Table Of ContentA Closed Class of Hydrodynamical Solutions for the Collective Excitations of a
Bose-Einstein Condensate
Pippa Storey∗1 and Maxim Olshanii1,2
1Lyman Laboratory, Harvard University, Cambridge, MA 02138, USA
2Department of Physics and Astronomy, University of Southern California, Los Angeles CA 90089-0484 USA
E-mail: [email protected], [email protected]
(February 1, 2008)
In the mean field approximation, the evolution of
A trajectory approach is taken to the hydrodynamical
the condensate wavefunction is determined by the time-
treatmentofcollectiveexcitationsofaBose-Einstein conden-
2 dependentGross-Pitaevskiequation,which,forzerotem-
sate in a harmonic trap. The excitations induced by linear
0 perature, is
deformationsofthetrapareshowntoconstituteabroadclass
0
ofsolutionsthatcan befullydescribed byasimplenonlinear
2 ¯h2
n mforattrhixeesqouluattiioonn.dAenscerixbaicntgctlohseedm-foodrme enxp=re0ss,imon=is+ob2taiinneda i¯h∂tΦ(r,t)= −2m△+U(r,t)+Ng|Φ(r,t)|2 Φ(r,t)
(cid:20) (cid:21)
a { }
cylindrically symmetric trap, and the calculated amplitude-
J (2)
dependentfrequencyshiftshowsgoodagreementwiththeex-
1
perimental results of theJILA group.
1 where N is the number of atoms in the condensate, and
g isa measureofthe strengthofthe atomicinteractions.
2 g is related to the s-wave scattering length a through
v g =4π¯h2a/m, and is assumed to be positive.
1
1 The recent realisation of Bose-Einstein condensation Defining the density of the condensate as ρ(r,t) =
2 in dilute trapped atomic vapours [1–3] has motivated an N Φ(r,t)2, and its velocity field as v(r,t) =
3 extensivetheoreticalstudyofthesesystems(forareview (h¯/|2im)∇|log[Φ(r,t)/Φ∗(r,t)],itiseasilyshownthatthe
0 see reference [4]). Many of the nonlinear features of the Gross-Pitaevski equation (2) is equivalent to the follow-
9 condensatesaremanifestedin theircollectiveexcitations ing equations for ρ and v
9
[5–8]. In this letter we consider the collective excita-
/
at tions of a condensed atomic gas in a harmonic trap in ∂tρ+∇(vρ) =0 (3)
m the regime in which the number of atoms is sufficiently 1
m∂ v+∇ U + mv2 =0, (4)
- large that the hydrodynamical treatment of Stringari is t eff 2
d valid [9]. We apply this treatment by means of a tra- (cid:18) (cid:19)
n jectory approach, which constitutes a generalisation of where
o
the scaling transformation of Castin and Dum [10]. To
c ¯h2
: firstorder in the excitationamplitude our results for the U =U +gρ √ρ (5)
v eff − 2m√ρ△
oscillatory modes in a stationary trap are identical to
i
X thoseofthe perturbativetreatmentofO¨hberget al.[11].
is an effective potential.
r Howeverforabroadclassofexcitations,ouranalysispro-
a Following Stringari [9] we consider the limit in which
duces a simple nonlinear matrix equation, which is valid
the number ofatomsN is sufficiently largethatthe den-
toallordersintheexcitationamplitude. Theexcitations
sity ρ becomes smooth, and it is valid to neglect the
to which this matrix equation applies are those induced kinetic energy pressure term ¯h2 √ρ /2m√ρ. The ra-
by time-varying linear deformations of the trap (that is, △
tio between the typical force caused by this force and
perturbationsofthe trapping potentialthat maintainits the force ∇(gρ) caused by in(cid:0)ter-pa(cid:1)rticle interactions
harmonicity). This class includes allthe excitationsthat can be sh−own to be of the order of (h¯2/mR2)/µ
have to date been studied experimentally. N−2/3(T /µ)2,andforthemostoftheBECexpe0rimen∼ts
c
We suppose that the condensed gas is confined by a this ratio is as small as 10−1 10−3. (Here and below
harmonic potential, which, in its most general form, is −
R µ/mω2 is the Thomas-Fermi radius of the con-
0
given by ∼
densate, T ¯hωN1/3 is the Bose-Einsteincondensation
p c ∼
1 temperature, µ gρ is the chemical potential, and ω is
U(r,t)= (t)[r r¯(t)][r r¯ (t)], (1) ∼
ij i i j j a typical trapping frequency.) In this regime the con-
2 K − −
Xi,j densate can be modeled as a classical gas in which each
where ¯r(t) is the position of the centre of the trap, particle is subject to a force
andK(t) is a symmetric matrixwith components Kij(t), F(r,t)= ∇r[U(r,t)+gρ(r,t)], (6)
which represent the spring ‘constants’ of the trap. −
1
andforwhichthevelocityfieldisirrotational. Weinclude irrotational, we can express it as the gradient of some
the subscript r on the gradient operator ∇ to indicate function Θ(r,t)
explicitly that all derivatives are taken with respect to
thecomponentsofr. Thisnotationisusedsinceweshall v(t)=∇rΘ(r,t), (12)
later introduce a change of coordinates.
The requirement that ∇r v(t) 0 imposes a con- in terms of which we obtain
× ≡
straint only on the initial condition, the reason being Φ (r )
Φ(r,t) e−iβ(t)eimΘ(r,t)/h¯ 0 0 , (13)
that since the only force involved (6) is the gradient of ≈ det(∂r/∂r )
a potential, the total derivative of ∇r v(t) vanishes. 0
×
This means that, provided the velocity field is initially wherer isuniquelydefinedintpermsofrandt,andβ(t)
0
irrotational everywhere,it will remain so with time. is some function of time.
Using expression (6) for the classical force, we obtain By applying this analysis to oscillatory modes in the
the following evolution equation for r(t) stationary harmonic potential U(r,t) = U (r) (that is,
0
(t) = , ¯r(t) = 0) in the limit of small excitation
m¨r(t)= (t)[r(t) ¯r(t)] g∇rρ(r,t). (7) KamplitudKes0, we recover the results of O¨hberg et al. [11],
−K − −
as we now show. For low amplitude oscillations
In order to solve this equation, it is necessary to deter-
minethedensityofthecondensateatpositionrandtime r(r ,t) r +ǫ δr(r )e−iωt+δr∗(r )eiωt , (14)
0 0 0 0
t, which in turn requires a knowledge of the trajectories ≈
of all the other particles in the condensate. To this end, where ǫ is a pertur(cid:2)bation parameter. Substitu(cid:3)ting this
we associate with eachtrajectoryr(t) a unique reference expressionintoequation(11),andkeepingonlyfirstorder
pointr0,correspondingtothepositionofaparticleinthe terms in ǫ we obtain the relation
ground state Φ of the condensate in the hydrodynami-
0
cal limit for a stationary harmonic trap U0(r), given by mω2δr=∇{δr·(K0r)−[µ−U0(r)]∇·δr}, (15)
equation(1)with¯r(t)=0and (t)= . Φ isobtained
from the solution ρ of equatiKon (6)Kw0ith F0 (r,t) = 0, in which we have dropped the distinction between r and
0 r , since this affects only higher orders. δr is clearly
through 0
expressible as a gradient
ρ (r )=N Φ (r )2, (8)
0 0 | 0 0 | δr=∇W, (16)
and is given by
where the function W satisfies the equation
Φ0(r0)=sµ−NU0g(r0), (9) mω2W =∇W ·(K0r)−[µ−U0(r)]△W. (17)
Expanding the solution (13) to first order, we obtain
where µ is the chemical potential. Note that this is just
the Thomas-Fermi solution. Φ(r,t) e−iµt/h¯ Φ (r)+ǫ ue−iωt v∗eiωt , (18)
0
≈ −
The density of the condensate at time t can now be
written as where (cid:2) (cid:0) (cid:1)(cid:3)
ρ(r,t)=det ∂∂rr −1ρ0(r0). (10) u= f++2 f−, v = f+−2 f− (19)
(cid:20) 0(cid:21)
and
where∂r/∂r isamatrixwhoseijthelementis∂r /∂r .
0 i 0,j
Given that ∇r = (∂r/∂r0)−1 T ∇r0, we can rewrite f± =C± 1 U0(r) ±1/2W, (20)
− µ
equation (7) as a pahrtial differenitial equation for r(r0,t) (cid:20) (cid:21)
m∂2r= (t)[r ¯r(t)] with C−/C+ = h¯ω/2µ. The expression (18), with the
t −K − differential equation (17) for W, is precisely the result
∂r −1 T ∂r −1 obtained by O¨hberg et al. [11]. We have thus shown
−g"(cid:18)∂r0(cid:19) # ∇r0"det(cid:18)∂r0(cid:19) ρ0(r0)#. tehxacittathtieoinrssoolfutthioenndoneslicnreibaershtyhderoedleymneanmtaicraylpmhoodneoln.-like
(11) The oscillatory modes for which the function W(r) is
of third or higher order in the components of r can-
The functions r satisfying this equation can be used
not be excited by linear deformations of the harmonic
to construct hydrodynamical solutions of the Gross-
trap. That is, they cannot be produced by applying a
Pitaevski equation (2). Given that the velocity field is
time-dependent harmonic potential of the form (1) to
2
the groundstate condensateΦ (r). However,the modes whereδ isaconstantmatrix,independentofbothposi-
0
for which W(r) is linear or quadratic can be produced tionandMtime. Itcompletelycharacterisesthemodesand
in this way, and for these modes it is possible to go be- their linear combinations in the small-amplitude limit.
yondthe perturbativelimit, toobtainasimplenonlinear Forthe scalingmodes describedby CastinandDum [10]
matrix equation that fully describes the dynamics of the (the modes n=2,m=0,( ) , and a symmetric super-
{ ± }
condensate in the hydrodynamical regime. positionofthe modes n=0,m= 2 )δ is diagonal.
IfthecondensateisinitiallyinthegroundstateΦ (r), For the mode n=0,{m=+2 , wh±ich}weMshall consider
0
{ }
then the effect of a time-dependent harmonic potential next, it is
(1)issimply todistortthe condensateinalinearfashion
1 i 0
r(r ,t)=R(t)+ (t)r . (21) δ = i 1 0 . (28)
0 M 0 M 0 −0 0
Here R(t) is the centre of mass of the condensate, and
Anexactsolutionforthismodecanbefoundbysetting
(t)describesthetime-dependentcontractionorshear-
M (t)= in equation (24), and is given by
ing of the condensate along various directions. The ma- 0
K K
trix ∂r/∂r is now independent of position
0 (t)=exp[λ (t)] (t), (29)
λ
M Q R
∂r
= (t)=constant(r ), (22) where
∂r M 0
0
cos(ωt) sin(ωt) 0
and equation (11) can be separated into a part that is (t)= sin(ωt) cos(ωt) 0 , (30)
Q −
independentofposition,andapartthatdependslinearly 0 0 0
on position, giving the following two relations
and ω is the mode frequency. From expression (30) for
mR¨ = (t)[R ¯r(t)] (23) (t) it can be seen that the transformation exp[λ (t)]
−K − Q Q
m ¨ = (t) +det( )−1 −1 T . (24) describesthe distortionofthe condensateintoanellipse,
M −K M M M K0 and the rotation of that ellipse about the z axis. The
From equation (23) it is clear that(cid:0)for a(cid:1)stationary trap parameter λ is a measure of the amplitude of excitation,
and is precisely the quantity we labelled ǫ in the small
( (t)= )thecentreofmassmotionisseparablealong
0
K K amplitude limit.
the three axial directions, and the frequencies of oscilla-
The matrix (t) describes a slow rotation about the
tionalongthesedirectionsareindependentofamplitude, λ
R
z axis, together with a centrifugal stretching in the x-y
in agreement with the Kohn Theorem (see [12] and ref-
plane, and a corresponding contraction along z
erences therein).
The velocity can be written in terms of r as
βcos(ω t) βsin(ω t) 0
s s
−
v=R˙ +M˙ M−1(r−R), (25) Rλ(t)= βsin0(ωst) βcos0(ωst) β−02/3 . (31)
and hence the requirement that the velocity field have
Thisrotationisrequiredinorderfor (t)tosatisfycon-
zero curl is equivalent to the constraint that the matrix M
dition(26). (t)dependsonλthroughboththeparam-
˙ −1 be symmetric Rλ
MM eter β and the slow frequency ωs.
T The aspect ratio α of the condensate is defined as
˙ −1 = ˙ −1. (26)
MM MM L L
l s
h i α − , (32)
Using equation (24) it is easy to verify that provided ≡ Ll+Ls
condition (26) is satisfied at the initial time it will be
where L /L is the ratio of the lengths of the long and
l s
satisfied at all later times.
short axes of the condensate in the x-y plane. It is
Given this constraint it is clear that for a stationary
related to the parameter λ by α = tanhλ. The con-
potential in the limit of small excitations, there must
stant β depends in turn on α through β = [(1 +
be exactly six linearly independent modes for which the α4)/(1 α4)]3/10. The slow frequency ω is given by
function W(r) is quadratic in the components of r. For − s
ωs = [√2α2/√1+α4]ω⊥, and the mode frequency ω
cylindrically symmetric traps (ωx = ωy ≡ ω⊥) they are scales with the aspect ratio as
(in the notation of reference [11]) n=2,m=0,( ) ,
{ ± }
n=0,m= 2 and n=1,m= 1 [13]. Inthissmall 1+α2
{ ± } { ± } ω =√2 ω⊥. (33)
amplitude limit we can write √1+α4
(t)=1+ǫ δ e−iωt+δ eiωt , (27) ω thus depends on the amplitude of excitation. For very
M M M
low amplitudes, λ 0, α 0, and the mode frequency
(cid:2) (cid:3) → →
3
tends to ω √2ω⊥. At very high amplitudes, λ , to a fraction of the transition temperature.
→ → ∞
α 1andthemodefrequencyapproachesthelimitω Inconclusion,wehaveconsideredthecollectiveexcita-
→ →
2ω⊥, which coincides with the frequency of oscillationof tions of a Bose-Einstein condensate in the hydrodynam-
a non-interacting gas. icalregime. We haveshownthat the excitations induced
This mode has been observed experimentally by Jin by arbitrary time-dependent linear deformations of the
et al. [5,6], who measured its frequency as a function of trapping potential form a broad class of solutions of the
response amplitude (aspect ratio) after opening the trap hydrodynamical equations, which remains closed under
andallowingthecondensatetoexpand. Due tothepres- evolution. The dynamics of the condensate for this class
ence of atomic interactions, the aspect ratio will change of excitations can be described by a nonlinear equation
during the expansion. In order to compare our results involving a 3 3 matrix. We have obtained an exact
×
with the experimental data it is therefore necessary to solution of this equation for the n=0,m=+2 mode,
{ }
model the evolutionof the condensate after the trap has which shows good agreement with experimental results.
been opened. We assume that the opening of the trap We would also like to note that described in the liter-
occurs instantaneously. Taking the solution (29) for the ature 3-fold class of nonlinear solutions [10] covers two
trapped condensate as an initial condition, we insert it n=2,m=0 modes and any linear combination of
{ }
into (24), setting (t) = 0, and integrate numerically. n=0,m= 2 modes with (generally complex) equal
K { ± }
Theaspectratioofthecondensateiscalculatedfromthe by the absolute value coefficients: the observed in the
elements of after the appropriate expansion time. experiment [6] n=0,m=+2 mode does not belong
M { }
to this class. The 6-fold closed class of nonlinear solu-
tions obtained in our work extends the treatment to two
other modes ( n=0,m= 2 respectively) as well as
{ ± }
y 1.12 toanarbitrarysuperpositionof n=0,m= 2 modes,
enc including n=0,m=+2 . { ± }
equ Inthe o{therwords,the}existing treatment[10]cande-
e fr 1.08 scribe any excitation obtained by a modulation of the
d
mo matrixelementsofthe spring-constanttensor ij(t)(see
ed 1.04 (1))provideditalwaysremainsdiagonal. ThisKmodelcan
s
ali not be applied to describe the rotation-like modulations
m
or such as the “m = 2” JILA experiment [6]. Instead, our
n 1.00
paper presents an extention for this model which can be
appliedtoanykindofthetimedependenceofthespring-
0 20 40 60
constant tensor, including the one used in the above ex-
response amplitude [%]
periment.
FIG.1. Thenormalisedmodefrequencyω/ω(0)isplotted M.O. was supported by the National Science Foun-
against response amplitude (aspect ratio α) after expansion. dation grant for light force dynamics #PHY-93-12572.
Our theoretical predictions (curve) are compared with the P.S. was supported by the University of Auckland. She
experimental data of Jin et al.[6]. is grateful to Professor R. Glauber for making possible
her visit to Harvard. This work was also partially sup-
Shown in Figure 1 is the experimental data obtained ported by the NSF through a grant for the Institute for
by Jin et al. [6] for the mode frequency as a function of Theoretical Atomic and Molecular Physics at Harvard
aspect ratio. These points were obtained with a radial University and the Smithsonian Astrophysical Observa-
trap frequency ω⊥/2π of 132 Hz and an expansion time tory. TheauthorsaregratefultoProfessorE.Cornellfor
of 7 ms [5], and the frequencies were normalised to the helpful comments.
smallamplitudevalueω(0)[14]. Theratiooftheaxialto
radialtrapfrequencieswasωz/ω⊥ =√8. The numberof
condensed atoms was approximately 4500, and temper-
ature was T = (0.50 0.08)T , where T is the critical
c c
±
temperature. The theoretical curve, calculated for the
sameparameters,showsgoodagreementwith the exper-
∗
imental data. Permanent address: Physics Department, University of
Auckland,Private Bag 92019, Auckland,NewZealand
Notethatinourtreatmentwetotallyneglectedthethe
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4
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[13] Note that in the limit of a spherically symmetric trap,
the ‘fast mode’ n=2,m=0,(+) reduces to the l=0
{ }
mode with frequency √5ωtrap, and the remaining five
modesform an l=2quintupletwith frequency√2ωtrap.
[14] E. A.Cornell and D.S. Jin, private communication
5
