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

Journal Article · Sat Jan 15 00:00:00 EST 2011 · Physical Review. A

DOI:https://doi.org/10.1103/PHYSREVA.83.013607· OSTI ID:21537022

Sakaguchi, Hidetsugu ^[1]; Malomed, Boris A ^[2]

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1. Department of Applied Science for Electronics and Materials, Interdisciplinary Graduate School of Engineering Sciences, Kyushu University, Kasuga, Fukuoka 816-8580 (Japan)
2. Department of Physical Electronics, School of Electrical Engineering, Faculty of Engineering, Tel Aviv University, Tel Aviv 69978 (Israel)

The quantum-mechanical collapse (alias fall onto the center of particles attracted by potential -r{sup -2}) is a well-known issue in quantum theory. It is closely related to the quantum anomaly, i.e., breaking of the scaling invariance of the respective Hamiltonian by quantization. We demonstrate that the mean-field repulsive nonlinearity prevents the collapse and thus puts forward a solution to the quantum-anomaly problem that differs 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, resulting in a redefinition of the scattering length which accounts for the contact repulsion between the bosons. In lieu of the collapse, the cubic nonlinearity creates a 3D ground state (GS), which does not exist in the respective linear Schroedinger equation. The addition of the harmonic trap gives rise to a tristability, in the case when the Schroedinger equation 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). Counterintuitively, such self-trapped 2D modes exist even in the case of a weakly repulsive potential r{sup -2}. The 2D vortical modes avoid the phase singularity at the pivot (r=0) by having the amplitude diverging at r{yields}0 instead of the usual situation with the amplitude of the vortical mode vanishing at r{yields}0 (the norm of the mode converges despite of the singularity of the amplitude at r{yields}0). In the presence of the harmonic trap, the 2D quintic model with a weakly repulsive central potential r{sup -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 in the form of an analytical approximation, which is asymptotically exact in the limit of the large norm.

• OSTI ID: 21537022
• Journal Information: Physical Review. A, Vol. 83, Issue 1; Other Information: DOI: 10.1103/PhysRevA.83.013607; (c) 2011 American Institute of Physics; ISSN 1050-2947
• Country of Publication: United States
• Language: English

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
74 ATOMIC AND MOLECULAR PHYSICS
AMPLITUDES
ANGULAR MOMENTUM
APPROXIMATIONS
BOSONS
CENTRAL POTENTIAL
HAMILTONIANS
INTERACTIONS
MAGNETIC DIPOLES
MATHEMATICAL SOLUTIONS
MEAN-FIELD THEORY
NONLINEAR PROBLEMS
QUANTIZATION
QUANTUM FIELD THEORY
QUANTUM MECHANICS
SCALE INVARIANCE
SCATTERING LENGTHS
SCHROEDINGER EQUATION
SINGULARITY
WAVE FUNCTIONS
CALCULATION METHODS
DIFFERENTIAL EQUATIONS
DIMENSIONS
DIPOLES
EQUATIONS
FIELD THEORIES
FUNCTIONS
INVARIANCE PRINCIPLES
LENGTH
MATHEMATICAL OPERATORS
MECHANICS
MULTIPOLES
PARTIAL DIFFERENTIAL EQUATIONS
POTENTIALS
QUANTUM OPERATORS
WAVE EQUATIONS