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.
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.
Supersolid is a special state of matter with both superfluid properties and spontaneous modulation of particle density. In this paper we focus on the supersolid stripe phase realized in a spin-orbit-coupled Bose-Einstein condensate and explore the properties of a class of metastable supersolids. In particular, we study a one-dimensional supersolid whose characteristic wave number
k (magnitude of wave vector) deviates from
km, i.e., the one in the ground state. In other words, the period of density modulation is shorter or longer than the one in the ground state. We find that this class of supersolids can still be stable if their wave numbers fall in the range
kc1<k<kc2, with two thresholds
kc1 and
kc2. Stripes with
k outside this range suffer from dynamical instability with a complex Bogoliubov excitation spectrum at long wavelength. Experimentally, these stripes with
k different from
km are accessible by exciting the longitudinal spin dipole mode, resulting in temporal oscillation of the stripe period as well as
k. Within the mean-field Gross-Pitaevskii theory, we numerically confirm that for a large enough amplitude of spin dipole oscillation, the stripe states become unstable through breaking periodicity, in qualitative agreement with the existence of thresholds of
k for stable stripes. Our work extends the concept of supersolid and uncovers an unconventional class of metastable supersolids to explore.
Spin-orbit-coupled Bose-Einstein condensates are a flexible experimental platform to engineer synthetic quantum many-body systems. In particular, they host the so-called stripe phase, an instance of a supersolid state of matter. The peculiar excitation spectrum of the stripe phase, a definite footprint of its supersolidity, has been difficult to measure experimentally. In this work, we performed in situ imaging of the stripes and directly observed both superfluid and crystal excitations. We investigated superfluid hydrodynamics and revealed a stripe compression mode, thus demonstrating that the system possesses a compressible crystalline structure. Through the frequency softening of this mode, we located the supersolid transition point. Our results establish spin-orbit-coupled supersolids as ideal systems to investigate supersolidity and its rich dynamics.
Supersolidity---a coexistence of superfluidity and crystalline or amorphous density variations---has been vividly debated ever since its conjecture. While the initial focus was on helium-4, recent experiments uncovered supersolidity in ultracold dipolar quantum gases. Here we propose a self-bound supersolid phase in a binary mixture of Bose gases with short-range interactions, making use of the nontrivial properties of spin-orbit coupling. We find that a first-order phase transition from a self-bound supersolid stripe phase to a zero-minimum droplet state of the Bose gas occurs as a function of the Rabi coupling strength. These phases are characterized using the momentum distribution, the transverse spin polarization, and the superfluid fraction. The critical point of the transition is estimated in an analytical framework. The predicted density-modulated supersolid stripe and zero-minimum droplet phase should be experimentally observable in a binary mixture of
39K with spin-orbit coupling.
Soon after its theoretical prediction, striped-density states in the presence of synthetic spin-orbit coupling were realized in Bose-Einstein condensates of ultracold neutral atoms [J.-R. Li et al., Nature 543, 91 (2017)]. The achievement opens avenues to explore the interplay of superfluidity and crystalline order in the search for supersolid features and materials. The system considered is essentially made of two linearly coupled Bose-Einstein condensates, that is, a pseudo-spin-
1/2 system, subject to a spin-dependent gauge field ${\ensuremath{\sigma}}_{z}\ensuremath{\hbar}{k}_{\ensuremath{\ell}}$. Under these conditions, the stripe phase is achieved when the linear coupling $\ensuremath{\hbar}\mathrm{\ensuremath{\Omega}}/2$ is small against the gauge energy $m\mathrm{\ensuremath{\Omega}}/\ensuremath{\hbar}{k}_{\ensuremath{\ell}}^{2}<1$. The resulting density stripes have been interpreted as a standing-wave interference pattern with approximate wave number $2{k}_{\ensuremath{\ell}}$. Here, we show that the emergence of the stripe phase is induced by an array of Josephson vortices living in the junction defined by the linear coupling. As happens in superconducting junctions subject to external magnetic fields, a vortex array is the natural response of the superfluid system to the presence of a gauge field. Also similar to superconductors, the Josephson currents and their associated vortices can be present as a metastable state in the absence of gauge field. We provide closed-form solutions to the 1D mean-field equations that account for such vortex arrays. The underlying Josephson currents coincide with the analytical solutions to the sine-Gordon equation for the relative phase of superconducting junctions [C. Owen and D. Scalapino, Phys. Rev. 164, 538 (1967)].