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.
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.
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.
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 the first part of this thesis, we present a unified kinetic theory that describes the finite-temperature, non-equilibrium dynamics of a Bose-Einstein condensed gas interacting with a thermal cloud in a trap. This theory includes binary interactions to second order in the interaction potential and reduces to a diagonal quantum Boltzmann equation for Bogoliubov quasiparticles. The Hartree-Fock-Bogoliubov interactions include the pairing field and are expressed as many-body T matrices to second order. The interactions thus include the correct renormalized scattering physics. This renormalized theory is automatically gapless. Thus, the excited Bogoliubov modes are naturally orthogonal to the condensate ground state. This kinetic theory is a complete second-order theory that reduces to the Gross-Pitaevskii equation and the quantum Boltzmann equation in the respective limits and thus is capable of describing the system over a wide temperature range. In the second part, we consider a many-body theory of a dilute Fermi gas near a Feshbach resonance. Experiments explore the crossover physics between the Bardeen-Cooper-Schrieffer (BCS) superfluidity of a two-spin Fermi gas, and the Bose-Einstein condensation (BEC) of composite bosons. We consider correlations between a composite boson and a fermion pair and show that such correlations are the minimal ingredients needed in a many-body theory to generate the correct boson-boson scattering length in the Bose-Einstein limit of the crossover. We also use imaginary-time propagation to find zero-temperature ground states in the BCS/BEC crossover. A cumulant expansion allows us to systematically include higher-order interactions between bosons and fermions. In particular, we calculate the Hartree term across the resonance. We further apply the cumulant-expansion method to thermal fermions and composite bosons interacting above the transition temperature in the normal phase. We numerically calculate the full time dependence in ramps across the resonance in this regime and find different two-body and many-body time scales in the system. We calculate molecular conversion efficiencies as a function of temperature and phase-space density, and find good agreement with results from JILA potassium experiments.
Research in the field of ultra-cold quantum gases provides a testbed to examine Hamiltonians from other areas of physics such as the Bose-Hubbard Hamiltonian, as well as interesting phenomena such as superconductivity and superfluidity. Of emerging interest in the field is the analysis of non-equilibrium behaviour. This thesis focuses on the non-equilibrium dynamics of Bose gases in two different elongated systems: a continuously-pumped atom laser, and the formation of solitons as a result of a rapid quench through the Bose-Einstein condensation phase transition. We begin by introducing the theoretical tools required to model these systems, including the Stochastic Projected Gross-Pitaevskii formalism that describes the dynamics of a condensate in contact with a thermal cloud. We apply the Stochastic Projected Gross-Pitaevskii formalism to model a Raman-outcoupled atom laser continuously outcoupled from a replenished condensate reservoir at finite temperature. We investigate the first and second order coherence of the output as a function of the temperature of the condensate reservoir. For large momentum kicks given to outcoupled atoms, we show for the first time that the coherence of the atom laser beam is linked to that of the condensate reservoir. We observe a decrease in first order coherence of the condensate reservoir as its temperature is increased, and show that this decrease is caused by the dephasing of thermal fluctuations. We also investigate the effects of varying the outcoupling rate and the momentum kick on the first order coherence of the beam. We demonstrate that the first order coherence of the atom laser beam deviates from that of the condensate reservoir for small momentum kicks which we identify as due to interactions between the condensate and atom laser beam. Thermal density fluctuations in the condensate cause the atom laser beam to experience a stochastic force and the temporal coherence of the atom laser beam is reduced. We find that the condensate reservoir is significantly depleted for large outcoupling rates, resulting in a reduction in the steady state condensate fraction and in the first order coherence of the atom laser beam. In a second study, the momentum linewidth of a guided atom laser outcoupled using a radio-frequency transition is investigated at zero and finite temperature. We examine the momentum distribution of the beam by measuring the transmission of atoms over a potential barrier following the experimental scheme of the group of Professor Alain Aspect at Institut d'Optique. We observe a noisy momentum distribution for outcoupling rates in the weak-outcoupling limit at zero temperature. We show that the noise is caused by bulk excitations within the condensate reservoir as a result of a density depression caused by the non-adiabatic switch-on of outcoupling. We show that these excitations can be prevented by increasing the outcoupling rate to its final value linearly over time. In addition, we demonstrate that a thermally-broadened linewidth could potentially be measured using this experimental technique, and discuss challenges associated with the measurement scheme. Finally, we focus on the non-equilibrium dynamics of condensate growth in elongated geometries. Following a rapid quench through the Bose-Einstein condensation phase transition, we observe the spontaneous appearance of density dips which we identify as solitons. We develop an algorithm for counting these solitons in order to observe the temporal dependence of the soliton number as a function of the quench rate. To compare with the predictions of the Kibble-Zurek mechanism describing topological defect formation as a result of the dynamics of phase transitions in homogeneous systems, we calculated the scaling exponent of the soliton number with respect to the quench rate. Our results were qualitatively consistent with the Kibble-Zurek predictions, but quantitative comparisons were inconclusive. In trapped systems the phase transition proceeds inhomogeneously, with a phase front that propagates in space with a velocity. If this is lower than a critical value it has been predicted that this can inhibit defect formation. However, our results do not show conclusive evidence of this in the parameter regime that we study. We find that the number of solitons reduces as the condensate reaches equilibrium as a result of the damping included in the Stochastic Projected Gross-Pitaevskii formalism.
We consider the atom-optical $\ensuremath{\delta}$-kicked accelerator when the initial momentum distribution is symmetric. We demonstrate the existence of quantum-resonant dynamics, and derive analytic expressions for the system evolution. In particular, we consider the dynamical evolution of the momentum moments and find that all even-ordered momentum moments exhibit a power-law growth. In the ultracold (zero-temperature) limit the exponent is determined by the order of the moment, whereas for a broad, thermal initial momentum distribution the exponent is reduced by 1. To demonstrate the power-law behavior explicitly we consider the evolutions of the second- and fourth-order momentum moments, and cumulants, for an initially Gaussian momentum distribution corresponding to the Maxwell-Boltzmann distribution of an ideal gas at thermal equilibrium.
After the first experimental realization of a Bose-Einstein condensate (BEC) in 1995, BECs have become a subject of intense experimental and theoretical study. In this dissertation, I present our results on the classical and quantum dynamics of BECs at zero temperature under different scenarios. First, I consider the analog of slow light in the collision of two BECs near a Feshbach resonance. The scattering length then becomes a function of the collision energy. I derive a generalization of the Gross-Pitaevskii equation for incorporating this energy dependence. In certain parameter regimes, the group velocity of a BEC traveling through another BEC decreases. I also study the feasibility of an experimental realization of this phenomena. Second, I analyze an experiment in which a BEC in a ring-shaped trap is stirred by a rotating barrier. The phase drop across and current flow through the barrier is measured from spiral-shaped density profiles created by interfering the BEC in the ring-shaped trap and a concentric reference BEC after release from all trapping potentials. I show that a free-particle expansion is sufficient to explain the origin of the spiral pattern and relate the phase drop to the geometry of a spiral. I also bound the expansion times for which the phase drop can be accurately determined and study the effect of inter-atomic interactions on the expansion time scales. Third, I study the dynamics of few-mode BECs when they become dynamically unstable after preparing an initial state at a saddle point of the Hamiltonian. I study the dynamics within the truncated Wigner approximation (TWA) and find that, due to phase-space mixing, the expectation value of an observable relaxes to a steady-state value. Using the action-angle formalism, we derive analytical expressions for the steady-state value and the time evolution towards this value. I apply these general results to two systems: a condensate in a double-well potential and a spin-1 (spinor) condensate. Finally, I study quantum corrections beyond the TWA in the semiclassical limit. I derive general expressions for the dynamics of an observable by using the van Vleck-Gutzwiller propagator and find that the interference of classical paths leads to non-perturbative corrections. As a case study, I consider a single-mode nonlinear oscillator; this system displays collapse and revival of observables. I find that the interference of classical paths, which is absent in the TWA, leads to revivals.