Self-bound droplet of Bose and Fermi atoms in one dimension: Collective properties in mean-field and Tonks-Girardeau regimes

0.36 MB·English

by  Luca Salasnich

Self-bound droplet of Bose and Fermi atoms in one dimension:
Collective properties in mean-field and Tonks-Girardeau regimes
Luca Salasnich1∗, Sadhan K. Adhikari2†, and Flavio Toigo3‡
1CNISM and CNR-INFM, Unit`a di Padova, Via Marzolo 8, 35131 Padova, Italy
2Instituto de F´ısica Te´orica, UNESP − S˜ao Paulo State University, 01.405-900 Sao Paulo, Sao Paulo, Brazil
3Dipartimento di Fisica “G. Galilei” and CNISM,
7 Universit`a di Padova, Via Marzolo 8, 35131 Padova, Italy
0
0 We investigate a dilute mixture of bosons and spin-polarized fermions in one-dimension. With
2 an attractive Bose-Fermi scattering length the ground-state is a self-bound droplet, i.e. a Bose-
Fermibrightsoliton wheretheBoseandFermicloudsaresuperimposed. Wefindthatthequantum
n
a fluctuations stabilize the Bose-Fermi soliton such that the one-dimensional bright soliton exists for
J anyfiniteattractiveBose-Fermiscatteringlength. Westudydensityprofileandcollectiveexcitations
of the atomic bright soliton showing that they depend on the bosonic regime involved: mean-field
0
or Tonks-Girardeau.
3

PACSnumbers: 03.75.Ss,03.75.Hh,64.75.+g
]
r
e
h
I. INTRODUCTION never be a true Bose-Einstein condensate due to phase
t
o fluctuations [1, 23]. For a repulsive 1D Bose gas, one
. mustdistinguishtworegimes: aquasi-Bose-Einsteincon-
at Ultracold vapors of alkali-metal atoms, like 87Rb, densate (BEC) regime, well described by the 1D Gross-
m 85Rb, 40K, 23Na, 6Li, 7Li, etc. are now actively stud-
Pitaevskii equation with positive nonlinearity [23], and
ied in the regime of deep Bose and Fermi degeneracy
- a Tonks-Girardeau (TG) regime at very low densities,
d [1, 2, 3, 4, 5]. Trapped Bose-Fermimixtures, with Fermi
wherethe1Dbosonsbehaveas1Didealfermions[24,25].
n atoms in a single hyperfine state, have been investigated
An attractive 1D Bose gas is instead well described by
o
by various authors both theoretically [6, 7, 8, 9, 10, 11,
c theHartreemean-fieldtheory[26,27],i.e. the1DGross-
12] and experimentally [13, 14, 15, 16, 17, 18]. Recently,
[ Pitaevskii equation with negative nonlinearity [28].
ithasbeenpredictedthatself-bounddroplets,alsocalled
The existence of the TG regime above has been re-
1 atomic bright solitons, can be formed within a mixture
v centlyexperimentallyconfirmed[29]inastudyofthe1D
of degenerate Bose-Fermi gases provided the gases at-
2 degenerate 87Rb system. In a subsequent study of this
tract each other strongly enough and that there is an
5 system [30], the 1D Bose gas in the TG regime has been
external transverse confinement [19, 20, 21]. Formation
7 found to possess the peculiar property of not attaining
1 of bright solitons in a dilute spin-polarized Fermi gas is
a thermal equilibrium even after thousands of collisions.
0 prevented by Pauli repulsion. The formation of bright
This is often termed fermionization of 1D bosons in the
7 solitoninaBose-Fermimixtureisrelatedtothefactthat
TG regime. It is well-known that due to Pauli princi-
0 the system can lower its energy by forming high-density
/ ple the spin-polarized trapped fermions do not interact
t regions (bright solitons) when the Bose-Fermiattraction
a at low temperature and hence fail to reach a thermal
issufficienttoovercomethePaulirepulsionamongFermi
m equilibrium necessary for evaporative cooling leading to
atomsandanypossiblerepulsionamongtheBoseatoms.
a degenerate state. The necessary thermal equilibrium
- A common point of these papers [19, 20, 21] is that the
d was attained only in Bose-Fermi [16, 17, 18] or Fermi-
n Fermi cloud is three-dimensional (3D). In fact, for not
Fermi[31]mixturesthroughcollisionbetweenbosonsand
o too strong Bose-Bose repulsion the transverse width of
fermionsorbetweenfermionsindifferentquantumstates,
c theFermicomponentsignificantlyexceedsthetransverse
v: width of the Bose component [19, 20, 21]. respectively.
In this paper we consider a Bose-Fermi mixture
Xi In the strict one-dimensional (1D) regime, the Bose- strongly confined by a 2D harmonic potential in the
r Fermi mixture requires an appropriate theoretical de- transverse cylindric radial coordinate. The ensuing ef-
a scription. Theexponentofthepower-lawwhichdescribes fective 1D system is described in the quantum hydrody-
thebulkenergyofaFermigasasafunctionofitsdensity namical approximation, i.e. the time-dependent density
depends on the dimensionality (see for instance [22]). In functional approach based on real hydrodynamic vari-
addition, even at zero temperature, the 1D Bose gas can ables or complex scalar fields. Quantum hydrodynam-
ics is very useful for the study of static and collective
properties of a Bose-Fermi mixture and it has been used
successfully in 3D [20, 21, 32] for a description of bright
∗Electronic address: [email protected]; URL: and dark solitons and collapse. We investigate the 1D
http://www.padova.infm.it/salasnich
mixture of bosons and spin-polarized fermions by using
†Electronic address: [email protected]; URL:
an effective 1D Lagrangian [1, 33, 34, 35]. A Gaus-
http://www.ift.unesp.br/users/adhikari
‡Electronicaddress: [email protected] sianvariationalapproachisadoptedtoderiveaxialstatic
2
and dynamical properties of the mixture with attractive We use a hydrodynamic effective Lagrangianto study
Bose-Fermi scattering length (a < 0). The solution the static and collective propertiesof the 1D Bose-Fermi
bf
of the variational scheme was found to be in satisfac- mixture. In the rest of the paper all quantities are di-
tory agreement with the accurate numerical solution of mensionless. In particular, lengths are in units of a ,
⊥b
the hydrodynamic equations. We find that a self-bound lineardensities inunits ofa−1,times inunits ofω−1 and
⊥b ⊥
droplet, i.e. a Bose-Fermi bright soliton, exists also for energies in units of h¯ω . The Lagrangian density of
⊥
L
very small values of a . In this case the axial width of the mixture reads
bf
| |
the Fermi component is very large while the axial width
of the Bose component depends on the sign and magni- = b+ f + bf . (1)
L L L L
tude of the Bose-Bose scattering length a . Remarkably,
b
the TG regime is essential to preserve a localized Bose- The term b is the bosonic Lagrangian,defined as
L
Fermi soliton for very small a ; in fact, for a repulsive
bf
Bose-Bose interaction in the| qu|asi-BEC 1D regime the =ψ∗ i∂ +∂2 ψ ψ 6G gb V ψ 2 , (2)
theory predicts a minimum value of |abf| below which Lb b t z b−| b| (cid:18)2|ψb|2(cid:19)− b| b|
the mixture is uniform, i.e. fully delocalized. For large (cid:0) (cid:1)
valuesof abf theBose-Fermisystemisselfconfinedina where ψb(z,t) is the hydrodynamic field of the Bose gas,
verynarro|wr|egionandthereforethe localaxialdensities such that nb(z,t) = ψb(z,t)2 is the 1D density and
| |
of bosons and fermions strongly increase. We must re- vb(z,t) = i∂zln(ψb(z,t)/ψb(z,t)) is its velocity. Here
| |
member, however,that above a critical axial density the gb =2ab/a⊥b is the scaled inter-atomic strength with ab
Fermi system is no more strictly one-dimensional, and the Bose-Bose scattering length. We take gb < 1 to
| |
the same happens for repulsive bosons. avoid the confinement-induced resonance [39]. Interact-
Thepaperisorganizedasfollows. InSec. IIwepresent ing bosons are one-dimensional if gbnb 1 [35, 36, 37].
≪
the model used to study the degenerate Bose-Fermi sys- For x>0 the function G(x) is the so-calledLieb-Liniger
tem. Then we derive a set of coupled equations for the function, defined as the solution of a Fredholm equation
mixture starting from a Lagrangian density. In Sec. III, and such that G(x) x for 0<x 1 and G(x) π2/3
≃ ≪ ≃
by using a Gaussian variational ansatz, we demonstrate for x 1 [24]. For x < 0 we set G(x) = x [28, 35].
≫
thatforanattractiveBose-Fermiinteraction,theground Vb(z)is the longitudinalexternal potential acting on the
state of the model is a self-bound bright soliton (in the bosons. In the static case the Lagrangian density b
L
absence of an external longitudinal trap). In Sec. IV we reduces exactly to the energy functional recently intro-
present a study of the system in the quasi-BEC regime ducedbyLieb,SeiringerandYngvason[40]. Inaddition,
and in the TG regime for attractive Bose-Fermi inter- b has beensuccessfully used to determine the collective
L
action. These results are further explored in Sec. V oscillationofthe1DBosegaswithlongitudinalharmonic
considering a single Fermi atom inside the Bose cloud. confinement [35].
In Sec. VI we consider the problem of coupled breath- The fermionic Lagrangian density f is given instead
L
ing oscillations of the Bose-Fermi system and calculate by
the frequencies of these oscillations. Finally, in Sec. VII
π2λ
we presenta brief summary of our investigationand dis- =ψ∗ ∂ +λ ∂2 ψ m ψ 6 V ψ 2, (3)
cusstheexperimentalconditionsnecessarytoachievethe Lf f t m z f − 3 | f| − f| f|
one-dimensional Bose-Fermi soliton. (cid:0) (cid:1)
where λ = m /m and ψ (z,t) is the hydrodynamic
m b f f
field of the 1D spin-polarized Fermi gas, such that
n (z,t) = ψ (z,t)2 is the 1D fermionic density and
f f
II. BOSE-FERMI LAGRANGIAN FOR 1D | |
v (z,t) = iλ ∂ ln(ψ (z,t)/ψ (z,t)) is the velocity of
f m z f f
HYDRODYNAMICS | |
the Fermi gas. The non-interacting fermions are one-
dimensionalif(π2λ /2)n2 1[10]. V (z)isthelongitu-
m f ≪ f
We consider a mixture of Nb bosons of mass mb and dinalexternalpotential acting on fermions. In the static
Nf spin-polarized fermions of mass mf at zero tempera- case and with Vf(z) = 0 the Lagrangian f gives the
L
turetrappedbyatightcylindricallysymmetricharmonic correct energy density of a uniform and non-interacting
potentialoffrequencyω⊥ inthetransversedirection. We 1D Fermi gas. More generally,the Euler-Lagrangeequa-
assume factorization of the transverse degrees of free- tion of yields the hydrodynamic equations of the 1D
f
L
dom. It is justified in 1D confinement where, regardless Fermi gas [34].
of the longitudinal behavior or statistics, the transverse Finally, the Lagrangiandensity of the Bose-Fermi
bf
L
spatial profile is that of the single-particle ground-state interaction reads
[10,35,36,37,38]. Thetransversewidthoftheatomdis-
tribution is given by the characteristic harmonic length = g ψ 2 ψ 2 , (4)
bf bf b f
L − | | | |
of the single-particle ground-state: a = ¯h/(2m ω ),
⊥j j ⊥
with j = b,f. The atoms have an effective 1D behavior whereg =2a /a is the scaledinter-atomicstrength
bf bf ⊥b
p
atzerotemperatureiftheirchemicalpotentialsaremuch between bosons and fermions, with a the Bose-Fermi
bf
smaller than the transverse energy h¯ω [10, 35]. scattering length [10].
⊥
3
Euler-Lagrangeequationsofthe Lagrangian provide We insert the Gaussian fields ψ (z,t) into the La-
j
L
the two coupled partial differential equations for ψ and grangian and integrate over the spatial variable z and
b
L
ψ : get an effective Lagrangian [42, 43] which depends on
f
σ (t), z (t), φ (t), θ (t) and their time derivatives. By
j j j j
g
i∂ ψ = ∂2+3ψ 4G b writingtheeightEuler-Lagrangeequationsonefindsthat
t b (cid:20)− z | b| (cid:18)2|ψb|2(cid:19) the slopes φj(t) and the curvatures θj(t) of the fields
1g ψ 2G′ gb +V +g ψ 2 ψ ,(5) ψj(z,t) can be obtained from the widths σj(t) and the
− 2 b| b| 2ψ 2 b bf| f| b center coordinates zj(t) through the equations
(cid:18) | b| (cid:19) (cid:21)
σ˙
j
φ = z˙ 2θ z , θ = , (8)
i∂tψf = −λm∂z2+π2λm|ψf|4+Vf +gbf|ψb|2 ψf . (6) j − j − j j j −2σj
For gbf =(cid:2)0 and 0 < gb < 1, the first partial di(cid:3)fferential withj =b,f. Theequationsofmotionoftheparameters
equation(5) reduces, in the regime gb/nb ≪1, to the fa- σj(t) and zj(t) do not depend on the phase parameters
miliarmean-field1DGross-Pitaevskiiequation[1],i.e. to φ (t) and θ (t) [42, 43]. They are the “classical” equa-
j j
the 1D cubic nonlinear Schr¨odinger equation describing tions of motion of a system with effective Lagrangian
a quasi-BEC. Instead, in the regime where everywhere
g /n 1, Eq. (5) for bosons becomes the quintic non- L=T E , (9)
b b ≫ −
linear Schr¨odinger equation proposed by Kolomeisky et
where
al. [33] for the dynamics of a TG gas, which is formally
equivalent to Eq. (6) describing the 1D noninteracting N N λ
T = b σ˙2+2z˙2 + f m σ˙2 +2z˙2 (10)
Fermi gas. Actually, Girardeau and Wright [41] have 2 b b 2 f f
shown that this quintic nonlinear Schr¨odinger equation (cid:0) (cid:1) (cid:0) (cid:1)
is the effective kinetic energy and
overestimates the coherence in interference patterns at
small number of particles. Nevertheless, Minguzzi et al.
E =E +E +E , (11)
b f bf
[34] have found that this quintic equation is quite accu-
rate in describing the density profile and the collective is the effective potential energy of the system. The term
oscillations of the 1D ideal Fermi gas with longitudinal E involves a complicated integral of the Lieb-Liniger
b
harmonic confinement. If we define G(x) = x for x < 0 function G(x), namely
then,wheng =0andg <0,theEq. (5)reducestothe
bf b
mean-field 1D Gross-Pitaevskii equation with attractive E = Nb + Nb3 +∞e−3y2G gbσb ey2 dy . (12)
(negative) nonlinearity, which describes quite accurately b 2σ2 π3/2σ2 2N
b b Z−∞ (cid:18) b (cid:19)
the attractive 1D Bose gas [27, 28].
The other two terms, E and E , are given by
f bf
III. SELF-BOUND SOLUTION: BOSE-FERMI Nfλm Nf3πλm
E = + , (13)
BRIGHT SOLITON f 2σ2 3√3σ2
f f
and
In the remaining part of the paper we set V (z) =
b
Vf(z) = 0 and investigate the case of a negative Bose- gbfNbNf (zb zf)2
Fdeeprmenidsecnattvtearriinagtiolnenalgtahns(agtbzffo<r th0e).fieldWseψju(sze,ta)ttoimdee-- Ebf = √πσbf exp(cid:16)− σ−b2+σf2(cid:17). (14)
termine the conditions under which a self-bound droplet q
of1Dbosonsandfermionsexists. Inparticularweinves- We stress that the potential energy (11) of the effective
tigatethetwomainregimesof1Dbosons: thequasi-BEC Lagrangian (9) can be easily obtained from ansatz (7)
regime and the TG regime. For the two fields ψj(z,t), without including the phase parameters φj(t) and θj(t).
with j =b,f, we use the following Gaussian ansatz Onthe contrary,to getthe kinetic energy term(10) it is
necessaryto include in the ansatz the four phase param-
N1/2 (z z )2 eters of Eq. (7) [42, 43]. The kinetic term is essential to
ψj = π1/4jσj1/2 exp − −2σj2j +iφjz+iθjz2!, (7) caasltchuelactoelltehcetidveynoasmcililcaatliopnrsopcoerntsiiedseroefdthineSmeicx.tuVrIe., such
The stable stationary state of the system is found by
wherethetime-dependentvariationalparametersarethe
minimizing the effective potential energy:
longitudinal widths σ (t), the centers of mass z (t), and
j j
the slopes φ (t) and curvatures θ (t) of the phase. It
j j ∂E
is obvious that the tails of the Gaussian n (z,t) = =0, j =b,f , (15)
b ∂z
ψ (z,t)2 given by Eq. (7) are locally in the TG regime j
b
| |
but, in our terminology, a non-uniform cloud of bosons
is in the TG regime only if everywhere its local density ∂E
=0, j =b,f . (16)
nb(z,t) satisfies the condition gb/nb(z,t)≫1. ∂σj
4
0.04
bosons
0.03
)
(z 0.02
ρj
Potential 0.01 fermions
0.5 0
-60 -40 -20 0 20 40 60
0.03
0
fermions
0.02
-0.5 )
z
(
ρj
0.01
-1 bosons
60
0
-1.5 40 -75 -50 -25 0 25 50 75
0 σ z
20 20 f
40
σb 60 0 F|ψIG(z.)|22:/N(Cooflobrosoonnslinaen)d. ferPmrioobnasbiilnitythedesneslfit-byouρnjd(z)mix=-
j j
ture with g = 0.01 and g = −0.2. In the upper panel:
b bf
N = 100 and N = 20; in the lower panel: N = 300 and
b f b
N =10. Solid lines: numerical results. Dashed lines: varia-
f
tional results. Units as in Fig. 1.
FIG. 1: (Color online). Effective potential energy E of Eq.
(11)asafunctionofsolitonwidthsσ andσ forparameters:
b f
λm = 1,Nb = 100,Nf = 10,gb = 0.01 and gbf = −0.2. The fermionic clouds, while for g = 0 and g < 0, the
bf b
potential has a minimum at σ = 16.90 and σ = 21.84.
b f fermionic 1D density is uniform while the bosonic cloud
Lengths are in units of a⊥b = ¯h/(2mbω⊥) and energy in is localized with σ = 2√2π/(g N ). For g < 0 the
b b b bf
unitsof ¯hω . | |
⊥ Eqs. (17) and (18) must be solved numerically. In the
p
numerical calculations the Lieb-Liniger function G(x) is
modeled by an efficient Pad`e approximant based on the
Equations (15) lead to z = z and without loss of gen-
b f exact numerical determination of G(x) [44].
erality we set z = z = 0. Equations (16) can then be
b f As the effective potential of the problem is E of Eq.
rewritten as
(11),theequation(16)togetherwithconditionimplicitin
1+ 2Nb2 +∞e−3y2G gbσb ey2 dy Eq. (19)minimizestheeffectivepotentialasafunctionof
π3/2 2N thetwowidthsσb andσf. Atypicalplotofthepotential
Z−∞ (cid:18) b (cid:19) for N = 100, N = 10, g = 0.01, g = 0.2, and
b f b bf
−
λ = 1 is shown in Fig. 1. Stable oscillations of the
m
gbNbσb +∞e−2y2G′ gbσb ey2 = −gbfNfσb4 , systemarepossiblearoundtheminimum. Weshallstudy
− 2π3/2 2N √πσ3 different features of these oscillations in the following.
Z−∞ (cid:18) b (cid:19) bf As previously stressed, bosons are one-dimensional
(17)
under the condition g n 1, which corresponds to
b b
≪
λ + 2Nf2πλm = −gbfNbσf4 , (18) σNb/≫(√gπbNgb)/,√thπe. FboorsoanqsueitnetelarrginewthidethT,Gnarmegeilmyefo,rwσhbe≫re
m 3√3 √πσ3 b b
bf g /n 1. The fermions are instead one-dimensional
b b
≫
underthecondition(π2λ /2)n2 1,whichcorresponds
whereσ = σ2+σ2. FromEqs. (17)and(18)onecan m f ≪
bf b f to σf Nf λmπ/2.
≫
determine thqe widths σb and σf at equilibrium. Stabil- We shall base the present study on the Gaussian vari-
p
ityrequiresthattheHessianmatrixofthesecondpartial ational approach described above, which, like any vari-
derivativesofthe effective potentialenergyE(σj) is pos- ational approach, should be reliable for the Bose-Fermi
itive definite; equivalently the Gaussian curvature ground state studied in this paper. Moreover, the an-
alytical variational solution provides interesting phys-
∂2E∂2E ∂2E 2 ical insight into the problem, as we shall see in the
K = (19)
G ∂σ2 ∂σ2 − ∂σ ∂σ following. Also, the actual numerical solution of the
b f (cid:18) b f(cid:19)
full coupled dynamics is pretty complicated to imple-
must be positive. An inspection of Eqs. (17) and (18) ment for all cases reported in this paper in the vari-
showsthattherearestablesolutionsonlyforg 0. For ous parameter ranges. Nevertheless, we find it worth-
bf
≤
g =0andg >0,thesolutionsareσ =σ =+ ,cor- while to compare the solution of the variational scheme
bf b b f
∞
responding to infinitely extended, uniform bosonic and with the accurate numerical solution of Eqs. (5) and
5
100 σ (solid line) and σ (dashed line) from Eqs. (17) and
σ b f
80 (18) as a function of g , with g < 0 for the values
σb | bf| bf
σj 60 f Nb = 100, Nf = 20, gb = 0.01, and λm = 1 of the pa-
40 rameters. In the upper panel the linear-linear scale is
20 employed while in the lower panel we report the same
0 results on a log-log scale, to better visualize the TG
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
regime and the quasi-BEC regimes. When their width
108 takes values between σ = g N /√π 0.56 (dot-dashed
b b b
≃
line) and σ = N /(g √π) 5600 ( dot-dot-dashed
σj104 b b b ≃
line) bosons are in the quasi-BEC1D regime, while they
100 are in the TG 1D regime if σb > 5600 (above the dot-
dot-dashed line). Fermions are in the 1D regime when
10-6 10-4 10-2 100 102 σ > N λ π/2 25, i.e. above the dotted line. For
|g | f f m ≃
bf large g , both the widths tend to zero corresponding
bf
| p|
to narrow soliton(s).
FIG. 3: (Color online). 1D Self-bound Bose-Fermi droplet
withrepulsivebosons(g >0). Axialwidthsσ andσ ofthe
b b f
bosonicandfermionicwidthsasafunctionoftheBose-Fermi IV. QUASI-BEC 1D REGIME AND
interaction strength|gbf|, with gbf <0. Mixtureparameters: TONKS-GIRARDEAU REGIME
N = 100, N = 20, λ = 1 and g = 0.01. Upper panel:
B F m b
linear-linear scale. Lower panel: log-log scale. Between the
TheexpressionfortheeffectiveLagrangianoftheBose-
dot-dashed line and the dot-dot-dashed line the bosons are
Fermi system given by Eqs. (11) (14) is greatly sim-
inthequasi-BEC1Dregime, whileabovethedot-dot-dashed
−
linetheyareintheTGregime. Fermionsareinthe1Dregime plified in the quasi-BEC 1D regime and also in the TG
above thedotted line. Unitsas in Fig. 1. regime. In the quasi-BEC 1D regime one has G(x) x
≃
and the expressionfor the effective energy E of Eq. (11)
can be evaluated analytically and written as
(6) in certain cases. We solved these numerically, us-
1 α β 1 γ δ
ing a imaginary-time integration method based on the E = + + , (20)
finite-differenceCrank-Nicholsonscheme,asdescribedin 2σb2 σb 2σf2 − σ2+σ2
b f
Ref. [45]. We discretize the mean-field equations using
q
a time step ∆t = 0.05 and a space step ∆z = 0.05, and where α = N , β = g N2/(2√2π), γ = λ N [1 +
z [ L/2,L/2] with L = 2000. The boundary con- b b b m f
dit∈ion−s are ψj(−L/2) = ψj(L/2) = 0, with j = b,f. 2TπhNenf2/a(t3e√q3u)il]i,barniudmδo=ne|gfibfn|dNsbfNrofm/√Eπq.w(it1h6)gbtfha=tσ−|gabnf|d.
We startwith broadGaussians as initial wavefunctions. f
δ can be written as functions of σ :
In the course of the imaginary-time evolution the self- b
boundmixtureisquicklyformedbut,duetostrongPauli σ
σ =γ1/4 b (21)
repulsion among identical spin-polarized fermions, the f (α+βσ )1/4
b
fermionic density profile extends to many hundredths of
length’s units. It is then essential to take a very large and also
spaceinterval[ L/2,L/2]ofintegrationtoseethatthese
long tails of th−e fermionic cloud are indeed decaying to α γ 3/2
δ =( +β) 1+ . (22)
zero. σ α+βσ
b (cid:18) r b(cid:19)
In Fig. 2 we plot two sets of numerical results for
Eq. (22)impliesthatforanyfiniteσ andσ onehasδ >
the probability density in the quasi-BEC 1D regime. b f
The figure shows that, for fixed values of the interaction β, i.e. gbf > gbf min =√2gbNb/(4Nf). Thus, for gb >
| | | |
strengthsgb andgbf,theaxialwidthofthebosonicprob- 0 the Bose-Fermi bright soliton exists only for |gbf| >
ability density becomes larger than the fermionic one by gbf min, while the system will not be bound (σb ,
| | → ∞
reducing the number Nf of fermions and increasing the σf → ∞) for |gbf| < |gbf|min. This last situation is
number Nb of bosons. The figure shows that the varia- howeverunphysicalbecauseforaverylargeσbthesystem
tionalapproachcan be usedto give a reasonableestima- enters in the TG regime and Eq. (20) is no more valid.
tion of the axial widths of the two clouds. When the Bose-Bosescattering length is zero (gb =0)
We now turn to a study of the Bose-Fermi system by then gbf min = 0. For gb < 0 (attractive Bose-Bose
| |
using the variational approach which enables us to ex- interaction), σb is finite even for gbf min = 0, but it
| |
plorequiteeasilyallregimesandextractphysicallyinter- cannot exceed the value σb = 2√2π/(gb Nb) with the
| |
esting analytical results. With the intention of illustrat- correspondingvalueofσf infinitelylarge. When gbf =0
| |
ing the TG regime and the quasi-BEC Gross-Pitaevskii thebosoniccloudislocalized(σ =2√2π/(g N ))while
b b b
| |
regime for bosons and 1D regime for fermions for a spe- thefermioniccloudisfullydelocalized(σ =+ )inthe
f
∞
cific set of Bose-Fermi parameters. In Fig. 3 we plot axial direction.
6
40 as previously shown, at g = 0, one finds σ = +
bf f
| | ∞
35 σ while σb =2√2π/(gb Nb ). For large values of gbf , the
| | | | |
Fermi width σ quickly decreases and reaches the value
30 σb f
σ = N λ π/2 12.5 (dotted line) below which the
f f m
25 f Fermi system is no≈more strictly one-dimensional.
p
σj20
15 V. A SINGLE FERMIONIC ATOM IN THE
BOSE CLOUD
10
5 TheexistenceoftheBose-Fermibrightsolitonforboth
attractive and repulsive Bose-Bose interaction, provided
0
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
|g | their is an attractive Bose-Fermi interaction, even van-
bf ishingly small,is due to the 1Deffect of quantumfluctu-
ations described by the Lieb-Liniger function G(x). To
FIG. 4: (Color online). 1D Self-bound Bose-Fermi droplet understand this result further we consider the case of
withattractivebosons(gb<0). Axialwidthsσbandσf ofthe a single fermionic atom (N = 1) interacting with the
f
bosonicandfermionicwidthsasafunctionoftheBose-Fermi
Bosecloud. InthiscasetheexactstationarySchr¨odinger
interaction strength|g |, with g <0. Mixtureparameters:
bf bf equationofthesingle-particlewavefunctionψ (z)ofthe
N =50, N =10, λ =1 and g =−0.01. Fermions are in f
b f m b fermion is given by
the1D regime abovethe dotted line. Unitsas in Fig. 1.
λ ∂2 g ψ 2 ψ =ǫψ , (26)
− m z −| bf|| b| f f
ComingbacktothecaseofarepulsiveBose-Bosescat- (cid:16) (cid:17)
teringlength(g >0),weobservethat,afterfixingg and where ǫ is the energy of the fermionic bound state under
b b
Nb, by reducing |gbf| the bosonic width σb increases and the effective potential well Veff(z) = −|gbf||ψb(z)|2 de-
the system enters in the TG regime, where G(x) π2/3 terminedbytheBosecloud. Weknowthatforgb <0the
≃
and the effective energy E of Eq. (11) can also be eval- pure 1D Bosesystemsupports a brightsolitondescribed
uated analytically as by ψ (z)= (|gb|Nb)1/2sech |gb|z . Guided by this result,
b 2 2
1 α 1 β˜ 1 γ δ inthe presentcaseofrepu(cid:18)lsiveb(cid:19)osons(g >0)attracted
E = + + . (23) b
2σb2 2σb2 2σf2 − σ2+σ2 byasinglefermiongbf <0weadoptforthebosonicfield
b f ψ (z) the following ansatz
b
q
The quantities α, γ and δ are the same as in Eq. (20) N1/2 z
while β˜ = 2Nb3π/(3√3). In this case the widths σb and ψb(z)= (2ξb)1/2sech ξ , (27)
σ attheequilibriumcanalsobedeterminedanalytically. (cid:18) (cid:19)
f
They are with ξ the variational width of the bosonic cloud. This
ansatzisalsosuggestedbytheexistenceoftheanalytical
3/2
α+β˜ γ solutionfor the eigenvalueofthe correspondingEq. (26)
σ = 1+ (24)
b δ (α+β˜)! for the fermionic bound state. One finds [46]:
r
2
1
ǫ(ξ)= 1+ g N ξ 1+2g N ξ . (28)
3/2 −2ξ2 | bf| b − | bf| b
(α+β˜)3/4 γ (cid:18) q (cid:19)
σ =γ1/4 1+ , (25)
f δ (α+β˜)! The energy ǫ(ξ) is the sum of the kinetic energy of the
r
fermionicatomandoftheinteractionenergybetweenthe
and both increase as δ decreases, and only when δ = fermionic atom and the bosonic cloud.
g = 0 the two widths become infinitely large. Thus The total energy of the system is then the effective
bf
| |
we conclude that for any finite value of g and g > 0 energyofthe Bosecloudcalculatedfromthe Lagrangian
bf b
| |
there exists a Bose-Fermi bright soliton, the existence of density (2) added to the energy ǫ(ξ) of the Fermi atom
this bright soliton being guaranteed by the behavior of and is given by
the bosonic energy term in the TG regime.
In Fig. 4 we show the axial widths σb and σf of the E = Nb +Nb3 +∞sech6(y)G gbξ dy+ǫ(ξ).
Bose-FermimixturewithanattractiveBose-Bosescatter- 3ξ2 8ξ2 N sech2(y)
Z−∞ (cid:18) b (cid:19)
ing length (g < 0). We choose N = 50, N = 10, and (29)
b b f
g = 0.01 and plot the widths as a function of g . The existence of the Bose-Fermi bright soliton depends
b bf
− | |
The figure shows that, as g goes to zero, the Fermi on the existence of a local minimum of this total en-
bf
| |
width σ is muchlargerthanthe Bosewidth σ . Infact, ergy E = E(ξ) of the system. From Eq. (29) we
f b
7
0.05 the eigenvalue equation
∂2E ∂2E
0.04 ωω12  ∂∂∂σ2σbEσb2f ∂∂∂σ2σbEσf2f −ω2(cid:18)N0b Nf0λm (cid:19)=0 (30)
0.03 Ω  
where the partial derivatives must be calculated at the
equilibrium (σ ,σ ), where σ and σ are the Bose and
b b b f
0.02
FermiwidthsobtainedfromEqs. (17)and(18). Thetwo
frequencies ω and ω then read
1 2
0.01
N ∂2E +λ N ∂2E √∆
b∂σ2 f f ∂σ2 ±
ω = f b , (31)
00 0.05 0.1 0.15 0.2 0.25 0.3 1,2 vu 2λmNbNf
|g | u
bf t
where
FIG. 5: (Color online). Collective breathing frequencies ω1 2
∂2E ∂2E
and ω2 of the self-bound Bose-Fermi droplet with attractive ∆= N +λ N 4λ N N K (32)
bosons(gb <0). Ωistheharmonicfrequencyofthedisplace- b∂σf2 m f ∂σb2! − m b f G
ment |z1−z2| of the centers of mass of the Bose and Fermi
clouds. Parameters as in Fig. 4. Frequencies are in units of and K is the Gaussian curvature of Eq. (19).
G
thefrequency ω⊥ of transverse harmonic confinement. Inaddition,bytakingintoaccountthe“reducedmass”
λ N N /(N +λ N ), one can derive the frequency Ω
m b f b m f
of harmonic oscillation of the relative distance z z
b f
find that E(ξ) |gbf|Nb as ξ + , since for large (displacement) of the two clouds from the equat|ion− |
∼ − 2ξ → ∞
ξ’s G Nbsegcbhξ2(y) ≃ π32 for any value of y. Thus, for λmNbNf Ω2 = ∂2E . (33)
ξue→s. +(cid:0)A∞ga,itnhefreonme(cid:1)rgEyq.goe(2s9t)owzeerosetehrtohuagthEn(eξg)ative+val- Nb+λmNf ∂(|zb−zf|)2
→ ∞
as ξ 0 and therefore we conclude that the function The result is
→
E(ξ), being positive at the origin and vanishingly small,
but negative, at large ξ’s, must possess a negative local 2gbf (Nb+λmNf)
Ω= | | , (34)
minimum that is also the global minumum of the en- s π(σb2+σf2)3/4
ergy. This implies that there is always a finite value of ξ
that minimizes the total energy. As previously stressed, whereagainσ andσ are the Boseand Fermiwidths at
b f
this behavior strictly depends on the properties of the equilibrium, where z =z .
b f
Lieb-Liniger function G(x) for large x and, as a conse- InFig. 5weplotthe frequenciesω andω ofthe cou-
1 2
quence, the Bose-Fermibright solitonexists for any neg- pled breathing modes of the Bose and Fermi clouds by
ative value g even for N =1. choosingthesameparametersofFig. 4,namelyN =50,
bf f b
N = 10, λ = 1 and g = 0.01. The frequencies are
f m b
−
plotted as a function of the Bose-Fermi strength g .
bf
| |
For g = 0, the frequencies are decoupled: ω is the
bf 1
VI. COLLECTIVE OSCILLATIONS OF THE frequ|enc|y of the fermionic axial breathing mode and ω
2
BOSE-FERMI BRIGHT SOLITON
is the frequency of the bosonic axial breathing mode.
Without a Bose-Fermiinteraction the Fermi cloud is de-
After having established the existence of stationary localized(σf = )anditsbreathingfrequencyisω1 =0,
∞
bright solitons in a degenerate Bose-Fermi mixture, we while the Bose cloud remains localized (due to the nega-
study two types of small oscillations of this system tive Bose-Bose strength gb) and its breathing frequency
aroundthestableequilibriumpositionandcalculatetheir ω2 remainsfiniteandisequaltoω2 =gb4Nb4/(16π2). Fig.
frequencies. The first is the stable breathing oscillation 5 shows that both the breathing frequency ω1 and the
of the system around its mean position and the second harmonic frequency Ω of the displacement z1 z2 start
| − |
describes the stable smalloscillations once the centersof from zero and grow as gbf increases.
| |
the FermiandBosecloudsareslightlydisplacedwithre-
spect to each other. Given the effective Lagrangian (9),
the problem of small oscillations is solved by expand- VII. CONCLUSION
ing the kinetic energy (10) and the potentialenergy (11)
around the equilibrium solution up to quadratic terms. We have studied a degenerate 1D Bose-Fermi mixture
Inthiswayweextract[43]thefrequenciesωofthecollec- by using the quantum hydrodynamics. We find that for
tive breathing modes of the Bose and Fermi clouds from attractive Bose-Fermi interaction (g < 0) the ground
bf
8
state of the system is a self-bound Bose-Fermi droplet. ture. The 1D Bose-Fermi mixture must be created in an
The nonexistence of a threshold in the strength of an axialharmonictrapandtheBose-Fermiinteractionmust
attractive Bose-Fermi interaction for the formation of a be turnedfromrepulsivetoattractivebymanipulating a
Bose-Fermi bright soliton in one dimension is confirmed background magnetic field. At the same time the axial
in the case of a single Fermi atom immersed in a de- harmonic trap on the system should be removed. Upon
generate Bose gas with repulsive Bose-Bose interaction. removal of the axial trap, the result is the formation of
We also calculate the frequencies of stable oscillation of a single or a trainof brightsolitons as in the experiment
the Bose-Fermibright soliton. Such a Bose-Fermibright with the degenerate Bose system of 7Li atoms [17] or
soliton is similar to a recently studied Bose-Bose bright as in a numerical simulation in a degenerate Bose-Fermi
soliton bound through an attractive interspecies interac- mixture [19]. By choosing numbers of atoms and inter-
tion [47]. atomic strengths as suggested in the present paper, one
Inviewoftherecentexperimentalstudiesofadegener- obtains a single Bose-Fermibrightsoliton andcan study
ate 1D87Rb gas[29,30]andthe successfulidentification its static and dynamical properties.
of the quasi-BEC and TG regime in it and the observa-
tionofdegenerateBose-Fermimixturein6Li-7Li[16,17],
40K-87Rb[13], 6Li-23Na [18]etc. bydifferentgroups,the
Acknowledgments
experimental realization of a Bose-Fermi bright soliton
seemspossiblewithpresenttechnology. Themostattrac-
tive procedure seems to make use of an experimentally This work is partially supported by the FAPESP and
observed Feshbach resonance [48] in a Bose-Fermi mix- CNPq of Brazil.
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