Dynamics of Collapse of a Confined Bose Gas

TL;DR

This paper rigorously analyzes the nonlinear dynamics and collapse conditions of a confined dilute Bose gas with negative scattering length, using the virial theorem applied to the Gross-Pitaevskii equations.

Contribution

It provides new rigorous results on collapse conditions and the temporal evolution of the Bose gas in harmonic traps, including explicit criteria for collapse in 2D and 3D cases.

Findings

In 2D, the gas oscillates with frequency 2ω₀ and collapses if energy is negative.

In 3D, the system also collapses after finite time for negative energy states.

Stable small-amplitude oscillations are possible under certain initial conditions.

Abstract

Rigorous results on the nonlinear dynamics of a dilute Bose gas with a negative scattering length in an harmonic magnetic trap are presented and sufficient conditions for the collapse of the system are formulated. By using the virial theorem for the Gross-Pitaevskii equations in an external field we analyze the temporal evolution of the mean square radius of the gas cloud. In the 2D case the quantity undergoes harmonic oscillation with frequency $2\omega _0$ It implies that for a negative value of energy of the system, the gas cloud will collapse after a finite time interval. For positive energy the cloud collapses if the initial conditions correspond to a large enough amplitude of oscillations. Stable oscillations with a small amplitude are possible. In the 3D case the system also collapsed after a finite time for a state with negative energy. A stringent condition for the collapse is also derived.