Supersolid phases and collective excitations in two-dimensional Rashba spin-orbit coupled spin-1 condensates

TL;DR

This work develops a comprehensive Bogoliubov–de Gennes analysis of a quasi-2D Rashba spin-orbit coupled spin-1 Bose–Einstein condensate with tunable Rabi coupling, spanning ferromagnetic and antiferromagnetic interaction regimes. By combining analytical BdG spectra with real-time Gross–Pitaevskii simulations, it maps out stability phases in the $k_L-\\Omega$ plane, revealing region-specific behaviors: stable phonon-dominated modes, multi-band dynamical instabilities, unstable avoided crossings, and roton-like features that herald stripe and supersolid ordering. A key finding is that supersolid phases in the spin-1 Rashba system are generically dynamically unstable, while antiferromagnetic interactions promote robust stripe/supersolid-like density modulations subject to SOC strength; these signatures provide experimentally accessible probes of nonequilibrium quantum phases. The results highlight the intricate coupling between spin-dependent interactions and synthetic gauge fields in determining collective excitations and pattern formation in nonequilibrium spinor quantum fluids, offering guidance for realizing and detecting supersolid and stripe states in ultracold gases.

Abstract

We investigate the collective excitation spectrum and dynamics of a quasi two-dimensional spin-1 Bose-Einstein condensate with Rashba type spin-orbit (SO) coupling. Employing Bogoliubov-de-Gennes analysis, we analytically compute the excitation spectra across a wide range of interaction strengths and coupling parameters. By systematically varying the SO and Rabi couplings, we uncover distinct dynamical signatures of quantum phase transitions, including mode softening, the appearance of roton-like minima, and miscibility-driven instabilities in both ferromagnetic and antiferromagnetic interaction regimes. In the antiferromagnetic case, these instabilities lead to a dynamically unstable supersolid phase characterized by the coexistence of density modulation and global phase coherence. To corroborate the analytical predictions, we numerically solve the coupled Gross-Pitaevskii equations and analyze the dynamical stability of the condensate. Our results provide experimentally accessible signatures for spinor condensates with tunable spin-orbit coupling and demonstrate the rich interplay between spin-dependent interactions and synthetic couplings in nonequilibrium quantum fluids.

Figures (22)

• Figure 1: Single-particle energy spectrum top row along $q_{x}$ direction and bottom row along $q_{y}$ direction for the different set of SO and Rabi coupling strengths $(k_L,\Omega)$: (a)-(f): $(0, 0)$, $(0.7, 0)$, $(0, 0.5)$, $(0.7, 0.5)$, $(0.7, 0.7)$, $(1.0, 0.5)$. The thick solid red, thin solid blue line, and dash-dotted green line represent the energy eigenspectrum for $\{-1, 0, +1\}$ components of the spin, respectively. Along $q_{x}$ direction, the double minimum appears in the eigenvalue dispersion, following the relation $k_{L}^{2} > \Omega$, which shows zero-momentum to stripe-wave phase transitions. While along the $q_{y}$ direction, the eigenvalue dispersion exhibits the asymmetric double minima in the presence of both SO and Rabi coupling.
• Figure 2: Stability phase diagram in the $k_{L}$–$\Omega$ plane for ferromagnetic interactions ($c_{0} = 50.0$ and $c_{2} = -2.5$). (a) Full range of $q_x$ with $q_y=0$. The dash--dotted white line with green dots ($k_L^2\simeq\Omega$) separates stable (I) and unstable (II) regions. Within region II, the solid white line with blue dots ($\Omega=-1.6329+0.13k_L^2$) distinguishes gapped (IIa) and gapless (IIb) spectra. The case $\Omega \sim 0$ lies in the unstable regime. The marker in different regions represents the coupling parameters points for which detailed collective excitation analysis has been reported in the paper.
• Figure 3: Eigenvalue spectrum and eigenvectors along the quasi-momentum directions $q_{x}$ and $q_{y}$ for $k_{L} = 0.5$, $\Omega = 1.0$, $c_{0} = 50$, and $c_{2} = -2.5$. Top row: low-lying brach. (a i) Eigenvalue spectrum, with $\mathrm{Re}(\omega_{ll})$ (solid magenta) and $\vert{\mathrm{Im}(\omega_{ll})}\vert$ (dashed dotted green). (a ii, a iii) Corresponding eigenvector components $\vert u_{+m}\vert^{2}_{ll}$ and $\vert v_{m}\vert^{2}_{ll}$ for $m=0,\pm1$. Bottom row: first excited branch. (b i) Eigenvalue spectrum with $\mathrm{Re}(\omega_{fe})$ (solid blue). (b ii, b iii) Corresponding eigenvector components $\vert u_{m}\vert^{2}_{fe}$ and $\vert u_{m}\vert^{2}_{fe}$. All eigenvalues are real and positive, and the low-lying branch exhibits a phonon mode, indicating both dynamical and energetic stability.
• Figure 4: Pseudocolor density plots showing the time evolution of the condensate following a quench of the harmonic trap strength from $\lambda = 1$ to $\lambda = 0.5$. Panel (a) shows the ground-state density profile, while panels (b-d) correspond to the evolution at $t = 100$, 200, and 1600, respectively, for coupling parameters $(k_L,\Omega) = (0.5,1.0)$ and interaction strengths $c_0 = 50$, $c_2 = -2.5$. Columns (i-iii) represent the $j = +1, 0, -1$ components of the spinor condensate. The persistence of the density profiles over long times demonstrates the dynamical stability of the condensate under the trap quench.
• Figure 5: Eigenvalue spectra [(a i), (b i)] and corresponding eigenvectors [(a ii, a iii) and (b ii, b iii)] for $k_L = 2.0$ and $\Omega = 3.0$ with interaction strengths $c_0 = 50.0$ and $c_2 = -2.5$, shown along the quasi-momentum directions $q_x$ and $q_y$, respectively. Line styles and symbols follow the conventions of Fig. \ref{['fig08a']}. A multiband instability develops along the $q_x$ direction, leading to spin--density--spin mixed modes in the eigenvector components, while the low-lying branch exhibits negative-energy excitations along the $q_y$ direction.