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arXiv:2208.07320 · Physical review. B./Physical review. B

Nonlocality as the source of purely quantum dynamics of BCS superconductors

Aidan Zabalo, Ang-Kun Wu, J. H. Pixley, Emil A. Yuzbashyan

Published Sep 22, 2022• 3 citations
arXivHTMLPDF

Abstract

We show that the classical (mean-field) description of far from equilibrium superconductivity is exact in the thermodynamic limit for local observables but breaks down for global quantities, such as the entanglement entropy or Loschmidt echo. We do this by solving for and comparing exact quantum and exact classical long-time dynamics of a BCS superconductor with interaction strength inversely proportional to time and evaluating local observables explicitly. Mean field is exact for both normal and anomalous averages (superconducting order) in the thermodynamic limit. However, for anomalous expectation values, this limit does not commute with adiabatic and strong coupling limits and, as a consequence, their quantum fluctuations can be unusually strong. The long-time steady state of the system is a gapless superconductor whose superfluid properties are only accessible through energy resolved measurements. This state is nonthermal but conforms to an emergent generalized Gibbs ensemble. Our study clarifies the nature of symmetry-broken many-body states in and out of equilibrium and fills a crucial gap in the theory of time-dependent quantum integrability.

DOI
10.1103/physrevb.106.104513

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We show that the classical (mean-field) description of far from equilibrium superconductivity is exact in the thermodynamic limit for local observables but breaks down for global quantities, such as the entanglement entropy or Loschmidt echo. We do this by solving for and comparing exact quantum and exact classical long-time dynamics of a BCS superconductor with interaction strength inversely proportional to time and evaluating local observables explicitly. Mean field is exact for both normal and anomalous averages (superconducting order) in the thermodynamic limit. However, for anomalous expectation values, this limit does not commute with adiabatic and strong coupling limits and, as a consequence, their quantum fluctuations can be unusually strong. The long-time steady state of the system is a gapless superconductor whose superfluid properties are only accessible through energy resolved measurements. This state is nonthermal but conforms to an emergent generalized Gibbs ensemble. Our study clarifies the nature of symmetry-broken many-body states in and out of equilibrium and fills a crucial gap in the theory of time-dependent quantum integrability.
Within the time-dependent Gutzwiller approximation applied to the negative-UUU Hubbard model, we investigate the dynamics of a superconductor after an interaction quench for different values of the final interaction from weak to strong coupling. The equilibrium BCS-BEC crossover becomes a sharp transition in the out-of-equilibrium dynamics between weak- and strong-coupling dynamical phases. Two different frequencies (${\mathrm{\ensuremath{\Omega}}}_{J}<{\mathrm{\ensuremath{\Omega}}}_{U}$) dominate the order parameter dynamics. In the weak coupling phase, ${\mathrm{\ensuremath{\Omega}}}_{J}$ follows twice the asymptotic average value of the gap (${\mathrm{\ensuremath{\Omega}}}_{J}\ensuremath{\approx}2{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\infty}}$) and ${\mathrm{\ensuremath{\Omega}}}_{U}$ is much larger but hardy visible in the anomalous density. At long times the BCS dynamics is recovered but surprisingly differences remain during the transient phase. In the strong coupling phase, the dynamics decouples from the asymptotic value of the order parameter except at exactly half filling and close to the dynamical transition where ${\mathrm{\ensuremath{\Omega}}}_{U}\ensuremath{\approx}2{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\infty}}$. The out-of-equilibrium transient spectral density and optical conductivity are presented and discussed in relation to pump probe experiments. Both ${\mathrm{\ensuremath{\Omega}}}_{J}$ and ${\mathrm{\ensuremath{\Omega}}}_{U}$ give rise to a complex structure of self-driven slow Rabi oscillations which are visible in the nonequilibrium optical conductivity where also sidebands appear due to the modulation of the double occupancy by superconducting amplitude oscillations. Analogous results apply to CDW and SDW systems. Our results show that in systems with long coherence times, pump-probe experiments allow us to characterize the regime (BCS vs preformed pairs) through very specific out-of-equilibrium fingerprints.
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.
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{\mathrm{MgB}}_{2}MgB2​, and SrTiO3{\mathrm{SrTiO}}_{3}SrTiO3​.
Stabilizing superconductivity beyond its equilibrium regime has been one of the quests of nonequilibrium state engineering. We report the presence of nonthermal pairing gap in BCS superconductors coupled to two thermal baths, where one of the baths can be at a temperature well above the equilibrium TcT_cTc​. Superconducting state is enabled by a highly non-equilibrium steady state stabilized by weak coupling to baths that break detailed balance condition. The steady-state is well described by a generalized Gibbs ensemble parametrized with chemical potentials for associated Bogoliubov quasiparticle. A strong, non-perturbative effect is reported for both sss-wave and ddd-wave superconductors, and its possible realization in cavity-based experiments is discussed.
A standard perturbative expansion around the mean-field solution is used to derive the low-energy effective action for superconductors at T=0.T=0.T=0. Taking into account the density fluctuations at the outset we get the effective action where the density $\ensuremath{\rho}$ is the conjugated momentum to the phase $\ensuremath{\theta}$ of the order parameter. In the hydrodynamic regime, the dynamics of the superconductor is described by a time-dependent nonlinear Schr\"odinger (TDNLS) equation for the field $\ensuremath{\Psi}(x)=\sqrt{\ensuremath{\rho}/2}{e}^{i\ensuremath{\theta}}.$ The evolution of the density fluctuations in the crossover from weak-coupling (BCS) to strong-coupling (Bose condensation of localized pairs) superconductivity is discussed for the attractive Hubbard model. In the bosonic limit, the TDNLS equation reduces to the Gross-Pitaevskii equation for the order parameter, as in the standard description of superfluidity. The conditions under which a phase-only action can be derived in the presence of a long-range interaction to describe the physics of the superconductivity of ``bad metals'' are discussed.
A theory of superfluidity (S.F.) is developed from first principles using two novel concepts, (1) that of a 'superfluid Ensemble' (S.E.) i.e. a 'Restricted Ensemble' constructed from a 'Separable Phase Space' (S.P.S.) admitting independent configurations, at least one set of which are statistically equivalent. (2) The notion of 'Dynamical Equivalence' (D.E.), satisfied if and only if (i) all dynamical symmetries are rearranged (not broken) for two Lagrangian formulations of the same problem and if (ii) the expectation values of all the constants of motion are the same, even if their functional expressions are not. The dynamical variables (d.v.) of the S.P.S. are defined from the ('q' and 'c' number) fields of the most general 'Linear Coherent State Representation', more general than those of Glauber and Bogoljubov-Valatin combined. Three independent pairs of d.v. are obtained. D.E. is proven for the Ideal Bose Gas and for a non-linear, interacting zero order Bose problem (I.Z.O.P.). An exact relation is obtained from the action principle, ensuring the cancellation of 'low and high order dangerous diagrams'. From this it follows that D.E. for the exact interacting problem must be demonstrable at infinite order of perturbation, in the finite volume limit. The I.Z.O.P. is posed in the Random Phase Approximation (R.P.A.), free from 'anomalous averages' and solved for the three branches of the excitation spectrum in a pure state description; the lowest branch is gapless, whilst the upper two coincide and show a gap. The standard strategy of linearization is found to be faulty. The partition functions for both superfluid and non-superfluid ensembles are obtained for the I.Z.O.P. in the R.P.A. The coincident upper two branches (in a pure state description) split into a band in thermal equilibrium for the superfluid ensemble, in agreement with an upper band recently observed experimentally. O.D.L.R.O. is found in the second reduced density matrix, but ruled out in the first. Integral equations are obtained in thermal equilibrium - for the I.Z.O.P. in the R.P.A.; which differ, however, from those of existing approaches for the same problem. Most existing theories of S.F. are in fact shown not to predict superfluid behaviour. The present theory is applicable to arbitrary Bose or Fermi systems, whether superfluid or not. O.D.L.R.O. is found to be sufficient for SF. No a priori assumption is made as to the occurrence of Bose-Einstein condensation, its existence being here contingent on the solution of the integral equation; in any case, it is not to be associated with O.D.L.R.O. or S.F.