Collisional oscillations of trapped boson-fermion mixtures approaching collapse

0.33 MB·English

by  P. Capuzzi

Collisional oscillations of trapped boson-fermion mixtures approaching collapse
P. Capuzzi, A. Minguzzi, and M. P. Tosi
NEST-INFM and Classe di Scienze, Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa, Italy
(Dated: February 2, 2008)
We study the collective modes of a confined gaseous cloud of bosons and fermions with mutual
attractiveinteractionsatzerotemperature. ThecloudconsistsofaBose-Einstein condensateanda
4 spin-polarizedFermigasinsideasphericalharmonictrapandthecouplingbetweenthetwospeciesis
0 variedbyincreasingeitherthemagnitudeoftheinterspeciess-wavescatteringlengthorthenumber
0 of bosons. The mode frequencies are obtained in the collisional regime by solving the equations of
2 generalizedhydrodynamicsandarecomparedwiththespectracalculatedinthecollisionless regime
within a random-phaseapproximation. We findthat, as the mixtureis driven towards the collapse
n
instability,thefrequenciesofthemodesoffermionicoriginshowablueshiftwhichcanbecomevery
a
significantforlargenumbersofbosons. Insteadthemodesofbosonicoriginshowasoftening,which
J
becomes most pronounced in the very proximity of collapse. Explicit illustrations of these trends
9
are given for the monopolar spectra, but similar trends are found for the dipolar and quadrupolar
1
spectra except for thesurface (n=0) modes which are essentially unaffected by theinteractions.

]
t PACSnumbers: 03.75.Kk,03.75.Ss,67.60.-g
f
o
s
. I. INTRODUCTION similarfrequencysofteningisfoundforapurecondensate
t
a withincreasinglylargebosonattractions[28,29]. Incon-
m trast, the numerical solution of the equations of motion
Boson-fermion mixtures have been produced with
- for boson-fermion mixtures in the collisionless Random-
atomic gases in several experiments [1, 2, 3, 4, 5, 6] and
d
Phase-Approximation(RPA)doesnotgiveindicationsof
n the deep quantum-degeneracy regime has been reached
frequencysoftening forthe choice ofparametersadopted
o for both the bosonic and the fermionic component [7, 8].
in the calculations [30]. Moreover,it is found that when
c Theaimoffermioncoolinginthesemixtureshasbeenthe
[ thenumberofbosonsismuchlargerthanthatoffermions
realization of a superfluid state [9, 10]. Even though the

somefamiliesofmodesarestronglyblue-shiftedasacon-
1 mixturesareverydilute,theinteractionsbetweenbosons
sequence of the increase of the particle densities in the
v and fermions play important roles and in particular the
cloud approaching collapse [31]. However, the RPA cal-
1 s-wave collisions between the two species are exploited
3 in the evaporative cooling process to obtain rethermal- culations could not be brought to the very proximity of
3 collapse because of very severe numerical difficulties.
ization. Mean-fieldinteractionsaffectthe expansionrate
1 In this paper we study the collective modes of a
of the cloud and its thermodynamic properties [5, 11].
0
4 Fora87Rb-40Kmixturetheboson-fermioninteractions trapped boson-fermion mixture with attractive interac-
tions using the equations of generalized hydrodynamics.
0 are largeandattractive [12], while in other systems they
Theseallowus to spanthe entirerangeofboson-fermion
/ could be tuned to become attractive by exploiting Fesh-
t interaction strength up to the collapse point. Although
a bach resonances [13, 14, 15]. Boson-fermion attractions
m thehydrodynamicequationsarestrictlyvalidonlyinthe
enhance the overlapbetween the two species, favoringin
collisionalregime,theirpredictionsalsoserveasa guide-
- principle a boson-mediated fermion pairing [16, 17], and
d line to interpret the RPA spectra. By evaluating the
ultimately leadtothe collapseofthe cloudasthe boson-
n frequencies of the first ten low-lying modes we find that
co fCeormllaiopnsechouasplbinegenstorbensegrtvhedoveexrcpoemrimesetnhtealFlyerimnitphree8s7suRrbe-. somemodesundergoablueshiftwhichcanbecomequite
: 40K mixture as a sudden disappearance of the fermion largeatlargenumbersofbosons,whilesomeothermodes
v display a softening which becomes pronounced in a very
Xi cloud when the number of bosons, and hence the cou- narrow window in the proximity to collapse.
pling between the two species, is increased [18]. The
r equilibrium properties and the phase diagram of a mix-
a
ture with attractive interactions have been investigated
by several authors [19, 20, 21, 22], finding good agree- II. EQUATIONS OF GENERALIZED
ment between the experimental data and a mean-field HYDRODYNAMICS
theoretical description [23, 24].
The study of the collective modes of a boson-fermion Weconsideramodelforadilutemixtureoftwospecies
mixture with repulsive interactions has proved to be a of alkali atoms inside harmonic traps, one species being
good indicator of the quantum transition to a spatially spin-polarizedfermionsofmassm andtheotherbosons
F
demixedstate[25]. Forthecaseofattractiveinteractions of mass m . The boson-boson and boson-fermion in-
B
some analyses using semi-analytical methods [26, 27] teractions are described by contact potentials involving
have suggested that the lowest-lying monopolar mode the coupling constants g =4π~2a /m and g =
BB BB B BF
should show a softening in the approach to collapse. A 2π~2a /m ,wherea anda arethecorresponding
BF r BB BF
2
s-wavescatteringlengthsandm =m m /(m +m ). the Fermi kinetic pressure and the boson-boson repul-
r B F F B
Thefermion-fermioninteractionsarenegligible,sincethe sions. Collapse is identified in the numerical search for
Pauli principle forbids collisions in the s-wave channel the equilibrium state as the point where it is no longer
between fermions of the same spin. In the following we possible to find a stable minimum for the energy func-
treat the case of boson-fermion attractions (a < 0) tional in Eq. (5). We have verified that the location of
BF
and boson-boson repulsions (a >0). collapse found in this way is well approximated by the
BB
Thedynamicsofthemixtureinthecollisionalregimeis estimate [19, 32]
describedbytheequationsofgeneralizedhydrodynamics
1/2
as in Ref. [25]. These are (i) the equations for particle a
ac = BB , (6)
conservation, | BF| (cid:18)αk (cid:19)
F
∂tρσ+∇·jσ =0, (1) where kF = (6π2ρF(0))1/3 is the Fermi wave number
estimated from the fermion density at the center of the
relating the particle density ρσ(r,t) to the current den- trap and α=[31/3/(2π)2/3](mF +mB)2/(4mFmB).
sity jσ(r,t) for each component (σ = B,F); and (ii) the In the following we characterize the approach of the
quantumNavier-Stokesequationsformomentumconser- mixture to the collapse instability by following the be-
vation, which are havior of its low-lying collective oscillation frequencies.
The normal modes of Eqs. (1) and (2) are found by ex-
mσ∂tjσ =ρσ(Fσ −∇Vσ−gBF ∇ρσ¯) (2) panding ρσ(r,t) around the equilibrium state ρ0σ(r) as
ρ (r,t) = ρ0(r)+δρ (r)eiωt, linearizing Eqs. (1) and
to linear order in the velocity fields. Here σ¯ denotes σ σ σ
the component different from σ, V (r) = m ω2r2/2 (2), and Fourier transforming with respect to the time
σ σ σ variable. This yields the coupled eigenvalue equations
are isotropic trapping potentials, and F (r,t) are self-
σ
cpornesssisiotennst internal forces. We take for the latter the ex- mσω2δρσ =∇·(ρ0σδFσ)−gBF ∇·(ρ0σ∇δρσ¯), (7)
where δF are the forces obtained as δF = F [ρ0 +
FB =−∇(cid:20)gBBρB− 2~m2B ∇√2√ρBρB(cid:21) (3) δ(fρoσr]t−heFirσσ[eρx0σp]rteossliionnesarseoerdReerf.in[2t5h])e.dWenesiatyσlsofludcettuσeartmiσoinnes
thesolutionsoftheeigenvalueequationsinthecasewhere
and the dynamical coupling between the two components is
neglected. Hereafter we shall call these solutions the un-
F = Aρ2/3 ~2 ∇2√ρF , (4) coupled boson and fermion modes and calculate them
F −∇(cid:20) F − 6mF √ρF (cid:21) by solving Eq. (7) for σ = B and F with the second
term in the rhs set to zero. This calculation will help
with A = ~2(6π2)2/3/2mF. The equations of general- us in attributing bosonic or fermionic character to the
ized hydrodynamics are thereby closed by means of a coupled modes of the mixture by comparison with the
local-density approximation on the kinetic stress ten- frequencies of the uncoupled modes. Even though the
sors, which also includes surface kinetic contributions bosonic and fermionic oscillations that are obtained in
throughthe lastterm onthe rhs ofEqs.(3)and (4) [25]. this way are dynamically uncoupled, the effect of the
These surface correction terms transcend the standard mutualboson-fermioninteractionisincludedintheircal-
Thomas-Fermiapproximationandforbosonsthepresent culation through the use of the density profiles of the
approachis in fact equivalent to the full time-dependent mixture at equilibrium.
Gross-Pitaevskiiequation. The numerical procedure that we have used can be
The equilibrium state of the mixture is given by the summarized as follows: (i) we find the equilibrium den-
stationarysolutions ofEq.(2). These also correspondto sityprofilesbyminimizingtheenergyfunctional(5)using
a minimum of the energy functional the steepest-descent method; (ii) we project Eq. (7) into
subspacesofdifferentangularmomentumlbyfactorizing
g
E[ρB,ρF] = Z d3r(cid:16)VBρB+ B2Bρ2B+ξB(cid:17) tYhe(drˆe)nsairtey flthuectsupahtieornicsaalshδaρrσm(orn)i=cs;δρalσn(dr)(Yiilmi)(rwˆ)ewshoelvree
lm
+ d3r V ρ + 3Aρ5/3+ξ the coupled equations in a given l-subspace by means of
Z (cid:18) F F 5 F F(cid:19) standard linear-algebra routines [33].
3
+g d rρ ρ , (5)
BF F B
Z
III. COLLECTIVE MODES
where the surface kinetic-energy terms are ξ =
B
~2 √ρB 2/2mB and ξF =~2 √ρF 2/6mF. We have solved Eq. (7) for two sets of experimentally
|A∇t incr|easingly large boson|∇-fermi|on attractions the relevant system parameters, corresponding to a 7Li-6Li
87 40
densities of the two species increase in their overlap re- mixture [4] and to a Rb- K mixture [18]. In both
gionandcollapseoccurswhentheseattractionsovercome cases we have set the frequency of the isotropic trap at
3
the geometric average of the frequencies in the experi- considerably as a consequence of the large difference in
mentalsetupandhaveadoptedtheexperimentalvalueof atomic masses. In the numerical calculations we take
theboson-bosonscatteringlength. Inourcalculationswe ω = 2π 90.9s−1 and ω = 2π 134s−1. Collapse is
B F
followtwodifferentroutestoreachcollapse. Wedrivethe approache×dbyvaryingN from4 ×103toapproximately
B
7Li-6Li mixture to collapse by varying the value of a 8 104, while keeping N =2 1×04 fixed.
BF F
× ×
from positive to strongly negative, as could be achieved In Fig. 2 we show the frequencies of the low-lying
bymeansofaFeshbachresonance. Inthe87Rb-40Kmix- monopolar modes for this mixture. At increasing val-
ture instead we keep a fixed and increase the number ues of the boson number we find that the frequencies of
BF
N of bosons,as is done in the experiments ofModugno the bosonic modes, identified as those that are closer to
B
et al. [18]. the uncoupled bosonic fluctuations, show a monotonic
decrease which becomes very pronounced very close to
collapse. On the other hand, the fermionic modes only
A. The 7Li-6Li mixture show a modest blue shift. This is not as large as in our
resultsforthe 7Li-6Limixture,sincethenumbersofpar-
ticles of the two species are here more similar.
Figure 1 shows the frequencies of the low-lying
monopole (l = 0) modes in the 7Li-6Li mixture as func-
tions of a , with the choice of parameters ω = ω =
BF F B
2π 1000s−1, a = 0.27nm, and particle numbers IV. SPECTRAL FUNCTIONS
BB
N ×=104 andN =106. Pronouncedblueshiftsareob-
F B
served at intermediate values of the boson-fermion scat- We have also calculated the whole spectral functions
tering length both in a set of eigenfrequencies of Eq. (7) undertheeffectoftheperturbingfieldsδUσ(r)eiωt,inor-
(circles)andinthefrequenciesoftheuncoupledfermionic der to make a comparison of the collisional spectra with
modes (dashed lines). This effect is a consequence of those of the mixtures in the RPA collisionless regime.
the increase of fermion density in the central part of the Peaks in the spectral functions indicate the location of
trap as the strength of the coupling is increased. For collective modes, while their height measures the prob-
ourchoiceofparametersthebosonicdensityprofileisal- ability of a transition from the equilibrium state to an
most unaffected in this range of boson-fermion coupling excited state with energy ~ω.
and essentially acts as aneffective attractive well for the Inthe collisionalregimethe spectralpeaksarelocated
fermions. The proximity of these modes to the uncou- in correspondence of the eigenfrequencies ωi of Eq. (7),
pled fermionic modes indicates that the effect of the dy- whilethestrengthofatransitionatafrequencyω isesti-
namical coupling is negligible, thus confirming the inter- matedfromthecorrespondingdensityfluctuationsδρiσ(r)
pretation in terms of static mean-field effects. At larger as
valuesof a the othereigenfrequenciesofEq.(7)(cir-
cles) show| BanF|increasing departure from the uncoupled Sσcoσl′l(ω)=−π1Im ω2Zσi′ω2 Z d3rδUσ∗(r)δρiσ′(r) (8)
bosonic modes (solid lines) and tend to rapidly soften as Xi i −
collapse is approached. We conclude that, in contrast
(see Appendix). Here the sum is performed over the
to the blue-shift of the fermionic modes, the softening of
whole spectrum of Eq. (7) and i is given by
“bosonic” modes is a truly dynamical signature of the Zσ
impending collapse.
Similar trends of the mode frequencies are also found i = 2A d3r(ρ0)−1/3δρi ρ0 δU (9)
ZF 3m Z F F ∇· F∇ F
for the dipole (l = 1) and quadrupole (l = 2) oscilla- F (cid:0) (cid:1)
tions, except that the lowest (n = 0) mode is in both and
cases essentially unaffected by the interactions. For the
g
dipole mode this behavioris anexactconsequenceofthe i = BB d3rδρi ρ0 δU . (10)
generalized Kohn theorem [34, 35], while for modes of ZB mB Z B∇·(cid:0) B∇ B(cid:1)
higher l it is strictly valid only in the Thomas-Fermi
Since we are using in Eq. (8) the equilibrium profiles
limit. This can be explicitly verified from Eq. (7) by
and the mode frequencies of the fully coupled dynamical
making the Ansatz δρ (r) rlY (rˆ) for the density
fluctuations and by expσloitin∝g thelmproperty 2δρ = 0 equations, we expect it to yield the spectral functions
∇ σ quite accurately.
for these modes.
More generally, the spectral functions are related
to the density-density responses χσσ′(r,r′,ω) by the
fluctuation-dissipation theorem,
B. The 87Rb-40K mixture
1
We evaluate the collective modes of a 87Rb-40K mix- Sσσ′(ω)=−πImZ d3rd3r′δUσ∗(r)χσσ′(r,r′,ω)δUσ′(r′).
ture with scattering lengths a = 5.5nm and a = (11)
BB BF
21.7nm. Although both components are inside the We evaluate the response functions in the collisionless
−
same magnetic trap, their trapping frequencies differ regimewithintheRPA,followingtheapproachdescribed
4
in Refs. [31, 36]. For a monopolar drive (δU (r) r2) V. SUMMARY AND CONCLUDING REMARKS
σ
wecalculateboththefermionicresponseSl=0(ω)an∝dthe
FF
bosonic response Sl=0(ω), corresponding to excitations
BB Insummary,wehaveusedtheequationsofgeneralized
inducedbyapplyingthedriveseparatelyonthefermions
hydrodynamics to study the collective modes of trapped
or on the bosons.
boson-fermion mixtures with mutual attractive interac-
tionsdrivingthemixturetowardsthecollapseinstability.
In Figs. 3 and 4 we report the spectral functions of
We havefocusedontwo specific systems ofexperimental
the7Li-6LimixtureasobtainedfromtheRPAformalism
interest,forwhichwehavechosentwodifferentroutesto
(top panels) and from Eq. (8) for the collisional regime
collapse.
(bottom panels) at various values of a and with the
BF In both cases we find that the collective spectra show
same choiceofparametersas inFig.1. Inthe absenceof
a frequency softening of a family of modes of bosonic
boson-fermion coupling a fermionic perturbing field ex-
nature asa signatureofthe incipientcollapse. This soft-
cites the pure n = 0 mode of the fermions, which gives
ening becomes most pronouncedin a very narrowregion
the only contribution to the FF spectra. In contrast,
of parameters near collapse. A second effect of increas-
the BB spectra contain in the same limit several weak
ing the attractive coupling between the two species is a
peaks at higher frequencies above the pure n = 0 mode
blueshiftofthemodesoffermionicorigin,whichbecomes
ofthebosons,owingtotheboson-bosoninteractions. For
moreevidentforlargenumbersofbosonsandreflectsthe
a =0theRPAspectraareverycomplex,fromthelift-
BF 6 compressionof the fermionic cloud from the interactions
ing of degeneracy in the discrete-level structure which is
with the bosons.
due to the interactions with the bosons. The fermionic
A comparison of the hydrodynamic spectra with the
peaks can be recognized as broad fragmented contribu-
spectra calculated in the collisionless regime within the
tionsoflargestheightinthefermionicspectralfunctions,
random-phaseapproximationsuggeststhatboththeblue
whilethebosonicpeaksareusuallyisolatedandaremost
shift of the “fermionic” modes and the softening of the
marked in the bosonic spectra [31].
“bosonic” modes are generalfeatures of the dynamics of
themixture. Bycomparingtheheightofthepeaksinthe
Quite interestingly,the spectrainthe tworegimesdis-
bosonicandfermionicspectralfunctionswealsoconclude
play similar gross features. At small values of a
BF
| | that the two families of modes can be most efficiently
the peaks are located near the frequencies of the higher-
excited by applying the driving fields separately on the
order monopolar modes, but at larger values they tend
species.
to spread over the whole range of frequency. This phe-
nomenon occurs for the same choice of parameters for Our analysis has been restricted to the linear regime,
which the blue shift of the collisional fermionic modes neglectingnonlinearandbeyond-mean-fieldeffectswhich
becomes pronounced and several mode crossings occur. may start to play a role close to collapse. We hope to
For a = 3 nm the first fermionic peak is strongly address these issues in future work.
BF
−
blue-shiftedwhilethelow-frequencypartofthespectrum
is dominated by bosonic modes having considerable os-
cillator strength in both bosonic and fermionic spectral Acknowledgments
functions. This indicates a strong dynamical coupling of
the density fluctuations of the two species in a regime of
This work was partially supported by INFM through
parameters where the uncoupled frequencies are close to
the PRA-Photonmatter Program.
the coupled solutions for the collisional fermionic modes
(see again Fig. 1). A careful analysis also shows that for
a = 3 nm the frequencies of the bosonic modes in
BF
− APPENDIX: SPECTRAL FUNCTIONS IN THE
theRPAspectraareslightlyred-shifted,showingthefirst
COLLISIONAL REGIME
indications of a trend towardssoftening onthe approach
to collapse.
The spectral functions of a mixture in the collisional
Finally, in Fig. 5 we compare the collisional results regime can be obtained by adding the perturbing fields
forthe monopolarspectralfunction(bottom panel)with δU (r)eiωttothetrappingpotentialV (r)inEqs.(1)and
σ σ
those of the RPA (top panel) for a 87Rb-40K mixture (2)andthenlinearizingtheequationsofmotioninterms
with the same choice of parameters as in Fig. 2. The of the induced density fluctuation δρ (r,ω). This leads
σ
RPA spectra show a fragmented fermionic contribution inFouriertransformwithrespecttotimetotheequation
around the frequency 2ω of the bare monopolar oscil-
F
lation as well as two main bosonic peaks at its sides.
The frequency of these two peaks tends to decrease with mσω2δρσ =∇·(ρ0σδFσ)−gBF ∇·(ρ0σ∇δρσ¯)−∇·(ρ0σ∇δUσ)
increasing N , similarly to what is found for the same (A.1)
B
modes in the collisional regime. This suggests that such for each component of the mixture. Here, at variance
frequency softening is a signature of the incipient col- from Eq. (7) ω is fixed by the frequency of the drive.
lapse. Equation (A.1) can be solved by expanding δρ (r,ω) in
σ
5
the basis of the eigenvectors δρi(r) of Eq. (7) as Here we take for the scalar product (A.4) that of a
σ
bosonic and of a fermionic cloud in the Thomas-Fermi
δρ (r,ω)= Ci(ω)δρi(r). (A.2) limit with vanishing mutual interactions. We thus set
σ Xi σ wFF = 2A(ρ0F)−1/3/3mF and wBB = gBB/mB and ne-
glecttheoff-diagonalweightfunctions. Thischoiceyields
In turn, the expansion coefficients Ci are obtained by
projecting Eq. (A.1) into the subspace generated by a
given fluctuation δρj(r). This procedure requires defin- i
ing a scalar productσbetween two eigenmodes of Eq. (7), Cσi(ω)= ω2Zσω2 (A.5)
which we shall denote by the braket δρi δρj . The pro- i −
h | i
jection yields
for fermions and bosons, where i is given in Eqs. (9)
Zσ
Ci(ω)= hδρi|∇ω·i2(cid:0)−ρ0σω∇2δUσ(cid:1)i (A.3) ahδnρTdih|δ(e1ρ0isi)p.=ecW1tr)ea.lhfauvnecttaioknesnaδrρeiσtahsennocramlcaulliazeteddtofroumnittyhe(
where ω is the frequency correspondingto the eigenvec- definition
i
tor δρi(r).
σ
A generic scalar product can be written as Sσσ′(ω)=−π1ImZ d3rδUσ∗(r)δρσ′(r,ω) (A.6)
hδρi|δρji=σX,σ′Z d3rδρiσ(r)wσσ′(r)δρjσ′(r) (A.4) where δρσ′ is evaluated by setting δUσ¯′ = 0. The use of
Eqs. (A.2) and (A.5) in Eq. (A.6) directly leads to Eq.
where wσσ′(r) are suitable weights to be determined. (8) in the main text.
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6
15 [36] P.CapuzziandE.S.Hern´andez,Phys.Rev.A64,043607
(2001).
10
B
ω
/
ω
5
0
−4 −3 −2 −1 0
a [nm]
BF
FIG.1: Monopolarfrequenciesω(inunitsofthebareboson-
trapfrequencyωB)ofthecollisionalcollectivemodesasfunc-
tions of the mutual scattering length aBF (in nm) for a 6Li-
7Li mixture with NF = 104 and NB = 106. The solid and
dashed lines show the frequencies of the dynamically uncou-
pled modes for bosons and fermions, respectively. The tri-
angles correspond to the values of aBF for which we have
performed the RPAcalculations of Figs. 3 and 4.
7
15
10

B
ω
/
ω
5
0
1 2 3 4 5 6 7
N × 104
B
FIG. 2: Monopolar frequencies ω (in units of ωB) of the
collisional collectivemodes asfunctionsof thenumberNB of
bosons (in log scale) for the 87Rb-40K mixture with NF =
2×104. The solid and dashed lines show the frequencies of
the dynamically uncoupled modes for bosons and fermions,
respectively. ThetrianglescorrespondtothevaluesofNB for
which we haveperformed theRPA calculations of Fig. 5.
8
a = 0.0 nm
BF
a = −1.0 nm
BF ↑ ↑ ↑ ↑ ↑ ↑
0F
=F
lS
↑↑ ↑ ↑ ↑
a = −2.0 nm
BF
a = −3.0 nm
BF
↑ ↑ ↑ ↑ ↑↑ ↑
a = 0.0 nm
BF
oll
c
0, F aBF=−1.0 nm↑↑ ↑↑ ↑↑ ↑↑ ↑ ↑ ↑ ↑
l=SF F B F B F B F B F B F B
a =−2.0 nm
BF
↑↑ ↑ ↑ ↑ ↑↑ ↑↑ ↑↑
(BF) B F B F B F B (BF)
a =−3.0 nm
BF
↑ ↑ ↑ ↑ ↑ ↑ ↑
B B B B B F B
0 2 4 6 8 10
ω/ω
B
FIG. 3: Monopolar spectrum (in log scale and arbitrary
units)forfermionsasfunctionsofω (inunitsofωB)ina6Li-
7Limixturewith NF =104,NB =106 andforvariousvalues
of aBF. The spectra are plotted for the sake of clarity with
a spectral width of order 10−5ωB. The vertical dotted lines
indicate the values of the bosonic and fermionic monopolar
modes in the absence of boson-fermion coupling. Top panel:
spectral response within the RPA; the arrows indicate the
frequencies of thecollisional modes with non-negligible spec-
tral weight for thesame parameters. Bottom panel: spectral
response in the collisional regime; the arrows denote all col-
lisional modes for the same parameters with label B or F to
indicate the type of mode as identified from its proximity to
an uncoupled collisional mode (the labels BF indicate am-
biguous cases).
9
a = 0.0 nm
BF
a = −1.0 nm
BF
↑↑ ↑↑ ↑↑ ↑ ↑ ↑
0B
=B
lS
a = −2.0 nm
BF ↑↑ ↑ ↑ ↑ ↑
aBF = −3.0 nm ↑ ↑ ↑ ↑ ↑↑ ↑
a = 0.0 nm
BF
a =−1.0 nm
l=0SBB BF F↑ ↑B F ↑↑B F ↑↑B F ↑↑B F↑ ↑B F ↑B ↑
a =−2.0 nm
BF
↑↑ ↑ ↑ ↑ ↑↑ ↑↑ ↑↑
(BF) B F B F B F B (BF)
a =−3.0 nm
BF
↑B ↑B ↑B ↑B ↑B ↑F ↑B
0 2 4 6 8 10
ω/ω
B
FIG. 4: The same as in Fig. 3 for thebosonic response.
ω/ω
1 F 2
N = 104 ↑ ↑ ↑
B B F B
N = 2×104
=0FF B ↑B ↑F ↑B
lS N = 4×104
B
↑ ↑ ↑
B F B
N = 5×104
B
↑ ↑ ↑
B F B
N = 104
B
↑ ↑ ↑
B F
B
N = 2×104
B
=0BB ↑B ↑F ↑
lS N = 4×104 B
B
↑B ↑ ↑B
F
N = 5×104
B
↑B ↑F ↑B
1 2 ω/ω 3 4
B
FIG. 5: Monopolar spectral response (in log scale and arbi-
traryunits)fromtheRPAforfermions(top)andbosons(bot-
tom) as functions of ω (in unitsof ωF or ωB) for a 87Rb-40K
mixturewithaBB =5.5nm,aBF =−21.7nm,NF =2×104,
and increasing values of NB. The arrows indicate the colli-
sional modes for the same parameters. The vertical dotted
lines are the values of the bosonic and fermionic monopolar
modes in theabsence of boson-fermion coupling.