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arXiv:2312.13391 · Physical review. B./Physical review. B

Integrable-to-thermalizing crossover in nonequilibrium superconductors

Andrey Grankin, Victor Galitski

Published Dec 12, 2025• 1 citation
arXivHTMLPDF

Abstract

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.

DOI
10.1103/nzyh-hgpr

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This thesis reports on theoretical investigations of the ultrafast nonequilibrium dynamics in superconductors. In particular, two nonequilibrium scenarios are modeled by combining several analytical and numerical methods: transient superconducting phase transition and collective excitations of a superconducting condensate. In the first part of the thesis the possibility to induce singlet superconducting correlations far from equilibrium is explored by applying time-dependent Lanczos algorithm to the one-dimensional extended Hubbard model. Here, the system is driven into a nonequilibrium state by (i) interaction quench and (ii) quench by pulse. In both cases the buildup of the superconducting correlations is studied in view of several correlation functions. Moreover, a transient Meissner effect, which reflects fingerprints of the induced superconductivity, is observed in calculations of the important for experiments time-dependent optical conductivity. In addition, the differences between nonequilibrium dynamics of the extended Hubbard model initiated by interaction quench and by pulse quench are discussed in detail. The second part is devoted to the excitation of collective modes in a superconductor out of equilibrium. First of all, focus is given to the properties of collective modes close to equilibrium conditions, where a linearized response can be expected. Within the framework of the (Nambu-) matrix kinetic theory in the clean limit a comprehensive gauge-invariant description of the collective modes associated with phase fluctuations of the order parameter is analytically formulated for the generalized case of two- band superconducting systems. Particular emphasis is on the application to the recently discovered noncentrosymmetric superconductors. A detailed analysis of the collective modes includes specification of the gauge mode (common to all superconductors) and the Leggett mode (collective fluctuations of the interband phase difference), and investigation of their subtle interplay in view of validity of the charge conservation law and the participation in the Anderson-Higgs mechanism. Further, the nonequilibrium excitation of the superconducting gaps for the case of two-band superconductors with singlet pairing correlations is investigated in a view of optical pump-probe experiments. For this purpose a semi-numerical method based on the density- matrix theory is formulated. In this case additional amplitude (Higgs) modes were identified, which allow to directly detect the properties of the superconducting condensates as a function of time. Moreover, a new interplay between the phase (Leggett) and amplitude (Higgs) modes is studied in detail. In particular it was found that the nonadiabatic excitation process leads to strong coupling between both modes. Finally, based on calculations of the time-dependent optical conductivity the prediction is made that the coupled Higgs and Leggett modes can be clearly visible in the pump-probe absorption spectra.
We investigate the dynamics of a phonon-mediated superconductor driven out of equilibrium. The electronic hopping amplitude is ramped down in time, resulting in an increased electronic density of states. The dynamics of the coupled electron-phonon model is investigated by solving Migdal-Eliashberg equations for the double-time Keldysh Green's functions. The increase of the density of states near the Fermi level leads to an enhancement of superconductivity when the system thermalizes to the new state at the same temperature. We provide a time- and momentum-resolved view on this thermalization process and show that it involves fast processes associated with single-particle scattering and much slower dynamics associated with the superconducting order parameter. The importance of electron-phonon coupling for the rapid enhancement and the efficient thermalization of superconductivity is demonstrated, and the results are compared to a BCS time-dependent mean-field approximation.
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We study the relaxation of the Holstein model after a sudden switch-on of the interaction by means of the nonequilibrium dynamical mean field theory, with the self-consistent Migdal approximation as an impurity solver. We show that there exists a qualitative change in the thermalization dynamics as the interaction is varied in the weak-coupling regime. On the weaker interaction side of this crossover, the phonon oscillations are damped more rapidly than the electron thermalization time scale, as determined from the relaxation of the electron momentum distribution function. On the stronger interaction side, the relaxation of the electrons becomes faster than the phonon damping. In this regime, despite long-lived phonon oscillations, a thermalized momentum distribution is realized temporarily. The origin of the ``thermalization crossover'' found here is traced back to different behaviors of the electron and phonon self-energies as a function of the electron-phonon coupling. In addition, the importance of the phonon dynamics is demonstrated by comparing the self-consistent Migdal results with those obtained with a simpler Hartree-Fock impurity solver that neglects the phonon self-energy. The latter scheme does not properly describe the evolution and thermalization of isolated electron-phonon systems.