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arXiv:quant-ph/0512072 · Physical Review A

Decay of quantum accelerator modes

Michael Sheinman, Shmuel Fishman, Italo Guarneri, Laura Rebuzzini

Published May 23, 2006• 37 citations
arXivHTMLPDF

Abstract

Experimentally observable quantum accelerator modes are used as a test case for the study of some general aspects of quantum decay from classical stable islands immersed in a chaotic sea. The modes are shown to correspond to metastable states, analogous to the Wannier-Stark resonances. Different regimes of tunneling, marked by different quantitative dependence of the lifetimes on $1∕\ensuremath{\hbar}$, are identified, depending on the resolution of KAM substructures that is achieved on the scale of $\ensuremath{\hbar}$. The theory of resonance assisted tunneling introduced by Brodier et al. [Ann. Phys. 300, 88 (2002)] is reexamined and found to well describe decay whenever applicable.

DOI
10.1103/physreva.73.052110

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Experimentally observable quantum accelerator modes are used as a test case for the study of some general aspects of quantum decay from classical stable islands immersed in a chaotic sea. The modes are shown to correspond to metastable states, analogous to the Wannier-Stark resonances. Different regimes of tunneling, marked by different quantitative dependence of the lifetimes on 1∕\ensuremathℏ1∕\ensuremath{\hbar}1∕\ensuremathℏ, are identified, depending on the resolution of KAM substructures that is achieved on the scale of \ensuremathℏ\ensuremath{\hbar}\ensuremathℏ. The theory of resonance assisted tunneling introduced by Brodier et al. [Ann. Phys. 300, 88 (2002)] is reexamined and found to well describe decay whenever applicable.
This thesis is devoted to the study of some theoretical aspects of the problem of Quantum Accelerator Modes (QAMs), which has received considerable interest in recent years both by experimentalists and theorists in the fields of Quantum Chaos, Quantum Optics and cold-ultracold atomic physics.\nQAMs, recently observed in cold atom optics, are formed by exposing cold alkali atoms to periodic kicks in the direction of the gravitational field. The kicks are generated by a pulsed standing wave of light. A QAM is characterized by a momentum transfer, which increases linearly with the number of pulses, to a substantial fraction of the atoms. QAMs arise for values of the pulse period close to a integer multiple of half of a characteristic time TB (the Talbot time), typical of the kind of atoms used.\nA detailed theoretical explanation of this quantum effect is presented. The system can be modelled by a variant of the well-known quantum Kicked Rotator, in which the effects of a static force, produced by the earth gravitational field, are taken into account. A pseudo-classical theory has been formulated, in which the role of Planck constant is played by the detuning of the kicking period from the resonant value: in the asymptotic limit for a vanishing detuning, QAMs are described by the stable periodic orbit of a “formally” classical map on the 2-torus. An analysis of parametric dependence of the map is performed perturbatively; the parameter regions where the motion is stable and periodic exhibit Arnol’d tongues-like structures. The ordering of Arnol’d tongues and hence the classification of experimentally observed accelerator modes is provided by a number-theoretic construction known as the Farey Tree.\nQuantum decay of the modes from classical stable islands immersed in a chaotic sea is analysed and its relations to the famous Wannier-Stark problem are investigated. Theoretical estimates for the lifetimes of the metastable states as a function of the effective Planck constant are obtained.\nQAM dynamics in a Bose-Einstein condensate is then considered. Modifications on transport phenomena, imposed by nonlinearities of Gross-Pitaevskii type, which describe the atomic interactions in a mean-field approach, are examined by computer-assisted analysis.\nFinally, the problem of QAMs in the vicinity of a generic quantum resonance, namely for kicking period sufficiently close to any rational multiple of the Talbot time, is studied.\nThe theoretical framework is nontrivially generalized in terms of spinor dynamics: an ansatz of the Born-Oppenheimer type allows to decouple the dynamics of spinor and orbital degrees of freedom and a description of the orbital motion by means of “formally” classical equations is achieved. New rich families of experimental observable QAMs are predicted.
The study of quantum tunneling dates back to the early days of quantum mechanics, and has been a major topic of investigation in both theory and experiment. In recent years, there has been significant effort to understand tunneling phenomena for systems that exhibit chaos in their classical counterpart [13,15,18,29,42]. In particular, tunneling was predicted for mixed systems where chaotic and stable regions are intermingled [14]. The tunneling in this case is between two stable regions (referred to as nonlinear resonances, or islands of stability) in the classical phase space. The classical transport between these islands is forbidden by “dynamical barriers” in phase space. In contrast, quantum tunneling can couple the two islands so that a wave packet oscillates coherently between the two symmetry-related regions. This phenomenon is called dynamical tunneling because it occurs in phase space, without a fixed potential barrier. In the case of a mixed phase space, the presence of the chaotic region can enhance the rate of dynamical tunneling, a phenomenon known as chaos-assisted tunneling [53]. In this chapter, we review our experimental observation of chaosassisted tunneling with ultracold cesium atoms [48-51]. We first give a brief overview of some basic concepts and of earlier work in order to provide a context for our results.
The recent pioneering experiments of the [Nature 412, 52 (2001)] and [Science, 293, 274 (2001)] groups have demonstrated the dynamical tunneling of cold atoms interacting with standing electromagnetic waves. It has been shown [Phys. Rev. Lett. 89, 253201 (2002)], that the tunneling oscillations observed in these experiments correspondingly stems from two- and three-Floquet quantum state mechanism and can be controlled by varying the experimental parameters. The question where are the fingerprints of the classical chaotic dynamics in a quantum dynamical process which is controlled by 2 or 3 quantum states remains open. Our calculations show that although the effective $\ensuremath{\hbar}$ associated with the two experiments is large, and the quantum system is far from its semiclassical limit, the quantum Floquet-Bloch quasienergy states still can be classified as regular and chaotic states. In both experiments the quantum and the classical phase-space entropies are quite similar, although the classical phase space is a mixed regular-chaotic space. It is also shown that as the wave packet which is initially localized at one of the two inner regular islands oscillates between them through the chaotic sea, it accumulates a random phase which causes the decay of the amplitude of the oscillating mean momentum, ⟨P(t)⟩⟨P(t)⟩⟨P(t)⟩, as measured in both experiments. The extremely high sensitivity of the rate of decay of the oscillations of ⟨P(t)⟩⟨P(t)⟩⟨P(t)⟩ to the very small changes in the population of different Floquet-Bloch states, is another type of fingerprint of chaos in quantum dynamics that presumably was measured in the NIST and AUSTIN experiments for the first time.
Using a freely falling cloud of cold cesium atoms periodically kicked by pulses from a vertical standing wave of laser light, we present the first experimental observation of high-order quantum accelerator modes. This confirms the recent prediction by Fishman, Guarneri, and Rebuzzini [Phys. Rev. Lett. 89, 084101 (2002)]]. We also show how these accelerator modes can be identified with the stable regions of phase space in a classical-like chaotic system, despite their intrinsically quantum origin.
In the presence of a complex classical dynamics associated with amixed phase space, a quantum wave function can tunnel between two stable islands through the chaotic sea, an effect that has no classical counterpart. This phenomenon, referred to as chaos-assisted tunneling, is characterized by large fluctuations of the tunneling rate when a parameter is varied. To date, the full extent of this effect as well as the associated statistical distribution have never been observed in a quantum system. Here, we analyze the possibility of characterizing these effects accurately in a cold-atom experiment. Using realistic values of the parameters of an experimental setup, we examine through analytical estimates and extensive numerical simulations a specific system that can be implemented with cold atoms, the atomic modulated pendulum. We assess the efficiency of three possible routes to observe in detail chaos-assisted tunneling properties. Our main conclusion is that due to the fragility of the symmetry between positive and negative momenta as a function of quasimomentum, it is very challenging to use tunneling between classical islands centered on fixed points with opposite momentum. We show that it is more promising to use islands symmetric in position space, and characterize the regime where it could be done. The proposed experiment could be realized with current state-of-the-art technology.
We formulate a general method for the study of semiclassical-like dynamics in stable regions of a mixed phase space, in order to theoretically study the dynamics of quantum accelerator modes. In the simplest case, this involves determining solutions, which are stable when constrained to remain pure-state Gaussian wave packets, and then propagating them using a cumulant-based formalism. Using this methodology, we study the relative longevity, under different parameter regimes, of quantum accelerator modes. Within this attractively simple formalism, we are able to obtain good qualitative agreement with exact wave-function dynamics.