Dark-Bright Solitons in a Superfluid Bose-Fermi Mixture

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by  Marek Tylutki

Dark-Bright Solitons in a Superfluid Bose-Fermi Mixture
Marek Tylutki,1,∗ Alessio Recati,1,2 Franco Dalfovo,1 and Sandro Stringari1
1INO-CNR BEC Center and Dipartimento di Fisica,
Universita` di Trento, Via Sommarive 14, I-38123 Povo, Italy
2Technische Universita¨t Mu¨nchen, James-Franck-Straße 1, 85748 Garching, Germany
(Dated: May 12, 2016)
The recent experimental realization of Bose-Fermi superfluid mixtures of dilute ultracold atomic
gases has opened new perspectives in the study of quantum many-body systems. Depending on
6 the values of the scattering lengths and the amount of bosons and fermions, a uniform Bose-Fermi
1 mixtureispredictedtoexhibitafullymixedphase,afullyseparatedphaseor,inaddition,apurely
0 fermionicphasecoexistingwithamixedphase. Theoccurrenceofthisintermediateconfigurationhas
2 interesting consequences when the system is nonuniform. In this work we theoretically investigate

y the case of solitonic solutions of coupled Bogoliubov-de Gennes and Gross-Pitaevskii equations
a for the fermionic and bosonic components, respectively. We show that, in the partially separated
M phase, a dark soliton in Fermi superfluid is accompanied by a broad bosonic component in the
soliton, forming a dark-bright soliton which keeps full spatial coherence.

1
1 PACSnumbers: 03.75.Ss,03.75.Hh,03.75.Lm,67.60.Fp,64.75.-g

]
s I. INTRODUCTION which reveals the crucial effects of nonlinearity caused
a
by interactions. The opposite limit of bright solitons in
g
- Alongstandingprobleminthecontextofquantumflu- a mixture with fermions as the minority component was
t discussed in [16].
n ids is the description of mixtures composed of two kinds
a
ofinteractingsuperfluidsbelongingtodifferentstatistics. In pure Fermi superfluids of dilute atomic gases, theo-
u
The first theoretical analysis of superfluid 4He mixed retical predictions of the structure and dynamics of dark
q
. with superfluid 3He dates back to the 70’s (see, for ex- solitons[17–24]haverecentlystimulatedexperimentalin-
t
a ample, [1, 2] and references therein). In experiments, vestigations [25]. The experiments confirm the theoreti-
m however, the simultaneous superfluidity of the two com- cal expectation that dark solitons in a three-dimensional
- ponents of liquid 3He-4He mixtures has never been re- fermionic superfluid quickly decay into vortical excita-
d alized, since the miscibility of 3He in 4He is very small tions due to snaking instability [26], as it was earlier ob-
n (a few percent) and the temperature needed to reach su- served with bosons [27, 28]. With bosons very long-lived
o
perfluidity of fermions in the mixture is too low to be dark solitons have been generated by filling the soliton
c
[ reached with available cryogenic techniques. Dilute ul- withatomsinanotherhyperfinestate[29–31],thuscreat-
tracold atomic gases are instead excellent candidates for ing a dark-bright solitonic structure of a two-component
2
studyingsuperfluidpropertiesofmixtures. Superfluidity Bose-Bose superfluid [32]. Here we theoretically inves-
v
has been recently obtained experimentally in a mixture tigate the analogue dark-bright soliton in a Bose-Fermi
1
ofaBosecondensedgasandasuperfluidFermigasoftwo superfluid mixture, the main difference between the two
7
4 lithium isotopes, 6Li and 7Li [3, 4], where a new mech- cases being that the Bose-Fermi phase diagram is known
1 anism for superfluid instability was observed, related to to admit, in addition to a fully mixed phase and a fully
0 thedynamicalinstabilityofthesupercurrentcounterflow separated phase, also a third phase consisting of pure
1. ratherthantothemorestandardLandaucriterion[5,6]. fermions in equilibrium with a mixture of fermions and
0 Inaddition, inultracoldatomicgasesthestrengthofthe bosons. The stability conditions of such an intermediate
6 interspeciesandintraspeciesinteractioncanbevariedby phase in a uniform system were studied in [7] by using
1 meansofanexternalmagneticfield,thankstotheoccur- the equation of state of an ideal Fermi gas weakly inter-
v: rence of Feshbach resonances. One can thus foresee the acting with a dilute Bose gas. Such a phase is predicted
i exploration of the whole phase diagram of the mixture, to occur also in the strongly interacting regime [11]. A
X
which is expected to be very rich [7–15]. more refined equation of state, including the interaction
ar In the present work, we focus on the case where the among fermions, was later applied also to nonuniform
bosonic superfluid is the minority component, while the configurations by treating the interaction energy in local
fermionic superfluid exhibits a dark soliton. We choose density approximation [8]. However, since a dark soli-
this case as a paradigmatic configuration in which the ton is localized on the length scale of the healing length
interplay between miscibility and immiscibility, together of the superfluid, which is of the order of the inverse
with superfluidity, gives rise to a peculiar behaviour Fermi wave vector for fermions at unitarity, its charac-
terizationrequiresatheorywhichproperlyincludesnon-
localeffects,beyondthelocaldensityapproximation. For
this purpose, we use coupled Bogoliubov-de Gennes and
∗Electronicaddress: [email protected] Gross-Pitaevskii equations for the fermionic and bosonic
2
components respectively. a function of the densities of the two components and
Ourpaperisorganizedasfollows: in IIwediscussthe the existence of three phases was predicted for a posi-
§
stability condition of the uniform phase of a Bose-Fermi tive Bose-Fermi scattering length: i) a uniform mixture,
mixturewhentheFermigasisatunitarity(infinitescat- where both components occupy the entire space at con-
teringlength);in IIIwewritethemean-fieldequations, stantdensities;ii)apartiallyseparatedphase,wherepart
§
which are subsequently used to find the stationary soli- of the space is occupied by pure fermions and part by a
tonic configurations of the system; in IV we analyse Bose-Fermi mixture; iii) a fully separated phase, where
§
the behaviour of solitons of a Bose-Fermi mixture when bosonsandfermionsarecompletelyseparated. Thesame
fermionsareatunitarity. Wepayspecialattentiontothe happens at unitarity, with the only difference that the
transition from the miscible state to the so called par- FermienergyisrenormalizedbytheuniversalBertschpa-
tially separated phase and to the fully separated state. rameterη. Itisworthnoticingthattheexistenceofthree
Wefindthat,inthepartiallyseparatedphase,thedensity phases is peculiar of the Bose-Fermi mixture. In fact a
depletion of fermions possesses a solitonic character and Bose-Bosemixtureonlyadmitsthemixeduniformphase
thephasecoherencebetweentheleftandrighthandsides and the fully separated phase, because of the different
is maintained; conversely, in the fully separated phase, power-lawdependenceonthedensitiesintheequationof
the depletion in the Fermi density is completely filled state.
by bosons (the density of the fermions vanishes) and the
The stability condition predicted by the energy den-
phase coherence between the two sides is lost. Finally
sity (1) for the uniform mixture is
in V we investigate the behaviour of solitons along the
§
crossover from the Bardeen-Cooper-Schrieffer phase to
Bose-Einstein condensation (BCS-BEC crossover), and n1/3 2(6π2)2/3η ¯h2 gbb . (2)
we conclude that, while on the BCS side the system ex- f ≤ 3 2m g2
f bf
hibits a behaviour similar to that at unitarity, on the
BEC side the partially separated configuration disap-
Forn largerthanthiscriticalvaluetheuniformmixture
pears as expected for a Bose-Bose mixture. f
is unstable and the system exhibits either partial or full
phase separation.
II. PHASE SEPARATION
Let us first discuss the conditions for the stability of
thehomogeneousphaseofthemixture,whichisacrucial III. MEAN-FIELD EQUATIONS
point in order to understand the numerical results of the
following sections. The phase diagram of a weakly in-
Inordertodescribebothuniformandnonuniformcon-
teracting Bose-Fermi mixture, at zero temperature, can
figurations of the mixture, including solitons, we use the
be derived starting from the following expression for the
Bogoliubov-deGennes(BdG)equations[35]fortheinter-
energy density:
actingsuperfluidfermionsandtheGross-Pitaevskii(GP)
1 3 equation [37] for the Bose-condensed bosons, which are
E[nf,nb]= 2gbbn2b +gbfnbnf + 5ηEFnf . (1) coupled via the interspecies interaction term, fixed by
g . Wealsouseasimplegeometryconsistingofathree-
bf
Here nb and nf are the densities of the Bose and Fermi dimensional rectangular box with hard walls in the lon-
gas respectively, and EF = h¯2kF2/2mf is the Fermi en- gitudinal direction z (box size L) and periodic boundary
ergy, with k = (3π2n )1/3. In the above expression conditions in the transverse directions x and y (box size
F f
we assume that the Fermi gas is at unitarity (infinite L ). The system is assumed to be uniform in the trans-
⊥
scatteringlength)andη isthedimensionlessBertschpa- verse directions and all spatial variations are along z.
rameter [33–35], which simply rescales the energy den- The Fermi gas is described in terms of a set of
sity of the Fermi gas with respect to the ideal gas ex- quasiparticle amplitudes u and v , solutions of
j j
pression. The quantities gbb and gbf are the bosonic in- the BdG equations, which{ in}our ca{se}take the form:
tsinrtaagsnpltesen,cgiwetshhsaicnhadccatorheredriBneolgastete-odFegtrbomb it=hinet4ecπro¯hsrp2raeecsbbpie/osmncdboiunapgnldisncggabtcftoen=r-- (ug√ej,nkke⊥Fr(i/crL)q⊥u=)aenikt(⊥u√·mrk⊥Fvn/ju,Lkm⊥⊥b()ezeri)k,a⊥ws·sri⊥othcuijar,tke=⊥d(z(tr)o⊥at,hnzed) alovnnjd,gki⊥jtu(irds)itnh=ael
4π¯h2abf(mb+mf)/(2mbmf), where mf and mb are the degreesoffreedom. Withthischoice, theBdGequations
masses of fermions and bosons. We assume that the become
Bose-Fermi scattering length does not depend on the in-
ternal state of the Fermi atoms, as in the case of the
H ∆ u u
r(1ec)ewnatsefixrpsetruimseednitns[w7]itahndlith[3i6u]mtoadteosmcrsib[3e,t4h]e.pEhqauseatdiioan- (cid:20)∆∗ −H(cid:21)(cid:20)vjj,,kk⊥⊥(cid:21)=εj,k⊥(cid:20)vjj,,kk⊥⊥(cid:21) (3a)
gramofadiluteBosegasinteractingwithanidealFermi
gas (η = 1). In [7] the phase diagram was explored as whereH =h¯2(k2 ∂2)/2m µ +g ψ 2isaneffective
⊥− z f− f bf| b|
3
n n
0 0 ∆
0
y n g =0.51E ,
t 0 bf F
nsin0/2 n0gbb=1.69EF n0/2 0 ∆
e
d
∆
0 0 − 0
n n
0 0 ∆
0
y n g =1.01E ,
t 0 bf F
nsin0/2 n0gbb=1.69EF n0/2 0 ∆
e
d
∆
0 0 − 0
n n
0 0 ∆
0
y n g =1.69E ,
t 0 bf F
nsin0/2 n0gbb=1.69EF n0/2 0 ∆
e
d
∆
0 0 − 0
-35 -25 -15 -5 5 15 25 35 -35 -25 -15 -5 5 15 25 35 -35 -25 -15 -5 5 15 25 35
zk zk zk
F F F
FIG. 1: (color online) Left column: Density profiles of the ground state of fermions (blue line) and bosons (red line), with
fermionsatunitarity(1/k a =0),forthreesetsofinteractionparameterssuchthatn g2 /(g E )=0.15,0.6and1.69,from
F ff 0 bf bb F
top to bottom. The number of bosons is 10% of the number of fermions. The three panels represent examples of fully mixed,
partially mixed and fully separated phases. Central columns: Same as before, but for stationary states with a dark soliton in
theFermicomponent. Inthetoppanelthedarksolitonisslightlymodifiedbybosons;inthecentralpanelthetwosuperfluids
form a dark-bright soliton; in the bottom panel the two components are fully separated. Right column: order parameter ∆ of
theFermisuperfluidforthesameconfigurationsofthecentralcolumn; thevalueinthebulkofpurefermionsis∆ =0.68E .
0 F
single quasiparticle grand-canonical Hamiltonian, while intherange 1 1/k a 1,inthecrossoverfromthe
F ff
− ≤ ≤
BCS regime (negative values) to the BEC regime (posi-
g
∆(z) = −Lf2f uj,k⊥(z)vj∗,k⊥(z) , (3b) tive values), passing through unitarity (1/kFaff =0).
⊥ j(cid:88),k⊥
2
n (z) = v (z)2 , (3c)
f L2 | j,k⊥ | IV. SOLITONIC SOLUTIONS IN THE
⊥ j(cid:88),k⊥ UNITARITY REGIME
are the order parameter (gap function) and the density
In the following we will consider a mixture where the
of the superfluid fermionic component. The quantity µ
f numberofbosonsisabout10%ofthenumberoffermions.
isthechemicalpotentialoffermions,andtheeigenvalues
For simplicity we also impose m = m , which is a rea-
ε are the quasiparticle energies. b f
j,k⊥ sonableapproximationformixturesoftwoisotopesofthe
The Bose-condensed component is instead described
same atomic species, but this assumption does not affect
by the Gross-Pitaevskii equation for the bosonic order
themainresultsofthework. WetypicallysolveEqs(3a)-
parameter (macroscopic wave function) ψ :
b (3d) with N 500 fermions and N 50 bosons in a
f b
≈ ≈
− 2¯hm2 ∇2ψb+gbb|ψb|2ψb+gbfnfψb =µbψb, (3d) tbroixalwfuitnhctLion=s 7∆0(kzF−)1aannddψLb⊥(z)=a1n5dkF−it1e.raWteetsiltlarctonfrvoemr-
b gence to a stationary solution which does not depend on
whereµ isthechemicalpotentialofbosonsandtheden- the initial choice.
b
sity is n (z)= ψ(z)2. ExamplesaregiveninFig.1forthecaseoffermionsat
b
| |
The four equations (3a)-(3d) must be solved self- unitarity. TheFermiwavevectork isthesameinallfig-
F
consistently. AnenergycutoffE =50E isusedtosolve ures and is related to the bulk density of fermions in the
c F
theBdGequations;thisimpliesaproperrenormalization pure phase, n , by k3 =3π2n . The corresponding bulk
0 F 0
of the Fermi-Fermi coupling constant, for which we use valueoftheorderparameteratunitarityis∆ =0.68E .
0 F
the relation 1/(k a )=8πE /(g k3)+2 E /E /π Letusconcentrate onthefirstcolumnwherewe plot the
F ff F ff F c F
[18]. The key parameter characterizing the interaction density profiles of fermions (blue lines) and bosons (red
(cid:112)
amongfermionsis1/k a ,andweperformcalculations lines) for the ground state in the box, for three different
F ff
4
1 in the fully mixed phase and vanishes in the fully sep-
1.0 immiscible 100 arated phase. The instability of the uniform mixture is
clearly visible as a sharp deviation from 1, which occurs
atthecriticalvalueofg2 n /(g E ) 0.4. Inaninfinite
u.]0.8 A pure 80 system (i.e., in the thebrfmo0dynbabmFic l(cid:39)imit, where surface
a. fermions and interface effects are ignored) this value is expected
[
f0.6 60 to be well approximated by the linear stability condition
n %
b (2); by using the mean-field value of the Bertsch param-
n
dz0.4 miscible B 40 eter, η =0.59 [35], this threshold is gb2fn0/(gbbEF)(cid:39)0.6
pure (vertical green dashed line). In the same figure we also
R
C bosons showthepercentageofspaceoccupiedbythepureFermi
0.2 20 gas(thickdashedline)andthepureBosegas(solidline).
Onecanseethat,byincreasingg2 n /(g E )abovethe
bf 0 bb F
critical value 0.4, pure fermions start occupying a sig-
0.0 0
(cid:39)
10 1 100 101 nificant part of the box while bosons remain still mixed
−
g2 n /g E with fermions, which is another indication of the occur-
bf 0 bb F
rence of the intermediate phase.
FIG.2: (coloronline)Transitionfrommiscibleto immiscible Havingdiscussedthegroundstateasatestcase,letus
phasesofthemixture. Theoverlapintegral(cid:82) dzn (z)n (z)is nowconcentrateonthesecondcolumnofFig.1,wherewe
f b
calculatedinthegroundstateobtainedfordifferentvaluesof showthedensityprofilesinthepresenceofadarksoliton
gb2fn0/(gbbEF),andtheresultsareshownasmarkers(scaleon intheFermicomponent. Theparametersarethesameas
the left axis). When this quantity deviates from 1 the fully in the first column. The dark soliton is imprinted in the
mixed phase becomes unstable. The vertical green dashed Fermisuperfluidbyimposinganode(∆=0)oftheorder
line indicates the instability threshold predicted by Eq. (2)
parameteratthecentreofthebox(z =0)andaπ phase
in the thermodynamic limit, with η =0.59. For large values
difference between the two sides. In a purely fermionic
of g2 n /(g E ) the two components fully separate and the
bf 0 bb F superfluid, the stationary solution of the BdG equation
overlap eventually vanishes. The three letters A, B and C
withsuchconstraintsexhibitsadeepdensitydepletionof
indicate the solutions reported in the left column of Fig. 1.
The thick dashed and solid lines show the percentage of vol- width of the order of kF−1 [18]. If bosons are present, de-
umeoccupiedbypurefermionsandpurebosons,respectively pendingonthevaluesoftheinteractionstrengthsgbf and
(scale on the right axis). Near the instability threshold, a fi- gbb, theycanbeattractedintothesoliton, thuschanging
nite interval of g2 n /(g E ) exists where a significant part its structure. The figure shows that, for small values of
bf 0 bb F
of the box is filled by pure fermions, while bosons are still in g (toppanel),thesystemfavoursauniformmixedcon-
bf
a mixed phase in the remaining volume; this corresponds to figurationinagreementwiththecondition(2)andonlya
thepartiallymixedphaseofthemixture. Theverticalshaded smallfractionofbosonsisattractedbythesoliton,which
area is the region where we find a stable dark-bright soliton.
remainsalmostunaltered. Theresultingstructureisana-
logtothedark–anti-darksolitonpairsthatarepredicted
for Bose-Bose mixtures [38, 39]. In the opposite limit of
values of the parameter gb2fn0/(gbbEF). The three val- large gbf (bottom panel) all bosons form a pure phase at
uesarechosenasrepresentativeofthedifferentphases: a the centre, pushing fermions apart, and the soliton is re-
fullymixedphaseforthesmallestvalueofgb2fn0/(gbbEF) placed by two domain walls separating pure bosons from
(top panel), a fully separated phase for the largest one pure fermions. The central case is the most interesting:
(bottom panel), and an intermediate partially separated bosonsliketostaymixedwithfermions,butfermionslike
phase (middle panel). In the latter case fermions occupy to form a pure phase. The net effect is that all bosons
thewholevolume, partlyasapurephaseandpartlyina are pushed into the soliton but with a broad overlap be-
mixed phase, and the interface between the two regions tween the two components. The overall structure keeps
is significantly wider than the domain wall found for the itssolitoniccharacterandbecomesabright-darksoliton.
fullyseparatedphase(bottompanel),wherethewidthof TheshadedareainFig.2istheintervalofg2 n /(g E )
bf 0 bb F
the interface is of the order of the healing lengths of the where we find stationary solutions having such a bright-
two superfluids. The healing length of bosons is much dark soliton structure, with both the node of the order
smaller than the size of the box in our case, because we parameter and the minimum of the fermionic density at
havechosenthevalueofgbb suchthatthesolutionofthe a single point, z = 0. The resulting scenario shares in-
GP equation is well approximated by the Thomas-Fermi teresting analogies with the dark-bright solitonic struc-
approximation nb(z)=(µb gbfnf(z))/gbb, except near ture exhibited by two-component Bose superfluids [32].
−
the box boundaries. We find similar solutions also for different values of the
To gain further insight, we perform calculations for bosonic fraction: N =20%, 30% and 50% of N .
b f
severalvaluesofgb2fn0/(gbbEF)and,ineachcase,wecal- For the three dark soliton solutions in the second col-
culate the overlap integral dzn (z)n (z). The results umn of Fig. 1 the phase of the order parameter is π
f b
are shown as markers in Fig. 2. The overlap integral is for negative z and 0 for positive z, which implies that
(cid:82)
5
In the BCS regime (1/k a < 0) fermions can
n F ff
0 also phase-separate from bosons either partially or com-
y
ensitn0/2 nn00ggbbff ==01..5619EEFF, kF1a=−1 ptalertyelrye.giTmheiss isshanroetmsuarnpyrissiinmgi,laarsftehaetuBrCesSaasnsdugthgeesutendi-,
d
for example, by the fact that the chemical potential is
0 positive in both cases. The qualitative picture of phase
separation is the same: a uniform mixture is stable for
n
0
small values of the the parameter g2 n /(g E ), a par-
y bf 0 bb F
ensitn0/2 nn00ggbbff ==11..0619EEFF, tpilaeltephpahsaesesespeapraartaiotnionocfcourrsanfoervleanrglearrgvearluceosu,palnindga. cAosma-
d
consequence, also the structure of the solitonic solutions
0 is expected to be similar. In Fig. 3 we show the results
n0 for 1/kFaff = 1, for the same parameters used at uni-
−
y tarity in the central column of Fig. 1. Apart from more
densitn0/2 nn00ggbbff ==11..6699EEFF, pdreopnleotuionnceidnFthrieeddeelnosistcyilldaitsitornibsuatniodnaofshfearllmowioenrss[o1l8it]o,nwice
find that the results look indeed very similar. If one
0
-35 -25 -15 -5 5 15 25 35 moves further into the BCS regime, however, the soli-
zkF tonic structure becomes broader and broader, and less
robust against instability mechanisms associated to the
FIG. 3: (color online) Density profiles of fermions (blue line) fermionic degrees of freedom [23].
andbosons(redline)forsolitonicstatesasinthecentralcol- In order to discuss the BEC regime (1/kFaff > 0)
umnofFig.1,withthesameinteractionparameters,butwith we have to recall that the fermionic superfluid of ultra-
fermions in the BCS regime (1/k a =−1). The structure cold atoms in the BCS-BEC crossover is actually made
F ff
is qualitatively the same as at unitarity, but the soliton is of an equally populated two-spin-component Fermi gas
shallower and the fermionic density exhibits Friedel oscilla- and the s-wave scattering length a accounts for the
ff
tions. The order parameter, not shown, is also similar to the
interaction between atoms with different spins. Due to
one in the right column of Fig. 1, but with ∆ =0.22E .
0 F pairing, this two-component gas behaves as superfluid
describedbyanorderparameter∆andthetotaldensity
n , as we have done so far. However, in the BEC regime
∆(z) is a real function. The third column of the same f
fermions with opposite spins form tightly bound bosonic
figure shows the corresponding profiles of ∆(z). How-
dimers, which in turn form a Bose-Einstein condensate.
ever, in the case of complete separation (bottom panel)
In this situation neither Eq. (1) nor Eq. (2) hold. The
the phase difference between the two sides is irrelevant,
Bose-Fermi mixture thus behaves as a Bose-Bose mix-
because the two regions of pure fermions, separated by
ture, where one of the two bosonic components is the
the central Bose gas, behave as independent superfluids
condensate of dimers, and the system can be described
with no phase coherence; in fact, we have numerically
by two coupled GP equations
checked that, by arbitrarily changing the phase differ-
uennccehaonfgethdeinortdheerspoaluratimoneteorf,thbeotchounpfleadnBdd|G∆|arnedmGaiPn i¯h∂tψd = −2¯hm2 ∇2ψd+gdd|ψd|2ψd+g˜|ψb|2ψd ,
d
equations. Moreover, adding more bosons would sim-
¯h2
plyresultinfurtherseparatingthefermionicsuperfluids, i¯h∂ ψ = 2ψ +g ψ 2ψ +g˜ψ 2ψ , (4)
keeping the same structure of the domain walls. This is t b −2mb∇ b bb| b| b | d| b
not the case of the partially mixed phase shown in the
central panel where, if we suddenly change the sign of where the mass of the dimer is m = 2m and the cou-
d f
∆ on the left side, for instance, the solitonic structure pling constants of the dimer-dimer and dimer-boson in-
is quickly lost and the solution converges to one with- teractions, in the first Born approximation, are given by
out soliton, as in the first column. This proves that the g = 2g and g˜ = 2g (note that, though the exact
dd ff bf
dark-bright soliton in the intermediate partially mixed many-bodyvaluesfortheseparametersaredifferent[35],
phase has indeed solitonic character, being an effect of we use the mean-field expressions for consistency with
nonlinearity and phase coherence. the BdG equations). Replacing the BdG equation for
the order parameter ∆ of paired fermions with a GP
equation for the order parameter ψ of bosonic dimers
d
V. FERMIONS IN THE BCS AND BEC is expected to be a good approximation for 1/(k a )
F ff
REGIMES of the order of, or larger than one. An important con-
sequence is that the region of the phase diagram of the
Nowwediscusswhathappenswhenthefermioniccom- mixturewhereonefindstheintermediate,partiallymixed
ponentofthemixtureisintheBCS-BECcrossover,away phase becomes narrower when moving from unitarity to-
from unitarity. wards the BEC regime [8] and eventually disappears in
6
on the solutions of coupled BdG and GP equations.
n0 We tune the interaction between fermions in the BCS-
n g =1.01E BEC crossover, and we also vary the interspecies and
y 0 bf F
sit n0gbb=1.69EF intraspecies interactions in order to explore the three
nn /2 regimes of fully mixed, partially mixed and fully sepa-
e 0
d
1 =1 rated phases. The focus of the work is on solitonic solu-
kFa tions,andwefindthat,intheregimeofpartialmixing,a
0 dark soliton in the Fermi component becomes wider and
deeper, taking all bosons in, but maintaining phase co-
n
0
herence. This dark-bright solitonic structure can serve
y to stabilize dark solitons in Fermi superfuids against
t
si snaking instability, but it is also interesting in itself as
nn /2
de 0 an example of many-body state where nonlinearity and
coherence play a relevant role. Here we have found that
bosons favour and amplify an inhomogeneous (soliton-
0
-35 -25 -15 -5 5 15 25 35 like) order parameter of fermions. A similar mechanism
zkF might also favour the realization of the Fulde-Ferrell-
Larkin-Ovchinnikov(FFLO)phaseintheunitaryregime
FIG. 4: (color online) Density profiles of fermions (blue line) oftheFermigas,whichinvolvesdensitymodulationsand
andbosons(redline)foruniform(top)andsolitonic(bottom) could be enhanced by the presence of bosons with spin-
statesasinthecentralrowofFig.1,withthesameinteraction dependent interaction, thus making the observation of
parameters,butwithfermionsintheBECregime(1/k a =
F ff such elusive phase easier.
1). Theblackdashedlinesshowtheresultsobtainedwiththe
coupled GP equations (4).
Wefinallynoticethattherangeofparametersinwhich
we find partial mixing may be experimentally accessible
using,forinstance,mixturesof6Li-7Li,6Li-87Rbor23Na-
thedeepBEClimit. AnexampleisgiveninFig.4,where
87Rb. Whenthemassofthetwocomponentsisdifferent,
we show the density profiles of fermions (blue lines) and
and particularly when m /m 1, the situation is ex-
bosons (red lines) of the Fermi-Bose mixture, obtained f b (cid:28)
pected to be even more favourable. In fact, by express-
by solving the coupled BdG and GP equations (3a)-(3d)
ing Eq. (2) in terms of masses and scattering lengths,
for 1/k a = 1, the other parameters remaining the
F ff one finds that the condition for the stability of the mis-
same as in the central row of Fig. 1. The solution in
the top panel corresponds to the case where the phase cible phase becomes (n1f/3)crit ∝ abb/a2bf(mf/mb)/[1 +
of the order parameter ∆ is constant, while the one in (mf/mb)]2 and phase separation can be reached for
the bottom panel is obtained by imposing a π phase dif- smaller densities, possibly accessible to current experi-
ference and a node at the centre. Two main comments ments.
are in order: i) the comparison with Fig. 1 shows that
the partially mixed phase, which was present at unitar-
ity for the same value of g2 n /(g E ), is lost in the
bf 0 bb F
BECregimeinfavourofafullyseparatedphase; ii)ifwe
calculate the density profile by solving the coupled GP
equations (4) (dashed lines) instead of the the coupled
BdG and GP equations (3a)-(3d) (solid lines), we find
very similar results. Acknowledgments
These considerations suggest that the unitary regime
is the most suitable and interesting for the investigation
of dark-bright solitons in Fermi-Bose mixtures. M.T. thanks Stefano Giorgini and Krzysztof Sacha
for useful discussions. This work is supported by ERC
throughtheQGBEgrant,bytheQUICgrantoftheHori-
VI. CONCLUSIONS zon2020 FET program and by Provincia Autonoma di
Trento. A.R. acknowledges support from the Alexander
In this work we theoretically study a mixture of Bose von Humboldt foundation. This work is also supported
andFermisuperfluidsbyusingamean-fieldtheorybased by the PL-Grid infrastructure (M.T.).
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