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Physical Review A

Impact of spontaneous spatial symmetry breaking on the critical atom number for two-component Bose-Einstein condensates

Brett D. Esry

Published Nov 1, 1998• 16 citations
Publisher

Abstract

The consequences of spatial symmetry breaking in two-component Bose-Einstein condensates are explored for the ${}^{85}{\mathrm{R}\mathrm{b}+}^{87}\mathrm{Rb}$ system. It is shown that the mean-field critical number of ${}^{85}\mathrm{Rb}$ atoms depends strongly on whether the single-particle wave function is symmetry preserving or symmetry breaking. This system thus provides a uniquely straightforward experimental means of observing spatial symmetry breaking.

DOI
10.1103/physreva.58.r3399

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The consequences of spatial symmetry breaking in two-component Bose-Einstein condensates are explored for the 85Rb+87Rb{}^{85}{\mathrm{R}\mathrm{b}+}^{87}\mathrm{Rb}85Rb+87Rb system. It is shown that the mean-field critical number of 85Rb{}^{85}\mathrm{Rb}85Rb atoms depends strongly on whether the single-particle wave function is symmetry preserving or symmetry breaking. This system thus provides a uniquely straightforward experimental means of observing spatial symmetry breaking.
We discuss the general features of spontaneous spatial symmetry breaking in trapped two-component alkali-metal--atom Bose-Einstein condensates, and give qualitative guidelines for when it will occur. We further show that the Hartree-Fock equations admit symmetry-broken solutions for as few as two trapped atoms, discussing the particular system of one 85Rb{}^{85}\mathrm{Rb}85Rb atom and one 87Rb{}^{87}\mathrm{Rb}87Rb atom. It is also shown that the critical value of the interspecies scattering length for a mixture of Na and Rb in an isotropic trap depends strongly on the number of atoms in a manner not described by the standard Thomas-Fermi phase separation condition.
We examine the stability properties of the ground state of two-component Bose-Einstein condensates as a function of the interspecies interactions. A stability criterion is identified from the curvature matrix of the Gross-Pitaevskii energy functional subject to the normalization conditions. By analyzing the stability signature, the characteristic features of the spontaneous spatial symmetry breaking are verified in various types of traps. The details of the differences between the continuous symmetry breaking and discrete symmetry breaking are discussed.
We study both mean-field and full quantum dynamics of symmetry-breaking transitions (SBTs) in a coupled two-component Bose-Einstein condensate. By controlling s-wave scattering lengths and the coupling strength, it is possible to stimulate SBTs between normal and spontaneously polarized ground states. In static transitions, the probability maxima of full quantum ground states correspond to the mean-field ground states. In dynamical transitions, due to the vanishing of excitation gaps, the mean-field dynamics shows universal scalings obeying the Kibble-Zurek mechanism. Both mean-field and full quantum defect modes appear as damped oscillations, but they appear at different critical points and undergo different oscillation regimes. The anomalous breakdown of mean-field dynamics induced by SBTs depends on the approaching direction.
The interplay between spontaneously broken gauge symmetries and Bose–Einstein condensation has long been controversially discussed in science, since the equations of motion are invariant under phase transformations. Within the present model, it is illustrated that spontaneous symmetry breaking appears as a non-local process in position space, but within disjoint subspaces of the underlying Hilbert space. Numerical simulations show that it is the symmetry of the relative phase distribution between condensate and non-condensate quantum fields which is spontaneously broken when passing the critical temperature for Bose–Einstein condensation. Since the total number of gas particles remains constant over time, the global U(1)-gauge symmetry of the system is preserved.
Abstract We consider the homogeneous Bose gas in the three-dimensional unit torus, where N particles interact via a two-body potential of the form N−1v(x)N^{-1} v(x)N−1v(x) N - 1 v ( x ) . The system is studied at inverse temperatures of order N−2/3N^{-2/3}N−2/3 N - 2 / 3 , which corresponds to the temperature scale of the Bose–Einstein condensation phase transition. We show that spontaneous U (1) symmetry breaking occurs if and only if the system exhibits Bose–Einstein condensation in the sense that the one-particle density matrix of the Gibbs state has a macroscopic eigenvalue.