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.
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.
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.
In this thesis, the dynamics of cold, trapped atomic gases are investigated, and the prospects for exploiting their nonlinear dynamics for inertial sensing are discussed. In the first part, the resonant and antiresonant dynamics of the atom-optical quantum delta-kicked accelerator with an initial symmetric momentum distribution are considered. The system is modelled as an ideal, non-interacting atomic gas, with a temperature-dependence governed by the width of the initial momentum distribution. The existence of resonant and antiresonant behaviour is established, and analytic expressions describing the dynamics of momentum moments of the time-evolved momentum distribution are derived. In particular, the momentum moment dynamics in both the resonant and antiresonant regimes depend strongly on the width of the initial momentum distribution. The resonant dynamics of all even-ordered momentum moments are shown to exhibit a power-law growth with an exponent given by the order of the moment in the zero-temperature regime, whereas for a broad, thermal initial momentum distribution the exponent is reduced by one. The cross-over in the intermediate regime is also examined, and a characteristic time is determined up to which the system exhibits dynamics associated with the zero-temperature regime. A similar analysis is made for the temperature-dependence of the antiresonant dynamics. This general behaviour is demonstrated explicitly by considering a Maxwell-Boltzmann and uniform momentum distribution, allowing exact expressions describing the dynamics of the second- and fourth-order momentum moments, and momentum cumulants, to be obtained. The relevance of these results to the potential of using this system in accurate determinations of the local gravitational acceleration is discussed. In the second part, the dynamics of one- and two-component Bose-Einstein Condensates prepared in a counter-rotating superposition of flows in a quasi-1D toroidal trap are studied. Particular attention is paid to the dynamical stability of the initial state in the presence of atom-atom interactions, included via a mean-field description within the Gross-Pitaevskii equation. A broad regime of dynamical stability using a two-component BEC is identified, in which a typical implementation using Rb-87 is predicted to lie. A proof-of-principle Sagnac atom-interferometer using a two-component Rb-87 BEC is then presented, and the accumulation of the Sagnac phase is shown to be possible via relative population measurement or, alternatively, through the continuous monitoring the precession of atomic density fringes. In contrast to conventional Sagnac interferometers, the accumulation of the Sagnac phase is independent of the enclosed area of the interferometer. The prospects of using this system for high-precision determinations of rotation is discussed.
We consider resonant dynamics in a dilute atomic gas falling under gravity through a periodically pulsed standing-wave laser field. Our numerical calculations are based on a Monte Carlo method for an incoherent mixture of noninteracting plane waves, and we show that quantum resonances are highly sensitive to the relative acceleration between the atomic gas and the pulsed optical standing wave. For particular values of the atomic acceleration, we observe fractional resonances. We investigate the effect of the initial atomic momentum width on the fractional resonances and quantify the sensitivity of fractional resonances to thermal effects.
In this thesis, the dynamics of cold, trapped atomic gases are investigated, and the prospects for exploiting their nonlinear dynamics for inertial sensing are discussed. In the first part, the resonant and antiresonant dynamics of the atom-optical quantum delta-kicked accelerator with an initial symmetric momentum distribution are considered. The system is modelled as an ideal, non-interacting atomic gas, with a temperature-dependence governed by the width of the initial momentum distribution. The existence of resonant and antiresonant behaviour is established, and analytic expressions describing the dynamics of momentum moments of the time-evolved momentum distribution are derived. In particular, the momentum moment dynamics in both the resonant and antiresonant regimes depend strongly on the width of the initial momentum distribution. The resonant dynamics of all even-ordered momentum moments are shown to exhibit a power-law growth with an exponent given by the order of the moment in the zero-temperature regime, whereas for a broad, thermal initial momentum distribution the exponent is reduced by one. The cross-over in the intermediate regime is also examined, and a characteristic time is determined up to which the system exhibits dynamics associated with the zero-temperature regime. A similar analysis is made for the temperature-dependence of the antiresonant dynamics. This general behaviour is demonstrated explicitly by considering a Maxwell-Boltzmann and uniform momentum distribution, allowing exact expressions describing the dynamics of the second- and fourth-order momentum moments, and momentum cumulants, to be obtained. The relevance of these results to the potential of using this system in accurate determinations of the local gravitational acceleration is discussed. In the second part, the dynamics of one- and two-component Bose-Einstein Condensates prepared in a counter-rotating superposition of flows in a quasi-1D toroidal trap are studied. Particular attention is paid to the dynamical stability of the initial state in the presence of atom-atom interactions, included via a mean-field description within the Gross-Pitaevskii equation. A broad regime of dynamical stability using a two-component BEC is identified, in which a typical implementation using Rb-87 is predicted to lie. A proof-of-principle Sagnac atom-interferometer using a two-component Rb-87 BEC is then presented, and the accumulation of the Sagnac phase is shown to be possible via relative population measurement or, alternatively, through the continuous monitoring the precession of atomic density fringes. In contrast to conventional Sagnac interferometers, the accumulation of the Sagnac phase is independent of the enclosed area of the interferometer. The prospects of using this system for high-precision determinations of rotation is discussed.
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.