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arXiv:2506.03505 · Physical Review Research

Stable supersolids and boselets in spin-orbit-coupled Bose-Einstein condensates with three-body interactions

Rajamanickam Ravisankar, Sanu Kumar Gangwar, Henrique Fabrelli, Yongping Zhang, Paulsamy Muruganandam, Pankaj Kumar Mishra, Emmanuel Kengne, Gao Xianlong, Boris A. Malomed

Published Jun 2, 2025• 1 citation
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Abstract

We explore the stability of supersolid striped waves, plane-wave boselets, and other extended states in one-dimensional spin-orbit-coupled Bose-Einstein condensates with repulsive three-body interactions (R3BIs), modeled by quintic terms in the framework of the corresponding Gross-Pitaevskii equations. In the absence of R3BIs, the extended states are susceptible to the modulational instability (MI) induced by the cubic attractive nonlinearity. Using the linearized Bogoliubov-de-Gennes equations, we identify multiple new types of MI, including baseband, passband, mixedband, and zero-wavenumber-gain ones, which give rise to deterministic rogue waves and complex nonlinear wave patterns. Our analysis reveals that R3BIs eliminate baseband and zero-wavenumber-gain MIs, forming, instead, phonon modes that enable stable boselets. Additionally, mixedband and passband MIs are suppressed, which results in a latticelike phonon-roton mode that supports a stable supersolid phase. These stable supersolids can be realized using the currently available ultracold experimental setup.

DOI
10.1103/gycf-981f

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We explore the stability of supersolid striped waves, plane-wave boselets, and other extended states in one-dimensional spin-orbit-coupled Bose-Einstein condensates with repulsive three-body interactions (R3BIs), modeled by quintic terms in the framework of the corresponding Gross-Pitaevskii equations. In the absence of R3BIs, the extended states are susceptible to the modulational instability (MI) induced by the cubic attractive nonlinearity. Using the linearized Bogoliubov-de-Gennes equations, we identify multiple new types of MI, including baseband, passband, mixedband, and zero-wavenumber-gain ones, which give rise to deterministic rogue waves and complex nonlinear wave patterns. Our analysis reveals that R3BIs eliminate baseband and zero-wavenumber-gain MIs, forming, instead, phonon modes that enable stable boselets. Additionally, mixedband and passband MIs are suppressed, which results in a latticelike phonon-roton mode that supports a stable supersolid phase. These stable supersolids can be realized using the currently available ultracold experimental setup.
Supersolid is a special state of matter with both superfluid properties and spontaneous modulation of particle density. In this paper we focus on the supersolid stripe phase realized in a spin-orbit-coupled Bose-Einstein condensate and explore the properties of a class of metastable supersolids. In particular, we study a one-dimensional supersolid whose characteristic wave number kkk (magnitude of wave vector) deviates from km{k}_{m}km​, i.e., the one in the ground state. In other words, the period of density modulation is shorter or longer than the one in the ground state. We find that this class of supersolids can still be stable if their wave numbers fall in the range kc1<k<kc2{k}_{c1}<k<{k}_{c2}kc1​<k<kc2​, with two thresholds kc1{k}_{c1}kc1​ and kc2{k}_{c2}kc2​. Stripes with kkk outside this range suffer from dynamical instability with a complex Bogoliubov excitation spectrum at long wavelength. Experimentally, these stripes with kkk different from km{k}_{m}km​ are accessible by exciting the longitudinal spin dipole mode, resulting in temporal oscillation of the stripe period as well as kkk. Within the mean-field Gross-Pitaevskii theory, we numerically confirm that for a large enough amplitude of spin dipole oscillation, the stripe states become unstable through breaking periodicity, in qualitative agreement with the existence of thresholds of kkk for stable stripes. Our work extends the concept of supersolid and uncovers an unconventional class of metastable supersolids to explore.
Beyond the mean-field theory, a new model of the Gross–Pitaevskii equation (GPE) that describes the dynamics of Bose–Einstein condensates (BECs) is derived using an appropriate phase-imprint on the old wavefunction. This modified version of the GPE in addition to the two-body interactions term, also takes into account effects of the three-body interactions. The three-body interactions consist of a quintic term and the delayed nonlinear response of the condensate system term. Then, the modulational instability (MI) of the new GPE confined in an attractive harmonic potential is investigated. The analytical study shows that the three-body interactions destabilize more the condensate system while the external potential alleviates the instability. Numerical results confirm the theoretical predictions. Further numerical investigations of the behavior of solitons reveal that the three-body interactions enhance the appearance of solitons, increase the number of solitons generated and deeply change the lifetime of solitons. Moreover, the external potential delays the appearance of solitons. Besides, a new initial condition is introduced which enables to increase the number of solitons created and deeply affects the trail of chains of solitons generated. Moreover, the MI of a condensate without the external potential, and in a repulsive potential is also investigated.
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.
The nonlinear dynamics induced by the modulation instability (MI) of a binary mixture in an atomic Bose-Einstein condensate (BEC) is investigated theoretically under the joint effects of higher-order residual nonlinearities and helicoidal spin-orbit (SO) coupling in a regime of unbalanced chemical potential. The analysis relies on a system of modified coupled Gross-Pitaevskii equations on which the linear stability analysis of plane-wave solutions is performed, from which an expression of the MI gain is obtained. A parametric analysis of regions of instability is carried out, where effects originating from the higher-order interactions and the helicoidal spin-orbit coupling are confronted under different combinations of the signs of the intra- and intercomponent interaction strengths. Direct numerical calculations on the generic model support our analytical predictions and show that the higher-order interspecies interaction and the SO coupling can balance each other suitably for stability to take place. Mainly, it is found that the residual nonlinearity preserves and reinforces the stability of miscible pairs of condensates with SO coupling. Additionally, when a miscible binary mixture of condensates with SO coupling is modulationally unstable, the presence of residual nonlinearity may help soften such instability. Our results finally suggest that MI-induced formation of stable solitons in mixtures of BECs with two-body attraction may be preserved by the residual nonlinearity even though the latter enhances the instability.
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.
We study modulation instability (MI) of flat states in two-component spin-orbit-coupled (SOC) Bose-Einstein condensates (BECs) in the framework of coupled Gross-Pitaevskii equations for two components of the pseudospinor wave function. The analysis is performed for equal densities of the components. Effects of the interaction parameters, Rabi coupling, and SOC on the MI are investigated. In particular, the results demonstrate that the SOC strongly alters the commonly known MI (immiscibility) condition, g122>g1g2{g}_{12}^{2}>{g}_{1}{g}_{2}g122​>g1​g2​, for the binary superfluid with coefficients g1,2{g}_{1,2}g1,2​ and g12{g}_{12}g12​ of the intra- and interspecies repulsive interactions. In fact, the binary BEC is always subject to the MI under the action of the SOC, which implies that the ground state of the system is plausibly represented by a striped phase.