We study the nonadiabatic dynamics of a two-dimensional
p+ip superfluid following an instantaneous quantum quench of the BCS coupling constant. The model describes a topological superconductor with a nontrivial BCS (trivial BEC) phase appearing at weak- (strong-) coupling strengths. We extract the exact long-time asymptotics of the order parameter $\ensuremath{\Delta}(t)$ by exploiting the integrability of the classical
p-wave Hamiltonian, which we establish via a Lax construction. Three different types of asymptotic behavior can occur depending upon the strength and direction of the interaction quench. We refer to these as the nonequilibrium phases {I, II, III}, characterized as follows. In phase I, the order parameter asymptotes to zero due to dephasing. In phase II, $\ensuremath{\Delta}\ensuremath{\rightarrow}{\ensuremath{\Delta}}_{\ensuremath{\infty}}$, a nonzero constant. Phase III is characterized by persistent oscillations of $\ensuremath{\Delta}(t)$. For quenches within phases I and II, we determine the topological character of the asymptotic states. We show that two different formulations of the bulk topological winding number, although equivalent in the BCS or BEC ground states, must be regarded as independent out of equilibrium. The first winding number
Q characterizes the Anderson pseudospin texture of the initial state; we show that
Q is generically conserved. For $Q\ensuremath{\ne}0$, this leads to the prediction of a ``gapless topological'' state when $\ensuremath{\Delta}$ asymptotes to zero. The presence or absence of Majorana edge modes in a sample with a boundary is encoded in the second winding number
W, which is formulated in terms of the retarded Green's function. We establish that
W can change following a quench across the quantum critical point. When the order parameter asymptotes to a nonzero constant, the final value of
W is well defined and quantized. We discuss the implications for the (dis)appearance of Majorana edge modes. Finally, we show that the parity of zeros in the bulk out-of-equilibrium Cooper-pair distribution function constitutes a
Z2-valued quantum number, which is nonzero whenever $W\ensuremath{\ne}Q$. The pair distribution can in principle be measured using rf spectroscopy in an ultracold-atom realization, allowing direct experimental detection of the
Z2 number. This has the following interesting implication: topological information that is experimentally inaccessible in the bulk ground state can be transferred to an observable distribution function when the system is driven far from equilibrium.
We study the dynamic response of an
s-wave BCS-BEC (atomic-molecular) condensate to detuning quenches within the two-channel model beyond the weak-coupling BCS limit. At long times after the quench, the condensate ends up in one of three main asymptotic states (nonequilibrium phases), which are qualitatively similar to those in other fermionic condensates defined by a global complex order parameter. In phase I the amplitude of the order parameter vanishes as a power law, in phase II it goes to a nonzero constant, and in phase III it oscillates persistently. We construct exact quench phase diagrams that predict the asymptotic state (including the many-body wave function) depending on the initial and final detunings and on the Feshbach resonance width. Outside of the weak-coupling regime, both the mechanism and the time dependence of the relaxation of the amplitude of the order parameter in phases I and II are modified. Also, quenches from arbitrarily weak initial to sufficiently strong final coupling do not produce persistent oscillations in contrast to the behavior in the BCS regime. The most remarkable feature of coherent condensate dynamics in various fermion superfluids is an effective reduction in the number of dynamic degrees of freedom as the evolution time goes to infinity. As a result, the long-time dynamics can be fully described in terms of just a few new collective dynamical variables governed by the same Hamiltonian only with ``renormalized'' parameters. Combining this feature with the integrability of the underlying (e.g., the two-channel) model, we develop and consistently present a general method that explicitly obtains the exact asymptotic state of the system.
Recent experimental progress in the fields of cold quantum gases and ultrafast optical spectroscopy of quantum materials allows us to controllably induce and probe nonadiabatic dynamics of superconductors and superfluids. The time evolution of the gap function before relaxation with the lattice is determined by the superposition of coherently evolving individual Cooper pairs within the manifold of the Bardeen-Cooper-Schrieffer (BCS) wave function. While dynamics following an abrupt quench of the pairing interaction strength in the single-band BCS model has been exactly solved due to the integrability of the model, the dynamics of postquench multiband superconductors remain under scrutiny. Here, we develop a generalization of the Volkov-Kogan Laplace-space perturbative method that allows us to determine the nonadiabatic gap dynamics of two-band fully gapped superconductors for a wide range of quench amplitudes. Our approach expands the long-time dynamics around the steady-state asymptotic value of the gap, which is self-consistently determined, rather than around the equilibrium value of the gap. We explicitly demonstrate that this method recovers the exact solution of the long-time gap dynamics in the single-band case and perfectly agrees with a numerical solution of the two-band model. We discover that dephasing of Cooper pairs from different bands leads to faster collisionless relaxation of the gap oscillation with a power law of ${t}^{\ensuremath{-}3/2}$ instead of the well-known ${t}^{\ensuremath{-}1/2}$ behavior found in the single-band case. Furthermore, the gap oscillations display beating patterns arising from the existence of two different asymptotic gap values. Our results have important implications to a variety of two-band superconductors driven out of equilibrium, such as iron-based superconductors,
MgB2, and
SrTiO3.
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.
I consider relaxation of the pairing amplitude in a disordered Bardeen-Cooper-Schrieffer (BCS) superconductor in the absence of the two-particle collisions. My main assumption is that nonmagnetic and magnetic disorder scattering rates are much smaller than the value of the superconducting pairing gap ${\mathrm{\ensuremath{\Delta}}}_{0}$. I derive a system of nonlinear equations which describe the collisionless relaxation of the pairing amplitude following a quench of the pairing strength. I find that in a superconductor in which scattering on paramagnetic impurities is dominant, the pairing amplitude in a steady state varies periodically with time even for small deviations from equilibrium. It is shown that such a steady state emerges due to scattering on paramagnetic impurities which leads to a decrease in the value of the resonant frequency of the amplitude mode below $2{\mathrm{\ensuremath{\Delta}}}_{0}$.
Fermi gas with time-dependent pairing interaction hosts several different dynamical states. Coupling between the collective BCS pairing mode and individual Cooper pair states can make the latter either synchronize or dephase. We describe transition from phase-locked undamped oscillations to Landau-damped dephased oscillations in the collisionless, dissipationless regime as a function of coupling strength. In the dephased regime, we find a second transition at which the long-time asymptotic pairing amplitude vanishes. Using a combination of numerical and analytical methods we establish a continuous (type II) character of both transitions.