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arXiv:2301.06094 · Physical Review A

Metastable supersolid in spin-orbit-coupled Bose-Einstein condensates

Wei-Lei Xia, Lei Chen, Tiantian Li, Yongping Zhang, Qizhong Zhu

Published May 2, 2023• 11 citations
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Abstract

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 ${k}_{m}$, 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 ${k}_{c1}<k<{k}_{c2}$, with two thresholds ${k}_{c1}$ and ${k}_{c2}$. 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 ${k}_{m}$ 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.

DOI
10.1103/physreva.107.053302

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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 kkk (magnitude of wave vector) deviates from km{k}_{m}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{k}_{c1}<k<{k}_{c2}kc1​<k<kc2​, with two thresholds kc1{k}_{c1}kc1​ and kc2{k}_{c2}kc2​. Stripes with kkk outside this range suffer from dynamical instability with a complex Bogoliubov excitation spectrum at long wavelength. Experimentally, these stripes with kkk different from km{k}_{m}km​ are accessible by exciting the longitudinal spin dipole mode, resulting in temporal oscillation of the stripe period as well as kkk. 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 kkk for stable stripes. Our work extends the concept of supersolid and uncovers an unconventional class of metastable supersolids to explore.
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.
Despite ground-breaking observations of supersolidity in spin-orbit-coupled Bose-Einstein condensates, until now the dynamics of the emerging spatially periodic density modulations has been vastly unexplored. Here, we demonstrate the nonrigidity of the density stripes in such a supersolid condensate and explore their dynamic behavior subject to spin perturbations. We show both analytically in infinite systems and numerically in the presence of a harmonic trap how spin waves affect the supersolid's density profile in the form of crystal waves, inducing oscillations of the periodicity as well as the orientation of the fringes. Both these features are well within reach of present-day experiments. Our results show that this system is a paradigmatic supersolid, featuring superfluidity in conjunction with a fully dynamic crystalline structure.
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.
Abstract Supersolid is an exotic state of matter characterized by both superfluid properties and periodic particle density modulation, due to spontaneous breaking of U(1) gauge symmetry and spatial translation symmetry, respectively. For conventional supersolids, continuous translation symmetry breaking is accompanied by one gapless Goldstone mode in the excitation spectra. An interesting question naturally arises: what is the consequence of breaking discrete translation symmetry for supersolids? In this work, we study the consequence of discrete symmetry breaking in a Z n supersolid resulting from spontaneous breaking of a discrete Z n symmetry, or equivalently, a discrete translation symmetry. This Z n supersolid is realized in the stripe phase of spin–orbit-coupled Bose–Einstein condensate under an external periodic potential with period 1 / n of intrinsic stripe period. For n ⩾ 2 , there are n degenerate ground states with spontaneously broken lattice translation symmetry. The low-energy excitations of Z n supersolid include a pseudo-Goldstone mode, whose excitation gap at long wavelength limit is found to decrease rapidly with n. We further numerically show that, when confined in a harmonic trap, a spin-dependent perturbation can result in the transition between degenerate ground states of Z n supersolid. With the integer n tunable using the experimental technique of generating subwavelength optical lattice, the Z n supersolid proposed here offers a cold atom platform to simulate physics related with generic Z n symmetry breaking in a highly controllable setting.
We explore the stability of supersolid striped waves, plane-wave boselets, and other extended states in one-dimensional spin-orbit-coupled Bose-Einstein condensates with repulsive three-body interactions (R3BIs), modeled by quintic terms in the framework of the corresponding Gross-Pitaevskii equations. In the absence of R3BIs, the extended states are susceptible to the modulational instability (MI) induced by the cubic attractive nonlinearity. Using the linearized Bogoliubov-de-Gennes equations, we identify multiple new types of MI, including baseband, passband, mixedband, and zero-wavenumber-gain ones, which give rise to deterministic rogue waves and complex nonlinear wave patterns. Our analysis reveals that R3BIs eliminate baseband and zero-wavenumber-gain MIs, forming, instead, phonon modes that enable stable boselets. Additionally, mixedband and passband MIs are suppressed, which results in a latticelike phonon-roton mode that supports a stable supersolid phase. These stable supersolids can be realized using the currently available ultracold experimental setup.
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.