Nonlocality as the source of purely quantum dynamics of BCS superconductors

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 longtime 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 non-thermal 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. ment entropy in Sec. XII. We conclude and outline possible directions for future research in Sec. XIII. II. SUMMARY OF THE PAPERThis section is a condensed version of the present paper. We first summarize our key results and then separately list several notable complementary findings. We obtain four key results in this paper:(1) The exact long-time solution of the non-stationary Schrödinger equation for the BCS Hamiltonian, see Eq. ( 11), with superconducting coupling g(t) = 1 νt . The initial condition is the exact ground state at t = 0 + .