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arXiv:cond-mat/0505372

Atomic Theory of Collective Excitations in Bose-Einstein Condensation and Spontaneously Broken Gauge Symmetry

S. J. Han

Published May 16, 2005• 0 citations
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Abstract

A theory of collective excitations in Bose-Einstein condensation in a trap is developed based on the quantum Hamilton-Jacobi equation of Bohm and the phase coherence along with the idea of off-diagonal long range order of Penrose and Onsager. First, we show that a free surface behaves like a normal fluid - a breakdown of superfluidity. Second, inside the free surface it is shown that the spectrum of phonons is of the form $ω=ck$ scaled with the external potential, where the speed of (first) sound, $c=[4πaρ\hbar^{2}]^{1/2}/M$ and $k$ is the wave number. Third, in the limit $a\to 0$, the hard spheres in the Bose-Einstein condensation collapse to a close-packed classical lattice with the zero-point vibrational motion about fixed points.

DOI
10.48550/arxiv.cond-mat/0505372

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A theory of collective excitations in Bose-Einstein condensation in a trap is developed based on the quantum Hamilton-Jacobi equation of Bohm and the phase coherence along with the idea of off-diagonal long range order of Penrose and Onsager. First, we show that a free surface behaves like a normal fluid - a breakdown of superfluidity. Second, inside the free surface it is shown that the spectrum of phonons is of the form ω=ckω=ckω=ck scaled with the external potential, where the speed of (first) sound, c=[4πaρℏ2]1/2/Mc=[4πaρ\hbar^{2}]^{1/2}/Mc=[4πaρℏ2]1/2/M and kkk is the wave number. Third, in the limit a→0a\to 0a→0, the hard spheres in the Bose-Einstein condensation collapse to a close-packed classical lattice with the zero-point vibrational motion about fixed points.
This paper presents an elementary theory of the phase-coherent collective excitations in both He II in the gravitational field and atomic Bose-Einstein condensation in a trap. The theory is based on the concept of off-diagonal long-range order by Penrose and Onsager and the quantum theory of Bohm, with emphasis on the broken symmetry in a Bose-Einstein gas with repulsive interactions. It is shown that a spontaneously broken symmetry that accompanies a phonon (Nambu-Goldstone mode) takes place at the surface layer of an inhomogeneous Bose system in the presence of an external field. The spontaneously broken symmetry in a Bose system is described and is shown to manifest itself in both He II and the Bose-Einstein condensation - a shell-like structure of Bose-Einstein condensation in a trap. The broken symmetry gives a coherent explanation for a number of long standing puzzles in He II.
This paper presents an elementary theory of the phase-coherent collective excitations in both He II in the gravitational field and atomic Bose-Einstein condensation in a trap. The theory is based on the concept of off-diagonal long-range order by Penrose and Onsager and the quantum theory of Bohm, with emphasis on the broken symmetry in a Bose-Einstein gas with repulsive interactions. It is shown that a spontaneously broken symmetry that accompanies a phonon (Nambu-Goldstone mode) takes place at the surface layer of an inhomogeneous Bose system in the presence of an external field. The spontaneously broken symmetry in a Bose system is described and is shown to manifest itself in both He II and the Bose-Einstein condensation - a shell-like structure of Bose-Einstein condensation in a trap. The broken symmetry gives a coherent explanation for a number of long standing puzzles in He II.
We discuss the collective excitations in a spatially inhomogeneous (cylindrically symmetric) Bose-Einstein condensation (BEC) at low temperature (T≪TλT \ll T_λT≪Tλ​). The main result is the dispersion relation for a (first) sound wave that is obtained by describing the perturbation as a Lagrangian coordinate. The dispersion curve is in good agreement with the Bogoliubov phonon spectrum ω=ckω=c kω=ck, where k=kθ=m/rk=k_θ=m/rk=kθ​=m/r, the wave number and c=[4πaρℏ2]1/2/Mc=[4 πa ρ\hbar^{2}]^{1/2}/Mc=[4πaρℏ2]1/2/M, the speed of first sound. Based on Bohm's quantum theory, a spontaneously broken gauge symmetry in a quantum fluid is discussed in terms of the quantum fluctuation-dissipation, from which it is shown that the symmetry breaking takes place at the free surface of BEC in an external field.
Bose–Einstein condensation represents a quintessential manifestation of quantum mechanics at macroscopic scales, where a dilute gas of bosonic atoms undergoes a phase transition to form a coherent matter wave. This study presents a comprehensive theoretical investigation of collective excitations and critical phenomena in interacting Bose–Einstein condensates (BECs). Beginning with the Gross–Pitaevskii equation , we develop the Bogoliubov theory of excitations yielding the spectrum , where is the free-particle energy and characterizes contact interactions. The spectrum reveals phonon behavior at low momenta with sound velocity , transitioning to free-particle dispersion at momenta exceeding the inverse healing length . Collective modes in harmonically trapped condensates are analyzed in the Thomas–Fermi regime, yielding breathing mode frequency and quadrupole frequency . Damping mechanisms including Landau damping ( ) and Beliaev damping ( ) are characterized. The Lee–Huang–Yang correction extends the equation of state beyond mean field. Critical phenomena near are analyzed within the 3D XY universality class, with critical exponents , , and the interaction-induced shift with . All predictions show excellent agreement with experimental measurements on ultracold atomic gases.
Collective excitations of a dilute Bose condensate have been observed. These excitations are analogous to phonons in superfluid helium. Bose condensates were created by evaporatively cooling magnetically trapped sodium atoms. Excitations were induced by a modulation of the trapping potential, and detected as shape oscillations in the freely expanding condensates. The frequencies of the lowest modes agreed well with theoretical predictions based on mean-field theory. Before the onset of Bose-Einstein condensation, we observed sound waves in a dense ultracold gas.