Spin-1 bosons with coupled ground states in optical lattices

Konstantin Krutitsky

Fachbereich Physik der Universität Duisburg-Essen, Campus Essen, Universität Duisburg-Essen, Universitätsstr. 5, 45117 Essen, Germany

We have theoretically investigated quantum phase transitions of ultracold spin-1 bosons with ferromagnetic and antiferromagnetic interactions in an optical lattice. Two counterpropagating linearly polarized laser beams with the angle $\theta$ between the polarization vectors (lin-$\theta$-lin configuration), driving an $F_g=1$ to $F_e=1$ internal atomic transition, create the optical lattice and at the same time couple atomic ground states with magnetic quantum numbers +1 and -1. As a result of the coupling there are only two lowest-energy Bloch bands which should be taken into account and the original three-component system is reduced to a two-component one. To our knowledge the Bose-Hubbard Hamiltonian we have derived has not been studied before.

The interaction between the atoms is characterized by two scattering lengths, the symmetric $g_s$ and the much smaller but very important anti-symmetric $g_a$. This set-up could be realized with Rb or Na atoms, for which $g_a$ is negative (ferromagnetic interaction) and positive (antiferromagnetic interaction), respectively. It turns out that an apropriate change of the easily tunable parameters of the set-up, the laser intensity and the angle $\theta$, permits a very rich scenario of phase transitions.

The symmetries of the Hamiltonian are discussed and it is shown that the properties are essentially different for the atoms with the ferromagnetic and antiferromagnetic interactions. Depending on the sign of the anti-symmetric scattering length, the isospin symmetry is or is not spontaneously broken in the superfluid phase. The corresponding collective modes, besides the always present Bogoliubov mode, are isospin waves and either gapless or gapped at $k=0$.

The phase diagram of the system is worked out. In the case of ferromagnetic interactions, the superfluid - Mott insulator phase transition is always second order, but in the case of antiferromagnetic interactions for some values of system parameters it is first order and the superfluid and Mott phases can coexist. Varying the angle $\theta$ one can control the populations of atomic components and continuously turn on and tune their asymmetry.