Density and Stability in Ultracold Dilute Boson-Fermion Mixtures

0.32 MB·English

by  Steffen Roethel

Density and Stability in Ultracold Dilute Boson-Fermion Mixtures
Steffen Ro¨thel∗
Westf¨alische Wilhelms-Universit¨at Mu¨nster, Wilhelm-Klemm-Straße 9, 48149 Mu¨nster, Germany
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0 Axel Pelster†
2
Universit¨at Duisburg-Essen, Campus Duisburg, Fachbereich Physik, Lotharstraße 1, 47048 Duisburg, Germany
n
a (Dated: February 5, 2008)
J

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We analyze in detail recent experiments on ultracold dilute 87Rb–40K mixtures in Hamburg and
1

in Florence within a mean-field theory. To this end we determine how the stationary bosonic and
]
h
fermionicdensityprofilesinthismixturedependintheThomas-Fermilimitontherespectiveparticle
c
e
numbers. Furthermore, we investigate how the observed stability of the Bose-Fermi mixture with
m
respect to collapse is crucially related to thevalueof theinterspecies s-wavescattering length.
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t PACSnumbers: 03.75.Hh
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I. INTRODUCTION
n
o
c
[ Six years after the first experimentalachievementof Bose-Einsteincondensation(BEC) of trapped atomic gases in

2 1995 fermionic atomic gases were brought together with bosonic atoms to quantum degeneracy in a 7Li–6Li mixture
v
0 [1, 2], 23Na–6Li mixture [3], and 87Rb–40K mixture [4]. In contrast to a pure Bose system, quantum degeneracy in
2
2 a Fermi system with only one spin component means for T ≪ T = E /k that all energy states below the Fermi
F F B
3
0 energy E are occupied with one fermion each, whereas all states above E remain empty. The main problem to
F F
7
0 achievequantumdegeneracyinaFermigasistheinabilityoffermionstobe directlyevaporativelycooled. Duetothe
/
t
a Pauli exclusion principle, fermions in the same spin polarized hyperfine state are not allowed to be close together, so
m
that they can not collide via short-range contact interaction to rethermalize the gas during the evaporative cooling.
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d
This handicap was circumvented in the experiment of Ref. [5], where a mixture with two different spin states of 40K
n
o
was simultaneously evaporated by mutual cooling. In combination with a Bose gas the fermions are sympathetically
c
:
v cooled by elastic interactions with the bosons in the overlapping region [4, 6].
i
X
Beside the exploration of quantum degeneracy, one is also interested in studying how the two-particle interaction
r
a
influences the system properties. Mixtures with a strong interspecies interaction, with the prominent example of
4He–3He liquid [7, 8], lead to new phenomena like phase separation or BEC-induced interactions between fermions.
∗Electronicaddress: [email protected]
†Electronicaddress: [email protected]
2
Depending onthe natureoftheinterspeciesinteraction,arepulsionbetweenbosonsandfermionstendstoademixing
in order to minimize the overlapping region [9], whereas in the case of an attraction the mixture can collapse as
long as the particle numbers are sufficiently large [6, 10]. The possibility of superfluidity in a Fermi gas, especially
the predicted BEC-BCS crossover between BCS-type superfluidity of Cooper pairs of fermionic atoms and BEC of
molecules was recently probed [11, 12]. A Feshbach resonance was used to tune the interaction strength between
fermionicatomsoftwodifferentspinstates,characterizedby thes-wavescatteringlengtha,fromeffectivelyrepulsive
(a>0) to attractive (a<0). On the BEC side (a>0) of the magnetic field resonance, there exists a weakly bound
molecularstate. Fermionicatomsareboundintobosonicmoleculeswhichcancondenseatsufficientlowtemperatures.
On the BCS side (a < 0) of the resonance two fermions with different spin states form a loosely bounded Cooper
pair. The condensation of fermionic atom pairs was observed on both the BEC and the BCS side of the Feshbach
resonanceintheexperiment[11,12]. Furthermore,thesystempropertieswereobservedtovarysmoothlyintheBEC-
BCS crossover regime. An alternative and complementary access to Fermi superfluidity is expected from quantum
degenerate Bose-Fermi mixtures where an effective interaction between fermions is mediated by the bosons [13, 14],
similarly to phonons in a solid-state superconductor.
Another recentandfastgrowingfieldisthe investigationofultracoldboson-fermionmixturestrappedinanoptical
lattice which is created by standing waves of the electric field of counterpropagating laser beams. The atoms can be
confined to different lattice sites and, by varying the laser intensity, their tunneling to neighboring sites as well as
the strength of their on-site repulsive interactions can be controlled. In the case of a pure ultracold Bose-Einstein
condensatewithrepulsiveinteraction,heldinathree-dimensionalopticallatticepotential,aquantumphasetransition
fromasuperfluidtoaMottinsulatorphasewasobservedasthedepthofthelatticeisincreasedleadingtoasuppression
ofthetunnelingbetweenneighboringlatticesites[15]. ThepresenceoffermionicatomstogetherwiththeBose-Einstein
condensatemakesthesystemmorecomplexandricherinitsbehavioratlowtemperatures. Ithasbeenpredictedthat
novel quantum phases in the strong-coupling regime occur which involve the pairing of fermions with one or more
bosonsorbosonicholes,respectively,whentheboson-fermioninteractionisattractiveorrepulsive[16]. Dependingon
thephysicalparametersofthesystem,thesecompositefermionsmayappearasanormalFermiliquid,adensitywave,
a superfluid, or an insulator with fermionic domains. Instead of varying the lattice potential depth the transition
froma superfluid to a Mott insulator in bosonic87Rb can be shifted towardslargerlattice depth by adding fermionic
40K which interacts attractively with rubidium and therefore increases the effective lattice depth [17, 18].
Ultracold trapped boson-fermion mixtures were investigated with respect to a demixing of the components [19–
21] and to a collapse due to the interspecies attraction [20–24]. Furthermore, the time-dependent dynamics of the
collapse [25] and finite temperature effects on the stability in a boson-fermion mixture were also studied [26]. Our
theoretical investigation is based on the experiments with a 87Rb–40K boson-fermion mixture in a harmonic trap,
which were performed in Hamburg [6] and in Florence [4, 10]. After cooling down at a temperature below 1 µK, a
condensateof106 (2·105)87Rbatomscoexistingwith7.5·105 (3·104)40Katomswithaquantumdegeneracyofabout
3
Hamburg Experiment Florence Experiment
mass of 87Rb atom mB =14.43·10−26 kg
mass of 40K atom mF =6.636·10−26 kg
s-wave scattering length aBB =(5.238±0.002) nm
(bosons ↔ bosons)
s-wave scattering length aBF =−15.0 nm aBF =(−20.9±0.8) nm
(bosons ↔ fermions)
radial trap frequency (bosons) ωB,r =2π·257 Hz ωB,r =2π·215 Hz
axial trap frequency (bosons) ωB,z =2π·11.3 Hz ωB,z =2π·16.3 Hz
radialtrapfrequency(fermions) ωF,r =2π·379 Hz ωF,r =2π·317 Hz
axial trap frequency (fermions) ωF,z =2π·16.7 Hz ωF,z =2π·24.0 Hz
numberof bosons NB =106 NB =2·105
numberof fermions NF =7.5·105 NF =3·104
TABLE I:List of parameters of theexperiments with a87Rb–40K boson-fermion mixture. The valuesare taken from
theexperiments in Hamburg[6] and in Florence [4, 10, 27].
T/T =0.1 (T/T =0.3) was achievedin the Hamburg (Florence) experiment. The parametersof both experiments
F F
aresummarizedinTableI. Thedistinctvaluesforthe interspeciess-wavescatteringlengtha employedto describe
BF
eachexperimentareworthadetailedexplanationsincethisparameterisofgreatimportanceforthesystem,especially
for the stability of the mixture against collapsing. An overview of different values for a and their determination
BF
method along with a reference are shown in Table II. A comparison of the incompatible values for a shows the
BF
need of further investigation in this field.
Now we give an outline of the contents of the paper. At first, we develop in Section II a mean-field theory for
an ultracold dilute gaseous boson-fermion mixture within the functional integral approach to many-body theory.
By splitting the Bose fields into background fields and fluctuation fields, we derive an effective action of the Bose
subsystemwithoutintegratingoutthe fermionicdegreesoffreedom. Its extremizationatzerotemperature yieldstwo
coupled equations of motion, one for the condensate wave function and another one for the Green function of the
fermions. By evaluating the fermionic Green function for a stationary BEC within the semiclassical approximation,
we obtain how the time-independent Gross-Pitaevskii equation for the condensate wave function is modified in such
a mixture.
In Section III we apply the Thomas-Fermi approximation by neglecting the kinetic energy of the bosons and
obtain analgebraicGross-Pitaevskiiequation. With the help of its solutionwe determine the density profiles of both
components in a 87Rb–40K mixture, where the contact interaction is repulsive between the bosons and attractive
between both components.
Finally,weinvestigateinSectionIVthestabilityofthe Bose-Fermimixturewithrespecttocollapsebynumerically
4
aBF/aBohr Method of determination Reference (year)
measurementoftheelasticcrosssectionforcollisionsbetween
−261+−117509 41K and 87Rb in different temperature regimes and following [28] (2002)
mass scaling to thefermionic 40K isotope
measurement of the rethermalization time in the mixture in
−330+160 [4](2002)
−100
[4, 10] after a selectively heating of 87Rb
measurementofthedampingoftherelativeoscillationsof40K
−410+81 [10] (2002)
−91
and 87Rb in a magnetic trap
−395±15 mean-field analysis of thestability of themixturein [4, 10] [27] (2003)
magnetic Feshbach spectroscopy of an ultracold mixture of
−281±15 [29] (2004)
40K and 87Rb atoms
cross dimensional thermalrelaxation in amixtureof40K and
250±30 87Rb atoms after a increase of the radial confinement of the [30] (2004)
magnetic trap, here only |aBF/a0|
mean-field analysis of the stability, based on [24], of themix-
−284 [6](2006)
ture in [6]
extensivemagneticFeshbachspectroscopyofanultracoldmix-
−205±5 [31] (2006)
ture of 40K and 87Rb atoms
TABLE II: List of several published values of the s-wave scattering length between 87Rb and 40K including their
determination method and their reference.
evaluating the effective action for a trial Gaussian density profile of the condensate. We compare our results, which
stronglydependonthevalueoftheBose-Fermis-wavescatteringlength,withtheexperimentson87Rb–40Kmixtures
in Hamburg and Florence. The good agreement of our theoretical results with the measurement in the Florence
experiment allows us to fit a which describes the Hamburg data quite well, but remains incompatible with the
BF
value used in Florence.
II. DERIVATION OF GROSS-PITAEVSKII EQUATION
In this section we summarize briefly the mean-field theory of boson-fermion mixtures [14, 24] and derive within
the functional integral formalism a coupled set of differential equations for the condensate wave function and the
fermionic Green function.
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A. Grand-Canonical Partition Function
We consider a dilute gaseous mixture of ultracold bosonic and fermionic atoms. In order to obtain quantum
statistical quantities for such a Bose-Fermi mixture, we use the grand-canonical partition function in the functional
integral formalism. Thus, we integrate over all possible Bose fields ψ∗(x,τ), ψ (x,τ) and Fermi fields ψ∗(x,τ),
B B F
ψ (x,τ), which are weighted by a Boltzmann factor [14, 32, 33]:
F
Z = DψB∗ DψB DψF∗ DψF e−A[ψB∗,ψB,ψF∗,ψF]/~. (1)
I I I I
The complex fields ψ∗(x,τ), ψ (x,τ) represent the bosons and are periodic on the imaginary time interval [0,~β],
B B
whereas the fermions are described by Grassmann fields ψ∗(x,τ), ψ (x,τ) which are antiperiodic on this interval:
F F
ψ∗(x,~β)=ǫψ∗(x,0), ψ(x,~β)=ǫψ(x,0). (2)
Here ǫ=±1 holds for bosons and fermions, respectively. The total euclidean action of a Bose-Fermimixture consists
of three parts:
A[ψ∗,ψ ,ψ∗,ψ ]=A [ψ∗,ψ ]+A [ψ∗,ψ ]+A [ψ∗,ψ ,ψ∗,ψ ]. (3)
B B F F B B B F F F BF B B F F
The first term describes the bosonic component of the mixture:
~β ∂ ~2 g
A [ψ∗,ψ ] = dτ d3xψ∗(x,τ) ~ − ∆+V (x)−µ + BB |ψ (x,τ)|2 ψ (x,τ). (4)
B B B B ∂τ 2m B B 2 B B
Z0 Z (cid:20) B (cid:21)
It contains the Legendre transform, the kinetic energy, the external trap potential V (x), the chemical potential µ
B B
to fix the boson number, and the strength g = 4π~2a /m of the contact interaction between two bosons with
BB BB B
the s-wave scattering length a . As we deal with a dilute gas, an interaction between more than two particles
BB
is negligible, and we can restrict ourselves to the short-range contact interaction. Since the Pauli principle forbids
fermionsinthe samehyperfine stateto be closetogetherandthereforeto collideviacontactinteraction,wecanwrite
the corresponding action term for the fermionic component of the mixture as
~β ∂ ~2
A [ψ∗,ψ ]= dτ d3xψ∗(x,τ) ~ − ∆+V (x)−µ ψ (x,τ), (5)
F F F F ∂τ 2m F F F
Z0 Z (cid:20) F (cid:21)
where V (x) and µ denote the external trap potential and the chemical potential for fermions. As we assume a
F F
situation, where the bosonic and fermionic atoms cannot be transformedinto eachother, eachof both species has its
own chemical potential. The last term of the euclidean action (3)
~β
A [ψ∗,ψ ,ψ∗,ψ ]=g dτ d3x|ψ (x,τ)|2|ψ (x,τ)|2 (6)
BF B B F F BF B F
Z0 Z
describes the contact interaction between bosons and fermions, where its strength g is related to the s-wave
BF
scattering lengths a via [34, 35]
BF
m +m
g =2π~2a B F. (7)
BF BF
m m
B F
6
B. Background Method
In order to account for the fact that the bosons in the mixture can condense, we apply the background method of
field theory [33, 36, 37] and split the bosonic fields ψ∗(x,τ), ψ (x,τ) into two parts:
B B
ψ∗(x,τ)=Ψ∗(x,τ)+δψ∗(x,τ), ψ (x,τ)=Ψ (x,τ)+δψ (x,τ). (8)
B B B B B B
ThefirstpartrepresentsthebackgroundfieldsΨ∗(x,τ), Ψ (x,τ)whoseabsolutesquareisidentifiedwiththedensity
B B
of the condensed bosons. The second part consists of the fluctuation fields δψ∗(x,τ), δψ (x,τ) of the Bose gas
B B
describing the excited bosons, which are not in the ground state. Using the decomposition (8), we expand the
euclidean action (3) in a functional Taylor series with respect to the Bose fields ψ∗(x,τ), ψ (x,τ) around the
B B
backgroundfieldsΨ∗(x,τ),Ψ (x,τ). InthisworkwerestrictourselvestotheGross-Pitaevskiitheory,i.e.weconsider
B B
the euclidean action A only up to the zeroth order in the fluctuation fields δψ∗(x,τ), δψ (x,τ) which is equivalent
B B
to evaluate the euclideanaction (3) at the backgroundfields: A[Ψ∗ +δψ∗,Ψ +δψ ,ψ∗,ψ ]≈A[Ψ∗,Ψ ,ψ∗,ψ ].
B B B B F F B B F F
Thus, the bosonic functional integration in Eq. (1), whose integration measure transforms according Dψ∗(x,τ) =
B
Dδψ∗(x,τ)andDψ (x,τ)=Dδψ (x,τ),canbedropped. InthiswayweobtainfortheeffectiveactionΓ[Ψ∗,Ψ ]=
B B B B B
−lnZ[Ψ∗,Ψ ]/β the result
B B
1 1
Γ[Ψ∗,Ψ ]= A [Ψ∗,Ψ ]− lnZ [Ψ∗,Ψ ], (9)
B B ~β B B B β F B B
where
ZF[Ψ∗B,ΨB]= DψF∗ DψF e−AF,eff[Ψ∗B,ΨB,ψF∗,ψF]/~ (10)
I I
represents the functional integral over the Fermi fields resulting in a pure functional of the Bose background fields
Ψ∗(x,τ), Ψ (x,τ). The effective euclidean action A [Ψ∗,Ψ ,ψ∗,ψ ] depending on the Fermi fields ψ∗(x,τ),
B B F,eff B B F F F
ψ (x,τ) is summarized by
F
~β ∂
A [Ψ∗,Ψ ,ψ∗,ψ ]= dτ d3xψ∗(x,τ) ~ +Hˆ (x,τ)−µ ψ (x,τ), (11)
F,eff B B F F F ∂τ F,eff F F
Z0 Z (cid:20) (cid:21)
where the effective one-particle Hamilton operator for fermions reads:
~2
Hˆ (x,τ)=− ∆+V (x)+g |Ψ (x,τ)|2. (12)
F,eff F BF B
2m
F
C. Coupled Equations of Motion
Thegrand-canonicalfreeenergyF istheextremumoftheeffectiveactionΓ[Ψ∗,Ψ ]withrespecttothebackground
B B
fields Ψ∗(x,τ), Ψ(x,τ). Using Eqs. (4) and (9)–(12), this yields the time-dependent Gross-Pitaevskiiequation:
∂ ~2
~ − ∆+V (x)−µ +g |Ψ (x,τ)|2−g G (x,τ;x,τ) Ψ (x,τ)=0, (13)
B B BB B BF F B
∂τ 2m
(cid:20) B (cid:21)
7
where the fermionic two-point function is defined as follows:
GF(x,τ;x′,τ′)≡ Z [Ψ1∗,Ψ ] DψF∗ DψF ψF(x,τ)ψF∗(x′,τ′)e−AF,eff[Ψ∗B,ΨB,ψF∗,ψF]/~. (14)
F B B I I
This imaginary time-dependent nonlinear Schr¨odingerequation(13) of the condensate wavefunction Ψ (x,τ) repre-
B
sentsapartialdifferentialequation,wherethenonlineartermsareduetobothinteractions. Thelasttermresultsfrom
the boson-fermioninteraction, whereas the other terms have the conventionalGross-Pitaevskiiform for a condensate
[38, 39]. The fermionic two-point function is at the same time a Green function for the fermion fields, thus, it obeys
the linear inhomogeneous Schr¨odinger equation for fermions
∂
~ +Hˆ (x,τ)−µ G (x,τ;x′,τ′)=~δ(x−x′)δ(a)(τ −τ′), (15)
F,eff F F
∂τ
(cid:20) (cid:21)
where the inhomogeneity consists of the delta function in space and the antiperiodic repetitive delta function in
imaginary time
∞
δ(a)(τ −τ′)= (−1)nδ(τ −τ′+n~β). (16)
n=−∞
X
TheGross-Pitaevskiiequation(13)forthecondensateandthe inhomogeneousSchr¨odingerequation(15)forfermions
form a set of coupled equations in imaginary time. Therein the condensate wave function Ψ (x,τ) in the Gross-
B
Pitaevskii equation (13) is modified by the fermionic Green function G (x,τ;x,τ) and, vice versa, the condensate
F
wave function Ψ (x,τ) influences the fermionic Green function G (x,τ;x′,τ′) in the inhomogeneous Schr¨odinger
B F
equation (15). A Wick rotation τ =it and the omission of the chemical potentials in Eqs. (13) and (15) leads to two
coupled equations of motion in real time, which describe the dynamics in the Bose-Fermi mixture.
D. Fermionic Green Function
In the following we obtain an approximative solution for the fermionic Green function G (x,τ;x′,τ′). To this
F
end we restrict ourselves to a stationary BEC so that the imaginary time-dependent Gross-Pitaevskii equation (13)
reduces to
~2
− ∆+V (x)−µ +g |Ψ (x)|2−g G (x,τ;x,τ) Ψ (x)=0, (17)
B B BB B BF F B
2m
(cid:20) B (cid:21)
and the effective fermionic Hamiltonian (12) no longer depends explicitly on the imaginary time τ:
~2
Hˆ (x)=− ∆+V (x)+g |Ψ (x)|2. (18)
F,eff F BF B
2m
F
Now we assume that both the fermionic trap potential V (x) and the condensate wave function Ψ (x) have only a
F B
weak spatial dependence so that the semiclassical approximation can be applied. Therein the eigenvalue problem
HˆF,eff(x)ψk(x)=E(k,x)ψk(x) (19)
8
is approximately solved by plane waves
eik·x
ψk(x)= (20)
(2π)3/2
and the semiclassical energy spectrum
~2k2
E(k,x)= +V (x)+g |Ψ (x)|2. (21)
F BF B
2m
F
Thus, the semiclassical solution of Eq. (15) yields for the fermionic Green function
G (x,τ;x′,τ′)= d3k eik·(x−x′)
F (2π)3
Z
Θ(τ −τ′)e−[E(k,(x+x′)/2)−µF](τ−τ′−~β/2)/~−Θ(τ′−τ)e−[E(k,(x+x′)/2)−µF](τ−τ′+~β/2)/~
× . (22)
2coshβ[E(k,(x+x′)/2)−µ ]/2
F
The limit of equal imaginary times follows from τ′ ↓τ:
d3k eik·(x−x′)
G (x,τ;x′,τ)=− . (23)
F (2π)3eβ[E(k,(x+x′)/2)−µF]+1
Z
Thus, the fermionic particle density n (x)=−G (x,τ;x,τ) is given by
F F
d3k 1
n (x)= (24)
F (2π)3 eβ[~2k2/2mF−µ˜F(x)]+1
Z
with the local chemical potential
µ˜ (x)=µ −V (x)−g |Ψ (x)|2, (25)
F F F BF B
whichrepresentsthekineticenergyofthefermioninthehighestenergeticstate,whenitislocatedatthespace point x.
In order to evaluate the integral (24) in momentum space, we apply spherical coordinates and the substitution
ε(k)=~2k2/2m :
F
1 m 3/2 ∞ dεε1/2
n (x)= F . (26)
F Γ(3/2) (cid:16)2π~2(cid:17) Z0 eβ[ε−µ˜F(x)]+1
In the low-temperature limit T ↓0, the Sommerfeld expansion [40] of Eq. (26) yields in the lowest order
n (x)=κΘ(µ˜ (x)) µ˜3/2(x) (27)
F F F
with the abbreviation κ = (2m )3/2/6π2~3. Finally, we insert the result for the fermionic particle density (27) into
F
the Gross-Pitaevskiiequation (17):
~2
− ∆+V (x)−µ +g |Ψ (x)|2+κg Θ(µ˜ (x)) µ˜3/2(x) Ψ (x)=0. (28)
2m B B BB B BF F F B
(cid:20) B (cid:21)
Note that this stationary Gross-Pitaevskiiequation (28) follows from extremizing the effective action
~2 g 2
Γ[Ψ∗,Ψ ]= d3x Ψ∗(x) − ∆+V (x)−µ + BB |Ψ (x)|2 Ψ (x)− κΘ(µ˜ (x))µ˜5/2(x) (29)
B B B 2m B B 2 B B 5 F F
Z (cid:26) (cid:20) B (cid:21) (cid:27)
9
with respect to Ψ∗(x). This effective action can also be obtained from Eqs. (4) and (9)–(12) by evaluating the
B
fermionic functional integral (10) in the semiclassical approximation for a stationary BEC and by performing the
Sommerfeld expansion in the low-temperature limit. Identifying the extremum of the effective action Γ with the
grand-canonical free energy F, the number of bosons and fermions are obtained from (29) via the normalization
condition
∂F
N =− = dDxn (x), j =B,F, (30)
j j
∂µ
j
Z
where the particle densities of bosons and fermions read
n (x) = |Ψ (x)|2, (31)
B B
n (x) = κΘ(µ˜ (x)) µ˜3/2(x). (32)
F F F
III. DENSITY PROFILES
In this section we solve the stationaryGross-Pitaevskiiequation(28) in the Thomas-Fermiapproximationin order
to calculate the boson and the fermion density distribution for the parameters of the experiment in Hamburg [6].
There the 87Rb–40K boson-fermion mixture is confined in a three-dimensional rotationally symmetric harmonic trap
m
V (x)= i ω2 r2+ω2 z2 , i=B,F, (33)
i 2 i,r i,z
(cid:0) (cid:1)
where we use cylindricalcoordinates {r,φ,z}. The frequencies for both species are related by ω = m /m ω
F,k B F B,k
for k =r,z, so that m ω2 /2=m ω2 /2. p
B B,k F F,k
A. Thomas-Fermi approximation
Assumingthatthepotentialandinteractionenergyarelargerthanthekineticenergy,wecanusetheThomas-Fermi
approximation, where the kinetic term in the Gross-Pitaevskii equation can be neglected. With this approximation
the Gross-Pitaevskiiequation (28) reduces with the help of Eqs. (31) and (32) to an algebraic equation with respect
to the bosonic particle density n (x) [41]:
B
V (x)−µ +g n (x)+g n (x)=0. (34)
B B BB B BF F
The last term in Eq. (34) stands for the fermionic particle density
n (x)=κΘ(µ −V (x)−g n (x))[µ −V (x)−g n (x)]3/2, (35)
F F F BF B F F BF B
which modulates the bosonic density profile n (x) and vice versa.
B
10
B. Vanishing Boson-Fermion Interaction
Nowwediscussthe specialcaseofvanishingboson-fermioninteractiong →0,wherewemarkallquantitieswith
BF
the upper index (0). Onthe one hand, Eq.(34) reduces then to the well-knownparticle density ofa pure BEC in the
Thomas-Fermi approximation
1
n(0)(x)= Θ µ(0)−V (x) µ(0)−V (x) , (36)
B g B B B B
BB
(cid:16) (cid:17)h i
providedthatg >0. IncaseofanattractiveinteractingBECwithnegativeg ,thethirdterminEq.(34)cannot
BB BB
be balanced in the trap center at x=0 by the remaining chemical potential µ(0) leading to a collapse of the BEC.
B
The particle density (35) of fermions, on the other hand, becomes independent of that of bosons:
3/2
n(0)(x)=κΘ µ(0)−V (x) µ(0)−V (x) . (37)
F F F F F
(cid:16) (cid:17)h i
Thus, the clouds of the BEC and the Fermi gas coexist undisturbed from each other. Setting the particle densities
(36) and (37) zero yields the Thomas-Fermi-radiiof the BEC and the Fermi cloud along the respective axes:
2µ(0)
R(0) = i , i=B,F. (38)
i,k vm ω2
u i i,k
u
t
Thechemicalpotentialsµ(0) andµ(0) aredeterminedbythe particlenumbersN andN viathe normalization(30):
B F B F
15a N 2/5 ~ω˜
µ(0) = BB B B, µ(0) =(6N )1/3~ω˜ , (39)
B L˜ 2 F F F
(cid:18) B (cid:19)
where L˜ = ~/m ω˜ denotes the geometrical average of the oscillator lengths L = ~/m ω . Inserting the
i i i i,k i i,k
chemical potepntials (39) into the respective particle densities (36) and (37), the maximumpof the latter in the trap
center results in
(15N )2/5 2N1/2
n(0)(0)= B , n(0)(0)= F . (40)
B 8πa3/5L˜12/5 F 31/2π2L˜3
BB B F
IntheThomas-FermiapproximationtheBECneedsanon-vanishingrepulsionbetweenthebosonsinordertoprevent
the density from becoming infinite or complex, which indicates a collapse of the BEC,whereas the quantum pressure
inthenoninteractingFermigaspreservesthelatterfromacollapse. Themaximumfermionicparticledensityincreases
faster with the particle number than the maximum bosonic particle density, whereas the latter depends not only on
the oscillator length but also on a , the parameter for the bosonic interaction strength. The particle densities of
BB
both species in an undisturbed BEC and Fermi gas are plotted for typical particle numbers N and N as thin lines
B F
in Figure 1.
C. Non-Vanishing Boson-Fermion Interaction
After having treated the case g = 0 in the previous section, we now discuss the algebraic equation (34) for a
BF
BEC which interacts with fermions. In order to plot the corresponding bosonic and fermionic densities in Figure 1,