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Chinese Physics Letters

Phase Diagram and Phase Separation of a Trapped Interacting Bose–Fermi Gas Mixture

Ma Yong-Li

Published Dec 1, 2004• 0 citations
Publisher

Abstract

In six different regimes for a spatial phase diagram of a trapped interacting Bose–Fermi gas mixture at low temperatures, we present the conditions for the spatial demixing and separation of bosons and fermions. Starting from a semiclassically thermodynamic model for the local density functional of thermal bosons and fermions, the explicit analytical expressions for the fugacities of bosons and fermions are derived in different regimes by means of a first-order perturbation method in a local-density approximation. The critical values of the fermion–boson interaction strength as a function of the fractional composition of fermions have a general feature: increase, extreme and decrease with increasing the fermionic composition slightly above Bose–Einstein critical temperature.

DOI
10.1088/0256-307x/21/12/010

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In six different regimes for a spatial phase diagram of a trapped interacting Bose–Fermi gas mixture at low temperatures, we present the conditions for the spatial demixing and separation of bosons and fermions. Starting from a semiclassically thermodynamic model for the local density functional of thermal bosons and fermions, the explicit analytical expressions for the fugacities of bosons and fermions are derived in different regimes by means of a first-order perturbation method in a local-density approximation. The critical values of the fermion–boson interaction strength as a function of the fractional composition of fermions have a general feature: increase, extreme and decrease with increasing the fermionic composition slightly above Bose–Einstein critical temperature.
We use a semiclassical three-fluid model to evaluate the phenomena of spatial demixing in mesoscopic clouds of fermionic and bosonic atoms at high dilution under harmonic confinement. We assume repulsive boson-boson and boson-fermion interactions and include a bosonic thermal cloud at finite temperature T.T.T. The finite system size allows three different regimes for the equilibrium density profiles at T=0:T=0:T=0: a fully mixed state, a partially mixed state in which the overlap between bosons and fermions is decreasing, and a fully demixed state where the two clouds have zero overlap. We propose simple analytical rules for the two crossovers between the three regimes as functions of the system parameters and support these rules by extensive numerical calculations. A universal ``phase diagram'' is shown to be valid for the transition to the regime of full demixing, where we identify several exotic configurations in addition to simple ones consisting of a core of bosons enveloped by fermions and vice versa. With increasing temperature some exotic configurations are transformed into more symmetric ones, until demixing is ultimately lost. For very high values of the boson-fermion repulsion we also report demixing between the fermions and the thermally excited bosons.
Experimental realizations of a variety of atomic binary Bose-Fermi mixtures have brought opportunities for studying composite quantum systems with different spin statistics. The binary atomic mixtures can exhibit a structural transition from a mixture into phase separation as the boson-fermion interaction increases. By using a path-integral formalism to evaluate the grand partition function and the thermodynamic grand potential, we obtain the effective potential of binary Bose-Fermi mixtures. Thermodynamic quantities in a broad range of temperatures and interactions are also derived. The structural transition can be identified as a loop of the effective potential curve, and the volume fraction of phase separation can be determined by the lever rule. For $^{6}\mathrm{Li}\ensuremath{-}^{7}\mathrm{Li}$ and $^{6}\mathrm{Li}\ensuremath{-}^{41}\mathrm{K}$ mixtures, we present the phase diagrams of the mixtures in a box potential at zero and finite temperatures. Due to the flexible densities of atomic gases, the construction of phase separation is more complicated when compared to conventional liquid or solid mixtures where the individual densities are fixed. For harmonically trapped mixtures, we use the local density approximation to map out the finite-temperature density profiles and present typical trap structures, including the mixture, partially separated phases, and fully separated phases.
The structure of trapped binary mixtures of bosonic and fermionic atoms at zero temperature is studied using a modified Gross-Pitaevskii equation for the bosons which self-consistently includes the mean-field interaction generated by the fermionic cloud. The density of the fermionic component is described within the Thomas-Fermi approximation. The influence of the boson-boson and the boson-fermion s-wave interaction on the density profiles and the stability of the mixture is investigated systematically. Critical particle numbers for the mean-field collapse caused either by attractive boson-boson or by attractive boson-fermion interactions and for the onset of spatial component separation are discussed. It is shown that the interplay between the boson-boson and the boson-fermion interaction generates a rich and complex phase diagram. Finally, the specific properties and prospects of the boson-fermion mixtures available in present experiments are addressed.
A cooling strategy that is currently being pursued for a gas of fermionic atoms uses sympathetic cooling via s-wave collisions against a Bose-condensed gas. At increasing values of the scattering lengths, the boson-fermion cloud undergoes demixing. Locating the onset of incipient spatial separation is relevant to fermion cooling, since at that point the diminishing overlap between the two clouds will start reducing the effectiveness of the collisional transfer processes. With full demixing, the system enters a new quantum phase, where its properties are expected to be substantially modified. In this perspective, we have used a semiclassical model to evaluate spatial demixing in mesoscopic clouds of fermionic and bosonic atoms at high dilution inside cylindrical harmonic traps. The finite system size allows three different regimes for the equilibrium density profiles: a fully mixed state, a partially mixed state in which the overlap between the boson and fermion clouds decreases, and a fully demixed state wherein the two clouds have zero overlap. Simple analytical rules account for the crossovers between these regimes as functions of the physical system parameters and are supported by extensive numerical calculations. A universal phase diagram involving simple scaling parameters is valid for the transition to full demixing, inside which we identify several exotic configurations for the two phase-separated clouds in addition to the simple ones consisting of a core of bosons enveloped by fermions and vice versa.