Nonequilibrium dynamics from BCS to the bosonic limit

Abstract

Within the time-dependent Gutzwiller approximation applied to the negative-$U$ Hubbard model, we investigate the dynamics of a superconductor after an interaction quench for different values of the final interaction from weak to strong coupling. The equilibrium BCS-BEC crossover becomes a sharp transition in the out-of-equilibrium dynamics between weak- and strong-coupling dynamical phases. Two different frequencies (${\mathrm{\ensuremath{\Omega}}}_{J}<{\mathrm{\ensuremath{\Omega}}}_{U}$) dominate the order parameter dynamics. In the weak coupling phase, ${\mathrm{\ensuremath{\Omega}}}_{J}$ follows twice the asymptotic average value of the gap (${\mathrm{\ensuremath{\Omega}}}_{J}\ensuremath{\approx}2{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\infty}}$) and ${\mathrm{\ensuremath{\Omega}}}_{U}$ is much larger but hardy visible in the anomalous density. At long times the BCS dynamics is recovered but surprisingly differences remain during the transient phase. In the strong coupling phase, the dynamics decouples from the asymptotic value of the order parameter except at exactly half filling and close to the dynamical transition where ${\mathrm{\ensuremath{\Omega}}}_{U}\ensuremath{\approx}2{\mathrm{\ensuremath{\Delta}}}_{\ensuremath{\infty}}$. The out-of-equilibrium transient spectral density and optical conductivity are presented and discussed in relation to pump probe experiments. Both ${\mathrm{\ensuremath{\Omega}}}_{J}$ and ${\mathrm{\ensuremath{\Omega}}}_{U}$ give rise to a complex structure of self-driven slow Rabi oscillations which are visible in the nonequilibrium optical conductivity where also sidebands appear due to the modulation of the double occupancy by superconducting amplitude oscillations. Analogous results apply to CDW and SDW systems. Our results show that in systems with long coherence times, pump-probe experiments allow us to characterize the regime (BCS vs preformed pairs) through very specific out-of-equilibrium fingerprints.