A spin-orbit-coupled two-dimensional Bose gas is shown to simultaneously possess quasi- and true long-range order in the total and relative phase sectors, respectively. The total phase undergoes a Berenzinskii-Kosterlitz-Thouless transition to a low-temperature phase with quasi-long-range order, as expected for a two-dimensional quantum gas. Additionally, the relative phase undergoes an Ising-type transition building up true long-range order, which is induced by the anisotropic spin-orbit coupling in combination with spin-dependent particle-particle interactions. Based on the Bogoliubov approach, expressions for the total- and relative-phase fluctuations are derived analytically for the low-temperature regime. Numerical simulations of the stochastic projected Gross-Pitaevskii equation give a good agreement with the analytical predictions.
A spin-orbit-coupled two-dimensional Bose gas is shown to simultaneously possess quasi- and true long-range order in the total and relative phase sectors, respectively. The total phase undergoes a Berenzinskii-Kosterlitz-Thouless transition to a low-temperature phase with quasi-long-range order, as expected for a two-dimensional quantum gas. Additionally, the relative phase undergoes an Ising-type transition building up true long-range order, which is induced by the anisotropic spin-orbit coupling in combination with spin-dependent particle-particle interactions. Based on the Bogoliubov approach, expressions for the total- and relative-phase fluctuations are derived analytically for the low-temperature regime. Numerical simulations of the stochastic projected Gross-Pitaevskii equation give a good agreement with the analytical predictions.
A spin-orbit-coupled two-dimensional Bose gas is shown to simultaneously possess quasi- and true long-range order in the total and relative phase sectors, respectively. The total phase undergoes a Berenzinskii-Kosterlitz-Thouless transition to a low-temperature phase with quasi-long-range order, as expected for a two-dimensional quantum gas. Additionally, the relative phase undergoes an Ising-type transition building up true long-range order, which is induced by the anisotropic spin-orbit coupling in combination with spin-dependent particle-particle interactions. Based on the Bogoliubov approach, expressions for the total- and relative-phase fluctuations are derived analytically for the low-temperature regime. Numerical simulations of the stochastic projected Gross-Pitaevskii equation give a good agreement with the analytical predictions.
A two-dimensional spin-orbit coupled Bose gas is shown to simultaneously possess quasi and true long-range orders in the total and relative phases, respectively. The total phase undergoes a conventional Berenzinskii- Kosterlitz-Thouless transition, where an quasi long-range order is expected. Additionally, the relative phase undergoes an Ising-type transition building up true long-range order, which is induced by the anisotropic spin- orbit coupling. Based on the Bogoliubov approach, expressions for the total- and relative-phase fluctuations are derived analytically for the low temperature regime. Numerical simulations of the stochastic projected Gross- Pitaevskii equation give a good agreement with the analytical predictions.
This paper develops and implements the stochastic projected Gross-Pitaevskii equation for spin-1 Bose gases, addressing key considerations for numerical simulations. As an application of the theory we explore equilibrium phases in a two-dimensional spin-1 gas, where quasi-long-range order emerges via a Berezinskii-Kosterlitz-Thouless transition. Our analysis includes the definition of superfluid densities for both mass and spin degrees of freedom in a manner suitable for implementation within a stochastic projected Gross-Pitaevskii equation simulation. We present a finite-temperature phase diagram for the ferromagnetic spin-1 Bose gas and identify three distinct superfluid phases: two exhibiting conventional Berezinskii-Kosterlitz-Thouless-like behavior and an additional phase that simultaneously supports independent mass and spin superflows. As temperature increases, the stability region of this additional phase shrinks. We provide a characterization of the phase transitions through consideration of the spin-component densities and the unbinding of multiple types of vortices. This work provides a foundation for further studies of nonequilibrium and finite-temperature phenomena in spinor Bose gases.
Spinor Bose gases may exhibit both phase and ferromagnetic order. Consequently, they may support superflow of both mass and spin currents. In this thesis we utilize a stochastic Gross-Pitaevskii model to investigate the superfluid properties of a two-dimensional spin-1 Bose gas with ferromagnetic interactions. Owing to the low dimensionality, the superfluidity of this system can not be attributed to the presence of a Bose-Einstein condensate, with thermal fluctuations precluding long-range order. Rather, mass and spin superfluidity emerge via respective Berezinskii-Kosterlitz-Thouless (BKT) transitions. Such transitions are characterized by the emergence of quasi-long-range order, driven by the binding of topological defect pairs. First, we consider an easy-plane ferromagnetic gas. We find the system superfluidity may be understood by direct application of existing BKT theory. In particular, mass and spin superfluidity are respectively associated with quasi-long-range order in the global phase and local magnetization. The former arises from the binding of mass vortex pairs, while the latter arises from the binding of polar-core spin vortex pairs. Following this, we introduce axial magnetization, and consider a gas in the broken-axisymmetric phase. We find axial magnetization acts to couple mass and spin superflows via a mechanism analogous to the superfluid drag exhibited by two-component superfluids. This coupling renders invalid the direct applicability of existing BKT theory: here mass and spin superfluidity may each be associated with both phase and magnetic order. We remedy this with a generalization making concrete the relation between superfluidity and quasi-long-range order in this system. In addition, through variations of the quadratic Zeeman energy and axial magnetization we identify three distinct superfluid phases, which serve as finite-temperature generalizations of the polar, broken-axisymmetric, and easy-axis ferromagnetic ground states.
We determine the finite-temperature phase diagram of two-dimensional bosons with two hyperfine (pseudospin) states coupled via a Rashba-Dresselhaus spin-orbit interaction using classical field Monte Carlo calculations. For anisotropic spin-orbit coupling, we find a transition to a Berezinskii-Kosterlitz-Thouless superfluid phase with quasi-long-range order. We show that the spin order of the quasicondensate is driven by the anisotropy of the interparticle interaction, favoring either a homogeneous plane-wave state or stripe phase with broken translational symmetry. Both phases show characteristic behavior in the algebraically decaying spin-density correlation function. For a fully isotropic interparticle interaction, our calculations indicate a fractionalized quasicondensate where the mean-field degeneracy of the plane-wave and stripe phase remains robust against critical fluctuations. In the case of fully isotropic spin-orbit coupling, the circular degeneracy of the single-particle ground state destroys the algebraic ordered phase in the thermodynamic limit, but a crossover remains for finite-size systems.
The coherence of a quantum system is one of the key signatures of many-body physics. While a three-dimensional Bose-Einstein condensate (BEC) exhibits true long-range coherence, a Bose gas confined in a one-dimensional (1D) potential displays quasi-long-range order. Experimentally, these coherence properties are typically probed via the momentum distribution, which corresponds to the Fourier transform of the one-body correlation function. In two-dimensional (2D) systems, for example, momentum distribution enables the observation of the Berezinskii-Kosterlitz-Thouless (BKT) transition from a thermal cloud to a superfluid. With trimodal distribution fitting, an intermediate quasicondensate phase is also revealed. In 1D systems, the momentum distribution of a quenched Tonks-Girardeau (TG) Bose gas can be mapped to that of a non-interacting Fermi gas at a specific evolution time, achieving the rapidity distribution. On the other hand, spin-orbit coupling (SOC) has emerged as one of the central topics in ultracold atomic physics. In additional to creating an artificial electric or magnetic field, researchers also observe negative-mass dynamics in a spin-orbit-coupled BEC by engineering the curvature of the band dispersion. This phenomenon is directly observable in the density distribution of the system. In 1D systems, SOC also introduces rich physical phenomena; while low-energy 1D quantum systems are conventionally described by Tomonaga-Luttinger liquid (TLL) theory, SOC notably alters the energy-momentum dispersion. This modification gives rise to unconventional Luttinger liquids whose properties are governed by the spin-orbit coupling strength.In this work, we first investigate the qualitative properties of a 1D Bose gas and demonstrate the temperature dependence of the momentum distribution, revealing how the thermal fluctuations destroy the quasi-long-range order. In the second part, we probe the effects of 1D SOC on a 1D Bose gas, focusing on how SOC modifies the energy-momentum dispersion and the resulting momentum distribution. We demonstrate that within a specific parameter regime, the momentum distribution and its associated coherence properties undergo distinct qualitative changes. The occupied higher momentum states induce phase fluctuations and destroy the coherence of the system, showing signals of the non-Luttinger liquid physics. We probe the coherence properties of a 1D system along these two parameter axes.