The absence of a characteristic momentum scale in the pseudopotential description of atomic interactions in ultracold (two-component Fermi) gases is known to lead to divergences in perturbation theory. Here we show that they also plague the calculation of the dynamics of the total energy following a quantum quench. A procedure to remove the divergences is devised, which provides finite answers for the time evolution of the total energy after a quench in which the interaction strength is ramped up according to an arbitrary protocol. An important result of this analysis is the time evolution of the asymptotic tail of the momentum distribution (related to Tan's contact) to leading order in the scattering length. Explicit expressions for the dynamics of the total energy and the contact for a linear interaction ramp are obtained, as a function of the interaction ramp time in the crossover from the sudden quench to the adiabatic limit. In the sudden quench limit, the contact, following a rapid oscillation, reaches a stationary value which is different from the equilibrium one. In the adiabatic limit, the contact grows quadratically in time and later saturates to the equilibrium value for the final value of the scattering length.
The absence of a characteristic momentum scale in the pseudopotential description of atomic interactions in ultracold (two-component Fermi) gases is known to lead to divergences in perturbation theory. Here we show that they also plague the calculation of the dynamics of the total energy following a quantum quench. A procedure to remove the divergences is devised, which provides finite answers for the time evolution of the total energy after a quench in which the interaction strength is ramped up according to an arbitrary protocol. An important result of this analysis is the time evolution of the asymptotic tail of the momentum distribution (related to Tan's contact) to leading order in the scattering length. Explicit expressions for the dynamics of the total energy and the contact for a linear interaction ramp are obtained, as a function of the interaction ramp time in the crossover from the sudden quench to the adiabatic limit. In the sudden quench limit, the contact, following a rapid oscillation, reaches a stationary value which is different from the equilibrium one. In the adiabatic limit, the contact grows quadratically in time and later saturates to the equilibrium value for the final value of the scattering length.
Recent advances in the tunability of ultracold atomic gases have created opportunities for studying interesting quantum many-body systems. Fano-Feshbach resonances, in particular, allow experimenters to freely adjust the scattering of atoms by controlling an external magnetic field. By rapidly changing this field near a resonance, it is possible to drive systems out of equilibrium towards novel quantum states where correlations between atoms change dynamically. In this thesis, we take a wave-function-based approach to theoretically examine the response of several interesting systems to suddenly-switched, or "quenched", interactions.\nWe first calculate the time evolution of a Bose-Einstein condensate that is quenched to the unitarity regime, where the scattering length 'a' diverges. Working within the time-dependent variational formalism, we find that the condensate does not deplete as quickly as the usual Bogoliubov theory would suggest. We also make a quantitative prediction for the dynamics of short-range pair correlations, encoded in Tan's contact. We then consider the dynamics of these correlations for quenches to small 'a', and we find that bound states can cause high-contrast oscillations of the contact. These dynamics can be modelled quantitatively at short times by using a properly-chosen two-body model. Finally, we characterize the nonlocal correlation waves that are generated by an interaction quench in arbitrary dimensionality. Our analysis demonstrates that the large-momentum limit of the post-quench momentum distribution can sometimes include contributions from both the short range and the long range, depending on the quench protocol.
We present an experimental study of a two component Fermi gas following an interaction quench into the superfluid phase. Starting with a weakly attractive gas in the normal phase, interactions are ramped to unitarity at a range of rates and we measure the subsequent dynamics as the gas approaches equilibrium. Both the formation and condensation of fermion pairs are mapped via measurements of the pair momentum distribution and can take place on very different timescales, depending on the adiabaticity of the quench. The contact parameter is seen to respond very quickly to changes in the interaction strength, indicating that short-range correlations, based on the occupation of high-momentum modes, evolve far more rapidly than the correlations in low-momentum modes necessary for pair condensation.
The study of the interplay between interaction-induced correlations and nonequilibrium initial conditions in many-body systems has recently attracted a lot of attention. New experimental techniques provide high control over many-body systems and systematic access to their nonequilibrium regime. In many systems the nonequilibrium properties of a Fermi liquid can be relevant. A first approach to their understanding is the main content of this thesis. At the beginning, nonequilibrium phenomena, Landau's theory of a Fermi liquid, the Hubbard model, time evolution in quantum mechanics, related experiments, and the flow equation method are reviewed. The key observation of this thesis, namely a characteristic mismatch of expectation values in equilibrium and nonequilibrium, is first illustrated for the squeezed oscillator. For this one-particle model Hamiltonian the perturbative approach can be compared with the exact solution and it is shown that the mismatch holds even beyond perturbation theory. Afterwards these observations are generalized to a larger class of one-particle models. Then the nonequilibrium behavior of a Fermi liquid is examined by analyzing the Fermi liquid phase of the Hubbard model in more than one dimension. After a sudden switch-on of a weak two-particle interaction to the noninteracting Fermi gas the relaxation of the many-body system is observed. For this purpose, the flow equation transformation is implemented for the Hubbard Hamiltonian. Then the discussion of the momentum distribution function and of the kinetic energy displays a three-step relaxation behavior of the Fermi liquid from the initial perturbation until thermalization is reached. Firstly, the sudden switch inserts excitation energy into the system which drives the following dynamics. Then a rapid initial build-up of correlations is caused by initial dephasing and leads to the establishment of a quasiparticle picture. By that the system enters into a quasi-stationary state which can be long-lasting for weak interaction. This state shows prethermalization of the kinetic energy which already has relaxed to its final values. However, the momentum distribution still resembles a zero temperature Fermi liquid. Its later relaxation on a second time scale is caused by a residual two-particle interaction which allows for scattering processes and can be described by a quantum Boltzmann equation. The physical origin of this delayed relaxation can be traced back to an interplay of translational invariance of the Hubbard model and the Pauli principle for fermions; together they restrict the phase space for scattering events. Comparing with similar work I conjecture on the generic nature of the findings made for the quenched Fermi liquid in other many-body systems. Finally I point out to the potential relevance of the delayed relaxation for the observation of further nonequilibrium phenomena, for instance in BCS systems. In order to extend the study of sudden switching to arbitrary switching processes the calculation is repeated using the Keldysh perturbation theory. First evaluations for a linear increase of the interaction strength are given. Extensions of this work are suggested and may motivate further research.
Non-equilibrium phenomena in many-body quantum systems involves the interplay of correlations and dynamics spanning different spatio-temporal scales. We present an experimental study of these dynamics in a two-component Fermi gas following a quench from weak to strong interactions. Starting with a weakly attractive gas in the normal phase, interactions are ramped to the unitarity limit at a range of rates, which involves crossing the normal to superfluid phase transition. We measure the subsequent dynamics of the pair momentum distribution which reveals the timescales for both the formation and condensation of pairs. Furthermore, we show that the contact parameter responds very rapidly to changes in the interaction strength, demonstrating that short-range correlations, based on the occupation of high-momentum modes, evolve far more rapidly than correlations in low-momentum modes necessary for pair condensation.
We study the dynamics of a zero-temperature Bose condensate following a sudden quench of the scattering length from noninteracting to unitarity (infinite scattering length). We apply three complementary approaches to understand the momentum distribution and loss rate. First, using a time-dependent variational ansatz for the many-body state, we calculate the dynamics of the momentum distribution. Second, we demonstrate that, at short times and large momenta compared to those set by the density, the physics can be understood within a simple, analytic two-body model. We make a quantitative prediction for the evolution of Tan's contact and find features in the momentum distribution that are absent in equilibrium. Third, we study three-body loss at finite density under the same dynamic scenario. We find lifetimes that are long compared to the saturation times of large-momentum modes, and we relate this result to the three-body inelasticity parameter.
The recent experimental implementation of condensed matter models in optical lattices has motivated research on their non-equilibrium behavior. Predictions about the dynamics of superconductors following a sudden quench of the pairing interaction have been made based on the effective Bardeen–Cooper–Schrieffer (BCS) Hamiltonian; however, their experimental verification requires the preparation of a suitable excited state of the Hubbard model along a twofold constraint: (i) a sufficiently non-adiabatic ramping scheme is essential to excite the non-equilibrium dynamics and (ii) overheating beyond the critical temperature of superconductivity must be avoided. For commonly discussed interaction ramps, there is no clear separation of the corresponding energy scales. Here we show that the matching of both conditions is simplified by the intrinsic relaxation behavior of ultracold fermionic systems: for the particular example of a linear ramp we examine the transient regime of prethermalization (Moeckel and Kehrein 2008 Phys. Rev. Lett. 100 175702) under the crossover from sudden to adiabatic switching using Keldysh perturbation theory. A real-time analysis of the momentum distribution exhibits a temporal separation of an early energy relaxation and its later thermalization by scattering events. For long but finite ramping times this separation can be large. In the prethermalization regime, the momentum distribution resembles a zero-temperature Fermi liquid as the energy inserted by the ramp remains located in high-energy modes. Thus ultracold fermions prove to be robust to heating, which simplifies the observation of non-equilibrium BCS dynamics in optical lattices.