Abstract
A one-dimensional Bose-Einstein condensate may experience nonlinear periodic modulations known as "cnoidal waves". We argue that such structures represent promising candidates for the study of supersolidity-related phenomena in a non-equilibrium state. A mean-field treatment makes it possible to rederive Leggett's formula for the superfluid fraction of the system and to estimate it analytically. We determine the excitation spectrum, for which we obtain analytical results in the two opposite limiting cases of (i) a linearly modulated background and (ii) a train of dark solitons. The presence of two Goldstone (gapless) modes -- associated with the spontaneous breaking of $\mathrm{U}(1)$ symmetry and of continuous translational invariance -- at large wavelength is verified. We also calculate the static structure factor and the compressibility of cnoidal waves, which show a divergent behavior at the edges of each Brillouin zone.
Highlights
Supersolid phases of matter have attracted increasing interest in the past few years
We note here that the spectrum of cnoidal waves has been studied by the mathematical physics community, which mainly addressed the problem of dynamic stability; our focus is different and concerns the energetic instability on one side and the relationship with the phenomenon of supersolidity on the other side
We have studied several relevant features of an ultracold Bose gas in a cnoidal-wave state
Summary
Supersolid phases of matter have attracted increasing interest in the past few years. Similar to 4He, a large number of excitations with momentum close to this value (called levons) are created when crossing the Landau velocity, which represents the onset of the transition to the layered phase The latter features a superfluid fraction smaller than 1. Cnoidal-wave-like solutions have been found for the extended Gross-Pitaevskii equation describing a self-trapped cigar-shaped Bose gas [35]. They are candidates for studying phenomena related to supersolidity within the most standard Bose-Einstein condensates The latter do not suffer from the strong three-body losses typical of dilute ultracold systems in which the stabilization is due to beyond-mean-field equation of states, as for dipolar gases and quantum mixtures. After taking the gradient on both sides, it becomes formally identical to the Euler equation for the potential flow of an inviscid fluid, with the addition of a quantum potential
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