z akdeniz et al- phase separation in a boson–fermion mixture of lithium atoms

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Phase separation in a boson-fermion mixture of lithium atoms

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2002 J. Phys. B: At. Mol. Opt. Phys. 35 L105

(http://iopscience.iop.org/0953-4075/35/4/102)

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INSTITUTE OF PHYSICS PUBLISHING JOURNAL OF PHYSICS B: ATOMIC, MOLECULAR AND OPTICAL PHYSICS

J. Phys. B: At. Mol. Opt. Phys. 35 (2002) L105L111 PII: S0953-4075(02)31667-5

LETTER TO THE EDITOR

Phase separation in a bosonfermion mixture oflithium atoms

Z Akdeniz1, P Vignolo2, A Minguzzi and M P Tosi

NEST-INFM and Classe di Scienze, Scuola Normale Superiore, Piazza dei Cavalieri 7,I-56126 Pisa, Italy

E-mail: [email protected]

Received 6 December 2001, in final form 23 January 2002

Published 11 February 2002

Online at stacks.iop.org/JPhysB/35/L105

Abstract

We use a semiclassical three-fluid model to analyse the conditions for spatial

phase separation in a mixture of fermionic 6Li and a (stable) BoseEinstein

condensate of7Li atoms under cylindrical harmonic confinement, both at zero

andfinitetemperature. We show thatwith the parametersof 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 bosonfermion coupling

we also find a new phase separation between the fermions and the bosonicthermal cloud at finite temperature.

1. Introduction

Following the achievement of BoseEinstein condensation in alkali gases [1], a great deal of

experimental effort is being devotedto trap andcool gases of fermionicisotopesof alkaliatoms.

The usual evaporative cooling scheme is ineffective, since the Pauli exclusion principle forbids

s-wave collisions between spin-polarized fermions. This problem has been circumvented by

recourse to sympathetic cooling, which relies on s-wave collisions with atoms belonging to

different species, and several groups have thus reached, in the Fermi cloud, temperatures as

low as T = 0.2 TF [24], TF being the Fermi temperature. In particular, an experiment in

Paris has produced a mixture of a degenerate Fermi gas and a stable BoseEinstein condensate,

made from lithium isotopes [4].

1 Permanent address: Department of Physics, University of Istanbul, Istanbul, Turkey.2 Author to whom any correspondence should be addressed.

0953-4075/02/040105+07$30.00 2002 IOP Publishing Ltd Printed in the UK L105
http://stacks.iop.org/jb/35/L105


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L106 Letter to the Editor

The condensatefermion mixture is an interesting system for the study of the effects of the

interactions, since it can show spatial phase separation of the two components on increasing

the value of the bosonboson and bosonfermion coupling constants [5]. This is an example

of a quantum phase transition [6]. Several configurations for phase-separated clouds have

been proposed in [7]. A schematic phase diagram for the trapped mixture consisting of similarnumbers of bosons and fermions at zero temperature has been given in [8], while the case

of low numbers of fermions has been evaluated in [9]. The phases and the stability of a

homogeneous bosonfermion mixture have been studied in [10]. In the experiments it might

be possible to investigate the phase-separated region by exploiting optically or magnetically

induced Feschbach resonances to increase the values of the bosonboson and/or the fermion

boson scattering length [11].

In this letter we have chosen the trap geometry and parameters typical of the Paris

experiment and have analysed the possible phase-separated configurations; we also indicate

how various equilibrium observables are affected by phase separation.

The letter is organized as follows: in section 2 we present the model which we have used

and discuss its limits of applicability; section 3 gives the conditions for phase separation at

zero temperature and illustrates the column-density profiles corresponding to different phase-separated configurations. The results are extended to finite temperature in section 4.

2. The model

We adopt a three-fluid model for a mixture of bosons and fermions at finite temperature T [12].

The model is used to evaluate the density profile of the condensate (nc) within the Thomas

Fermi approximation and those of the fermionic (nf) and thermal bosonic component (nnc)

within the HartreeFock approximation.

We shall assume that the number of bosons in the trap is large enough that the kinetic

energy term in the GrossPitaevskii equation for the wavefunction(r) of the condensate can

be neglected [13]. The density profile of the condensed atoms is

nc(r) = 2(r) = [b Vextb (r) 2gnnc(r) f nf(r)]/g (1)

for positive values of the function in the brackets and zero otherwise. Here, g = 4h2abb/mband f = 2h2abf/mr with abb and abf the bosonboson and bosonfermion s-wave scattering

lengths and mr = mbmf/(mb +mf) with mb and mf the atomic masses; b is the chemical

potential of the bosons. The system is confined by external axially symmetric potentials

Vextb,f(r) = mb,f2b,f(r

2 + 2b,fz2)/2 (2)

where b,f are the frequencies and b,f the anisotropies of the traps.

As already proposed in early work on the confined Bose fluid [14], we treat both the

thermal bosons and the fermions as ideal gases in effective potentials Veff

b,f (r) involving the

relevant interactions. We write

Veff

b

(r) = Vext

b

(r) + 2gnc(r) + 2gnnc(r) + f nf(r) (3)

and

Veff

f (r) = Vextf (r) + f nc(r) + f nnc(r). (4)

We are taking the fermionic component as a dilute, spin-polarized Fermi gas: the fermion

fermion interactions are then associated at leading order with p-wave scattering and are

negligible at the temperatures of present interest [15]. Here and in equation (1), the factor

2 arises from exchange.


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Letter to the Editor L107

We may then evaluate the thermal averages by means of standard BoseEinstein and

FermiDirac distributions, taking the thermal bosons to be in thermal equilibrium with the

condensate at the same chemical potential and the fermions at chemical potential f. In this

semiclassical approximation the densities of the cold-atom clouds are

nnc,f(r) =

d3p

(2h)3

exp

p2

2mb,f+ V

eff

b,f (r) b,f

kBT

1

1. (5)

The chemical potentials are determined from the total numbers of bosons and fermions.

The above model is valid for bosons when the diluteness condition nca3bb 1 holds. For

the Fermi cloud in the mixed regime the conditionkfabf 1with kf = (48NF)1/6/aho should

hold, but this is not a constraint in the regime of phase separation where the bosonfermion

interaction energy drops rapidly (see later).

3. Conditions for phase separation

Phase separation is complete when the overlap between the bosonic and the fermionic density

profiles vanishes. The transition to this regime depends on the confinement and on thenumbers, masses and interaction strengths of the trapped fermions and bosons. In the case of

equal numbers of bosons and fermions and equal trapping frequencies the condition of phase

separation can be written as

kfabb >3

4

abb

abf

2(6)

at zero temperature [8, 10].

Let us focus on the geometry and on typical values of the numbers of atoms in the

Paris experiment [4] for the 7Li6Li mixture: b = 2 4000 Hz, f = 2 3520 Hz,

b f 1/60, andNb Nf 10 000. Since the two trap frequencies are of the same order

of magnitude, we can use equation (6) to estimate at which values of the scattering lengths

the phase transition occurs. For the above experimental parameters condition (6) becomes

a2bf/abb > 7000 a0. Using the predicted values for the scattering lengths of the7Li6Li

mixture, that is abb = 5.1 a0 and abf = 38 a0 [4], we have 283 a0 and the gases are

not as yet in the phase-separated regime. However, in a Feschbach-resonance experiment the

scattering lengths could be increased. There is therefore the possibility to fulfil equation (6)

and to enter the phase-separated regime.

Since the positions of the Feschbach resonances for abf and abb would not necessarily

coincide, in order to explore the phase diagram for the bosonfermion mixture we have first

of all kept abb fixed at its non-resonant value and changed abf. In this case, in order to reach

phase separation it is sufficient to increase the value of the bosonfermion scattering length by

a factor five, namely abf > abf 200 a0.

To check this estimate we have evaluated the bosonfermion interaction energy

Eint = f

d3

r [nc(r) + nnc(r)]nf(r), (7)

using the model described in section 2. The behaviour of this quantity at T = 0 as a function

ofabf/abb, with abb = 5.1 a0, is shown in figure 1 (solid curve).

Because of finite size effects the transition to the phase-separated regime shows a large

crossover region andthe interaction energy goes smoothly to zero when separation is complete3.

3 In an infinite system this quantum phase transition is instead characterized by a discontinuity in the interactionenergy as a function ofabf/abb .


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L108 Letter to the Editor

0 5 100

0.01

0.02

0.03

X 1/20

a bf/a bbln( )

hb

N

Eint/(

)

Figure 1. Bosonfermion interaction energy (in units ofNhb with N = 20 000 being the totalnumber of particles), as a function of ln(abf/abb). The curves are obtained for abb = 5.1 a0 and

equal numbers of bosons and fermions at various values of the temperature: T = 0 (solid curve),

T = 0.2TF (dashed curve) andT = 0.6TF (dotted-dashed curve,on a scale reduced by a factor20).

The arrow indicates the location of the Paris experiment.

For the value abf = 38 a0 corresponding to the current experiments (marked by the arrow

in figure 1), the bosonfermion interaction energy is close to its maximum value. We find that

the overlap between the two clouds vanishes ifabf > 200 a0, in accord with the estimate given

previously.

To make close contact with experiments we have calculated the column densities of the

fermionic and condensate vapours in the phase-separated regime for various values of abf.

At the threshold value we have found only one possible configuration, with a central hole

in the fermionic cloud which is occupied by the condensed bosons (see figure 2). For this

configuration there is only a quantitative difference in the value of the contrast at the centre of

the fermionic cloud as compared to the mixed state.

The regime of phase separation is univocally characterized by the presence of other

configurations at higher values ofabf. Starting from different initial conditions, for 1000 a0

z akdeniz et al- phase separation in a boson–fermion mixture of lithium atoms

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