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arXiv:1403.1267 · Physical Review A

Superfluidity breakdown and multiple roton gaps in spin-orbit-coupled Bose-Einstein condensates in an optical lattice

Daniele Toniolo, Jacob Linder

Published Jun 30, 2014• 23 citations
arXivHTMLPDF

Abstract

We investigate the superfluid phases of a Rashba spin-orbit-coupled Bose-Einstein condensate residing on a two-dimensional square optical lattice in the presence of an effective Zeeman field $\ensuremath{\Omega}$. At a critical value $\ensuremath{\Omega}={\ensuremath{\Omega}}_{c}$, the single-particle spectrum ${E}_{\mathbf{k}}$ changes from having a set of four degenerate minima to a single minimum at $\mathbf{k}=0$, corresponding to condensation at finite or zero momentum, respectively. We describe this quantum phase transition and the symmetry breaking of the condensate phases. We use the Bogoliubov theory to treat the superfluid phases and determine the phase diagram, the excitation spectrum, and the sound velocity of the phonon excitations. A dynamically unstable superfluid regime occurring when $\ensuremath{\Omega}$ is close to ${\ensuremath{\Omega}}_{c}$ is analytically identified and the behavior of the condensate quantum depletion is discussed. Moreover, we show that there are two types of roton excitations occurring in the $\ensuremath{\Omega}<{\ensuremath{\Omega}}_{c}$ regime and obtain explicit values for the corresponding energy gaps.

DOI
10.1103/physreva.89.061605

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We investigate the superfluid phases of a Rashba spin-orbit-coupled Bose-Einstein condensate residing on a two-dimensional square optical lattice in the presence of an effective Zeeman field \ensuremathΩ\ensuremath{\Omega}\ensuremathΩ. At a critical value \ensuremathΩ=\ensuremathΩc\ensuremath{\Omega}={\ensuremath{\Omega}}_{c}\ensuremathΩ=\ensuremathΩc​, the single-particle spectrum Ek{E}_{\mathbf{k}}Ek​ changes from having a set of four degenerate minima to a single minimum at k=0\mathbf{k}=0k=0, corresponding to condensation at finite or zero momentum, respectively. We describe this quantum phase transition and the symmetry breaking of the condensate phases. We use the Bogoliubov theory to treat the superfluid phases and determine the phase diagram, the excitation spectrum, and the sound velocity of the phonon excitations. A dynamically unstable superfluid regime occurring when \ensuremathΩ\ensuremath{\Omega}\ensuremathΩ is close to \ensuremathΩc{\ensuremath{\Omega}}_{c}\ensuremathΩc​ is analytically identified and the behavior of the condensate quantum depletion is discussed. Moreover, we show that there are two types of roton excitations occurring in the \ensuremathΩ<\ensuremathΩc\ensuremath{\Omega}<{\ensuremath{\Omega}}_{c}\ensuremathΩ<\ensuremathΩc​ regime and obtain explicit values for the corresponding energy gaps.
A weakly interacting, spin-orbit-coupled, ultracold, dilute Bose gas on a two-dimensional square lattice with an external Zeeman field is studied. We explore the plane- and stripe-wave phases of the system involving nonzero condensate momenta, which occur when the Zeeman field is below a critical value. Their excitation spectra are found using Bogoliubov theory and by two different routes. The validity of each method to obtain the excitation spectrum is discussed, and it is found that projection on the lowest single-particle band is an excellent approximation in the plane-wave phase, while it is a poor approximation in the stripe-wave phase. While the plane-wave phase has a phonon minimum at its single condensate momentum, revealing a nonzero sound velocity of the excitations, the stripe-wave phase has quadratic minima at its two condensate momenta showing zero sound velocity of the excitations. We discuss how the presence of more than one condensate momentum is essential for these differences between the two phases. Additionally, it is emphasized that the zero sound velocity in the stripe-wave phase is a lattice effect, since continuum studies of the same phase have shown nonzero sound velocity.
We investigate the collective excitation spectrum and dynamics of a quasi two-dimensional spin-1 Bose-Einstein condensate with Rashba type spin-orbit (SO) coupling. Employing Bogoliubov-de-Gennes analysis, we analytically compute the excitation spectra across a wide range of interaction strengths and coupling parameters. By systematically varying the SO and Rabi couplings, we uncover distinct dynamical signatures of quantum phase transitions, including mode softening, the appearance of roton-like minima, and miscibility-driven instabilities in both ferromagnetic and antiferromagnetic interaction regimes. In the antiferromagnetic case, these instabilities lead to a dynamically unstable supersolid phase characterized by the coexistence of density modulation and global phase coherence. To corroborate the analytical predictions, we numerically solve the coupled Gross-Pitaevskii equations and analyze the dynamical stability of the condensate. Our results provide experimentally accessible signatures for spinor condensates with tunable spin-orbit coupling and demonstrate the rich interplay between spin-dependent interactions and synthetic couplings in nonequilibrium quantum fluids.
By calculating the density response function we identify the excitation spectrum of a Bose-Einstein condensate with equal Rashba and Dresselhaus spin-orbit coupling. We find that the velocity of sound along the direction of spin-orbit coupling is deeply quenched and vanishes when one approaches the second-order phase transition between the plane-wave and the zero momentum quantum phases. We also point out the emergence of a roton minimum in the excitation spectrum for small values of the Raman coupling, providing the onset of the transition to the stripe phase. Our findings point out the occurrence of a strong anisotropy in the dynamic behavior of the gas. A hydrodynamic description accounting for the collective oscillations in both uniform and harmonically trapped gases is also derived.
We investigate the ground state of two-dimensional Bose-Einstein condensates with Rashba spin-orbit coupling in square optical lattices and demonstrate the existence of rich phases with different lattice structures, which is closely related to the degenerate structure of single particle energy spectrum induced by the competition of spin-orbit coupling and optical lattices. We find that the ground state is in the phase with either parity-time-reversal or parity symmetry by direct numerical simulation. We show the phase diagram of ground state in the whole regime of spin-orbit coupling strength, and particularly find that the system supports multi-fold vortex lattices, in which ground state holds half-quantum vortex lattices, vortex-antivortex pair lattices and fundamental vortex lattices, simultaneously, when single particle energy minimums touch the boundary of the first Brillouin zone.
A weakly interacting, spin-orbit coupled, two-component, ultracold Bose gas bound to a Bravais lattice is studied. Motivated by recent experimental advances in the field of synthetically spin-orbit coupled, ultracold, neutral atomic gases showing Bose-Einstein condensation, an analytic framework with which to describe such systems in the superfluid regime is presented. This is applied to a Rashba spin-orbit-coupled Bose gas in a two-dimensional optical lattice. The exotic nature of Bose-Einstein condensation in the presence of spin-orbit coupling is an interesting study by itself. Additionally, when the optical lattice is introduced, the system provides a highly controllable experimental testing ground for numerous condensed matter physics phenomena. Five phases of the system are considered, and their excitation spectra, critical superfluid velocities and free energies are found. In obtaining the free energy, the effects of terms in the Hamiltonian that are linear in excitation operators are included, and such terms have not been studied previously in this context. Minimization of the free energy at zero temperature is used to confirm the phase diagrams reported in the literature, where it has usually been obtained by neglecting the effect of excitations. The plane and stripe wave phases in the phase diagram are bosonic analogues of Fulde-Ferrell-Larkin-Ovchinnikov states in superconductors involving nonzero condensate momenta.
Roton-type excitations usually emerge from strong correlations or long-range interactions, as in superfluid helium or dipolar ultracold atoms. However, in a weakly short-range interacting quantum gas, the recently synthesized spin-orbit (SO) coupling can lead to various unconventional phases of superfluidity and give rise to an excitation spectrum of roton-maxon character. Using Bragg spectroscopy, we study a SO-coupled Bose-Einstein condensate of ^{87}Rb atoms and show that the excitation spectrum in a "magnetized" phase clearly possesses a two-branch and roton-maxon structure. As Raman coupling strength Ω is decreased, a roton-mode softening is observed, as a precursor of the phase transition to a stripe phase that spontaneously breaks spatially translational symmetry. The measured roton gaps agree well with theoretical calculations. Furthermore, we determine sound velocities both in the magnetized and in the nonmagnetized phases, and a phonon-mode softening is observed around the phase transition in between. The validity of the f-sum rule is examined.