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 of the BCS coupling constant. We show that if initially the fermions are in a normal phase, the amp
litude of the BCS order parameter averaged over the Boltzman 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 quench dynamics of a topological $p$-wave superfluid with two competing order parameters, $Delta_pm(t)$. When the system is prepared in the $p+ip$ ground state and the interaction strength is quenched, only $Delta_+(t)$ is nonzero. Howev
er, we show that fluctuations in the initial conditions result in the growth of $Delta_-(t)$ and chaotic oscillations of both order parameters. We term this behavior phase III. In addition, there are two other types of late time dynamics -- phase I where both order parameters decay to zero and phase II where $Delta_+(t)$ asymptotes to a nonzero constant while $Delta_-(t)$ oscillates near zero. Although the model is nonintegrable, we are able to map out the exact phase boundaries in parameter space. Interestingly, we find phase III is unstable with respect to breaking the time reversal symmetry of the interaction. When one of the order parameters is favored in the Hamiltonian, the other one rapidly vanishes and the previously chaotic phase III is replaced by the Floquet topological phase III that is seen in the integrable chiral $p$-wave model.
Non-analyticities in the logarithm of the Loschmidt echo, known as dynamical quantum phase transitions [DQPTs], are a recently introduced attempt to classify the myriad of possible phenomena which can occur in far from equilibrium closed quantum syst
ems. In this work, we analytically investigate the Loschmidt echo in nonequilibrium $s$-wave and topological $p_x+ip_y$ fermionic superfluids. We find that the presence of non-analyticities in the echo is not invariant under global rotations of the superfluid phase. We remedy this deficiency by introducing a more general notion of a grand canonical Loschmidt echo. Overall, our study shows that DQPTs are not a good indicator for the long time dynamics of an interacting system. In particular, there are no DQPTs to tell apart distinct dynamical phases of quenched BCS superconductors. Nevertheless, they can signal a quench induced change in the topology and also keep track of solitons emerging from unstable stationary states of a BCS superconductor.
We study the collisionless dynamics of two classes of nonintegrable pairing models. One is a BCS model with separable energy-dependent interactions, the other - a 2D topological superconductor with spin-orbit coupling and a band-splitting external fi
eld. 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 $Delta(t)$ can either 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 time scale emerges in the relaxation to Phase III, such that short-time numerical simulations risk erroneously classifying the asymptotic state. This time scale diverges near integrable points. Second, an entirely new Phase IV of quasiperiodic oscillations of $|Delta|$ 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.
We predict synchronization of the chaotic dynamics of two atomic ensembles coupled to a heavily damped optical cavity mode. The atoms are dissipated collectively through this mode and pumped incoherently to achieve a macroscopic population of the cav
ity photons. Even though the dynamics of each ensemble are chaotic, their motions repeat one another. In our system, chaos first emerges via quasiperiodicity and then synchronizes. We identify the signatures of synchronized chaos, chaos, and quasiperiodicity in the experimentally observable power spectra of the light emitted by the cavity.
We study the dynamics of two ensembles of atoms (or equivalently, atomic clocks) coupled to a bad cavity and pumped incoherently by a Raman laser. Our main result is the nonequilibrium phase diagram for this experimental setup in terms of two paramet
ers - detuning between the clocks and the repump rate. There are three main phases - trivial steady state (Phase I), where all atoms are maximally pumped, nontrivial steady state corresponding to monochromatic superradiance (Phase II), and amplitude-modulated superradiance (Phase III). Phases I and II are fixed points of the mean-field dynamics, while in most of Phase III stable attractors are limit cycles. Equations of motion possess an axial symmetry and a $mathbb{Z}_{2}$ symmetry with respect to the interchange of the two clocks. Either one or both of these symmetries are spontaneously broken in various phases. The trivial steady state loses stability via a supercritical Hopf bifurcation bringing about a $mathbb{Z}_{2}$-symmetric limit cycle. The nontrivial steady state goes through a subcritical Hopf bifurcation responsible for coexistence of monochromatic and amplitude-modulated superradiance. Using Floquet analysis, we show that the $mathbb{Z}_{2}$-symmetric limit cycle eventually becomes unstable and gives rise to two $mathbb{Z}_{2}$-asymmetric limit cycles via a supercritical pitchfork bifurcation. Each of the above attractors has its own unique fingerprint in the power spectrum of the light radiated from the cavity. In particular, limit cycles in Phase III emit frequency combs - series of equidistant peaks, where the symmetry of the frequency comb reflects the symmetry of the underlying limit cycle. For typical experimental parameters, the spacing between the peaks is several orders of magnitude smaller than the monochromatic superradiance frequency, making the lasing frequency highly tunable.
In a recent preprint [arXiv:1803.04118v2] Chern and Barros report numerical simulations of the mean-field interaction quench dynamics, $U_ito U_f$, of the attractive Hubbard model that confirm our earlier prediction [Europhys. Lett. 85, 20004 (2008),
arXiv:0805.2798] of spontaneous eruption of spatial inhomogeneities in the post-quench state with periodically oscillating superconducting order. Chern and Barros attribute this instability with respect to spatial fluctuations to the large magnitude of the final Hubbard coupling $U_f$. We point out that this interpretation is inaccurate and discuss further work necessary to numerically verify the mechanism of the instability and the nature of the steady state.
We solve the non-stationary Schrodinger equation for several time-dependent Hamiltonians, such as the BCS Hamiltonian with an interaction strength inversely proportional to time, periodically driven BCS and linearly driven inhomogeneous Dicke models
as well as various multi-level Landau-Zener tunneling models. The latter are Demkov-Osherov, bow-tie, and generalized bow-tie models. We show that these Landau-Zener problems and their certain interacting many-body generalizations map to Gaudin magnets in a magnetic field. Moreover, we demonstrate that the time-dependent Schrodinger equation for the above models has a similar structure and is integrable with a similar technique as Knizhnikov-Zamolodchikov equations. We also discuss applications of our results to the problem of molecular production in an atomic Fermi gas swept through a Feshbach resonance and to the evaluation of the Landau-Zener transition probabilities.
We formulate a set of conditions under which dynamics of a time-dependent quantum Hamiltonian are integrable. The main requirement is the existence of a nonabelian gauge field with zero curvature in the space of system parameters. Known solvable mult
istate Landau-Zener models satisfy these conditions. Our method provides a strategy to incorporate time-dependence into various quantum integrable models, so that the resulting non-stationary Schrodinger equation is exactly solvable. We also validate some prior conjectures, including the solution of the driven generalized Tavis-Cummings model.
We study level statistics in ensembles of integrable $Ntimes N$ matrices linear in a real parameter $x$. The matrix $H(x)$ is considered integrable if it has a prescribed number $n>1$ of linearly independent commuting partners $H^i(x)$ (integrals of
motion) $left[H(x),H^i(x)right] = 0$, $left[H^i(x), H^j(x)right]$ = 0, for all $x$. In a recent work, we developed a basis-independent construction of $H(x)$ for any $n$ from which we derived the probability density function, thereby determining how to choose a typical integrable matrix from the ensemble. Here, we find that typical integrable matrices have Poisson statistics in the $Ntoinfty$ limit provided $n$ scales at least as $log{N}$; otherwise, they exhibit level repulsion. Exceptions to the Poisson case occur at isolated coupling values $x=x_0$ or when correlations are introduced between typically independent matrix parameters. However, level statistics cross over to Poisson at $ mathcal{O}(N^{-0.5})$ deviations from these exceptions, indicating that non-Poissonian statistics characterize only subsets of measure zero in the parameter space. Furthermore, we present strong numerical evidence that ensembles of integrable matrices are stationary and ergodic with respect to nearest neighbor level statistics.
