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Physical Review B

Phase-jump nucleation in a one-dimensional model of a supersolid

Peter Mason, Christophe Josserand

Published Dec 13, 2013• 2 citations
Publisher

Abstract

We consider the activation of phase jumps in a one-dimensional Bose-Einstein condensate. Our system includes a nonlocal interaction term which mimics, for example, a dipole-blockaded Rydberg system. In the mean-field limit the condensate can form superfluid droplets, arranged periodically on a line, thus displaying a supersolidlike ground state. Under an imposed velocity, phase jumps will develop. We study these phase jumps numerically and analytically, and are able to write down a relationship between the velocity, the width of the density peaks, the number of phase jumps, and $\ensuremath{\Lambda}$, a parameter that determines the number of peaks in the condensate.

DOI
10.1103/physrevb.88.224506

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We consider the activation of phase jumps in a one-dimensional Bose-Einstein condensate. Our system includes a nonlocal interaction term which mimics, for example, a dipole-blockaded Rydberg system. In the mean-field limit the condensate can form superfluid droplets, arranged periodically on a line, thus displaying a supersolidlike ground state. Under an imposed velocity, phase jumps will develop. We study these phase jumps numerically and analytically, and are able to write down a relationship between the velocity, the width of the density peaks, the number of phase jumps, and \ensuremathΛ\ensuremath{\Lambda}\ensuremathΛ, a parameter that determines the number of peaks in the condensate.
We investigate theoretically and numerically a model of a supersolid in a dipole-blockaded Bose-Einstein condensate. The dependence of the superfluid fraction with an imposed thermal bath and a uniform boost velocity on the condensate is considered. Specifically, we observe a critical velocity for the nucleation of vortices in our system that is strongly linked to a steplike decrease in the superfluid fraction. We are able to use a scaling argument based on the energy required to activate a vortex, relating the critical temperature to the critical velocity, and find that this relationship is in good agreement with the numerical simulations carried out on the nonlocal Gross-Pitaevskii equation.
Motivated by a recent experiment [L. Chomaz et al., Nat. Phys. 14, 442 (2018)], we perform numerical simulations of a dipolar Bose-Einstein condensate (BEC) in a tubular, periodic confinement at T=0T=0T=0 within density functional theory, where the beyond-mean-field correction to the ground-state energy is included in the local density approximation. We study the excitation spectrum of the system by solving the corresponding Bogoliubov--de Gennes equations. The calculated spectrum shows a roton minimum, and the roton gap decreases by reducing the effective scattering length. As the roton gap disappears, the system spontaneously develops a periodic linear structure formed by denser clusters of atomic dipoles immersed in a dilute superfluid background. This structure shows the hallmarks of a supersolid system, i.e., (i) a finite nonclassical translational inertia along the tube axis and (ii) the appearance of two gapless modes, i.e., a phonon mode associated with density fluctuations and resulting from the translational discrete symmetry of the system, and a Nambu-Goldstone gapless mode corresponding to phase fluctuations, resulting from the spontaneous breaking of the gauge symmetry. A further decrease in the scattering length eventually leads to the formation of a periodic linear array of self-bound droplets.
We theoretically investigate a supersolid structure taking the form of stacked, disk-shaped superfluid droplets connected via a dilute superfluid, in an antidipolar condensate. A phase diagram is determined for varying the particle number and scattering length, identifying the regions of a regular dipolar superfluid, supersolid stacks, and isolated stacked disk-shaped droplets in an experimentally realizable trapping potential. The collective Bogoliubov excitation spectrum across the superfluid-supersolid phase transition is studied, and the transition point is found to be associated with the breaking of the degeneracy of the two lowest-lying modes. The dynamical generation of the supersolid stacks is also investigated by ramping down the scattering length across the phase transition. Moreover, we have studied the impact of vortex-line penetration on the phase transition. We have found that the presence of a vortex line causes the supersolid region to move towards weaker contact interactions. Our detailed numerical simulations highlight that an antidipolar condensate can create such supersolid stacks within an experimentally reachable parameter regime.
Recently, experimental and theoretical studies have been actively conducted on the supersolid state of a Bose-Einstein condensate (BEC) with magnetic dipole-dipole interaction (DDI).However, most of these studies have employed a single-component BEC, whose stabilization relies on the beyondmean-field effect with high density, and thus its lifetime is restricted.In this work, we propose a supersolid state of a two-component dipolar BEC, in which a weaker DDI can form the supersolid structure, stabilizing the system without the beyond-mean-field effect.As a result, such a supersolid has a longer lifetime, which enables the observation of long-time dynamics.Here we numerically demonstrate the dynamics of periodic droplet-structure formation triggered by sudden or gradual change of a parameter.We find that the generated droplets have the phase coherence, confirming dynamical formation of supersolid.
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.