Suppression of quantum collapse in an anisotropic gas of dipolar bosons

Abstract

In recent work [Sakaguchi and Malomed, Phys. Rev. A 83, 013607 (2011)], a solution to the problem of the quantum collapse (fall onto the center) in the three-dimensional space with the attractive potential $\ensuremath{-}({U}_{0}/2){r}^{\ensuremath{-}2}$ was proposed, based on the replacement of the linear Schr\odinger equation by the Gross-Pitaevskii (GP) equation with the repulsive cubic term. The model applies to a quantum gas of molecules carrying permanent electric dipole moments, with the attraction center representing a fixed electric charge. It was demonstrated that the repulsive nonlinearity suppresses the quantum collapse and creates the corresponding spherically symmetric ground state (GS), which was missing in the case of the linear Schr\odinger equation. Here, we aim to extend the analysis to the cylindrical geometry and to eigenstates carrying angular momentum. The cylindrical anisotropy is imposed by a uniform dc field, which fixes the orientation of the dipole moments, thus altering the potential of the attraction to the center. First, we analyze the modification of the condition for the onset of the quantum collapse in the framework of the linear Schr\odinger equation with the cylindrically symmetric potential for the states with azimuthal quantum numbers $m=0$ (the GS) and $m=1$, 2. The corresponding critical values of the strength of the attractive potential ${({U}_{0})}_{\mathrm{cr}}(m)$ are found. Next, a numerical solution of the nonlinear GP equation is developed, which demonstrates the replacement of the quantum collapse by the originally missing eigenstates with $m=0,1,2$. Their dynamical stability is verified by means of numerical simulations of the perturbed evolution. For $m=0$, the Thomas-Fermi approximation is presented too, in an analytical form. Crucially important for the solution is the proper choice of the boundary conditions at $r\ensuremath{\rightarrow}0$.