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arXiv:0806.4337

Low-density, one dimensional quantum gases in the presence of a localised attractive potential

John Goold, Donal O Donoghue, Thomas Busch

Published Jun 26, 2008• 0 citations
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Abstract

We investigate low-density, quantum-degenerate gases in the presence of a localised attractive potential in the centre of a one-dimensional harmonic trap.The attractive potential is modelled using a parameterised delta-function, allowing us to determine all single particle eigenfunctions analytically. From these we calculate the ground state many-body properties for a system of spin-polarised fermions and, using the Bose-Fermi mapping theorem, extend the results to strongly interacting bosonic systems. We discuss the single particle densities, the pair correlation functions, the reduced single particle density matrices and the momentum distributions as a function of particle number and strength of the attractive point potential. As an important experimental observable, we place special emphasis on spatial coherence properties of such samples.

DOI
10.48550/arxiv.0806.4337

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We investigate low-density, quantum-degenerate gases in the presence of a localised attractive potential in the centre of a one-dimensional harmonic trap.The attractive potential is modelled using a parameterised delta-function, allowing us to determine all single particle eigenfunctions analytically. From these we calculate the ground state many-body properties for a system of spin-polarised fermions and, using the Bose-Fermi mapping theorem, extend the results to strongly interacting bosonic systems. We discuss the single particle densities, the pair correlation functions, the reduced single particle density matrices and the momentum distributions as a function of particle number and strength of the attractive point potential. As an important experimental observable, we place special emphasis on spatial coherence properties of such samples.
We investigate low-density, quantum-degenerate gases in the presence of a localized attractive potential in the centre of a one-dimensional harmonic trap. The attractive potential is modelled using a parameterized δ-function, allowing us to determine all single-particle eigenfunctions analytically. From these we calculate the ground-state many-body properties for a system of spin-polarized fermions and, using the Bose–Fermi mapping theorem, extend the results to strongly interacting bosonic systems. We discuss the single-particle densities, the pair-correlation functions, the reduced single-particle density matrices and the momentum distributions as a function of the particle number and strength of the attractive point potential. As an important experimental observable, we place special emphasis on spatial coherence properties of such samples.
We investigate degenerate quantum gases in one dimension, trapped in a harmonic potential which is split centrally by a point-like potential. Since the single-particle eigenfunctions of such a system are known for all strengths of the central potential, the dynamics for non-interacting fermionic gases and low-density, strongly interacting bosonic gases can be investigated exactly using the Fermi–Bose mapping theorem. We calculate the exact many-particle ground state wavefunctions for both particle species, investigate soliton-like solutions, and compare the bosonic system to the well known physics of Bose gases described by the Gross–Pitaevskii equation. We also address the experimentally important questions of the creation and detection of such states.
We consider a strongly interacting one-dimensional (1D) Bose-Fermi mixture confined in a harmonic trap. It consists of a Tonks-Girardeau (TG) gas (1D Bose gas with repulsive hard-core interactions) and of a noninteracting Fermi gas (1D spin-aligned Fermi gas), both species interacting through hard-core repulsive interactions. Using a generalized Bose-Fermi mapping, we determine the one-body density matrices, exact particle density profiles, momentum distributions, and behavior of the mixture under 1D expansion when opening the trap. In real space, bosons and fermions do not display any phase separation: the respective density profiles extend over the same region and they both present a number of peaks equal to the total number of particles in the trap. In momentum space the bosonic component has the typical narrow TG profile, while the fermionic component shows a broad distribution with fermionic oscillations at small momenta. Due to the large boson-fermion repulsive interactions, both the bosonic and the fermionic momentum distributions decay as $C{p}^{\ensuremath{-}4}$ at large momenta, like in the case of a pure bosonic TG gas. The coefficient CCC is related to the two-body density matrix and to the bosonic concentration in the mixture. When opening the trap, both momentum distributions ``fermionize'' under expansion and turn into that of a Fermi gas with a particle number equal to the total number of particles in the mixture.
We investigate the ground-state properties of two-component Bose gases confined in one-dimensional harmonic traps in the scheme of density-functional theory. The density-functional calculations employ a Bethe-ansatz-based local-density approximation for the correlation energy, which accounts for the correlation effect properly from the weakly interacting regime to the strongly interacting regime. For the binary Bose mixture with spin-independent interaction, the homogeneous reference system is exactly solvable by the Bethe-ansatz method. Within the local-density approximation, we determine the density distribution of each component and study its evolution from Bose distributions to Fermi-like distribution with the increase in interaction. For the binary mixture of Tonks-Girardeau gases with a tunable interspecies repulsion, with a generalized Bose-Fermi transformation we show that the Bose mixture can be mapped into a two-component Fermi gas, which corresponds to exact soluble Yang-Gaudin model for the homogeneous system. Based on the ground-state energy function of the Yang-Gaudin model, the ground-state density distributions are calculated for various interspecies interactions. It is shown that with the increase in interspecies interaction, the system exhibits composite-fermionization crossover.
We use a BCS-type variational wave function to study attractively interacting quasi-one-dimensional fermionic atomic gases, motivated by cold-atom experiments that access the one-dimensional regime using an anisotropic harmonic trapping potential (with trapping frequencies ${\ensuremath{\omega}}_{x}={\ensuremath{\omega}}_{y}\ensuremath{\gg}{\ensuremath{\omega}}_{z}$) that confines the gas to a cigar-shaped geometry. To handle the presence of the trap along the zzz direction, we construct our variational wave function from the harmonic oscillator Hermite functions, which are the eigenstates of the single-particle problem. Using an analytic determination of the effective interaction among harmonic oscillator states along with a numerical solution of the resulting variational equations, we make specific experimental predictions for how pairing correlations would be revealed in experimental probes, such as the local density and the momentum correlation function.
Motivated by the realization of hard-wall boundary conditions in experiments with ultracold atoms, we investigate the ground-state properties of spin-1/2 fermions with attractive interactions in a one-dimensional box.We use lattice Monte Carlo methods to determine essential quantities like the energy, which we compute as a function of coupling strength and particle number in the regime from few to many particles.Many-fermion systems bound by hard walls display non-trivial density profiles characterized by so-called Friedel oscillations (which are similar to those observed in harmonic traps).In non-interacting systems, the characteristic length scale of the oscillations is set by (2k F ) -1 , where k F is the Fermi momentum, while repulsive interactions tend to generate Wignercrystal oscillations of period (4k F ) -1 .Based on the non-interacting result, we find a remarkably simple parametrization of the density profiles of the attractively interacting case, which we generalize to the one-body density matrix.While the total momentum is not a conserved quantity in the presence of hard walls, the magnitude of the momentum does provide a good quantum number.We are therefore able to provide a detailed characterization of the (quasi-)momentum distribution, which displays rather robust discontinuity at the Fermi surface.In addition, we determine the spatially varying on-site density-density correlation, which in turn yields Tan's contact density and, upon integration, Tan's contact.As is well known, the latter fully determines the short-range correlations and plays a crucial role in a multitude of equilibrium and non-equilibrium sum rules.