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arXiv:1309.3224 · Physical Review A

Nambu-Goldstone modes in segregated Bose-Einstein condensates

Hiromitsu Takeuchi, Kenichi Kasamatsu

Published Oct 9, 2013• 32 citations
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Abstract

Nambu-Goldstone modes in immiscible two-component Bose-Einstein condensates are studied theoretically. In a uniform system, a flat domain wall is stabilized and then the translational invariance normal to the wall is spontaneously broken in addition to the breaking of two U(1) symmetries in the presence of two complex order parameters. We clarify the properties of the low-energy excitations and identify that there exist two Nambu-Goldstone modes: an in-phase phonon with a linear dispersion and a ripplon with a fractional dispersion. The signature of the characteristic dispersion can be verified in segregated condensates in a harmonic potential.

DOI
10.1103/physreva.88.043612

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Nambu-Goldstone modes in immiscible two-component Bose-Einstein condensates are studied theoretically. In a uniform system, a flat domain wall is stabilized and then the translational invariance normal to the wall is spontaneously broken in addition to the breaking of two U(1) symmetries in the presence of two complex order parameters. We clarify the properties of the low-energy excitations and identify that there exist two Nambu-Goldstone modes: an in-phase phonon with a linear dispersion and a ripplon with a fractional dispersion. The signature of the characteristic dispersion can be verified in segregated condensates in a harmonic potential.
Abstract We present new features of low energy Bogoliubov quasiparticle excitations of a two-component Bose–Einstein condensate (TBEC) in quasi-2D geometry at zero temperature using Hartree–Fock–Bogoliubov formalism. We, in particular, consider the TBECs of 133 Cs– 87 Rb and 85 Rb– 87 Rb, and show specific features in the low energy excitation spectrum as a function of the interaction strength. For 85 Rb– 87 Rb TBEC, the appearance of a new zero energy mode is observed. Whereas for 133 Cs– 87 Rb TBEC we report a bifurcation of the slosh mode at the point of transition from miscible to immiscible domain. The lower energy mode, after the bifurcation, goes soft and becomes a new Nambu–Goldstone mode of the system.
We theoretically study the properties of Nambu-Goldstone bosons in an interacting single-component Bose-Einstein condensate (BEC). We first point out that the proofs of Goldstone's theorem by Goldstone, et al. [Phys. Rev. {\bf 127} (1962) 965] may be relevant to distinct massless modes of the BEC: whereas the first proof deals with the poles of the single-particle Green's function G^\hat{G}G^, the second one concerns those of the two-particle Green's function. Thus, there may be multiple Nambu-Goldstone bosons even in the single-component BEC with broken U(1) symmetry. The second mode turns out to have an infinite lifetime in the long-wavelength limit in agreement with the conventional viewpoint. In contrast, the first mode from G^\hat{G}G^, i.e., the Bogoliubov mode in the weak-coupling regime, is shown to be a "bubbling" mode fluctuating temporally out of and back into the condensate. The substantial lifetime originates from an "improper" structure of the self-energy inherent in the BEC, which has been overlooked so far and will be elucidated here, and removes various infrared divergences pointed out previously.
The ripplon modes of two segregated Bose-Einstein condensates (BECs) confined by one and two hard walls are respectively studied by means of the hydrodynamic approach within the Gross-Pitaevskii (GP) theory. For the system at rest we find that due to the spatial restriction the dispersion relations are of the form ω∼k2\omega \sim {k^2}ω∼k2 in low momentum limit for both cases, while for the system in motion parallel to the interface the dispersion relations for both cases are ω∼k\omega \sim kω∼k at low momentum limit and, furthermore, the system becomes unstable.
The localized low-energy interfacial excitations, or interfacial Nambu-Goldstone modes, of phase-segregated binary mixtures of Bose-Einstein condensates are investigated analytically. To this end a double-parabola approximation (DPA) is performed on the Lagrangian density in Gross-Pitaevskii theory for a system in a uniform potential. This DPA entails a model in which analytic expressions are obtained for the excitations underlying capillary waves or ripplons for arbitrary strength KKK (>1>1>1) of the phase segregation. The dispersion relation $\ensuremath{\omega}(k)\ensuremath{\propto}{k}^{3/2}$ is derived directly from the Bogoliubov--de Gennes equations in the limit that the wavelength $2\ensuremath{\pi}/k$ is much larger than the interface width. The proportionality constant in the dispersion relation provides the static interfacial tension. A correction term in $\ensuremath{\omega}(k)$ of order k5/2{k}^{5/2}k5/2 is calculated analytically within the DPA model. The combined result is tested against numerical diagonalization of the exact Bogoliubov--de Gennes equations. Satisfactory agreement is obtained in the range of physically relevant wavelengths. The ripplon dispersion relation is relevant to state-of-the-art experiments using (quasi)uniform optical-box traps. Furthermore, within the DPA model explicit expressions are obtained for the structural deformation of the interface due to the passing of the capillary wave. It is found that the amplitude of the wave is enhanced by an amount that is quadratic in the ratio of the phase velocity $\ensuremath{\omega}/k$ to the sound velocity ccc. For generic mixtures consisting of condensates with unequal healing lengths, an additional modulation is predicted of the common value of the condensate densities at the interface.