Collective motions of a quantum gas confined in a harmonic trap

TL;DR

This paper explores the collective motions of a quantum gas in a harmonic trap, showing that center of mass oscillates classically and that non-interacting gases exhibit large radial breathing modes, with implications for understanding atomic interactions.

Contribution

It demonstrates the invariance-based derivation of collective motions in quantum gases, including center of mass oscillations and large breathing modes, extending understanding of non-interacting and long-range interacting systems.

Findings

Center of mass oscillates along a classical harmonic trajectory.

Non-interacting gases exhibit large radial breathing modes at twice the trap frequency.

Large breathing motions imply absence of short-range interactions, possibly indicating long-range inverse-square interactions.

Abstract

Single-component quantum gas confined in a harmonic potential, but otherwise isolated, is considered. From the invariance of the system of the gas under a displacement-type transformation, it is shown that the center of mass oscillates along a classical trajectory of a harmonic oscillator. It is also shown that this harmonic motion of the center has, in fact, been implied by Kohn's theorem. If there is no interaction between the atoms of the gas, the system in a time-independent isotropic potential of frequency $\nu_c$ is invariant under a squeeze-type unitary transformation, which gives collective {\it radial} breathing motion of frequency $2\nu_c$ to the gas. The amplitudes of the oscillating and breathing motions from the {\it exact} invariances could be arbitrarily large. For a Fermi system, appearance of $2\nu_c$ mode of the large breathing motion indicates that there is no interaction between the atoms, except for a possible long-range interaction through the inverse-square-type potential.