Abstract
The recent experimental implementation of condensed matter models in optical lattices has motivated research on their non-equilibrium behavior. Predictions about the dynamics of superconductors following a sudden quench of the pairing interaction have been made based on the effective Bardeen–Cooper–Schrieffer (BCS) Hamiltonian; however, their experimental verification requires the preparation of a suitable excited state of the Hubbard model along a twofold constraint: (i) a sufficiently non-adiabatic ramping scheme is essential to excite the non-equilibrium dynamics and (ii) overheating beyond the critical temperature of superconductivity must be avoided. For commonly discussed interaction ramps, there is no clear separation of the corresponding energy scales. Here we show that the matching of both conditions is simplified by the intrinsic relaxation behavior of ultracold fermionic systems: for the particular example of a linear ramp we examine the transient regime of prethermalization (Moeckel and Kehrein 2008 Phys. Rev. Lett. 100 175702) under the crossover from sudden to adiabatic switching using Keldysh perturbation theory. A real-time analysis of the momentum distribution exhibits a temporal separation of an early energy relaxation and its later thermalization by scattering events. For long but finite ramping times this separation can be large. In the prethermalization regime, the momentum distribution resembles a zero-temperature Fermi liquid as the energy inserted by the ramp remains located in high-energy modes. Thus ultracold fermions prove to be robust to heating, which simplifies the observation of non-equilibrium BCS dynamics in optical lattices.
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GENERAL SCIENTIFIC SUMMARY Introduction and background. The interplay of quantum correlations and nonequilibrium initial conditions in interacting many-body quantum systems can be studied by following the temporal evolution of an excited state. Initialization in such a state is possible by changing the two-particle interaction nonadiabatically in time. For fermions, strong correlations are imposed by the Pauli principle; below the critical temperature of superconductivity attractive interactions lead to pairing of fermions around the Fermi energy. Their nonequilibrium behavior can be observed only if two competing constraints can be met; a sufficiently strong excitation must be possible without overheating the superconductor beyond its critical temperature. Then one expects, for instance, temporal oscillations in the spectral gap parameter.
Main results. We find an interaction switching process which allows to meet both the above requirements simultaneously. This is possible since two time scales separate such that a transient time window of opportunity (called the prethermalization regime) arises. First, a nonequilibrium superconducting state builds up shortly after the interaction switching. However, the excitation energy is mostly inserted into high energy modes which do not participate in the pairing mechanism. Therefore the observation of nonequilibrium superconductivity is possible before the second time scale (thermalization) is reached. It is characterized by the redistribution of the excitation energy into the pairing modes. Since that happens by two-particle scattering events which are suppressed by phase space constraints due to the Pauli principle, overheating can be deferred.
Wider implications. The result of this work is relevant for implementing nonequilibrium superfluidity in ultracold atom gases which are loaded in optical lattices. However, Pauli phase space blocking is most effective at very low temperatures, and this still poses a substantial challenge in such experiments.
Figure. Comparison of two possible interaction switching scenarios. A sudden interaction quench (Q; case 2) is not followed by a sufficiently long prethermalization regime 
to observe nonequilibrium BCS superconductivity but thermalization (Th) is reached soon. Ramping up the interaction linearly on a finite time scale given by the gap parameter 
leads to a prethermalization regime which is sufficiently extended in time (case 1).