One-dimensional superfluid Bose-Fermi mixture: Mixing, demixing, and bright solitons Sadhan K. Adhikari1,* and Luca Salasnich2,† 1Instituto de Física Teórica, UNESP, São Paulo State University, 01.405-900 Sao Paulo, Sao Paulo, Brazil 2CNISM and CNR-INFM, Unità di Padova, Dipartimento di Fisica “G. Galilei,” Università di Padova, Via Marzolo 8, 35131 Padova, Italy �Received 30 May 2007; published 28 August 2007� We study an ultracold and dilute superfluid Bose-Fermi mixture confined in a strictly one-dimensional �1D� 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 gbf and 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 gbf �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�s�: 03.75.Ss, 03.75.Hh, 64.75.�g I. INTRODUCTION The effects of quantum statistics are strongly enhanced in strictly one-dimensional �1D� systems �1�. 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 �2�. Recently two experi- mental groups �3,4� have reported the observation of the crossover between a 1D quasi-Bose-Einstein condensate �quasi-BEC� in the weak-coupling mean-field Gross- Pitaevskii �GP� domain and a Tonks-Girardeau �TG� gas �5,6� with ultracold 87Rb 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 �LL� 44 years ago �6�. An extension of the LL theory for finite and inhomogeneous 1D Bose gases has been proposed on the basis the local density approximation �LDA� �7�. The LDA is improved by including a gradient term that repre- sents additional kinetic energy associated with the inhomo- geneity of the gas �8–13�. In the last few years several experimental groups have observed the crossover from the Bardeen-Cooper-Schrieffer �BCS� state of Cooper Fermi pairs to the BEC of molecular dimers with ultracold two-hyperfine-component Fermi va- pors of 40K atoms �14–16� and 16Li �17,18� atoms. It is well known that purely attractive potentials have bound states in 1D and 2D for any strength, contrary to the 3D case �19�. A rigorous theoretical analysis of the ground-state properties of a uniform 1D Fermi gas was performed by Gaudin and Yang �GY� 40 years ago �20�. For repulsive interaction the GY model gives a Tomonaga-Luttinger liquid �21�, while for at- tractive interaction it describes a Luther-Emery �superfluid� liquid �22�. 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 �TG gas� or, equivalently, like 1D noninteract- ing fermions �23,24�. Degenerate Bose-Fermi mixtures of alkali-metal atoms have been experimentally observed in 6,7Li �25,26�, 6Li-23Na �27�, and 40K-87Rb �28�. In these mixtures, the theoretical investigation of phase separation �29–35� and solitonlike structures has drawn significant attention. Bright solitons have been observed in BECs of Li �36� and Rb �37� atoms and studied subsequently �38�. It has been demonstrated us- ing microscopic �39� and mean-field hydrodynamic �40� 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 �41� 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 �solitons� when the in- terspecies attraction is large enough to overcome the Pauli repulsion in the degenerate Fermi gas �and any possible re- pulsion in the BEC� �42�. There have also been studies of the mixing-demixing transition in degenerate Bose-Fermi �43� and Fermi-Fermi �44� 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 �45�, the BCS-Bose crossover �46� in it seems to be under control �14–18� by manipulating the Fermi-Fermi in- teraction near a Feshbach resonance �47� by varying a uni- form external magnetic field. Naturally, the question of the *adhikari@ift.unesp.br; URL: www.ift.unesp.br/users/adhikari †salasnich@pd.infn.it; URL: www.padova.infm.it/salasnich PHYSICAL REVIEW A 76, 023612 �2007� 1050-2947/2007/76�2�/023612�11� ©2007 The American Physical Society023612-1 http://dx.doi.org/10.1103/PhysRevA.76.023612 possibility of bright solitons in a superfluid fermionic con- densate assisted by a BEC �with attractive interspecies Bose- Fermi interaction� 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 �48–50� that an attractive BEC with periodic boundary conditions, which can be experimentally produced with a quasi-1D ring �51�, displays a quantum phase transition from a uniform state to a symmetry-breaking state characterized by a localized bright-soliton condensate �50�. 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 scattering length or with the increase of the number of Fermi atoms leading to large repulsion we find a phase separation between bosons and fermions �a mixing-demixing transition�, in anal- ogy with previous theoretical �29–35� and experimental stud- ies �25� with 3D trapped 6Li-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 �for a sufficiently strong Bose-Fermi attraction� and for the exis- tence of mixing—i.e., states with constant density in space �for weak Bose-Fermi attraction and for Bose-Fermi repul- sion�. 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 equations 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 �radial cylindric� 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 �34,52�. The transverse width of the atom distribution is given by the characteristic harmonic length of the single- particle ground state: a�j =�� / �mj���, 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 ��� �34,52�. The boson-boson interaction is described by a contact pseudopotential with repulsive �posi- tive� scattering length ab. The fermion-fermion interaction is modeled by a contact pseudopotential with attractive �nega- tive� 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 �53� we take ab , �af� , �abf��a�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 density L of the mixture reads L = Lb + L f + Lbf . �1� The term Lb is the bosonic Lagrangian, defined as Lb = i��b * � �t �b + �2 2mb �b * �2 �z2�b − �2 2mb ��b�6G� mbgb �2��b�2� , �2� where �b�z , t� is the hydrodynamic field of the Bose gas along the longitudinal axis, such that nb�z , t�= ��b�z , t��2 is the 1D local probability density of bosonic atoms and gb =2���ab is the 1D boson-boson interaction strength �gb �0�. G�x� is the Lieb-Liniger function �54�, defined as the solution of a Fredholm equation for x�0 and such that �6� G�x� = x − 4 3� x3/2 + ¯ for 0 � x � 1, �2 3 �1 − 2 x + ¯ � for x 1. �3� In the extreme weak-coupling limit x→ +0, and G�x�→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 �9�. In addition, Lb has been successfully used to determine the collective oscillation of the 1D Bose gas with longitudi- nal harmonic confinement �13�. In the strong-coupling limit x→ + and G�x�→�2 /3 while the Lagrangian above re- duces to the strongly repulsive bosonic Lagrangian in the TG limit �5�. As x changes from +0 to + the above Lagrangian shows a continuous transition from the GP BEC to TG phase. SADHAN K. ADHIKARI AND LUCA SALASNICH PHYSICAL REVIEW A 76, 023612 �2007� 023612-2 The fermionic Lagrangian density L f is given instead by L f = i�� f * � �t � f + �2 2mf � f * �2 �z2� f − �2 2mf �� f�6F� mfgf �2�� f�2 � , �4� where � f�z , t� is the field of the 1D superfluid Fermi gas, such that nf�z , t�= �� f�z , t��2 is the 1D fermionic local density along the longitudinal axis and gf =2���af is the 1D fermion-fermion interaction strength �gf �0�. F�x� is the Gaudin-Yang function �55�, defined as the solution of a Fred- holm equation for x�0 and such that �20,23� F�x� = �2 48 �1 − 1 x + 3 4x2 + ¯ � for x � − 1, �2 12 �1 + 6 �2x + ¯ � for − 1 � x � 0. �5� In the static and uniform case the Lagrangian density L f reduces exactly to Gaudin-Yang energy functional �20�, which has been recently used by Fuchs et al. �23�. The physical content of the fermionic Lagrangian �4� is easy to understand. For x→−0 we are in the domain of weak Fermi-Fermi attraction �af ,gf →0, corresponding to the BCS limit�, while F�x�→�2 /12 from Eq. �5�. Consequently, the fermionic interaction term in Eq. �4� involving the F�x� func- tion reduces to ��2 /2mf���2 /12��� f�6 �independent of Fermi- Fermi interaction strength gf or Fermi-Fermi scattering length af�, which is the Pauli repulsive term of noninteract- ing fermions considered in Refs. �52,56� 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 �in nonsuperfluid phase� studied in Ref. �52� corresponding to x→ +0. �Although the probability densities in the x→ +0 �nonsuperfluid degenerate Fermi gas� and x→−0 �superfluid BCS condensate� 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.� 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. �The unitarity limit corresponds to infi- nitely large Fermi-Fermi attraction: af →− .� As the Fermi- Fermi scattering length �af� increases from the BCS to the unitarity limit, the interaction energy in Eq. �4� 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. �4� behaves like that of a Tonks-Girardeau gas of bosons—bosonic molecules—and again becomes in- dependent of af �23�. The system then becomes a Bose gas of molecules and its interaction energy is no longer a function of af. Finally, the Lagrangian density Lbf of the Bose-Fermi in- teraction taken to be of the following standard zero-range form �43,52� Lbf = − gbf��b�2�� f�2, �6� where gbf =2���abf is the 1D boson-fermion interaction strength. The Euler-Lagrange equations of the Lagrangian L are the two following coupled partial differential equations: i��t�b = �− �2 2mb �z 2 + 3 �2 2mb nb 2G�mbgb �2nb � − 1 2 gbnbG��mbgb �2nb � + gbfnf��b, �7� i��t� f = �− �2 2mf �z 2 + 3 �2 2mf nf 2F�mfgf �2nf � − 1 2 gfnfF��mfgf �2nf � + gbfnb�� f , �8� where nj = �� j�2, j=b , f , are probability densities for bosons and fermions, respectively, with the normalization − njdz =Nj. It is convenient to work in terms of dimensionless vari- ables defined in terms of a frequency � and length l ��� / �mb�� by � j = �̂ j /�l, t=2t̂ /�, z= ẑl, gj = ĝj� 2 / �2mbl�, and gbf = ĝbf� 2 / �2mbl�. In terms of these new variables Eqs. �7� and �8� can be written as i�t�b = �− �z 2 + 3nb 2G� gb 2nb � − 1 2 gbnbG�� gb 2nb � + gbfnf��b, �9� i�t� f = �− ��z 2 + 3�nf 2F� gf 2nf � − 1 2 gfnfF�� gf 2nf � + gbfnb�� f , �10� where we have dropped the carets over the variables and where �=mb /mf, nj = �� j�2, j=b , f , are probability densities for bosons and fermions, respectively, with the normalization − njdz=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 7Li-6Li and 39K-40K of experimental interest. The bosonic and fermionic nonlinearities in Eqs. �9� and �10�, respectively, have complex structures in general. How- ever, in the weak-coupling GP limit �gb→0� the bosonic nonlinearity in Eq. �9� is cubic and turns quintic in the strong-coupling TG limit �gb→ + �. The fermionic nonlin- earity in Eq. �10� becomes quintic in both weak �gF→−0� and strong �gf →− � coupling but with coefficients �2 /12 and �2 /48, respectively. For stationary states the solutions of Eqs. �9� and �10� have the form � j =� j exp�−i jt� where j are the respective chemical potentials. Consequently, these equations reduce to ONE-DIMENSIONAL SUPERFLUID BOSE-FERMI… PHYSICAL REVIEW A 76, 023612 �2007� 023612-3 b�b = �− �z 2 + 3nb 2G� gb 2nb � − 1 2 gbnbG�� gb 2nb � + gbfnf��b, �11� f� f = �− �z 2 + 3nf 2F� gf 2nf � − 1 2 gfnfF�� gf 2nf � + gbfnb�� f . �12� 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., R a�j, j=b , f—then effects of curvature can be neglected �50� and one can safely use the Lagrangian density �1� with z=R�, where � the azimuthal angle �50,51�. The energy density of the uniform mixture is immediately derived from the La- grangian density �1�, with Eqs. �2�, �4�, and �6�, dropping out space-time derivatives. It is given by E = nb 3G� gb 2nb � + nf 3F� gf 2nf � + gbfnbnf . �13� 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 / �2�R� and nf =Nf / �2�R�. 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 Ẽ�nb,nf� = E�nb,nf� − b�nb *,nf *�nb − f�nb *,nf *�nf , �14� where b and f are Lagrange multipliers �imposing that the numbers of bosons and fermions are fixed� which may be identified with the Bose and Fermi chemical potentials and nb * and nf * are the values of nb and nf at the minimum. Setting the derivatives of Ẽ with respect to nf and nb equal to zero, one finds b = �E �nb = 3nb 2G� gb 2nb � − 1 2 gbnbG�� gb 2nb � + gbfnf , �15� f = �E �nf = 3nf 2F� gf 2nf � − 1 2 gfnfF�� gf 2nf � + gbfnb. �16� The solution of Eqs. �16� and �15�, which are exactly Eqs. �11� and �12� if one drops out the space derivatives, gives a minimum if the corresponding Hessian of Ẽ is positive—i.e., if �2Ẽ �nb 2 �2Ẽ �nf 2 − � �2Ẽ �nb�nf �2 � 0. �17� 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 �gray� area in the plane �nb ,nf� for different values of gbf. Note that the sign of gbf is not important because, in Eq. �17�, 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 �34� 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. �Color online� Region of stability of the homogeneous mixture denoted by shaded �gray� area in the plane �nb ,nf�. Differ- ent panels correspond to different values of the Bose-Fermi inter- action strength gbf. We set gb=0.1 and �a� gf =−0.1 �corresponding to a weak Fermi-Fermi attraction in the BCS phase� and �b� gf =−30 �corresponding to a strong Fermi-Fermi attraction, with fermions in the unitarity regime�. SADHAN K. ADHIKARI AND LUCA SALASNICH PHYSICAL REVIEW A 76, 023612 �2007� 023612-4 fermions. In fact, a uniform 3D mixture is stable only for small values of the fermionic density �29–33,35�. Figure 1 also shows that when bosons enter in the TG regime �i.e., gb /nb 1� 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. 1 suggest 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�z�, with j=b , f , for different values of gbf calculated by directly solving Eqs. �11� and �12� 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 gbf induces a strong localization while a large and positive gbf produces demixing. These density profiles have been obtained by solv- ing Eqs. �9� and �10� with a finite-difference Crank- Nicholson algorithm and imaginary time �57�. In the regime gb /nb�1 �BEC limit� and �gf� /nf �1 �BCS limit�, the above analysis yields simple analytic results. In this case E=gbnb 2 /2+�2nf 3 /12+gbfnbnf, consequently, Eq. �17� leads to the condition 1 2 �2gbnf � gbf 2 �18� for the stability of the uniform mixture. The condition �2gbnf /2�gbf 2 is qualitatively consistent with Fig. 1�a�. For example, for gb=0.1 and gf =−0.1 the condition of stability of uniform mixture becomes nf �2gbf 2 and for �gbf�=0, 0.2, 0.4, and 0.6 leads to nf �0, 0.08, 0.32, and 0.72, respectively. In the regime gb /nb�1 �BEC limit� and �gf /nf� 1 �uni- tarity or tightly bound molecule formation limit� E=gbnb 2 /2 +�2nf 3 /48+gbfnbnf, consequently, Eq. �17� leads to the con- dition 1 8 �2gbnf � gbf 2 �19� 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 �gbf�=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�b�. The time-independent condi- tions �18� and �19� are also derived in following subsection using rigorous time-dependent modulational instability analysis of the uniform mixture �constant-amplitude solu- tions� in the weak and strong Fermi-Fermi coupling limits. Finally, in the regime gb /nb 1 �TG limit� and �gf /nf� �1 �BCS limit� E=�2nb 3 /3+�2nf 3 /12+gbfnbnf, conse- quently, Eq. �17� leads to the condition �4nbnf � gbf 2 �20� for the stability of the uniform mixture. The functional de- pendence of condition �20� on nb, nf, gb, and gf is qualita- tively different from conditions �18� and �19�. The inequality �17� can be written as cb 2cf 2 � 4gbf 2 nbnf , �21� where cf =�2nf�� f /�nf� is the sound velocity of the super- fluid Fermi component and cb=�2nb�� b /�nb� 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 �58� for a two-component BEC. One finds cbf = 1 �2 �cb 2 + cf 2 ± ��cb 2 − cf 2�2 + 16gbf 2 nbnf . �22� 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 �59,60� of Eqs. �9� and �10� we consider the special case of weak Bose-Bose �small positive gb� and both weak �small �gf /nf��1 corresponding to BCS limit� and strong �large �gf /nf� 1 corresponding to molecular unitarity limit� Fermi-Fermi interactions while these equations reduce to i�t�b = �− �z 2 + gb��b�2 − gbf�� f�2��b, �23� i�t� f = �− �z 2 + ��2�� f�4 − gbf��b�2�� f , �24� where �=1/4 for �gf /nf��1 and 1/16 for �gf /nf� 1. 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 �23� and �24� 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�z� of bosons �dashed line� and nf�z� of fermions �solid line� of the Bose-Fermi mixture with Nb=100 bosons and Nf =10 fermions and periodic boundary condi- tions �axial length L�2�R=10�. We choose gb=0.01 and gf =−0.025 and calculate the profiles for different values of the Bose-Fermi interaction strength gbf. ONE-DIMENSIONAL SUPERFLUID BOSE-FERMI… PHYSICAL REVIEW A 76, 023612 �2007� 023612-5 �b0 = Ab0 exp�i�b� � Ab0eit�gbfAf0 2 −gbAb0 2 �, �25� � f0 = Af0 exp�i� f� � Af0eit�gbfAb0 2 −��2Af0 4 �, �26� of Eqs. �23� and �24�, respectively, where Aj0 is the ampli- tude and � j a phase for component j=b , f . The constant- amplitude solution develops an amplitude-dependent phase on time evolution. We consider a small perturbation Aj exp�i� j� to these solutions via � j = �Aj0 + Aj�exp�i� j� , �27� where Aj =Aj�z , t�. Substituting these perturbed solutions in Eqs. �23� and �24� and for small perturbations retaining only the linear terms in Aj, we get i�tAb + �z 2Ab − gbAb0 2 �Ab + Ab *� + gbfAb0Af0�Af + Af *� = 0, �28� i�tAf + �z 2Af − 2��2Af0 4 �Af + Af *� + gbfAb0Af0�Af + Af *� = 0. �29� We consider the complex plane-wave perturbation Aj�z,t� = A j1 cos�Kt − �z� + iA j2 sin�Kt − �z� , �30� with j=b , f , where A j1 and A j2 are the amplitudes for the real and imaginary parts, respectively, and K and � are fre- quency and wave numbers. Substituting Eq. �30� into Eqs. �28� and �29� and separat- ing the real and imaginary parts we get − Ab1K = Ab2�2, �31� − Ab2K = Ab1�2 − 2gbfAb0Af0A f1 + 2gbAb0 2 Ab1, �32� for j=b, and − Af1K = Af2�2, �33� − A f2K = A f1�2 − 2gbfAb0Af0Ab1 + 4��2Af0 4 Af1, �34� for j= f . Eliminating Ab2 between Eqs. �31� and �32� we get Ab1�K2 − �2��2 + 2gbAb0 2 �� = − 2A f1gbfAb0Af0�2 �35� and eliminating A f2 between �33� and �34�, we have A f1�K2 − �2��2 + 4��2Af0 4 �� = − 2Ab1gbfAb0Af0�2. �36� Finally, eliminating Ab1 and A f1 from Eqs. �35� and �36�, we obtain the dispersion relation K = ± ����2 + gbAb0 2 + 2��2Af0 4 � ± ��gbAb0 2 − 2��2Af0 4 �2 + 4gbf 2 Ab0 2 Af0 2 �1/2�1/2. �37� For stability of the plane-wave perturbation, K has to be real. For any � this happens for �gbAb0 2 + 2��2Af0 4 �2 � �gbAb0 2 − 2��2Af0 4 �2 + 4gbf 2 Ab0 2 Af0 2 �38� or for 2gb��2Af0 2 � gbf 2 . �39� However, for 2gb��2Af0 2 �gbf 2 , 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 �uniform mixture of Sec. III A�, signaling the possibility of coupled Bose-Fermi bright soli- ton to appear. Noting that Af0 2 =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. �39� reduces to Eq. �18� obtained from energetic considerations. In the strong-coupling unitarity limit �=1/16 and Eq. �39� reduces to Eq. �19� for the stability of the uniform mixture. Finally, we comment that Eq. �20� can also be derived in a straightforward fashion from the modulational instability analysis of an appropriate set of dynamical equations, with gb��b�2 replaced by �2��b�4 in Eq. �23� and �=1/4 in Eq. �24�. 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., L a�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 �1� with z� �− , + �. With a finite number of particles �bosons and fermions� the uniform mixture in a infinite cylinder cannot exist, but localized solutions are instead possible with an attractive Bose-Fermi strength �gbf �0� �52�. A. Variational results Here we develop a variational localized solution to Eqs. �11� and �12�, noting that these equations can be derived from the Lagrangian �61� L = � − � b�b 2 + f� f 2 − ��b�� 2 − �� f�� 2 − �b 6G�gb/2�b 2� − � f 6F�gf/2� f 2� − gbf�b 2� f 2�dz − bNb − fNf , �40� by demanding �L /��b=�L /�� f =�L /� b=�L /� f =0. To develop the variational approximation we use the Gaussian ansatz �62� �b�z� = �−1/4�NbAb wb exp�− z2 2wb 2� , �41� � f�z� = �−1/4�NfAf wf exp�− z2 2wf 2� , �42� where the variational parameters are Aj, the solitons’ norm and wj the widths, in addition to j. Note that this Gaussian ansatz can be extended to include time dependence �63�, as done recently to investigate the collective oscillation of a quasi-1D mixture made of condensed bosons and normal fer- mions �52�. SADHAN K. ADHIKARI AND LUCA SALASNICH PHYSICAL REVIEW A 76, 023612 �2007� 023612-6 The substitution of this variational ansatz in Lagrangian �40� yields L = bNbAb + fNfAf − NbAb 2wb 2 − NfAf 2wf 2 − Ab 3Nb 3 ��3wb 2 G� cgbwb 2NbAb � − Af 3Nf 3 �3�wf 2 F� dgfwf 2NfAf � − gbfNbNfAbAf ���wf 2 + wb 2� − bNb − fNf , �43� with c=�3� /2. The integrals over the G�x� and F�x� func- tions cannot be evaluated exactly for all x. However, the term involving the LL G�x� function can be evaluated analytically in the GP �x→0� and TG �x 1� limits. The analytic term involving the G�x� function in Eq. �43� 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 =�3� /2. The fermionic integral over the GY function F�x� 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=�3� /2 which provides a faithful analytical representation of the numerical result. The first variational equations emerging from Eq. �43� �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 variational equations �L /�wj =0 lead to 1 + 2Nb 2 ��3 G� cgbwb 2Nb � − Nbgbwb 2�2� G�� cgbwb 2Nb � + gbfNfwb 4 ���wb 2 + wf 2�3/2 = 0, �44� 1 + 2Nf 2 �3� F� cgfwf 2Nf � − Nfgfwf 2�2� F�� cgfwf 2Nf � + gbfNbwf 4 ���wb 2 + wf 2�3/2 = 0. �45� 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 �2� G�� cgbwb 2Nb � + �3Nb 2 �wb 2 G� cgbwb 2Nb � + gbfNf ���wf 2 + wb 2� , �46� f = 1 2wf 2 + �3Nf 2 �wf 2 F� cgfwf 2Nf � − Nfgf 2wf �2� F�� cgfwf 2Nf � + gbfNb ���wf 2 + wb 2� . �47� Equations �44�–�47� 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 �11� and �12� by using an imaginary-time propagation method based on the finite-difference Crank-Nicholson dis- cretization scheme of time-dependent equations �9� and �10�. The nonequilibrium dynamics from an initial stationary state is studied by solving the time-dependent equations �9� and �10� 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 �64�. 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 �self-defocusing� 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 �self-defocusing� 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 �mixed� 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. �11� and �12� by the tech- nique of imaginary-time propagation and compared them with the Gaussian variational results obtained from Eqs. �44� and �45�. 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. 3 we present typical soliton profiles illustrating the change in the results during BCS-unitarity crossover as well as the demixed profiles. In Figs. 3�a�–3�c� we show the soli- ton profiles for weak �BCS phase�, moderate, and strong �unitarity regime� 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. Figure 3�d� illustrates a demixed state ob- tained by increasing the fermion number from the configu- ration of Fig. 3�c� from Nf =20 to 100. In this case the �nu- merically obtained� 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�a� �obtained by the imaginary- time propagation method� and subject it to the perturbation ONE-DIMENSIONAL SUPERFLUID BOSE-FERMI… PHYSICAL REVIEW A 76, 023612 �2007� 023612-7 � j�z , t�=1.1� j�z , t� and observe the resultant dynamics �ob- tained by the real-time propagation method�, 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 �gf� 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 �gbf� from 0.4 to 0.8. In Fig. 5 the small-�gf� limit corresponds to the BCS phase of bosons whereas the large-�gf� limit corresponds to the unitarity regime of fermions. The intermediate values of �gf� denote the crossover from BCS to the unitarity regime. 0 0.06 0.12 0.18 0.24 -8 -4 0 4 8 n i (z ) z Nf = 20 Nb = 300 gb = 0.01 gbf = -0.4 gf = -0.1 bos (num) 0 0.06 0.12 0.18 0.24 -8 -4 0 4 8 n i (z ) z Nf = 20 Nb = 300 gb = 0.01 gbf = -0.4 gf = -0.1 bos (num) fer (num) 0 0.06 0.12 0.18 0.24 -8 -4 0 4 8 n i (z ) z Nf = 20 Nb = 300 gb = 0.01 gbf = -0.4 gf = -0.1 bos (num) fer (num) bos (var) 0 0.06 0.12 0.18 0.24 -8 -4 0 4 8 n i (z ) z Nf = 20 Nb = 300 gb = 0.01 gbf = -0.4 gf = -0.1 bos (num) fer (num) bos (var) fer (var) 0 0.06 0.12 0.18 0.24 -8 -4 0 4 8 n i (z ) z Nf = 20 Nb = 300 gb = 0.01 gbf = -0.4 gf = -1 bos (num) 0 0.06 0.12 0.18 0.24 -8 -4 0 4 8 n i (z ) z Nf = 20 Nb = 300 gb = 0.01 gbf = -0.4 gf = -1 bos (num) fer (num) 0 0.06 0.12 0.18 0.24 -8 -4 0 4 8 n i (z ) z Nf = 20 Nb = 300 gb = 0.01 gbf = -0.4 gf = -1 bos (num) fer (num) bos (var) 0 0.06 0.12 0.18 0.24 -8 -4 0 4 8 n i (z ) z Nf = 20 Nb = 300 gb = 0.01 gbf = -0.4 gf = -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 n i (z ) z Nf = 20 Nb = 300 gb = 0.01 gbf = -0.4 gf = -10 bos (num) 0 0.06 0.12 0.18 0.24 0.3 0.36 -6 -4 -2 0 2 4 6 n i (z ) z Nf = 20 Nb = 300 gb = 0.01 gbf = -0.4 gf = -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 n i (z ) z Nf = 20 Nb = 300 gb = 0.01 gbf = -0.4 gf = -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 n i (z ) z Nf = 20 Nb = 300 gb = 0.01 gbf = -0.4 gf = -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 n i (z ) z Nf = 100 Nb = 300 gb = 0.01 gbf = -0.4 gf = -10 bos (num) 0 0.06 0.12 0.18 0.24 -50 -40 -30 -20 -10 0 10 20 30 40 50 n i (z ) z Nf = 100 Nb = 300 gb = 0.01 gbf = -0.4 gf = -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 n i (z ) z Nf = 100 Nb = 300 gb = 0.01 gbf = -0.4 gf = -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 n i (z ) z Nf = 100 Nb = 300 gb = 0.01 gbf = -0.4 gf = -10 bos (num) fer (num) bos (var) fer (var) (a) (c) (b) (d) FIG. 3. �Color online� Probability densities from numerical so- lution of Eqs. �9� and �10� �here normalized to unity: − ni�z�dz =1� compared with variational results given by Eqs. �44� and �45� for Nb=300, gb=0.01, gbf =−0.4, and �a� Nf =20, gf =−0.1, �b� Nf =15, gf =−1, �c� Nf =20, gf =−10, and �d� Nf =100, gf =−10. Of these, �a� represents fermions in the BCS regime, �b� represents fermions in the BCS-to-unitarity crossover, �c� represents fermions in the unitarity regime, and �d� represents bosons and fermions in a partially demixed configuration. -12 -8 -4 0 4 8 12z 0 20 40 60 80 t 0 0.04 0.08 0.12 0.16 0.2 0.24 0.28 nb(z,t) -12 -8 -4 0 4 8 12z 0 20 40 60 80 t 0 0.04 0.08 0.12 0.16 0.2 nf(z,t) (a) (b) FIG. 4. �Color online� Dynamics of the probability density pro- files of �a� Bose and �b� Fermi solitons of Fig. 3�a� when at t=20 they are subject to a quite strong perturbation by setting � j�z , t� =1.1� j�z , t�. 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: − nj�z , t�dz=1. SADHAN K. ADHIKARI AND LUCA SALASNICH PHYSICAL REVIEW A 76, 023612 �2007� 023612-8 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 �Tonks-Girardeau� 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 �52�. We consider two distinct situations for the superfluid Bose-Fermi mixture: �i� in a ring with periodic boundary condition realizable from a toroidal trap in the limit of strong transverse confinement and �ii� 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 �gbf� 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 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 �45� and observation of solitons in a pure BEC �36,37�, 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 �47� 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 µ b gf Nf = 20 Nb = 300 gb = 0.01 gbf= -0.4 var -6 -4 -2 0 -50 -40 -30 -20 -10 0 µ b gf Nf = 20 Nb = 300 gb = 0.01 gbf= -0.4 var num -6 -4 -2 0 -50 -40 -30 -20 -10 0 µ b gf Nf = 20 Nb = 300 gb = 0.01 gbf= -0.4 var num gbf= -0.8 var -6 -4 -2 0 -50 -40 -30 -20 -10 0 µ b gf Nf = 20 Nb = 300 gb = 0.01 gbf= -0.4 var num gbf= -0.8 var num (a) (b) -20 -15 -10 -5 0 -50 -40 -30 -20 -10 0 µ f gf Nf = 20 Nb = 300 gb = 0.01 gbf= -0.4 var -20 -15 -10 -5 0 -50 -40 -30 -20 -10 0 µ f gf Nf = 20 Nb = 300 gb = 0.01 gbf= -0.4 var num -20 -15 -10 -5 0 -50 -40 -30 -20 -10 0 µ f gf Nf = 20 Nb = 300 gb = 0.01 gbf= -0.4 var num gbf= -0.8 var -20 -15 -10 -5 0 -50 -40 -30 -20 -10 0 µ f gf Nf = 20 Nb = 300 gb = 0.01 gbf= -0.4 var num gbf= -0.8 var num FIG. 5. �Color online� Chemical potentials for �a� bosons and �b� fermions vs Fermi-Fermi interaction strength gf for Nf =20, Nb=300, gb=0.01 for gbf =−0.4 and −0.8 calculated numerically �labeled “num” shown by symbols� compared with variational re- sults �labeled “var” shown by solid lines�. ONE-DIMENSIONAL SUPERFLUID BOSE-FERMI… PHYSICAL REVIEW A 76, 023612 �2007� 023612-9 �1� M. Takahashi, Thermodynamics of One-Dimensional Solvable Models �Cambridge University Press, Cambridge, England, 1999�. �2� A. J. Leggett, Quantum Liquids: Bose Condensation and Coo- per Pairing in Condensed-Matter Systems �Oxford University Press, Oxford, 2006�. �3� T. Kinoshita, T. Wenger, and D. S. Weiss, Science 305, 1125 �2004�. �4� B. Paredes et al., Nature �London� 429, 277 �2004�. �5� M. Girardeau, J. Math. Phys. 1, 516 �1960�. �6� E. H. Lieb and W. Liniger, Phys. Rev. 130, 1605 �1963�. �7� D. S. Petrov, G. V. Shlyapnikov, and J. T. M. Walraven, Phys. Rev. Lett. 85, 3745 �2000�; V. Dunjko, V. Lorent, and M. Olshanii, ibid. 86, 5413 �2001�. �8� P. Öhberg and L. Santos, Phys. Rev. Lett. 89, 240402 �2002�. �9� E. H. Lieb, R. Seiringer, and J. Yngvason, Phys. Rev. Lett. 91, 150401 �2003�; Commun. Math. Phys. 244, 347 �2004�. �10� L. Salasnich, A. Parola, and L. Reatto, Phys. Rev. A 72, 025602 �2005�. �11� L. Salasnich, Laser Phys. 12, 198 �2002�; L. Salasnich, A. Parola, and L. Reatto, Phys. Rev. A 65, 043614 �2002�. �12� L. Salasnich, A. Parola, and L. Reatto, Phys. Rev. A 69, 045601 �2004�; L. Salasnich, J. Phys. B 39, 1743 �2006�. �13� L. Salasnich, A. Parola, and L. Reatto, Phys. Rev. A 70, 013606 �2004�. �14� M. Greiner, C. A. Regal, and D. S. Jin, Nature �London� 426, 537 �2003�. �15� C. A. Regal, M. Greiner, and D. S. Jin, Phys. Rev. Lett. 92, 040403 �2004�. �16� J. Kinast, S. L. Hemmer, M. E. Gehm, A. Turlapov, and J. E. Thomas, Phys. Rev. Lett. 92, 150402 �2004�. �17� M. W. Zwierlein, C. A. Stan, C. H. Schunck, S. M. F. Raupach, A. J. Kerman, and W. Ketterle, Phys. Rev. Lett. 92, 120403 �2004�; M. W. Zwierlein, C. H. Schunck, C. A. Stan, S. M. F. Raupach, and W. Ketterle, Phys. Rev. Lett. 94, 180401 �2005�. �18� C. Chin et al., Science 305, 1128 �2004�; M. Bartenstein, A. Altmeyer, S. Riedl, S. Jochim, C. Chin, J. H. Denschlag, and R. Grimm, ibid. 92, 203201 �2004�. �19� L. D. Landau and E. M. Lifshitz, Quantum Mechanics—Non Relativistic Theory, Course of Theoretical Physics, Vol. 3 �Per- gamon Press, New York, 1989�. �20� M. Gaudin, Phys. Lett. 24A, 55 �1967�; C. N. Yang, Phys. Rev. Lett. 19, 1312 �1967�. �21� S. Tomonaga, Prog. Theor. Phys. 5, 544 �1950�; J. M. Lut- tinger, J. Math. Phys. 4, 1154 �1963�. �22� A. Luther and V. J. Emery, Phys. Rev. Lett. 33, 589 �1974�. �23� J. N. Fuchs, A. Recati, and W. Zwerger, Phys. Rev. Lett. 93, 090408 �2004�. �24� I. V. Tokatly, Phys. Rev. Lett. 93, 090405 �2004�. �25� A. G. Truscott et al., Science 291, 2570 �2001�; G. Modugno, G. Roati, F. Riboli, F. Ferlaino, R. J. Brecha, and M. Inguscio, ibid. 297, 2240 �2002�. �26� F. Schreck, L. Khaykovich, K. L. Corwin, G. Ferrari, T. Bour- del, J. Cubizolles, and C. Salomon, Phys. Rev. Lett. 87, 080403 �2001�. �27� Z. Hadzibabic, C. A. Stan, K. Dieckmann, S. Gupta, M. W. Zwierlein, A. Gorlitz, and W. Ketterle, Phys. Rev. Lett. 88, 160401 �2002�. �28� B. DeMarco and D. S. Jin, Science 285, 1703 �1999�; G. Roati, F. Riboli, G. Modugno, and M. Inguscio, Phys. Rev. Lett. 89, 150403 �2002�. �29� K. Mölmer, Phys. Rev. Lett. 80, 1804 �1998�; R. Roth, Phys. Rev. A 66, 013614 �2002�; P. Capuzzi, A. Minguzzi, and M. P. Tosi, ibid. 67, 053605 �2003�; M. Modugno, F. Ferlaino, F. Riboli, G. Roati, G. Modugno, and M. Inguscio, ibid. 68, 043626 �2003�. �30� N. Nygaard and K. Mølmer, Phys. Rev. A 59, 2974 �1999�. �31� M. J. Bijlsma, B. A. Heringa, and H. T. C. Stoof, Phys. Rev. A 61, 053601 �2000�. �32� H. Heiselberg, C. J. Pethick, H. Smith, and L. Viverit, Phys. Rev. Lett. 85, 2418 �2000�. �33� L. Viverit, C. J. Pethick, and H. Smith, Phys. Rev. A 61, 053605 �2000�; L. Viverit, ibid. 66, 023605 �2002�. �34� K. K. Das, Phys. Rev. Lett. 90, 170403 �2003�. �35� L. Salasnich and F. Toigo, Phys. Rev. A 75, 013623 �2007�. �36� K. E. Strecker, G. B. Partridge, A. G. Truscott, and R. G. Hulet, Nature �London� 417, 150 �2002�; L. Khaykovich, F. Schreck, G. Ferrari, T. Bourdel, J. Cubizolles, L. D. Carr, Y. Castin, and C. Salomon, Science 256, 1290 �2002�; V. M. Pérez-García, H. Michinel, and H. Herrero, Phys. Rev. A 57, 3837 �1998�. �37� S. L. Cornish, S. T. Thompson, and C. E. Wieman, Phys. Rev. Lett. 96, 170401 �2006�. �38� K. E. Strecker, G. B. Partridge, A. G. Truscott, and R. G. Hulet, New J. Phys. 5, 73 �2003�; V. A. Brazhnyi and V. V. Konotop, Mod. Phys. Lett. B 18, 627 �2004�; F. Kh. Abdul- laev, A. Gammal, A. M. Kamchatnov, and L. Tomio, Int. J. Mod. Phys. B 19, 3415 �2005�; V. I. Yukalov, Laser Phys. Lett. 1, 435 �2004�; A. Minguzzi, S. Succi, F. Toschi, M. P. Tosi, and P. Vignolo, Phys. Rep. 395, 223 �2004�. �39� T. Karpiuk, K. Brewczyk, S. Ospelkaus-Schwarzer, K. Bongs, M. Gajda, and K. Rzazewski, Phys. Rev. Lett. 93, 100401 �2004�; T. Karpiuk, M. Brewczyk, and K. Rzazewski, Phys. Rev. A 73, 053602 �2006�. �40� S. K. Adhikari, Phys. Rev. A 72, 053608 �2005�. �41� S. K. Adhikari, J. Phys. A 40, 2673 �2007�; Eur. Phys. J. D 40, 157 �2006�; Laser Phys. Lett. 3, 605 �2006�; J. Phys. B 38, 3607 �2005�; I. Kourakis et al., Eur. Phys. J. B 46, 381 �2005�. �42� S. K. Adhikari, Phys. Lett. A 346, 179 �2005�; V. M. Pérez- García and J. B. Beitia, Phys. Rev. A 72, 033620 �2005�. �43� S. K. Adhikari and L. Salasnich, Phys. Rev. A 75, 053603 �2007�. �44� S. K. Adhikari, Phys. Rev. A 73, 043619 �2006�; S. K. Adhikari and B. A. Malomed, ibid. 74, 053620 �2006�. �45� M. W. Zwierlein, J. R. Abo-Shaeer, A. Schirotzek, C. H. Schunck, and W. Ketterle, Nature �London� 435, 1074 �2005�. �46� D. M. Eagles, Phys. Rev. 186, 456 �1969�; A. J. Leggett, J. Phys. �Paris�, Colloq. 41, C7-19 �1980�; M. Casas, J. M. Ge- tino, M. de Llano, A. Puente, R. M. Quick, H. Rubio, and D. M. van dev Walt, Phys. Rev. B 50, 15945 �1994�; S. K. Adhikari, M. Casas, A. Puente, A. Rigo, M. Fortes, M. A. Solis, M. de Llano, A. A. Valladaves, and O. Rajo, ibid. 62, 8671 �2000�; Physica C 453, 37 �2007�. �47� S. Inouye, M. R. Andrews, J. Stenger, H. J. Miesner, D. M. Stamper-Kurn, and W. Ketterle, Nature �London� 392, 151 �1998�. �48� G. M. Kavoulakis, Phys. Rev. A 67, 011601�R� �2003�; R. Kanamoto, H. Saito, and M. Ueda, ibid. 67, 013608 �2003�. �49� R. Kanamoto, H. Saito, and M. Ueda, Phys. Rev. A 68, 043619 �2003�; G. M. Kavoulakis, ibid. 69, 023613 �2004�. SADHAN K. ADHIKARI AND LUCA SALASNICH PHYSICAL REVIEW A 76, 023612 �2007� 023612-10 �50� L. Salasnich, A. Parola, and L. Reatto, Phys. Rev. A 59, 2990 �1999�; A. Parola, L. Salasnich, R. Rota, and L. Reatto, ibid. 72, 063612 �2005�; L. Salasnich, A. Parola, and L. Reatto, ibid. 74, 031603�R� �2006�. �51� S. Gupta, K. W. Murch, K. L. Moore, T. P. Purdy, and D. M. Stamper-Kurn, Phys. Rev. Lett. 95, 143201 �2005�. �52� L. Salasnich, S. K. Adhikari, and F. Toigo, Phys. Rev. A 75, 023616 �2007�. �53� M. Olshanii, Phys. Rev. Lett. 81, 938 �1998�. �54� The Lieb-Liniger function G�x� is such that for x�1 one finds G�x��x+Bx3/2+�x2, where B=4/ �3�� and �=0.0648 �from numerics�. Instead for x 1 one has G�x����2 /2�x2 / �x+2�2. G�x� can be very well described by the following Padè approx- imant: G�x�= �x+Ax3� / �1+Bx1/2+Cx�x+2�2�, where A =�2�B2−�� /12 and C= �B2−�� /4. �55� The Gaudin-Yang function F�x� is conveniently parametrized by the following Padè approximant: F�x�= ��2 /48��x2−x +3/4� / �x2+q1x+q2�, where q1=−�9/ �8�2�+1/4�=−0.3633 and q2=3/16. �56� L. Salasnich, J. Math. Phys. 41, 8016 �2000�; L. Salasnich, B. Pozzi, A. Parola, and L. Reatto, J. Phys. B 33, 3943 �2000�. �57� E. Cerboneschi, R. Mannella, E. Arimondo, and L. Salasnich, Phys. Lett. A 249, 495 �1998�; L. Salasnich, A. Parola, and L. Reatto, Phys. Rev. A 64, 023601 �2001�. �58� A. S. Alexandrov and V. V. Kabanov, J. Phys.: Condens. Mat- ter 14, L327 �2002�. �59� V. V. Konotop and M. Salerno, Phys. Rev. A 65, 021602�R� �2002�; L. Salasnich, A. Parola, and L. Reatto, Phys. Rev. Lett. 91, 080405 �2003�. �60� L. Salasnich, N. Manini, F. Bonelli, M. Korbman, and A. Pa- rola, Phys. Rev. A 75, 043616 �2007�. �61� B. A. Malomed, in Progress in Optics edited by E. Wolf �North-Holland, Amsterdam, 2002�, Vol. 43, p. 71; V. M. Pérez-García, H. Michinel, J. I. Cirac, M. Lewenstein, and P. Zoller, Phys. Rev. A 56, 1424 �1997�. �62� L. Salasnich, Mod. Phys. Lett. B 11, 1249 �1997�; A. Parola, L. Salasnich, and L. Reatto, Phys. Rev. A 57, R3180 �1998�; L. Salasnich, Mod. Phys. Lett. B 12, 649 �1998�. �63� L. Reatto, A. Parola, and L. Salasnich, J. Low Temp. Phys. 113, 195 �1998�; L. Salasnich, Phys. Rev. A 61, 015601 �2000�; Int. J. Mod. Phys. B 14, 1 �2000�; S. K. Adhikari, Phys. Rev. E 71, 016611 �2005�; J. Phys. B 38, 579 �2005�. �64� S. K. Adhikari and P. Muruganandam, J. Phys. B 35, 2831 �2002�; S. K. Adhikari, Phys. Rev. A 69, 063613 �2004�; P. Muruganandam and S. K. Adhikari, J. Phys. B 36, 2501 �2003�; S. K. Adhikari, Phys. Lett. A 265, 91 �2000�; Phys. Rev. E 62, 2937 �2000�;Phys. Rev. C 19, 1729 �1979�. ONE-DIMENSIONAL SUPERFLUID BOSE-FERMI… PHYSICAL REVIEW A 76, 023612 �2007� 023612-11