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Physical Review A

Mean-field interaction-induced dimensional crossover from two to three dimensions in a weakly interacting Bose gas

Jiho Noh, Jeongwon Lee, Jongchul Mun

Published Feb 25, 2020• 1 citation
Publisher

Abstract

We study an interaction-induced dimensional crossover from two to three dimensions in a weakly interacting $^{87}\mathrm{Rb}$ Bose-Einstein condensate (BEC). A highly oblate harmonic optical dipole trap was used where we independently controlled the chemical potential and the temperature of the gas. The dimensional crossover was studied from the thermal state occupancy of atoms. In particular, the growth rate of the thermal component was measured, which contradicted the expected behavior in a fixed dimension. Crossover chemical potential and temperature were obtained at the crossover point. We further validate our observation by numerical calculations based on mean-field Hartree-Fock theory and local density approximation, which not only provided a qualitative picture of the underlying mechanism for the dimensional crossover, but also matched well with the experimentally measured crossover parameter.

DOI
10.1103/physreva.101.023619

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We study an interaction-induced dimensional crossover from two to three dimensions in a weakly interacting 87Rb^{87}\mathrm{Rb}87Rb Bose-Einstein condensate (BEC). A highly oblate harmonic optical dipole trap was used where we independently controlled the chemical potential and the temperature of the gas. The dimensional crossover was studied from the thermal state occupancy of atoms. In particular, the growth rate of the thermal component was measured, which contradicted the expected behavior in a fixed dimension. Crossover chemical potential and temperature were obtained at the crossover point. We further validate our observation by numerical calculations based on mean-field Hartree-Fock theory and local density approximation, which not only provided a qualitative picture of the underlying mechanism for the dimensional crossover, but also matched well with the experimentally measured crossover parameter.
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 present a detailed beyond-mean-field analysis of a weakly interacting Bose gas in the crossover from three to low dimensions. We find an analytical solution for the energy and provide a clear qualitative picture of the crossover in the case of a box potential with periodic boundary conditions. We show that the leading contribution of the confinement-induced resonance is of beyond-mean-field order and calculate the leading corrections in the three- and low-dimensional limits. We also characterize the crossover for harmonic potentials in a model system with particularly chosen short- and long-range interactions and show the limitations of the local-density approximation. Our analysis is applicable to Bose-Bose mixtures and gives a starting point for developing the beyond-mean-field theory in inhomogeneous systems with long-range interactions such as dipolar particles or Rydberg-dressed atoms.
We investigate theoretically the transition from two to three dimensions in a weakly interacting ultracold Fermi gas. This transition was recently observed by Dyke et al. [Phys. Rev. Lett. 106, 105304 (2011)] in a cloud of fermionic 6{}^{6}6Li atoms trapped in an oblate harmonic potential. We calculate the density distribution and cloud size of the Fermi cloud by numerically solving three-dimensional Hartree-Fock mean-field equations. Local density approximation is used to ease the calculations in the weakly confined radial direction. Our theoretical results are shown to be in good agreement with the experimental measurements.
We investigate the dimensionally induced phase transition from the normal to the Bose-Einstein-condensed phase for a weakly interacting Bose gas in an optical lattice. To this end we make use of the Hartree-Fock-Bogoliubov-Popov theory, where we include numerically exact hopping energies and effective interaction strengths. At first we determine the critical chemical potential, where we find a much better agreement with recent experimental data than a pure Hartree-Fock treatment. This finding originates from an important correction due to quantum fluctuations, which are enhanced in lower dimensions. Furthermore, we determine for the one-dimensional to three-dimensional transition the power-law exponent of the critical temperature for two different noninteracting Bose gas models yielding the same value of 1/2, which indicates that they belong to the same universality class. For the weakly interacting Bose gas we find for both models that this exponent is robust with respect to finite interaction strengths.
The fact that two-dimensional interacting trapped systems do not undergo Bose-Einstein Condensation (BEC) in the thermodynamic limit, though rigorously proved, is somewhat mysterious because all relevant limiting cases (zero temperature, small atom numbers, noninteracting particles) suggest otherwise. We study the possibility of condensation in finite trapped two-dimensional systems. We first consider the ideal gas, which incorporates the inhomogeneity and finite size of experimental systems and can be solved exactly. A semiclassical self-consistent approximation gives us a feel for the temperature scales; diagonalization of the one-body density matrix confirms that the condensation is into a single state. We squeeze a three-dimensional system and see how it crosses over into two dimensions. Mean-field theory, our main tool for the study of interacting systems, prescribes coupled equations for the condensate and the thermal cloud: the condensate receives a full quantum-mechanical treatment, while the noncondensate is described by different schemes of varying sophistication. We analyze the T = 0 case and its approach to the thermodynamic limit, finding a criterion for the dimensionality crossover and obtaining the coupling constant of the two-dimensional system that results from squeezing a three-dimensional trap. We next apply a semiclassical Hartree-Fock approximation to purely two-dimensional finite gases and find that they can be described either with or without a condensate; this continues to be true in the thermodynamic limit. The condensed solutions have a lower free energy at all temperatures but neglect the presence of phonons within the system and cease to exist when we allow for this possibility. The uncondensed solutions, in turn, are valid under a more rigorous scheme but have consistency problems of their own. Path-integral Monte Carlo simulations provide an essentially exact description of finite interacting gases and enable us to study highly anisotropic systems at finite temperature. We find that our two-dimensional Hartree-Fock solutions accurately mimic the surface density profile and predict the condensate fraction of these systems; the equivalent interaction parameter is smaller than that dictated by the T = 0 analysis. We conclude that, in two-dimensional isotropic finite trapped systems and in highly compressed three-dimensional gases, there is a phenomenon resembling a condensation into a single state.
We discuss the transition from a fully decoherent to a (quasi)condensate regime in a harmonically trapped weakly interacting one-dimensional (1D) Bose gas. By using analytic approaches and verifying them against exact numerical solutions, we find a characteristic crossover temperature and crossover atom number that depend on the interaction strength and the trap frequency. We then identify the conditions for observing either an interaction-induced crossover scenario or else a finite-size Bose-Einstein condensation phenomenon characteristic of an ideal trapped 1D gas.