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arXiv:cond-mat/0404401 · Low Temperature Physics

Wigner distribution function formalism for superconductors and collisionless dynamics of the superconducting order parameter

M. H. S. Amin, E. V. Bezuglyi, A. S. Kijko, A. N. Omelyanchouk

Published Jul 1, 2004• 28 citations
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Abstract

A technique for studying collisionless dynamics of a homogeneous superconducting system is developed which is based on Riccati parametrization of the Wigner distribution function. The quantum evolution of the superconducting order parameter, initially deviating from the equilibrium value, is calculated using this technique. The effect of a time-dependent BCS paring interaction on the dynamics of the order parameter is also studied.

DOI
10.1063/1.1789939

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A technique for studying collisionless dynamics of a homogeneous superconducting system is developed which is based on Riccati parametrization of the Wigner distribution function. The quantum evolution of the superconducting order parameter, initially deviating from the equilibrium value, is calculated using this technique. The effect of a time-dependent BCS paring interaction on the dynamics of the order parameter is also studied.
I consider relaxation of the pairing amplitude in a disordered Bardeen-Cooper-Schrieffer (BCS) superconductor in the absence of the two-particle collisions. My main assumption is that nonmagnetic and magnetic disorder scattering rates are much smaller than the value of the superconducting pairing gap ${\mathrm{\ensuremath{\Delta}}}_{0}$. I derive a system of nonlinear equations which describe the collisionless relaxation of the pairing amplitude following a quench of the pairing strength. I find that in a superconductor in which scattering on paramagnetic impurities is dominant, the pairing amplitude in a steady state varies periodically with time even for small deviations from equilibrium. It is shown that such a steady state emerges due to scattering on paramagnetic impurities which leads to a decrease in the value of the resonant frequency of the amplitude mode below $2{\mathrm{\ensuremath{\Delta}}}_{0}$.
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.
Non-equilibrium aspects of the BCS model have fascinated physicists for decades, from the seminal works of Eliashberg to modern realizations in cold atom experiments. The latter scenarios have lead to a great deal of interest in the quench dynamics of fermions with pairing interactions. The recently introduced notion of a dynamical quantum phase transition is an attempt to classify the myriad of possible phenomena which can result in such far from equilibrium systems. These are defined as non-analytic points of the logarithm of the Loschmidt echo and are linked to oscillations in the dynamics a systems order parameter. In this work we analytically investigate the relation between DQPTs and oscillation of the superconducting order parameter in quenches of the BCS model. We find that each oscillation of the order parameter is accompanied by a DQPT which is first order in nature. We show this for a variety of initial states and furthermore find that when the order parameter attains a constant steady state then no DQPTS occur.
We investigate the quantum dynamics of a conventional superconductor, described by the attrac- tive Hubbard model, driven far from equilibrium by quenching the Hubbard attraction. We evolve the full density matrix of the system in time using the Heisenberg equation of motion, tracking the spatiotemporal dynamics of the pairing and density fields. We use very weakly inhomogeneous initial conditions to probe the stability of the homogeneous dynamics to spatial perturbations. At initial times after the quench the pairing and density fields remain homogeneous in space while fluctuating in time. However, depending on the quench strength the spatially uniform dynamics can become unstable beyond a threshold time, with emergence of spatial textures. The onset time for the instability depends on the strength of the quench and the initial inhomogeneity. At long times the system evolves towards a state with significantly reduced spatial heterogeneity. The time dependence of the global superconducting order parameter suggests two broad categories of re- sponse to a quench. One of these involves strongly inhomogeneous pairing and bond currents in the nonequilibrium superconductor.
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​.