Recently a spin-orbit-coupled Bose condensate has been realized experimentally using two-photon Raman coupling, and this condensate exhibits two distinct equilibrium phases known as the plane wave phase and the stripe phase at equilibrium. In this Rapid Communication we show that such condensate exhibits unique dynamic behaviors, which originate from interactions and are absent in a noninteracting system. In the plane wave phase, a dipole mode will induce a breathing mode in its perpendicular plane and vice versa. This is a cooperation effect between spin-velocity locking and spin-dependent interaction. While in the stripe phase, a breathing mode will induce the sliding mode of the density stripe along its perpendicular direction. This reflects the fact that the sliding mode is the gapless excitation in this phase due to spontaneous spatial translation symmetry breaking.
Recently a spin-orbit-coupled Bose condensate has been realized experimentally using two-photon Raman coupling, and this condensate exhibits two distinct equilibrium phases known as the plane wave phase and the stripe phase at equilibrium. In this Rapid Communication we show that such condensate exhibits unique dynamic behaviors, which originate from interactions and are absent in a noninteracting system. In the plane wave phase, a dipole mode will induce a breathing mode in its perpendicular plane and vice versa. This is a cooperation effect between spin-velocity locking and spin-dependent interaction. While in the stripe phase, a breathing mode will induce the sliding mode of the density stripe along its perpendicular direction. This reflects the fact that the sliding mode is the gapless excitation in this phase due to spontaneous spatial translation symmetry breaking.
We study analytically and numerically the nonlinear collective dynamics of quasi-one-dimensional spin–orbit coupled spin-1 Bose–Einstein condensates trapped in harmonic potential. The ground state of the system is determined by minimizing the Lagrange density, and the coupled equations of motions for the center-of-mass coordinate of the condensate and its width are derived. Then, two low energy excitation modes in breathing dynamics and dipole dynamics are obtained analytically, and the mechanism of exciting the anharmonic collective dynamics is revealed explicitly. The coupling among spin–orbit coupling, Raman coupling and spin-dependent interaction results in multiple external collective modes, which leads to the anharmonic collective dynamics. The cooperative effect of spin momentum locking and spin-dependent interaction results in coupling of dipolar and breathing dynamics, which strongly depends on spin-dependent interaction and behaves distinct characters in different phases. Interestingly, in the absence of spin-dependent interaction, the breathing dynamics is decoupled from spin dynamics and the breathing dynamics is harmonic. Our results provide theoretical evidence for deep understanding of the ground sate phase transition and the nonlinear collective dynamics of the system.
Abstract Since their first realization more than a decade ago spin-orbit-coupled Bose-Einstein condensates have been the subject of intense theoretical and experimental investigations. Spin-orbit coupling deeply modifies the equilibrium properties of the condensate, giving rise to novel configurations such as a supersolid stripe phase and a phase-separated plane-wave state. At the level of dynamics, both the frequency and the nature of the collective modes are significantly affected by the coupling with the spin degree of freedom. Here we review some of the most relevant advances in the field and provide our perspective on possible future research directions.
We studied the collective modes of a Bose-Einstein condensate (BEC) with spin-orbit coupling. We developed the hydrodynamic equations for spin-orbit coupled BECs and used them to study collective modes in the plane-wave phase and large Rabi coupling regime for both a uniform BEC and a BEC in a harmonic trap. In the homogeneous situation, we obtained energy spectra of elementary excitations and found that the spin-orbit coupling can increase the effective mass of the atoms, which will suppress the sound velocity. The spin-orbit coupling can also change the spin mixing, which will modify the interaction energy, and may lead to an enhancement of sound velocity. The competition between these two effects gives the behavior of sound velocity. In a harmonic trap, we found that the dipole mode and the breathing mode are coupled together in the plane-wave phase, and these two modes have a $\ensuremath{\pi}/2$ phase difference, because the spin-orbit coupling and the interaction are not invariant under spin rotation. However, in the large Rabi coupling regime, the dipole mode and the breathing mode are decoupled due to the symmetry restriction.
The recent realization of synthetic spin-orbit coupling represents an outstanding achievement in the physics of ultracold quantum gases. In this review we explore the properties of a spin-orbit-coupled Bose-Einstein condensate with equal Rashba and Dresselhaus strengths. This system presents a rich phase diagram, which exhibits a tricritical point separating a zero-momentum phase, a spin-polarized plane-wave phase, and a stripe phase. In the stripe phase translational invariance is spontaneously broken, in analogy with supersolids. Spin-orbit coupling also strongly affects the dynamics of the system. In particular, the excitation spectrum exhibits intriguing features, including the suppression of the sound velocity, the emergence of a roton minimum in the plane-wave phase, and the appearance of a double gapless band structure in the stripe phase. Finally, we discuss a combined procedure to make the stripes visible and stable, thus allowing for a direct experimental detection.
The recent realization of synthetic spin-orbit coupling represents an outstanding achievement in the physics of ultracold quantum gases. In this thesis we explore the properties of spin-orbit-coupled Bose-Einstein condensates with equal Rashba and Dresselhaus strengths. These systems present a rich phase diagram, which exhibits a tricritical point separating a single-minimum phase, a spin-polarized plane-wave phase, and a stripe phase. In the stripe phase translational invariance is spontaneously broken, in analogy with supersolids. Spin-orbit coupling also strongly affects the dynamics of the system. In particular, the excitation spectrum exhibits intriguing features, including the suppression of the sound velocity, the emergence of a roton minimum in the plane-wave phase, and the appearance of a double gapless band structure in the stripe phase. Finally, we discuss a combined procedure to make the stripes visible and stable, thus allowing for a direct experimental detection.
The phase diagram of a Bose-Einstein condensate with Raman-induced spin-orbit coupling includes a stripe phase with supersolid features. In this work we develop a perturbation approach to study the ground state and the Bogoliubov modes of this phase, holding for small values of the Raman coupling. We obtain analytical predictions for the most relevant observables (including the periodicity of stripes, sound velocities, compressibility, and magnetic susceptibility) which are in excellent agreement with the exact (non perturbative) numerical results, obtained for significantly large values of the coupling. We further unveil the nature of the two gapless Bogoliubov modes in the long-wavelength limit. We find that the spin branch of the spectrum, corresponding in this limit to the dynamics of the relative phase between the two spin components, describes a translation of the fringes of the equilibrium density profile, thereby providing the crystal Goldstone mode typical of a supersolid configuration. Finally, using sum-rule arguments, we show that the superfluid density can be experimentally accessed by measuring the ratio of the sound velocities parallel and perpendicular to the direction of the spin-orbit coupling.