Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

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

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

Published Jan 1, 2004• 13 citations
OpenAlex

Abstract

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

Related papers

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

M. H. S. Amin, E. V. Bezuglyi, A. S. Kijko +1 more

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.

arXiv:cond-mat/0404401•Low Temperature Physics•Jul 1, 2004•28 citations

Collisionless dynamics of the pairing amplitude in disordered superconductors

Maxim Dzero

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 $2{\mathrm{\ensuremath{\Delta}}}_{0}$.

arXiv:2303.06750•Physical review. B./Physical review. B•Mar 8, 2024•10 citations

Dynamical Quantum Phase transitions and Recurrences in the Non-Equilibrium BCS model

Colin Rylands, Victor M. Galitski

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.

arXiv:2001.10084•Jan 27, 2020•4 citations

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

H. P. Ojeda Collado, Gonzalo Usaj, J. Lorenzana +1 more

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.

arXiv:1901.08607•Physical review. B./Physical review. B•May 15, 2019•21 citations

Distribution of the order parameter in strongly disordered superconductors: An analytic theory

Anton V. Khvalyuk, M. V. Feigel’man

We developed an analytic theory of inhomogeneous superconducting pairing in strongly disordered materials, which are moderately close to superconducting-insulator transition. Single-electron eigenstates are assumed to be Anderson localized, with a large localization volume. Superconductivity develops due to coherent delocalization of originally localized preformed Cooper pairs. The key assumption of the theory is that each such pair is coupled to a large number $Z\ensuremath{\gg}1$ of similar neighboring pairs. We derived integral equations for the probability distribution $P(\mathrm{\ensuremath{\Delta}})$ of local superconducting order parameter $\mathrm{\ensuremath{\Delta}}(\mathbit{r})$ and analyzed their solutions in the limit of small dimensionless Cooper coupling constant $\ensuremath{\lambda}\ensuremath{\ll}1$. The shape of the order-parameter distribution is found to depend crucially upon the effective number of nearest neighbors ${Z}_{\text{eff}}=2{\ensuremath{\nu}}_{0}{\mathrm{\ensuremath{\Delta}}}_{0}Z$, with ${\ensuremath{\nu}}_{0}$ being the single-particle density of states at the Fermi level. The solution we provide is valid both at large and small ${Z}_{\text{eff}}$; the latter case is nontrivial as the function $P(\mathrm{\ensuremath{\Delta}})$ is heavily non-Gaussian. One of our key findings is the discovery of a broad range of parameters where the distribution function $P(\mathrm{\ensuremath{\Delta}})$ is non-Gaussian but also noncritical (in the sense of superconductor-insulator transition criticality). The analytic results are supplemented by numerical data and good agreement between them is observed.

arXiv:2106.11848•Physical review. B./Physical review. B•Dec 15, 2021•3 citations

Time-dependent Ginzburg-Landau theory for a weak-coupling superconductor

Henk T. C. Stoof

Using a functional approach to the Keldysh formalism we develop a time-dependent Ginzburg-Landau equation for the long-wavelength and low-frequency dynamics of a weak-coupling superconductor, which is also valid in situations that deviate significantly from thermal equilibrium. The theory takes the interactions between quasiparticles and Cooper pairs into account exactly and is explicitly Galilean-invariant at zero temperature. However, the temporal and spatial dependence of the order parameter is assumed to be such that the effects of Landau damping can be neglected. We also consider the screening properties of the superconductor and show that the inclusion of the polarizability of the electron gas is necessary for a complete description.

Physical review. B, Condensed matter•Apr 1, 1993•30 citations
A technique for studying collisionless dynamics of a homogeneous superconducting system is \ndeveloped which is based on Riccati parametrization of the Wigner distribution function. The \nquantum evolution of the superconducting order parameter, initially deviating from the equilibrium \nvalue, is calculated using this technique. The effect of a time-dependent BCS paring \ninteraction 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 $2{\mathrm{\ensuremath{\Delta}}}_{0}$.
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.
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.
We developed an analytic theory of inhomogeneous superconducting pairing in strongly disordered materials, which are moderately close to superconducting-insulator transition. Single-electron eigenstates are assumed to be Anderson localized, with a large localization volume. Superconductivity develops due to coherent delocalization of originally localized preformed Cooper pairs. The key assumption of the theory is that each such pair is coupled to a large number $Z\ensuremath{\gg}1$ of similar neighboring pairs. We derived integral equations for the probability distribution $P(\mathrm{\ensuremath{\Delta}})$ of local superconducting order parameter $\mathrm{\ensuremath{\Delta}}(\mathbit{r})$ and analyzed their solutions in the limit of small dimensionless Cooper coupling constant $\ensuremath{\lambda}\ensuremath{\ll}1$. The shape of the order-parameter distribution is found to depend crucially upon the effective number of nearest neighbors ${Z}_{\text{eff}}=2{\ensuremath{\nu}}_{0}{\mathrm{\ensuremath{\Delta}}}_{0}Z$, with ${\ensuremath{\nu}}_{0}$ being the single-particle density of states at the Fermi level. The solution we provide is valid both at large and small Zeff{Z}_{\text{eff}}Zeff​; the latter case is nontrivial as the function $P(\mathrm{\ensuremath{\Delta}})$ is heavily non-Gaussian. One of our key findings is the discovery of a broad range of parameters where the distribution function $P(\mathrm{\ensuremath{\Delta}})$ is non-Gaussian but also noncritical (in the sense of superconductor-insulator transition criticality). The analytic results are supplemented by numerical data and good agreement between them is observed.
Using a functional approach to the Keldysh formalism we develop a time-dependent Ginzburg-Landau equation for the long-wavelength and low-frequency dynamics of a weak-coupling superconductor, which is also valid in situations that deviate significantly from thermal equilibrium. The theory takes the interactions between quasiparticles and Cooper pairs into account exactly and is explicitly Galilean-invariant at zero temperature. However, the temporal and spatial dependence of the order parameter is assumed to be such that the effects of Landau damping can be neglected. We also consider the screening properties of the superconductor and show that the inclusion of the polarizability of the electron gas is necessary for a complete description.