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arXiv:2502.18792 · Physical Review A

Stochastic Gross-Pitaevskii theory for a spin-1 Bose gas: Application to superfluidity in two dimensions

Andrew P. C. Underwood, P. B. Blakie

Published Jul 17, 2025• 3 citations
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Abstract

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.

DOI
10.1103/slxb-df37

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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.
In this work we investigate the unique properties of ultracold Bose gases in one and two dimensions. In two dimensions, we present simulations of the Berezinskii-Kosterlitz-Thouless (BKT) phase transition using the projected Gross-Pitaevskii equation formalism. We confirm the presence of the BKT phase by observing algebraic decay in the first-order correlation function, unbinding of vortex pairs, and behaviour of the superfluid fraction. In one dimension we use a transverse Gaussian ansatz and the Lagrangian approach to derive an effective equation of motion for the one dimensional Bose gas at zero temperature.
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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 stochastic Gross-Pitaevskii equation is derived for partially condensed Bose gas systems subject to binary contact interactions. The theory we present provides a classical-field theory suitable for describing dissipative dynamics and phase transitions of spinor and multicomponent Bose gas systems composed of an arbitrary number of distinct interacting Bose fields. A class of dissipative processes involving distinguishable particle interchange between coherent and incoherent regions of phase space is identified. The formalism and its implications are illustrated for two-component mixtures and spin-1 Bose-Einstein condensates. For systems composed of atoms of equal mass, with thermal reservoirs that are close to equilibrium, the dissipation rates of the theory are reduced to analytical expressions that may be readily evaluated. The unified treatment of binary contact interactions presented here provides a theory with broad relevance for quasiequilibrium and far-from-equilibrium Bose-Einstein condensates.