Suppression of the quantum-mechanical collapse by repulsive interactions in a quantum gas

TL;DR

This paper demonstrates that repulsive nonlinear interactions can prevent quantum-mechanical collapse in dipolar quantum gases, creating stable ground states in 2D and 3D, and offers analytical and numerical solutions to this phenomenon.

Contribution

It introduces a nonlinear mean-field approach to prevent quantum collapse, providing new stable states in 2D and 3D quantum gases with repulsive interactions.

Findings

Repulsive nonlinearity prevents collapse in 3D and 2D quantum gases.

Stable ground states are formed by cubic and quintic nonlinearities.

Harmonic trapping leads to tristability and mode dynamics.

Abstract

The quantum-mechanical collapse (alias fall onto the center of particles attracted by potential -1/r^2), or "quantum anomaly", is a well-known issue in the quantum theory. We demonstrate that the mean-field repulsive nonlinearity prevents the collapse and thus puts forward a solution to the quantum-anomaly problem different from that previously developed in the framework of the linear quantum-field theory. This solution may be realized in the 3D or 2D gas of dipolar bosons attracted by a central charge, and in the 2D gas of magnetic dipoles attracted by a current filament. In the 3D setting, the dipole-dipole interactions are also taken into regard, in the mean-field approximation. In lieu of the collapse, the cubic nonlinearity creates a 3D ground state (GS), which does not exist in the respective linear Schroedinger equation (SE). The addition of the harmonic trap gives rise to a tristability, in the case when the SE still does not lead to the collapse. In the 2D setting, the cubic nonlinearity is not strong enough to prevent the collapse; however, the quintic term does it, creating the GS, as well as its counterparts carrying the angular momentum (vorticity). Counter-intuitively, such self-trapped 2D modes exist even in the case of a weakly repulsive potential 1/r^2. In the presence of the harmonic trap, the 2D quintic model with a weakly repulsive central potential 1/r^2 gives rise to three confined modes, the middle one being unstable, spontaneously developing into a breather. In both the 3D and 2D cases, the GS wave functions are found in a numerical form, and also in the form of an analytical approximation, which is asymptotically exact in the limit of the large norm.