Supersolid stripes enhanced by correlations in a Raman spin-orbit-coupled system

Abstract

A Bose gas under the effect of Raman Spin-Orbit Coupling (SOC) is analyzed using the Discrete Spin T-moves Diffusion Monte Carlo method [1,2]. By computing the energy as well as the static structure factor and the superfluid fraction of the system, the emergence of an energetically favorable supersolid stripe state is observed, which is in agreement with recent observations. A significant enhancement of the stability of the stripe phase with respect to the mean-field prediction is observed when the strength of the inter-atomic correlations is increased. We also quantify and characterize the degree of superfluidity of the stripes and show that this quantity is mostly determined by the ratio between the Raman coupling and the square of the momentum difference between the pair of SOC inducing laser beams.Spin-Orbit Coupling (SOC), which denotes the interplay between a particle's momentum and its spin, has been a subject of interest in the recent years, both theoretically and experimentally. This is due to the wide variety of exotic quantum states induced by this kind of interaction, which include topological insulators [3], topological superconductors [4], and Majorana fermions [5]. SOC is a relativistic effect that emerges naturally in electronic systems, and that is also synthetically engineered [6] in ultracold atomic gases. These recent realizations in dilute gases represent an important achievement in the study of the physics of SOC due to the high controllability and tunability of these systems. In the particular case of Raman SOC, its implementation was first achieved experimentally by inducing a Raman coupling via two laser beams on an atomic Lambda type configuration. SOC is then generated by the simultaneous driving of a spin flip transition and transferring of momentum [7][8][9][10][11]. Under this scheme, Raman SOC has been realized with 87 Rb bosons, both in the continuum [7] and in a lattice [12,13], and also with other species: 6 Li [14], 40 K [15], 87 Sr [16], 173 Yb [17, 18], and 161 Dy [19]. In this context, two hyperfine states of the atom are labeled as the spin states.In this Letter, we focus in Raman SOC, which couples the linear momentum of an atom with its spin according toŴwith m the mass of the particle,P x the x-component of the momentum,σ x andσ z the Pauli matrices, Ω the Raman coupling, and k 0 the magnitude of the wave vector difference between the two laser beams. Some striking features induced by the SOC interaction can be observed already at the single particle level. The coupling between momentum and spin implies that the minimum of the energy dispersion relation is in a non-zero momentum, degenerate state for a given range of values of the Raman coupling [20]. This degeneracy involves states of equal magnitude but opposite sign in momentum space, enabling the possibility of a stripe phase ground state. Supersolid stripes arise from the breaking of two symmetries: a gauge symmetry, giving rise to off-diagonal long-range order, and spatial symmetry, seen as a periodic density modula...

Supersolid phase of a spin-orbit-coupled Bose-Einstein condensate: A perturbation approach

“…Notice that in the plane-wave phase the sound velocity c 2 d,x should be actually replaced by the product c x,+ c x,− of the sound velocities along the +x and −x directions. The general behavior of the superfluid density as a function of the Raman coupling Ω R is qualitatively similar to the results derived in [51], where the phase twist method was employed using a variational Ansatz for the order parameter in the stripe phase; first studies based on Monte Carlo methods have shown that quantum fluctuations do not significantly affect the value of the superfluid density, which in three-dimensional dilute systems remains close to the mean-field prediction [54]. As pointed out in [62] the effects of the spin-orbit coupling have a dramatic consequence near the second-order transition between the plane-wave and zeromomentum phases, characterized by the divergence of the magnetic susceptibility χ M .…”

“…We conclude by mentioning that for g ss > 0 in the large-density regime the plane-wave phase is no longer energetically favored, and one has a direct first-order transition from the stripe to the zero-momentum phase [53,54].…”

Supersolid striped droplets in a Raman spin-orbit-coupled system

“…In contrast to other systems featuring spatial ordering like dipoles [11,12], this property is present in SOC systems even in ultradilute conditions. The stripe phase has been both predicted theoretically [5,10] and detected experimentally [13], and the resulting stripes have been shown to be superfluid [10,14,15], therefore being often referred as supersolid stripes or superstripes. It has also been shown that the increase of interatomic correlations enhances the domain of the stripe phase in the phase diagram of Raman SOC systems [15].…”

Coherently Coupled Mixtures of Bose-Einstein Condensed Gases

“…It is also important to point out that quantum fluctuations have a negligible consequence on the depletion of the condensate (70), thereby revealing that the superfluid density crucially differs from Bose-Einstein condensation in these systems. The behavior of the superflud density has been also the object of theoretical calculations in the stripe phase (85,86) where the occurrence of density modultaions and the consequent emergence of crystal like effects is a further source of reduction of the ratio ρs/ρ. A closely related quantity emphasizing the effects of superfluidity is the moment of inertia of a trapped gas.…”