Using Bogoliubov theory we calculate the excitation spectrum of a spinor Bose-Einstein condensed gas with equal Rashba and Dresselhaus spin-orbit coupling in the stripe phase. The emergence of a double gapless band structure is pointed out as a key signature of Bose-Einstein condensation and of the spontaneous breaking of translational invariance symmetry. In the long wavelength limit the lower and upper branches exhibit, respectively, a clear spin and density nature. For wave vectors close to the first Brillouin zone, the lower branch acquires an important density character responsible for the divergent behavior of the structure factor and of the static response function, reflecting the occurrence of crystalline order. The sound velocities are calculated as functions of the Raman coupling for excitations propagating orthogonal and parallel to the stripes. Our predictions provide new perspectives for the identification of supersolid phenomena in ultracold atomic gases. . It is characterized by the coexistence of two spontaneously broken symmetries. The breaking of gauge symmetry gives rise to off-diagonal long-range order yielding superfluidity, while the breaking of translational invariance yields diagonal long-range order characterizing the crystalline structure. The First experimental efforts toward the search of supersolidity were carried out in solid helium [5]. The strongly interacting nature of this system makes, however, the effects due to Bose-Einstein condensation (BEC) extremely small and no conclusive proof of supersolidity is still available in such a system [6]. More recently, systematic attempts to predict the occurrence of a supersolid phase have been carried out in atomic gases with dipolar [7][8][9] and soft core, finite range interactions [10][11][12][13][14][15]. However, these configurations have not yet been experimentally realized in the quantum degenerate phase required to observe the new effects.The recent realization of spinor BECs with spin-orbit coupling [17][18][19][20] is opening new perspectives in the field. In systems with equal Rashba and Dresselhaus couplings and for small values of the Raman coupling, theory in fact predicts the occurrence of a stripe phase where translational invariance is spontaneously broken [21][22][23]. Actually these systems are periodic only in one direction and can be considered as superfluid nematic liquid crystals. Experiments are already available in the relevant range of parameters, but no direct evidence of the density modulations is still available, due to the smallness of the contrast and the microscopic distance separating consecutive stripes. A phase transition has been nevertheless detected [20] at values of the Raman coupling below which theory predicts the occurrence of the stripe phase.The purpose of this work is to show that the excitation spectrum of the gas in the stripe phase exhibits typical supersolid features, like the occurrence of two gapless bands and the divergent behavior of the static structure factor for wave vectors approach...

By calculating the density response function we identify the excitation spectrum of a Bose-Einstein condensate with equal Rashba and Dresselhaus spin-orbit coupling. We find that the velocity of sound along the direction of spin-orbit coupling is deeply quenched and vanishes when one approaches the second-order phase transition between the plane wave and the zero momentum quantum phases. We also point out the emergence of a roton minimum in the excitation spectrum for small values of the Raman coupling, providing the onset of the transition to the stripe phase. Our findings point out the occurrence of a strong anisotropy in the dynamic behavior of the gas. A hydrodynamic description accounting for the collective oscillations in both uniform and harmonically trapped gases is also derived.

Using a sum rule approach we investigate the dipole oscillation of a spin-orbit coupled Bose-Einstein condensate confined in a harmonic trap. The crucial role played by the spin polarizability of the gas is pointed out. We show that the lowest dipole frequency exhibits a characteristic jump at the transition between the stripe and spin-polarized phase. Near the second order transition between the spin-polarized and the single minimum phase the lowest frequency is vanishingly small for large condensates, reflecting the divergent behavior of the spin polarizability. We compare our results with recent experimental measurements as well as with the predictions of effective mass approximation.

We study the zero-temperature phase diagram of a spin-orbit-coupled Bose-Einstein condensate of spin 1, with equally weighted Rashba and Dresselhaus couplings. Depending on the antiferromagnetic or ferromagnetic nature of the interactions, we find three kinds of striped phases with qualitatively different behaviors in the modulations of the density profiles. Phase transitions to the zero-momentum and the plane-wave phases can be induced in experiments by independently varying the Raman coupling strength and the quadratic Zeeman field. The properties of these transitions are investigated in detail, and the emergence of tricritical points, which are the direct consequence of the spin-dependent interactions, is explicitly discussed.

The striped phase exhibited by a spin-1/2 Bose-Einstein condensate with spin-orbit coupling is characterized by the spontaneous breaking of two continuous symmetries: gauge and translational symmetry. This is a peculiar feature of supersolids and is the consequence of interaction effects. We propose an approach to produce striped configurations with high-contrast fringes, making their experimental detection in atomic gases a realistic perspective. Our approach, whose efficiency is directly confirmed by three-dimensional Gross-Pitaevskii simulations, is based on the space separation of the two spin components into a two-dimensional bilayer configuration, causing the reduction of the effective interspecies interaction and the increase of the stability of the striped phase. We also explore the effect of a π/2 Bragg pulse, causing the increase of the fringe wavelength, and of a π/2 rf pulse, revealing the coherent nature of the order parameter in the spin channel.The experimental realization of synthetic gauge fields and spin-orbit-coupled configurations in spinor Bose-Einstein condensates [1-4] has opened interesting perspectives for the realization of novel quantum phases. Among them the striped phase represents one of the most challenging configurations, being characterized by the spontaneous breaking of two continuous symmetries: gauge and translational symmetry. The simultaneous breaking of these symmetries is a typical feature of supersolids, a phase of matter not yet realized experimentally, despite systematic efforts made in solid 4 He [5]. The realization of supersolidity is presently the object of several theoretical proposals, focusing on atomic gases interacting with dipolar [6-8] or soft-core, finite-range forces [9][10][11][12][13][14][15].The occurrence of a striped phase in spin-orbit-coupled two-component Bose gases has been the object of several recent theoretical investigations [17][18][19][20][21][22][23][24][25][26][27]. The experiment of Ref.[4] was not, however, able to reveal the typical features of this phase, characterized by the occurrence of periodic modulations of the density profile. The main reason is that the contrast and the wavelength of the modulations in the experimental conditions of [4] are too small. When written in a locally spin-rotated frame [28], the single-particle spin-orbit Hamiltonian realized in [4] takes the following form:This Hamiltonian is the result of the application of two counterpropagating polarized lasers with wave vector difference k 0 , chosen along the x direction, providing Raman transitions between two different hyperfine states, in the presence of a nonlinear Zeeman field. The strength of the Raman coupling is fixed by the parameter . Equation (1) also includes the coupling with an effective magnetic field δ given by the sum of the true external magnetic field and of the frequency detuning between the two lasers (see, for example, [28]). The spin matrices entering the single-particle Hamiltonian are the usual 2 × 2 Pauli matrices. The operator p = −i ∇ i...

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We consider a weakly interacting uniform atomic Bose gas with a time-dependent nonlinear coupling constant. By developing a suitable Bogoliubov treatment we investigate the time evolution of several observables, including the momentum distribution, the degree of coherence in the system, and their dependence on dimensionality and temperature. We rigorously prove that the low-momentum Bogoliubov modes remain frozen during the whole evolution, while the high-momentum ones adiabatically follow the change in time of the interaction strength. At intermediate momenta we point out the occurrence of oscillations, which are analogous to Sakharov oscillations. We identify two wide classes of time-dependent behaviors of the coupling for which an exact solution of the problem can be found, allowing for an analytic computation of all the relevant observables. A special emphasis is put on the study of the coherence property of the system in one spatial dimension. We show that the system exhibits a smooth "light-cone effect," with typically no prethermalization. arXiv:1810.01362v2 [cond-mat.quant-gas]

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