Nonlinear Schroedinger equation for a superfluid Bose gas from weak coupling to unitarity: Study of vortices

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by  Sadhan K. Adhikari

Nonlinear Schr¨odinger equation for a superfluid Bose gas
from weak coupling to unitarity: Study of vortices
S. K. Adhikari1∗ and L. Salasnich2†
1Instituto de F´ısica Te´orica, UNESP - S˜ao Paulo State University, 01.405-900 S˜ao Paulo, S˜ao Paulo, Brazil
2CNR-INFM and CNISM, Unit`a di Padova, Dipartimento di Fisica “Galileo Galilei”,
Universit`a di Padova, Via Marzolo 8, 35131 Padova, Italy
WeintroduceanonlinearSchr¨odingerequationtodescribethedynamicsofasuperfluidBosegas
in the crossover from the weak-coupling regime, where an1/3 1 with a the inter-atomic s-wave
≪
8 scattering length and n the bosonic density, to the unitarity limit, where a + . We call this
→ ∞
0 equationtheunitarity Schr¨odinger equation (USE).Thezero-temperaturebulkequationof stateof
0 this USE is parametrized by the Lee-Yang-Huang low-density expansion and Jastrow calculations
2 at unitarity. With the help of the USE we study the profiles of quantized vortices and vortex-core
radiusinauniformBosegas. WealsoconsiderquantizedvorticesinaBosegasundercylindrically-
n
symmetricharmonicconfinementandstudytheirprofileandchemicalpotentialusingtheUSEand
a
comparetheresultswiththoseobtainedfromtheGross-Pitaevskii-typeequationsvalidintheweak-
J
coupling limit. Finally, the USE is applied to calculate the breathing modes of the confined Bose
8
gas as a function of thescattering length.
2

PACSnumbers: 03.75.Lm,03.75.Kk,03.75.Hh
]
r
e
h I. INTRODUCTION equation and is related to the energy per particle of the
t
o system.) This result is independent of the atomic scat-
. tering length. In the present work the equation of state
t For the theoretical investigation of the Bose-Einstein
a of the bulk system is parametrizedwith a Pad`eapproxi-
condensate (BEC) of an ultra-cold bosonic gas the main
m
mantbyusingthefirsttwotermsoftheLee-Yang-Huang
tool is the mean-field Gross-Pitaevskii equation (GPE)
- [1, 2], that is reliable for small values of the gas parame- expansion [5] and the Jastrow calculations at unitarity
d
n ter an1/3, where a is the s-wave scattering length of the [8, 9]. In this way we obtain a time-dependent highly-
nonlinear Schr¨odinger equation, that we call unitarity
o inter-atomic potential and n is the bosonic density. By
c manipulating a background magnetic field near a Fes- Schr¨odingerequation(USE).TheUSEingeneralformis
[ time dependentandcanbe usedtostudy non-stationary
hbach resonance the s-wave scattering length a can be

dynamics, whereas its time-independent form is appro-
1 modified and can reach very large values corresponding
priate to study stationary states. The USE gives the
v to a strong atomic interaction [3]. To take into account
2 the effect of a large scattering length, a modified GPE hydrodynamic equations of bosonic superfluids at zero
0 (MGPE) has been introduced by Fabrocini and Polls [4] temperature,andenablesonetostudycollectivedynam-
3 ical properties of the system in the full crossover from
by using the first two terms in the Lee-Yang-Huang ex-
4 weak-coupling to unitarity.
pansion [5] of the energy of a uniform Bose gas, which
.
1 include terms of the order of √na3. Within the density AsanapplicationoftheUSE,herewestudythestruc-
0 functional approach of Fabrocini and Polls [4], the lead- ture of quantized vortices in both uniform Bose gas and
8
ingtermofexpansiongivestheGPEwhilethenextterm Bose gas under axially-symmetric harmonic confinement
0
: is responsible for corrections due to moderate atomic in- and compare and contrast the results with those ob-
v teraction. These two leading terms have been used to tained with the MGPE [4] and GPE. Quantized vortices
i
X calculate beyond-mean-field corrections to properties of in superfluids area manifestationof quantum mechanics
trapped BECs [6]. The contribution terms of the order at the macroscopic level [10]. Recently quantized vor-
r
a of na3 has also been discussed in the literature [7]. tices have been observed in rotating ultra-cold atomic
In this paper we generalize the MGPE [4] by consid- BECs [11] and also in atomic Fermi gases in the BCS-
ering also the behavior of the bosonic system in the uni- BEC crossover near a Feshbach resonance [12]. Using
tarity limit, where a + . Jastrow calculations [8, 9] the GPE, quantized vortices in a harmonic trap have
suggest that in the u→nitar∞ity limit the zero-temperature been analyzed by Dalfovo and Stringari [13]. More re-
bulk chemical potential µ of the Bose system is given cently vortices have been investigated with GPE in var-
by µ = ξn2/3¯h2/m, where ξ is a constant and m is the ious problems; for instance, collective modes of a vor-
mass ofa single atom. (Bulk chemicalpotentialis essen- tex [14], vortex under toroidal confinement [15, 16], vor-
tially the nonlinear term that appear in the mean-field tex lattice [17], vortex with attractive scattering length
[18, 19], collision dynamics of vortices [20], collapse of a
vortexstate[21], freeexpansionofvortices[22], andvor-
tex in Bose-Fermi mixtures [23]. Nilsen et al. [24] used
∗[email protected];URL:www.ift.unesp.br/users/adhikari theMGPE[4]toinvestigatethestructureofvortices,and
†[email protected];URL:www.padova.infm.it/salasnich found a good agreement between the MGPE and varia-
2
tional Monte Carlo results. Using the USE, we extend chemical potential µ(n,a) of the system is a function of
the study of Nilsen et al. [24] andfind that the radiusof the density n and of the inter-atomic scattering length
the vortex core decreases with the increase of the scat- a. For a superfluid the velocity field is irrotational, i.e.
tering length. At the unitarity limit it reaches a critical v=0, and the circulation is quantized, i.e.
∇∧
minimal radius. The properties of this minimal radius
¯h
dependonthetrappinggeometry: inthecaseofavortex v dr=2πL , (3)
I · m
in a uniform Bose gas it is a decreasing function of the
uniform density at large distances; in the case of a vor- where L is an integer (angular momentum) quantum
texintheharmonictrapitisadecreasingfunctionofthe number [10].
totalnumberofatoms. UsingtheUSE,wealsocalculate We can introduce the complex order parameter [2, 32]
the radial and axial frequencies of collective oscillations Ψ(r,t) = n(r,t)1/2eiS(r,t) such that the phase S(r,t) of
from weak-coupling to unitarity in a cigar-shapedtrap. the order parameter fixes the superfluid velocity field
In Sec. II we introduce the present model and relate
¯h
it to superfluid hydrodynamics. Formulation for quan- n(r,t)v(r,t)= i (Ψ∗ Ψ Ψ Ψ∗) (4)
− 2m ∇ − ∇
tized vortices in the present model is considered in Sec.
III. Section IV is devoted to vortex in a uniform Bose so that v(r,t)=(h¯/m) S(r,t). In this way we can map
∇
gas, where we numerically study the vortex profile and Eqs. (1)and(2)intothefollowingtime-dependenthighly
vortex-core radius for different scattering lengths. In nonlinear Schr¨odinger equation
this case the USE is solved by the fourth-order Runge-
∂ ¯h2
Kutta method [25, 26]. The vortex-coreradius decreases i¯h Ψ= 2+U(r)+µ(n,a) Ψ, (5)
with increasingscattering length and saturates to a con- ∂t h− 2m∇ i
stant value in the unitarity limit. In Sec. V we con- In general, one can use the hydrodynamic equations (1)
sideravortexinaBosegasunderaxially-symmetrichar- and (2), or equivalently Eq. (5), to study the global
monic pancake-shaped confinement by solving the USE properties of the superfluid, like the stationary density
by imaginary time propagation using the semi-implicit profile, the free expansion and the collective oscillations.
Crank-Nicholson rule [26, 27, 28]. In this case we study For a Bose gas the following two leading terms of the
theprofilesofvorticesandcorrespondingchemicalpoten- low-densityexpansionofthe bulk chemicalpotentialcan
tials and compare the results with those obtained from be obtained [24] from the expression for energy per par-
the MGPE [4] and GPE. The interesting feature of the ticle as obtained by Lee, Yang and Huang [5]
results of the USE is that they saturate in the unitarity
4π¯h2 32
limit as a , whereas the results of MGPE and GPE µ(n,a)= an 1+ (n1/3a)3/2+ ... , (6)
donotsatu→ra∞teinthislimit. Thefrequenciesofcollective m (cid:18) 3π1/2 (cid:19)
breathing oscillatons in a cigar-shaped trap are consid-
where n1/3a is the dimensionless gas parameter [32].
ered in Sec. VI. Finally, in Sec. VII. we present the
Note that in this expansionthe scattering length a must
concluding remarks.
be positive (a>0) corresponding to a repulsive interac-
tion. Higher order correctionterms to the bulk chemical
potential have also been considered in the literature [7].
II. SUPERFLUID HYDRODYNAMICS AND
The lowest order term of expansion (6) was derived by
NONLINEAR SCHRO¨DINGER EQUATION
Lenz [33]. Considering only this term, Eq. (5) becomes
the familiar GPE [1, 2]
The zero-temperature collective properties of a dilute
bosonic superfluid under the externalpotentialU(r) can i¯h ∂ Ψ= ¯h2 2+U(r)+ 4π¯h2aΨ2 Ψ. (7)
be described by the quantum hydrodynamic equations ∂t h− 2m∇ m | | i
[29, 30]:
By taking into account also the second term of the ex-
pansion (6), Eq. (5) becomes the so-called modified
∂n
+ (nv)=0, (1) Gross-Pitaevskii equation (MGPE) introduced by Fab-
∂t ∇· rocini and Polls [4]:
∂v ¯h2 2√n m
m + ∇ + v2+U +µ(n,a) =0(,2) ∂ ¯h2 4π¯h2
∂t ∇(cid:20)−2m √n 2 (cid:21) i¯h Ψ = 2+U(r)+ aΨ2
∂t h− 2m∇ m | |
where n(r,t) is the local density and v(r,t) is the local 32
1+ a3/2 Ψ Ψ. (8)
velocityofthebosonicsystem. Inthesezero-temperature × (cid:18) 3√π | |(cid:19)i
hydrodynamical equations statistics enters the equation
Inthe unitaritylimit, wherea + ,fordimensional
of state through the bulk chemical potential µ(n,a) and → ∞
reasons [8, 9] the bulk chemical potential must be of the
the quantum-pressure term [h¯2/(2m√n)] 2√n, which
− ∇ form [8]
is absent in the classical hydrodynamic equations [31].
Thishydrodynamicregimeisachievedinthelimitofvery ¯h2
largenumberofatomsN. Atzerotemperature,thebulk µ(n,a)=ξ n2/3 (9)
m
3
where ξ is a universal coefficient. Thus, in the unitarity III. QUANTIZED VORTICES
limit the bulk chemical potential is proportional to that
of a non-interacting Fermi gas [34]. Recent numerical The structure of vortices in superfluid 4He at zero-
calculations basedonJastrowvariationalwavefunctions
temperaturewasinvestigatedmanyyearsagobyChester,
give ξ =22.22[8] and we shallconsider this value ofξ in
Metz and Reatto [39] by using a many-body variational
our calculation.
wave function, and more recently by Dalfovo [40] by us-
In the full crossover from the small-gas-parameter
ing the Orsay-Trento density-functional [41]. The two
regimetothelarge-gas-parameterregime,wesuggestthe
approaches,whichgive verysimilar results,are basedon
followingexpressionasthebulkchemicalpotentialofthe
the assumption that the superfluid velocity of the quan-
Bose superfluid
tized vortex rotating aroundthe cylindric z axis is given
¯h2 by
µ(n,a)= n2/3 f(n1/3a) (10)
m
¯h L
v= u , (13)
where f(x) is an unknown dimensionless universal func- m ρ φ
tion of the gas parameter x = n1/3a. A general Pad`e
approximant for the function f(x) consistent with the where L is the quantum number of circulation, ρ is
expansion of Lee, Huang, and Yang (6) [5] and Eq. (10) the cylindric radial coordinate and u is the unit az-
φ
should have the following form imuthal vector, with φ the azimuthal angle. In 1997
Sadd, Chester and Reatto [42] suggested that the veloc-
x+αx5/2 ity of superfluid 4He is not truly singular at the vortex
f(x)=4π , (11)
1+γx3/2+βx5/2 line and the vorticity is distributed over a finite region.
Nevertheless, the deviations from Eq. (13) in 4He seem
where α, β, and γ are yet undetermined parameters.
to be very small [42].
Withoutfurtherinformationaboutµ(n,a)wecannotde-
Werecallthatthereareremarkabledifferencesbetween
termine all these parameters consistently. In this paper
superfluid 4He and the dilute superfluid we are consid-
we consider the minimal form of this function consistent
ering here. In 4He the effective radius R of the inter-
withEqs. (6)and(10)obtainedbysettingγ =0andα= 0
atomic potential is of the order of the average distance
32/(3√π) and β =4πα/ξ, with ξ =22.22. The function
f(x)ofEq. (11)issuchthatf(x)=4π[x+32x5/2/(3√π)] n−1/3 betweenatoms, i.e. R0n1/3 1. Instead, in dilute
≃
ultra-cold gases the effective radius R is always much
for x 1 and the present model reduces to the MGPE 0
≪ smaller than the average distance n−1/3 between atoms,
(8) [4]. In the opposite extreme x 1, f(x) = ξ, and
≫ i.e R n1/3 1. For a dilute gas the coupling regime de-
the present model reduces to the unitarity limit (9). We 0
≪
pends on the scattering length a, namely on the gas pa-
call Eq. (5) equipped with Eqs. (10) and (11), e.g.,
rameter an1/3: in the weak-coupling regime an1/3 1,
≪
∂ ¯h2 4π¯h2 while in the strong-coupling regime an1/3 1. Another
i¯h Ψ = 2+U(r)+ aΨ2 ≫
remarkable difference between liquid helium and quan-
∂t h− 2m∇ m | |
tum gases of alkali-metal atoms is the width of the vor-
1+αa3/2 Ψ
| | Ψ. (12) tex core. The core of a quantized vortex is only few
× (cid:18)1+βa5/2|Ψ|5/3(cid:19)i angstroms in superfluid 4He while it is of the order of
by the name USE, i.e. unitarity Schr¨odinger equa- sub-micron in a dilute atomic gas [43].
tion. As, by construction, the USE has the proper weak In our USE we get Eq. (13) by setting Ψ(r,t) =
[24] and strong [8] coupling limits, it is appropriate for ψ(ρ,z)exp[i(Lφ−µh¯0t)],whereµ0 isthechemicalpoten-
the study of weak-to-strong coupling crossover. This is tial of the inhomogeneous superfluid, fixed by the nor-
the model equation we use in the following sections to malization. In this way, Eq. (5) with Eq. (10) becomes
study quantized vortices in a Bose superfluid from weak
to strong coupling. In this Sec. Ψ is normalized by ¯h2 L2 ∂2
|Ψ|2dr=N. h−2m(cid:18)∇2ρ− ρ2 + ∂z2(cid:19)+U(ρ,z)+µ(n,a)iψ =µ0 ψ ,
R It is important to stress that the present approach, (14)
based on the quantum hydrodynamics, could be applied where 2 = 1 ∂ (ρ ∂ ) is the Laplacian operator in the
to both superfluid bosons and fermions [29, 35]. This radial d∇irρectioρn∂ρand∂ρh¯2L2 is the centrifugal term which
hydrodynamic approach is a time-dependent local den- 2mρ2
determinesthesizeofthevortexcore,thatisoftheorder
sity approximationwith gradientcorrections. In the last
of the healing length l =h¯L/√2mµ [44]. This expres-
few years we have succesfully applied it to investigate h 0
sion is obtained by equating the centrifugal term to the
the collective properties of different dilute systems, like
chemical potential µ .
Bose-Fermimixtures[36],thesuperfluidFermigasinthe 0
BCS-BEC crossover[37], and the 1D Lieb-Liniger liquid We introduce scaled variables by using the character-
[38]. Here we have obtained the USE valid from weak- istic length lc of the system. In particular, we make the
coupling to unitarity based on the general quantum hy- following transformations: ρ¯= ρ/lc, z¯= z/lc, a¯ = a/lc,
drodynamical scheme. U¯ = U(ml2)/¯h2, µ¯ = µ (ml2)/¯h2. ψ¯ = ψ l3/2. In this
c 0 0 c c
4
1111
IV. VORTEX IN A UNIFORM BOSE GAS
0000....8888
yyyy
sitsitsitsit Let us first consider the case U(ρ,z)=0. We suppose
DenDenDenDen 0000....6666 aaaa////llllcccc ==== 0000....00001111 that the Bose gas is asymptotically uniform, i.e.
d d d d
eeee 0000....4444
calcalcalcal NNNNuuuuEEEmmmmqqqeeee... rrrr(((iiii222cccc000aaaa)))llll LLLLLLL=======1111111 ψ¯(ρ¯,z¯) 1 (16)
SSSS 0000....2222 NNuummeerriiccaall LL==22 →
Eq. (20) L=2
for ρ¯,z¯ . This condition fixes the characteristic
0000
→ ∞
0000 5555 ρρρρ 1111////llll0000 11115555 22220000 length lc, that must be lc =n∞−1/3 with n∞ the uniform
cccc density at infinity. In this case the chemical potential is
1111
fixed by the asymptotic condition, from which one finds
0000....8888
yyyy
sitsitsitsit µ¯ =f(a¯). (17)
enenenen 0000....6666 aaaa////llllcccc ==== 0000....00003333 0
DDDD
ed ed ed ed 0000....4444 Inaddition,wecanchooseawavefunctionwhichdepends
calcalcalcal NNNNuuuuEEEmmmmqqqeeee... rrrr(((iiii222cccc000aaaa)))llll LLLLLLL=======1111111 only on ρ¯, and Eq. (15) becomes
SSSS 0000....2222 NNuummeerriiccaall LL==22
Eq. (20) L=2 1 ∂2 1 ∂ L2
0000 + +ψ¯4/3f(ψ¯2/3a¯) f(a¯) ψ¯=0.
0000 2222 4444 6666 8888 11110000 11112222 (cid:20)− 2∂ρ¯2 − 2ρ¯∂ρ¯ 2ρ¯2 − (cid:21)
ρρρρ////llll
cccc (18)
1111
This second-orderordinarydifferentialequationmust be
yyyy 0000....8888 solved numerically. Clearly with L = 0 the vortex func-
DensitDensitDensitDensit 0000....6666 aaaa////llllcccc ==== 0000....1111 tbieohnavψ¯io(ρr¯)ofhψa¯s(ρ¯a)fcoorrρ¯e arounidsoρ¯bt=ain60e.dbTyhneegalseyctminpgtotthiec
d d d d →∞
eeee 0000....4444 spatialderivativesinEq. (18). Inthiswayweobtainthe
alalalal NNNNuuuummmmeeeerrrriiiiccccaaaallll LLLL====1111
cccc EEEqqq... (((222000))) LLL===111 algebraic equation
SSSS 0000....2222 NNuummeerriiccaall LL==22
Eq. (20) L=2
0000 ψ¯(ρ¯)4/3f(ψ¯(ρ¯)2/3a¯)=f(a¯) L2 . (19)
0000 1111 2222 3333 4444 5555 6666 − 2ρ¯2
ρρρρ////llll
cccc
1111
In the weak-coupling regime, a¯ 1 where f(x) = 4πx,
yyyy 0000....8888 Eq. (19) gives ≪
sitsitsitsit
DenDenDenDen 0000....6666 aaaa////llllcccc ==== 11110000 L2 1/2
aled aled aled aled 0000....4444 NNNNuuuummmmeeeerrrriiiiccccaaaallll LLLL====1111 ρ¯l→im∞ψ¯(ρ¯)=(cid:18)1− 8πa¯ρ¯2(cid:19) . (20)
cccc EEEqqq... (((222111))) LLL===111
SSSS 0000....2222 NNuummeerriiccaall LL==22
Eq. (21) L=2 Instead, in the strong-coupling regime, a¯ 1 where
0000 f(x)=ξ, Eq. (19) gives ≫
0000 1111 2222 3333
ρρρρ////llll
cccc L2 3/4
lim ψ¯(ρ¯)= 1 . (21)
ρ¯→∞ (cid:18) − 2ξρ¯2(cid:19)
FIG.1: (Coloronline)Numericalresultsforscaleddensityof
vortices ψ¯2(ρ¯) with L = 1 and 2 in a uniform Bose gas ob- Althoughthe limit (20) depends onthe scaledscattering
tainedbysolvingtheUSEforscaledscatteringlengthsa/lc = length a¯, the limit (21) is independent of a¯. The scaled
(a) 0.01, (b) 0.03, (c) 0.1, and (d) 10 compared with the an- healing length ¯l =l /l , which estimates the size of the
alytic asymptotic results (20) or (21). h h c
vortex core [44], is given by
¯l =L/ 2f(a¯). (22)
h
way Eq. (14) becomes p
Now we consider Eq. (18) in the limit ρ¯ 0. Because
→
of the regularity of the wave function and the function
1 ∂2 1 ∂ 1 ∂2 L2 f(x) in this limit the last two terms [the terms involv-
+ +U¯(ρ¯,z¯)
(cid:20)− 2∂ρ¯2 − 2ρ¯∂ρ¯− 2∂z¯2 2ρ¯2 ingthefunctionf(x)]inthisequationremainsfiniteand
can be neglected in comparison to the angular momen-
+ψ¯4/3f(ψ¯2/3a¯) ψ¯=µ¯ ψ¯, (15) tumtermL2/(2ρ¯2). Consequently,wehavethefollowing
0
(cid:21)
condition at small ρ¯
where ψ¯(ρ¯,z¯) is assumed to be real. ρ¯li→m0ψ¯(ρ¯)=ρ¯L×ϕ(ρ¯), (23)
5
6666 555000000000
nnnnuuuummmmeeeerrrriiiiccccaaaallll LLLL====2222
eeee yyy
rtex corrtex corrtex corrtex cor 4444 nnuuEEEEmmqqqqee.... rr((((ii2222cc2222aa))))ll LLLLLL======112221 onlinearitonlinearitonlinearit 343434000000000000000000 NNN === 111000000000
oooo nnn
us of vus of vus of vus of v 2222 ective ective ective 121212000000000000000000 MMGGUUUPPSSSEEEEE
didididi EffEffEff GPE
aaaa
RRRR 000
0000 000 000...111 000...222 000...333 000...444
aaa///lll
----2222 ----1111 0000 1111 ccc
111000000000000
lllloooogggg ((((aaaa////llll ))))
11110000 cccc
yyy
aritaritarit 888000000000
eee aaa///lll === 000...000555
nnn ccc
FIG. 2: (Color online). The numerical values of vortex core nlinlinli 666000000000
radiusforL=1and2vs. scaledscatteringlengthcalculated ooo
nnn
by the USE compared with the analytic results of healing e e e 444000000000
vvv
length given by Eq. (22). cticticti UUUSSSEEE
eee 222000000000 MMGGPPEE
EffEffEff GPE
000
TABLE I: The parameters for L = 1 used in the numerical
solutionofEq. (18)asplottedinFig. 1;inthiscaseψ¯(0)=0. 000 222000000000 444000000000 666000000000 888000000000 111000000000000
For L=2, ψ¯(0)=ψ¯′(0)=0. NNN
a¯ ψ¯′(0) f(a¯) ¯l =1/ 2f(a¯)
h
0.01 0.2934232 0.126415651 1.989 p FIG. 3: (Color online). The numerical values of effective
nonlinearity N2/3f(N1/3a¯) of the USE, 4πNa¯ of the GPE,
0.03 0.5157240 0.388573686 1.134
and 4πNa¯(1+αa¯3/2√N) of the MGPE for ψ¯ = 1 for (a)
0.1 1.0186620 1.47985617 0.581
N =1000 vs. a/l and (b) a/l =0.05 vs. N.
c c
10 3.4622586006 22.3160243 0.149
asit canbe comparedwith the chemicalpotentialofEq.
where ϕ(ρ¯) is a smooth function.
(18). Hence the numerical values of f(a¯) for different a¯
We next solve Eq. (18) for L = 1 and 2 using the
shouldgiveanideaofthe variationofchemicalpotential
classic fourth-order Runge-Kutta method [25, 26]. This
with coupling. The quantity f(a¯) is also related to the
methodproducesveryaccurateconvergence. We employ scaled healing length ¯l given by Eq. (22) as shown in
a space step of ∆ = 0.0001 and integrate up to a scaled h
Table I.
distance of ρ¯max = 20. The integration is started with
the initial boundary condition (23) with a trial ψ¯(0) and Itisinterestingtocalculatenumericallythevortexcore
ψ¯′(0). For L=1 this condition is taken as ψ¯(0)=0 and radiusdefinedasthevalueofρ¯forwhichψ¯2(ρ¯)=0.5. In
ψ¯′(0) = constant and the integration started at ρ¯ = 0. Fig. 2 we plot the numericalvalues of vortexcore radius
ForL=2bothψ¯(0)=0andψ¯′(0)=0andthenumerical versusscaledscatteringlengthandcomparewiththecor-
integrationcannot be started at ρ¯=0, as then ψ¯as well responding analytical results of healing length given by
asψ¯′ atsubsequentsitesbecomezero. ForL=2,theinte- Eq. (22). The figure shows that the vortex-core radius
has the same trend of the analytical healing length [44].
gration is started at ρ¯=∆ with the boundary condition
ψ¯(∆)=0andψ¯′(∆)equaltoasmallconstant. Theinte- From Figs. 1 and 2 we find that the vortex-core radius
decreaseswithincreasingcouplingbutincreaseswithan-
grationispropagatedtoρ¯=ρ¯max wheretheasymptotic
gular momentum L. However, for very large coupling it
conditions (20) or (21) remain valid. If after integration
saturates. This saturation of the vortex core is properly
with a trial guess, the boundary conditions (20) or (21)
given by the USE.
cannot be satisfied, the method is implemented with a
newtrialguess. Theprocessiscontinueduntilasolution
satisfying the proper boundary conditions at small and
large ρ¯is obtained. We obtain the solution for different V. VORTEX IN A BOSE GAS IN A HARMONIC
values of a¯. TRAP
The numerical results for the scaled density [ ψ¯2(ρ¯)]
≡
for different scaled scattering length a¯ a/lc is plotted Now we consider the bosonic fluid in an axially-
≡
in Fig. 1 and compared with the asymptotic forms (20) symmetric harmonic trapping potential, i.e.
or (21) for L = 1 and 2. The corresponding parameters
ψ¯′(0)andf(a¯)usedinthenumericalsolutionaregivenin 1
TableI. Thequantityf(a¯)showninTableIisinteresting U = 2mωρ2(ρ2+λ2z2) (24)
6
wthheetrreaλns=verωsez/fωreρquisenthcye aanndisoωtrotphye apxairaalmfreetqeur,enwciyt.hWωρe ntialntialntial 111111242424000000 UUUSSSEEE
z eee
choose as characteristic length of the system the trans- PotPotPot 111000000 MMGGGPPPEEE
ivnegrsethhearsmcaolnedicvlaenrigatbhl,esi.et.helcco=nfinpin¯hg/(pmoωteρn)t.ialByreaudss- cal cal cal 888000
U¯ = 1(ρ¯2 + λ2z¯2) . With this confining potential the mimimi 666000 NNN === 111000000000000
mean-fi2eld equation can be written explicitly as hehehe λλλ222 === 888
CCC 444000
d d d
1 ∂2 1 ∂ 1 ∂2 + L2 + 1(ρ¯2+λ2z¯2) alealeale 222000
(cid:20)− 2∂ρ¯2 − 2ρ¯∂ρ¯− 2∂z¯2 2ρ¯2 2 ScScSc 000
1+αa¯3/2N1/2ψ¯ ---444 ---333...555 ---333 ---222...555 ---222 ---111...555 ---111 ---000...555 000 000...555
+4πNa¯ψ¯21+βa¯5/2N5/6ψ¯5/3(cid:21)ψ¯=µ¯0 ψ¯, (25) llloooggg111000(((aaa///lllccc)))
alalal 111444000
Now the normalization condition is given by entientienti 111222000 MMGGUUUSSSPPEEEEE
∞ ∞ ∞ PotPotPot 111000000 GPE
2πZ0 ρ¯dρ¯Z−∞dz¯ψ¯2(ρ¯,z¯)≡2πZ0 ρ¯dρ¯nc(ρ¯)=1(.26) mical mical mical 888000 aaa///lllccc === 000...111555111555555
666000
hehehe λλλ222 === 888
Here we defined nc(ρ¯) as the column density (as in [24]) CCC 444000
appropriate for the study of radial distribution of mat- d d d
eee 222000
ter inthe vortex. Equations(25)and(26) constitute the alalal
ccc
USE for an axially trapped Bose gas valid from weak- SSS 000
coupling (a¯ 0 ) to the unitarity limit (a¯ ) and 000 111 222 333 444 555
→ → ∞
(in spite of a slightly complicated numerical structure) llloooggg (((NNN)))
111000
it is no more difficult to solve than the usual GP equa-
tion valid in the weak-coupling limit. In the extreme
weak-coupling limit, the non-linear term of Eq. (25) be- FIG. 4: (Color online). The numerically calculated scaled
comestheusualGPterm4πNa¯ψ¯2. Intheunitaritylimit chemicalpotentialµ¯0 (inunitsofoscillator energy¯hωρ)from
GPE,MGPEandUSEvs. (a)scaledscatteringlengtha/l for
this term becomes independent of scattering length and c
equalsξN2/3ψ¯4/3,ξ =22.22. Werecallthatifweusethe N =10000,λ2=8,and vs. (b) N for a/lc =0.15155,λ2=8.
GPE(7),aswellastheMGPE(8),intheunitaritylimit,
it will lead to an inappropriate dependence of the non-
linearity on scattering length, as well as on the number of Eq. (25) by imaginary time propagation method af-
of atoms N. terdiscretizingitwiththesemi-implicitCrank-Nicholson
It is appropriate to study the behavior of the nonlin- rule[26,27,28]. Inthe processofdiscretizationweuse a
earity of the USE (25), that is N2/3ψ¯4/3f(N1/3ψ¯2/3a¯). timestepof0.0005andaspacestepof0.03. Welimitour
To study the dependence of this nonlinearity on N and numerical study to the L = 1 vortex, as L > 1 vortices
a¯ we set ψ¯= 1 in this expression. The resultant nonlin- areunstableanddecaytoseveralL=1vorticesconserv-
earities are plotted in Fig. 3 (a) and (b) as a function ing the angularmomentum. In the numericalsimulation
of a¯ and N for constant N = 1000 and a/l = 0.05, re- of a vortex with L = 1 we employ a pizza-shaped con-
c
spectively. The interesting feature of the nonlinearity of densatewithtrapanisotropyλ2 =8asinthe theoretical
this equation is exhibited in Fig. 3 (a), where we find study of Nilsen et al. [24] and the experiment at JILA
that, for a fixed N, the nonlinearityof this equationsat- [45]. Afterthewavefunctionψ¯(ρ¯)ofEq. (25)isobtained
urates at large scattering lengths a¯. In the GP model by the imaginary time propagation, the chemical poten-
the nonlinearity increases linearly for all a¯, whereas for tialisobtainedbymultiplyingthis equationbyψ¯(ρ¯)and
theMGPmodelitincreasesindefinitely,butwithamore integrating over all space.
complicated dependence on a¯. From Fig. 3 (b) we find In Fig. 4 (a) we plot the numerical results of scaled
that for a fixed scattering length the nonlinearity of Eq. chemical potential vs. scaled scattering length a/l as
c
(25) as well as the GP equation increases with N. For obtained from the USE (25) as well as the GPE (7)
theGPequationitincreaseslinearlywithN,whereasfor for N = 10000. In Fig. 4 (b) we plot the same vs.
Eq. (25) it increases as N2/3 for large N. N for a/l = 0.15155. The scaled scattering length
c
TofurtherstudytheUSE(25),wesolveitnumerically. a/l = 0.15155 is the value studied by Nilsen et al. and
c
Fornumericalconveniencewe transformthis equationto isappropriateforRbatomsinanexperimentaltrapused
its time-dependent form by replacing the term µ¯ ψ¯ by at JILA for a scattering length a = 35a(Rb), where
0
its time-dependent counterpart i∂ψ¯/∂t appropriate for a(Rb)= 100a (a the Bohr radius) is the experimen-
0 0
non-stationary states. However, in this paper we limit tal atomic scattering length of Rb. In Table II we plot
ourselves to a study of stationary states. Nevertheless, someofthe resultsforchemicalpotentialµ¯ fordifferent
0
theintroductionoftime-dependenceallowsforasolution a¯ and N values. The first row in this Table agrees with
7
0000000.......000000022222227777777
GGGGGGGPPPPPPPEEEEEEE,,,,,,, MMMMMMMGGGGGGGPPPPPPPEEEEEEE aaaaaaa///////lllllll =======0000000.......00000001111111
TABLEII:Scaledchemicalpotentialµ¯0 ofGPE,MGPE,and ccccccc
UUUUUUSSSSSSEEEEEE aaaaaa//////llllll ======000000......000000111111
USEfor different a¯=a/l , N, and GP nonlinearity a¯N. cccccc
c GGGGGPPPPPEEEEE,,,,, MMMMMGGGGGPPPPPEEEEE aaaaa/////lllll =====00000.....0000055555
a0¯.15155 N500 7a¯5N.775 µ13G.P1E9 µ15M.6G2PE 1µ5U.S2E7 densitydensitydensitydensitydensitydensitydensity 0000000.......000000011111118888888 UUUUSSSSGGEEEEUUU SSPSPaaaaEEEEE////llllccccccccc aaaaa====/////0000lllllccccc....=====0000333335555
0.01 10000 100 14.63 14.88 14.88 mn mn mn mn mn mn mn MGPE a/lc=3
0.01 100000 1000 35.73 36.74 36.70 ColuColuColuColuColuColuColu 0000000.......000000000000009999999 NλNλNλNλNλNλNλ2222222 ============== 111111188888880000000000000000000000000000
0.01 1000000 10000 89.25 93.16 92.99
0.1 10000 1000 35.73 43.32 42.10
1 10000 10000 89.25 196.13 93.83 0000000
3 10000 30000 138.40 460 95.26 0000000 2222222 4444444 6666666 8888888 11111110000000 11111112222222 11111114444444 11111116666666 11111118888888 22222220000000 22222222222222 22222224444444
ρρρρρρρ///////lllllll
ccccccc
00000000........0000000033333333
GGGGGGGGPPPPPPPPEEEEEEEE NNNNNNNN========555555550000000000000000
UUUUUUUSSSSSSSEEEEEEE NNNNNNN=======555555500000000000000
P(ρ/l ,z/l ) yyyyyyyy 00000000........000000002222222244444444 MMMMMMGGGGGGPPPPPPEEEEEE NNNNNN======555555000000000000
0.0009 c c sitsitsitsitsitsitsitsit GGGGGPPPPPEEEEE NNNNN=====55555000000000000000
nnnnnnnn 00000000........000000001111111188888888 UUUUSSSSEEEE NNNN====5555000000000000
eeeeeeee
dddddddd GGGPPPEEE NNN===111000000000000000
n n n n n n n n UUSSEE NN==110000000000
0.0006 mmmmmmmm 00000000........000000001111111122222222 MGPE N=100000
uuuuuuuu
ColColColColColColColCol aaaaaaaa////////llllllllcccccccc ======== 00000000........1111111155555555111111115555555555555555
6 00000000........000000000000000066666666 λλλλλλλλ22222222 ======== 88888888
0.0003
3
00000000
0 z/l 00000000 22222222 44444444 66666666 88888888 1111111100000000 1111111122222222 1111111144444444 1111111166666666 1111111188888888
c ρρρρρρρρ////////llllllll
0 -3 cccccccc
15
10 5 -6
0
ρ/l FIG. 6: (Color online). The numerically calculated column
c
density n (ρ¯) = ∞ dz¯ψ¯2(ρ¯,z¯) from the USE, GPE, and
c −∞
MGPE vs. ρ/l Rfor (a) N = 10000,λ2 = 8 and different
c
FIG. 5: (Color online) A typical profile of the probability a/l , and (b) a/l =0.15155,λ2 =8 and different N.
density P(ρ/l ,z/l ) = ψ¯2(ρ/l ,z/l ) for a vortex obtained c c
c c c c
with theUSEfor N =10000 and a¯=1.
nonlinearity of Na¯ = 10000) the chemical potentials of
the calculation of Nilsen et al. for the GP and the MGP theMGPEandtheUSEdifferbylessthan0.2%whereas
models. the chemical potentials of these two models differ from
From Fig. 4 (a) and Table II we find a saturation of that of the the GP model by about 3%. (We recall that
the present mean-field result for large scattering length a¯ = 0.01 is the typical experimental value of this quan-
in the unitarity limit. This is consistent with the satu- tity in the experiment at JILA [2, 45].) The chemical
ration of the nonlinearity of the USE (25) in this limit. potentials of MGPE and USE start to differ for large a¯
After this saturation is obtained, any further increase in values(largerthantheexperimentalvaluebutattainable
the scattering length at a fixed N does not change the by the Feshbach resonance technique [3]): the chemical
chemical potential µ¯ . Figure 4 must be compared with potential of the USE model exhibits saturation,whereas
0
Fig. 3. For a fixed scattering length, as N is increased the chemical potential of the MGP model increases very
the nonlinearityofthe USE (25)keeps onincreasing. As rapidlywithincreasinga¯valueswhichhasbeenmadeex-
aconsequencethechemicalpotentialalsoincreasesindef- plicit in the data in the last two rows of Table II, where
initely asN. In Fig. 4 (b) the USE chemicalpotential is the results of the MGPE [4] and USE show the main
always greater than the GP one. On the other hand, for differences.
a fixed N, as a¯ is increased the nonlinearity of the USE In Fig. 5 we demonstrate a typicalprofile of the prob-
saturates above a certain value of a¯. This has a con- ability density of a vortex with L = 1 as obtained from
sequence in the results: With the increase of scattering the USE for N = 10000 and a¯ = 1, corresponding to a
length, as the system tends to the unitarity limit, there large GP nonlinearity of Na¯ = 10000. The density is
are a saturation of the present chemical potential and a manifestly zero for ρ = 0 for the vortex state. However,
crossover,beyond which the USE chemical potential be- both for theoretical and experimental analysis it is ap-
comes smaller than the GP one. From Figs. 4 we find propriate and more convenient to calculate the column
that for moderate to small a¯ values the results for chem- density n (ρ¯) as defined in Eq. (26) to study the ra-
c
ical potential of the MGPE (8) [4] and USE (25) remain dialdistributionofmatter ina vortexforthe USE,GPE
very close to each other. From Table II we find that for and MGPE. The results for column density are plotted
N = 106, and a¯ = 0.01 (corresponding to a large GP in Figs. 6 for (a) N = 10000 for different l/l and for
c
8
111111666666 4.4
MMMMMMGGGGGGGGGGGUUUUPPPPPPPPPPPSSSSEEEEEEEEEEEEEEE <<<<<<<<<<<<<<<ρρρρρρρρρρρρρρρ///////////////lllllllllllllllccccccccccc>>>>>>>>>>>>>>> 4.2 USE
111111222222 cccc
MMMGGGPPPEEE <<<zzz///lllccc>>> 1) 4
eseseseseses GGPPEE <<zz//llcc>> +
zzzzzz USE <z/l > Γ
sisisisisisi 888888 c 2( 3.8
s s s s s s =
rmrmrmrmrmrm 444444 NλNλNλNλNλNλ222222 ============ 111111888888000000000000000000000000 2νρ 3.6
3.4
000000 3.2
------444444 ------333333 ------222222 ------111111 000000 111111 0 0.4 0.8 1.2 1.6
lllllloooooogggggg ((((((aaaaaa//////llllll )))))) atan(a/l )
111111000000 cccccc c
USE
2.52
FIG.7: (Coloronline). Thenumericallycalculatedscaledrms
)
sizeshρ/lciandhz/lciofthecondensatefromUSE,GPE,and +1 2.48
MGPE vs. scaled scattering length a¯=a/l Γ
c (
1/
- 2.44
3
=
2
z
ν 2.4
2.36
(b) a/l = 0.15155 for different N. The density profile
c
0 0.4 0.8 1.2 1.6
of Fig. 6 (b) for N = 500 is also reported in [24] and
atan(a/l )
the two agree with each other for the GP equation. An c
interesting feature of the plots of Figs. 6 is that with
an increaseof nonlinearity (either via an increase ofa/l
c FIG. 8: (Color online) Square of the reduced frequencies of
for a fixed N, or via an increase of N for a fixed a/lc), collective radial and axial oscillations (a) ν2 and (b) ν2 in a
ρ z
the columndensity extends up to a largerradius. Ofthe cigar-shaped trap vs. atan(a/l ).
c
columndensities,thatobtainedwiththeUSEextendsto
a larger radius than the GP column density in general.
This trendis reversedinthe unitaritylimitoflargescat- VI. COLLECTIVE OSCILLATIONS IN
tering length as can be seen from the plot of Fig. 6 (a) HARMONIC CONFINEMENT
for a/l = 3. For small values of a/l the MGPE results
c c
are close to the GPE results. For medium values of a/l We consider the effect of confinement due to an exter-
c
theMGPEresultsareclosetotheUSEresults. However, nalanisotropicharmonicpotential(24)onfrequenciesof
forlargevaluesofa/l the MGPEwavefunctionsextend collectiveoscillationofthesystemfromweak-couplingto
c
to very large distances, compared to the other models, unitarity using the USE. It has been shown by Cozzini
due to an unphysically large nonlinearity in this model andStringari[46]thatassumingapower-lawdependence
(MGPE). µ=AnΓ forthechemicalpotential(polytropicequation
of state [47]), from Eqs. (1) and (2) without the quan-
tumpressureterm,one finds analyticexpressionsfor the
A manifestation of the saturation of nonlinearity of
collectivebreathingfrequenciesofthesuperfluid. Inpar-
the USE is explicit in the rms (root mean square) ra-
ticular, for the very elongated cigar-shaped traps, the
dial ( ρ/l ) and axial ( z/l ) sizes of the condensate as
the schattecriing length ahis inccireased. The results for the collective radial breathing mode frequency Ωρ is given
by [46]
rms sizes calculated with the GPE keeps on increasing
indefinitely as the scattering length is increased. The Ω = 2(Γ+1)ω , (27)
ρ ρ
rmssizescalculatedwiththeMGPEincreasesevenmore p
while the collective longitudinal breathing mode Ω is
rapidly than the results of the GPE. However, the rms z
sizes calculated with the USE saturates after a certain
3Γ+2
value of a, beyond which they remain constant. This Ωz =r Γ+1 ωz . (28)
is illustrated in Fig. 7 where we plot the rms sizes vs.
log(a/l )ascalculatedbytheUSE,GPE,andMGPE.As
c Here we introduce an effective polytropic index Γ as
we are considering a pizza-shaped condensate the radial
the logarithmic derivative of the bulk chemical potential
size ρ/l is largerthanthe axialsizez/l . The theoret-
h ci c µ, that is
ical results for the rms sizes exhibited in Fig. 7 can be
verified experimentally with present technology and this n∂µ 2 1 f′(x)
Γ= = + x , (29)
will provide a test for the USE proposed in this paper. µ∂n 3 3 f(x)
9
where x = n1/3a is the gas parameter. This formula for time-independent form the USE is useful to study the
the local polytropic equation is useful to have a simple stationaryproperties of a BEC.The full time-dependent
analytical prediction of the collective frequencies. In the USE can be used to study non-equilibrium properties of
weak-couplingregime(x 1)one finds xf′(x)/f(x)=1 a dilute BEC, such as, dynamical oscillations [28], col-
≪
and Γ = 1, consequently, the reduced frequency square lapse[48],free expansionafterreleasefromthe trap[22],
ν2 (Ω /ω )2 = 2(Γ+1)) = 4 and ν2 (Ω /ω )2 = soliton formation and soliton dynamics [20, 49] etc.
ρ ≡ ρ ρ z ≡ z z
3 1/(Γ+1)=5/2;whileintheunitarityregime(x 1)
InthispaperweappliedtheUSEtostudyvorticesina
it−holds xf′(x)/f(x)=0 and Γ=2/3 and conseque≫ntly,
superfluidBosegas. First,westudyvorticesinauniform
ν2 = 10/3 and ν2 = 12/5 . The analytical predictions
ρ z Bose gas as the scattering length is varied from weak to
of Eqs. (27) and (28) with Eq. (29) are shown in Fig.
strong coupling values. The vortexcore radius decreases
8, where we plot ν2 and ν2 vs. atan(a/l ), so that the
ρ z c with increasing scattering length eventually attaining a
entireregion >a 0ismappedintothefiniteinterval
saturation value. The vortex core radius is comparable
∞ ≥
π/2 atan(a/l ) 0.
c to the healing length in this case.
≥ ≥
Next, we study vortices in a BEC confined by an
axially-symmetricharmonictrapbysolvingthe USE nu-
VII. CONCLUSION
merically. It is found that the effective nonlinearity of
the USE saturates with the increase of scattering length
In this paper we have proposed a new nonlinear
a in the strong-coupling limit for a fixed number of par-
Schr¨odiger equation Eq. (12) or (25) to study the
− − ticles N. In this limit the chemical potential as well as
propertiesofaBECorageneralsuperfluidBosegasvalid
the rmssizesofthe BCSsaturateattainconstantvalues.
in both the weak-coupling and strong-coupling (or uni-
However, the rms sizes obtained from the USE and the
tarity) limit. We call this equation the USE (unitary
MGPE grows indefinitely as the scattering length is in-
Schr¨odiger equation). This equation has a complicated
creasedtowardstheunitaritylimit. Intheweak-coupling
nonlinearity structure in its dependence on scattering
limittheresultsoftheUSEarecompatiblewiththoseob-
length a and number of atoms N. In the extreme weak-
tainedfromtheGPEandtheMGPE.Finally,wepresent
coupling limit the USE (12) reduces to the usual mean-
results of axial and radial frequencies of collective oscil-
fieldGPequation(7)[2];formediumcouplingitbecomes
lation of a BEC in a cigar-shapedtrap using the USE.
the modified GPE (MGPE) (8) introduced by Fabrocini
The results of this paper for a BEC confined in a
andPolls[4],whichisageneralizationoftheGPequation
axially-symmetric harmonic trap can be tested experi-
validformediumcouplingincorporatingthecorrectionto
mentally with present technology specially in the uni-
the bulk chemical potential of a Bose gas as introduced
tarity limit near a Feshbach resonance [3] and this will
by Lee, Yang, and Huang [5]. However, for very strong
provide a stringent test for the proposed USE.
coupling, both the GP and the MGP equations break
down and the bulk chemical potential attains a satura- S.K.A.waspartiallysupportedbyFAPESPandCNPq
tion [8]. Consideringthis bulk chemicalpotentialCowell (Brazil), and the Institute for Mathematical Sciences of
et al. [8] derived a nonlinear equation for a superfluid National University of Singapore. Research was (par-
Bose gas valid in the strong coupling limit. The USE tially) completed while S.K.A. was visiting the Insti-
reduces to the equation by Cowell et al. in the strong tute for Mathematical Sciences, National University of
couplinglimitandthushavethecorrectforminboththe Singapore in 2007. L.S. was partially supported by
weak and strong-coupling regimes. Hence this equation GNFM-INdAMandFondazioneCARIPAROandthanks
should be useful to study the crossover of a superfluid Francesco Ancilotto, Arturo Polls, Luciano Reatto and
Bose gasfrom the weakto strong-couplinglimits. In the Grigori Volovik for useful discussions.
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