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arXiv:cond-mat/0203025 · Physical Review A

Demixing in mesoscopic boson-fermion clouds inside cylindrical harmonic traps: Quantum phase diagram and role of temperature

Z. Akdeniz, Anna Minguzzi, Patrizia Vignolo, Monica Tosi

Published Jul 30, 2002• 20 citations
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Abstract

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 $T.$ The finite system size allows three different regimes for the equilibrium density profiles at $T=0:$ 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.

DOI
10.1103/physreva.66.013620

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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.
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.
Abstract A short review is given of recent progress in understanding the demixing transition in a mesoscopic gaseous cloud of bosonic and fermionic alkali atoms under the effect of their mutual repulsions at zero temperature. The mixture is confined inside a magnetic trap exerting external harmonic potentials on the atoms and the bosonic component is taken in a Bose-Einstein condensed state while the fermions are in a spin-polarized state. Main emphasis is placed on the role of the confinement: in a mesoscopic system (i) the phase transition becomes smooth as a function of the driving parameter, which is the boson-fermion s-wave scattering length; and (ii) various configurations of the phase-separated gaseous cloud are allowed, including some very exotic ones in an anisotropic trap. The transition inside a spherical trap leads to an egg-type structure, with the bosonic condensate being surrounded by a shell of fermions, and is best identified from a sharp upturn of the frequency of low-lying monopolar and dipolar oscillations of fermionic character. This dynamical signature of the transition appears when the overlap between the two components has already started to decrease, but is still quite appreciable. A system of specific experimental interest is the mixture of 6Li and 7Li isotopes, whose parameters could be driven towards the phase transition.
We calculate the spectrum of low-lying collective excitations in a gaseous cloud formed by a Bose-Einstein condensate and a spin-polarized Fermi gas over a range of the boson-fermion coupling strength extending from strongly repulsive to strongly attractive. Increasing boson-fermion repulsions drives the system towards spatial separation of its components (``demixing''), whereas boson-fermion attractions drives it towards implosion (``collapse''). The dynamics of the system is treated in the experimentally relevant collisionless regime by means of a random-phase approximation, and the behavior of a mesoscopic cloud under isotropic harmonic confinement is contrasted with that of a macroscopic mixture at given average particle densities. In the latter case the locations of both the demixing and the collapse phase transitions are sharply defined by the same stability condition, which is determined by the softening of an eigenmode of either fermionic or bosonic origin. In contrast, the transitions to either demixing or collapse in a mesoscopic cloud at fixed confinement and particle numbers are spread out over a range of boson-fermion coupling strength, and some initial decrease of the frequencies of a set of collective modes is followed by hardening, as evidenced by blue shifts of most eigenmodes. The spectral hardening can serve as a signal of the impending transition and is most evident when the number of bosons in the cloud is relatively large. We propose physical interpretations for these dynamical behaviors with the help of suitably defined partial compressibilities for the gaseous cloud under confinement.
We calculate the spectrum of low-lying collective excitations in a mesoscopic cloud formed by a Bose-Einstein condensate and a spin-polarized Fermi gas as a function of the boson-fermion repulsions. The cloud is under isotropic harmonic confinement and its dynamics is treated in the collisional regime by using the equations of generalized hydrodynamics with inclusion of surface effects. For large numbers of bosons, we find that as the cloud moves towards spatial separation (demixing) with increasing boson-fermion coupling, the frequencies of a set of collective modes show a softening followed by a sharp upturn. This behavior permits a clear identification of the quantum phase transition. We propose a physical interpretation for the dynamical transition point in a confined mixture, leading to a simple analytical expression for its location.
We use a semiclassical three-fluid model to analyse the conditions for spatial phase separation in a mixture of fermionic 6 Li and a (stable) Bose-Einstein condensate of 7 Li atoms under cylindrical harmonic confinement, both at zero and finite temperature. We show that with the parameters of the Paris experiment (Schreck F et al 2001 Phys. Rev. Lett. 87 080403) an increase of the boson-fermion scattering length by a factor five would be sufficient to enter the phase-separated regime. We give examples of configurations for the density profiles in phase separation and estimate that the transition should persist at temperatures typical of current experiments. For higher values of the boson-fermion coupling we also find a new phase separation between the fermions and the bosonic thermal cloud at finite temperature.
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.