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arXiv:1310.5272 · Physical Review A

Molecular branch of a small highly elongated Fermi gas with an impurity: Full three-dimensional versus effective one-dimensional description

Seyed Ebrahim Gharashi, Xiangyu Yin, D. Blume

Published Feb 6, 2014• 8 citations
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Abstract

We consider an impurity immersed in a small Fermi gas under highly elongated harmonic confinement. The impurity interacts with the atoms of the Fermi gas through an isotropic short-range potential with three-dimensional free-space $s$-wave scattering length ${a}_{3\text{D}}$. We investigate the energies of the molecular branch, i.e., the energies of the state that corresponds to a gas consisting of a weakly bound diatomic molecule and ``unpaired'' atoms, as a function of the $s$-wave scattering length ${a}_{3\text{D}}$ and the ratio $\ensuremath{\eta}$ between the angular trapping frequencies in the tight and weak confinement directions. The energies obtained from our three-dimensional description that accounts for the dynamics in the weak and tight confinement directions are compared with those obtained within an effective one-dimensional framework, which accounts for the dynamics in the tight confinement direction via a renormalized one-dimensional coupling constant. Our theoretical results are related to recent experimental measurements.

DOI
10.1103/physreva.89.023603

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We consider an impurity immersed in a small Fermi gas under highly elongated harmonic confinement. The impurity interacts with the atoms of the Fermi gas through an isotropic short-range potential with three-dimensional free-space sss-wave scattering length a3D{a}_{3\text{D}}a3D​. We investigate the energies of the molecular branch, i.e., the energies of the state that corresponds to a gas consisting of a weakly bound diatomic molecule and ``unpaired'' atoms, as a function of the sss-wave scattering length a3D{a}_{3\text{D}}a3D​ and the ratio \ensuremathη\ensuremath{\eta}\ensuremathη between the angular trapping frequencies in the tight and weak confinement directions. The energies obtained from our three-dimensional description that accounts for the dynamics in the weak and tight confinement directions are compared with those obtained within an effective one-dimensional framework, which accounts for the dynamics in the tight confinement direction via a renormalized one-dimensional coupling constant. Our theoretical results are related to recent experimental measurements.
Full three-dimensional (3D) calculations of small two-component Fermi gases under highly elongated confinement, in which unlike fermions interact through short-range potentials with variable atom-atom sss-wave scattering length, are performed using the correlated Gaussian approach. In addition, microscopic one-dimensional (1D) calculations are performed for effective ``atomic'' and ``molecular'' 1D model Hamiltonian. Comparisons of the 3D and 1D energies and excitation frequencies establish the validity regimes of the effective 1D Hamiltonian. Our numerical results for three- and four-particle systems suggest that the effective 1D atom-dimer and dimer-dimer interactions are to a good approximation determined by simple analytical expressions. Implications for the description of quasi-1D Fermi gases within strict 1D frameworks are discussed.
The transition from ``few to many'' has recently been probed experimentally in an ultracold harmonically confined one-dimensional lithium gas, in which a single impurity atom interacts with a background gas consisting of one, two, or more identical fermions [A. N. Wenz et al., Science 342, 457 (2013)]. For repulsive interactions between the background or majority atoms and the impurity, the interaction energy for relatively moderate system sizes was analyzed and found to converge toward the corresponding expression for an infinitely large Fermi gas. Motivated by these experimental results, we apply perturbative techniques to determine the interaction energy for weak and strong coupling strengths and derive approximate descriptions for the interaction energy for repulsive interactions with varying strength between the impurity and the majority atoms and any number of majority atoms.
Based on the recently developed interaction renormalization for the one-dimensional ppp-wave interaction, we study the problem of a single impurity immersed in a highly polarized Fermi sea. They interact through a narrow ppp-wave Feshbach resonance, so the effective range r0{r}_{0}r0​ naturally appears in the system. We use the variational approach limited to single-particle--hole excitations on top of the unperturbed Fermi sea. The polaron exhibits two branches of solutions, namely, the attractive and repulsive branches, with varying scattering length across the resonance. We calculate the energy spectrum, residue, and effective mass for each of the branches. We compare the polaronic energy with the energy of the dressed molecule and find that the molecular state is energetically favored when increasing the interaction strength. The critical interaction strength for the polaron-to-molecule transition will shift to the BCS side of the ppp-wave resonance as the effective range increases.
Quasi-one-dimensional two-component Fermi gases with effectively attractive and repulsive interactions are characterized for arbitrary interaction strength. The ground-state properties of the gas confined in highly elongated harmonic traps are determined within the local density approximation. For strong attractive effective interactions the existence of a molecular Tonks-Girardeau gas is predicted. The frequency of the lowest breathing mode is calculated as a function of the coupling strength for both attractive and repulsive interactions.
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 ${\ensuremath{\omega}}_{z}\ensuremath{\rightarrow}0$. Furthermore, our results suggest that Tc{T}_{c}Tc​ can be enhanced by relaxing the confinement and perturbing away from the 2D limit.
Ground-state properties of a single impurity in a one-dimensional Fermi gas are investigated in uniform and trapped geometries. The energy of a trapped system is obtained (i) by generalizing the McGuire expression from a uniform to trapped system (ii) within the local density approximation (iii) using the perturbative approach in the case of a weakly interacting impurity and (iv) diffusion Monte Carlo method. We demonstrate that there is a closed formula based on the exact solution of the homogeneous case which provides a precise estimation for the energy of a trapped system even for a small number of fermions and arbitrary coupling constant of the impurity. Using this expression, we analyze energy contributions from kinetic, interaction, and potential components, as well as spatial properties such as the system size and the pair-correlation function. Finally, we calculate the frequency of the breathing mode. Our analysis is directly connected and applicable to the recent experiments in microtraps.