Exciting the Goldstone Modes of a Supersolid Spin-Orbit-Coupled Bose Gas

Supersolidity is deeply connected with the emergence of Goldstone modes, reflecting the spontaneous breaking of both phase and translational symmetry. Here, we propose accessible signatures of these modes in harmonically trapped spin-orbit-coupled Bose-Einstein condensates, where supersolidity appears in the form of stripes. By suddenly changing the trapping frequency, an axial breathing oscillation is generated, whose behavior changes drastically at the critical Raman coupling. Above the transition, a single mode of hybridized density and spin nature is excited, while below it, we predict a beating effect signaling the excitation of a Goldstone spin-dipole mode. We further provide evidence for the Goldstone mode associated with the translational motion of stripes. Our results open up new perspectives for probing supersolid properties in experimentally relevant configurations with both symmetric as well as highly asymmetric intraspecies interactions.

Figure 1

Collective modes for symmetric intraspecies interactions. (a),(b) Oscillations of the observables $x_{\mathrm{rms}}=\sqrt{{⟨x⟩}^2}$ and $d_x={⟨x⟩}_{\uparrow }-{⟨x⟩}_{\downarrow }$ after suddenly removing the perturbation $H_{\mathrm{pert}}=\lambda m{\omega }_x^2{x}^2$ with $\lambda =0.2$. In the stripe phase (a), a clear beating of two frequencies ${\omega }_{\text{B}}\approx 1.49{\omega }_x$ and ${\omega }_{\mathrm{SD}}\approx 0.32{\omega }_x$ is visible in the observable $d_x$, which is absent in the plane-wave phase (b), where $d_x$ oscillates only at a single frequency ${\omega }_{\text{B}}\approx 1.02{\omega }_x$. (c) Dispersion $\omega (\mathrm{\Omega })$ of the breathing mode (B), the spin-dipole mode (SD), and the center-of-mass (dipole) mode (D), calculated for $\lambda \ll 1$. The breathing and the spin-dipole modes are fully hybridized above the critical coupling ${\mathrm{\Omega }}_{\mathrm{cr}}\approx 2.5E_r$, while below ${\mathrm{\Omega }}_{\mathrm{cr}}$, a new Goldstone mode of spin nature appears. The dipole frequencies ${\omega }_{\text{D}}$ have been obtained from the center-of-mass oscillation $x_{\mathrm{c}.\mathrm{m}.}=⟨x⟩$ after a sudden shift of the trap center. For $\mathrm{\Omega }>{\mathrm{\Omega }}_{\mathrm{cr}}$, they practically coincide with the bound ${\omega }_x/\sqrt{1+2E_r\chi }$. The violation of this upper bound by ${\omega }_{\text{D}}$ for $\mathrm{\Omega }<{\mathrm{\Omega }}_{\mathrm{cr}}$ implies the emergence of a new low-energy mode.

Figure 2

Evidence for the zero-frequency Goldstone mode associated with the translation of the stripes. (a) Time evolution of the polarization $⟨{\sigma }_z⟩$ with respect to its equilibrium value $⟨{{\sigma }_z}_0⟩$ after removing the perturbation $H_{\mathrm{pert}}=-\lambda E_r{\sigma }_z$ with $\lambda =0.02$ in the stripe phase ($\mathrm{\Omega }=2E_r$) and $\lambda =0.1$ in the plane-wave phase ($\mathrm{\Omega }=3E_r$). In the latter case, the polarization oscillates around equilibrium at the dipole frequency ${\omega }_{\text{D}}\approx 0.55{\omega }_x$. By contrast, in the stripe phase, the polarization remains locked for a time much longer than $2\pi /{\omega }_x$, providing evidence for the zero-frequency Goldstone mode. The low-amplitude oscillations at the dipole frequency ${\omega }_{\text{D}}\approx 0.89{\omega }_x$ shown in the inset indicate a weak excitation of the center-of-mass mode. (b) Time evolution of the density profile $n(x)$ in the stripe phase for the same scenario as in (a), explicitly revealing the excitation of the translational motion of the stripes. In the linear regime, the velocity of the stripes is proportional to the perturbation strength $\lambda$ according to $v(\lambda )\approx 0.48\lambda \hslash k_0/m$.

Figure 3

Same as Fig. 1, but for strongly asymmetric intraspecies interactions, as relevant for ${}^{39}\mathrm{K}$. The asymmetry leads to a smooth crossover between the plane-wave and the single-minimum regime. Nevertheless, in the stripe phase, the beating effects (a) and the additional frequencies (c) characteristic of the Goldstone mode are evident, while they are absent in the plane-wave phase (b).