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arXiv:1408.0476 · Physical Review B

Quasi-two-dimensional Fermi gases at finite temperatures

Andrea M. Fischer, Meera M. Parish

Published Dec 1, 2014• 19 citations
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Abstract

We consider a Fermi gas with short-range attractive interactions that is confined along one direction by a tight harmonic potential. For this quasi-two-dimensional (quasi-2D) Fermi gas, we compute the pressure equation of state, radiofrequency spectrum, and the superfluid critical temperature ${T}_{c}$ using a mean-field theory that accounts for all the energy levels of the harmonic confinement. Our calculation for ${T}_{c}$ provides a natural generalization of the Thouless criterion to the quasi-2D geometry, and it correctly reduces to the 3D expression derived from the local density approximation in the limit where the confinement frequency ${\ensuremath{\omega}}_{z}\ensuremath{\rightarrow}0$. Furthermore, our results suggest that ${T}_{c}$ can be enhanced by relaxing the confinement and perturbing away from the 2D limit.

DOI
10.1103/physrevb.90.214503

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We consider a Fermi gas with short-range attractive interactions that is confined along one direction by a tight harmonic potential. For this quasi-two-dimensional (quasi-2D) Fermi gas, we compute the pressure equation of state, radiofrequency spectrum, and the superfluid critical temperature Tc{T}_{c}Tc​ using a mean-field theory that accounts for all the energy levels of the harmonic confinement. Our calculation for Tc{T}_{c}Tc​ provides a natural generalization of the Thouless criterion to the quasi-2D geometry, and it correctly reduces to the 3D expression derived from the local density approximation in the limit where the confinement frequency \ensuremathωz\ensuremath→0{\ensuremath{\omega}}_{z}\ensuremath{\rightarrow}0\ensuremathωz​\ensuremath→0. Furthermore, our results suggest that Tc{T}_{c}Tc​ can be enhanced by relaxing the confinement and perturbing away from the 2D limit.
We consider a two-component gas of fermionic atoms confined to a quasi-two-dimensional (quasi-2D) geometry by a harmonic trapping potential in the transverse direction. We construct a mean-field theory of the BCS-BEC crossover at zero temperature that allows us to extrapolate to an infinite number of transverse harmonic oscillator levels. In the extreme BEC limit, where the confinement length exceeds the dimer size, we recover 3D dimers confined to two dimensions with weak repulsive interactions. However, even when the interactions are weak and the Fermi energy is less than the confinement frequency, we find that the higher transverse levels can substantially modify fermion pairing. We argue that recent experiments on pairing in quasi-2D Fermi gases [Y. Zhang et al., Phys. Rev. Lett. 108, 235302 (2012)] have already observed the effects of higher transverse levels.
A two-dimensional (2D) harmonically trapped interacting Fermi gas is anticipated to exhibit a quantum anomaly and possesses a breathing mode at frequencies different from a classical scale-invariant value ω_{B}=2ω_{⊥}, where ω_{⊥} is the trapping frequency. The predicted maximum quantum anomaly (∼10%) has not been confirmed in experiments. Here, we theoretically investigate the zero-temperature density equation of state and the breathing mode frequency of an interacting Fermi superfluid at the dimensional crossover from three to two dimensions. We find that the simple model of a 2D Fermi gas with a single s-wave scattering length is not adequate to describe the experiments in the 2D limit, as commonly believed. A more complete description of quasi-2D leads to a much weaker quantum anomaly, consistent with the experimental observations. We clarify that the reduced quantum anomaly is due to the significant confinement-induced effective range of interactions.
We investigate the ground-state properties of ultracold two-component Fermi gases in the presence of a transverse harmonic potential, focusing on the strongly interacting regime in which pairs of fermions form tightly bound molecules. Using the fixed-node diffusion Monte Carlo method, we calculate the equation of state and density profiles for the full fermionic system, which allows us to address the importance of finite-range corrections arising from the internal fermionic structure of the dimers. We interpret the results in terms of a molecular Bose gas in quasi-two-dimensional confinement and compare them with theoretical predictions for a weakly interacting two-dimensional Bose gas, identifying the range of validity of mean-field and beyond-mean-field descriptions. We also develop an analytical theory for the transverse density profile, capturing its broadening with increasing interaction strength. This work provides a benchmark for an effective bosonic description of strongly bound fermionic dimers and offers insights into confinement-induced dimensional effects.
Using the finite-temperature path integral Monte Carlo method, we investigate dilute, trapped Bose gases in a quasi-two-dimensional (quasi-2D) geometry. The quantum particles have short-range, sss-wave interactions described by a hard-sphere potential whose core radius equals its corresponding scattering length. The effect of both the temperature and the interparticle interaction on the equilibrium properties such as the total energy, the density profile, and the superfluid fraction is discussed. We compare our accurate results with both the semiclassical approximation and the exact results of an ideal Bose gas. Our results show that for repulsive interactions (i) the minimum value of the aspect ratio, where the system starts to behave quasi-2D, increases as the two-body interaction strength increases; (ii) the superfluid fraction for a quasi-two-dimensionally Bose gas is distinctly different from that for both a quasi-1D Bose gas and a true 3D system, i.e., the superfluid fraction for a quasi-2D Bose gas decreases faster than that for a quasi-1D system and a true 3D system with increasing temperature, and shows a stronger dependence on the interaction strength; (iii) the superfluid fraction for a quasi-2D Bose gas lies well below the values calculated from the semiclassical approximation; and (iv) the Kosterlitz-Thouless transition temperature decreases as the strength of the interaction increases.
We present a closed-form expression for the finite temperature first-order density matrix of an isotropic harmonically trapped ideal Fermi gas in any dimension. This constitutes a much sought after generalization of the recent results in the literature, where analytical expressions have been limited to quantities derived from the diagonal first-order density matrix. We compare our results with the Thomas-Fermi approximation (TFA) and demonstrate numerically that the TFA provides an excellent description of the first-order density matrix in the large-NNN limit. As an interesting application, we derive a closed-form expression for the finite temperature Hartree-Fock exchange energy of a two-dimensional (2D) parabolically confined quantum dot. We numerically test this result against the 2D TF exchange functional, and comment on the applicability of the local-density approximation to the exchange energy of an inhomogeneous 2D Fermi gas.
We review the current understanding of the uniform two-dimensional (2D) Fermi gas with short-range interactions. We first outline the basics of two-body scattering in 2D, including a discussion of how such a 2D system may be realized in practice using an anisotropic confining potential. We then discuss the thermodynamic and dynamical properties of 2D Fermi gases, which cold-atom experiments have only just begun to explore. Of particular interest are the different pairing regimes as the interparticle attraction is varied; the superfluid transition and associated finite-temperature phenomenology; few-body properties and their impact on the many-body system; the Fermi polaron problem; and the symmetries underlying the collective modes. Where possible, we include the contributions from 2D experiment. An underlying theme throughout is the effect of the quasi-2D geometry, which we view as an added richness to the problem rather than an unwanted complication.