The extended Bose-Hubbard model for a double-well potential with pair
tunneling is studied through both exact diagonalization and mean field theory
(MFT). When pair tunneling is strong enough, the ground state wavefunction
predicted by the MFT is complex and doubly degenerate while the quantum ground
state wavefunction is always real and unique. The time reversal symmetry is
spontaneously broken when the system transfers from the quantum ground state
into one of the mean field ground states upon a small perturbation. As the gap
between the lowest two levels decreases exponentially with particle number, the
required perturbation inducing the spontaneous symmetry breaking (SSB) is
infinitesimal for particle number of typical cold atom systems. The quantum
ground state is further analyzed with the Penrose-Onsager criterion, and is
found to be a fragmented condensate. The state also develops the pair
correlation and has non-vanishing pair order parameter instead of the
conventional single particle order parameter. When this model is generalized to
optical lattice, a pair superfluid can be generated. The mean field ground
state can be regarded as effective ground state in this simple model. The
detailed computation for this model enables us to offer an in-depth discussion
of the relation between SSB and effective ground state, giving a glimpse on how
nonlinearity arises in the SSB of a quantum system.Comment: 6 pages, 6 figure
