Fate of dynamical phases of a BCS superconductor beyond the dissipationless regime

Abstract

The BCS model of an isolated superconductor initially prepared in a nonequilibrium state, predicts the existence of interesting dynamical phenomena in the time-dependent order parameter as decaying oscillations, persistent oscillations and overdamped dynamics. To make contact with real systems remains an open challenge as one needs to introduce dissipation due to the environment in a self-consistent computation. Here, we reach this goal with the use of the Keldysh formalism to treat the effect of a thermal bath. We show that, contrary to the dissipationless case, all dynamical phases reach the equilibrium order parameter in a characteristic time that depends on the coupling with the bath. Remarkably, as time evolves, the overdamped phase shows a fast crossover where the superconducting order parameter recovers to reach a state with a well-developed long range order that tends towards equilibrium with the damped Higgs mode oscillations. Our results provide a benchmark for the description of the dynamics of real out-of-equilibrium superconductors relevant for quantum technological applications.

Nonlinear dynamics of driven superconductors with dissipation

“…In the presence of inelastic scattering (γ) the superconducting order parameter is reduced. As already mentioned, another important result is that at equilibrium the present formalism presents a rigorous justification for the Dynes formula for the density of states [28]. As will be shows explicitly below, this provides a simple way to estimate the γ parameter close to equilibrium directly from tunneling experiments [27,29].…”

“…The method to treat the bath was explained in detail in Ref. [28]. Here we summarize the main results.…”

“…There have been several theoretical proposals to provide a formal justification including Eliashberg physics [49], inelastic tunneling [50] and magnetic impurities [51,52]. In our previous work we have shown that the coupling with a bath [28] provides a mechanism to justify Dynes formula and allows to link directly equilibrium tunneling with the γ parameter. Of course since we are not providing a microscopic theory of the bath our justification is still semi-phenomenological.…”

“…A simple estimate can be obtained through the Dynes parameter [27] in tunneling experiments which, as explicitly shown in Ref. [28], is directly related to the damping of quasiparticles by the bath ultimately limiting the coherent dynamics. In aluminium samples, an inverse Dynes parameter 1 γ ≡ τ ∼ 10 6 ∆ ∼ µs has been measured [29] which suggest that there is a large time window for coherent dynamics.…”

Adiabatic transition from a BCS superconductor to a Fermi liquid and phase dynamics

“…The experiments mentioned above have motivated the analysis of the BCS pairing problem with time-dependent interactions and several proposals based on the realization of a suitable out-of equilibrium dynamics (pump) which is then measured by a probe pulse. [8][9][10][11][12][13][14][15][16][17][18] Within the pseudospin formulation of Anderson 19 the problem can be mapped onto an effective spin Hamiltonian for which the Bloch dynamics can be solved exactly. [20][21][22][23][24] In the present paper we investigate the case of adia-batically switching off the interaction in a BCS superconductor.…”