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arXiv:1303.7007 · Physical Review A

Trapped one-dimensional ideal Fermi gas with a single impurity

G. E. Astrakharchik, Ioannis Brouzos

Published Aug 2, 2013• 54 citations
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Abstract

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.

DOI
10.1103/physreva.88.021602

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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.
A one-dimensional world is very unusual as there is an interplay between quantum statistics and geometry, and a strong short-range repulsion between atoms mimics Fermi exclusion principle, fermionizing the system. Instead, a system with a large number of components with a single atom in each, on the opposite acquires many bosonic properties. We study the ground-state properties of a multicomponent repulsive Fermi gas trapped in a harmonic trap by a fixed-node diffusion Monte Carlo method. The interaction between all components is considered to be the same. We investigate how the energetic properties (energy, contact) and correlation functions (density profile and momentum distribution) evolve as the number of components is changed. It is shown that the system fermionizes in the limit of strong interactions. Analytical expressions are derived in the limit of weak interactions within the local density approximation for an arbitrary number of components and for one plus one particle using an exact solution.
A one-dimensional world is very unusual as there is an interplay between quantum statistics and geometry, and a strong short-range repulsion between atoms mimics Fermi exclusion principle, fermionizing the system. Instead, a system with a large number of components with a single atom in each, on the opposite acquires many bosonic properties. We study the ground-state properties of a multicomponent repulsive Fermi gas trapped in a harmonic trap by a fixed-node diffusion Monte Carlo method. The interaction between all components is considered to be the same. We investigate how the energetic properties (energy, contact) and correlation functions (density profile and momentum distribution) evolve as the number of components is changed. It is shown that the system fermionizes in the limit of strong interactions. Analytical expressions are derived in the limit of weak interactions within the local density approximation for an arbitrary number of components and for one plus one particle using an exact solution.
The ground-state properties of a single-component one-dimensional Coulomb gas are investigated. We use Bose-Fermi mapping for the ground-state wave function which permits solution of the Fermi sign problem in the following respects: (i) the nodal surface is known, permitting exact calculations; and (ii) evaluation of determinants is avoided, reducing the numerical complexity to that of a bosonic system and, thus, allowing simulation of a large number of fermions. Due to the mapping, the energy and local properties in one-dimensional Coulomb systems are exactly the same for Bose-Einstein and Fermi-Dirac statistics. The exact ground-state energy is calculated in homogeneous and trapped geometries using the diffusion Monte Carlo method. We show that in the low-density Wigner crystal limit an elementary low-lying excitation is a plasmon, which is to be contrasted with the high-density ideal Fermi gas/Tonks-Girardeau limit, where low-lying excitations are phonons. Exact density profiles are compared to the ones calculated within the local density approximation, which predicts a change from a semicircular to an inverted parabolic shape of the density profile as the value of the charge is increased.
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.
Abstract We discuss the local density approximation approach to calculating the ground state energy of a one-dimensional Fermi gas containing a single impurity, and compare the results with exact numerical values that we have for up to 11 particles for general interaction strengths and up to 30 particles in the strongly interacting case. We also calculate the contact coefficient in the strongly interacting regime. The different theoretical predictions are compared to recent experimental results with few-atom systems. Firstly, we find that the local density approximation suffers from great ambiguity in the few-atom regime, yet it works surprisingly well for some models. Secondly, we find that the strong interaction theories quickly break down when the number of particles increase or the interaction strength decreases.
We study the ground state and excitations of a one-dimensional trapped polarized Fermi gas interacting with a single impurity. First, we study the tunneling dynamics of the impurity through a potential barrier, such as one effectively created by a double-well trap. To this end, we perform an exact diagonalization of the full few-body Hamiltonian and analyze the results in a local-density approximation. Off-diagonal and one-particle correlation matrices are studied and are shown to be useful for discerning between different symmetries of the states. Second, we consider a radio-frequency spectroscopy of our system and the resulting spectral function. These calculations can motivate future experiments, which can provide further insight into the physics of a Fermi polaron.