We study the time evolution of a system of fermions with pairing interactions at a finite temperature. The dynamics is triggered by an abrupt increase in the BCS coupling constant. We show that if initially the fermions are in a normal phase, the amplitude of the BCS order parameter averaged over the Boltzmann distribution of initial states exhibits damped oscillations with a relatively short decay time. The latter is determined by the temperature, the single-particle level spacing, and the ground-state value of the BCS gap for the new coupling. In contrast, the decay is essentially absent when the system was in a superfluid phase before the coupling increase.
We study the time evolution of a system of fermions with pairing interactions at a finite temperature. The dynamics is triggered by an abrupt increase in the BCS coupling constant. We show that if initially the fermions are in a normal phase, the amplitude of the BCS order parameter averaged over the Boltzmann distribution of initial states exhibits damped oscillations with a relatively short decay time. The latter is determined by the temperature, the single-particle level spacing, and the ground-state value of the BCS gap for the new coupling. In contrast, the decay is essentially absent when the system was in a superfluid phase before the coupling increase.
We revisit the study of amplitude oscillations in a pair condensate of fermions after an interaction quench, and generalize it to nonzero temperature. For small variations of the order parameter, we show that the energy transfer during the quench determines both the asymptotic pseudo-equilibrated value of the order parameter and the magnitude of the oscillations, after multiplication by, respectively, the static response of the order parameter and spectral weight of the pair-breaking threshold. Since the energy transferred to the condensed pairs decreases with temperature as the superfluid contact, the oscillations eventually disappear at the critical temperature. For deeper quenches, we generalize the regimes of persistent oscillations and monotonic decay to nonzero temperatures, and explain how they become more abrupt and are more easily entered at high temperatures when the ratio of the initial to final gap either diverges, when quenching towards the normal phase, or tends to zero, when quenching towards the superfluid phase. Our results are directly relevant for existing and future experiments on the non-equilibrium evolution of Fermi superfluids near the phase transition.
We determine the limiting dynamics of a fermionic condensate following a sudden perturbation for various initial conditions. We demonstrate that possible initial states of the condensate fall into two classes. In the first case, the order parameter asymptotes to a constant value. The approach to a constant is oscillatory with an inverse square root decay. This happens, e.g., when the strength of pairing is abruptly changed while the system is in the paired ground state and more generally for any nonequilibrium state that is in the same class as the ground state. In the second case, the order parameter exhibits persistent oscillations with several frequencies. This is realized for nonequilibrium states that belong to the same class as excited stationary states. Our classification of initial states extends the concept of excitation spectrum to nonequilibrium regime and allows one to predict the evolution without solving equations of motion.
We analyze the dynamics of a condensate of ultracold atomic fermions following an abrupt change of the pairing strength. At long times, the system goes to a nonstationary steady state, which we determine exactly. The superfluid order parameter asymptotes to a constant value. We show that the order parameter vanishes when the pairing strength is decreased below a certain critical value. In this case, the steady state of the system combines properties of normal and superfluid states -- the gap and the condensate fraction vanish, while the superfluid density is nonzero.
Quench dynamics of fermionic superfluids are an active topic both experimentally and theoretically. Using the BCS theory, such non-equilibrium problems can be reduced to nearly independent spin dynamics, only with a time-dependent mean-field pairing term. This results in persisting oscillations of the pairing strength in certain parameter regimes. However, experiments have observed that the oscillations decay rapidly when the interaction becomes strong, such as in the unitary Fermi gas [Phys. Rev. Res. 3, 023205 (2021)]. Theoretical analysis on this matter is still absent. In this work, we construct an SYK-like model to analyze the effect of strong interactions in a one-dimensional BCS system. We employ the large- N N approximation and a Green’s function-based technique to solve the equilibrium problem and quench dynamics. Our findings reveal that a strong SYK interaction suppresses the pairing order. Additionally, we verify that the system quickly thermalizes with SYK interactions, whether it involves intrinsic pairing order or proximity effect, resulting in a rapid decay of the oscillation strength. The decay rates exhibit different scaling laws against SYK interaction, which can be understood in terms of the Boltzmann equation. This work represents a first step towards understanding the attenuating dynamics of strongly interacting fermionic superfluids.
In this paper we study the BCS Bose-Einstein condensation (BEC) crossover scenario within the superconducting state, using a T-matrix approach which yields the ground state proposed by Leggett. Here we extend this ground state analysis to finite temperatures T and interpret the resulting physics. We find two types of bosoniclike excitations of the system: long lived, incoherent pair excitations and collective modes of the superconducting order parameter, which have different dynamics. Using a gauge invariant formalism, this paper addresses their contrasting behavior as a function of T and superconducting coupling constant g. At a more physical level, our paper emphasizes how, at finite T, BCS-BEC approaches introduce an important parameter ${\ensuremath{\Delta}}_{\mathrm{pg}}^{2}={\ensuremath{\Delta}}^{2}\ensuremath{-}{\ensuremath{\Delta}}_{\mathrm{sc}}^{2}$ into the description of superconductivity. This parameter is governed by the pair excitations and is associated with particle-hole asymmetry effects that are significant for sufficiently large g. In the fermionic regime, ${\ensuremath{\Delta}}_{\mathrm{pg}}^{2}$ represents the difference between the square of the excitation gap ${\ensuremath{\Delta}}^{2}$ and that of the superconducting order parameter ${\ensuremath{\Delta}}_{\mathrm{sc}}^{2}.$ The parameter ${\ensuremath{\Delta}}_{\mathrm{pg}}^{2},$ which is necessarily zero in the BCS (mean field) limit increases monotonically with the strength of the attractive interaction g. It follows that there is a significant physical distinction between this BCS-BEC crossover approach (in which g is the essential variable which determines ${\ensuremath{\Delta}}_{\mathrm{pg}})$ and the widely discussed phase fluctuation scenario in which the plasma frequency is the tuning parameter. Finally, we emphasize that in the strong coupling limit, there are important differences between the composite bosons that arise in crossover theories and the usual bosons of the (interacting) Bose liquid. Because of constraints imposed on the fermionic excitation gap and chemical potential, in crossover theories, the fermionic degrees of freedom can never be fully removed from consideration.
We investigate the nonequilibrium behavior of dilute, attractively interacting Fermi gases subjected to finite-duration ramps of their internal interaction strength. We identify three dynamical regimes as a function of ramp duration using a time-dependent version of the Bardeen-Cooper-Schrieffer theory of superconductivity to model these systems. For short ramp durations, these systems become nonsuperconducting; however, fermions with opposite momenta remain paired albeit with reduced amplitudes, and the associated pair amplitude distribution is nonthermal. In this first regime, the disappearance of superconductivity is due to the loss of phase coherence between pairs. By contrast, for intermediate ramp durations, superconductivity survives but the magnitude of the order parameter is reduced and presents long-lived oscillations. Finally, for long ramp durations, phase coherence is almost fully retained during the finite-duration interaction quench, and the steady state is characterized by a thermal-like pair amplitude distribution. Using this approach, one can therefore dynamically tune the coherence between pairs to control the magnitude of the superconducting order parameter and even engineer a nonequilibrium state made of preformed pairs.