Self-bound droplet of Bose and Fermi atoms in one dimension:
Collective properties in mean-field and Tonks-Girardeau regimes
Luca Salasnich,1,*Sadhan K. Adhikari,2,† and Flavio Toigo3,‡ 1CNISM and CNR-INFM, Unità di Padova, Via Marzolo 8, 35131 Padova, Italy
2Instituto de Física Teórica, UNESP-São Paulo State University, 01.405-900 São Paulo, São Paulo, Brazil 3
Dipartimento di Fisica “G. Galilei” and CNISM, Università di Padova, Via Marzolo 8, 35131 Padova, Italy
共Received 10 January 2007; published 20 February 2007兲
We investigate a dilute mixture of bosons and spin-polarized fermions in one dimension. With an attractive Bose-Fermi scattering length the ground state is a self-bound droplet, i.e., a Bose-Fermi bright soliton where the Bose and Fermi clouds are superimposed. We find that the quantum fluctuations stabilize the Bose-Fermi soliton such that the one-dimensional bright soliton exists for any finite attractive Bose-Fermi scattering length. We study density profile and collective excitations of the atomic bright soliton showing that they depend on the bosonic regime involved: mean-field or Tonks-Girardeau.
DOI:10.1103/PhysRevA.75.023616 PACS number共s兲: 03.75.Ss, 03.75.Hh, 64.75.⫹g
I. INTRODUCTION
Ultracold vapors of alkali-metal atoms, such as 87Rb,
85
Rb,40K,23Na,6Li,7Li, etc. are now actively studied in the regime of deep Bose and Fermi degeneracy关1–5兴. Trapped
Bose-Fermi mixtures, with Fermi atoms in a single hyperfine state, have been investigated by various authors both theo-retically关6–12兴 and experimentally 关13–18兴. Recently, it has
been predicted that self-bound droplets, also called atomic bright solitons, can be formed within a mixture of degenerate Bose-Fermi gases provided the gases attract each other strongly enough and that there is an external transverse con-finement 关19–21兴. Formation of bright solitons in a dilute
spin-polarized Fermi gas is prevented by Pauli repulsion. The formation of bright soliton in a Bose-Fermi mixture is related to the fact that the system can lower its energy by forming high-density regions 共bright solitons兲 when the Bose-Fermi attraction is sufficient to overcome the Pauli re-pulsion among Fermi atoms and any possible rere-pulsion among the Bose atoms. A common point of these papers 关19–21兴 is that the Fermi cloud is three-dimensional 共3D兲. In
fact, for not too strong Bose-Bose repulsion the transverse width of the Fermi component significantly exceeds the transverse width of the Bose component关19–21兴.
In the strict one-dimensional共1D兲 regime, the Bose-Fermi mixture requires an appropriate theoretical description. The exponent of the power-law, which describes the bulk energy of a Fermi gas as a function of its density depends on the dimensionality 共see, for instance, Ref. 关22兴兲. In addition,
even at zero temperature, the 1D Bose gas can never be a true Bose-Einstein condensate due to phase fluctuations 关1,23兴. For a repulsive 1D Bose gas, one must distinguish
two regimes: a quasi-Bose-Einstein condensate 共BEC兲 re-gime, well described by the 1D Gross-Pitaevskii equation
with positive nonlinearity关23兴, and a Tonks-Girardeau 共TG兲
regime at very low densities, where the 1D bosons behave as 1D ideal fermions关24,25兴. An attractive 1D Bose gas is
in-stead well described by the Hartree mean-field theory 关26,27兴, i.e., the 1D Gross-Pitaevskii equation with negative
nonlinearity关28兴.
The existence of the TG regime above has recently been experimentally confirmed关29兴 in a study of the 1D
degener-ate 87Rb system. In a subsequent study of this system关30兴,
the 1D Bose gas in the TG regime has been found to possess the peculiar property of not attaining a thermal equilibrium even after thousands of collisions. This is often termed fer-mionization of 1D bosons in the TG regime. It is well known that due to Pauli principle the spin-polarized trapped fermi-ons do not interact at low temperature and hence fail to reach a thermal equilibrium necessary for evaporative cooling leading to a degenerate state. The necessary thermal equilib-rium was attained only in Bose-Fermi 关16–18兴 or
Fermi-Fermi 关31兴 mixtures through collision between bosons and
fermions or between fermions in different quantum states, respectively.
In this paper we consider a Bose-Fermi mixture strongly confined by a 2D harmonic potential in the transverse cylin-dric radial coordinate. The ensuing effective 1D system is described in the quantum hydrodynamical approximation, i.e., the time-dependent density-functional approach based on real hydrodynamic variables or complex scalar fields. Quantum hydrodynamics is very useful for the study of static and collective properties of a Bose-Fermi mixture and it has been used successfully in 3D关20,21,32兴 for a description of
bright and dark solitons and collapse. We investigate the 1D mixture of bosons and spin-polarized fermions by using an effective 1D Lagrangian 关1,33–35兴. A Gaussian variational
approach is adopted to derive axial static and dynamical properties of the mixture with attractive Bose-Fermi scatter-ing length 共abf⬍0兲. The solution of the variational scheme
was found to be in satisfactory agreement with the accurate numerical solution of the hydrodynamic equations. We find that a self-bound droplet, i.e., a Bose-Fermi bright soliton, exists also for very small values of兩abf兩. In this case the axial
width of the Fermi component is very large while the axial *URL: http://www.padova.infm.it/salasnich. Electronic address:
salasnich@pd.infn.it
†
URL: http://www.ift.unesp.br/users/adhikari. Electronic address: adhikari@ift.unesp.br
‡Electronic address: toigo@pd.infn.it
width of the Bose component depends on the sign and mag-nitude of the Bose-Bose scattering length ab. Remarkably,
the TG regime is essential to preserve a localized Bose-Fermi soliton for very small 兩abf兩; in fact, for a repulsive
Bose-Bose interaction in the quasi-BEC 1D regime the theory predicts a minimum value of 兩abf兩 below which the
mixture is uniform, i.e., fully delocalized. For large values of 兩abf兩 the Bose-Fermi system is self-confined in a very narrow
region and therefore the local axial densities of bosons and fermions strongly increase. We must remember, however, that above a critical axial density the Fermi system is no more strictly one dimensional, and the same happens for re-pulsive bosons.
The paper is organized as follows. In Sec. II we present the model used to study the degenerate Bose-Fermi system. Then we derive a set of coupled equations for the mixture starting from a Lagrangian density. In Sec. III, by using a Gaussian variational ansatz, we demonstrate that for an at-tractive Bose-Fermi interaction, the ground state of the model is a self-bound bright soliton 共in the absence of an external longitudinal trap兲. In Sec. IV we present a study of the system in the quasi-BEC regime and in the TG regime for attractive Bose-Fermi interaction. These results are fur-ther explored in Sec. V considering a single Fermi atom inside the Bose cloud. In Sec. VI we consider the problem of coupled breathing oscillations of the Bose-Fermi system and calculate the frequencies of these oscillations. Finally, in Sec. VII we present a brief summary of our investigation and discuss the experimental conditions necessary to achieve the 1D Bose-Fermi soliton.
II. BOSE-FERMI LAGRANGIAN FOR ONE-DIMENSIONAL HYDRODYNAMICS
We consider a mixture of Nb bosons of mass mb and Nf
spin-polarized fermions of mass mf at zero temperature
trapped by a tight cylindrically symmetric harmonic potential of frequency⬜in the transverse direction. We assume fac-torization of the transverse degrees of freedom. It is justified in 1D confinement where, regardless of the longitudinal be-havior or statistics, the transverse spatial profile is that of the single-particle ground state关10,35–38兴. The transverse width
of the atom distribution is given by the characteristic har-monic length of the single-particle ground state: a⬜j =
冑
ប/共2mj⬜兲, with j=b, f. The atoms have an effective 1Dbehavior at zero temperature if their chemical potentials are much smaller than the transverse energyប⬜关10,35兴.
We use a hydrodynamic effective Lagrangian to study the static and collective properties of the 1D Bose-Fermi mix-ture. In the rest of the paper all quantities are dimensionless. In particular, lengths are in units of a⬜b, linear densities in units of a⬜b−1, times in units of ⬜−1, and energies in units of ប⬜. The Lagrangian density L of the mixture reads
L = Lb+Lf+Lbf. 共1兲
The termLbis the bosonic Lagrangian, defined as
Lb=b*共it+2z兲b−兩b兩6G
冉
gb2兩b兩2
冊
− Vb兩b兩2, 共2兲
where b共z,t兲 is the hydrodynamic field of the Bose gas,
such that nb共z,t兲=兩b共z,t兲兩2 is the 1D density and vb共z,t兲
= izln关b共z,t兲/兩b共z,t兲兩兴 is its velocity. Here gb= 2ab/ a⬜bis
the scaled interatomic strength with ab the Bose-Bose
scat-tering length. We take 兩gb兩⬍1 to avoid the
confinement-induced resonance 关39兴. Interacting bosons are
one-dimensional if gbnbⰆ1 关35–37兴. For x⬎0 the function G共x兲
is the so-called Lieb-Liniger function, defined as the solution of a Fredholm equation and such that G共x兲⯝x for 0⬍xⰆ1 and G共x兲⯝2/ 3 for xⰇ1 关24兴. For x⬍0 we set G共x兲=x
关28,35兴. Vb共z兲 is the longitudinal external potential acting on
the bosons. In the static case the Lagrangian densityLb
re-duces exactly to the energy functional recently introduced by Lieb, Seiringer, and Yngvason关40兴. In addition, Lbhas been
successfully used to determine the collective oscillation of the 1D Bose gas with longitudinal harmonic confinement 关35兴.
The fermionic Lagrangian densityLfis given instead by Lf=f *共 t+mz 2兲 f− 2 m 3 兩f兩 6− V f兩f兩2, 共3兲
wherem= mb/ mf and f共z,t兲 is the hydrodynamic field of
the 1D spin-polarized Fermi gas, such that nf共z,t兲
=兩f共z,t兲兩2 is the 1D fermionic density and vf共z,t兲
= imzln关f共z,t兲/兩f共z,t兲兩兴 is the velocity of the Fermi gas.
The noninteracting fermions are 1D if共2
m/ 2兲n2fⰆ1 关10兴. Vf共z兲 is the longitudinal external potential acting on
fermi-ons. In the static case and with Vf共z兲=0 the Lagrangian Lf
gives the correct energy density of a uniform and noninter-acting 1D Fermi gas. More generally, the Euler-Lagrange equation ofLf yields the hydrodynamic equations of the 1D
Fermi gas关34兴.
Finally, the Lagrangian densityLbfof the Bose-Fermi
in-teraction reads
Lbf= − gbf兩b兩2兩f兩2, 共4兲
where gbf= 2abf/ a⬜b is the scaled interatomic strength
be-tween bosons and fermions, with abfthe Bose-Fermi
scatter-ing length关10兴.
Euler-Lagrange equations of the Lagrangian L provide the two coupled partial differential equations forb andf,
itb=
冋
−z 2+ 3兩 b兩4G冉
gb 2兩b兩2冊
−1 2gb兩b兩 2G⬘
冉
gb 2兩b兩2冊
+ Vb+ gbf兩f兩2册
b, 共5兲 itf=关− mz2+2m兩f兩4+ Vf+ gbf兩b兩2兴f. 共6兲For gbf= 0 and 0⬍gb⬍1, the first partial differential
equa-tion 共5兲 reduces, in the regime gb/ nbⰆ1, to the familiar
mean-field 1D Gross-Pitaevskii equation关1兴, i.e., to the 1D
cubic nonlinear Schrödinger equation describing a quasi-BEC. Instead, in the regime where everywhere gb/ nbⰇ1,
Schrödinger equation proposed by Kolomeisky et al.关33兴 for
the dynamics of a TG gas, which is formally equivalent to Eq.共6兲 describing the 1D noninteracting Fermi gas. Actually,
Girardeau and Wright关41兴 have shown that this quintic
non-linear Schrödinger equation overestimates the coherence in interference patterns at a small number of particles. Never-theless, Minguzzi et al. 关34兴 have found that this quintic
equation is quite accurate in describing the density profile and the collective oscillations of the 1D ideal Fermi gas with longitudinal harmonic confinement. If we define G共x兲=x for x⬍0 then, when gbf= 0 and gb⬍0 Eq. 共5兲 reduces to the
mean-field 1D Gross-Pitaevskii equation with attractive 共negative兲 nonlinearity, which describes quite accurately the attractive 1D Bose gas关27,28兴.
III. SELF-BOUND SOLUTION: BOSE-FERMI BRIGHT SOLITON
In the remaining part of the paper we set Vb共z兲=Vf共z兲
= 0 and investigate the case of a negative Bose-Fermi scat-tering length共gbf⬍0兲. We use a time-dependent variational
ansatz for the fieldsj共z,t兲 to determine the conditions under
which a self-bound droplet of 1D bosons and fermions ex-ists. In particular, we investigate the two main regimes of 1D bosons: the quasi-BEC regime and the TG regime. For the two fieldsj共z,t兲, with j=b, f, we use the following
Gauss-ian ansatz: j= Nj 1/2 1/4 j 1/2exp
冉
− 共z − zj兲2 2j 2 + ijz + ijz2冊
, 共7兲where the time-dependent variational parameters are the lon-gitudinal widths j共t兲, the centers of mass zj共t兲, and the
slopes j共t兲 and curvaturesj共t兲 of the phase. It is obvious
that the tails of the Gaussian nb共z,t兲=兩b共z,t兲兩2given by Eq.
共7兲 are locally in the TG regime but, in our terminology, a
nonuniform cloud of bosons is in the TG regime only if everywhere its local density nb共z,t兲 satisfies the condition gb/ nb共z,t兲Ⰷ1.
We insert the Gaussian fieldsj共z,t兲 into the Lagrangian L and integrate over the spatial variable z and get an effec-tive Lagrangian关42,43兴, which depends onj共t兲, zj共t兲,j共t兲,
j共t兲 and their time derivatives. By writing the eight
Euler-Lagrange equations one finds that the slopes j共t兲 and the
curvaturesj共t兲 of the fieldsj共z,t兲 can be obtained from the
widths j共t兲 and the center coordinates zj共t兲 through the
equations
j= − z˙j− 2jzj, j= −
˙j
2j
, 共8兲
with j = b , f. The equations of motion of the parametersj共t兲
and zj共t兲 do not depend on the phase parameters j共t兲 and
j共t兲 关42,43兴. They are the “classical” equations of motion of
a system with effective Lagrangian
L = T − E, 共9兲 where T =Nb 2 共˙b 2+ 2z˙ b 2兲 +Nfm 2 共˙f 2+ 2z˙ f 2兲 共10兲
is the effective kinetic energy and
E = Eb+ Ef+ Ebf 共11兲
is the effective potential energy of the system. The term Eb
involves a complicated integral of the Lieb-Liniger function G共x兲, namely, Eb= Nb 2b2 + Nb 3 3/2 b 2
冕
−⬁ +⬁ e−3y2G冉
gbb 2Nb ey2冊
dy . 共12兲 The other two terms, Ef and Ebf, are given byEf= Nfm 2f2 +Nf 3 m 3
冑
32f , 共13兲 and Ebf= gbfNbNf冑
bf exp冋
− 共zb− zf兲 2冑
b 2 +f 2册
. 共14兲We stress that the potential energy共11兲 of the effective
La-grangian共9兲 can be easily obtained from ansatz 共7兲 without
including the phase parametersj共t兲 and j共t兲. On the
con-trary, to get the kinetic energy term共10兲 it is necessary to
include in the ansatz the four phase parameters of Eq. 共7兲
关42,43兴. The kinetic term is essential to calculate the
dynami-cal properties of the mixture, such as the collective oscilla-tions considered in Sec. VI.
The stable stationary state of the system is found by mini-mizing the effective potential energy
E zj = 0, j = b, f , 共15兲 E j = 0, j = b, f . 共16兲
Equations共15兲 lead to zb= zf and without loss of generality
we set zb= zf= 0. Equations共16兲 can then be rewritten as
1 +2Nb 2 3/2
冕
−⬁ +⬁ e−3y2G冉
gbb 2Nb ey2冊
dy −gbNbb 23/2冕
−⬁ +⬁ e−2y2G⬘
冉
gbb 2Nb ey2冊
=− gbfNfb 4冑
bf3 , 共17兲 m+ 2Nf 2 m 3冑
3 = − gbfNbf 4冑
bf 3 , 共18兲wherebf=
冑
b2+2f. From Eqs.共17兲 and 共18兲 one candeter-mine the widthsbandf at equilibrium. Stability requires
that the Hessian matrix of the second partial derivatives of the effective potential energy E共j兲 is positive definite;
KG= 2E b 2 2E f 2−
冉
2E bf冊
2 共19兲 must be positive. An inspection of Eqs.共17兲 and 共18兲 showsthat there are stable solutions only for gbf艋0. For gbf= 0 and gb⬎0, the solutions areb=f= +⬁, corresponding to
infi-nitely extended, uniform bosonic and fermionic clouds, while for gbf= 0 and gb⬍0, the fermionic 1D density is
uni-form while the bosonic cloud is localized with b
= 2
冑
2/共兩gb兩Nb兲. For gbf⬍0, Eqs. 共17兲 and 共18兲 must besolved numerically. In the numerical calculations the Lieb-Liniger function G共x兲 is modeled by an efficient Padé ap-proximant based on the exact numerical determination of G共x兲 关44兴.
As the effective potential of the problem is E of Eq.共11兲,
Eq. 共16兲 together with the condition implicit in Eq. 共19兲
minimizes the effective potential as a function of the two widths b and f. A typical plot of the potential for Nb
= 100, Nf= 10, gb= 0.01, gbf= −0.2, and m= 1 is shown in
Fig.1. Stable oscillations of the system are possible around the minimum. We shall study different features of these os-cillations in the following.
As previously stressed, bosons are 1D under the condition gbnbⰆ1, which corresponds to bⰇgbNb/
冑
. For a quitelarge width, namely, forbⰇNb/共
冑
gb兲, the bosons enter inthe TG regime, where gb/ nbⰇ1. The fermions are instead
1D under the condition 共2m/ 2兲nf
2Ⰶ1, which corresponds
tofⰇNf
冑
m/ 2.We shall base the present study on the Gaussian varia-tional approach described above, which, like any variavaria-tional approach, should be reliable for the Bose-Fermi ground state studied in this paper. Moreover, the analytical variational so-lution provides interesting physical insight into the problem, as we shall see in the following. Also, the actual numerical solution of the full coupled dynamics is pretty complicated to implement for all cases reported in this paper in the various parameter ranges. Nevertheless, we find it worthwhile to compare the solution of the variational scheme with the
ac-curate numerical solution of Eqs.共5兲 and 共6兲 in certain cases.
We solved these numerically, using a imaginary-time integra-tion method based on the finite-difference Crank-Nicholson scheme, as described in Ref. 关45兴. We discretize the
mean-field equations using a time step⌬t=0.05 and a space step ⌬z=0.05, and z苸关−L/2,L/2兴 with L=2000. The boundary conditions are j共−L/2兲=j共L/2兲=0, with j=b, f. We start
with broad Gaussians as initial wave functions. In the course of the imaginary-time evolution the self-bound mixture is quickly formed but, due to strong Pauli repulsion among identical spin-polarized fermions, the fermionic density pro-file extends to many hundredths of length’s units. It is then essential to take a very large space interval关−L/2,L/2兴 of integration to see that these long tails of the fermionic cloud are indeed decaying to zero.
In Fig. 2 we plot two sets of numerical results for the probability density in the quasi-BEC 1D regime. The figure shows that, for fixed values of the interaction strengths gb
and gbf, the axial width of the bosonic probability density
becomes larger than the fermionic one by reducing the num-ber Nfof fermions and increasing the number Nbof bosons.
The figure shows that the variational approach can be used to give a reasonable estimation of the axial widths of the two clouds.
We now turn to a study of the Bose-Fermi system by using the variational approach, which enables us to explore quite easily all regimes and extract physically interesting analytical results. With the intention of illustrating the TG regime and the quasi-BEC Gross-Pitaevskii regime for bosons and 1D regime for fermions for a specific set of Bose-Fermi parameters. In Fig.3we plotb共solid line兲 and
f 共dashed line兲 from Eqs. 共17兲 and 共18兲 as a function of
兩gbf兩, with gbf⬍0 for the values Nb= 100, Nf= 20, gb= 0.01,
andm= 1 of the parameters. In the upper panel the
linear-linear scale is employed while in the lower panel we report the same results on a log-log scale, to better visualize the TG regime and the quasi-BEC regimes. When their width takes values between b= gbNb/
冑
⯝0.56 共dot-dashed line兲 and0
20
40
60
σ
b0
20
40
60
σ
f-1.5
-1
-0.5
0
0.5
Potential
FIG. 1. 共Color online兲 Effective potential energy E of Eq. 共11兲
as a function of soliton widths b and f for parameters: m= 1,
Nb= 100, Nf= 10, gb= 0.01, and gbf= −0.2. The potential has a
mini-mum at b= 16.90 and f= 21.84. Lengths are in units of a⬜b
=
冑
ប/共2mb⬜兲 and energy is in units of ប⬜.-60 -40 -20 0 20 40 60 0 0.01 0.02 0.03 0.04 ρ j (z ) -75 -50 -25 0 25 50 75 z 0 0.01 0.02 0.03 ρ j (z ) bosons fermions fermions bosons
FIG. 2. 共Color online兲 Probability densityj共z兲=兩j共z兲兩2/ Njof
bosons and fermions in the self-bound mixture with gb= 0.01 and
gbf= −0.2. In the upper panel, Nb= 100 and Nf= 20; in the lower
panel, Nb= 300 and Nf= 10. Solid lines: numerical results. Dashed
b= Nb/共gb
冑
兲⯝5600 共dot-dot-dashed line兲 bosons are inthe quasi-BEC 1D regime, while they are in the TG 1D re-gime ifb⬎5600 共above the dot-dot-dashed line兲. Fermions
are in the 1D regime whenf⬎Nf
冑
m/ 2⯝25, i.e., abovethe dotted line. For large兩gbf兩, both the widths tend to zero
corresponding to narrow soliton共s兲.
IV. QUASI-BOSE-EINSTEIN CONDENSATE ONE-DIMENSIONAL REGIME AND TONKS-GIRARDEAU
REGIME
The expression for the effective Lagrangian of the Bose-Fermi system given by Eqs.共11兲–共14兲 is greatly simplified in
the quasi-BEC 1D regime and also in the TG regime. In the quasi-BEC 1D regime one has G共x兲⯝x and the expression for the effective energy E of Eq.共11兲 can be evaluated
ana-lytically and written as
E =1 2 ␣ b2 +  b +1 2 ␥ 2f −
冑
␦ b 2 +f 2, 共20兲 where ␣= Nb, = gbNb 2/共2冑
2兲, ␥= mNf关1+2Nf 2/共3冑
3兲兴,and ␦=兩gbf兩NbNf/
冑
with gbf= −兩gbf兩. Then at equilibriumone finds from Eq. 共16兲 that f and ␦ can be written as
functions ofb, f=␥1/4 b 共␣+b兲1/4 , 共21兲 and also ␦=
冉
␣ b +冊
冉
1 +冑
␥ ␣+b冊
3/2 . 共22兲Equation共22兲 implies that for any finite b andf one has
␦⬎, i.e.,兩gbf兩⬎兩gbf兩min=
冑
2gbNb/共4Nf兲. Thus, for gb⬎0 theBose-Fermi bright soliton exists only for 兩gbf兩⬎兩gbf兩min,
while the system will not be bound 共b→⬁, f→⬁兲 for
兩gbf兩⬍兩gbf兩min. This last situation is, however, unphysical
be-cause for a very largebthe system enters in the TG regime
and Eq.共20兲 is no more valid.
When the Bose-Bose scattering length is zero共gb= 0兲 then
兩gbf兩min= 0. For gb⬍0 共attractive Bose-Bose interaction兲, b
is finite even for兩gbf兩min= 0, but it cannot exceed the value
b= 2
冑
2/共兩gb兩Nb兲 with the corresponding value offinfi-nitely large. When 兩gbf兩 =0 the bosonic cloud is localized
关b= 2
冑
2/共兩gb兩Nb兲兴 while the fermionic cloud is fullyde-localized共f= +⬁兲 in the axial direction.
Coming back to the case of a repulsive Bose-Bose scat-tering length共gb⬎0兲, we observe that, after fixing gband Nb,
by reducing兩gbf兩 the bosonic widthbincreases and the
sys-tem enters in the TG regime, where G共x兲⯝2/ 3 and the
effective energy E of Eq.共11兲 can also be evaluated
analyti-cally as E =1 2 ␣ b 2+ 1 2  ˜ b 2+ 1 2 ␥ f 2− ␦
冑
b 2 +f 2. 共23兲The quantities␣,␥, and␦ are the same as in Eq.共20兲 while 
˜ =2Nb3/共3
冑
3兲. In this case the widths b and f at theequilibrium can also be determined analytically. They are
b= ␣+˜ ␦
冉
1 +冑
␥ 共␣+˜ 兲冊
3/2 , 共24兲 f=␥1/4 共␣+˜ 兲3/4 ␦冉
1 +冑
␥ 共␣+˜ 兲冊
3/2 , 共25兲and both increase as ␦ decreases, and only when ␦=兩gbf兩
= 0 the two widths become infinitely large. Thus we conclude that for any finite value of 兩gbf兩 and gb⬎0 there exists a
Bose-Fermi bright soliton, the existence of this bright soliton being guaranteed by the behavior of the bosonic energy term in the TG regime.
In Fig. 4 we show the axial widths b and f of the
Bose-Fermi mixture with an attractive Bose-Bose scattering length 共gb⬍0兲. We choose Nb= 50, Nf= 10, and gb= −0.01
and plot the widths as a function of兩gbf兩. The figure shows
that, as兩gbf兩 goes to zero, the Fermi widthf is much larger
than the Bose width b. In fact, as previously shown, at
兩gbf兩 =0, one finds f= +⬁ while b= 2
冑
2/共兩gb兩Nb兩兲. Forlarge values of兩gbf兩, the Fermi width f quickly decreases
and reaches the valuef= Nf
冑
m/ 2⬇12.5 共dotted line兲be-low which the Fermi system is no more strictly one dimen-sional.
V. SINGLE FERMIONIC ATOM IN THE BOSE CLOUD
The existence of the Bose-Fermi bright soliton for both attractive and repulsive Bose-Bose interaction, provided there is an attractive Bose-Fermi interaction, even vanish-ingly small, is due to the 1D effect of quantum fluctuations described by the Lieb-Liniger function G共x兲. To understand 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 20 40 60 80 100
σ
j σb σf 10-6 10-4 10-2 100 102|g
bf|
100 104 108σ
jFIG. 3. 共Color online兲 1D self-bound Bose-Fermi droplet with repulsive bosons共gb⬎0兲. Axial widths b and f of the bosonic
and fermionic widths as a function of the Bose-Fermi interaction strength兩gbf兩, with gbf⬍0. Mixture parameters: NB= 100, NF= 20,
m= 1, and gb= 0.01. Upper panel: linear-linear scale. Lower panel:
log-log scale. Between the dot-dashed line and the dot-dot-dashed line the bosons are in the quasi-BEC 1D regime, while above the dot-dot-dashed line they are in the TG regime. Fermions are in the 1D regime above the dotted line. Units are as in Fig.1.
this result further we consider the case of a single fermionic atom共Nf= 1兲 interacting with the Bose cloud. In this case the
exact stationary Schrödinger equation of the single-particle wave functionf共z兲 of the fermion is given by
共− mz 2
−兩gbf兩兩b兩2兲f=⑀f, 共26兲
where⑀is the energy of the fermionic bound state under the effective potential well Vef f共z兲=−兩gbf兩兩b共z兲兩2 determined by
the Bose cloud. We know that for gb⬍0 the pure 1D Bose
system supports a bright soliton described by b共z兲
=共兩gb兩Nb兲
1/2
2 sech
共
兩gb兩z
2
兲
. Guided by this result, in the present caseof repulsive bosons 共gb⬎0兲 attracted by a single fermion gbf⬍0 we adopt for the bosonic field b共z兲 the following
ansatz: b共z兲 = Nb 1/2 共2兲1/2sech
冉
z 冊
, 共27兲withthe variational width of the bosonic cloud. This ansatz is also suggested by the existence of the analytical solution for the eigenvalue of the corresponding Eq.共26兲 for the
fer-mionic bound state. One finds关46兴 ⑀共兲 = − 1
22共1 + 兩gbf兩Nb−
冑
1 + 2兩gbf兩Nb兲2. 共28兲
The energy⑀共兲 is the sum of the kinetic energy of the mionic atom and of the interaction energy between the fer-mionic atom and the bosonic cloud.
The total energy of the system is then the effective energy of the Bose cloud calculated from the Lagrangian density共2兲
added to the energy⑀共兲 of the Fermi atom and is given by
E = Nb 32+ Nb 3 82
冕
−⬁ +⬁ sech6共y兲G冉
gb Nbsech2共y兲冊
dy +⑀共兲. 共29兲 The existence of the Bose-Fermi bright soliton depends on the existence of a local minimum of this total energy E = E共兲 of the system. From Eq. 共29兲 we find that E共兲⬃−兩gbf兩Nb
2 as→ +⬁, since for large’s G
共
gbNbsech2共y兲
兲
⯝2
3 for
any value of y. Thus, for → +⬁, the energy goes to zero through negative values. Again, from Eq. 共29兲 we see that
E共兲→ +⬁ as→0 and therefore we conclude that the func-tion E共兲, being positive at the origin and vanishingly small but negative, at large’s, must possess a negative local mini-mum that is also the global minimini-mum of the energy. This implies that there is always a finite value ofthat minimizes the total energy. As previously stressed, this behavior strictly depends on the properties of the Lieb-Liniger function G共x兲 for large x and, as a consequence, the Bose-Fermi bright soliton exists for any negative value gbf, even for Nf= 1.
VI. COLLECTIVE OSCILLATIONS OF THE BOSE-FERMI BRIGHT SOLITON
After having established the existence of stationary bright solitons in a degenerate Bose-Fermi mixture, we study two types of small oscillations of this system around the stable equilibrium position and calculate their frequencies. The first is the stable breathing oscillation of the system around its mean position and the second describes the stable small os-cillations once the centers of the Fermi and Bose clouds are slightly displaced with respect to each other. Given the ef-fective Lagrangian共9兲, the problem of small oscillations is
solved by expanding the kinetic energy共10兲 and the potential
energy 共11兲 around the equilibrium solution up to quadratic
terms. In this way we extract关43兴 the frequencies of the collective breathing modes of the Bose and Fermi clouds from the eigenvalue equation,
冢
2E b 2 2E bf 2E bf 2E 2f冣
−2冉
Nb 0 0 Nfm冊
= 0, 共30兲where the partial derivatives must be calculated at the equi-librium 共b,b兲, where b and f are the Bose and Fermi
widths obtained from Eqs. 共17兲 and 共18兲. The two
frequen-cies1 and2then read
1,2=
冑
Nb 2E 2f +fNf 2E b2 ±冑
⌬ 2mNbNf , 共31兲 where ⌬ =冉
Nb 2E f2 +mNf 2E b2冊
2 − 4mNbNfKG, 共32兲and KGis the Gaussian curvature of Eq. 共19兲.
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
|g
bf|
0 5 10 15 20 25 30 35 40σ
jσ
bσ
fFIG. 4. 共Color online兲 1D self-bound Bose-Fermi droplet with attractive bosons共gb⬍0兲. Axial widthsband f of the bosonic and fermionic widths as a function of the Bose-Fermi interaction strength 兩gbf兩, with gbf⬍0. Mixture parameters: Nb= 50, Nf= 10,
m= 1, and gb= −0.01. Fermions are in the 1D regime above the
In addition, by taking into account the “reduced mass” mNbNf/共Nb+mNf兲, one can derive the frequency ⍀ of
har-monic oscillation of the relative distance 兩zb− zf兩
共displace-ment兲 of the two clouds from the equation mNbNf Nb+mNf ⍀2= 2E 共兩zb− zf兩兲2 . 共33兲 The result is ⍀ =
冑
2兩gbf兩共Nb+mNf兲 共b 2 +f 2兲3/4 , 共34兲where again b and f are the Bose and Fermi widths at
equilibrium, where zb= zf.
In Fig.5we plot the frequencies1and2of the coupled breathing modes of the Bose and Fermi clouds by choosing the same parameters of Fig.4, namely, Nb= 50, Nf= 10,m
= 1, and gb= −0.01. The frequencies are plotted as a function
of the Bose-Fermi strength兩gbf兩. For 兩gbf兩 =0, the frequencies
are decoupled: 1 is the frequency of the fermionic axial
breathing mode and2is the frequency of the bosonic axial
breathing mode. Without a Bose-Fermi interaction the Fermi cloud is delocalized共f=⬁兲 and its breathing frequency is
1= 0, while the Bose cloud remains localized 共due to the
negative Bose-Bose strength gb兲 and its breathing frequency
2remains finite and is equal to2= gb 4
Nb 4
/共162兲. Figure5
shows that both the breathing frequency1and the harmonic
frequency⍀ of the displacement 兩z1− z2兩 start from zero and
grow as兩gbf兩 increases.
VII. CONCLUSION
We have studied a degenerate 1D Bose-Fermi mixture by using the quantum hydrodynamics. We find that for attractive Bose-Fermi interaction共gbf⬍0兲 the ground state of the
sys-tem is a self-bound Bose-Fermi droplet. The nonexistence of a threshold in the strength of an attractive Bose-Fermi inter-action for the formation of a Bose-Fermi bright soliton in one dimension is confirmed in the case of a single Fermi atom immersed in a degenerate Bose gas with repulsive Bose-Bose interaction. We also calculate the frequencies of stable oscillation of the Bose-Fermi bright soliton. Such a Bose-Fermi bright soliton is similar to a recently studied Bose-Bose bright soliton bound through an attractive inter-species interaction关47兴.
In view of the recent experimental studies of a degenerate 1D 87Rb gas关29,30兴 and the successful identification of the
quasi-BEC and TG regime in it and the observation of the degenerate Bose-Fermi mixture in6Li-7Li关16,17兴,40K-87Rb 关13兴,6Li-23Na关18兴, etc. by different groups, the
experimen-tal realization of a Bose-Fermi bright soliton seems possible with present technology. The most attractive procedure seems to make use of an experimentally observed Feshbach resonance关48兴 in a Bose-Fermi mixture. The 1D Bose-Fermi
mixture must be created in an axial harmonic trap and the Bose-Fermi interaction must be turned from repulsive to at-tractive by manipulating a background magnetic field. At the same time the axial harmonic trap on the system should be removed. Upon removal of the axial trap, the result is the formation of a single or a train of bright solitons as in the experiment with the degenerate Bose system of 7Li atoms 关17兴 or as in a numerical simulation in a degenerate
Bose-Fermi mixture关19兴. By choosing numbers of atoms and
in-teratomic strengths as suggested in the present paper, one obtains a single Bose-Fermi bright soliton and can study its static and dynamical properties.
ACKNOWLEDGMENTS
This work is partially supported by the FAPESP and CNPq of Brazil.
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