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.
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.
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.
Motivated by recent experiments in ultracold atomic gases that explore the nonequilibrium dynamics of interacting quantum many-body systems, we investigate the opposite limit of Landau's Fermi-liquid paradigm: We study a Hubbard model with a sudden interaction quench, that is, the interaction is switched on at time t=0. Using the flow equation method, we are able to study the real time dynamics for weak interaction U in a systematic expansion and find three clearly separated time regimes: (i) An initial buildup of correlations where the quasiparticles are formed. (ii) An intermediate quasi-steady regime resembling a zero temperature Fermi liquid with a nonequilibrium quasiparticle distribution function. (iii) The long-time limit described by a quantum Boltzmann equation leading to thermalization of the momentum distribution function with a temperature T proportional, variantU.
We study the crossover between the sudden quench limit and the adiabatic dynamics of superconducting states in the attractive Hubbard model. We focus on the dynamics induced by the change of the attractive interaction during a finite ramp time. The ramp time is varied in order to track the evolution of the dynamical phase diagram from the sudden quench to the equilibrium limit. Two different dynamical regimes are realized for quenches towards weak and strong coupling interactions. At weak coupling the dynamics depends only on the energy injected into the system, whereas a dynamics retaining memory of the initial state takes place at strong coupling. We show that this is related to a sharp transition between a weak and a strong coupling quench dynamical regime, which defines the boundaries beyond which a dynamics independent from the initial state is recovered. Comparing the dynamics in the superconducting and nonsuperconducting phases, we argue that this is due to the lack of an adiabatic connection to the equilibrium ground state for nonequilibrium superconducting states in the strong coupling quench regime.
Motivated by recent experiments on interacting cold atoms, we analyze interaction quenches in Luttinger liquids (LLs), where the interaction is ramped from zero to a finite value within a finite time. The fermionic single particle density matrix reveals several regions of spatial and temporal coordinates relative to the quench time, termed as Fermi liquid, sudden quench LL, adiabatic LL regime, and a LL regime with a time-dependent exponent. The various regimes can also be observed in the momentum distribution of the fermions, directly accessible through time of flight experiments. Most of our results apply to arbitrary quench protocols.
Understanding nonequilibrium dynamics of strongly interacting quantum systems represents one of the most challenging problems in many-body physics. Here, quantum thermalization dynamics are explored in real-time in an ultracold Fermi gas suddenly quenched to the BEC-BCS crossover. When quenched to unitarity, it is observed that the cloud size remains unchanged in the early evolution while the momentum distribution emerges two prethermal states with a lifetime difference of two orders of magnitude in the early and intermediate stage before very slowly evolving to the final stationary state. It is revealed that a crossover momentum, at which the momentum distribution remains nearly unchanged, is determined by the thermal wavelength at high temperatures and the Fermi momentum while at low temperatures. It is identified that the universal prethermal dynamics scaling where momentum distributions with different temperatures collapse onto one curve. When quenched to the BEC side, the thermalization rapidly relaxes into a prethermal state and exhibits the low energy oscillation related to the molecular bound states. This work provides benchmarks for the study of quantum thermalization in strongly interacting fermionic many-body systems.
Motivated by the experiment by M. Budden {\em et al.} [Nature Physics {\bf 17}, 611 (2021)], who observed signatures of long-lived photo-induced superconductivity, we develop an accurate analytical/computational approach to non-equilibrium superconductivity following a quench. We consider the BCS-Holstein model, which includes both integrable local electron-electron interactions and integrability-breaking electron-phonon coupling. We develop Keldysh-Eliashberg theory on the Kadanoff-Baym contour, which enables us to describe non-equilibrium dynamics of the superconductor. We consider a quench in interactions, which results in a dynamic transition from the initial superconducting state to a normal thermal state in the end of the evolution. It is shown that the dynamics contain two stages: The early-time integrable behavior, involving coherent oscillations of the superconducting order parameter, crosses over to the late-time ergodic dynamics exhibiting a thermal decay into an equilibrium state. In the former regime, our computational approach both reproduces exact analytical results on the integrable dynamics of the order parameter and generalizes those to the case of an initial thermal state. The method also succeeds for the first time in describing both integrable-to-thermalizing crossover and the late-time thermal decay, which is shown to be consistent with the time-dependent Ginzburg-Landau theory (with the exponential decay time dependent on the density of quasiparticle excitations). We observe the electron distribution function approaching the Fermi-Dirac thermal distribution at final stages. The details of two-time non-equilibrium dynamics depend on the density of quasiparticles in the initial state and the integrability-breaking parameters, which under certain conditions may result in a long-lived transient superconductivity consistent with experiment.