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arXiv:cond-mat/9603077 · Physical Review Letters

Local Spin-Gauge Symmetry of the Bose-Einstein Condensates in Atomic Gases

Tin-Lun Ho, V. B. Shenoy

Published Sep 23, 1996• 76 citations
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Abstract

The Bose-Einstein condensates of alkali atomic gases are spinor fields with local ``spin-gauge'' symmetry. This symmetry is manifested by a superfluid velocity ${\mathbf{u}}_{s}$ (or gauge field) generated by the Berry phase of the spin field. In ``static'' traps, ${\mathbf{u}}_{s}$ splits the degeneracy of the harmonic energy levels, breaks the inversion symmetry of the vortex nucleation frequency ${\mathbf{\ensuremath{\Omega}}}_{c1}$, and can lead to vortex ground states.

DOI
10.1103/physrevlett.77.2595

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The Bose-Einstein condensates of alkali atomic gases are spinor fields with local spin-gauge'' symmetry. This symmetry is manifested by a superfluid velocity ${\mathbf{u}}_{s}$ (or gauge field) generated by the Berry phase of the spin field. In static'' traps, us{\mathbf{u}}_{s}us​ splits the degeneracy of the harmonic energy levels, breaks the inversion symmetry of the vortex nucleation frequency \ensuremathΩc1{\mathbf{\ensuremath{\Omega}}}_{c1}\ensuremathΩc1​, and can lead to vortex ground states.
Quantum mechanically, static or applied gauge fields manifest themselves by directly coupling to a system's phase. Artificial gauge field techniques manipulate the phase of a Bose-Einstein condensate's order parameter, yielding effects present in condensed matter systems subject to magnetic fields, electric fields, or spin-orbit coupling. In this chapter, we explore the impact of these artificial gauge fields on Bose- Einstein condensates and explain the consequences. Several analogues to traditional condensed matter systems are enabled by these techniques, including the Hall effect, superfluid vortex nucleation in systems with artificial magnetic fields, and magnetic-like phase transitions in systems with spin-orbit coupling. Introduction Since the first experiments with Bose-Einstein condensates (BECs), researchers have exploited the universality of physical laws to draw analogies between the behavior of ultracold quantum gases and systems ranging from material to astrophysical. These analogous situations are deliberately engineered, giving cold-atom “quantum simulators” [1] that exhibit behavior either observed or predicted in analogous quantum systems. Superfluidity was one of the first many-body quantum phenomena demonstrated in BECs. For example, several experiments showed superfluid behavior by rapidly rotating the condensates and observing the formation of vortices [2, 3, 4, 5] – quantized point-defects in the BEC's order parameter. In the rotating frame of these experiments, this nucleation of vortices could be understood as the consequence of an artificial magnetic field [6] that arises via the formal equivalence of the rotating-frame Coriolis force and the Lorentz force, which we describe in Section 15.4.1. Together with other measurements, these results proved to be in excellent agreement with the predictions of the Gross-Pitaevskii equation (GPE) [7], a decades-old idealization of superfluid helium. Buoyed by the prospect for studying magnetic field–driven strongly correlated systems using these nearly ideal quantum gases, proposals for alternate techniques to simulate gauge fields were developed. These methods primarily employ an internal state–selective atom–light coupling, where the photon momentum provides the forces necessary to mimic a Lorentz force [8, 9, 10, 11, 12, 13, 14, 15, 16]. The first demonstration of this type of technique used a spatially dependent Ramantransition scheme to effect a magnetic field [17], and experiments demonstrating artificial electric fields [18] and spin-orbit coupling [19, 20, 21, 22, 23, 24] soon followed.
The Bogoliubov theory is extended to a Bose-Einstein condensation with internal degrees of freedom, realized recently in 23^{23}23Na gases where several hyperfine states are simultaneously cooled optically. Starting with a Hamiltonian constructed from general gauge and spin rotation symmetry principles fundamental equations for condensate are derived. The ground state where time reversal symmetry is broken in some case and low-lying collective modes, e.g. spin and density wave modes, are discussed. Novel vortex as a topological defect can be created experimentally.
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Spinor Bose gases form a family of quantum fluids manifesting both magnetic order and superfluidity. This article reviews experimental and theoretical progress in understanding the static and dynamic properties of these fluids. The connection between system properties and the rotational symmetry properties of the atomic states and their interactions are investigated. Following a review of the experimental techniques used for characterizing spinor gases, their mean-field and many-body ground states, both in isolation and under the application of symmetry-breaking external fields, are discussed. These states serve as the starting point for understanding low-energy dynamics, spin textures and topological defects, effects of magnetic dipole interactions, and various non-equilibrium collective spin-mixing phenomena. The paper aims to form connections and establish coherence among the vast range of works on spinor Bose gases, so as to point to open questions and future research opportunities.