Fermionic bright soliton in a boson-fermion mixture
Sadhan K. Adhikari*Instituto de Física Teórica, Universidade Estadual Paulista, 01.405-900 São Paulo, São Paulo, Brazil
共Received 3 May 2005; revised manuscript received 28 June 2005; published 10 November 2005兲 We use a time-dependent dynamical mean-field-hydrodynamic model to study the formation of fermionic bright solitons in a trapped degenerate Fermi gas mixed with a Bose-Einstein condensate in a quasi-one-dimensional cigar-shaped geometry. Due to a strong Pauli-blocking repulsion among spin-polarized fermions at short distances there cannot be bright fermionic solitons in the case of repulsive boson-fermion interactions. However, we demonstrate that stable bright fermionic solitons can be formed for a sufficiently attractive boson-fermion interaction in a boson-fermion mixture. We also consider the formation of fermionic solitons in the presence of a periodic axial optical-lattice potential. These solitons can be formed and studied in the laboratory with present technology.
DOI:10.1103/PhysRevA.72.053608 PACS number共s兲: 03.75.Lm, 03.75.Ss
I. INTRODUCTION
There cannot be an effective evaporative cooling leading to a trapped quantum degenerate Fermi gas共DFG兲 due to a strong repulsive Pauli-blocking interaction at low tempera-ture among spin-polarized fermions 关1兴. However, it has been possible to achieve a DFG by sympathetic cooling in the presence of a second boson or fermion component. Re-cently, there have been successful observations 关1–4兴 and associated experimental关5–7兴 and theoretical 关8–12兴 studies of mixtures of a trapped DFG and a Bose-Einstein conden-sate 共BEC兲 by different experimental groups 关1–4兴 in the following systems: 6,7Li 关3兴, 23Na-6Li 关4兴, and 87Rb-40K 关5,6兴. Also, there have been studies of mixtures of two-component trapped DFGs in40K关1兴 and 6Li 关2兴 atoms.
The formation and collapse of a DFG in a boson-fermion mixture 87Rb-40K have been observed and studied by Modugno et al.关5,10,12兴. Although, the fermion-fermion in-teraction at short distances is repulsive due to strong Pauli blocking and hence incapable of leading to collapse, a suffi-ciently attractive boson-fermion interaction could overcome the Pauli repulsion and could result in a collapse of a DGF. Bright solitons in a BEC are formed due to an attractive nonlinear atomic interaction. As the interaction in a pure DFG at short distances is repulsive, there cannot be bright solitons in a DFG.
In this paper we study the possibility of the formation of stable fermionic bright solitons in a mixture of a DFG with a BEC in the presence of a sufficiently attractive boson-fermion interaction which can overcome the Pauli repulsion among fermions. The formation of a fermionic soliton is re-lated to the fact that the system can lower its energy by forming high density regions共the solitons兲 when the attrac-tion between the bosons and fermions is large enough to overcome the Pauli repulsion in the DFG and any possible repulsion in the BEC. In particular we consider the formation of fermionic bright solitons, which can freely move in the
axial direction, in such a mixture for a quasi-one-dimensional cigar-shaped geometry using a coupled time-dependent mean-field-hydrodynamic model where the bosonic component is treated by the mean-field Gross-Pitaevskii equation 关13兴 and the fermionic component is treated by a hydrodynamic model关11兴. This time-dependent mean-field-hydrodynamic model was suggested recently by the present author 关12兴 to study the collapse dynamics of fermions and is a dependent extension of a time-independent model used for the stationary states by Capuzzi
et al.关11兴.
Bright solitons are really eigenfunctions of the one-dimensional nonlinear Schrödinger equation. However, the experimental realization of bright solitons in trapped attrac-tive cigar-shaped BECs has been possible under strong trans-verse binding which, in the case of weak or no axial binding, simulates the ideal one-dimensional situation for the forma-tion of bright solitons. The dimensionless nonlinear Schrödinger共NLS兲 equation in the attractive or self-focusing case关14兴
iut+ uxx+兩u兩2u = 0 共1.1兲
sustains the following bright soliton关14兴:
u共x,t兲 =
冑2B sech关冑
B共x −␦+ 2vt兲兴⫻ exp关− iv共x −␦兲 + i共B − vv兲t + i兴 共1.2兲 with four parameters. The parameter B represents the ampli-tude as well as pulse width,v represents velocity, the
param-eters␦ andare phase constants. The bright soliton profile is easily recognized forv =␦= 0 as 兩u共x,t兲兩=
冑
2Bsech关x冑
B兴.There have been experimental关15兴 and theoretical 关16兴 stud-ies of the formation of bright solitons in a BEC. In view of this, here we study the possibility of the formation of a stable fermionic bright soliton in a boson-fermion mixture.
In recent times there have been routine experimental stud-ies on the formation of BEC in the presence of a periodic axial optical-lattice potential 关17兴. This leads to a different condition of trapping from the harmonic trap and generates a *Electronic address: adhikari@ift.unesp.br; URL: http://
www.ift.unesp.br/users/adhikari/
BEC of distinct modulation. Hence we also consider in this paper the modulations of the fermionic bright solitons in the presence of an optical-lattice potential.
In Sec. II we present the time-dependent mean-field model consisting of a set of coupled partial differential equa-tions involving the bosonic and fermionic probability densi-ties. In the case of a cigar-shaped geometry with stronger radial trapping, the above model is reduced to an effective one-dimensional form appropriate for the study of bright solitons. In Sec. III we present our results for stationary axially-free fermionic bright solitons as well as those formed on an axial periodic optical-lattice potential. We also demon-strate the stability of the bright solitons after a perturbation is applied. The bright solitons are found to execute stable breathing oscillation upon perturbation. Finally, a summary of our findings is given in Sec. IV.
II. NONLINEAR MEAN-FIELD-HYDRODYNAMIC MODEL
The time-dependent Bose-Einstein condensate wave func-tion⌿共r,t兲 at position r and time t may be described by the following mean-field nonlinear Gross-Pitaevskii equation 关13兴
冋
− iប t− ប2 r 2 2mB + VB共r兲 + gBBnB册
⌿B共r,t兲 = 0, 共2.1兲with normalization兰dr兩⌿B共r,t兲兩2= NB. Here mB is the mass
and NB the number of bosonic atoms in the condensate, nB
⬅兩⌿B共r,t兲兩2 is the boson probability density, gBB
= 4ប2aBB/ mB the strength of interatomic interaction, with
aBB the boson-boson scattering length. The trap potential
with axial symmetry may be written as VB共r兲=
1
2mB2共2
+2z2兲, where and are the angular frequencies in the radial共兲 and axial 共z兲 directions with the anisotropy pa-rameter. The probability density of an isolated DFG in the Thomas-Fermi approximation is given by关10兴
nF=
关max共0,兵⑀F− VF共r兲其兲兴3/2
A3/2 , 共2.2兲
where A =ប2共62兲2/3/共2m
F兲,⑀Fis the Fermi energy, mFis the
fermionic mass, and the function max denotes the larger of the arguments. The confining trap potential VF共r兲
=12mFF
2共2+2z2兲 has axial symmetry as the bosonic poten-tial VB共r兲, whereFis the radial frequency. The anisotropy
parameter will be taken to be zero for axially free solitons in the following. The number of fermionic atoms NFis given
by the normalization兰drnF共r兲=NF.
We developed a set of practical time-dependent mean-field-hydrodynamic equations for the interacting boson-fermion mixture starting from the following Lagrangian den-sity关12兴: L = i 2ប
冋
⌿B ⌿B * t −⌿B *⌿B t册
+ i 2ប冋
冑
nF 冑
nF * t −冑
nF *冑
nF t册
+冉
ប 2兩 r⌿B兩2 2mB + VB兩⌿B兩2+ 1 2gBB兩⌿B兩 4冊
+冉
ប 2兩 r冑
nF兩2 6mF + VF兩nF兩 + 3 5A兩nF兩 5/3冊
+ g BFnF兩⌿B兩2, 共2.3兲 where gBF= 2ប2aBF/ mR with the boson-fermion reducedmass mR= mBmF/共mB+ mF兲, where aBFis the boson-fermion
scattering length.
The terms in the first round bracket on the right-hand side of Eq.共2.3兲 are the standard Gross-Pitaevskii terms for the bosons and are related to a Schrödinger-like equation关13兴. However, terms in the second round bracket are derived from the hydrodynamic equation of motion of the fermions 关11兴. Hence, the second kinetic energy term has a different mass factor 6mFand not the conventional Schrödinger mass factor
2mBas in the first integral. Finally, the last term in this
equa-tion corresponds to an interacequa-tion between bosons and fermi-ons. The interaction between fermions in spin polarized state is highly suppressed due to Pauli blocking and has been ne-glected in Eq. 共2.3兲 and will be neglected throughout this paper.
Recently, Jezek et al. 关18兴 used the Thomas-Fermi-Weizsäcker kinetic energy term TF of fermions in their
for-mulation which, in our notation, will correspond to a fermi-onic kinetic energy ofប2兩r
冑
nF兩2/共9mF兲 in Eq. 共2.3兲 in placeof the present term ប2兩
r
冑
nF兩2/共6mF兲. This kinetic energyterm contributes little to this problem compared to the domi-nating 3A兩nF兩5/3/ 5 term in Eq.共2.3兲 and is usually neglected
in the Thomas-Fermi approximation. However, its inclusion leads to an analytic solution for the probability density ev-erywhere关18兴. For a discussion of these two fermionic ki-netic energy terms we refer the reader to Refs.关11,18,19兴.
With the Lagrangian density 共2.3兲, the Euler-Lagrange equations of motion become关12兴
冋
− iប t− ប2 r 2 2mB + VB共r兲 + gBBnB+ gBFnF册
⌿B共r,t兲 = 0, 共2.4兲冋
− iប t− ប2 r 2 6mF + VF共r兲 + A兩nF兩2/3+ gBFnB册
冑
nF共r,t兲 = 0. 共2.5兲 When the nonlinearity in Eq.共2.5兲 is very large, the ki-netic energy term in this equation can be neglected and the time-independent stationary form of this equation becomesnF=
关max„0,兵⑀F− VF共r兲 − gBFnB其…兴3/2
A3/2 , 共2.6兲
which is the generalization of Eq. 共2.2兲 in the presence of boson-fermion coupling. Equation 共2.6兲 has been used by Modugno et al.关10兴 for an analysis of a DFG-BEC mixture. In actual experimental condition the nonlinearity in Eq.共2.5兲
is quite large and Eq.共2.6兲 is a good approximation. For the study of bright solitons we shall reduce Eqs.共2.4兲 and 共2.5兲 to the minimal one-dimensional form under the action of stronger radial trapping. The one-dimensional form is appropriate for studying bright solitons in the so-called cigar-shaped quasi-one-dimensional geometry where Ⰶ1. For radially-bound and axially-free solitons we eventually set= 0. In this case the dynamical equations can be reduced to strict one-dimensional coupled NLS equations without any trap. We perform this reduction below where we take VB共r兲
= VF共r兲=
1
2mB2共2+2z2兲 which corresponds to a reduction
ofF andFin VF共r兲 by a factor
冑
mB/ mF as in the studyby Modugno et al.关10兴 and Jezek et al. 关18兴.
For= 0, Eqs.共2.4兲 and 共2.5兲 can be reduced to an effec-tive one-dimensional form by considering solutions of the type ⌿B共r,t兲=B共z,t兲B 共0兲共兲 and
冑
n F共r,t兲=F共z,t兲F 共0兲共兲, where 兩i共0兲共兲兩2⬅ Mi ប exp冉
− Mi2 ប冊
, 共2.7兲and i = B , F represents bosons and fermions and MB= mBand
MF=
冑
3mBmF. The expression 共2.7兲 corresponds to there-spective ground-state wave function in the absence of non-linear interactions and satisfies
− ប 2 2mB ⵜ2 B 共0兲+1 2mB 22 B 共0兲=ប B 共0兲, 共2.8兲 − ប 2 6mF ⵜ2 F 共0兲+1 2mB 22 F 共0兲=
冑
mB 3mF បF共0兲, 共2.9兲with normalization 2兰0⬁兩共0兲i 共兲兩2d= 1. Now the dynamics is carried byi共z,t兲 and the radial dependence is frozen in
the ground state i共0兲共兲. The factorization of ⌿B and
冑
nFabove follows from the structure of the mathematical equa-tions共2.4兲 and 共2.5兲. Although nF gives the probability
den-sity of DFG it may not be to the point to associateF0andF
to physical fermionic one-particle wave functions. The true fermionic wave function has the form of a many-particle Slater determinant. Nevertheless, the functions F
0
and F
could be regarded as mathematical functions related to fer-mionic density 关8兴. In the quasi-one-dimensional cigar-shaped geometry the linear fermionic and bosonic probabil-ity densities are given by 兩F共z,t兲兩2 and 兩B共z,t兲兩2,
respectively.
Averaging over the radial modei共0兲共兲, i.e., multiplying Eqs. 共2.4兲 and 共2.5兲 by i共0兲*共兲 and integrating over , we obtain the following one-dimensional dynamical equations 关20兴:
冋
− iប t− ប2 2mB 2 z2+ FBB兩B兩 2+ F BF兩F兩2册
B共z,t兲 = 0, 共2.10兲冋
− iប t− ប2 6mF 2 z2+ FFF兩F兩 4/3+ F BF兩B兩2册
F共z,t兲 = 0, 共2.11兲 where FBB= gBB冕
0 ⬁ 兩B共0兲兩4d冕
0 ⬁ 兩B共0兲兩2d = gBB mB 2ប, 共2.12兲 FBF= gBF冕
0 ⬁ 兩F共0兲兩2兩B共0兲兩2d冕
0 ⬁ 兩B共0兲兩2d = gBF MBF ប , 共2.13兲 FFF= A冕
0 ⬁ 兩F共0兲兩2+4/3d冕
0 ⬁ 兩B共0兲兩2d =3A 5冋
MF ប册
2/3 . 共2.14兲In Eq. 共2.13兲 MBF= MBMF/共MB+ MF兲. In Eqs. 共2.10兲 and
共2.11兲 the normalization is given by 兰−⬁⬁兩i共z,t兲兩2dz = Ni. In
these equations we have set the anisotropy parameter= 0 to remove the axial trap and thus to generate axially free quasi-one-dimensional solitons.
For calculational purposes it is convenient to reduce the set of Eqs.共2.10兲 and 共2.11兲 to dimensionless form by intro-ducing convenient dimensionless variables. Although the al-gebra is quite straightforward, the expressions become messy with different factors of masses. As we shall not be interested in a particular boson-fermion system in this paper, but will be concerned with the formation of fermionic bright solitons in general, we take in the rest of this paper mB= 3mF
= m共87Rb兲, where m共87Rb兲 is the mass of87
Rb, and whence
mR= 3mF/ 4, MB= MF= mB, and MBF= mB/ 2. In the two
ex-perimental situations of Refs.关4,5兴 mB⬇3mF.
In Eqs.共2.10兲 and 共2.11兲, we consider the dimensionless variables= t/ 2, y = z / l,i=
冑
共l/Ni兲i, with l =冑
ប/共mB兲, so that冋
− i − d2 dy2+ NBB兩B兩2+ NBF兩F兩2册
B共y,兲 = 0, 共2.15兲冋
− i − d2 dy2+ NFB兩B兩 2+ N FF兩F兩4/3册
F共y,兲 = 0, 共2.16兲 where NBB= 4aBBNB/ l, NBF= 8aBFNF/ l, NFB= 8aBFNB/ l, andNFF= 9共6NF兲2/3/ 5. In Eqs.共2.15兲 and 共2.16兲, the
normaliza-tion condinormaliza-tion is given by 兰−⬁⬁ 兩i共y,兲兩2dy = 1. Equations
共2.15兲 and 共2.16兲 are the coupled one-dimensional NLS equations describing the formation of solitons in the
DFG-BEC mixture in a cigar-shaped quasi-one-dimensional geometry.
In Eqs.共2.15兲 and 共2.16兲 the term NFF兩F兩4/3represents a
very strong Pauli repulsion which increases with the fermion number NF. The purpose of this study is to show that a
suf-ficiently strong attractive boson-fermion coupling term
NFB兩B兩2 can overcome this repulsion and form the bright
solitons.
III. NUMERICAL RESULT
We solve the coupled mean-field-hydrodynamic equations 共2.15兲 and 共2.16兲 for bright solitons numerically using a time-iteration method based on the Crank-Nicholson dis-cretization scheme elaborated in Ref.关21兴. We discretize the mean-field-hydrodynamic equation using time step 0.0005 and space step 0.025.
We performed the time evolution of the set of equations 共2.15兲 and 共2.16兲 introducing harmonic oscillator potentials
y2in these equations and setting the nonlinear terms to zero:
NBB= NBF= NFB= NFF= 0 and starting with the eigenfunctions
of the linear harmonic oscillator problem, e.g., with
B共y,兲=F共y,兲=−1/4exp共−y2/ 2兲exp共−i兲. The
introduc-tion of the extra harmonic oscillator potential in these equa-tions only aids in starting the time evolution with an exact analytic solution. In the end the harmonic oscillator poten-tials will be set equal to zero and will have no effect on the final wave function for solitons. During the course of time evolution the nonlinear terms are switched on very slowly and the resultant solutions iterated until convergence was obtained. Then the time evolution is continued and the har-monic oscillator potential terms in both bosonic and fermi-onic equations are slowly switched off by reducing the y2 term to zero in 10 000 steps of time evolution. Then the resultant solutions are iterated about 50 000 times for vergence without any harmonic oscillator potential. If con-verged solutions are obtained, they correspond to the re-quired axially free bright solutions in the absence of any axial potential. In our numerical investigation as in the the-oretical study of Refs.关12,18兴 we use= 2⫻100 Hz, and take mBas the mass of87Rb. Consequently, the unit of length
l⬇1m and unit of time 2 /⬇3 ms.
First we solve Eqs.共2.15兲 and 共2.16兲 with NF= NB= 1000,
aBB= 5 nm and aBF= −20 nm. This value of aBBis the
experi-mental scattering length of Rb atoms 关13兴, and aBF=
−20 nm is the experimental scattering length of the Rb-K system关5,6兴. With these parameters the nonlinearities in Eqs. 共2.15兲 and 共2.16兲 are NBB= 20, NBF= −160, NFB= −160, and
NFF= 274.6. The converged bright solitons are plotted in Fig.
1. In this case the fermionic and bosonic functionsF and
B, respectively, have similar spatial extentions. It is possible
to have solitons with different extensions in space by varying the parameters of the system. We took the experimental values for the scattering lengths in Fig. 1. However, the scattering length can be manipulated in the boson-fermion 6
Li-23Na and40K-87Rb systems near the recently discovered Feshbach resonances in them关22兴 by varying a background magnetic field. Thus by varying the scattering length and the number of atoms we could arrive at different values of non-linearity than in Fig. 1.
To simulate a different situation of nonlinearity param-eters next we take aBB= −1 nm, aBF= −1.875 nm, NF= 1000,
and NB= 10 000. So that NBB= −40, NBF= −15, NFB= −150,
and NFF= 274.6. In this case the profiles of the bright solitons
shown in Fig. 2 are very different from those in Fig. 1. In Fig. 1 both the solitons are localized to a small region in space, whereas in Fig. 2 only the bosonic soliton is localized to a small region in space whereas the fermionic soliton ex-tends to a very large region of space.
Next we consider NF= 1000, NB= 10 000, aBB= 0.5 nm,
and aBF= −3.75 nm, so that the nonlinearity parameters are
NBB= 20, NBF= −30, NFB= −300, and NFF= 274.6. The
pro-files of the solitons in Fig. 3 are different from those in Figs. 1 and 2. In this case the bosonic function extends over a longer region in space than the fermionic function. In Figs. 1 and 2 it was the fermionic function that extends over a longer region in space.
In Fig. 2 the nonlinearity NFB appearing in the fermion
equation is less attractive compared to that in Figs. 1 and 3. FIG. 1. 共Color online兲 The stationary function 兩i共z,t兲兩 for
axially-free bosonic 共dotted line兲 and fermionic 共full line兲 bright solitons vs z for NF= NB= 1000, aBB= 5 nm, aBF= −20 nm, harmonic
oscillator length l⬇1m and =0. The arrows in dotted and full lines indicate the bosonic and fermionic axes, respectively. The nonlinearity parameters are NBB= 20, NBF= −160, NFB= −160, and
NFF= 274.6.
FIG. 2.共Color online兲 The stationary function 兩i共z,t兲兩 for axi-ally free bosonic共dotted line兲 and fermionic 共solid line兲 bright soli-tons vs z for NF= 1000, NB= 10 000, aBB= −1 nm, aBF= −1.875 nm, harmonic oscillator length l⬇1m and =0. The arrows in dotted and full lines indicate the bosonic and fermionic axes, respectively. The nonlinearity parameters are NBB= −40, NBF= −15, NFB= −150, and NFF= 274.6.
Hence the resultant nonlinear interaction in the fermion com-ponent is less attractive and hence the fermionic soliton ex-tends to a large distance in space. If the attraction in NFB is
further reduced the fermionic soliton ceases to bind. Hence by manipulating the parameters one could have different situations of localization of the solitons. One could either have both the solitons extending up to similar distance in space as in Fig. 1 or one of the solitons extending over a longer region in space as in Figs. 2 and 3. It is worth em-phasizing that in the fermionic equation共2.16兲 the diagonal nonlinearity NFFis repulsive, hence the binding solely comes
from the attractive off-diagonal nonlinearity NFB
correspond-ing to an attractive boson-fermion interaction. Hence for a fermionic soliton to appear NFBand aBFare always taken to
be negative or attractive. Consequently, NBFis also negative.
Finally, terms NBBand aBBcould be either positive or
nega-tive. When these terms are positive or repulsive the bosonic solitons are formed due to an attractive or negative NBF, as in
Fig. 3.
Next we study the stability of the bright solitons numeri-cally. We consider the soliton of Fig. 3 and during time evo-lution we suddenly jump at t = 100 ms the nonlinearity NBF
from −30 to −33 and the nonlinearity NFB from −300 to
−330. This can be achieved by manipulating a background magnetic field near a Feshbach resonance关22兴 in the boson-fermion interaction and thus varying the boson-boson-fermion scat-tering length by 10%. Due to the sudden change in the non-linearity the bosonic and fermionic bright solitons are set into stable breathing oscillation. The evolution of the wave function profile of the two solitons are shown in Fig. 4. The solitons are found to execute stable non-periodic breathing oscillation. In Fig. 5 we plot the root-mean-square size 具z典r.m.s.of the bosonic and fermionic solitons of Fig. 4 as a function of time. The breathing oscillation of the two solitons after the perturbation is applied results in the stable nonperi-odic oscillation of the root-mean-square sizes illustrated in Fig. 5. As, after applying the perturbation, the boson-fermion attraction has been increased this corresponds to a reduction in the root-mean-square size具z典r.m.s.as we find in Fig. 5. The
steady propagation of the solitons in Fig. 4 and the stable oscillation of their root-mean-square sizes in Fig. 5 after the perturbation is applied will demonstrate the stability of the solitons.
Finally, we consider the fermionic bright solitons formed on a periodic optical-lattice potential. For that purpose we include in Eqs.共2.15兲 and 共2.16兲 the following optical-lattice potential formed by a standing-wave laser beam关17兴: FIG. 3.共Color online兲 The stationary function 兩i共z,t兲兩 for
axi-ally free bosonic共dotted line兲 and fermionic 共solid line兲 bright soli-tons vs z for NF= 1000, NB= 10 000, aBB= 0.5 nm, aBF= −3.75 nm,
=0 and harmonic oscillator length l⬇1m. The arrows in dotted and full lines indicate the bosonic and fermionic axes, respectively. The nonlinearity parameters are NBB= 20, NBF= −30, NFB= −300, and NFF= 274.6.
FIG. 4.共Color online兲 The function 兩i共z,t兲兩 for 共a兲 bosonic and
共b兲 fermionic bright solitons vs z and t for the solitons of Fig. 3. At
t = 0 NF= 1000, NB= 10 000, aBB= 0.5 nm, aBF= −3.75 nm,=0 and
harmonic oscillator length l⬇1m. The nonlinearity parameters at
t = 0 are NBB= 20, NBF= −30, NFB= −300, and NFF= 274.6. At t
= 100 ms共marked by arrows兲 the bright solitons are set into small breathing oscillations by suddenly jumping the nonlinearities NBF
and NFBto −33 and −330, respectively.
FIG. 5.共Color online兲 The root-mean-square size 具z典r.m.s.of the bosonic共dotted line兲 and fermionic 共full line兲 solitons of Fig. 4 vs time.
VOP= V0sin2共2y/兲, 共3.1兲 where V0 is the strength, and is the wave length of the laser. In our calculation we take V0= 100 and =/ 2. To solve Eqs.共2.15兲 and 共2.16兲 with this optical-lattice potential and desired nonlinearities, we again start the time evolution with the solution of the linear oscillator problem. In the course of time evolution the nonlinearities and the optical-lattice potential are slowly introduced and eventually the har-monic oscillator potential is slowly removed. Then the final solutions are iterated for convergence. The resultant soliton wave functions are plotted in Fig. 6 for NF= 1000, NB
= 10 000, aBB= 0.3 nm, and aBF= −2.375 nm. The
optical-lattice potential introduces modulations in the solitonic wave function. For the parameters of Fig. 6 the modulations are more prominent on the fermionic soliton than the bosonic one. By changing the parameters it is possible to have modu-lations on the bosonic soliton as well.
IV. SUMMARY
We use a coupled set of time-dependent mean-field-hydrodynamic equations for a boson-fermion mixture to
study the formation of fermionic bright soliton in a DFG as a stationary state. In this study we take the boson-boson inter-action to be both attractive and repulsive and the fermion interaction to be attractive. An attractive boson-fermion interaction is necessary for the formation of a fermionic bright soliton as the diagonal nonlinearity NFFin
the fermion-fermion system is always repulsive.
In the present study we demonstrate that stable solitons can be formed in coupled NLS equations for the boson-fermion mixture with repulsive diagonal nonlinearities and attractive off-diagonal nonlinearities above some cutoff val-ues. In another study关23兴 we showed the possibility of the formation of bright solitons in coupled bosonic condensates with intraspecies repulsion supported by interspecies attrac-tion. The stability of the present fermionic and bosonic soli-tons is demonstrated through their sustained breathing oscil-lation initiated by a sudden jump in the boson-fermion scattering length. Bright solitons have been created experi-mentally in attractive BECs in three dimensions in the pres-ence of radial trapping only without any axial trapping关15兴. In view of this, fermionic bright solitons can be observed in the laboratory in the presence of radial trapping only in a mixture of a DFG and BEC. We also suggest the possibility of the formation of fermionic solitons on a periodic optical-lattice potential. In the present investigation we used a set of mean-field equations for the DFG-BEC mixture. A proper treatment of the DFG should be performed using a fully antisymmetrized many-body Slater determinant wave func-tion 关8兴, as in the case of atomic and molecular scattering involving many electrons关24兴. However, in view of the suc-cess of the hydrodynamic model in other contexts 关12,25兴, we do not believe that the present conclusion about the ex-istence of robust fermionic solitons in a DFG-BEC mixture to be so peculiar as to have no general validity.
Note added in proofs: Recently, we became aware of
reaching a similar conclusion from a numerical solution of a model using an antisymmetrized wave function for the fer-mion wave function关26兴. We apologize for missing this ar-ticle before.
ACKNOWLEDGMENTS
The work is supported in part by the CNPq of Brazil.
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