Conductance fluctuations in microwave-driven Rydberg atoms

EUROPHYSICS LETTERS

15 October 1998

Europhys. Lett., 44 (2), pp. 162-167 (1998)

Conductance fluctuations in microwave-driven Rydberg atoms A. Buchleitner1,2 , I. Guarneri3,4,5 and J. Zakrzewski6 1

Max-Planck-Institut f¨ ur Quantenoptik Hans-Kopfermann-Str. 1, D-85748 Garching, Germany 2 The Queen’s University of Belfast - Belfast BT7 1NN, Northern Ireland 3 Universit´ a di Milano, Sede di Como - Via Lucini 3, I-22100 Como, Italy 4 Istituto Nazionale di Fisica Nucleare, Sezione di Pavia Via Bassi 6, I-27100 Pavia, Italy 5 Istituto Nazionale di Fisica della Materia, Unit` a di Milano Via Celoria 16, I-20134 Milano, Italy 6 Instytut Fizyki imienia Mariana Smoluchowskiego, Uniwersytet Jagiello´ nski ulica Reymonta 4, PL-30-059 Krak´ ow, Poland (received 8 June 1998; accepted in final form 27 August 1998) PACS. 05.45+b – Theory and models of chaotic systems. PACS. 32.80Rm – Multiphoton ionization and excitation to highly excited states (e.g., Rydberg states). PACS. 72.15Rn – Quantum localization.

Abstract. – We draw and quantitatively support a parallel between conductance fluctuations in one-dimensional disordered solids and fluctuations of lifetimes of hydrogen atoms in microwave fields. In particular, we numerically analyze the statistics of scaled atomic resonance widths in the localized regime and we find a lognormal distribution, quantitatively consistent with the theory of Anderson localization.

Introduction. – The ionization of hydrogen atoms by linearly polarized microwaves has been a subject of intensive studies for a quarter of this century [1]. Classically, ionization occurs due to the chaotic excitation of the electron in sufficiently strong microwave fields (exceeding the “classical chaos border” [2]). When the microwave frequency ω is smaller than the Kepler frequency of the initial electronic motion ωK , classical and quantum ionization thresholds (i.e. microwave field amplitudes allowing for non-negligeable ionization for sufficiently long microwave pulses) agree. When ω0 = ω/ωK > 1, this is no longer true, because quantum thresholds are then significantly higher than the corresponding classical ones. This experimentally observed effect [3] was beforehand predicted by the so-called photonic localization theory [2]. According to this theory, for fields below a “delocalization border”, the atomic population distribution remains exponentially localized over the ladder of quasi-resonant bound states which are connected to the initial state via a chain of one-photon transitions, such that the population of a bound state which is reached from the initial one by absorbing M photons decreases, on the average, proportionally to exp[−2M/L]. A quantitative prediction c EDP Sciences

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for the “localization length” L then leads to an estimate of the delocalization border, typically higher than the classical chaos border. The theory is based on an analogy between atomic excitation and quantum transport in a disordered one-dimensional (1D) solid, as described by tight-binding models of Anderson type [2, 4-7]. A tight-binding-like formulation of the ionization problem is in fact achieved either by means of a “Quantum Kepler Map” [2, 8], or by resorting to a resonant approximation for the Floquet Hamiltonian [6, 7, 9]. Theory. – Referring the reader to the original papers [2, 5], let us quote only the results used later on. For a 1D hydrogen atom exposed to a linearly polarized microwave field of amplitude F and frequency ω, ionization will be essentially suppressed if the localization length L is small with respect to the number of photons N required to equalize the ionization potential of the initial atomic state | n0 i (n0 the principal quantum number). L measures the width of the exponentially localized bound space population distribution around | n0 i, in units of photon energies ω of the driving field, and is given by −10/3 2 n0 ,

L = 3.33F02 ω0

(1)

where F0 = F n40 and ω0 = ωn30 are the scaled amplitude and frequency of the driving field, respectively [2]. The photon number reads   n0 n20 N= 1− 2 , (2) 2ω0 nc where we additionally allow for a shift of the atomic continuum threshold to finite “cut-off” values nc of the principal quantum number. Such shifts occur in real laboratory experiments, e.g. as a consequence of unavoidable electric stray fields experienced by the Rydberg atoms during their interaction with the microwave field [1, 3]. The ionization yield for fixed values of F0 , ω0 and n0 , is now expected to be “tunable” by the “localization parameter”   L 6.66F02n0 n20 `= = 1− 2 . (3) 7/3 N nc ω 0

Thus, the atomic excitation process is determined by the ratio L/N , much in the same way as the conduction through a disordered 1D solid is determined by the ratio between the localization length and the length of the sample. This analogy can be used to import from the latter field other results and concepts than just the estimate for ionization thresholds. In this letter we follow this strategy, and analyze the atomic counterpart of conductance fluctuations in the localized regime. Even at a fixed value of L/N most of the physically measurable quantities related to quantum transport in a disordered solid wildly fluctuate from sample to sample [10, 11]. In fact the decay of localized wave functions over a given, finite-length N is not given by the realization-independent exponential factor exp[−2N/L], but rather by exp[−2N/LN ]. The finite-sample decay rate LN tends to the nonfluctuating, sample-independent value L as N → ∞, but, at any finite N , it is a fluctuating quantity, the statistics of which —according to the scaling theory of localization [10, 11]— are again determined by the ratio L/N alone. In particular, the fluctuations of the conductance g (defined either via transmission [12] or via level curvature [13]) follow, in the localized regime, a log-normal distribution [14, 15], that is ln g is normally distributed. Roughly speaking, this is so because the probability of crossing the sample is proportional to exp[−2N/LN ], and L−1 N is normally distributed. Moreover, the variance of the distribution, Var(ln g), is proportional to the mean hln gi, with a coefficient of the order of unity.

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In order to find an analog in the ionization problem, we first generate different realizations of “disorder”, by slightly changing parameters of the system (F0 and ω0 ), keeping the ratio L/N fixed. Doing this, numerically computed ionization probabilities do indeed show considerable fluctuations [5,8,16,17]. Next, we have to identify quantities that can be meaningfully defined on both sides. It is not obvious how to define, in a way amenable to effective computation, an analog of the dimensionless conductance g of a disordered sample for the ionization process, because in the latter case one does not deal with a transmission problem, but rather with one in which a particle initially localized inside the sample diffuses out of it. A clue is offered by Thouless’ definition of conductance [13], which relates g to the width that bound states in a closed solid acquire when the solid is connected to open leads; that width is then measured by the (scaled) curvature of levels of the closed system on changing boundary conditions. No fully formal connection can be made, (there seems to be no equivalent for the curvature of levels which enters Thouless’ formula), but still level widths naturally correspond to widths of scattering resonances. In the light of this heuristic connection, in the following we focus our attention on the dimensionless quantity Γ/∆, where Γ is related to scattering resonance widths (see below), and ∆ is the mean spacing between resonances (in real energy). Before explaining how a sort of dimensionless “atomic conductance” can be built on this quantity, we need a brief discussion of resonances in the ionization problem and say how we compute them. To study ionization, we use an almost exact description of the quantum dynamics of the hydrogen atom in the field, which in particular comprises the exact description of the coupling to the atomic continuum, induced by the driving field. Our only approximations will be a) the confinement of the Rydberg electron to motion along the polarization axis of the field, and b) the assumption of a constant amplitude of the driving field, i.e. the neglect of pulse-induced effects on the population distribution [18]. Whereas a) has been shown to be a reasonable approximation for the ionization of quasi–one-dimensional Rydberg states [2,19], b) is actually adapted to the framework of dynamical localization theory, which is focussed on general spectral properties of the quantum problem under study, for a well-defined structure of the classical phase space, i.e. for fixed values of ω0 and F0 [2]. Of course, both approximations are also very convenient from the computational point of view, since already the results presented below are extremely expensive in terms of CPU time. Under these assumptions, the Hamiltonian of a Rydberg state confined to the positive z-axis and driven by a linearly polarized electromagnetic field reads, in atomic units, p2 1 F pz − − sin(ωt), (4) 2 z ω where we treat the driving as a classical field in dipole approximation, employ the velocity gauge (dropping the constant energy shift due to the ponderomotive energy of the electron in the driving field), neglect relativistic effects, and assume an infinite mass of the nucleus [19]. Due to the coupling to the atomic continuum induced by the driving field, all atomic bound states turn into resonances, for the exact problem described by (4). These resonances | χν i are eigensolutions of the associated scattering problem, which satisfy purely outgoing boundary conditions. Also known as Gamow states [20], they are characterized by quasienergies Eν and resonance widths Γν . Those can be determined by a combined application of the Floquet theorem [21] and of complex dilation [22]. In our approach they are given by complex eigenvalues Eν − iΓν /2 of the dilated Floquet Hamiltonian Hθ , with discrete, 2π/ω-periodic and square-integrable eigenstates |χνθ i. Hθ is obtained from the Floquet Hamiltonian H = H(t) − i∂t by a non-unitary, complex scaling transformation R(θ), depending on the positive angle θ [19, 22]. The widths Γν and the associated time-averaged overlaps (weights) on the initial state wν = hχνθ | R(θ) | n0 i2 (the bar denotes an average over the driving field period 2π/ω [19]) [23] determine the decay of the chosen initial state | n0 i into the atomic continuum, H=

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(b) 2

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0.1 0.0 -15

-10

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Fig. 1. – Distribution of the logarithm of the dimensionless atomic conductance G (defined as the ratio of the resonance width to the mean real spacing between resonances) for the localization parameter (see (3)) (a) L/N = 0.2, (b) L/N = 2, and principal quantum number n0 = 70 of the initial atomic state. The smooth line is the best Gaussian fit. In (b), the deviation of the conductance statistics from the log-normal distribution, which we attribute to the transition from the localized to the diffusive regime, is significant.

and are in fact the basis of recent work which has yielded the hitherto closest ab initio reproduction of experimental ionization threshold fields over a large interval of driving field frequencies [19]. In those numerical experiments, the transition from the localized to the delocalized regime has been observed as a threshold behaviour of the ionization probability as a function of F0 , at fixed values of ω0 and t. To extract the widths Γν , as well as the weights wν , we diagonalize the complex dilated Floquet Hamiltonian Hθ in a Sturmian basis [19]. The finite size of the basis, as well as a scaling parameter α which essentially determines the length scale on which the basis functions live, introduces an effective numerical ionization limit which can be identified with the cut-off value nc encountered in laboratory experiments, as mentioned earlier. The cut-off was fixed to nc = 2n0 for all the numerical results presented hereafter. Convergence of eigenvalues and eigenvectors is achieved by choosing a sufficiently large basis size, and can be controlled by the stability of the eigenvalues Eν − iΓν /2 under changes of α and the dilation angle θ. Once the widths are known, one can look for their statistics. In doing that, one has to take account of the fixed “sample length” N , because we are interested in the decay of an initial state which is located N photons away from the ionization threshold. Therefore, each resonance is accounted for with its weight wν , so that the “atomic conductance” from a given initial state to the atomic continuum reads 1 X G= Γν wν . (5) ∆ ν Numerical results. – To collect sufficient statistics for the analysis we accumulate the widths and the overlaps for a fixed L/N ratio by varying simultaneously F0 and ω0 , with ω0 ∈ [2, 2.5]. Typically 500 slightly different F0 and ω0 values are taken for each L/N value, yielding circa 25 000 resonances with nonnegligible wν . For the spacing ∆ we have used the estimate ∆ ' 2ω 2 n20 , which comes from assuming a number N of “effective” resonances (i.e., of significant weight) in a single Floquet zone of length ω. The statistics of G are found to be robust with respect to the definition of the statistical sample space spanned by F0 and ω0 , provided the underlying classical dynamics are dominantly chaotic and no more subject to local regions of regular dynamics. By implementing the just described numerical apparatus, we have found that, for small L/N ' 0.2, our parameter G computed for a given initial state (here we choose n0 = 70) obeys the log-normal distribution characteristic for the conductance of a disordered quasi–one-

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Var(lnG)

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Fig. 2. – Relation between the mean and the variance of fitted log-normal distributions of the atomic conductance from n0 = 70, in the interval L/N ∈ [0.15, 0.25] of the localization parameter. The straight line fit yields Var(ln G) = −5.7 − 0.75hln Gi and is represented by the dashed line.

dimensional system in the localized regime —see fig. 1(a). This can be qualitatively understood from the standpoint of photonic localization theory, because, in the localized regime, decay into the atomic continuum occurs mainly via one-photon jumps from the last quasi-resonant state, hence the decay rate is proportional to the population of that state, which goes like exp[−2N/LN ]. Fits of a similar quality as in fig. 1(a) are obtained in the range of L/N ∈ (0.15, 0.25) interval. This allows for the analysis of the relation between the average conductance and its variance in fig. 2. A linear fit to the numerical data yields Var(ln G) = a − khln Gi with k = 0.75 and a ' −5.7. The value of k is in rough agreement with k ' 1 observed for Thouless’ conductance in the localized regime [15]. Due to the difference between the standard definition of g in a scattering problem and our G describing the ionization process, one can hardly expect a perfect agreement. The offset a depends, e.g., on the way the dimensionless conductance is defined, and in particular on how the spacing ∆ is estimated. The “atomic conductance” G is of course strongly dependent on the length N , hence on the initial state | n0 i via the overlap weights wν . However, we have verified that the distribution of G is well approximated by the log-normal distribution also for n0 = 60 provided L/N ' 0.2. Further extensive studies are needed to verify the range of n0 values where the linear relation between Var(ln G) and hln Gi holds, as well as to check the possible weak dependence of k on n0 . For L/N < 0.1, the resonance widths become very small on the average, on the level of double-precision arithmetics. For L/N > 0.25 deviations from the log-normal distribution become clearly visible (compare fig. 1(b)) indicating the beginning of the transition to the diffusive regime; at the same time the dependence of Var(ln G) on hln Gi deviates from linear, see the right-hand part of fig. 2. From localization theory one would expect a Gaussian distribution in that regime [14, 15], which in our computations is not fully attained even as high as L/N = 2 (corresponding to F0 ' 0.17 . . . 0.22), which is roughly at the limit of a reasonable cost in CPU. Hence, on the basis of the present data it remains an open question whether the analogy between ionization and conductance observed in the localized regime also holds in the case of diffusive transport. Summary. – In summary, our findings indicate that, at least in the limited range of L/N which we have explored, the “atomic conductance” exhibits fluctuations strongly reminiscent of conductance fluctuations in quasi–one-dimensional disordered samples. These atomic conductance fluctuations are observed in a regime of parameters where the classical transport mechanism is dominantly chaotic and no more subject to remaining regular structures in

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phase space such as, e.g., elliptic islands (in the latter situation, for quantum states localized on the elliptic islands the fluctuations of the corresponding “atomic conductance” have very distinct properties [24]). Our results (the first quantitative statistical study of fluctuations of the ionization rates in this parameter regime) provide significant quantitative support for the validity of photonic localization theory for the description of the microwave ionization of atomic Rydberg states. The more so, in that our numerical method exactly accounts for the coupling to the atomic continuum, in contrast to previous simulations of the ionization process [5, 8]. Therefore, our results confirm the deep intrinsic link between microwave ionization of Rydberg atoms and transport properties in quasi–one-dimensional disordered solids. Future work will extend the present study to a larger domain in parameter space, to test the universality of our above observations. One interesting question will be whether Var(ln G) will be of the same order, independently of the (atomic) system under consideration. *** We thank D. Delande for fruitful discussions. Partial support by EU (AB) and KBN (JZ, project No. 2P03B 03810) is gratefully acknowledged. CPU time has been provided by RZG and Pozna´ n SCNC. REFERENCES [1] Koch P. M. and van Leeuven K. A. H., Phys. Rep., 255 (1995) 289, and references therein. [2] Casati G., Guarneri I. and Shepelyansky D. L., IEEE J. Quantum Electron., 24 (1988) 1420. [3] Galvez E. J., Sauer B. E., Moorman L., Koch P. M. and Richards D., Phys. Rev. Lett., 61 (1988) 2011; Bayfield J. E., Casati G., Guarneri I. and Sokol D. W., Phys. Rev. Lett., 63 ¨ mel R., Graham R., Sirko L., Smilansky U., Walther H. and Yamada K., (1989) 364; Blu Phys. Rev. Lett., 62 (1989) 341; Arndt M., Buchleitner A., Mantegna R. N. and Walther H., Phys. Rev. Lett., 67 (1991) 2435. [4] Fishman S., Grempel D. R. and Prange R. E., Phys. Rev. Lett., 49 (1982) 509. [5] Casati G., Guarneri I. and Shepelyansky D. L., Physica A, 163 (1990) 205. [6] Casati G. and Guarneri I., preprint ITP, Santa Barbara (1989). [7] Jensen R. V., Susskind S. M. and Sanders M. M., Phys. Rep., 201 (1991) 1. [8] Leopold J. G. and Richards D., J. Phys. B, 23 (1990) 2911. [9] Brenner N. and Fishman S., Phys. Rev. Lett., 77 (1996) 3763. [10] Lee P. A. and Ramakrishnan T. V., Rev. Mod. Phys., 57 (1985) 287. [11] Pichard J. L., J. Phys. C, 19 (1986) 1519. [12] Landauer R., Philos. Mag., 21 (1970) 873. [13] Thouless D., Phys. Rep., 13 (1974) 93. [14] Pichard J. L., Zanon N., Imry Y. and Stone A. D., J. Phys. (Paris), 51 (1990) 587. ˙ [15] Casati G., Guarneri I., Izrailev F. M., Molinari L. and Zyczkowski K., Phys. Rev. Lett., 72 (1994) 2697. [16] Sirko L., Arndt M., Koch P. M. and Walther H., Phys. Rev. A, 49 (1994) 3831. [17] Buchleitner A. and Delande D., Chaos, Solitons & Fractals, 5 (1995) 1125. [18] Breuer H. P., Holthaus M. and Dietz K., Z. Phys. D, 8 (1988) 349. [19] Buchleitner A., Delande D. and Gay J.-C., J. Opt. Soc. Am., 12 (1995) 507. ¨ ndas E. and Elander N. (Editors), Resonances (Springer-Verlag, Berlin) 1989. [20] Bra [21] Shirley J. H., Phys. Rev., 138 (1965) B979. [22] Balslev E. and Combes J. M., Commun. Math. Phys., 22 (1971) 280; Yajima K., Commun. Math. Phys., 87 (1982) 331. [23] Note that the weights wν are formally complex, though effectively exhibit vanishing imaginary parts. Only resonances close to the real axis contribute to the spectral decomposition of a bound initial state | n0 i. [24] Zakrzewski J., Delande D. and Buchleitner A., Phys. Rev. Lett., 75 (1995) 4015; Phys. Rev. E, 57 (1998) 1458.