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arXiv:2412.19285 · New Journal of Physics

Thermal amplification and melting of phases in spin–orbit-coupled spin-1 Bose–Einstein condensates

Ritu Ritu, Rajat Rajat, Arko Roy, Sandeep Gautam

Published Mar 28, 2025• 2 citations
arXivHTMLPDF

Abstract

Abstract We implement the Hartree–Fock–Bogoliubov theory with the Popov approximation for a homogeneous Raman-induced spin–orbit-coupled spin-1 Bose–Einstein condensate and investigate the effects of finite temperature (T) on the ground-state phase diagram. We calculate the roton gap as a function of Raman coupling (Ω) or quadratic Zeeman field strength (ε) to extract the critical points separating the supersolid stripe phase from the plane wave or zero-momentum phase at finite temperatures. We present a few representative finite-temperature phase diagrams for the system in the T − Ω and T − ϵ planes. Our observations indicate that the supersolid stripe phase melts at finite temperatures. We also discuss the contrasting roles of quantum and thermal fluctuations in shifting the phase boundary separating the supersolid stripe from the plane-wave phase.

DOI
10.1088/1367-2630/adc6ae

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Abstract We implement the Hartree–Fock–Bogoliubov theory with the Popov approximation for a homogeneous Raman-induced spin–orbit-coupled spin-1 Bose–Einstein condensate and investigate the effects of finite temperature (T) on the ground-state phase diagram. We calculate the roton gap as a function of Raman coupling (Ω) or quadratic Zeeman field strength (ε) to extract the critical points separating the supersolid stripe phase from the plane wave or zero-momentum phase at finite temperatures. We present a few representative finite-temperature phase diagrams for the system in the T − Ω and T − ϵ planes. Our observations indicate that the supersolid stripe phase melts at finite temperatures. We also discuss the contrasting roles of quantum and thermal fluctuations in shifting the phase boundary separating the supersolid stripe from the plane-wave phase.
Close to the superfluid plane-wave (PW)--supersolid stripe (ST) phase transition point of a zero-temperature quasi-one-dimensional spin-orbit-coupled Bose gas, we find that an increase in temperature induces a phase transition to the supersolid phase with a broken translational symmetry from the superfluid plane-wave phase. We use the Hartree-Fock-Bogoliubov theory with the Popov approximation to investigate the effect of thermal fluctuations on the collective excitation spectrum and investigate the softening of the spin-dipole mode corresponding to the shift in the quantum critical point. This is in stark contrast to the PW-ST phase transition in a homogeneous system where nonzero temperatures facilitate the melting of the stripe phase.
We theoretically study a three-dimensional weakly interacting Bose gas with Raman-induced spin-orbit coupling at finite temperature. By employing a generalized Hartree-Fock-Bogoliubov theory with Popov approximation, we determine a complete finite-temperature phase diagram of three exotic condensation phases (i.e., the stripe, plane-wave, and zero-momentum phases), against both quantum and thermal fluctuations. We find that the plane-wave phase is significantly broadened by thermal fluctuations. The phonon mode and sound velocity at the transition from the plane-wave phase to the zero-momentum phase are thoughtfully analyzed. At zero temperature, we find that quantum fluctuations open an unexpected gap in sound velocity at the phase transition, in stark contrast to the previous theoretical prediction of a vanishing sound velocity. At finite temperature, thermal fluctuations continue to significantly enlarge the gap, and simultaneously shift the critical minimum. For a Bose gas of 87Rb^{87}\mathrm{Rb}87Rb atoms at the typical experimental temperature, T=0.3T0T=0.3{T}_{0}T=0.3T0​, where T0{T}_{0}T0​ is the critical temperature of an ideal Bose gas without spin-orbit coupling, our results of gap opening and critical minimum shifting in the sound velocity are qualitatively consistent with the recent experimental observation [Ji et al., Phys. Rev. Lett. 114, 105301 (2015)].
We study the ground-state phase diagram and the quantum phase transitions in spin-1 Bose gases with Raman-induced spin-orbit coupling. In addition to the Bose-Einstein condensates with uniform density, three types of stripe condensation phases that simultaneously break the U(1) symmetry and the translation symmetry are identified. The transitions between these phases are investigated, and the occurrences of the various tricritical points are predicted. The excitation spectra in the plane-wave phase and the zero-momentum phase show rich roton-maxon structures, and their instabilities indicate the tendency to develop the crystalline order. We propose the atomic gas of 23Na^{23}\mathrm{Na}23Na could be a candidate for observing the stripe condensate with high contrast fringes.
We study two-component Bose gases with Raman induced spin-orbit coupling via a perturbation approach at finite temperature. For weak coupling, free energy is expanded in terms of Raman coupling strength up to the second order, where the coefficient (referred to as Raman susceptibility) is determined according to linear response theory. The equation of state for the stripe phase and the plane-wave phase are obtained in Popov approximation, and the first order transition between these two phases is investigated. As temperature increases, we find the phase boundary bends toward the stripe phase side in most temperature regions, which implies the ferromagnetic order is more robust than the crystalline order in presence of thermal fluctuations. Our results qualitatively agree with the recent experimental observation in rubidium atomic gases. A method to measure Raman susceptibility through the two-photon Bragg scattering experiment is also discussed.
The phase diagram of a Bose-Einstein condensate with Raman-induced spin-orbit coupling includes a stripe phase with supersolid features. In this work we develop a perturbation approach to study the ground state and the Bogoliubov modes of this phase, holding for small values of the Raman coupling. We obtain analytical predictions for the most relevant observables (including the periodicity of stripes, sound velocities, compressibility, and magnetic susceptibility) which are in excellent agreement with the exact (non perturbative) numerical results, obtained for significantly large values of the coupling. We further unveil the nature of the two gapless Bogoliubov modes in the long-wavelength limit. We find that the spin branch of the spectrum, corresponding in this limit to the dynamics of the relative phase between the two spin components, describes a translation of the fringes of the equilibrium density profile, thereby providing the crystal Goldstone mode typical of a supersolid configuration. Finally, using sum-rule arguments, we show that the superfluid density can be experimentally accessed by measuring the ratio of the sound velocities parallel and perpendicular to the direction of the spin-orbit coupling.
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.