Nonequilibrium dynamics of superconductivity in the attractive Hubbard model

Abstract

We present a framework of semiclassical superconductivity (SC) dynamics that properly includes effects of spatial fluctuations for the attractive Hubbard model. We consider both coherent and adiabatic limits. To model the coherent SC dynamics, we develop a real-space von Neumann equation based on the time-dependent Hartree-Fock-Bogoliubov theory. Applying our method to interaction quenches in the negative-U Hubbard model, we show that the relaxation of SC order at weak coupling is dominated by Landau-damping. At strong coupling, we find a two-stage relaxation of the pairing field: a collapse of the synchronized oscillation of Cooper pairs due to spatial inhomogeneity, followed by a slow relaxation to a quasi-stationary state. SC dynamics in adiabatic limit is described by a quantum Landau-Lifshitz equation with Ginzburg-Landau relaxation. Numerical simulations of the pump-probe process show that long time recovery of the pairing field is dominated by defects dynamics. Our results demonstrate the important role of spatial fluctuations in both limits.The nonequilibrium dynamics of superconductivity (SC) subject to an external stimulation has been intensively studied for some time [1][2][3][4][5][6][7][8]. This interest has recently been renewed by remarkable pump-probe experiments reporting the observation of the collective amplitude mode [9,10]. Prior work largely studies two physical limits determined by the relation between the quasiparticle relaxation time τ and the relaxation time τ ∆ of SC order parameter. The SC dynamics in the collisionless limit, τ τ ∆ , can be described using a timedependent self-consistent field approach. An interesting phenomenon in this regime is the collective Rabi oscillations of the order parameter [3,4]. In the opposite, adiabatic limit, τ τ ∆ , the dynamics of the pairing field is usually described by time-dependent Ginzburg-Landau (TDGL) equation [11,12]. TDGL has recently been employed to simulate the out-of-equilibrium dynamics of superconductors in pump-probe experiments [13,14].Numerical simulations based on TDGL also provide useful insights on the dynamical inhomogeneity of nonequilibrium superconductivity [15,16], for example, the formation of topological defects by rapid thermal quenches as described in the Kibble-Zurek scenario [17,18]. However, contrary to numerous large-scale TDGL simulation studies, effects of spatial fluctuations in the collisionless limit are rarely addressed in most numerical studies, even though earlier calculations [19,20] have demonstrated dramatic effects of spatial inhomogeneity in out-of-equilibrium superconductivity.In this paper, we present a theoretical framework for SC dynamics that properly includes the effects of spatial fluctuations in both the collisionless and adiabatic limits. To model the collisionless (coherent) limit, we develop a real-space formulation of the time-dependent Hartree-Fock-Bogoliubov (TDHFB) theory, which is particularly suitable for the negative-U Hubbard model. Numerical solution shows two distinct dynamical regim...

Generation and decay of Higgs mode in a strongly interacting Fermi gas

“…The mode decays into a new state with spatially nonuniform order parameter. Numerical simulations performed for the attractive Fermi-Hubbard model [15,22] indicated that the mode is indeed dynamically unstable, which was consistent with the theoretical predictions.…”

Dynamical phase transitions in the collisionless pre-thermal states of isolated quantum systems: theory and experiments

“…V A) is the fact that practical sample sizes typically have linear dimension L ξ, where ξ = v F /π∆ is the coherence length. It can be shown that phase III is unstable in this case to the spontaneous generation of spatial fluctuations ("Cooper pair turbulence") [141,164]. The mechanism is parametric resonance.…”

Consequences of integrability breaking in quench dynamics of pairing Hamiltonians

“…Here T ∆ is the characteristic period of ∆(t) oscillations (T ∆ is of the order of the inverse equilibrium gap ∆ 0f in our separable BCS models). Another limitation is the parametric instability of Phase III with respect to spontaneous eruptions of spatial inhomogeneities [89][90][91][92] . To avoid this instability, the system size has to be smaller than the superconducting coherence length.…”