One-dimensional superfluid Bose-Fermi mixture: Mixing, demixing, and bright solitons

One-dimensional superfluid Bose-Fermi mixture: Mixing, demixing, and bright solitons

Sadhan K. Adhikari1,*and Luca Salasnich2,†

1_{Instituto de Física Teórica, UNESP, São Paulo State University, 01.405-900 Sao Paulo, Sao Paulo, Brazil} 2

CNISM and CNR-INFM, Unità di Padova, Dipartimento di Fisica “G. Galilei,” Università di Padova, Via Marzolo 8, 35131 Padova, Italy

共Received 30 May 2007; published 28 August 2007兲

We study an ultracold and dilute superfluid Bose-Fermi mixture confined in a strictly one-dimensional共1D兲 atomic waveguide by using a set of coupled nonlinear mean-field equations obtained from the Lieb-Liniger energy density for bosons and the Gaudin-Yang energy density for fermions. We consider a finite Bose-Fermi interatomic strength gbfand both periodic and open boundary conditions. We find that with periodic boundary

conditions—i.e., in a quasi-1D ring—a uniform Bose-Fermi mixture is stable only with a large fermionic density. We predict that at small fermionic densities the ground state of the system displays demixing if g_bf ⬎0 and may become a localized Bose-Fermi bright soliton for gbf⬍0. Finally, we show, using variational and

numerical solutions of the mean-field equations, that with open boundary conditions—i.e., in a quasi-1D cylinder—the Bose-Fermi bright soliton is the unique ground state of the system with a finite number of particles, which could exhibit a partial mixing-demixing transition. In this case the bright solitons are demon-strated to be dynamically stable. The experimental realization of these Bose-Fermi bright solitons seems possible with present setups.

DOI:10.1103/PhysRevA.76.023612 PACS number共s兲: 03.75.Ss, 03.75.Hh, 64.75.⫹g

I. INTRODUCTION

The effects of quantum statistics are strongly enhanced in strictly one-dimensional共1D兲 systems 关1兴. These effects can

be investigated with ultracold and dilute gases of alkali-metal atoms, which are now actively studied in the regime of deep Bose and Fermi degeneracies关2兴. Recently two

experi-mental groups 关3,4兴 have reported the observation of the

crossover between a 1D quasi-Bose-Einstein condensate 共quasi-BEC兲 in the weak-coupling mean-field Gross-Pitaevskii 共GP兲 domain and a Tonks-Girardeau 共TG兲 gas 关5,6兴 with ultracold87Rb atoms in highly elongated traps. A rigorous theoretical analysis of the ground-state properties of a uniform 1D Bose gas was performed by Lieb and Liniger 共LL兲 44 years ago 关6兴. An extension of the LL theory for

finite and inhomogeneous 1D Bose gases has been proposed on the basis the local density approximation共LDA兲 关7兴. The

LDA is improved by including a gradient term that repre-sents additional kinetic energy associated with the inhomo-geneity of the gas关8–13兴.

In the last few years several experimental groups have observed the crossover from the Bardeen-Cooper-Schrieffer 共BCS兲 state of Cooper Fermi pairs to the BEC of molecular dimers with ultracold two-hyperfine-component Fermi va-pors of40K atoms关14–16兴 and16Li关17,18兴 atoms. It is well

known that purely attractive potentials have bound states in 1D and 2D for any strength, contrary to the 3D case关19兴. A

rigorous theoretical analysis of the ground-state properties of a uniform 1D Fermi gas was performed by Gaudin and Yang 共GY兲 40 years ago 关20兴. For repulsive interaction the GY

model gives a Tomonaga-Luttinger liquid关21兴, while for

at-tractive interaction it describes a Luther-Emery共superfluid兲

liquid关22兴. The ground state of a weakly attractive 1D Fermi

gas is a BCS-like state with Cooper pairs, whose size is much larger than the average interparticle spacing. The strong-coupling regime with tightly bound dimers is reached by increasing the magnitude of the attractive interatomic strength. In this regime the fermion pairs behave like a hard-core Bose gas共TG gas兲 or, equivalently, like 1D noninteract-ing fermions关23,24兴.

Degenerate Bose-Fermi mixtures of alkali-metal atoms have been experimentally observed in6,7Li关25,26兴,6Li-23Na 关27兴, and 40K-87Rb 关28兴. In these mixtures, the theoretical

investigation of phase separation 关29–35兴 and solitonlike

structures has drawn significant attention. Bright solitons have been observed in BECs of Li关36兴 and Rb 关37兴 atoms

and studied subsequently关38兴. It has been demonstrated

us-ing microscopic 关39兴 and mean-field hydrodynamic 关40兴

models that the formation of stable fermionic bright and dark solitons is possible in a degenerate Bose-Fermi mixture as well as in a Fermi-Fermi mixture关41兴 with the fermions in

the normal state in the presence of a sufficiently attractive interspecies interaction which can overcome the Pauli repul-sion among fermions. The formation of a soliton in these cases is related to the fact that the system can lower its en-ergy by forming high-density regions共solitons兲 when the in-terspecies attraction is large enough to overcome the Pauli repulsion in the degenerate Fermi gas共and any possible re-pulsion in the BEC兲 关42兴. There have also been studies of the

mixing-demixing transition in degenerate Bose-Fermi 关43兴

and Fermi-Fermi关44兴 mixtures with fermions in the normal

state.

After observing the degenerate Fermi gas and the realiza-tion of BCS condensed superfluid phase of the fermionic system关45兴, the BCS-Bose crossover 关46兴 in it seems to be

under control关14–18兴 by manipulating the Fermi-Fermi

in-teraction near a Feshbach resonance关47兴 by varying a

uni-form external magnetic field. Naturally, the question of the *adhikari@ift.unesp.br; URL: www.ift.unesp.br/users/adhikari

†

possibility of bright solitons in a superfluid fermionic con-densate assisted by a BEC共with attractive interspecies Bose-Fermi interaction兲 needs to be revisited. Although the fermi-onic BCS or molecular dimer phase is possible with an attractive Fermi-Fermi attraction, once formed such a phase is inherently repulsive and self-defocussing and will not lead to a bright soliton spontaneously.

In this paper we investigate a 1D superfluid Bose-Fermi mixture composed of bosonic atoms, well described by the LL theory, and superfluid fermionic atoms, well described by the GY theory with the special intention of studying the lo-calized structures or bright solitons in this superfluid mix-ture. We derive a set of coupled nonlinear mean-field equa-tions for the Bose-Fermi mixture which we use in the present study. The solution of this mean-field equation is considered two types of boundary conditions: periodic and open. It has been recently shown 关48–50兴 that an attractive BEC with

periodic boundary conditions, which can be experimentally produced with a quasi-1D ring 关51兴, displays a quantum

phase transition from a uniform state to a symmetry-breaking state characterized by a localized bright-soliton condensate 关50兴. Here we show that a similar phenomenon appears in the

1D Bose-Fermi mixture with an attractive Bose-Fermi scat-tering length. Instead, with a repulsive Bose-Fermi scatscat-tering length or with the increase of the number of Fermi atoms leading to large repulsion we find a phase separation between bosons and fermions共a mixing-demixing transition兲, in anal-ogy with previous theoretical关29–35兴 and experimental

stud-ies关25兴 with 3D trapped6Li-7Li and 39K-40K mixtures. Fi-nally, we predict that with open boundary conditions—i.e., in a infinite quasi-1D cylinder—the ground state of the mixture with sufficiently attractive Bose-Fermi interaction is always a localized bright soliton.

The paper is organized as follows. In Sec. II we describe the 1D model we use in our investigation of the superfluid Bose-Fermi mixture. The Lagrangian density for bosons is appropriate for a TG to BEC crossover through the use of a quasianalytic LL function. The fermions are treated using the GY model through a quasianalytic GY function. The Bose-Fermi interaction is taken to be a standard contact interac-tion. The Euler-Lagrange equations for this system are a set of coupled nonlinear mean-field equations which we use in the present study. In Sec. III we consider the system with periodic boundary conditions and obtain the thresholds for demixing for the formation of localized bright solitons共for a sufficiently strong Bose-Fermi attraction兲 and for the exis-tence of mixing—i.e., states with constant density in space 共for weak Bose-Fermi attraction and for Bose-Fermi repul-sion兲. We also present a modulational instability analysis of a uniform solution of the mixture and obtain the condition for the appearance of bright solitons by modulational instability. In Sec. IV we consider the mixture with open boundary con-ditions and derive a variational approximation for the solu-tion of the mean-field equasolu-tions of the model using a Gaussian-type ansatz. We investigate numerically the bright Bose-Fermi solitons, demonstrate their stability under pertur-bation, and compare the numerical results with those of the variational approach. Finally, in Sec. V we present a sum-mary and discussion of our study.

II. MODEL

We consider a mixture of Nb atomic bosons of mass mb

and Nf superfluid atomic fermions of mass mf at zero

tem-perature trapped by a tight cylindrically symmetric harmonic potential of frequency␻_⬜in the transverse共radial cylindric兲 direction. We assume factorization of the transverse degrees of freedom. This is justified in 1D confinement where, re-gardless of the longitudinal behavior or statistics, the trans-verse spatial profile is that of the single-particle ground state 关34,52兴. The transverse width of the atom distribution is

given by the characteristic harmonic length of the single-particle ground state: a_⬜j=

冑

ប/共mj␻⬜兲, with j=b, f. The

at-oms have an effective 1D behavior at zero temperature if their chemical potentials are much smaller than the trans-verse energy ប␻_⬜ 关34,52兴. The boson-boson interaction is

described by a contact pseudopotential with repulsive 共posi-tive兲 scattering length ab. The fermion-fermion interaction is

modeled by a contact pseudopotential with attractive 共nega-tive兲 scattering length af. The boson-fermion interaction is

instead characterized by a contact pseudopotential with scat-tering length abf, which can be repulsive or attractive. To

avoid the confinement-induced resonance 关53兴 we take

ab,兩af兩,兩abf兩Ⰶa_⬜b, a_⬜f.

We use a mean-field effective Lagrangian to study the static and collective properties of the 1D Bose-Fermi mix-ture. The Lagrangian densityL of the mixture reads

L = Lb+Lf+Lbf. 共1兲

The termLbis the bosonic Lagrangian, defined as

Lb= iប␺b *⳵ ⳵t␺b+ ប2 2mb ␺b *⳵2 ⳵z2␺b− ប2 2mb 兩␺b兩6G

冉

mbgb ប2_兩␺ b兩2

冊

, 共2兲 where ␺b共z,t兲 is the hydrodynamic field of the Bose gas

along the longitudinal axis, such that nb共z,t兲=兩␺b共z,t兲兩2is the

1D local probability density of bosonic atoms and gb

= 2ប␻_⬜ab is the 1D boson-boson interaction strength 共gb

⬎0兲. G共x兲 is the Lieb-Liniger function 关54兴, defined as the

solution of a Fredholm equation for x⬎0 and such that 关6兴

G共x兲 =

冦

x − 4 3␲x 3/2_{+¯ for 0 ⬍ x Ⰶ 1,} ␲2 3

冉

1 − 2 x+ ¯

冊

for xⰇ 1.

冧

共3兲

In the extreme weak-coupling limit x→ +0, and G共x兲→x and, consequently, the bosonic Lagrangian above reduces to the standard mean-field GP Lagrangian. In the static case the Lagrangian density Lb reduces exactly to the energy

func-tional recently introduced by Lieb, Seiringer, and Yngvason 关9兴. In addition, Lbhas been successfully used to determine

the collective oscillation of the 1D Bose gas with longitudi-nal harmonic confinement关13兴. In the strong-coupling limit

x→ +⬁ and G共x兲→␲2_{/ 3 while the Lagrangian above} re-duces to the strongly repulsive bosonic Lagrangian in the TG limit关5兴. As x changes from +0 to +⬁ the above Lagrangian

The fermionic Lagrangian densityLfis given instead by Lf= iប␺f *⳵ ⳵t␺f+ ប2 2mf ␺f *⳵2 ⳵z2␺f− ប2 2mf 兩␺f兩6F

冉

mfgf ប2_兩␺ f兩2

冊

, 共4兲 where ␺f共z,t兲 is the field of the 1D superfluid Fermi gas,

such that nf共z,t兲=兩␺f共z,t兲兩2is the 1D fermionic local density

along the longitudinal axis and gf= 2ប␻_⬜af is the 1D

fermion-fermion interaction strength 共gf⬍0兲. F共x兲 is the

Gaudin-Yang function关55兴, defined as the solution of a

Fred-holm equation for x⬍0 and such that 关20,23兴

F共x兲 =

冦

␲2 48冉1 −1 x+ 3 4x2+ ¯

冊

for xⰆ − 1, ␲2 12

冉

1 + 6 ␲2x + ¯

冊

for − 1Ⰶ x ⬍ 0.

冧

共5兲 In the static and uniform case the Lagrangian density Lf

reduces exactly to Gaudin-Yang energy functional 关20兴,

which has been recently used by Fuchs et al.关23兴.

The physical content of the fermionic Lagrangian共4兲 is

easy to understand. For x→−0 we are in the domain of weak Fermi-Fermi attraction共af, gf→0, corresponding to the BCS

limit兲, while F共x兲→␲2_{/ 12 from Eq.共5兲. Consequently, the} fermionic interaction term in Eq.共4兲 involving the F共x兲

func-tion reduces to共ប2_{/ 2m}

f兲共␲2/ 12兲兩␺f兩6共independent of

Fermi-Fermi interaction strength gf or Fermi-Fermi scattering

length af兲, which is the Pauli repulsive term of

noninteract-ing fermions considered in Refs. 关52,56兴 arising due to the

Pauli exclusion principle and not due to the fundamental Fermi-Fermi interaction implicit in the scattering length af

via gf. This is quite expected as by taking the x→−0 limit

we switch off the Fermi-Fermi attraction and pass from the BCS domain to a degenerate noninteracting Fermi gas 共in nonsuperfluid phase兲 studied in Ref. 关52兴 corresponding to

x→ +0. 关Although the probability densities in the x→ +0

共nonsuperfluid degenerate Fermi gas兲 and x→−0 共superfluid BCS condensate兲 limits are identical, they correspond to dif-ferent physical states. The superfluid phase gas a “gap” as-sociated with it describable by the BCS equation.兴 As x passes from −0 to −⬁, the Fermi-Fermi attraction increases and we move from the superfluid BCS regime of weakly bound Cooper pairs to the unitarity regime of strongly-bound molecular fermions.共The unitarity limit corresponds to infi-nitely large Fermi-Fermi attraction: af→−⬁.兲 As the

Fermi-Fermi scattering length 兩af兩 increases from the BCS to the

unitarity limit, the interaction energy in Eq.共4兲 becomes a

term dependent on af. However, in the unitarity limit af→

−⬁ and all the fermions will be paired to form strongly bound noninteracting molecules and the Fermi-Fermi inter-action term in Eq.共4兲 behaves like that of a Tonks-Girardeau

gas of bosons—bosonic molecules—and again becomes in-dependent of af关23兴. The system then becomes a Bose gas of

molecules and its interaction energy is no longer a function of af.

Finally, the Lagrangian densityLbfof the Bose-Fermi

in-teraction taken to be of the following standard zero-range form关43,52兴

Lbf= − gbf兩␺b兩2兩␺f兩2, 共6兲

where gbf= 2ប␻⬜abf is the 1D boson-fermion interaction

strength.

The Euler-Lagrange equations of the Lagrangian L are the two following coupled partial differential equations:

iប⳵t␺b=

冋

− ប2 2mb ⳵z 2 + 3 ប 2 2mb nb 2 G

冉

mbgb ប2_n b

冊

− 1 2gbnbG

⬘

冉

mbgb ប2_n b

冊

+ gbfnf

册

␺b, 共7兲 iប⳵t␺f=

冋

− ប2 2mf ⳵z 2 + 3 ប 2 2mf nf 2 F

冉

mfgf ប2_n f

冊

−1 2gfnfF

⬘

冉

mfgf ប2_n f

冊

+ gbfnb

册

␺f, 共8兲

where nj=兩␺j兩2, j = b , f, are probability densities for bosons

and fermions, respectively, with the normalization 兰_−⬁⬁ njdz

= Nj.

It is convenient to work in terms of dimensionless vari-ables defined in terms of a frequency ␻ and length l

⬅

冑

ប/共mb␻兲 by ␺j=␺ˆj/

冑l, t = 2tˆ/

␻, z = zˆl, gj= gˆjប2/共2mbl兲,

and gbf= gˆbfប2/共2mbl兲. In terms of these new variables Eqs.

共7兲 and 共8兲 can be written as

i⳵t␺b=

冋

−⳵z2+ 3nb2G

冉

gb 2nb

冊

−1 2gbnbG

⬘

冉

gb 2nb

冊

+ gbfnf

册

␺b, 共9兲 i⳵t␺f=

冋

−␭⳵z 2_{+ 3␭n} f 2 F

冉

gf 2nf

冊

−1 2gfnfF

⬘

冉

gf 2nf

冊

+ gbfnb

册

␺f, 共10兲 where we have dropped the carets over the variables and where ␭=mb/ mf, nj=兩␺j兩2, j = b , f, are probability densities

for bosons and fermions, respectively, with the normalization 兰_−⬁⬁ _n

jdz = Nj. However, in the present study we set ␭=1 in

the following. All results reported in this paper are for this case. This should approximate well the Bose-Fermi mixtures 7

Li-6Li and39K-40K of experimental interest.

The bosonic and fermionic nonlinearities in Eqs.共9兲 and

共10兲, respectively, have complex structures in general.

How-ever, in the weak-coupling GP limit 共gb→0兲 the bosonic

nonlinearity in Eq. 共9兲 is cubic and turns quintic in the

strong-coupling TG limit共gb→ +⬁兲. The fermionic

nonlin-earity in Eq. 共10兲 becomes quintic in both weak 共gF→−0兲

and strong 共gf→−⬁兲 coupling but with coefficients ␲2/ 12

and␲2/ 48, respectively.

For stationary states the solutions of Eqs. 共9兲 and 共10兲

have the form␺j=␾jexp共−i␮jt兲 where␮jare the respective

␮b␾b=

冋

−⳵z2+ 3nb2G

冉

gb 2nb

冊

− 1 2gbnbG

⬘

冉

gb 2nb

冊

+ gbfnf

册

␾b, 共11兲 ␮f␾f=

冋

−⳵z 2 + 3nf 2 F

冉

gf 2nf

冊

− 1 2gfnfF

⬘

冉

gf 2nf

冊

+ gbfnb

册

␾f. 共12兲 A repulsive Bose-Bose interaction is produced by a positive

gb, while an attractive Fermi-Fermi interaction corresponds

to a negative gf.

III. MIXTURE WITH PERIODIC BOUNDARY CONDITIONS

A. General considerations

Here we consider the 1D Bose-Fermi mixture with peri-odic boundary conditions. These boundary conditions can be used to model a quasi-1D ring of radius R. If the radius is much larger than the transverse width—i.e., RⰇa_⬜j,

j = b , f—then effects of curvature can be neglected 关50兴 and

one can safely use the Lagrangian density 共1兲 with z=R␪, where␪ the azimuthal angle关50,51兴. The energy density of

the uniform mixture is immediately derived from the La-grangian density共1兲, with Eqs. 共2兲, 共4兲, and 共6兲, dropping out

space-time derivatives. It is given by

E = nb3G

冉

gb 2nb

冊

+ n3fF

冉

gf 2nf

冊

+ gbfnbnf. 共13兲

In this case one can have a uniform mixture or the formation of solitonlike structures and in the following we study the condition for these possibilities to happen. Obviously, for the 1D uniform mixture in the ring of radius R, we have nb

= Nb/共2␲R兲 and nf= Nf/共2␲R兲.

The uniformly mixed phase is energetically stable if its energy is a minimum with respect to small variations of the densities nf and nb, while the total number of fermions and

bosons are held fixed. To get the equilibrium densities one must then minimize the function

E˜共nb,nf兲 = E共nb,nf兲 −␮b共nb * ,nf *_兲n b−␮f共nb * ,nf *_兲n f, 共14兲

where␮band␮fare Lagrange multipliers共imposing that the

numbers of bosons and fermions are fixed兲 which may be identified with the Bose and Fermi chemical potentials and

nb

* and nf

*

are the values of nband nfat the minimum. Setting

the derivatives ofE˜ with respect to nf and nb equal to zero,

one finds ␮b= ⳵E ⳵nb = 3nb 2 G

冉

gb 2nb

冊

−1 2gbnbG

⬘

冉

gb 2nb

冊

+ gbfnf, 共15兲 ␮f= ⳵E ⳵nf = 3nf 2 F

冉

gf 2nf

冊

−1 2gfnfF

⬘

冉

gf 2nf

冊

+ gbfnb. 共16兲

The solution of Eqs.共16兲 and 共15兲, which are exactly Eqs.

共11兲 and 共12兲 if one drops out the space derivatives, gives a

minimum if the corresponding Hessian ofE˜ is positive—i.e., if ⳵2_E˜ ⳵nb 2 ⳵2_E˜ ⳵nf 2−

冉

⳵2_E˜ ⳵nb⳵nf

冊

2 ⬎ 0. 共17兲

The solution of this inequality gives the region in the param-eters’ space where the homogeneous mixed phase is energeti-cally stable.

In Fig.1 we show the region of stability of the homoge-neous mixture by the shaded共gray兲 area in the plane 共nb, nf兲

for different values of gbf. Note that the sign of gbf is not

important because, in Eq.共17兲, gbf

2

appears. The figure shows that, by increasing gbf, the uniform mixture is stable only at

large fermionic densities nf. This result, which is in

agree-ment with previous 1D predictions关34兴 on a 1D mixture of

Bose-condensed atoms and normal Fermi atoms, is exactly the opposite of what happens in a 3D mixture of bosons and

0 0.25 0.5 0 0.5 1

n

f 0 0.25 0.5 0 0.5 1 0 0.25 0.5

n

b 0 0.5 1

n

f 0 0.25 0.5

n

b 0 0.5 1 g_bf= 0 g_bf= 0.2 g_bf= 0.4 _g bf= 0.6 0 0.25 0.5 0 0.5 1

n

f 0 0.25 0.5 0 0.5 1 0 0.25 0.5

n

b 0 0.5 1

n

f 0 0.25 0.5

n

b 0 0.5 1 g_bf= 0 g_bf= 0.1 g_bf= 0.2 g_bf= 0.3 (a) (b)

FIG. 1. 共Color online兲 Region of stability of the homogeneous mixture denoted by shaded共gray兲 area in the plane 共n_b, n_f兲. Differ-ent panels correspond to differDiffer-ent values of the Bose-Fermi inter-action strength g_bf. We set g_b= 0.1 and共a兲 g_f= −0.1共corresponding to a weak Fermi-Fermi attraction in the BCS phase兲 and 共b兲 gf= −30共corresponding to a strong Fermi-Fermi attraction, with

fermions. In fact, a uniform 3D mixture is stable only for small values of the fermionic density 关29–33,35兴. Figure 1

also shows that when bosons enter in the TG regime 共i.e.,

gb/ nbⰇ1兲 the uniform mixture is much less stable: see the

behavior of the stability line at very small nb, with gbf⫽0.

The panels of Fig.1suggest that, with a finite gbf, at small

fermionic densities the uniform mixture is unstable: the ground state of the system displays demixing if gbf⬎0 and

becomes a localized Bose-Fermi bright soliton if gbf⬍0.

These effects are clearly shown in Fig.2, where we plot the density profiles nj共z兲, with j=b, f, for different values of gbf

calculated by directly solving Eqs.共11兲 and 共12兲 numerically

with appropriate boundary conditions.

The profiles of probability densities plotted in Fig.2 dem-onstrate the uniform-to-localized and the uniform-to-demixed transitions. In fact, a large and negative gbfinduces

a strong localization while a large and positive gbf produces

demixing. These density profiles have been obtained by solv-ing Eqs. 共9兲 and 共10兲 with a finite-difference

Crank-Nicholson algorithm and imaginary time关57兴.

In the regime gb/ nbⰆ1 共BEC limit兲 and 兩gf兩/nfⰆ1 共BCS

limit兲, the above analysis yields simple analytic results. In this case E=gbnb

2

/ 2 +␲2nf

3

/ 12+ gbfnbnf, consequently, Eq.

共17兲 leads to the condition

1 2␲

2_g

bnf⬎ gbf

2 _共18兲

for the stability of the uniform mixture. The condition ␲2_g

bnf/ 2⬎gbf

2 _{is qualitatively consistent with Fig.1共a兲. For} example, for gb= 0.1 and gf= −0.1 the condition of stability

of uniform mixture becomes nf⬎2gbf2 and for 兩gbf兩=0, 0.2,

0.4, and 0.6 leads to nf⬎0, 0.08, 0.32, and 0.72, respectively.

In the regime gb/ nbⰆ1 共BEC limit兲 and 兩gf/ nf兩Ⰷ1

共uni-tarity or tightly bound molecule formation limit兲 E=gbnb2/ 2

+␲2_n

f

3_{/ 48+ g}

bfnbnf, consequently, Eq.共17兲 leads to the

con-dition

1 8␲

2_g

bnf⬎ gbf2 共19兲

for the stability of the uniform mixture. For example, for

gb= 0.1 and gf= −30 the condition of stability of uniform

mixture becomes nf⬎8gbf

2 _{and for兩g}

bf兩=0, 0.1, 0.2, and 0.3

leads to nf⬎0, 0.08, 0.32, and 0.72, respectively, in

qualita-tive agreement with Fig.1共b兲. The time-independent condi-tions共18兲 and 共19兲 are also derived in following subsection

using rigorous time-dependent modulational instability analysis of the uniform mixture 共constant-amplitude solu-tions兲 in the weak and strong Fermi-Fermi coupling limits.

Finally, in the regime gb/ nbⰇ1 共TG limit兲 and 兩gf/ nf兩

Ⰶ1 共BCS limit兲 E=␲2_n b 3_{/ 3 +␲}2_n f 3_{/ 12+ g} bfnbnf,

conse-quently, Eq.共17兲 leads to the condition

␲4_n

bnf⬎ gbf2 共20兲

for the stability of the uniform mixture. The functional de-pendence of condition 共20兲 on nb, nf, gb, and gf is

qualita-tively different from conditions共18兲 and 共19兲.

The inequality共17兲 can be written as

cb 2 cf 2_{⬎ 4g} bf 2 nbnf, 共21兲

where cf=

冑

2nf共⳵␮f/⳵nf兲 is the sound velocity of the

super-fluid Fermi component and cb=

冑

2nb共⳵␮b/⳵nb兲 is the sound

velocity of the Bose gas. The sound velocity cbf of the 1D

Fermi-Bose mixture can be easily obtained following a pro-cedure similar to the one suggested by Alexandrov and Ka-banov关58兴 for a two-component BEC. One finds

cbf= 1

冑

2

冑

cb 2 + cf 2 ±

冑

共cb 2 − cf 2_兲2_{+ 16g} bf 2 nbnf. 共22兲

Thus the sound velocity has two branches and the homoge-neous mixture becomes dynamically unstable when the lower branch becomes imaginary.

B. Modulational instability

We show in the following that for attractive Bose-Fermi interaction, the transition from uniform mixture to localized solitonlike structures considered in Sec. III A is due to modulational instability. To study analytically the modula-tional instability关59,60兴 of Eqs. 共9兲 and 共10兲 we consider the

special case of weak Bose-Bose共small positive gb兲 and both

weak 共small 兩gf/ nf兩Ⰶ1 corresponding to BCS limit兲 and

strong共large 兩gf/ nf兩Ⰷ1 corresponding to molecular unitarity

limit兲 Fermi-Fermi interactions while these equations reduce to i⳵t␺b=关−⳵z 2 + gb兩␺b兩2− gbf兩␺f兩2兴␺b, 共23兲 i⳵t␺f=关−⳵z 2 +␬␲2兩␺f兩4− gbf兩␺b兩2兴␺f, 共24兲

where␬= 1 / 4 for 兩gf/ nf兩Ⰶ1 and 1/16 for 兩gf/ nf兩Ⰷ1. Here

we have taken the interspecies interaction to be attractive by inserting an explicit negative sign in gbf.

We analyze the modulational instability of a constant-amplitude solution corresponding to a uniform mixture in coupled equations共23兲 and 共24兲 by considering the solutions

-5 -2.5 0 2.5 5 0 5 10 15 n j (z) -5 -2.5 0 2.5 5 0 15 30 -5 -2.5 0 2.5 5 z 0 100 200 n j (z) -5 -2.5 0 2.5 5 z 0 15 30 g_bf= -10-3 g_bf= -10-2 g_bf= +10-2 g_bf= -0.05

FIG. 2. Probability density profiles nb共z兲 of bosons 共dashed line兲

and n_f共z兲 of fermions 共solid line兲 of the Bose-Fermi mixture with

Nb= 100 bosons and Nf= 10 fermions and periodic boundary

condi-tions 共axial length L⬅2␲R=10兲. We choose g_b= 0.01 and

gf= −0.025 and calculate the profiles for different values of the

␸b0= Ab0exp共i␦b兲 ⬅ Ab0eit共gbfAf0 2 −gbAb0 2 兲_{, 共25兲} ␸f0= Af0exp共i␦f兲 ⬅ Af0eit共gbfAb0 2 −␬␲2_A f0 4 兲_{, 共26兲} of Eqs. 共23兲 and 共24兲, respectively, where Aj0 is the

ampli-tude and ␦j a phase for component j = b , f. The

constant-amplitude solution develops an constant-amplitude-dependent phase on time evolution. We consider a small perturbation

Ajexp共i␦j兲 to these solutions via

␸j=共Aj0+ Aj兲exp共i␦j兲, 共27兲

where Aj= Aj共z,t兲. Substituting these perturbed solutions in

Eqs.共23兲 and 共24兲 and for small perturbations retaining only

the linear terms in Aj, we get

i⳵tAb+⳵z2Ab− gbAb02 共Ab+ Ab*兲 + gbfAb0Af0共Af+ Af*兲 = 0, 共28兲 i⳵tAf+⳵z 2 Af− 2␬␲2Af0 4_共A f+ Af *_{兲 + g} bfAb0Af0共Af+ Af *_{兲 = 0.} 共29兲 We consider the complex plane-wave perturbation

Aj共z,t兲 = Aj1cos共Kt − ⍀z兲 + iAj2sin共Kt − ⍀z兲, 共30兲

with j = b , f, whereAj1 and Aj2 are the amplitudes for the

real and imaginary parts, respectively, and K and⍀ are fre-quency and wave numbers.

Substituting Eq.共30兲 into Eqs. 共28兲 and 共29兲 and

separat-ing the real and imaginary parts we get

−A_b1K =A_b2⍀2_{, 共31兲} −Ab2K =Ab1⍀2− 2gbfAb0Af0Af1+ 2gbAb0 2 _A b1, 共32兲 for j = b, and − Af1K = Af2⍀2, 共33兲 −A_f2K =A_f1⍀2_{− 2g} bfAb0Af0Ab1+ 4␬␲2Af0 4_A f1, 共34兲

for j = f. EliminatingAb2between Eqs.共31兲 and 共32兲 we get

A_b1关K2−⍀2共⍀2+ 2gbAb02 兲兴 = − 2Af1gbfAb0Af0⍀2 共35兲

and eliminatingA_f2between共33兲 and 共34兲, we have

Af1关K2−⍀2共⍀2+ 4␬␲2Af0

4_{兲兴 = − 2A}

b1gbfAb0Af0⍀2.

共36兲 Finally, eliminatingA_b1andA_f1from Eqs.共35兲 and 共36兲, we

obtain the dispersion relation

K = ±⍀关共⍀2+ gbAb02 + 2␬␲2A4f0兲 ± 兵共gbAb02 − 2␬␲2A4f0兲2

+ 4gbf

2

A_b02 A_f02其1/2兴1/2. 共37兲

For stability of the plane-wave perturbation, K has to be real. For any⍀ this happens for

共gbAb0 2 + 2␬␲2A4_f0兲2⬎ 共gbAb0 2 − 2␬␲2A4_f0兲2+ 4gbf 2 A_b02 A_f02 共38兲 or for 2gb␬␲2Af0 2 _{⬎ g} bf 2 . 共39兲

However, for 2gb␬␲2A2f0⬍gbf2, K can become imaginary and

the plane-wave perturbation can grow exponentially with time. This is the domain of modulational instability of a constant-amplitude solution共uniform mixture of Sec. III A兲, signaling the possibility of coupled Bose-Fermi bright soli-ton to appear. Noting that A2_f0= nf for the uniform mixture,

this result is consistent with the stability analysis of Sec. III A. In the weak-coupling BCS limit␬= 1 / 4 and Eq.共39兲

reduces to Eq. 共18兲 obtained from energetic considerations.

In the strong-coupling unitarity limit ␬= 1 / 16 and Eq. 共39兲

reduces to Eq.共19兲 for the stability of the uniform mixture.

Finally, we comment that Eq.共20兲 can also be derived in a

straightforward fashion from the modulational instability analysis of an appropriate set of dynamical equations, with

gb兩␺b兩2 replaced by ␲2兩␺b兩4 in Eq. 共23兲 and ␬= 1 / 4 in Eq.

共24兲.

IV. MIXTURE WITH OPEN BOUNDARY CONDITIONS In this section we consider the 1D mixture with open boundary conditions. These boundary conditions can be used to model a quasi-1D cylinder. In practice, if the axial length

L of the atomic waveguide is much larger than the transverse

width—i.e., LⰇa_⬜j, j = b, f—then one has a quasi-1D cylin-der. The quasi-1D cylinder becomes infinite for L→⬁. It this case we can use the Lagrangian density 共1兲 with z苸共−⬁,

+⬁兲. With a finite number of particles 共bosons and fermions兲 the uniform mixture in a infinite cylinder cannot exist, but localized solutions are instead possible with an attractive Bose-Fermi strength共gbf⬍0兲 关52兴.

A. Variational results

Here we develop a variational localized solution to Eqs. 共11兲 and 共12兲, noting that these equations can be derived

from the Lagrangian关61兴

L =

冕

−⬁ ⬁ 关␮b␾b2+␮f␾2f−共␾b

⬘

兲2−共␾f

⬘

兲2−␾6bG共gb/2␾b2兲 −␾f 6 F共gf/2␾f 2_{兲 − g} bf␾b 2_␾ f 2_{兴dz −␮} bNb−␮fNf, 共40兲 by demanding␦L /␦␾b=␦L /␦␾f=␦L /␦␮b=␦L /␦␮f= 0.

To develop the variational approximation we use the Gaussian ansatz关62兴 ␾b共z兲 =␲−1/4

冑

NbAb wb exp

冉

− z 2 2wb 2

冊

, 共41兲 ␾f共z兲 =␲−1/4

冑

NfAf wf exp

冉

− z 2 2w2f

冊

, 共42兲

where the variational parameters are Aj, the solitons’ norm

and wjthe widths, in addition to␮j. Note that this Gaussian

ansatz can be extended to include time dependence关63兴, as

done recently to investigate the collective oscillation of a quasi-1D mixture made of condensed bosons and normal fer-mions关52兴.

The substitution of this variational ansatz in Lagrangian 共40兲 yields L =␮bNbAb+␮fNfAf− NbAb 2wb2 −NfAf 2w2f − Ab 3_N b 3 ␲

冑

3wb2 G

冉

cgbwb 2NbAb

冊

− Af 3 Nf 3

冑

3␲wf 2F

冉

dgfwf 2NfAf

冊

−g

_冑

bfNbNfAbAf ␲共wf 2 + wb 2_兲−␮bNb−␮fNf, 共43兲 with c =

冑

3␲/ 2. The integrals over the G共x兲 and F共x兲 func-tions cannot be evaluated exactly for all x. However, the term involving the LL G共x兲 function can be evaluated analytically in the GP共x→0兲 and TG 共xⰇ1兲 limits. The analytic term involving the G共x兲 function in Eq. 共43兲 is exact in both the

limits and provides a good description of the integral at other values of argument provided we take for the constant c =

冑

3␲/ 2. The fermionic integral over the GY function F共x兲 is evaluated similarly. There is no obvious reason for choos-ing the constant d in this integral. After a little experimenta-tion it is taken to be d = c =

冑

3␲/ 2 which provides a faithful analytical representation of the numerical result.

The first variational equations emerging from Eq. 共43兲

⳵L /⳵␮b=⳵L /⳵␮f= 0 yield Ab= Af= 1. Therefore the

condi-tions Ab= Af= 1 will be substituted in the subsequent

varia-tional equations. The variavaria-tional equations⳵L /⳵wj= 0 lead to

1 + 2Nb 2 ␲

冑

3G

冉

cgbwb 2Nb

冊

−Nbgbwb 2

冑

2␲ G

⬘

冉

cgbwb 2Nb

冊

+ gbfNfwb 4

冑

␲共wb 2 + wf 2_兲3/2 = 0, 共44兲 1 + 2Nf 2

冑

3␲F

冉

cgfwf 2Nf

冊

−Nfgfwf 2

冑

2␲F

⬘

冉

cgfwf 2Nf

冊

+ gbfNbwf 4

冑

␲共wb2+ w2f兲3/2 = 0. 共45兲

The remaining variational equations are⳵L /⳵Aj= 0, which

yield␮as a function of wj’s, and g’s:

␮b= 1 2wb 2− Nbgb 2wb

冑

2␲ G

⬘

冉

cgbwb 2Nb

冊

+

冑

3Nb 2 ␲wb 2G

冉

cgbwb 2Nb

冊

+

_冑

gbfNf ␲共wf2+ wb2兲 , 共46兲 ␮f= 1 2wf 2+

冑

3N2f ␲wf 2F

冉

cgfwf 2Nf

冊

− Nfgf 2wf

冑

2␲ F

⬘

冉

cgfwf 2Nf

冊

+

_冑

gbfNb ␲共wf2+ wb2兲 . 共47兲

Equations 共44兲–共47兲 are the variational results which we

shall use in our study of bright Bose-Fermi solitons. B. Numerical results

For stationary solutions we solve time-independent equa-tions共11兲 and 共12兲 by using an imaginary-time propagation

method based on the finite-difference Crank-Nicholson

dis-cretization scheme of time-dependent equations共9兲 and 共10兲.

The nonequilibrium dynamics from an initial stationary state is studied by solving the time-dependent equations 共9兲 and

共10兲 with real-time propagation by using as initial input the

solution obtained by the imaginary-time propagation method. The reason for this mixed treatment is that the imaginary-time propagation method deals with real variables only and provides a very accurate solution of the stationary problem at low computational cost关64兴. In the finite-difference

discreti-zation we use a space step of 0.025 and time step of 0.0005. First we report results for stationary profiles of the local-ized Bose and Fermi solitons formed in the presence of at-tractive Bose-Fermi and Fermi-Fermi interactions and repul-sive Bose-Bose interactions. In the presence of weak attractive Fermi-Fermi interactions, the fermions form a BCS state which satisfies a nonlinear Schrödinger equation with repulsive 共self-defocusing兲 quintic nonlinearity. As the strength of the attractive Fermi-Fermi interaction increases the fermions pass from the BCS regime to the unitarity re-gime which is described by another nonlinear Schrödinger equation with repulsive共self-defocusing兲 nonlinearity. Hence fermions cannot form a bright soliton by itself. However, they can form a bright soliton in the presence of an attractive Bose-Fermi interaction.

If the fermionic repulsive nonlinearity is not very large, bosons and fermions form mostly overlapping共mixed兲 soli-tons both in the BCS and unitarity regimes. Note that in the BCS and unitarity regimes the fermionic system becomes repulsive in the presence of a Fermi-Fermi attraction. How-ever, as the fermionic repulsive nonlinearity turns large, the fermionic profile comes out of the bosonic profile and par-tially demixed solitons are created. We studied the numeri-cally calculated soliton profiles for a wide range of Bose-Bose, Bose-Fermi, and Fermi-Fermi interactions and boson and fermion numbers solving Eqs.共11兲 and 共12兲 by the

tech-nique of imaginary-time propagation and compared them with the Gaussian variational results obtained from Eqs.共44兲

and共45兲. Except in the case of strong demixing, when the

fermion profile strongly deviates from Gaussian shape, the agreement between variational and numerical profiles is quite good.

In Fig.3we present typical soliton profiles illustrating the change in the results during BCS-unitarity crossover as well as the demixed profiles. In Figs.3共a兲–3共c兲we show the soli-ton profiles for weak 共BCS phase兲, moderate, and strong 共unitarity regime兲 Fermi-Fermi attraction corresponding to strengths gf= −0.1, −1, and −10, respectively, for gb= 0.01,

gbf= −0.4, Nf= 20, and Nb= 300 and compare these with the

variational results. Figure3共d兲illustrates a demixed state ob-tained by increasing the fermion number from the configu-ration of Fig.3共c兲from Nf= 20 to 100. In this case the

共nu-merically obtained兲 fermion profile stretches far beyond the bosonic profile and is poorly represented by a Gaussian shape, which is the cause of the deviation of the variational result from the numerical result.

After illustrating the soliton profiles in different states it is now pertinent to verify if these solitons are dynamically stable under perturbation. To this end we consider the typical stationary soliton of Fig. 3共a兲 共obtained by the imaginary-time propagation method兲 and subject it to the perturbation

␾j共z,t兲=1.1␾j共z,t兲 and observe the resultant dynamics

共ob-tained by the real-time propagation method兲, which is illus-trated in Fig.4. The solitons under this perturbation execute breathing oscillations and propagate for as long as the nu-merical simulation was continued without being destroyed. This demonstrates the stability of the solitons under pertur-bation.

Finally, in Fig.5 we show the chemical potential for␮b

bosons and ␮f for fermions as a function of Fermi-Fermi

interaction strength gf for Nf= 20, Nb= 300, gb= 0.01, and

gbf= −0.4 and −0.8 obtained from numerical solution and

Gaussian variational analysis. The agreement between the two results for␮b is good whereas for ␮f is only fair. The

increase of the Fermi-Fermi attraction strength兩gf兩 for a fixed

gbf corresponds to a reduction in both chemical potentials,

signaling strongly bound solitons. The same happens for the increase of Bose-Fermi attraction strength 兩gbf兩 from 0.4 to

0.8. In Fig. 5 the small-兩gf兩 limit corresponds to the BCS

phase of bosons whereas the large-兩gf兩 limit corresponds to

the unitarity regime of fermions. The intermediate values of 兩gf兩 denote the crossover from BCS to the unitarity regime. 0 0.06 0.12 0.18 0.24 -8 -4 0 4 8 ni (z) z N_f= 20 N_b= 300 g_b= 0.01 g_bf= -0.4 g_f= -0.1 bos (num) 0 0.06 0.12 0.18 0.24 -8 -4 0 4 8 ni (z) z N_f= 20 N_b= 300 g_b= 0.01 g_bf= -0.4 g_f= -0.1 bos (num) fer (num) 0 0.06 0.12 0.18 0.24 -8 -4 0 4 8 ni (z) z N_f= 20 N_b= 300 g_b= 0.01 g_bf= -0.4 g_f= -0.1 bos (num) fer (num) bos (var) 0 0.06 0.12 0.18 0.24 -8 -4 0 4 8 ni (z) z N_f= 20 N_b= 300 g_b= 0.01 g_bf= -0.4 g_f= -0.1 bos (num) fer (num) bos (var) fer (var) 0 0.06 0.12 0.18 0.24 -8 -4 0 4 8 ni (z) z N_f= 20 N_b= 300 g_b= 0.01 g_bf= -0.4 g_f= -1 bos (num) 0 0.06 0.12 0.18 0.24 -8 -4 0 4 8 ni (z) z N_f= 20 N_b= 300 g_b= 0.01 g_bf= -0.4 g_f= -1 bos (num) fer (num) 0 0.06 0.12 0.18 0.24 -8 -4 0 4 8 ni (z) z N_f= 20 N_b= 300 g_b= 0.01 g_bf= -0.4 g_f= -1 bos (num) fer (num) bos (var) 0 0.06 0.12 0.18 0.24 -8 -4 0 4 8 ni (z) z N_f= 20 N_b= 300 g_b= 0.01 g_bf= -0.4 g_f= -1 bos (num) fer (num) bos (var) fer (var) 0 0.06 0.12 0.18 0.24 0.3 0.36 -6 -4 -2 0 2 4 6 ni (z) z N_f= 20 N_b= 300 g_b= 0.01 g_bf= -0.4 g_f= -10 bos (num) 0 0.06 0.12 0.18 0.24 0.3 0.36 -6 -4 -2 0 2 4 6 ni (z) z N_f= 20 N_b= 300 g_b= 0.01 g_bf= -0.4 g_f= -10 bos (num) fer (num) 0 0.06 0.12 0.18 0.24 0.3 0.36 -6 -4 -2 0 2 4 6 ni (z) z N_f= 20 N_b= 300 g_b= 0.01 g_bf= -0.4 g_f= -10 bos (num) fer (num) bos (var) 0 0.06 0.12 0.18 0.24 0.3 0.36 -6 -4 -2 0 2 4 6 ni (z) z N_f= 20 N_b= 300 g_b= 0.01 g_bf= -0.4 g_f= -10 bos (num) fer (num) bos (var) fer (var) 0 0.06 0.12 0.18 0.24 -50 -40 -30 -20 -10 0 10 20 30 40 50 ni (z) z N_f= 100 N_b= 300 g_b= 0.01 g_bf= -0.4 g_f= -10 bos (num) 0 0.06 0.12 0.18 0.24 -50 -40 -30 -20 -10 0 10 20 30 40 50 ni (z) z N_f= 100 N_b= 300 g_b= 0.01 g_bf= -0.4 g_f= -10 bos (num) fer (num) 0 0.06 0.12 0.18 0.24 -50 -40 -30 -20 -10 0 10 20 30 40 50 ni (z) z N_f= 100 N_b= 300 g_b= 0.01 g_bf= -0.4 g_f= -10 bos (num) fer (num) bos (var) 0 0.06 0.12 0.18 0.24 -50 -40 -30 -20 -10 0 10 20 30 40 50 ni (z) z N_f= 100 N_b= 300 g_b= 0.01 g_bf= -0.4 g_f= -10 bos (num) fer (num) bos (var) fer (var) (a) (c) (b) (d)

FIG. 3.共Color online兲 Probability densities from numerical so-lution of Eqs.共9兲 and 共10兲 关here normalized to unity: 兰₋⬁_⬁ni共z兲dz

= 1兴 compared with variational results given by Eqs. 共44兲 and 共45兲

for Nb= 300, gb= 0.01, gbf= −0.4, and共a兲 Nf= 20, gf= −0.1,共b兲 Nf

= 15, g_f= −1, 共c兲 N_f= 20, g_f= −10, and 共d兲 N_f= 100, g_f= −10. Of these, 共a兲 represents fermions in the BCS regime, 共b兲 represents fermions in the BCS-to-unitarity crossover,共c兲 represents fermions in the unitarity regime, and共d兲 represents bosons and fermions in a partially demixed configuration.

-12 -8 -4

0

₄

₈

₁₂

z

0

20

40

60

80

t

0

0.04

0.08

0.12

0.16

0.2

0.24

0.28

n

_b

(z,t)

-12 -8 -4

₀

₄

₈

12

z

0

20

40

60

80

t

0

0.04

0.08

0.12

0.16

0.2

n

_f

(z,t)

(a)

(b)

FIG. 4. 共Color online兲 Dynamics of the probability density pro-files of共a兲 Bose and 共b兲 Fermi solitons of Fig.3共a兲when at t = 20 they are subject to a quite strong perturbation by setting ␾_j共z,t兲 = 1.1␾j共z,t兲. The solitons undergo stable propagation as long as we

could continue the numerical simulation. The initial soliton profile is calculated with an imaginary-time propagation algorithm and the dynamics studied with a real-time propagation algorithm. The soli-ton profiles are normalized to unity:兰_−⬁⬁ n_j共z,t兲dz=1.

V. SUMMARY AND CONCLUSION

In this paper we have studied a one-dimensional super-fluid Bose-Fermi mixture using a set of coupled mean-field equations derived as the Euler-Lagrange equation employing the Lieb-Liniger energy density of bosons and the Gaudin-Yang energy density of fermions and an interaction term pro-portional to the product of probability density of bosons and fermions. This set of coupled equations has a complex non-linearity structure for fermions and bosons and shows the proper transition from a cubic Bose nonlinearity in the weak-coupling GP BCS limit to a quintic nonlinearity in the strong-coupling TG共Tonks-Girardeau兲 limit. In addition, for fermions with attractive interactions considered in this paper, it shows the proper transition from the weak-coupling BCS regime to unitarity limit: both limits are described by a quin-tic nonlinearity with different coefficients. In this model, in the extreme weak-coupling BCS limit the superfluid fermi-onic energy density is identical to that of a noninteracting

degenerate Fermi gas in the normal state关52兴.

We consider two distinct situations for the superfluid Bose-Fermi mixture: 共i兲 in a ring with periodic boundary condition realizable from a toroidal trap in the limit of strong transverse confinement and 共ii兲 in an infinite cylinder with open boundary condition realizable from an axially symmet-ric trap in the limit of strong transverse and zero axial con-finement.

For the mixture in a ring, from energetic considerations, we obtain the condition of stability of a uniform mixture with a constant probability density. The uniform mixture is energetically stable for interspecies attraction strength 兩gbf兩

below a critical value, above which stable lowest-energy states are bright Bose-Fermi solitons. For the repulsive inter-species interaction the stable lowest-energy states are de-mixed states of Bose-Fermi mixture, where the region of a maximum of boson probability density corresponds to a minimum of fermion probability density. It is also demon-strated algebraically that for attractive Bose-Fermi interac-tion the bright solitons can be created from the uniform mix-ture above the critical Bose-Fermi attraction by modulational instability of the uniform mixture under a weak perturbation. In the one-dimensional infinite cylinder we solved the coupled set of equations for the superfluid Bose-Fermi mix-ture numerically and using a Gaussian variational approxi-mation. We calculated numerically the probability density profiles of the bright solitons as well as their chemical po-tentials and compared them with the respective Gaussian variational approximations. The agreement between the two is good to fair. In this case a partial demixing of the Bose-Fermi solitons is possible, when the Bose-Fermi soliton extends over a very large region in space while the Bose soliton remains fairly localized. We also established numerically the dynamical stability of the Bose-Fermi solitons by inflicting a perturbation on the solitons by multiplying the wave-function profiles by 1.1. The system is then found to propa-gate over a very long period of time executing breathing oscillation without being destroyed, which demonstrated the stability of the solitons. Finally, we comment that in view of the experimental realization of the superfluid Bose-Fermi mixture 关45兴 and observation of solitons in a pure BEC

关36,37兴, the achievement of a bright Bose-Fermi soliton

seems possible through a controlled manipulation of strengths of atomic interactions by varying an external back-ground magnetic field near a Feshbach resonance关47兴 and by

adopting the set of parameters we have used in this paper. ACKNOWLEDGMENTS

S.K.A. thanks FAPESP and CNPq for partial financial support. L.S. acknowledges partial financial support by the Italian GNFM-INdAM through the project “Giovani Ricer-catori” and by Fondazione CARIPARO.

-6 -4 -2 0 -50 -40 -30 -20 -10 0 µb g_f N_f= 20 N_b= 300 g_b= 0.01 g_bf= -0.4 var -6 -4 -2 0 -50 -40 -30 -20 -10 0 µb g_f N_f= 20 N_b= 300 g_b= 0.01 g_bf= -0.4 var num -6 -4 -2 0 -50 -40 -30 -20 -10 0 µb g_f N_f= 20 N_b= 300 g_b= 0.01 g_bf= -0.4 var num g_bf= -0.8 var -6 -4 -2 0 -50 -40 -30 -20 -10 0 µb g_f N_f= 20 N_b= 300 g_b= 0.01 g_bf= -0.4 var num g_bf= -0.8 var num

(a)

(b)

-20 -15 -10 -5 0 -50 -40 -30 -20 -10 0 µf g_f N_f= 20 N_b= 300 g_b= 0.01 g_bf= -0.4 var -20 -15 -10 -5 0 -50 -40 -30 -20 -10 0 µf g_f N_f= 20 N_b= 300 g_b= 0.01 g_bf= -0.4 var num -20 -15 -10 -5 0 -50 -40 -30 -20 -10 0 µf g_f N_f= 20 N_b= 300 g_b= 0.01 g_bf= -0.4 var num g_bf= -0.8 var -20 -15 -10 -5 0 -50 -40 -30 -20 -10 0 µf g_f N_f= 20 N_b= 300 g_b= 0.01 g_bf= -0.4 var num g_bf= -0.8 var num

FIG. 5. 共Color online兲 Chemical potentials for 共a兲 bosons and 共b兲 fermions vs Fermi-Fermi interaction strength gf for Nf= 20, N_b= 300, g_b= 0.01 for g_bf= −0.4 and −0.8 calculated numerically 共labeled “num” shown by symbols兲 compared with variational re-sults共labeled “var” shown by solid lines兲.

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