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.