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Quantum Phases and Collective Excitations of a Spin-Orbit-Coupled Bose–Einstein Condensate in a One-Dimensional Optical Lattice

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Abstract

The ground state of a spin-orbit-coupled Bose gas in a one-dimensional optical lattice is known to exhibit a mixed regime, where the condensate wave function is given by a superposition of multiple Bloch-wave components, and an unmixed one, in which the atoms occupy a single Bloch state. The unmixed regime features two unpolarized Bloch-wave phases, having quasimomentum at the center or at the edge of the first Brillouin zone, and a polarized Bloch-wave phase at intermediate quasimomenta. By calculating the critical values of the Raman coupling and of the lattice strength at the transitions among the various phases, we show the existence of a tricritical point where the mixed, the polarized and the edge-quasimomentum phases meet, and whose appearance is a consequence of the spin-dependent interaction. Furthermore, we evaluate the excitation spectrum in the unmixed regime and we characterize the behavior of the phonon and the roton modes, pointing out the instabilities occurring when a phase transition is approached.

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Notes

  1. Notice that, in order to perform the numerical calculation, one has to truncate the two summations in Eq. (9) to a finite number of terms with \(|l| \le 2 N_s + 1\) and \(|l^{\prime }| \le N_\mathrm{L}\), where \(N_s\) and \(N_\mathrm{L}\) must be chosen large enough such that all the relevant contributions to the wave function be retained.

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Acknowledgements

Useful discussions with T. Ozawa, D. Papoular, N. Pavloff, C. Qu, and S. Stringari are acknowledged. The research leading to these results has received funding from the European Research Council under European Community’s Seventh Framework Programme (FP7/2007-2013 Grant Agreement No. 341197).

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  1. LPTMS, CNRS, Univ. Paris-Sud, Université Paris-Saclay, 91405, Orsay, France

    G. I. Martone

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Martone, G.I. Quantum Phases and Collective Excitations of a Spin-Orbit-Coupled Bose–Einstein Condensate in a One-Dimensional Optical Lattice. J Low Temp Phys 189, 262–275 (2017). https://doi.org/10.1007/s10909-017-1816-9

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