Traveling supersolid stripe patterns in spin-orbit-coupled Bose-Einstein condensates

TL;DR

We address traveling supersolid stripe patterns in a spin-orbit-coupled Bose-Einstein condensate, showing that a population imbalance between the two dressed-spin minima induces a constant fringe velocity $v = rac{\mu_s}{\hbar k_1}$ and a spin-polarized moving density modulation. We solve for traveling stripe order parameters using a two-harmonic Ansatz and an exact Bloch-wave formulation, and evaluate them with variational and perturbative methods. The Bogoliubov spectrum in the comoving frame exhibits an inversion asymmetry and, at large $s_p$, energetic and dynamical instabilities driven by the spin-phonon branch, with the stripe melting into a plane-wave phase where the roton minimum emerges. The work outlines experimental routes to observe traveling stripes and highlights their connection to supersolidity and roton physics in spin-orbit-coupled quantum gases.

Abstract

We consider a traveling supersolid stripe pattern in a spin-orbit-coupled Bose gas. This configuration is associated with an unequal population of the two single-particle energy minima, giving rise to a chemical potential difference that sets the fringe velocity. Unlike stationary stripes, the moving pattern is spin-polarized, with decreasing contrast as the population imbalance increases, eventually leading to stripe melting and transition to the uniform plane-wave phase. The Bogoliubov spectrum of the moving stripes exhibits asymmetry under inversion of the excitation quasimomentum. At high population imbalance, we identify energetic and dynamical instabilities in the spin-phonon mode which transforms to the roton mode of the plane-wave phase as the stripe structure vanishes.

Figures (6)

• Figure 1: Schematic illustration of background flow and lattice motion scenarios in the stripe phase of a spin-orbit-coupled BEC. (a1) Supercurrent state in the laboratory frame: the superfluid background (horizontal black line) flows, while the Raman lasers generating the spin-orbit coupling (blue and red arrows) and the density modulations (green wavy line) are at rest. (a2) The same supercurrent state in the rest frame of the superfluid, where the Raman lasers and the fringes translate together. (b1) Traveling stripe pattern in the laboratory frame, characterized by the absence of net mass transport at the coarse-grained level. (b2) Traveling stripe pattern in the comoving frame, where the fringes are at rest (see Sec. \ref{['sec:exc_spectrum']}). In all scenarios, the normal component is pinned to the Raman lasers.
• Figure 2: Density profiles of spin-orbit-coupled BECs in the stripe phase for several values of the population imbalance between dressed spins: (a) $s_p = 0$ (stationary stripes), (b) $s_p = 0.5$, and (c) $s_p = 0.99$. The numerically obtained profiles of the bare spin-up (blue solid lines) and spin-down (yellow dashed lines) components are shown. Close to each curve the prediction of the perturbative approach is also plotted (black dotted lines). For definiteness, the fringe offset has been chosen such that a density minimum occurs at $x'=0$. The other parameters are $\hbar\Omega_R / E_R = 1.0$, $G_{dd} / E_R = 1.2$, and $G_{ss} / E_R = 0.32$.
• Figure 3: The fringe contrast [(a1)-(a2)], the mean condensation wave vector [(b1)-(b2)], the half stripe wave vector [(c1)-(c2)], and the bare spin polarization density [(d1)-(d2)] as functions of $s_p$. In panels [(a1)-(a2)], we show the fringe contrast of the total density (red), as well as the contrast of the majority (blue) and minority (yellow) bare spin components. Each observable is plotted for two different values of the Raman coupling: $\hbar\Omega_R / E_R = 1.0$ (left panels) and $\hbar\Omega_R / E_R = 2.6$ (right panels). Solid lines correspond to the numerical results. In the left column, close to each of these solid lines, we add a black dotted line showing the predictions of the perturbative approach. The interaction parameters are the same as in Fig. \ref{['fig:dens_profiles']}: $G_{dd} / E_R = 1.2$ and $G_{ss} / E_R = 0.32$.
• Figure 4: The energy per particle [(a1)-(a2)], the mean chemical potential [(b1)-(b2)], the chemical potential semi-difference [(c1)-(c2)], and the fringe velocity [(d1)-(d2)] as functions of $s_p$. As in Fig. \ref{['fig:observables']}, each quantity is shown for two different values of the Raman coupling: $\hbar\Omega_R / E_R = 1.0$ (left panels) and $\hbar\Omega_R / E_R = 2.6$ (right panels). Solid lines represent the results of numerical calculations. In the left panels, close to each of these lines, we add a black dotted line indicating the perturbative predictions. The interaction parameters are the same as in the previous figures: $G_{dd} / E_R = 1.2$ and $G_{ss} / E_R = 0.32$.
• Figure 5: Excitation spectrum of a spin-orbit-coupled BEC in the stripe phase for various values of $s_p$: (a1)-(a2) $s_p = 0$ (stationary stripes), (b1)-(b2) $s_p = 0.5$, (c1) $s_p = 0.95$, (c2) $s_p = 0.75$, and (d1)-(d2) $s_p = 0.99$. The top-row panels correspond to $\hbar\Omega_R / E_R = 1.0$, while those in the bottom row correspond to $\hbar\Omega_R / E_R = 2.6$. The excitation quasimomentum is taken along the $x$ direction. Only the lowest-lying excitation bands are shown; the band index $b$ is indicated next to each curve. In (c1), the inset shows a magnified view in the vicinity of $k_x = 0$ on the positive-$k_x$ side, where the lowest gapless band exhibits a negative frequency. In all panels, we plot only the real part of the frequency, $\omega_{b,k_x}^{\mathrm{R}}$; the imaginary part, $\omega_{b,k_x}^{\mathrm{I}}$, is zero except for the lowest gapless band in (c2) (see inset) and in (d2), where it is nonzero but very small and omitted for clarity. The red dashed line in (c2) displays the solution $\omega_{1,k_x}^*$ in the dynamically unstable region and $- \omega_{1,-k_x}$ in the dynamically stable region, both obtained from Eq. \ref{['eq:exc_eq_UV']}. In panels (d1)-(d2), the black dashed curves represent the two branches of the excitation spectrum in the plane-wave phase, computed using the same parameters as in the case of the stripe-phase spectrum, and with an added Doppler shift (see text for details). The interaction parameters are as in previous figures: $G_{dd} / E_R = 1.2$ and $G_{ss} / E_R = 0.32$.