Finite-rate quenches of site bias in the Bose-Hubbard dimer

For a Bose-Hubbard dimer, we study quenches of the site energy imbalance, taking a highly asymmetric Hamiltonian to a fully symmetric one. The ramp is carried out over a finite time that interpolates between the instantaneous and adiabatic limits. We provide results for the excess energy of the final state compared to the ground state energy of the final Hamiltonian, as a function of the quench rate. This excess energy serves as the analog of the defect density that is considered in the Kibble-Zurek picture of ramps across phase transitions. We also examine the fate of quantum `self-trapping' when the ramp is not instantaneous.