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arXiv:quant-ph/0409145

Quantum and Classical Effects in the Two-Frequency Kicked Rotor with Variable Initial Phase

Terry Mullins, AJ Hilliard, MP Sadgrove, Maarten D. Hoogerland, AS Parkins, R. Leonhardt

Published Sep 21, 2004• 0 citations
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Abstract

We present an investigation into effects exhibited by the two-frequency kicked rotor. Experiments were performed and in addition quantum and classical dynamics were simulated and compared with the experimental results. The experiments involved pulsing the optical standing wave with two pulsing periods of differing frequencies and variable initial phase offset. The ratio of pulsing periods was sampled for rational and irrational values for different experimental runs. In this paper we present these results and examine the measured momentum distributions for the cause of any structures that are seen in the energy as the initial phase offset is changed. Irrational ratios exhibit no significant quantum effects, whereas rational ratios show dynamical localisation (DL) for certain values of the initial phase. However, most of the observed structure is found to be due to classical effects, in particular KAM boundaries, and is therefore not of uniquely quantum origin.

DOI
10.48550/arxiv.quant-ph/0409145

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We present an investigation into effects exhibited by the two-frequency kicked rotor. Experiments were performed and in addition quantum and classical dynamics were simulated and compared with the experimental results. The experiments involved pulsing the optical standing wave with two pulsing periods of differing frequencies and variable initial phase offset. The ratio of pulsing periods was sampled for rational and irrational values for different experimental runs. In this paper we present these results and examine the measured momentum distributions for the cause of any structures that are seen in the energy as the initial phase offset is changed. Irrational ratios exhibit no significant quantum effects, whereas rational ratios show dynamical localisation (DL) for certain values of the initial phase. However, most of the observed structure is found to be due to classical effects, in particular KAM boundaries, and is therefore not of uniquely quantum origin.
We examine the effect of the initial atomic momentum distribution on the dynamics of the atom-optical realisation of the quantum kicked rotor. The atoms are kicked by a pulsed optical lattice, the periodicity of which implies that quasi-momentum is conserved in the transport problem. We study and compare experimentally and theoretically two resonant limits of the kicked rotor: in the vicinity of the quantum resonances and in the semiclassical limit of vanishing kicking period. It is found that for the same experimental distribution of quasi-momenta, significant deviations from the kicked rotor model are induced close to quantum resonance, while close to the classical resonance (i.e. for small kicking period) the effect of the quasi-momentum vanishes.
This thesis presents an account of experimental and numerical investigations of two quantum systems whose respective classical analogues are chaotic. These are the delta-kicked rotor, a paradigm in classical chaos theory, and the novel delta-kicked accelerator, created by application of a constant external acceleration or torque to the rotor. The experimental realisation of these systems has been achieved by the exposure of laser-cooled caesium atoms to approximate delta-kicks from a pulsed, high-intensity, vertical standing wave of laser light. Gravity’s effect on the atoms can be controlled by appropriate shifting of the profile of the standing wave. Numerical simulations of the systems are based on a diffractive model of the potential’s effect. Each system’s dynamics are characterised by the final form of the momentum distribution and the dependence of the atoms’ mean kinetic energy on the number and time period of the delta-kicks. The phenomena of dynamical localisation and quantum resonances in the delta-kicked rotor, which have no counterparts in the system’s classical analogue, are observed and investigated. Similar experiments on the delta-kicked accelerator reveal the striking phenomenon of the quantum accelerator mode, in which a large momentum is transferred to a substantial fraction of the atomic ensemble. This feature, absent in the system’s classical analogue, is characterised and an analytic explanation is presented. The effect on each quantum system of decoherence, introduced through spontaneous emission in the atoms, is examined and comparison is made with the results of classical simulations. While having little effect on the classical systems, the level of decoherence used is found to degrade quantum signatures of behaviour. Classical-like behaviour is, to some extent, restored, although significant quantum features remain. Possible applications of the quantum accelerator mode are discussed. These include use as a tool in atom optics and interferometry, a technique for measuring gravity, and a method of preparing atoms in a particular region of phase space. This may allow measurement of quantum phase space stability, and hence investigation of quantum chaos and quantum-classical correspondence.
We present experimental observations of diffusion resonances for the quantum kicked rotor with weak decoherence. Cold caesium atoms are subject to a pulsed standing wave of near-resonant light, with spontaneous emission providing environmental coupling. The mean energy as a function of the pulse period is determined during the late-time diffusion period for a constant probability of spontaneous emission. Structure in the late-time energy is seen to increase with physical kicking strength. The observed structure is related to Shepelyansky's predictions of the initial quantum diffusion rates. Additional results of diffusion rates as a function of the effective Planck's constant are given, showing non-trivial behaviour in the quantum-to-classical transition regime.
In this thesis we investigate quantum resonant effects in the atom-optical ὁ-kicked accelerator. Using Floquet analysis, we theoretically study the time evolution of quantum systems which have a classical counterpart that exhibits chaotic dynamics. We introduce quantum resonance and quantum antiresonance features of the quantum ὁ -kicked rotor by setting the pulse period to an integer multiple of the half-Talbot time. The model is generalised to the atom-optical ὁ -kicked accelerator by considering thermal alkali atoms subject to a periodically pulsed standing wave potential formed from counter-propagating laser beams. The dynamics of the momentum distribution is analysed by evaluating the momentum moments and momentum cumulants. We derive analytic solutions for these observables for the ultracold and thermal limiting cases, and observe fractional quantum resonant phenomena. Simulations have been developed to examine the time evolution for individual momentum eigenstates, which we use to construct a non-interacting finite temperature gas, based upon a Monte Carlo method. We investigate the temperature dependence of the ὁ-kicked rotor, neglecting gravitational effects, and show that the atomic dynamics is highly sensitive to the initial momentum width of the gas. A generalisation of the model to quantify the transition between the ultracold and thermal temperature regimes of the atom-optical ὁ -kicked accelerator is examined using linear regression analysis. High order quantum resonance features are found to be sensitive to the relative acceleration between the atomic gas and the pulsed optical standing wave. We assess the dependence of the ὁ -kicked accelerator upon gravitational acceleration, quantifying the width of the high order quantum resonance features, which we use to assess the prospect for precision measurement using a finite temperature gas.