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 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 study the collisionless dynamics of two classes of nonintegrable pairing models. One is a Bardeen-Cooper-Schrieffer model with separable energy-dependent interactions, and the other is a two-dimensional topological superconductor with spin-orbit coupling and a band-splitting external field. The long-time quantum quench dynamics at integrable points of these models are well understood. Namely the squared magnitude of the time-dependent order parameter $\mathrm{\ensuremath{\Delta}}(t)$ can vanish (Phase I), reach a nonzero constant (Phase II), or periodically oscillate as an elliptic function (Phase III). We demonstrate that nonintegrable models, too, exhibit some or all of these nonequilibrium phases. Remarkably, elliptic periodic oscillations persist, even though both their amplitude and functional form change drastically with integrability breaking. Striking new phenomena accompany loss of integrability. First, an extremely long timescale emerges in the relaxation to Phase III, such that short-time numerical simulations risk erroneously classifying the asymptotic state. This timescale diverges near integrable points. Second, an entirely new Phase IV of quasiperiodic oscillations of $|\mathrm{\ensuremath{\Delta}}|$ emerges in the quantum quench phase diagrams of nonintegrable pairing models. As integrability techniques do not apply for the models we study, we develop the concept of asymptotic self-consistency and a linear stability analysis of the asymptotic phases. With the help of these new tools, we determine the phase boundaries, characterize the asymptotic state, and clarify the physical meaning of the quantum quench phase diagrams of BCS superconductors. We also propose an explanation of these diagrams in terms of bifurcation theory.
The exact description of the time evolution of open correlated quantum systems remains one of the major challenges of condensed matter theory, especially for asymptotic long times where most numerical methods fail. Here, the post-quench dynamics of the
N-component Bose-Anderson impurity model is studied in the $N\ensuremath{\rightarrow}\ensuremath{\infty}$ limit. The equilibrium phase diagram is similar to that of the Bose-Hubbard model in that it contains local versions of Mott and Bose phases. Using a numerically exact procedure, we are able to study the real-time evolution including asymptotic long-time regimes. The formation of long-lived transient phases is observed for quench paths crossing foreign phases. For quenches inside the local Bose condensed phase, a dynamical phase transition is reported that separates the evolution towards a new equilibrium state and a regime characterized at large times by a persistent phase rotation of the order parameter. We explain how such nondecaying modes can exist in the presence of a dissipative bath. We discuss the extension of our results to the experimental relevant finite-
N case and their implication for the existence of nondecaying modes in generic quantum systems in the presence of a bath.
We present a pseudospin formulation for the post-quench dynamics of charge-density-wave (CDW) order in the half-filled spinless Holstein model on a square lattice, assuming spatially homogeneous evolution. This Anderson pseudospin description captures the coherent nonequilibrium dynamics of the coupled electron-lattice system. Numerical simulations reveal three distinct dynamical regimes of the CDW order parameter following a quench-locked oscillations, Landau-damped dynamics, and overdamped relaxation-closely paralleling quench dynamics in BCS superconductors and other electronically driven symmetry-breaking phases. Crucially, however, the presence of dynamical lattice degrees of freedom leads to qualitatively different long-time behavior. In particular, while the oscillation amplitude is reduced in the damped regimes, CDW oscillations do not fully decay but instead persist indefinitely due to feedback from the lattice field. We further show that these persistent oscillations are characterized by a nonequilibrium electronic distribution, which provides an intuitive understanding of both their amplitude and the renormalization of the oscillation frequency relative to the bare Holstein phonon frequency. Our results highlight the essential role of lattice dynamics in nonequilibrium ordered phases and establish a clear distinction between electron-lattice-driven CDW dynamics and their purely electronic counterparts.