ(x), which means (s) N different configurations {αk }N s=1 . The canonical average is obtained in the limit of infinite Markov chain N → ∞, if the members of the ensemble with energy Eψ are proportional to the Boltzmann factor. The single step s of the random walk is defined: (s)

1. Each set of amplitudes {αk } uniquely determines a state ψs (x), selected to be a member of the canonical ensemble at the s-th step of the walk. Its energy Eψs is computed according to (3.8). As initial configuration s = 1 one can choose any state satisfying the condition (3.10). (s)

(s)

(s)

(s)

2. One generates a trial new configuration {˜ αk } = {αk + δk }, where δk is a random disturbance. The new configuration has to be normalised according to ˜˜ . (3.10), and generates a trial classical field ψ˜s and the corresponding energy E ψs ˜ ˜ − Eψ and At this point one is able to compute the energy difference ∆s = E s ψs the Boltzmann factor ps = e−β∆s . 3. The new member of the walk is then selected according to: • if ∆s < 0 the trial state is accepted as new member of the ensemble, and so (s+1) (s) {αk } = {˜ αk }. • if ∆s > 0 a number 0 < u < 1 is randomly generated, and compared to (s+1) (s) ps : if ps > u the trial state is accepted, so {αk } = {˜ αk }; if, instead, (s) ps < u the disturbance is refused, and the “initial” state {αk } is once more (s+1) (s) included in the ensemble, so {αk } = {αk }. The Metropolis algorithm selects the states which minimise the energy functional. Without the apparently involved procedure of step 3 one could fall in a sneaky trick: a local minimum of the energy would keep the walk blocked, avoiding the exploration of relevant portions of the phase space. The choice of the maximal value Dk of the disturbance δk deserves some discussion. At first, the value of Dk should depend on k = |k|. Picking it uniformly, in fact, would produce weird results, since the low-k mode displacements change the energy of the system much less than the high-k ones. This would lead to a high refuse rate of any displacement in the most energetic αk , letting them substantially unvaried with respect to the initial configuration and introducing a sort of

42


“effective cut-off” which is lower than the desired one. In our computations, hence, we considered ( ) k δk = Dk rk where Dk = 1 − 0.99 h Kmax (3.12) rk ∈ [0, 1] is a random number, h is arbitrary. Another aspect has to be considered: the procedure convergence is much faster if the acceptance ratio, defined as the ratio between the accepted displacements and the total trials N , is approximately of 0.5. This quantity depends on Dk , which has to be set accordingly by varying the value of h. It is also useful to discard some number of the initial members of the ensemble, so to avoid any correlation with the arbitrarily selected initial configuration. Using this method one is able to compute mean values of observables. In the noninteracting case (g = 0), an interesting quantity to be evaluated is the probability distribution of having Nex excited atoms. This can be easily computed by considering the condition (3.10), and that, for each Markov step s: (s)

(s) Nex = N − N0

(3.13)

where (s)

N0

(s)

= |α0 |2 .

4

(3.14)

By piling these results in a histogram one is able to produce the probability distribution.

3.3

Non-interacting Bose gas

The first system one could analyse is the ideal gas, hence the case in which g is set to zero in (3.1) and (3.8). In this section we show some results for the excitation probability distribution for both a box and a harmonic confinement, and the evolution in time of a non interacting gas.

3.3.1

Box confinement

As first result we considered a ideal Bose gas in a 1D box confinement of length L. In this case it is useful to perform the expansion (3.7) in terms of the plane wave basis ψ(x) =

∑

1 αk √ eikx . L |k|≤Kmax

(3.15)

where the wave vectors are k= 4

2π n, L

n = −nmax , ..., +nmax .

(3.16)

Note this relation to be only valid if the system is non-interacting. The correct estimation for the g ̸= 0 condensed fraction of atoms is explained later.

43

3.3. Non-interacting Bose gas

The energy functional is given by equation (1.12), and is ∫

ℏ2 Eψ = − dxψ ∗ (x)∇2 ψ(x) 2m V =L ) ( n max ∑ (2π)2 ℏ2 2 2 |αn | n , = 2 2mL2 n=0

(3.17)

where we used the integral representation of the Kronecker delta δnm =

1 L

∫

dxei(n−m)x .

(3.18)

We were then able to produce the probability distribution of having Nex excited atoms for different temperatures. These results can be seen in Figure 3.1. The optimal cut-off choice was done accordingly to [4], and, for a 1D geometry was 2 n2max = Kmax /(2π/L)2 = 0.58/β˜

(3.19)

ℏ2 where5 β˜ = (2π)2 mL 2 β.

In [4] a different definition for β˜ is reported , which however is inconsistent with the calculations and with the theoretical result in [19]. 5

44


K=4 K=5 K=8

0.025

P(Nex )

0.02 0.015 0.01 0.005 0 0

20

40

60

80

100

Nex

(a) Our results

(b) Results by Witkowska et al.

Figure 3.1: (3.1a)Probability distribution of the number of excited atoms Nex for a non-interacting gas in a box. The dimensionless parameter β = (ℏ2 /2mL2 )/KB T represents the inverse temperature of the system. The optimal cut-off used is 2 /(2π/L)2 = 0.58/β, hence, respectively K Kmax max /(2π/L) = 4 for β = 0.036 (full circle), Kmax /(2π/L) = 5 for β = 0.023 (full squares) and Kmax /(2π/L) = 8 for β = 0.009 (full triangles). The parameters used in the simulation are number of atoms N = 100 and number of Monte Carlo steps N = 3 · 106 . (3.1b) Results with the same parameters obtained by Witkowska et al. in [4].

45


3.3.2

Harmonic confinement

After this first test we passed to consider an ideal Bose gas of particles with mass m confined in a 1D harmonic confinement of pulsation ω. The distances √ are conveniently ℏ expressed in units of the harmonic oscillator natural length aHO = mω . This time we performed the expansion (3.7) in terms of the well known quantum harmonic oscillator eigenfunctions: ∑

ψ(x) =

αk ϕHO k (x)

k = 0, 1, 2, ..., Kmax ,

(3.20)

k≤Kmax

where L is the selected volume of the space, chosen6 in order to be L ≫ aHO , and the ϕHO k (x) are k 1 2 2 k√ x2 /2a2ho d e−x /aho (3.21) ϕHO (x) = (−1) e k √ k dx 2k k!aho π obeying the orthonormality condition ∫ HO dxϕHO k (x)ϕq (x) = δkq .

(3.22)

The energy functional can hence be rewritten as ∫

Eψ =

]

[

ℏ2 ∇2 1 + mω 2 x2 ψ(x) dxψ ∗ (x) − 2m 2

= ℏω

∑

|αk |2 k.

(3.23)

k≤Kmax

The optimal cut-off was chosen according to [19] to be Kmax =

1 ℏωβ

(3.24)

for a 1D geometry. The results are reported in Figure (3.2).

6

In computations the space interval has to be segmented in a number M of steps d, in order to treat both discredited versions of functions and procedures (see subsection 2.1.3). The quantity M has to be chosen appropriately, so to allow d ≪ min{aHO , λth }, where λth is the de Broglie wavelength. In section 3.4 we will also have to consider interactions in the determination of L, which will have to be L ≫ max{aHO , RTF }, where RTF is the Thomas-Fermi radius.

46


P(Nex )

0.015

K=25

0.01

0.005

0 50

100 Nex

150

200

(a) Our results

(b) Results by Witkowska et al.

Figure 3.2: (3.2a)Probability distribution of the number of excited atoms Nex . The dimensionless parameter β = ℏω/KB T represents the inverse temperature of the system. The optimal cut-off used is βKmax = 1, hence, since β = 0.04 is Kmax = 25. The parameters used in the simulation are number of atoms N = 1000 and number of Monte Carlo steps N = 3 · 106 . (3.2b) Results with the same parameters obtained by Witkowska et al. in [19].

47


3.3.3

Schr¨ odinger evolution

An interesting aspect to be investigated is the g = 0 evolution in time, according to the Split-Step method explained in Chapter 2. An analysis of the results at the end of this section shows the presence of some kind of oscillations even in the non-interacting regime, where no solitons may exist. In fact they are not solitons. Looking at the period of these oscillations, one could observe it to be of order of 2π (in units of the inverse of the harmonic trap frequency ωHO ), and so: Tosc = 2π

1 ωHO

≡ THO

(3.25)

One could explain the origin of such fluctuations by mean of a simple reasoning: the finiteness of the cut-off introduces a dominant period in the system. In fact, if one considers the Schrödinger equation ∂ iℏ ψ(x, t) = Hψ(x, t) = ∂t

(

)

ℏ2 2 ∇ + VHO (x) ψ(x, t), 2m

(3.26)

for the (finite) decomposition in harmonic oscillator eigenstates (3.20) ψ(x, t) =

∑

αk (t)ϕHO k (x)

(3.27)

k≤Kmax

one gets: αk (t) = αk (0) exp (−iωHO kt) HO where HϕHO k (x) = kℏωHO ϕk (x)

(3.28)

The resulting function is hence ψ(x, t) =

∑

αk (0) exp (−iωHO kt)ϕHO k (x)

(3.29)

k≤Kmax

which will have as dominant period the one of the k = 1 component, say ωHO T = 2π

⇒

T = THO = Tosc

(3.30)

A useful way to prove the previous statement is to verify that this kind of oscillations only depend on the lowest k component in the harmonic oscillator expansion. At this purpose we compute the momenta distribution and show the results in Figures (3.3) and (3.4). Figure (3.4) in particular is indicative of the nature of these oscillations: we imposed an artificial cut-off in the system so to discard the high-k modes. If the fluctuations period is only determined by the low-k component, it should not be affected by this trick. This is precisely the case shown in the picture.

48


KB T = 5 ℏωho

KB T = 5 ℏωho

700

|α|2 distribution cutoff

1

6 600 4 500 2

|α|2 /N

x [aho ]

0.1

400 0 300 -2

10−2

200 -4 100 -6

10−3

0 0

1

2

3

4

5

0

6

5

10

(a) component distribution

20

25

30

(b) time evolution

KB T = 10 ℏωho

KB T = 10 ℏωho 700


1

15 −1 time [ωho ]

number of one-particle wavefunctions

6 600 4 500 2

|α|2 /N

x [aho ]

0.1

400 0 300 -2

10−2

200 -4 100 -6

10−3

0 0

2

4

6

8

0

10

5

10


time

(c) component distribution

|α|2

20

25

30

−1 [ωho ]

(d) time evolution KB T = 20 ℏωho

KB T = 20 ℏωho 1

15

900

distribution cutoff

6

|α|2 /N

x [aho ]

0.1

10−2

800

4

700

2

600 500

0 400 -2

300

-4

200 100

-6

10−3

0 0

5

10

15


(e) component distribution

20

0

5

10

15 time

20

25

30

−1 [ωho ]

(f) time evolution

Figure 3.3: (left pictures) Distribution in terms of single-particle wavefunctions of a system of N = 1000 bosons at a temperature KB T = 5ℏωho (3.3a), KB T = 10ℏωho (3.3c) and KB T = 20ℏωho (3.3e). Note the most relevant to be the low-n ones. (right pictures) Time evolution of the non interacting Bose gas. Note the period of the g = 0 oscillations to be of the order of 2π.

49


KB T = 50 ℏωho

KB T = 50 ℏωho 800


1

6 700 4

|α|2 /N

x [aho ]

0.1

10−2

600

2

500

0

400

-2

300 200

-4

100 -6

10−3

0 0

10

20

30

40

50

0

5

10


time

(a) component distribution

20

25

30

−1 [ωho ]

(b) time evolution

KB T = 50 ℏωho with artificial cutoff

KB T = 50 ℏωho with artificial cutoff 700

|α|2 distribution artificial cutoff

1

15

6 600 4 500 2

|α|2 /N

x [aho ]

0.1

400 0 300 -2

10−2

200 -4 100 -6

10−3

0 0

10

20

30

40


(c) component distribution

50

0

5

10

15 time

20

25

30

−1 [ωho ]

(d) time evolution

Figure 3.4: (left pictures) Comparison between the “full” system at KB T = 50ℏω, and the one at the same temperature, but with an artificial cut-off imposed at n = 10. (right pictures) Time evolution with and without the artificial cut-off. One could see the number of thermal perturbations to be reduced by the block, as expected by the exclusion of high energy modes. The period of oscillations, however, is not affected, showing it to be completely determined by the low-n one-particle wavefunctions, as explained in the text.

50


3.4

Interacting Bose gas: spontaneous soliton creation

Up to now we treated non-interacting gases. In this case g was set to 0 and the GPE simply reduced to the Schrödinger equation. We will now follow the path indicated in [3], and study the problem of a (repulsively) interacting N -particle bosonic system at finite temperature T > 0, confined in a harmonic trap. Interactions change dramatically the statistical properties of the system, and we have to modify some of the quantities mentioned before.

3.4.1

Momentum cut-off and condensate fraction

The first remark regards the momentum cut-off Kmax . In [20], following the fact that in the Bogoliubov approximation, energies of excitations are counted starting from the chemical potential µ rather than from zero, the authors postulated to modify it in the form ℏωKmax = µ + KB T . (3.31) The chemical potential µ is determined in an iterative way, from the Hartree-Fock approximation of a Bose gas at finite temperature. In this case the chemical potential µ can be proven7 to satisfy: µ = gn0 + 2gnT , (3.32) where n0 is the condensed fraction and nT = n − n0 . Hence, at first we evolve the system in imaginary time, and find the T = 0 density of the system, which is fully condensed. Then we use its central value n0 (0) to build the first approximation of the chemical potential µ0 = gn0 (0). This result is then used to perform the first Metropolis simulation. At this point we need a way to compute the condensed and the thermal density at T > 0, starting from the whole density. After the Monte-Carlo sampling, the total wavefunction is (3.20): ψ(x) =

∑

αk ϕHO k (x)

k = 0, 1, 2, ..., Kmax ,

(3.33)

k≤Kmax

The density matrix of the system is defined: ρ1 (x, x′ ) = ⟨ψ ∗ (x)ψ(x′ )⟩N =

K∑ max

HO ′ ρnm ϕHO n (x)ϕm (x ) ,

(3.34)

n,m=0

the (Kmax + 1) × (Kmax + 1) matrix elements being ρnm = ⟨αn∗ αm ⟩N =

N ∑

j (αnj )∗ αm ,

(3.35)

j=1

where N is the number of Monte-Carlo realisations. The density matrix so built is then diagonalised8 , and one gets the new basis com7

See for example [10].

8

In order to diagonalise the hermitian density matrix A+iB we first wrote the real symmetric matrix ( A −B ) B A

and then diagonalise it using the standard routines reported in [18]; then one gets (Kmax + 1)

51

3.4. Interacting Bose gas: spontaneous soliton creation

posed by the functions: φn (x) =

√

λn

∑

βnk ϕHO k (x) ,

(3.36)

k≤Kmax

so that ψ(x) =

∑√

∑

λn

n

βnk ϕHO k (x) and

αk =

∑√

λn βnk .

(3.37)

n

k≤Kmax

The condensed density is then computed by identifying the highest eigenvalue9 max (λn ) ≡ λMAX and by evaluating n0 (x) = |φMAX (x)| = λMAX 2

∑

2 k HO βMAX ϕk (x) ,

(3.38)

k≤Kmax

while the thermal density is determined as nT (x) = n(x) − n0 (x). The new cutoff is estimated according to (3.32) and (3.31), and the procedure is repeated, until the chemical potential value converges. This condition is usually achieved in a few realisations, since the number of one-particle wavefunctions has to be an integer number, and hence its change can at least be 1. This procedure is also useful to determine the behaviour of the condensate fraction N0 = λMAX in function of the temperature. In free, non-interacting one-dimensional systems, true condensation is not possible. It has been shown, however, condensation to be possible in confined systems10 . As a reference, we considered the N -dependent critical temperature for a 1D ideal Bose gas determined by Van Druten and Ketterle in [22]: ( ) KB Tc0 KB Tc0 N= ln 2 , (3.39) ℏω ℏω and we produce Figure (3.5). In particular KB Tc0 (N = 1000) ≃ 171ℏωho .

3.4.2

Energy functional and results

The second difference consists in considering also interactions in the energy functional, according to equation (1.12) which becomes: [

∫

]

ℏ2 ∇2 1 g Eψ = dxψ ∗ (x) − + mω 2 x2 ψ(x) + 2m 2 2 ∫ ∑ g |αk |2 k + = ℏω dx|ψ(x)|4 2 k≤K

∫

dx|ψ(x)|4 (3.40)

max

 couples of identical positive eigenvalues λn and the corresponding eigenvectors βn = 

0 βn

K

.. .

 , which

βn max

are also the eigenvalues and eigenvectors of the original density matrix. 9 The identification of the condensed fraction with the highest eigenvalue of the diagonalised density matrix was first proposed by Penrose and Onsager in [21]. 10 See [22] and the references therein.

52


1

N = 100 N = 500 N = 1000

0.9 0.8 0.7 N0 /N

0.6 0.5 0.4 0.3 0.2 0.1 0 0

0.5

1

1.5

2

T /Tc0

(a)

(b)

Figure 3.5: (3.5a) Condensed fraction in function of the reduced temperature for an interacting 1D system of N particles with g¯ = 0.31. For each curve KB Tc0 is computed according to (3.39), and refers to an ideal gas. (3.5b) Original picture from [22] of the same quantity but in an ideal Bose gas. The difference with the previous case is crucial. Solitons, in fact are a solution of a non-linear Schrödinger equation of the kind: (

)

∂ 1 i ψ = − ∂x2 + V (x) + κ|ψ|2 ψ ∂t 2

(3.41)

for which the dispersive and the non-linear term compensate, letting the solitary wave propagate without change in shape. As already determined in Section 2.3.6, the period of the soliton in a harmonically trapped BEC is Tsol =

√ 2THO .

(3.42)

Using the Monte Carlo method explained before, then, one is able to produce the thermalised wavefunction of a system of N bosons at temperature T . Letting it evolve in time according to the Gross-Pitaevskii equation using the Split-Step method of Chapter 2, we obtain results shown in Figure 3.6. By studying the results of Figure 3.6, one is able to recognise the period of the oscillations to be the one predicted in (3.42). We are thus in condition to reproduce a state of N bosons confined in a harmonic trap at temperature T > 0, exhibiting spontaneous soliton creation, induced by no other mechanisms than thermal fluctuations. What’s next? So far we built all the machinery needed to produce the wavefunction of a thermal state of N bosons. From now on, the work will be split in two paths. In the next chapter we will describe and study how to simulate a key experimental technique: evaporative cooling. Then we will change topic and show some nice pictures about the effects caused by an external perturbation moving through our system.

53

3.4. Interacting Bose gas: spontaneous soliton creation

KB T = 15ℏωho

KB T = 15ℏωho

single real. density average dens. potential

140 120

140

10

120 100

5 x [aho ]

100

15

80 60

80 0 60 -5

40

40 -10

20 0 -20

20

-15 -15

-10

-5

0

5

10

15

0 0

20

5

x [aho ]

20

(b) time evolution

KB T = 25ℏωho

KB T = 25ℏωho


120

15

time [1/ωho ]

(a) wavefunction

140

10

140

10

120 100

5 x [aho ]

100

15

80 60

80 0 60 -5

40

40 -10

20 0 -20

20

-15 -15

-10

-5

0

5

10

15

20

0 0

5

x [aho ]

15

20

time [1/ωho ]

(c) wavefunction

(d) time evolution

KB T = 260ℏωt extho

KB T = 260ℏωho


200

10

x [aho ]

150

100

50

0

15

300

10

250

5

200

0

150

-5

100

-10

50

-15 -20

-10

0 x [aho ]

(e) wavefunction

10

20

0 0

5

10

15

20

time [1/ωho ]

(f) time evolution

Figure 3.6: (left pictures) Wavefunctions of a system of N = 1000 bosons at a temperature KB T = 15ℏω (3.6a), KB T = 25ℏω (3.6c), KB T = 260ℏω (3.6e). Note that, although the plotted wavefunctions are not symmetric, the average over the last 70% of the Monte Carlo realisations (N = 107 ) is symmetric at each T . (right pictures) Spontaneous solitonic oscillations. The results for KB T = 15ℏω (3.6b) and KB T = 260ℏω (3.6f) can be compared with the ones reported in [3]. Note that for the two lowest temperature the solitons oscillate with a frequency which, within a few percent, agrees √ with the value ωsol = ω/ 2 computed in (1.43). For the high temperature result, intersoliton collisions start to dominate the oscillatory behaviour.

54


KB T = 15ℏωho

KB T = 15ℏωho

condensate dens. average dens. potential

140 120

120

100

100

80

80

60

60

40

40

20

20

0 -20

-15

-10

-5

0

5

10

thermal dens. average dens. potential

140

15

20

0 -20

-15

-10

-5

x [aho ]

(a) wavefunction

120 100

80

80

60

60

40

40

20

20 -15

-10

-5

0

5

10

15

20

0 -20

-15

-10

-5

x [aho ]

0

5

(d) time evolution




120

20

10

15

20

x [aho ]

(c) wavefunction

140

15


140

100

0 -20

10

KB T = 25ℏωho


120

5

(b) time evolution

KB T = 25ℏωho 140

0 x [aho ]


140 120

100

100

80

80

60

60

40

40

20

20

0

0 -20

-10

0 x [aho ]

(e) wavefunction

10

20

-20

-10

0

10

20

x [aho ]

(f) time evolution

Figure 3.7: (left pictures) Condensate density nC (x) (computed as explained in text) for a system of N = 1000 bosons at a temperature KB T = 15ℏω (3.7a), KB T = 25ℏω (3.7c), KB T = 260ℏω (3.7e). (right pictures) Thermal density nT (x) = n(x) − nC (x) for KB T = 15ℏω (3.7b), KB T = 25ℏω (3.7d), and KB T = 260ℏω (3.7f). The average has again be performed on the last 70% of N = 107 Metropolis realisations.

Chapter 4

Evaporative cooling One of the greatest achievement in the long path to get the Bose-Einstein condensate was the development of techniques to cool down atoms. This was the true problem which prevented researchers to produce it for almost 70 years. The cooling of a gas to ultra-cold temperatures may seem a conceptually simple problem, but is not, and requires many passages and experimental tricks. One of them, the last, is the so called evaporative cooling, and the intuition behind it lead to the Nobel Prize for Wolfgang Ketterle, Eric A. Cornell and Carl E. Wieman in 2001. In the following we will review briefly the key features of evaporative cooling, and show some numerical simulations involving the thermal state produced in Chapter 3.

4.1

Cooling atoms: not a simple work

Having a fresh drink is no longer a problem from the invention of the refrigerator machine, in the first half of the 20th century. The basics behind such a technological masterpiece are very simple: a liquefied refrigerant is expanded and suddenly evaporates; the latent heat required by evaporation is taken from the still liquefied part of the refrigerant, which passes through the coil, and a fan blows air from the refrigerator on the tubes. Then air is cooled down, and you can enjoy a fresh lemonade in the middle of summer. So, why not to use such a giant refrigerator to cool down gases to ultracold temperature? For a number of reasons. First of all the efficiency of such machines, decreasing with temperature. Then the fact that the fan is actually heating the gas while moving it. Temperatures of microkelvin would be unreachable with similar apparatus, and for BEC one needs nanokelvin. Several steps have to be performed in order to meet BEC conditions, increasingly complicated and fascinating 1 . Laser cooling. Despite the common representation of lasers cutting metal doors and armed droids, they can be actually used to cool down low density gases. The example briefly explained here is the so-called Doppler cooling, and is by far the most common and first method of slowing down atoms using light. Its mechanism is based on the absorption of a photon by the atom, and by the momentum transfer involved in the process. From the point of view of momentum conservation, when an atom is travelling 1

A wonderful and extraordinarily well written explanation of cold atom tools can be found in [23].

55

56

Chapter 4. Evaporative cooling

towards a laser beam and absorbs one of the beam’s photons, its momentum mv is reduced by the amount of the momentum of the photon, causing a decrease of velocity ∆v =

pphoton , m

where pphoton =

h . λ

(4.1)

In Doppler cooling, the frequency of light is slightly red-detuned from an electronic transition of the atom, having hence a frequency which is a little lower than the one requested for the transition (thus the absorption) to take place. For that reason, atoms will absorb more photons if they approach the light source, thanks to up-shift by the Doppler effect, and tend to ignore the ones from other directions. This reduces their velocity component in the direction of the laser beam. The atom, which is now in the excited state, may re-emit the photon spontaneously, but in a random direction. The momentum recovered by this “recoil” will then vanish on average over a large number of these absorption-emission loops, and its kinetic energy is reduced. For having the cooling process in all directions, one usually illuminates the atom with 3 orthogonal laser beams, which are then reflected back along the same direction. Doppler cooling apparatus are frequently coupled with a magnetic trap, so to form a Magneto-Optical Trap (MOT). The magnetic trap consists in a spatially varying magnetic quadrupole field added to the red-detuned optical field of the Doppler cooling. This leads to a Zeeman shift in the magnetic sensitive mf sublevels, which increases with the radial distance from the trap centre. Hence, if an atom is moving away from the centre of MOT setup, from [24]. the trap, the atomic transition is shifted closer to the frequency of the laser light, and the probability of getting a kick from a photon towards the centre increases. Using this combination, atoms which initially moved at hundreds of metres per second are slowed down to speeds of centimetres per second. The process is limited by physical reasons, and cannot reach temperatures below hundreds of microkelvin. Unfortunately this is still not enough for Bose-Einstein condensation. Experimentalist had therefore to contrive another hat-trick.

RF-induced evaporative cooling. To further cool down the system, one has to introduce somehow another dissipative force. It turns out that the right choice is to selectively throw fast-moving atoms out of the trap. At first atoms coming from the laser cooling are trapped in a magnetic trap. This is simply built by the superposition of a strong uniform and a varying magnetic field. The energy shift due to the resulting magnetic field B is ∆E = −µ · B ,

(4.2)

where µ is the atomic magnetic moment. This will cause atoms with magnetic moment

4.2. Numerical simulation of evaporative cooling

57

aligned with the field to occupy those regions where the field is stronger, so to decrease their energy (they are eloquently called “high-field seeking” atoms). Of course, atoms with counteraligned µ will be conversely pushed towards regions of low B (“low-field seeking” atoms). While it is impossible to create a stable maximum of magnetic field in empty space, the opposite is feasible. Just like in the MOT, it is only necessary a set of two (or four) coils with opposite currents: the so-called anti-Helmholtz configuration. This way it is possible to confine the low-field seeking atoms in the centre of the trap2 , which may be assumed harmonic. The existence of trapped and untrapped states is fundamental for the evaporative cooling process. Looking from a classical point of view, the mechanism is very simple. Suppose to have a cup of hot tea, at a certain temperature. At equilibrium, the tea molecules will have all the same average kinetic energy. In a collision between two molecules, however, it may happen that one of the two acquires the great part of the kinetic energy, leaving the other slowed down. If the “parasite” molecule happens to have a high enough energy, it may escape from the attraction of the others and “jump” out of the cup, as steam. In a certain sense, evaporative cooling is the same process (and hence, the name). Collisions between atoms in the trap may cause one to acquire high kinetic energy. This will cause it to oscillate at greater distances from the centre with respect to “cold” atoms. Hence, to remove “hot” atoms it is sufficient to remove the ones which are able to reach the outer edge of the trap. At this point the untrapped states come in. If one is able to flip the selected atom from the trapped to the untrapped state, it will feel a repulsive force pushing it away from the centre. The transition is induced by using radio-frequency (RF) light, tuned so that only the trapped and the untrapped state of the atoms at the edge are coupled. Since the energy separation of the states depends on the magnetic field, which depends on the position, one could select precisely which zones to “clean out”. What results is an effective lowering of the walls of the trap. Hence, proceeding with a little patience, one could progressively reduce the temperature of the system until it reaches the critical value for Bose-Einstein condensation.

4.2

Numerical simulation of evaporative cooling

Heretofore we learned the experimental aspects of evaporative cooling. Since in the previous chapters we were able to produce a thermal state for a system on N bosons in a harmonic potential, it can be worth to try the simulation of the evaporative cooling 2

To be precise, this is not sufficient. Atoms in the counteraligned configuration, being in a zero-field region, may flip their magnetic moment and decay into the untrapped state. In order to avoid this possibility, Cornell and Wieman used a “Time Orbiting Potential”, an extra field depending on time, which pushed the zero point out of the centre and spun it in a circle. Ketterle, instead, avoided the possibility for the atoms to lie in the centre of the trap by physically pushing them away with a laser. Brilliant.

58


process. An interesting paper about this can be found in [25]. The time evolution we consider is no longer the usual GP equation (2.29), but the following (all the quantities are assumed as dimensionless): i

∂ ϕ(x, t) = [H(x, t) − iΓ(x, t)]ϕ(x, t) , ∂t

with Hamiltonian H(x, t) = −

(4.3)

1 ∂2 + V (x, t) + g|ϕ(x, t)|2 . 2 ∂x2

(4.4)

The time dependent external potential V (x, t) is shown in Figure (4.1) and is V (x, t) = U (t)[1 − e−(x

2 /2U (t))

{

U0 + (Ur − U0 ) ttr , t−tr Ur + (Umax − Ur ) tmax −tr ,

where U (t) =

Umax

(4.5)

] if t ≤ tr . otherwise.

(4.6)

initial potential V (x, 0) end of cooling ramp V (x, tr ) final potential V (x, tmax ) harmonic potential

tmax

U0 Ur

tr −L/2

0

L/2

position [aho ]

Figure 4.1: Evaporative cooling potential. The initial potential V (x, 0) evolves linearly in tr to the evaporative potential V (x, tr ), and then ends at tmax to the final potential V (x, tmax ), which can be assumed harmonic in the region occupied by the gas. The first stage of the process, hence for t ≤ tr , is the evaporative cooling ramp, the effective lowering of the potential walls √ explained in the previous section. V (x, t) is a Gaussian dip of standard deviation U (t) and near to the centre is effectively harmonic with the same frequency of the harmonic oscillator. The depth of the dip decreases linearly during the cooling ramp from an initial value U0 = 100 to a final Ur = U0 /3. For t > tr evaporation is stopped, and the system thermalises for a longer period up to tmax = tr + 1000 in a deeper potential, which can be considered harmonic in the region occupied by the gas (as shown in Figure (4.1)).


59

Another crucial aspect to consider is the loss of “hot” particles, hence the one at the edges of the trap. This particles must carry energy away from the system and so we must avoid them to come back. This is realised by introducing the imaginary term of (4.3), where ) ( V (x, t) γ Γ(x, t) = Γ∞ (4.7) U (t) and Γ∞ = 10, γ = 50. Γ(x, t) acts as a knife in the high-potential regions. We performed simulations for a system of N = 1000 atoms, initially prepared at a temperature of KB T = 200ℏωho , at which, according to Figure (3.5a), the gas is not condensed. The cooling process has been simulated by imposing different values of the cooling ramp time tr . The results are shown in Figures (4.2), (4.3) and (4.4). We noticed that a great number of defects emerge in the beginning of evaporation, showing all the characteristics of solitons already discussed: they have a period of the order √ of T = 2Tho , collide passing through each other and turnaround near to the edges of the cloud when the density reaches zero. The progressive decrease of the system’s temperature is expressed by a reduction of the number of solitons inside the condensate. The ramp time tr plays a crucial role in the process, because the condensate fraction depends on it: a hasty evaporation does not allow all the solitons to escape, and the system is not fully condensed. Summary and future developments. The simulation of this kind of defects’ creation in evaporative cooling process was the purpose with which we started this Master’s Thesis work. This was just the first step. What we have done, is to use the methods devised by other groups (mainly [3] and [25]) in order to test their effectiveness to our purposes. The natural future development of this kind of simulation is to expand them to include higher-dimensional systems, and to better study the role of cooling time on the defects’ formation.

60


x [aho ]

250 20 15 10 5 0 -5 -10 -15 -20

200 150 100 50 0

-20

0

20

40

60 time

80

100

120

140

−1 [ωho ]

x [aho ]

90 20 15 10 5 0 -5 -10 -15 -20

80 70 60 50 40 30 20 10 0 136

138

140

142

144

146

148

150

x [aho ]

−1 time [ωho ]

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3

20 15 10 5 0 -5 -10 -15 -20 136

138

140

142 time

144

146

148

150

−1 [ωho ]

Figure 4.2: Figure (4.2a) shows evaporative cooling of a N = 1000 gas at temperature KB T = 200ℏωho . Evaporation starts at t = 0. Figure (4.2b) and (4.2c) show the density and the phase (modulo 2π) of the last surviving solitons, after a ramp time of −1 . tr = 150ωho

61


x [aho ]

120 20 15 10 5 0 -5 -10 -15 -20

100 80 60 40 20 0 300

310

320

330

340

350 time

360

370

380

390

400

−1 [ωho ]

x [aho ]

90 20 15 10 5 0 -5 -10 -15 -20

80 70 60 50 40 30 20 10 0 386

388

390

392

394

396

398

400

x [aho ]

−1 time [ωho ]

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3

20 15 10 5 0 -5 -10 -15 -20 386

388

390

392 time

394

396

398

400

−1 [ωho ]

Figure 4.3: Figure (4.3a) shows the last 100 time units of evaporative cooling of a N = 1000 gas at temperature KB T = 200ℏωho . Figure (4.3b) and (4.3c) show the density and the phase (modulo 2π) of the last surviving solitons, after a ramp time of −1 . tr = 400ωho

62


x [aho ]

90 20 15 10 5 0 -5 -10 -15 -20

80 70 60 50 40 30 20 10 0 750

760

770

780

790

800

−1 ] time [ωho

x [aho ]

80 20 15 10 5 0 -5 -10 -15 -20

70 60 50 40 30 20 10 0 786

788

790

792

x [aho ]

time

794

796

798

800

−1 [ωho ]

0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 -0.1 -0.2 -0.3

20 15 10 5 0 -5 -10 -15 -20 786

788

790

792

794

796

798

800

−1 time [ωho ]

Figure 4.4: Figure (4.4a) shows the last 100 time units of evaporative cooling of a N = 1000 gas at temperature KB T = 200ℏωho . Figure (4.4b) and (4.4c) show the density and the phase (modulo 2π) of the last surviving solitons, after a ramp time of −1 . tr = 800ωho

Chapter 5

Superfluidity and critical velocity in a BEC Bose-Einstein condensation and superfluidity have always been strictly interconnected phenomena1 . Superfluidity is a state of matter in which matter behaves like a fluid with null viscosity and entropy. A superfluid liquid, then, flows without dissipation: this leads to amazing effects, like the circulation over obstructions and the “climbing” of container walls. The first experimental observation was performed in liquid helium by Pyotr Kapitsa2 , John F. Allen and Don Misener3 in 1938. Some years later Lars Onsager predicted the existence of quantised vortices in superfluid helium, which was experimentally observed in the 1950s by Henry Hall and Joe Vinen. Quantum vortices are a typical feature of superfluids, and allowed the discovery of superfluidity in Bose-Einstein condensates: in the first years of the 2000s quantum vortices were observed in ultracold bosonic ([29]) and fermionic ([28]) gases. A criterion to define superfluidity was first proposed by Lev Landau, stating that the dissipation does not occur if the speed of the flow is lower than a certain critical value4 . Many experiments performed on superfluid helium showed that indeed it exists a critical velocity, but in many instances it is much lower than the one predicted by Landau. Richard Feynman ([30]) suggested this incongruity Vortices in a BEC, from [28]. to be due to vortex formation, hence to nonlinear perturbation of the fluid. The same behaviour for the critical temperature has been found in ultracold atoms. Regarding this aspect, it is useful to cite an experimental and a theoretical work. The first one is the reference [31] by Peter Engels. In this work a penetrable repulsive barrier is swept across an elongated cigar-shaped BEC with a 1

See Section 1.1 for a short historical review. Viscosity of Liquid Helium below the λ-Point, Nature, 1938 [26] 3 Flow Phenomena in Liquid Helium II, Nature, 1938 [27] 4 See Appendix B for a simple review over its predictions. 2

63

64

Chapter 5. Superfluidity and critical velocity in a BEC

constant speed v, which can be tuned in order to test the different behaviours. The condensate is trapped with a strong confinement in the radial direction, and weak in the axial one, resulting then quasi one-dimensional. The results, reported in Figure (5.1), suggest the existence of three different flow regimes: for slow and fast velocities the condensate remains unperturbed, while in an intermediate velocity regime, solitons appear. The low-speed case is clearly understandable because of superfluidity, while the high-speed soliton suppression is discussed later. The generation of solitons in the intermediate regime is the 1D equivalent to the generation of quantised vortices in 3D.

Figure 5.1: Soliton creation after a barrier sweep, from [31]. The white arrow in Fig. (c) indicates the final position of the barrier. Sweep speeds are (c) 0.4mm/s, (d) 0.5mm/s, (e) 0.6mm/s, (f) 0.7mm/s, (g) 0.8mm/s, (h) 1mm/s, (i) 1.3mm/s, (j) 2mm/s, (k) 3.3mm/s. The interesting theoretical work is [32], where Nicolas Pavloff showed the existence of a reduced critical velocity for a repulsive obstacle. It also contains some results about the direction and the nature of the excitations in the regime v > vc , and the high velocity unperturbed flow already met in the experiment. The analysis of [32] is performed for a uniform BEC, while we consider it to be harmonically trapped. This influences some of the results, like the form of the drag force exerted by the perturbation on the fluid. In the following we want to simulate something similar to the systems reported in the cited works, although a direct comparison is difficult. In Pavloff’s paper the system is uniform, while our condensate is inhomogeneous due to the harmonic trap. In Engels’ experiment, the number of atoms as well as the dimension of the condensate were large, and their direct simulation requires a noticeable computational effort. Being this a test to acquire familiarity with the problem, we decided to simulate a smaller

65

5.1. Numerical simulations

system, similar to the one of the previous chapters (hence considering as parameters N = 1000, g¯ = 0.31 and a spatial length of L = 100aho ). We will use to this purpose many of the methods we showed earlier.

5.1

Numerical simulations

Let us now pass to the description of the numerical simulation. We want to mimic the behaviour of a flow of BEC passing around a Gaussian obstacle with a certain velocity v. At this purpose is more convenient to consider the BEC as stationary, and to let the perturbation move across it, again with velocity v. We thus define a time-dependent potential (

V (x, t) = max Vho (x), V0 e

−

(x−vt)2 l2

)

,

(5.1)

where V0 and l2 define respectively the height and the width of the Gaussian pulse. The choice of these two parameters may influence dramatically the outcome of the simulation. An important quantity to compute is the drag force exerted by the perturbation on the condensate. According to [32], and taking care of our reverse reference perspective with respect to that paper, we postulate a definition for such a quantity to be ∫

Fd (t) =

(

n(x, t) −

dV (x) dx

)

dx .

(5.2)

This definition seems perfectly reasonable for physical explanation, since the force exerted by the obstacle is the mean value of the operator dV (x)/dx computed over the condensate wave function5 . The Landau critical velocity’s prediction for a BEC is the sound speed, computed according to (B.9). Given the fact that it depends on the density, in the nonuniform case one should consider the local critical velocity √

c(x) =

gn(x) . m

(5.3)

Taking it dimensionless means to consider c¯(x) =

5

√ √ c(x) = g¯n ¯ (x) = g¯|ϕ(x)| . aho ωho

(5.4)

Apart from this clever observation, the author was also able to give a rigorous analytical determination for it in [32].

66


5.1.1

Zero temperature

At first we consider the case of T = 0, in which we first impose

and

V0 = 10 , ng

(5.5)

l = 0.88 , ξ

(5.6)

√ where n ¯ g¯ = 3.1 ξ = ℏ/ 2mng is the condensate healing length of a N = 1000 particles condensate. This choice has been done in order to have a relatively small perturbation of the usual harmonic oscillator. The results are reported in Figure (5.3), (5.2) and (5.4). Like in the references, we find that three regimes arise, depending on the value of the extrapolated critical velocity vc0 : for v < vc0 , the Gaussian perturbation does not produce any excitation in the gas, which remains fully superfluid; for vc0 < v < vc1 , excitations are created at the barrier and we observe the emission of solitons; for v > vc1 , the fluid is again unperturbed. It is worth noticing that the value of the critical √ velocity, estimated to be 0 of the order of vc ≃ 0.65 ℏωho /m, is found to be much lower than the one predicted by Landau on the basis of the local sound speed of the unperturbed condensate. In order to explain this fact, one could consider that due to the repulsion exerted by the obstacle, the density in the perturbation region is decreased. The depletion is greater in the high density region, and this has a double effect: the local sound velocity is reduced, and due to flux conservation the local fluid velocity increases. This could lead eventually to have the latter to be greater than the former, allowing the superfluidity breakdown and the soliton creation. As for the value of√the upper critical velocity vc1 , we roughly estimated it to be of the order of vc1 ≃ 30 ℏωho /m, but a precision method to establish such a quantity has to be developed in order to give accurate measurements. In Figure (5.2) one can see the generation of a single soliton when the speed of the perturbation coincide with vc0 . A positive peak in the drag force corresponds precisely to the soliton emission6 . Together with it also a series of sound waves is emitted in the opposite direction (this is of particularly evidence in the v = 1 case of Figure (5.3)). The coincidental emission of downstream soliton and upstream sound waves is an interesting aspect already studied in [32] and in some references therein. As we can see in Figure (5.3), the harmonic confinement also causes the occurrence of centre-of-mass dipolar oscillations of the √ condensate (see for instance the case v = 3). For v ≳ 30 ℏωho /m both solitons and phonons are suppressed. As for solitons, this may be explained by the fact that at high velocities, the drag force is proportional 6 The drag shape deserves some discussion. One can see it shows a double behaviour: at first it goes negative, and then it turns positive. This is due to the repulsion of the fluid with respect to the perturbation cross, in a certain sense is the same as the Archimedes’ principle.

67


to the reflection coefficient. This is well known to decrease at high energy. Regarding phonons, an interesting explanation can be found in [33]. The author, by working in analogy with the radiation of capillary-gravity waves7 in a classical fluids, was able to show that the emission rate of phonons of wavelength k due to the crossing of a Gaussian obstacle of width σ, height W0 and speed v is P˜ (σ, k) ∝ exp(−βσv) ,

(5.7)

√

where β = ln(v 2 /4W0 ). For (relatively) high speeds the value of β depends very weakly on v, and so the phonon emission results exponentially suppressed.

7

A capillary wave is a wave travelling along the phase boundary of a fluid, whose dynamics are dominated by the effects of surface tension. In general, since waves are also affected by gravity, they may be referred as capillary-gravity waves. A common example is the ripple produced on the interface between water and air by the impact of a droplet.

68


10

120 100

x [aho ]

5

80

0

60 40

-5

20

-10

0 10

20

30

40

50

60

70

80

time [ωho ]

(a) v immediately before vc0 10

120 100

x [aho ]

5

80

0

60 40

-5

20

-10

0 10

20

30

40

50

60

70

80

time [ωho ]

(b) v = vc0 800

800

drag force for v < vc0

600

600

400

400

200

200

0

0

-200

-200

-400

-400

-600

-600

-800 0

10

20

30

40 50 60 −1 time [ωho ]

70

80

-800 0 90

drag force for v = vc0 Soliton emission

10

20

30

40

50

60

70

80

90

−1 time [ωho ]

Figure 5.2: Creation of solitons and appearance of a critical velocity for a T = 0 BEC. Figures√(5.2a) and (5.2b) show respectively the case v < vc0 (superfluid) and v = vc0 ≃ 0.65 ℏωho /m (dissipative). Note in (5.2b) the creation of both a downstream soliton and an upstream density fluctuation (sound waves), compatibly with the results of [32]. Figures (5.2c) and (5.2d) show the exerted drag force in the two cases. Note the positive force peak at soliton creation.

69

x [aho ]


20 15 10 5 0 -5 -10 -15 -20

120 100 80 v = 0.5

60 40 20 0

10

20

30

40

50

60

70

80

x [aho ]

time [ωho ] 20 15 10 5 0 -5 -10 -15 -20

140 120 100 80 60 40 20 0

v=1

10

20

30

40

50

60

x [aho ]

time [ωho ] 20 15 10 5 0 -5 -10 -15 -20

180 160 140 120 100 80 60 40 20 0

v=3

5

10

15

20

25

30

35

x [aho ]

time [ωho ] 250

20 15 10 5 0 -5 -10 -15 -20

200 150 v = 10 100 50 0 5

10

15

20

25

x [aho ]

time [ωho ] 20 15 10 5 0 -5 -10 -15 -20

120 100 80 v = 20

60 40 20 0

5

10

15

20

25

x [aho ]

time [ωho ] 20 15 10 5 0 -5 -10 -15 -20

100 90 80 70 60 50 40 30 20 10 0

v = 50

5

10

15

20

25

time [ωho ]

Figure 5.3: Different regimes for different values of v, reported on the figures.

70


7

local sound speed c(x) estrapolated critical velocity vc0

6 5 4 3 2 1 0 -10

-5

0

5

10

x [aho ]

(a) Local sound speed c¯(x) vs estrapolated critical velocity vc0 . density t=42.01 potential

100

80

80

60

60

40

40

20

20

0 -10

-5

0

5

density t=42.25 potential

100

10

0 -10

-5

x [aho ]


100

80

60

60

40

40

20

20

-5

0 x [aho ]

5

10

5


100

80

0 -10

0 x [aho ]

10

0 -10

-5

0

5

10

x [aho ]

Figure 5.4: Figure (5.4a): comparison√between the local sound speed c¯(x) and the extrapolated critical velocity vc0 ≃ 0.65 ℏωho /m. The latter results much lower than the first one (apart from the negligible borders of the cloud), as explained in text. The last four pictures show the soliton creation and propagation.

71


5.1.2

Finite temperature

Heretofore we studied the case of zero temperature, and we also made some comparisons with other theoretical and experimental works. We want now to see how the effect of temperature may influence the critical velocity. As already seen in Chapter 3, the condensed density fraction depends on temperature, and of course is lower at higher temperature. This means that, qualitatively, the critical density should decrease according to relation (5.4), accounting for the condensed density reduction. We hence repeat the previous simulations of a N = 1000 bosonic gas in a harmonic trap, and we considered a Gaussian perturbation with parameters given in (5.5) and (5.6), but this time the temperature of the system is set to be different from zero. We obtain again a lower critical velocity vc , above which solitons appear. Some results for different temperatures are reported in Figure (5.5). One should note that in order to be able to distinguish the spontaneous solitons from the ones induced by the barrier, we only investigated the low temperature case. We find that indeed the critical velocity decrease with temperature. The determination of the upper critical velocity is in this case very complicated, due to the excitations naturally present in the condensate which “cover” the highspeed barrier effects. In the development of a precision technique to measure the upper critical velocity, then, one should also consider how to extract the information about the excitation induced by the perturbation, and to distinguish it from the noise caused by thermal fluctuations.

7

T = 0 local sound speed vc0 vc (T = 5) vc (T = 10)

6 5 4 3 2 1 0 -10

-5

0

5

10

x [aho ]

Figure 5.5: Comparison between the local sound speed c¯(x), and the extrapolated √ √ /m for zero temperature, vc (T = 5) ≃ 0.57 ℏωho /m critical velocities vc0 ≃ 0.65 ℏωho√ for T = 5 and vc (T = 10) ≃ 0.50 ℏωho /m for T = 10. The temperature effect gives a lowering in the critical velocity.

72


120

20 15

100

10 80

x [aho ]

5 0

60

-5

40

-10 20

-15 -20

0 50

60

70

80

90

100

110

120

−1 time [ωho ]

(a) v < vc (T ) 20

140

15

120

10

100

x [aho ]

5

80

0 60

-5

40

-10

20

-15 -20

0 50

60

70

80

90

100

−1 time [ωho ]

(b) v = vc0 1000

1000

drag force for v < vc

drag force for v = vc solitons

500

500

0

0

-500

-500

-1000 0

20

40

60 80 −1 time [ωho ]

100

120

-1000 140 0

20

40

60 80 −1 time [ωho ]

100

120

Figure 5.6: Creation of solitons and appearance of a critical velocity for a KB T = 0 10ℏωho BEC. Figures √ (5.6a) and (5.6b) show respectively the case v < vc (superfluid) 0 and v = vc ≃ 0.65 ℏωho /m (dissipative). Figures (5.6c) and (5.6d) show the exerted drag force in the two cases. Note again the positive force peaks at soliton creation, in spite of the great noise due to the thermal fluctuations.

73


5.1.3

Sound speed determination

Apart from the theoretical determination of the sound speed (B.9), one could perform experimental and numerical simulations in order to extract its value. On the experimental side, a nice work has been carried out in a quasi one-dimensional system8 by Ketterle et al. in [34]. In this paper, a harmonic trapped Bose-Einstein condensate is perturbed by the sudden switch of a blue-detuned9 laser beam in its centre. This produce small density fluctuations, which propagate outward at the speed of sound.

Figure 5.7: Original pictures from [34]. −1 The experiment has been simulated by imposing a sudden switch at t0 = 5ωho of a stationary Gaussian perturbation

V (x) = V0 exp(−x2 /l2 ) , where we chose

(5.8)

V0 l2 = 5 and = 0.88 . ng ξ

The repulsive disturbance expels atoms from the centre of the condensate, creating two density peaks propagating symmetrically outward. The position of the peaks vary linearly with time10 , and hence it is possible to estimate the speed of sound. The results are plotted√in Figure (5.8). The sound speed has been estimated to be of the order of v˜ ≃ 7.5 ℏω m , hence of the same order of magnitude of the experimental one. The results appear to be the same even in case of different values of the simulated laser intensity V0 , in accordance with [34]. 8

One noticeable √ difference between our pure 1D system and the quasi-1D system on the paper is that the c1D = 2cq-1D . 9 The laser is far-off-resonant detuned from the relevant resonances in order to avoid heating from spontaneous emission. 10 This is only true near to the central region of the condensate, where one can suppose the density to be slow-varying. In time, this means to consider t − t0 < 0.5.

74


10

120 100

x [a−1 ho ]

5 80 0

60 40

-5 20 -10

0 5

5.1

5.2

5.3

5.4

5.5

5.6

5.7

5.8

−1 time [ωho ]

(a)

-10

-5

0

5

10

x [aho ]

(b)

(c)

Figure 5.8: Figure (5.8a): sound propagation in a N = 1000 BEC with g¯ = 0.31. Figure (5.8b) and (5.8c): Comparison between the numerical result and the experimental result of [34].

Summary and future developments. In Section 5.1 we saw the emergence of different regimes in the creation of excitations, depending on the perturbation’s velocity. A precise characterisation of these regimes in function of temperature, and how the role of dimensionality influences the relevant quantities involved may be a future interesting research topic. In particular, one could try to reproduce the critical velocity behaviour evaluated in [34], and to verify its dependence on the condensate density. Regarding Section 5.1.3, again a precise estimation of the effect of temperature on the sound speed may be of great interest, but this requires the development of techniques for measuring it. Its direct observation becomes, in fact, increasingly difficult with temperature rising, due to the presence of the thermal excitations.

Conclusions We are at the end of this long path, and is therefore useful to summarise the results we got. Our purpose was to acquire skills in the methods for the simulation of the dynamics of a Bose-Einstein condensate. At this end we performed some test simulations on the zero temperature 1D case, and explored some possible extensions to the finite temperature case. We focused in particular on the experimentally relevant case of a harmonic trapping. At the beginning, we briefly reviewed some of the key aspects of Bose-Einstein Condensation, introduced the Gross-Pitaevskii equation, and clarified the properties of its solitonic solution. In Chapter 2 we presented a method to solve numerically the GPE, the Split-Step method, and we applied it to different test situations, included the one of phase-imprinted solitons. In Chapter 3 we showed how to use the Classical Fields Approximation to produce a finite temperature quantum gas, and we were able to observe spontaneously created solitons. The last two chapters contain some applications of the machinery we built to some relevant experimental cases. In Chapter 4 we reproduced evaporative cooling, and we found the reduction of solitons, hence of temperature, to be influenced by the cooling ramp time. In the last chapter we observed the breakdown of superfluidity in a BEC, due to the crossing of a Gaussian perturbation at constant speed. We were able to observe three different regimes, depending on the barrier’s speed, and to approach the study of the temperature’s effect on the extrapolated critical velocity. In the very end we studied the propagation of sound inside a BEC at zero temperature. At this point, the analysis we performed is still incomplete, and further discussions are required on some aspects. At first, using a Metropolis simulation, one should be sure the detailed balance condition to be respected. We noticed, however, that some of the exposed results may suffer from a dependence on the way11 in which the configurations {αk } are displaced. This means that the detailed balance condition may be not fully respected, leading to measurements imprecision. This aspect may be crucial, and will deserve in the future a careful analysis. In [25] simulations with N = 104 particles at a temperature KB T = 360ℏωho have been performed, and the system was presented (and appeared) to be “well above the critical temperature”. In our simulations, using the same parameters, the system is almost completely condensed for such a temperature. The critical temperature for the ideal case is computed according to (3.39) to be Tc0 ≃ 1275ℏωho . One should then better clarify the role of interactions as well as the one of the number of particles N in the determination of the critical temperature. 11

See Section 3.2

75

Appendix A

Second quantisation formalism In this appendix we build the mathematical basis for the theoretical treatment of quantum many particle systems, and some concepts indispensable to the derivation of the Gross-Pitaevskii equation in Section 1.2. This is usually represented by the so-called Second Quantisation formalism1 , which is reported here. It may be most assuredly very boring to read, but, in my intentions, should help the unfamiliar reader to face the formal results of Chapter 1 with some confidence. The first idea in treating quantum many particle systems could be to work in a N -particle Hilbert space, for which a complete basis is |α, β, . . . , ω⟩ = |α⟩ ⊗ |β⟩ ⊗ . . . |ω⟩

(A.1)

and to solve the Schrödinger equation for a wave function ψ(r1 , r2 , . . . rN ) depending on the N variables corresponding to the N particles. This approach seems (and is) perfectly reasonable when dealing with a small number of particle, but results to be ill suited to work in the case of many interacting bosons or fermions. The first problem one has to engage is related to the indiscernibility of the particles. Even if they are non interacting, hence in the free case, not all states are acceptable as wave functions. In fact, only the total symmetric and total antisymmetric wave functions happen to be allowed. This means that, also for non interacting particles, indistinguishability somehow introduces correlations in the wave function, and we cannot use directly wave functions in the form (A.1). They become very complicated, since the number of terms of the wave function for a N -particle system grows as N !. The second problem is related to the way in which the operators are represented in the standard expression of quantum mechanics. Consider as an example the operator describing the total momenta of particles: it has to be the sum of operators acting on 1

The following treatment is reported almost entirely from [10] and from [35].

76

77

A.1. Introduction to first quantisation

each particle individually. Then: p tot =

N ∑

pi

(A.2)

i=1

being pi the operator acting on the i-th particle. We recognise this to be an abuse of notation, since in fact one should write: pi = 1 ⊗ 1 ⊗ · · · ⊗ p ⊗ · · · ⊗ 1

(A.3)

where 1 is the identity and p is inserted at the i-th position. The operators and the wave functions, then, depend explicitly on the total number of particles N . This means that the calculation changes completely when dealing with 2 or 2000 particles, and that the N → ∞ limit is also problematic. This results to be very useful when imposing the thermodynamic limit, when the volume of the system also goes to infinity. Taking this limit is higly desirable, and simplifies a lot of calculations. Moreover, one would also like to be able to consider the case in which the total number of particles is not fixed, but changes during the time of the experiment. For these reasons the standard quantum mechanical representation, also known as first quantisation is not the best choiche for systems of many indistinguishable particles. One desires to be provided with a representation that is able to take care automatically of the following points: 1. The antisymmetrization is implemented in a suitable way, so to avoid to deal explicitly with all N ! terms. 2. The description of the system is not explicitly dependent on the number of particles in the system. Such a representation is the so-called second quantisation method, that we build in the following. Let us at first summarize some of the basic concepts of the first quantisation.

A.1

Introduction to first quantisation

Single-particle Hilbert space. The single-particle Hilbert space, denoted by H1 , is the space of quantum states describing one particle. Consider a complete orthonormal basis of H1 ; it can be: • discrete: formed by states |α⟩ such that ⟨α|β⟩ = δαβ and

∑

α |α⟩ ⟨α|

• continuous: formed by states |r⟩ such that ⟨r|r′ ⟩ = δ(r − r′ ) and

∫

= 1.

dr |r⟩ ⟨r′ | = 1.

Then one defines the single-particle wave function in coordinate space as: φα (r) = ⟨r|α⟩

(A.4)

78

Appendix A. Second quantisation formalism

Two-particle Hilbert space. The two-particle Hilbert space is the tensor product of two single-particle Hilbert space H1 , and is denoted H2 = H1 ⊗ H1 . If |αj ⟩ refers to particle j, a complete basis for H2 is formed by the states |α1 α2 ⟩ = |α1 ⟩ ⊗ |α2 ⟩

⟨α1 α2 | = ⟨α1 | ⊗ ⟨α2 |

and

(A.5)

They satisfy the following properties: • orthogonality

⟨α1 α2 |α1′ α2′ ⟩ = ⟨α1 |α1′ ⟩⟨α2 |α2′ ⟩

(A.6)

• completeness ∑

|α1 α2 ⟩ ⟨α1 α2 | =

α1 α2

∑

|α1 ⟩ ⟨α1 |

α1

∑

|α2 ⟩ ⟨α2 | = 1

(A.7)

α2

An important observation has to be done regarding the simmetry of the system. If the two particle are indistinguishable, the relevant physical properties of the system must remain invariant under particle exchange. These are connected with the modulus square of the wave function, which means that the modulus of it must be the same as well. Then states can only be symmetric or antisymmetric with respect to particle exchange: 1 S |α1 α2 ⟩ = (|α1 α2 ⟩ + |α2 α1 ⟩) 2 1 A |α1 α2 ⟩ = (|α1 α2 ⟩ − |α2 α1 ⟩) 2

(A.8) (A.9)

where we call S and A, respectively, the simmetrizing and the antisimmetrizing operator. Let H2S be the symmetric subspace of H2 . Then the following holds: • S is the identity in H2S . • S is a projector on H2S : SS = S. • S is hermitian: S = S † . Then, using the previous relations, one may conclude that S |α1 α2 ⟩ is a complete system in H2S : ∑ S |α1 α2 ⟩ ⟨α1 α2 | S † (A.10) 1 = S = SS = SS † = α1 α2

The analog results can be obtained for the antisymmetrizing operator A on the antisymmetric subspace H2A . States S |α1 α2 ⟩ and A |α1 α2 ⟩ are orthogonal but not orthonormal. Orthonormal basis in H2S and H2A are given by: • Symmetric states

√

|α1 α2 ⟩S =

2 S |α1 α2 ⟩ 1 + δα1 ,α2

(A.11)

79

A.1. Introduction to first quantisation

• Antisymmetric states

√

|α1 α2 ⟩A =

2 A |α1 α2 ⟩

(A.12)

Then the normalized wave functions in coordinate space are:

• Symmetric states ⟨r1 r2 |α1 α2 ⟩S = √

)

(

1 2(1 + δα1 ,α2 )

φα1 (r1 )φα2 (r2 ) + φα1 (r2 )φα2 (r1 )

(A.13)

• Antisymmetric states ) 1 ( ⟨r1 r2 |α1 α2 ⟩A = √ φα1 (r1 )φα2 (r2 ) − φα1 (r2 )φα2 (r1 ) 2

(A.14)

N -particle Hilbert space. A generalization to a N particle system can be done considering a N -particle Hilbert space HN = H1 ⊗ H1 ⊗ . . . H1 . Then a complete basis |

{z

}

N times

for HN is formed by the states:

|α1 α2 . . . αN ⟩ = |α1 ⟩ |α2 ⟩ . . . |αN ⟩

(A.15)

Now, defined P the generic permutation of two particle labels and (−1)P the parity of the permutation (being P even or odd), the general symmetrizing and antisymmetrizing operators in HN are defined: S= A=

1 ∑ P N! P

1 ∑ (−1)P P N! P

(A.16) (A.17)

S , and for Then, complete orthonormal basis for the completely symmetric subspace HN A the completely antisymmetric subspace HN are:

• Symmetric states √

N! S |α1 α2 . . . αN ⟩ nα !nβ ! . . . nω ! ∑ 1 =√ |αP1 αP2 . . . αPN ⟩ N !nα !nβ ! . . . nω ! P

|α1 α2 . . . αN ⟩S =

≡ |nα nβ . . . nω ⟩

(A.18)

80


• Antisymmetric states √ N ! A |α1 α2 . . . αN ⟩ 1 ∑ =√ (−1)P |αP1 αP2 . . . αPN ⟩ N! P

|α1 α2 . . . αN ⟩A =

(A.19)

≡ |nα nβ . . . nω ⟩ In the previous we introduced the representation of the N -particle states in terms of the important quantities of the occupation numbers nα , nβ , counting the number of particles which occupy the given single-particle states |α⟩ and |β⟩, and which satisfy the normalization condition ∑ nα = N . (A.20) α

This representation will be convenient in the definition of the Fock space (see later). Then, completely symmetric and antisymmetric wave functions of N particles are given by: • Symmetric states ⟨r1 . . . rN |α1 . . . αN ⟩S = √

∑ 1 φαP1 (r1 ) . . . φαPN (rN ) N !nα !nβ ! . . . nω ! P

(A.21)

• Antisymmetric states 1 ∑ ⟨r1 . . . rN |α1 . . . αN ⟩A = √ (−1)P φαP1 (r1 ) . . . φαPN (rN ) N! P 

φα1 (r1 ) . . . 1  .. = √ det  . N! φαN (r1 ) . . .



φα1 (rN )  ..  .

(A.22)

φαN (rN )

The above sum over permutations of the particle labels is a determinant, and in particular is called Slater determinant. If two of the particles occupy the same state, the determinant will vanish, according to the Pauli exclusion principle. Hence, the occupation number of a particle in a single-particle state α of a completely antisymmetric wave function can either be nα = 0 or nα = 1. S describe bosonic States belonging to the completely symmetric Hilbert subspace HN A particles, while the ones beloning to the completely antisymmetric Hilbert subspace HN describe fermionic particles.

A.2. Fock Space and second quantisation

A.2

81

Fock Space and second quantisation

We have finally collected all the ingredients needed to introduce the second quantisation formalism. As we already saw, it is possible to know the whole state of a system by knowing the occupation numbers. We define the Fock space F as the infinite direct sum of N -particle Hilbert spaces for N = 0, 1, 2, . . . : F = H0 ⊕ H1 ⊕ H2 ⊕ . . . (A.23) where we define the space H0 to be the generate of the state |0⟩, the vacuum state. For it, we require the normalization condition ⟨0|0⟩ = 1. We can define creation and annihilation operators over the Fock space as: 1 |nα nβ . . . nω ⟩ = √ (a† )nα . . . (a†ω )nω |0⟩ nα ! . . . n ω ! α 1 ⟨nα nβ . . . nω | = √ ⟨0| (aα )nα . . . (aω )nω nα ! . . . n ω !

(A.24)

aα |0⟩ = ⟨0| a†α = 0 .

(A.25)

and

The following relations for the commutator of bosonic operators and for the anticommutator of fermionic operators hold: [

a†α , a†β

[

a α , aβ [

aα , a†β

] ∓

]

∓

]

∓

≡ a†α a†β ∓ a†β a†α = 0 ≡ aα aβ ∓ aβ aα = 0

(A.26)

= δα,β

Note that the above relations imply that, for fermions, (aα )2 = (a†α )2 = 0. Moreover, thanks to (A.26), one may verify the normalization condition for the states defined in (A.24): ⟨nα nβ . . . nω |nα nβ . . . nω ⟩ = 1 . (A.27) When applied to a state |nα . . . nω ⟩ the creation and annihilation operators give the following result for bosons (+ sign) and for fermions (− sign): √ a†λ |nα . . . nλ . . . nω ⟩ = (±1)m nλ + 1 |nα . . . nλ + 1 . . . nω ⟩ √ aλ |nα . . . nλ . . . nω ⟩ = (±1)m nλ |nα . . . nλ − 1 . . . nω ⟩

(A.28)

In particular, for both fermions and bosons: a†λ aλ |nα . . . nλ . . . nω ⟩ = nλ |nα . . . nλ . . . nω ⟩

(A.29)

The states |nα . . . nλ . . . nω ⟩ are then eigenstates of the operator a†λ aλ , with eigenvalue given by the occupation number nλ . Hence the operator a†λ aλ represents the number of particle in a defined single-particle state and is therefore called occupation number

82

Appendix A. Second quantisation formalism ∑

operator. Of course, states |nα . . . nω ⟩ are also eigenstates of the combination α a†α aα ∑ with eigenvalue α nα = N total number of particles. It is worth to see the behaviour of the creation and annihilation operators under change of a single-particle basis: |α⟩ =

∑

∑

i

i

⟨i|α⟩ |i⟩ ←→ a†α |0⟩ =

Then:

a†α =

⟨i|α⟩a†i |0⟩

∑

⟨i|α⟩a†i

i

aα =

(A.30)

∑

(A.31)

⟨α|i⟩ai

i

And now an important definition: the creation/annihilation operators of a particle at space position r, called field operators, are: ˆ † (r) = Ψ ˆ Ψ(r) =

∑

∑

∑

α

∑

α

α

⟨α|r⟩a†α =

ψα∗ (r)a†α

α

⟨r|α⟩aα =

(A.32) ψα (r)aα

Using the previous equations of the creation/annihilation operators, one may prove the following relations for the commutators and anticommutators of fields: [ [

A.3

]

ˆ ˆ ′) Ψ(r), Ψ(r

]

ˆ ˆ † (r′ ) Ψ(r), Ψ

[

∓ ∓

]

ˆ † (r), Ψ ˆ † (r′ ) = Ψ = δ(r − r′ )

∓

=0 (A.33)

Representation of operators

For now we have operators that allow to construct the whole Fock space. What remains to be done is to express the physical observables we want to compute in terms of these operators. Since they have to act on indistinguishable particles, we have some constraints on what they can be. To write observable in the second quantisation formalism, we have to classify them in terms of how many particles are involved. Indeed, there exist physical observables only measuring the quantum numbers of a single particle at a time (e.g. particle momentum, density, etc.) and others that need to deal with the quantum numbers of two of the particles to build the matrix elements, like operators measuring the particle interactions. The first kind is called one-body operators, while the latter is called is two-body operators. One can also have operators of higher correlations, but in treating our problems we will not need them.

A.3.1

One-body operators

Let us call F a certain physical observable described by a one-body operator, hence which only involves one particle at a time. Let then FN act on a space of N particles; it must do the measurement on each particle of the system. Let F1 be the operator

83

A.3. Representation of operators

acting on the single-particle Hilbert space H1 and let the states |i⟩ be its eigenstates with the corresponding eigenvalues fi : F1 =

∑

fi |i⟩⟨i|

(A.34)

i

In terms of a general basis |α⟩ of H1 we can write ∑

F1 =

|α⟩⟨α| F1 |β⟩⟨β| .

(A.35)

αβ

The particles being indistinguishable, the operator must be invariant with respect to a permutation of the particle indices, and can therefore only be defined as a sum with equal coefficients: (1)

FN = F1

(n)

where F1

(2)

⊗ 12 ⊗ · · · ⊗ 1N + 11 ⊗ F1

(N )

⊗ · · · ⊗ 1N + · · · + ⊗11 ⊗ · · · ⊗ F1

(A.36)

is the operator acting on the n-th particle: (n)

F1

|β1 . . . βn . . . βN ⟩ =

∑

⟨α| F1 |βn ⟩ |β1 . . . α . . . βN ⟩ .

(A.37)

α

We can rewrite equation (A.36) committing again a notation abuse as:

FN =

N ∑

(n)

F1

(A.38)

n=1

This is the most general form, for indistinguishable particles, of a one-body operator. When acting on a state, then: FN |β1 . . . βn . . . βN ⟩ =

N ∑ ∑

⟨αn | F1 |βn ⟩ |β1 . . . αn . . . βN ⟩

(A.39)

n=1 αn

Note that equation (A.39) also holds when FN is applied to a ”permuted state” P |β1 . . . βN ⟩, and so also for symmetric and antisymmetric physical states |β1 . . . βN ⟩S/A . The corresponding operator acting on the Fock space F is then F =

∑

⟨α| F1 |β⟩ a†α aβ

(A.40)

αβ

We can prove this assertion by proceeding as follows: • The generic symmetric (antisymmetric) states building a basis in the Fock space F can be written as |β1 . . . βn . . . βN ⟩S(A) = CS(A) a†β1 . . . a†βN |0⟩ where CS(A) is a normalization factor. • The commutator between F given in (A.39) and a generic creation operator is

84


given by:

∑

]

[

F, a†β =

⟨α| F1 |β⟩ a†α

(A.41)

α S(A)

• Then, the action of F on the generic basis state |β1 . . . βn . . . βN ⟩S(A) of HN ([

is:

]

F |β1 . . . βn . . . βN ⟩S(A) = CS(A) F, a†β1 a†β2 . . . a†βN |0⟩ + . . . [

)

]

+ a†β1 . . . a†βN −1 F, a†βN |0⟩ = CS(A)

N ∑ ∑ n=1 αn

=

N ∑ ∑

⟨αn | F1 |βn ⟩ a†β1 . . . a†βn a†βN |0⟩

(A.42)

⟨αn | F1 |βn ⟩ |β1 . . . αn . . . βN ⟩S(A)

n=1 αn

The last result is the exact behaviour of FN , as stated in equation (A.39), implying the two operators to be identical.

A.3.2

Two-body operators

Let us now have a look at operators involving two particles to define their matrix elements. Let V2 be a symmetric two-body operator acting on H2 . Its general form is given by ∑ V2 = |αβ⟩⟨αβ| V2 |γδ⟩⟨γδ| (A.43) αβγδ

where the matrix elements do not depend on the order of the two particles: ⟨αβ| V2 |γδ⟩ = ⟨βα| V2 |δγ⟩. On the N -particle Hilbert space HN we define the symmetric combination: (1,2)

VN = V2

(1,3)

+ V2

+ ··· =

N ∑

(n,m)

V2

(A.44)

n

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