Ground state and excitations of a Bose gas: From a harmonic trap to a double well

We determine the low-energy properties of a trapped Bose gas split in two by a potential barrier over the whole range of barrier heights and asymmetry between the wells. For either weak or strong coupling between the wells, our two-mode theory yields a two-site Bose-Hubbard Hamiltonian with the tunneling, interaction, and bias parameters calculated simply using an explicit form of two mode functions. When the potential barrier is relatively low, most of the particles occupy the condensate mode and our theory reduces to a two-mode version of the Bogoliubov theory, which gives a satisfactory estimate of the spatial shape and energy of the lowest collective excitation. When the barrier is high, our theory generalizes the standard two-site Bose-Hubbard model into the case of asymmetric modes, and correctly predicts a full separation of the modes in the limit of strong separation of the wells. We provide explicit analytic forms for the number squeezing and coherence as a function of particle number and temperature. We compare our theory to other two-mode theories for bosons in a double well and discuss their validity in different parameter regimes.