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arXiv:2112.13417

Information theoretic approach to effects of spin-orbit coupling in Bose-Einstein condensates

Golam Ali Sekh, B. Talukdar, Supriya Chatterjee, Basir Ahamed Khan

Published Dec 26, 2021• 0 citations
arXivHTMLPDF

Abstract

We make use of Shannon entropy ($S$) and Fisher information ($I$) to study the response of atomic density profiles of a spin-orbit coupled Bose-Einstein condensate to changes in the wave number ($κ_L$) of the Raman laser that couples two hyperfine states of atoms in the condensate. The choice for values of $κ_L$, the so-called spin-orbit parameter, and Rabi frequency ($Ω$) leads to two distinct regions in the system's energy spectrum with different order parameters and/or probability densities. In addition, we can have a spatially modulated density profile, reminiscent of the so called stripe phase. Our numbers for $S$ and $I$ demonstrate that for $κ_L^2Ω$ (region 2) for increasing values of $κ_L$. In the stripe phase the nature of $S$ and $I$ to changes in $κ_L$ is similar to that found for the condensate in region 2. The results for information theoretic quantities in the stripe phase are, in general, augmented compared to those of region 2. In particular, the highly enhanced values of position-space Fisher information imply an extremely concentrated atomic density distribution to provide an evidence for supersolid properties of Bose-Einstein condensates in the presence of spin-orbit coupling.

DOI
10.48550/arxiv.2112.13417

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Abstract We consider a spin-orbit coupled (SOC) one dimensional Bose–Einstein condensate (BEC) with attractive atom-atom interaction and make use of Shannon entropy ( S ), and Fisher information ( I ) to study the response of atomic density profiles to changes in κ L , the strength of the spin–orbit coupling for fixed values of Rabi frequency (Ω). Density profiles of our interest are characterized by (i) κ L 2 < Ω (region 1) and (ii) κ L 2 > Ω (region 2). We also pay attention to a spatially modulated density profile representing stripe phase of the condensate. Our numbers for S and I show that in region 1 the profile becomes localized as κ L increases while in region 2 we observe delocalization. Values of I in the stripe phase provide an evidence for the supersolid properties of the SOC BEC and indicate that supersolidity is a purely quantum phenomenon.
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.
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.
Quantum degenerate gases have provided rich systems to simulate engineered Hamiltonians and to explore quantum many-body problems in laboratory-scale experiments. In this work, I focus on spin-orbit-coupled (SOC) Bose-Einstein condensates (BECs) of Rubidium-87 atoms realized using two-photon Raman coupling in which various novel phases are predicted to exist due to competing energies from the atomic internal structure, coupling strength, and many-body collisions. BECs are observed primarily using the interaction between light and matter, where it is common to probe the atoms with near-resonant light and image their shadow on a camera. This absorption imaging technique measures the integrated column density of the atoms and it is crucial to focus the imaging system. I present a systematic method to bring the ultracold atom systems into an optimal focus using the power spectral density (PSD) of the atomic density-density correlation function. The spatial frequency at which the defocus-induced artifacts first appear in the PSD is maximized at the focus. The focusing process thus identifies the range of spatial frequencies over which the PSD is uncontaminated by finite-thickness effects. Next, I describe magnetic phases which exist in spin-1 spin-orbit-coupled condensates at a near-zero temperature. I observe ferromagnetic and unmagnetized phases which are stabilized by the locking between the spin and linear momentum of the system. Our measurements of both the first- and second-order transitions are in agreement with theory. Finally, I discuss the stripe-ordered phase that occurs in SOC Bose gases favoring the miscibility configuration. The stripe phase is theoretically predicted to have an excitation spectrum analogous to that of a supersolid and to exhibit spatial density modulation within specific regions of parameter space. I used optical Bragg scattering to probe the small density modulation present in the atomic spatial distribution. I present for the very first time observation of the stripe phase in a Raman SOC Bose gas and its phase diagram in various parameter space. Our observations of the phase boundaries are consistent with theory and previous work.
We theoretically investigate a three-dimensional weakly interacting Bose gas with one-dimensional Raman-type spin-orbit coupling at zero temperature. By employing an improved ansatz, including high-order harmonics in the stripe phase, we show that the critical transition from the stripe to the plane-wave phases is shifted to a relatively larger Rabi frequency compared to the prediction by previous work [Li et al., Phys. Rev. Lett. 108, 225301 (2012)] using a first-order stripe ansatz. We also determine the quantum depletion and superfluid density over a large range of Rabi frequency in different phases. The depletion exhibits an intriguing behavior with a discontinuous jump at the transition between the stripe and plane-wave phases, and a maximum at the transition between the plane-wave and zero-momentum phases. The superfluid density is derived through a phase-twist method. In the plane-wave and zero-momentum phases, it is significantly suppressed along the spin-orbit-coupling direction and vanishes at the transition, consistent with a recent work [Zhang et al., Phys. Rev. A 94, 033635 (2016)], while in the stripe phase, it smoothly decreases with increasing Rabi frequency. Our predictions would be useful for further theoretical and experimental studies of the exotic supersolid stripe phase.
Abstract Since their first realization more than a decade ago spin-orbit-coupled Bose-Einstein condensates have been the subject of intense theoretical and experimental investigations. Spin-orbit coupling deeply modifies the equilibrium properties of the condensate, giving rise to novel configurations such as a supersolid stripe phase and a phase-separated plane-wave state. At the level of dynamics, both the frequency and the nature of the collective modes are significantly affected by the coupling with the spin degree of freedom. Here we review some of the most relevant advances in the field and provide our perspective on possible future research directions.