Quantum Joule expansion for the one-dimensional Bose Hubbard model

We study the quantum Joule expansion for a Bose Hubbard model in a one dimensional chain initially at unity filling. Particles are first confined to the left half of the system by an infinite barrier in the middle. At t = 0, the barrier is removed and particles begin to expand to the other half of the system. We consider both pure and thermal initial states. We observe revivals in small systems and these are quickly pushed to long times as the size of the system is increased. As an example, we study a system with 10 sites by exact diagonalization. Particle density and entanglement entropy equilibrate quickly during the expansion for comparable values for the hopping amplitude J and on-site repulsion U. We argue that the diagonal ensemble of the density matrix is similar to the canonical ensemble with an effective temperature. This effective temperature can be negative if the energy of the initial system is large enough. The finite size effects of this effective temperature are also studied by finite temperature density matrix renormalization group.