Beyond-mean-field energy corrections for a dipolar gas of transversally oriented dipoles at 3D-1D crossover

We study a dimensional crossover of a dipolar quantum gas of bosons trapped in the x-y plane with dipoles oriented transversally to the z-direction. In particular, we compute beyond-mean-field Lee-Huang-Yang (LHY) correction to the energy density, which stabilizes the gas in the case of collapse predicted by the mean-field theory. Confinement introduces quantization in the x-y plane and significantly alters the LHY energy density. To simplify the problem, we introduce periodic boundary conditions in the x-y plane, i.e. we consider dipolar atoms trapped on a 2D torus and unconfined in the z-direction. For such a model we calculate a confined dipole-dipole interaction potential that is compatible with periodic boundary conditions and is qualitatively different from the corresponding 3D potential. In particular, the Fourier transform of the confined potential does not vanish for zero momentum, which makes the critical value of the dipole interaction strength dependent on the strength of the confinement. Moreover, we find that, with confined potential, the mean-field collapse always has a rotonic character, which allows for a possibility of entering a supersolid phase, a feature that is absent in the case of longitudinally oriented dipoles.