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arXiv:2204.10120 · Physical Review Research

Phase transition dimensionality crossover from two to three dimensions in a trapped ultracold atomic Bose gas

N. A. Keepfer, I-Kang Liu, F. Dalfovo, Nick P. Proukakis

Published Aug 16, 2022• 10 citations
arXivHTMLPDF

Abstract

The equilibrium properties of a weakly interacting atomic Bose gas across the Berezinskii-Kosterlitz-Thouless (BKT) and Bose-Einstein condensation (BEC) phase transitions are numerically investigated through a dimensionality crossover from two to three dimensions. The crossover is realized by confining the gas in an experimentally feasible hybridized trap which provides homogeneity along the planar $xy$ directions through a box potential in tandem with a harmonic transverse potential along the transverse $z$ direction. The dimensionality is modified by varying the frequency of the harmonic trap from tight to loose transverse trapping. Our findings, based on a stochastic (projected) Gross-Pitaevskii equation, showcase a continuous shift in the character of the phase transition from BKT to BEC, and a monotonic increase of the identified critical temperature as a function of dimensionality, with the strongest variation exhibited for small chemical potential values up to approximately twice the transverse confining potential.

DOI
10.1103/physrevresearch.4.033130

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The equilibrium properties of a weakly interacting atomic Bose gas across the Berezinskii-Kosterlitz-Thouless (BKT) and Bose-Einstein condensation (BEC) phase transitions are numerically investigated through a dimensionality crossover from two to three dimensions. The crossover is realized by confining the gas in an experimentally feasible hybridized trap which provides homogeneity along the planar xyxyxy directions through a box potential in tandem with a harmonic transverse potential along the transverse zzz direction. The dimensionality is modified by varying the frequency of the harmonic trap from tight to loose transverse trapping. Our findings, based on a stochastic (projected) Gross-Pitaevskii equation, showcase a continuous shift in the character of the phase transition from BKT to BEC, and a monotonic increase of the identified critical temperature as a function of dimensionality, with the strongest variation exhibited for small chemical potential values up to approximately twice the transverse confining potential.
We investigate systems of interacting bosonic particles confined within slablike boxes of size ${L}^{2}\ifmmode\times\else\texttimes\fi{}Z$ with $Z\ensuremath{\ll}L$, at their three-dimensional (3D) Bose-Einstein-condensation (BEC) transition temperature Tc{T}_{c}Tc​, and below Tc{T}_{c}Tc​ where they experience a quasi-two-dimensional (quasi-2D) Berezinskii-Kosterlitz-Thouless (BKT) transition (at TBKT<Tc{T}_{\mathrm{BKT}}<{T}_{c}TBKT​<Tc​ depending on the thickness ZZZ). The low-temperature phase below TBKT{T}_{\mathrm{BKT}}TBKT​ shows quasi-long-range order: the planar correlations decay algebraically as predicted by the 2D spin-wave theory. This dimensional crossover, from a 3D behavior for $T\ensuremath{\gtrsim}{T}_{c}$ to a quasi-2D critical behavior for $T\ensuremath{\lesssim}{T}_{\mathrm{BKT}}$, can be described by a transverse finite-size scaling limit in a slab geometry. Numerical evidence of the $3\mathrm{D}\ensuremath{\rightarrow}2\mathrm{D}$ dimensional crossover is presented for the Bose-Hubbard model defined in anisotropic ${L}^{2}\ifmmode\times\else\texttimes\fi{}Z$ lattices with $Z\ensuremath{\ll}L$. We extend this scaling analysis to the case the slab geometry of the gas is effectively realized by a transverse (inhomogeneous) harmonic trap. Finally, we discuss off-equilibrium behaviors arising from slow time variations of the temperature across the BEC transition of gases confined within a slab geometry. We argue that the system develops an off-equilibrium transverse finite-size scaling under these time-dependent protocols.
We present an overview of our recent work describing the treatment of ultracold atomic Bose gases confined to two dimensions in a harmonic trap. We show how the gas goes through a number of regimes as the axially confining potential is increased and discuss the scattering properties of the gas in each of these regimes. We then use the formalism developed to investigate the phase-coherent Bose-Einstein condensate that is present in the trapped two-dimensional gas at temperatures that are low compared to the ideal gas critical temperature and to analyze the first-order coherence as a function of temperature. We identify the regime in which the gas loses its global phase coherence, although a macroscopic occupation of the ground state persists at a temperature well below the ideal gas critical temperature. We comment on the possible connection between this fluctuating regime and the Berezinskii-Kosterlitz-Thouless phase.
The dimensionality of an ultracold quantum gas can vastly affect its properties, leading to a great deal of interest in lower dimensional systems. In two dimensions, the transition to superfluidity is a Berezinskii-Kosterlitz-Thouless (BKT) [1,2] phase transition. This is identified by the pairing of vortices with opposite circulation and has been observed in Bose gases [3]. Other phenomena that occur in two-dimensional systems include confinement-induced resonances [4] and the emergence of a two-body bound state for arbitrarily weak attractive interactions [5].
Current experiments on atomic gases in highly anisotropic traps present the opportunity to study in detail the low temperature phases of two-dimensional inhomogeneous systems. Although, in an ideal gas, the trapping potential favors Bose-Einstein condensation at finite temperature, interactions tend to destabilize the condensate, leading to a superfluid Kosterlitz-Thouless-Berezinskii phase with a finite superfluid mass density but no long-range order, as in homogeneous fluids. The transition in homogeneous systems is conveniently described in terms of dissociation of topological defects (vortex-antivortex pairs). However, trapped two-dimensional gases are more directly approached by generalizing the microscopic theory of the homogeneous gas. In this paper, we first derive, via a diagrammatic expansion, the scaling structure near the phase transition in a homogeneous system, and then study the effects of a trapping potential in the local density approximation. We find that a weakly interacting trapped gas undergoes a Kosterlitz-Thouless-Berezinskii transition from the normal state at a temperature slightly below the Bose-Einstein transition temperature of the ideal gas. The characteristic finite superfluid mass density of a homogeneous system just below the transition becomes strongly suppressed in a trapped gas.
In this thesis we study the dimensional crossover of a nonrelativistic Bose gas with pointlike repulsive interactions. Our focus is on the dimensional crossover from two to three dimensions, which is of particular relevance in ultracold experiments. Both the two-dimensional and the three-dimensional Bose gas feature a phase transition to a superfluid state. We investigate how the phase transition temperature changes as a function of the spatial extent of the transverse dimension, which is confined in a potential well with periodic boundary conditions. For these calculations we employ Functional Renormalization Group equations. Further, we relate the 3D and 2D s-wave scattering lengths and show how the lower-dimensional system inherits its properties from the higher-dimensional system during the renormalization group flow. In these calculations we introduce the crossover function, which is of central importance in the dimensional crossover. We also investigate other compactifications. We calculate the crossover function for a box confinement and introduce approximate flow equations for a confinement in a harmonic trap.
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.