Condition of stochasticity in quantum nonlinear systems

Berman, G. P.; Zaslavsky, G. M.

doi:10.1016/0378-4371(78)90190-5

Cited by 236 publications

(202 citation statements)

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“…As was shown in Refs. [116,117], for classically chaotic systems quantum effects are extremely strong and reveal themselves on a very short (logarithmic) time scale.…”

Section: Nse and Becmentioning

confidence: 99%

The Fermi–Pasta–Ulam problem: Fifty years of progress

Berman

Izrailev

2005

Chaos: An Interdisciplinary Journal of Nonlinear Science

307

292

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A brief review of the Fermi-Pasta-Ulam (FPU) paradox is given, together with its suggested resolutions and its relation to other physical problems. We focus on the ideas and concepts that have become the core of modern nonlinear mechanics, in their historical perspective. Starting from the first numerical results of FPU, both theoretical and numerical findings are discussed in close connection with the problems of ergodicity, integrability, chaos and stability of motion. New directions related to the Bose-Einstein condensation and quantum systems of interacting Boseparticles are also considered.

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“…As was shown in Refs. [116,117], for classically chaotic systems quantum effects are extremely strong and reveal themselves on a very short (logarithmic) time scale.…”

Section: Nse and Becmentioning

confidence: 99%

The Fermi–Pasta–Ulam problem: Fifty years of progress

Berman

Izrailev

2005

Chaos: An Interdisciplinary Journal of Nonlinear Science

307

292

View full text Add to dashboard Cite

show abstract

“…2, can be further confirmed by a general yet straightforward calculation based on Eq. (15). To simplify notation we consider the case when δ = (0, δ p ), i.e., when the net shift is aligned with one of the axes.…”

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confidence: 99%

Sub-Planck structure in phase space and its relevance for quantum decoherence

Zurek

2001

Nature

337

365

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Heisenberg's principle 1 states that the product of uncertainties of position and momentum should be no less than Planck's constanth. This is usually taken to imply that phase space structures associated with sub-Planck (≪h) scales do not exist, or, at the very least, that they do not matter. I show that this deeply ingrained prejudice is false: Non-local "Schrödinger cat" states of quantum systems confined to phase space volume characterized by 'the classical action' A ≫h develop spotty structure on scales corresponding to sub-Planck a =h 2 /A ≪h. Such structures arise especially quickly in quantum versions of classically chaotic systems (such as gases, modelled by chaotic scattering of molecules), that are driven into nonlocal Schrödinger cat -like superpositions by the quantum manifestations of the exponential sensitivity to perturbations 2 . Most importantly, these sub-Planck scales are physically significant: a determines sensitivity of a quantum system (or of a quantum environment) to perturbations. Therefore sub-Planck a controls the effectiveness of decoherence and einselection caused by the environment 3−8 . It may also be relevant in setting limits on sensitivity of Schrödinger cats used as detectors.One of the characteristic features of classical chaos is the evolution of the small scale structure in phase space probability distributions: As a consequence of the exponential sensitivity to initial conditions, an initially regular "patch" in phase space with a characteristic size ∆ will, after a time t, develop structure on the scales:where Λ is, in effect, the Lyapunov exponent 9,10 . This is a consequence of the exponential stretching and of the conservation of phase space volume in reversible evolutions. What happens with small scale structure in quantum versions of classically chaotic systems? We shall investigate this question using Wigner function W (x, p) -the closest quantum analogue of the classical phase space distribution 11 :Above, ρ is the density operator. When the quantum state is a pure |ψ >, the integrand becomes exp(ipy/h)ψ * (x + y/2)ψ(x − y/2). In a quantum system the smallest spatial structures in the wave function ψ(x) will be set by the highest phase space frequencies available. Given a finite energy, ψ(x), and, hence, W (x, p) cannot reach scales as small as the exponential squeezing of Eq. (1) would eventually imply. Thus, in a quantum system, there must be a scale below which structure should not appear. That some limit must exist was apparent already some time ago: Berry and Balazs (who, as we shall see below in connection with Eq. (13)) have an enviable record of correctly anticipating various aspects of quantum chaos) have conjectured 12,13 that structure saturates on scales given by Planck constant. If this were the case, W should be smooth on scales small compared to 2πh.I shall show that copious structure appears in the Wigner function W on much smaller sub-Planck scales associated with the action of the order of:and explain its origin. Most importantly, I shall demonstra...

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“…For a generic quantum mapping, it was first shown by Berman and Zaslavski [20,21], that quantum corrections become comparable to the classical limit at the time t E . This is the time needed for a minimal quantum wavepacket, δθ 0 δl 0 ≃k, to spread uniformly over the angular direction.…”

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confidence: 99%

“…It is widely believed [12,20,21,22] that this intermediate, 1 < t E < t L , time scale is indeed relevant for the quantum evolution of classically chaotic systems. This observation was put on the quantitative basis in Ref.…”

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confidence: 99%

Weak Dynamical Localization in Periodically Kicked Cold Atomic Gases

2004

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Quantum kicked rotor was recently realized in experiments with cold atomic gases and standing optical waves. As predicted, it exhibits dynamical localization in the momentum space. Here we consider the weak localization regime concentrating on the Ehrenfest time scale. The later accounts for the spread-time of a minimal wavepacket and is proportional to the logarithm of the Planck constant. We show that the onset of the dynamical localization is essentially delayed by four Ehrenfest times and give quantitative predictions suitable for an experimental verification.PACS numbers: 42.50.Vk, 72.15. Rn Unprecedented degree of control reached in experiments with ultra-cold atomic gases [1] allows to investigate various fundamental quantum phenomena. A realization of quantum kicked rotor (QKR) is one such possibility that recently attracted a lot of attention [2,3,4,5]. To this end cold atoms are placed in a spatially periodic potential V 0 cos(2k L x) created by two counterpropagated optical beams. The potential is switched on periodically for a short time τ p ≪ T , giving a kick to the atoms; here T is a period of such kicks. The evolution of the atomic momenta distribution may be monitored after a certain number of kicks. If the gas is sufficiently dilute [6], one may model it with the single-particle Hamiltonian, that upon the proper rescaling takes the form [7] of the QKR:Here θ ≡ 2k L x and time is measured in units of the kick period, T . The momentum operator is defined asl = ik ∂ θ , where the dimensionless Planck constant is given byk = 8hT k 2 L /(2m). Finally, the classical stochastic parameter is K =kV 0 τ p /h . The classical kicked rotor is known to have the rich and complicated behavior [8]. In particular, for sufficiently large K ( > ∼ 5), it exhibits the chaotic diffusion in the space of angular momentum [8]. The latter is associated with the diffusive expansion of an initially sharp momenta distribution: δ l 2 (t) ≡ (l(t) − l(0)) 2 = 2D cl t (dashed line on Fig. 1). For sufficiently large K, the classical diffusion constant may be approximated by K 2 /4 [8]. The higher order correction is an oscillatory function of the stochastic parameter, i.e., [9,10]. It was realized a while ago [11,12] that quantum interference destroys the diffusion in the long time limit and leads to localization:where the localization length is given by ξ = D cl /k. For a large localization length ξ ≫k, there is a long crossover regime, 1 < t < t L , between the classical diffusion and quantum localization. It was suggested in Ref. [13], that the QKR may be mapped onto the one-dimensional Anderson localization with the long range disorder. The universal long-time behavior of the latter is described by the non-linear sigmamodel [14], resulting in the standard weak-localization correction [15,16] Fig. 1). Notice, that the correction is linear ink and non-analytic in time. In an apparent contradiction with this fact, explicit studies [17,18] of the first few kicks show only renormalization of the diffusion constant starting fr...

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Condition of stochasticity in quantum nonlinear systems

Cited by 236 publications

References 1 publication

The Fermi–Pasta–Ulam problem: Fifty years of progress

The Fermi–Pasta–Ulam problem: Fifty years of progress

Sub-Planck structure in phase space and its relevance for quantum decoherence

Weak Dynamical Localization in Periodically Kicked Cold Atomic Gases

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