Sudden-quench dynamics of Bardeen-Cooper-Schrieffer states in deep optical lattices

We determine the exact dynamics of an initial Bardeen-Cooper-Schrieffer (BCS) state of ultra-cold atoms in a deep hexagonal optical lattice. The dynamical evolution is triggered by a quench of the lattice potential, such that the interaction strength $U_f$ is much larger than the hopping amplitude $J_f$. The quench initiates collective oscillations with frequency $|U_f|/(2\pi)$ in the momentum occupation numbers and imprints an oscillating phase with the same frequency on the BCS order parameter $\Delta$. The oscillation frequency of $\Delta$ is not reproduced by treating the time evolution in mean-field theory. In our theory, the momentum noise (i.e. density-density) correlation functions oscillate at frequency $|U_f|/2\pi$ as well as at its second harmonic. For a very deep lattice, with zero tunneling energy, the oscillations of momentum occupation numbers are undamped. Non-zero tunneling after the quench leads to dephasing of the different momentum modes and a subsequent damping of the oscillations. The damping occurs even for a finite-temperature initial BCS state, but not for a non-interacting Fermi gas. Furthermore, damping is stronger for larger order parameter and may therefore be used as a signature of the BCS state. Finally, our theory shows that the noise correlation functions in a honeycomb lattice will develop strong anti-correlations near the Dirac point.