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arXiv:quant-ph/0306146 · Physical Review A

Semiclassical catastrophes and cumulative angular squeezing of a kicked quantum rotor

M. Leibscher, Ilya Sh. Averbukh, P. Rozmej, Robert Arvieu

Published Mar 11, 2004• 37 citations
arXivHTMLPDF

Abstract

We present a detailed theory of spectacular semiclassical catastrophes happening during the time evolution of a kicked quantum rotor [Phys. Rev. Lett. 87, 163601 (2001)]. Both two- and three-dimensional rotational systems are analyzed. It is shown that the wave function of the rotor develops a cusp at certain delay after a kick, which results in a sharply focused rotational wave packet. The cusp is followed by a fold-type catastrophe manifested in the rainbow-type moving singularities. In the three-dimensional case, the rainbows are accompanied by additional singular features similar to the glory structures known in the wave optics. These catastrophes in the time-dependent angular wave function are well described by the appropriate tools of the quasiclassical wave mechanics, i.e., by Airy and Bessel approximations and Pearcey's functions. A scenario of ``cumulative squeezing'' is also presented in which a specially designed train of short kicks produces an unlimited narrowing of the rotor angular distribution. This scenario is relevant to the molecular alignment by short laser pulses, and also to the atom lithography schemes in which cold atoms are focused by an optical standing wave.

DOI
10.1103/physreva.69.032102

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We present a detailed theory of spectacular semiclassical catastrophes happening during the time evolution of a kicked quantum rotor [Phys. Rev. Lett. 87, 163601 (2001)]. Both two- and three-dimensional rotational systems are analyzed. It is shown that the wave function of the rotor develops a cusp at certain delay after a kick, which results in a sharply focused rotational wave packet. The cusp is followed by a fold-type catastrophe manifested in the rainbow-type moving singularities. In the three-dimensional case, the rainbows are accompanied by additional singular features similar to the glory structures known in the wave optics. These catastrophes in the time-dependent angular wave function are well described by the appropriate tools of the quasiclassical wave mechanics, i.e., by Airy and Bessel approximations and Pearcey's functions. A scenario of ``cumulative squeezing'' is also presented in which a specially designed train of short kicks produces an unlimited narrowing of the rotor angular distribution. This scenario is relevant to the molecular alignment by short laser pulses, and also to the atom lithography schemes in which cold atoms are focused by an optical standing wave.
Semiclassical catastrophes in the dynamics of a quantum rotor (molecule) driven by a strong time-varying field are considered. We show that for strong enough fields, a sharp peak in the rotor angular distribution can be achieved via a time-domain focusing phenomenon, followed by the formation of rainbowlike angular structures. A strategy leading to the enhanced angular squeezing is proposed that uses a specially designed sequence of pulses. The predicted effects can be observed in many processes, ranging from molecular alignment (orientation) by laser fields to heavy-ion collisions, and the trapping of cold atoms by a standing light wave.
Summary form only given. Recently, we have proposed an aperiodic kicked-rotor excitation strategy leading to unlimited angular squeezing of rotational wave packets. It has been pointed out that this approach should be useful in several contexts, but it is in the setting of a pulsed optical lattice that we discuss in the present paper. We consider both classical and quantum regimes of preparing ultra-narrow (squeezed) atomic spatial distribution by a proper sequence of laser pulses forming the optical lattice. It is shown that after a linear focusing phenomenon caused by a single pulse, the rainbow-like structures appear in atomic spatial density.
Atom-optics kicked rotor represents an experimentally realizable version of the paradigmatic quantum kicked rotor system. After a short initial diffusive phase the cloud settles down to a stationary state due to the onset of dynamical localization. In this work we realise an enhancement of localization by modification of the kick sequence. We experimentally implement the modification to this system in which the sign of the kick sequence is flipped by allowing for a free evolution of the wavepackets for half the Talbot time after every MMM kicks. Depending on the value of MMM, this modified system displays a combination of enhanced diffusion followed by asymptotic localization. This is explained as resulting from two competing processes -- localization induced by standard kicked rotor type kicks, and diffusion induced by half Talbot time evolution. The evolving states display a localized but non-exponential wave function profiles. This provides another route to quantum control in kicked rotor class of systems. The numerical simulations agree well with the experimental results.
This thesis deals with an investigation of the chaotic dynamics of an atom optics δ-kicked rotor (AOKR) system, a paradigm of classical and quantum chaos. The AOKR rotor is realized by subjecting a Bose-Einstein condensate (BEC) of ⁸⁷Rb atoms to a periodic potential. The potential in the form of a standing wave, obtained from a far detuned laser beam is turned on and off periodically in time, producing short pulses (kicks) of laser light. For certain values of the period between the kicks, resonant effects known as quantum resonances are observed that are purely quantum effects. Quantum resonances have been examined in the AOKR for a series of kicks. For large number of kicks, the width of these resonances have been shown to become narrow. Higher order resonances corresponding to fractional Talbot times have been observed in the system. These quantum resonant effects have been found to depend sensitively on the initial velocity of the atoms. The first three primary resonances have been studied in detail, by measuring the momentum distribution and the kinetic energy of the atoms, for two and four kicks. A sinusoidal dependence of the energy on the initial momentum has been observed for two kicks. By increasing the number of kicks, the system becomes extremely sensitive to the initial momentum of the atoms. For four kicks, a more complex behavior is observed in the system energy. A scheme for investigating Loschmidt time reversal of chaotic dynamics of the atomic deBroglie waves have been described in the AOKR system. It has been demonstrated that the initial wave function of an ensemble of ultracold atoms is restored, after a series of delta-kicks in what is essentially a multiple beam interferometry experiment. During the process, the evolution of the system for a fraction of the atoms with a very small initial momentum is reversed, which results in a narrowing of the momentum distribution. It is hoped that the research presented in this thesis will significantly contribute to the understanding of the connection between the classical and quantum worlds.
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.
Atom-optics kicked rotor represents an experimentally reliable version of the paradigmatic quantum kicked rotor system. In this system, a periodic sequence of kicks are imparted to the cold atomic cloud. After a short initial diffusive phase the cloud settles down to a stationary state due to the onset of dynamical localization. In this paper, to explore the interplay between localized and diffusive phases, we experimentally implement a modification to this system in which the sign of the kick sequence is flipped after every M kicks. This is achieved in our experiment by allowing free evolution for half the Talbot time after every M kicks. Depending on the value of M, this modified system displays a combination of enhanced diffusion followed by asymptotic localization. This is explained as resulting from two competing processes-localization induced by standard kicked rotor type kicks, and diffusion induced by the half Talbot time evolution. The experimental and numerical simulations agree with one another. The evolving states display localized but nonexponential wave function profiles. This provides another route to quantum control in the kicked rotor class of systems.