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arXiv:2110.01385 · Physical Review A

Quantum-geometric perspective on spin-orbit-coupled Bose superfluids

A. Levent Subaşı, M. Iskin

Published Feb 1, 2022• 10 citations
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Abstract

We employ the Bogoliubov approximation to study how the quantum geometry of the helicity states affects the superfluid properties of a spin-orbit-coupled Bose gas in continuum. In particular we derive the low-energy Bogoliubov spectrum for a plane-wave condensate in the lower helicity band and show that the geometric contributions to the sound velocity are distinguished by their linear dependences on the interaction strength; that is, they are in sharp contrast to the conventional contribution which has a square-root dependence. We also discuss the roton instability of the plane-wave condensate against the stripe phase and determine their phase-transition boundary. In addition we derive the superfluid density tensor by imposing a phase twist on the condensate order parameter and study the relative importance of its contribution from the interband processes that is related to the quantum geometry.

DOI
10.1103/physreva.105.023301

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We employ the Bogoliubov approximation to study how the quantum geometry of the helicity states affects the superfluid properties of a spin-orbit-coupled Bose gas in continuum. In particular we derive the low-energy Bogoliubov spectrum for a plane-wave condensate in the lower helicity band and show that the geometric contributions to the sound velocity are distinguished by their linear dependences on the interaction strength; that is, they are in sharp contrast to the conventional contribution which has a square-root dependence. We also discuss the roton instability of the plane-wave condensate against the stripe phase and determine their phase-transition boundary. In addition we derive the superfluid density tensor by imposing a phase twist on the condensate order parameter and study the relative importance of its contribution from the interband processes that is related to the quantum geometry.
A weakly interacting, spin-orbit-coupled, ultracold, dilute Bose gas on a two-dimensional square lattice with an external Zeeman field is studied. We explore the plane- and stripe-wave phases of the system involving nonzero condensate momenta, which occur when the Zeeman field is below a critical value. Their excitation spectra are found using Bogoliubov theory and by two different routes. The validity of each method to obtain the excitation spectrum is discussed, and it is found that projection on the lowest single-particle band is an excellent approximation in the plane-wave phase, while it is a poor approximation in the stripe-wave phase. While the plane-wave phase has a phonon minimum at its single condensate momentum, revealing a nonzero sound velocity of the excitations, the stripe-wave phase has quadratic minima at its two condensate momenta showing zero sound velocity of the excitations. We discuss how the presence of more than one condensate momentum is essential for these differences between the two phases. Additionally, it is emphasized that the zero sound velocity in the stripe-wave phase is a lattice effect, since continuum studies of the same phase have shown nonzero sound velocity.
We investigate the superfluidity of a three-dimensional weakly interacting Bose gas with a one-dimensional Raman-type spin-orbit coupling at both zero and finite temperatures. Using the imaginary-time Green's function within the Bogoliubov approximation, we explicitly derive analytic expressions of the current-current response functions in the plane-wave and zero-momentum phases, from which we extract the superfluid density in the limits of long wavelength and zero frequency. At zero temperature, we check that the resultant superfluid density agrees exactly with our previous analytic prediction obtained from a phase-twist approach. Both results also satisfy a generalized Josephson relation in the presence of spin-orbit coupling. At finite temperature, we find a significant nonmonotonic temperature dependence of superfluid density near the transition from the plane-wave phase to the zero-momentum phase. We show that this nontrivial behavior might be understood from the sound velocity, which has a similar temperature dependence. The nonmonotonic temperature dependence is also shared by Landau critical velocity, above which the spin-orbit-coupled Bose gas loses its superfluidity. Our results would be useful for further theoretical and experimental studies of superfluidity in exotic spin-orbit-coupled quantum gases.
We consider a weakly interacting Bose-Einstein condensate that is loaded into an optical lattice with a two-point basis and described by a two-band Bose-Hubbard model with generic one-body and two-body terms. By first projecting the system onto the lower Bloch band and then applying the Bogoliubov approximation to the resultant Hamiltonian, we show that the inverse effective-mass tensor of the superfluid carriers in the Bogoliubov spectrum has two physically distinct contributions. In addition to the usual inverse band-mass tensor that is originating from the intraband processes within the lower Bloch band, there is also a quantum-geometric contribution that is induced by the two-body interactions through the interband processes. We also discuss the conditions under which the latter contribution is expressed in terms of the quantum-metric tensor of the Bloch states, i.e., the natural Fubini-Study metric on the Bloch sphere.
By calculating the density response function we identify the excitation spectrum of a Bose-Einstein condensate with equal Rashba and Dresselhaus spin-orbit coupling. We find that the velocity of sound along the direction of spin-orbit coupling is deeply quenched and vanishes when one approaches the second-order phase transition between the plane-wave and the zero momentum quantum phases. We also point out the emergence of a roton minimum in the excitation spectrum for small values of the Raman coupling, providing the onset of the transition to the stripe phase. Our findings point out the occurrence of a strong anisotropy in the dynamic behavior of the gas. A hydrodynamic description accounting for the collective oscillations in both uniform and harmonically trapped gases is also derived.
The recent realization of synthetic spin-orbit coupling represents an outstanding achievement in the physics of ultracold quantum gases. In this thesis we explore the properties of spin-orbit-coupled Bose-Einstein condensates with equal Rashba and Dresselhaus strengths. These systems present a rich phase diagram, which exhibits a tricritical point separating a single-minimum phase, a spin-polarized plane-wave phase, and a stripe phase. In the stripe phase translational invariance is spontaneously broken, in analogy with supersolids. Spin-orbit coupling also strongly affects the dynamics of the system. In particular, the excitation spectrum exhibits intriguing features, including the suppression of the sound velocity, the emergence of a roton minimum in the plane-wave phase, and the appearance of a double gapless band structure in the stripe phase. Finally, we discuss a combined procedure to make the stripes visible and stable, thus allowing for a direct experimental detection.
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.