Sep 7, 2005 - cies have also been provided by Fetter 3, who used .... becomes equivalent to a time-dependent Thomas-Fermi so- ..... weight factors wsk u.
PHYSICAL REVIEW A 72, 033604 共2005兲
Quantum and thermal fluctuations of trapped Bose-Einstein condensates 1
V. I. Kruglov,1 M. K. Olsen,1,2 and M. J. Collett1
Department of Physics, University of Auckland, Private Bag 92019, Auckland, New Zealand Instituto de Física da Universidade Federal Fluminense, Boa Viagem 24210-340, Niterói-RJ, Brazil 共Received 29 March 2005; published 7 September 2005兲
2
We quantize a semiclassical system defined by the Hamiltonian obtained from the asymptotic self-similar solution of the Gross-Pitaevskii equation for a trapped Bose-Einstein condensate with a linear gain term. On the basis of a Schrödinger equation derived in a space of ellipsoidal parameters, we analytically calculate the quantum mechanical and thermal variance in the ellipsoidal parameters for Bose-Einstein condensates in various shapes of trap. We show that, except for temperatures close to zero, dimensionless dispersions do not depend on the frequencies of the trap and they have the same dependence on dimensionless temperatures. DOI: 10.1103/PhysRevA.72.033604
PACS number共s兲: 03.75.Kk, 42.50.Ct, 32.80.⫺t, 42.65.⫺k
I. INTRODUCTION
A large number of theoretical works have investigated the elementary excitations of confined interacting Bose-Einstein condensates 共BEC兲. These analyses have generally been based on zero-temperature mean-field theory. To calculate the low energy excitations of a dilute atomic Bose gas confined in a harmonic trap, Stringari 关1兴 developed an approach based on solutions to the linearized Gross-Pitaevskii equation 共GPE兲 关2兴 in the Thomas-Fermi approximation. Alternative approaches to the analytical study of excitation frequencies have also been provided by Fetter 关3兴, who used variational wave functions in the generalized Bogoliubov approach, and by Pérez-García et al. 关4兴, who also employed a variational technique, using a Gaussian trial wave function, to analyze the time-dependent GPE. Kagan et al. 关5兴 and Castin and Dum 关6兴 have used scaling transformations to study the collective excitations in the GPE approach. This approach has also been used by Dalfovo et al. 关7兴 to consider the nonlinear regime of excitation and to examine the effects of mode coupling and harmonic generation, which have also been considered by Graham et al. 关8兴. A finite temperature field theory has also been applied to trapped Bose condensates to study density profiles, collective excitations, and damping rates. These works use finitetemperature Hartree-Fock-Bogoliubov 共HFB兲 approximations, as well as other approximations originally introduced by Popov 关9兴, Beliaev 关10兴, and other authors 共see for example the Hohenberg and Martin paper 关11兴兲. We mention here as examples only a few papers 关12–16兴 based on the treatment of finite temperature BEC. The results we present in this article are particularly topical due to the fact that the direct numerical calculation of the quantum statistics of a trapped BEC is an extremely difficult problem. A quantum mechanical one-dimensional calculation has been performed using phase-space methods for the coherence of a trapped BEC 关17兴, but this approach was only successful over a short time. Quantum calculations of the evaporative cooling process have also been performed 关18兴, but these were not viable once the condensate had formed. Fully quantum calculations have been performed for the processes of photoassociation and dissociation of atomic and molecular condensates 关19–22兴, but the interesting dynamics 1050-2947/2005/72共3兲/033604共13兲/$23.00
here occur over short times. It should also be noted that in all these quantum treatments apart from that of evaporative cooling, the initial quantum state was always decreed rather than calculated from first principles. As a quantity which will be of interest is the coherence or, alternatively, the linewidth of a continuous wave atom laser in the steady state, it is of some importance to develop methods of dealing with the multimode problem in the long time limit. In this article, working in a space of the ellipsoidal parameters of a trapped BEC, we derive a Schrödinger equation from a semiclassical Hamiltonian which itself comes from the GPE with an added gain term. This then enables us to calculate the excitation frequencies and quantum expectation values for the radial coordinate of axially symmetric trapped condensates. An important requirement for achieving a continuous wave atom laser using trapped Bose-Einstein condensates is a pumped lasing mode of the condensate. In an early work in the field, Kneer et al. 关23兴 modeled the trapped lasing mode semiclassically, using a generalized Gross-Pitaevskii equation with added loss and gain terms. They found that, by analogy to optical lasers, there was a threshold value of pumping above which the lasing mode became macroscopically occupied. They also found that, depending on the spatial dependence of the pumping, the lasing mode could undergo number, and hence spatial, oscillations. Drummond and Kheruntsyan 关24兴 have also solved the GPE with gain to model the growth of a BEC. They found significant differences from the Thomas-Fermi solution, and also found that, as the condensate grows, it can develop a center of mass oscillation in the trap. What we will calculate here are the quantum uncertainties present in the radii of a trapped condensate, which will always be present along with the center of mass oscillations found by Drummond and Kheruntsyan. We will show that the quantum fluctuations of a trapped BEC with a harmonic potential in the general case, when the number of condensed atoms Nc is a function of time, can be found using an asymptotic self-similar solution to the GPE with a linear gain term 关25,26兴, thus obtaining a semiclassical Hamiltonian which lends itself to quantization. On the basis of a Schrödinger equation derived in a space of ellipsoidal parameters, we analytically calculate the quantum mechanical and thermal variance in the ellipsoidal parameters for Bose-Einstein condensates in various shapes of trap. In
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©2005 The American Physical Society
PHYSICAL REVIEW A 72, 033604 共2005兲
KRUGLOV, OLSEN, AND COLLETT
this paper we do not take into account collisional interactions of the BEC with the thermal cloud. We show that, except for temperatures close to zero, dimensionless transverse and longitudinal dispersions of the ellipsoidal parameters of a trapped BEC have the same dependence on dimensionless temperatures and do not depend on the frequencies of the trap. This scaling is an approximate but nevertheless very accurate law and it was found by numerical calculations.
differential equations for the functions f k共t兲 , ck共t兲, and the explicit expression F共兲 = 冑1 − 兺kk2k for 兺kk2k 艋 1 and zero otherwise. Here the k are positive constants. The normalization condition gives us the equation 1 / 冑123 = 15Nc共0兲 / 8 for the constants k, which allows us to exclude these indeterminate parameters in the solutions for amplitude and phase. Introducing the ellipsoidal parameters of the trapped Bose gas by
冉冕
冉
冊
m2k x2k 4បa0 2 ig共t兲 − បⵜ2 = +兺 + 兩兩 + . 共1兲 i 2 t 2m 2ប m k=1 In the above, m is the atomic mass, the k are the trap frequencies along each axis, and a0 is the atomic s-wave scattering length. The gain function is determined by the logarithmic derivative, g共t兲 = d ln Nc共t兲 / dt. We now rewrite the wave function in Eq. 共1兲 with a real amplitude and phase 共x , t兲 = A共x , t兲exp关i共x , t兲兴. In the amplitude equation we can then neglect the term បⵜ2A / 2m because 兩បⵜ2A / 2m兩 Ⰶ 4បa0A3 / m for the condition Ec Ⰷ 1, where the large posi2 , with lm being tive dimensionless parameter is Ec = 8nca0lm the minimum of the typical lengths along the three axes and nc = Nc / Vc the average condensate density. The condition Ec Ⰷ 1 means that we are considering the strongly nonlinear hydrodynamic time-dependent regime. The GPE given by Eq. 共1兲 has asymptotically self-similar solutions 关25,26兴 for large positive values of Ec. When g共t兲 = 0 this approximation becomes equivalent to a time-dependent Thomas-Fermi solution, coinciding with the normal Thomas-Fermi approximation in the stationary case. Under the above condition, the asymptotic self-similar solution for the amplitude variable can be written as A共x,t兲 = f 1共t兲f 2共t兲f 3共t兲F共1, 2, 3兲,
共2兲
A共x,t兲 =
15Nc共t兲 8a1共t兲a2共t兲a3共t兲
冊冉 1/2
3
1−兺
k=1
x2k a2k 共t兲
冊
1/2
共3兲
where
冉
3
q共x,t兲 = 1 − 兺
k=1
x2k a2k 共t兲
冊
.
ck共t兲 =
0共t兲 = 0共0兲 −
m d ln ak共t兲, 2ប dt
15បa0 2m
冕
t
0
Nc共t⬘兲dt⬘ , a1共t⬘兲a2共t⬘兲a3共t⬘兲
冊
gk共t⬘兲dt⬘ ,
0
冉
and the gk are functions satisfying the condition 3
gk共t兲 = g共t兲. 兺 k=1
共5兲
Applying a generalization of the method used to find solutions of the one-dimensional nonlinear Schrödinger equation with gain 关26兴, we find 共Appendix A兲 a system of ordinary
共10兲
冊
共11兲
with initial conditions ak兩t=0 = ak共0兲 and dak dt
共4兲
共9兲
where the ellipsoidal parameters are solutions of the equations 共k = 1,2,3兲:
where the k are self-similar dimensionless variables t
共8兲
In the above, S共q兲 is the Heaviside step function which equals one at positive q, and zero otherwise. In this case it serves to select when q共x , t兲 is positive. The phase variable 共x , t兲 in Eq. 共3兲 is defined by the functions 0共t兲 and ck共t兲 via the relations
冏 冏
k=1
S关q共x,t兲兴, 共7兲
3
冉冕
冉
15a0ប2 Nc共t兲 d2 2 a + a = , k k k dt2 m2 a 1a 2a 3a k
共x,t兲 = 0共t兲 + 兺 ck共t兲x2k ,
共6兲
after some transformations we derive 共more detail is given in Appendix A兲 an explicit self-similar asymptotic solution for the amplitude when the parameter Ec tends to infinity,
and that for the phase variables as
k = xk f 2k 共t兲exp −
gk共t⬘兲dt⬘ ,
0
We begin with the Gross-Pitaevskii gain equation 关23,24兴 for the semiclassical “wave function” of the condensate, normalized to the number, Nc共t兲, of condensed atoms, 3
冊
t
f −2 ak共t兲 = −1/2 k k 共t兲exp
II. SCHRÖDINGER EQUATION IN A SPACE OF ELLIPSOIDAL PARAMETERS
= t=0
2ប ck共0兲ak共0兲. m
共12兲
In the case when the gain term g共t兲 in the GPE is equal to zero 共Nc = const兲, and using some approximations, the solutions of Eq. 共1兲 were obtained in 关4–6兴 by different methods. It should however be noted that Refs. 关5,6,8兴 were interested in fluctuations of the condensate due to external trapping perturbations, while we will be calculating the quantum fluctuations which exist even for a completely stable trap. We will now consider the case where Nc is constant. The stationary solution of the system of Eqs. 共11兲 for the ellipsoidal parameters are ak = ¯ak, where ¯ak given for 共k = 1,2,3兲 by
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PHYSICAL REVIEW A 72, 033604 共2005兲
QUANTUM AND THERMAL FLUCTUATIONS OF TRAPPED… 1/5 ¯ak = −1 k 共G0123/m兲 .
G共t兲 = 15a0ប2Nc共t兲/7m.
共13兲
We defined here a new constant G0 = 15ប2a0Nc / m. One can set y k = ak − ¯ak and linearize the system of equations 共11兲 for the y k,
The Hamiltonian equations, dak H共t兲 = , dt pk
3
d2 y k + 兺 ⌳ksy s = 0, dt2 s=1
共14兲
⌳ks = ks + 22k ␦ks ,
共15兲
where
共21兲
H共t兲 dpk =− , dt ak
共22兲
lead directly to Eq. 共11兲, hence ak and pk = maa˙k are the generalized coordinate and the canonical momentum of the classical system defined by the Hamiltonian of Eq. 共20兲. Applying the standard quantization method to the Hamiltonian of Eq. 共20兲 results in the Schrödinger type equation 关27,29兴:
冉
3
3
冊
2 ma ប2 G共t兲 = − 2k a2k + , 兺 兺 2 + t 2ma k=1 ak 2 k=1 a 1a 2a 3
and ␦ks is the Kronecker delta function. We note here that we are not using the standard linearization procedure, which deals with fluctuations about a stationary solution, but are treating fluctuations about a deterministic bulk motion, in a manner similar to that previously used to treat the selfpulsing regime of optical second harmonic generation 关28兴. The characteristic frequencies ⍀k of the small oscillations given by Eq. 共14兲 are the roots of the dispersion relation
where the wave function 共a , t兲 has unit normalization. In the case when Nc is constant we can use also alternative variables: ak and k = Nc pk = a˙k, where the effective mass of the condensate is
det共⌳ − ⍀2k I兲 = 0,
= Ncma = Ncm/7.
iប
共23兲
共16兲
where the matrix ⌳ is given by Eq. 共15兲 and I is the identity matrix. In the case of an axially symmetric trapping potential, with 1 = 2 = , and setting 3 = 0, Eq. 共16兲 leads directly to the characteristic frequencies ⍀6k − 3共20 + 22兲⍀4k + 82共2 + 220兲⍀2k − 20204 = 0.
In this case the Hamiltonian of the whole system H = NcH is: 3
H=兺
k=1
共17兲 We note that Eq. 共1兲 with is completely equivalent to the ordinary, gainless GPE with a timedependent self-interaction parameter,
␦H . N共x,t兲 = * t ␦N共x,t兲
H共t兲 =
冕
冉
N* 共x兲 −
3
បⵜ +兺 2m k=1 2
冊
3
k=1
冉
冊
p2k G共t兲 ma 2 2 k ak + + , 2ma 2 a 1a 2a 3
15a0ប2N2c . 7m
共26兲
It is evident that the Hamiltonian equations 共22兲 for the new variables have the form dak H = , dt k
H dk =− . dt ak
共27兲
冉
共19兲
共20兲
where pk = maa˙k , ma = m / 7 is the renormalized boson mass, and the self-interaction parameter is
3
3
冊
G ⌿ 2 ប2 = − ⌿. 共28兲 2k a2k + iប 兺 兺 2 + t 2 k=1 ak 2 k=1 a 1a 2a 3
2
where we have used unit normalization of the mean field density. The above expression follows directly from Eq. 共1兲 and the time-dependent normalization condition N = / 冑Nc共t兲. By use of the above solutions for the variables A共x , t兲 and 共x , t兲 we can calculate the integrals in Eq. 共19兲. In the case where Ec Ⰷ 1, this allows for the explicit formulation of the Hamiltonian as H共t兲 = 兺
共25兲
Hence this quantization yields the Schrödinger-type equation for the new variables in the form
m2k x2k
2a0ប2Nc共t兲 + 兩N共x兲兩2 N共x兲dx, m
冊
G 2k 2 2 + k ak + , 2 2 a 1a 2a 3
G = N cG =
共18兲
Here H = H共t兲 is the real Hamiltonian function, 2
冉
where G is a new interaction constant given by
g共t兲 = N−1 c 共dNc / dt兲
iប
共24兲
We note here that Eq. 共28兲 is not the usual Schrödinger equation as it describes a wave function ⌿共a , t兲 of the trapped BEC in the space of ellipsoidal parameters ak 共k = 1,2,3兲, taking into account the self-interaction of this many-body system. This is evident from the fact that the strength parameter of the self-interaction G is proportional to the square number of condensed atoms, Nc. Thus, using the nonlinear GPE with added gain 关Eq. 共1兲兴 we have derived 共under the condition ⑀c ⬅ 1 / Ec Ⰶ 1兲 a linear Schrödinger-type equation for the wave function ⌿共a , t兲. We should emphasize here the linear character of the wave equation derived by this method. We also note that Eqs. 共23兲 and 共28兲 are different because the first one describes the contribution of one particle to the distribution of the ellipsoidal parameters ak but the second wave equation takes into account all particles. In next section we discuss this distinction in detail.
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PHYSICAL REVIEW A 72, 033604 共2005兲
KRUGLOV, OLSEN, AND COLLETT III. ASYMPTOTIC SOLUTION OF THE SCHRÖDINGER EQUATION
In this section we consider the solution of equations 共23兲 and 共28兲 in the quadratic approximation with respect to small deviations, y k = ak − ¯ak, of the ellipsoidal parameters. The Schrödinger equation in the space of ellipsoidal parameters, Eq. 共23兲, can be solved by an expansion of the internal and external potential energies,
共z1,z2,z3,t兲 =
兺 n n n 1 2
冉
共38兲 where nk = 0,1,2,… at k = 1,2,3 are the quantum numbers of a 3D harmonic oscillator, and An1n2n3 are the arbitrary constants. The n1n2n3 are the eigenfunctions of a 3D harmonic oscillator,
3
G ma 2k a2k + , V= 兺 2 k=1 a 1a 2a 3
n1n2n3共z1,z2,z3兲 =
共29兲
冉
m3a⍀1⍀2⍀3 ប 3 3 3
in a series with the variable y k, subject to the condition 兩y k兩 / ¯ak Ⰶ 1. In the quadratic approximation, and taking into account the stability condition
冉 冊 V yk
=
ma2k¯ak
y k=0
G − = 0, ¯a1¯a2¯a3¯ak
共30兲
3
共31兲
5 G , 2 ¯a1¯a2¯a3
共32兲
and the matrix ⌳ks is given by Eq. 共15兲. The eigenvectors of ⌳ks obey the equation
共34兲
so that the coordinate transformation 3
y s = as − ¯as = 兺 us共k兲zk
共35兲
冉
3
冊
3 1 2 = 22 + 20 ± 冑4共22 − 20兲2 + 540 , ⍀1,2 2 2
共42兲
⍀23 = 22 ,
共43兲
2 ⍀21 ⯝ 320 + 2, 3 ⍀21 = 52,
共37兲 This allows us to write the general solution of Eq. 共36兲 as
10 2 , 3
⍀23 = 22 .
共44兲
⍀22 = ⍀23 = 22 .
共45兲
Case 3 共Cigar form兲 Ⰷ 0 1 ⍀21 ⯝ 42 + 20, 2
where the eigenfrequencies are given by Eq. 共16兲, or, in explicit form,
− 2123共⍀2k − 322兲 − 2122共⍀2k − 323兲 − 2212223 = 0.
⍀22 ⯝
Case 2 共Spherical form兲 0 =
共36兲
共⍀2k − 321兲共⍀2k − 322兲共⍀2k − 323兲 − 2223共⍀2k − 321兲
共41兲
where the positive 共negative兲 sign in front of the square root belongs to ⍀21 共⍀22兲. Note that these are the same frequencies as given previously in Ref. 关1兴. We will now consider some important limiting cases of the above solutions for different trapping geometries. Case 1 共Disk form兲 0 Ⰷ
is orthogonal. Using Eqs. 共31兲–共35兲, we can now write Eq. 共23兲 as 3
1 . 2
Equations 共37兲–共41兲 now represent the general solution of the Schrödinger type equation 共36兲 in the space of ellipsoidal parameters. In the axially symmetric case with 共1 = 2 = , 3 = 0兲, the solution of Eq. 共17兲 is
k=1
2 ma ប2 iប = − + 兺 兺 ⍀ 2z 2 + V 0 , t 2ma k=1 z2k 2 k=1 k k
共39兲
共40兲
冉 冊
k=1
with the normalization conditions 3
m a⍀ k zk . ប
E n1n2n3 = V 0 + ប 兺 ⍀ k n k +
共33兲
us共i兲us共k兲 = ␦ik , 兺 s=1
冑
3
3
⌳isus共k兲 = ⍀2k u共k兲 兺 i , s=1
1/2
The eigenenergies En1n2n3 in Eq. 共38兲 are
In the above, V0 is the potential energy 关Eq. 共29兲兴 at y k = 0: V0 =
2−n1−n2−n3 n1!n2!n3!
where Hn共兲 is a Hermite polynomial and the variable k is given by
k =
3
冊冉 冊 冉 冊 1/4
1 ⫻ 兿 Hnk共k兲exp − 2k , 2 k=1
for Nc constant, we find ma V = V0 + 兺 兺 ⌳ksykys . 2 k=1 s=1
冊
i An1n2n3n1n2n3共z1,z2,z3兲exp − En1n2n3t , ប 3
5 ⍀22 ⯝ 20, 2
⍀23 = 22 .
共46兲
We note here that the frequencies given above are not those along the orthogonal Cartesian axes, but refer to the orthogonal axes which come from diagonalization of the matrix of Eq. 共15兲. As an example we will present here the explicit form of the quantized eigenfrequencies, ⍀n1n2n3 = En1n2n3 / ប, of a cigar shaped condensate in the case where 1 = 2 = .
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PHYSICAL REVIEW A 72, 033604 共2005兲
QUANTUM AND THERMAL FLUCTUATIONS OF TRAPPED…
Using Eq. 共30兲 and setting ¯a1 = ¯a2 = ¯a, we find ប 冑2ga , a = 2ma
共47兲
¯2
where ga is the dimensionless constant ga =
30a0Nc . ¯3 49a
共48兲
5 冑2gaប . 4
ˆ 兲, = Z共兲−1exp共− H 共49兲
Note that this expression has been derived assuming only axial symmetry of the trap 共i.e., 1 = 2兲, hence Eq. 共49兲 is valid for all three condensate shapes considered above. In the case of a cigar shaped trap, and using the quadratic approximation for the potential energy, Eqs. 共41兲, 共46兲, and 共49兲 result in ⍀n1n2n3 = ⍀0 + 共2n1 + 冑2n3兲 +
冑
5 0n 2 , 2
冉
1
冑2
+
冊 冑
5冑2ga 1 + 4 2
5 0 . 2
ˆ 兲兴, where Z共兲 = Tr关exp共− H
where  = 共kBT兲−1. The probability density w共z1 , z2 , z3兲 then has the form w共z1,z2,z3兲 = 具z1,z2,z3兩兩z1,z2,z3典 =
兺
n1n2n3
Pn1n2n3共兲兩⌿n1n2n3共z1,z2,z3兲兩2 . 共54兲
Note that here and in what follows zk and ak are c-number variables and that in Eq. 共54兲 we have used the definitions
共50兲
共51兲
It is evident that the appropriate solution of Eq. 共28兲 follows from the above equations by the substitution ma → , G → G, and ga → gc, where gc = gaN2c . Hence Eqs. 共23兲 and 共28兲 yield the same average ellipsoidal parameters ¯ak by Eq. 共30兲 or, in explicit form, by Eq. 共13兲 and the same eigenfrequencies ⍀k given by Eq. 共37兲. Moreover, Eq. 共50兲 for the quantized eigenfrequencies in these two cases differs only by the first term ⍀0, hence physically the differences of the quantized eigenfrequencies ⍀n1n2n3 are also the same. However, as seen in Eq. 共40兲, the wave functions in these cases differ by the scale factor N1/2 c , which should be taken into account if we use Eq. 共23兲. Below we will consider the case of constant Nc and use Eq. 共28兲 which automatically takes into account the total mass of the trapped BEC.
共52兲
共53兲
where the constant ⍀0 is given by the expression ⍀0 = 1 +
冊
We do not assume here axial symmetry of the trap; the new ˜ k = −iប / zk are connected with the canonical operators zk , canonical operators ak , k = −iប / ak by a unitary transform given by Eqs. 共33兲–共35兲, and the eigenfrequencies are given by Eq. 共37兲. According to quantum statistical mechanics the density matrix at thermal equilibrium is
Combining Eqs. 共32兲, 共47兲, and 共48兲, we find that V0 =
冉
3
5G ˜ 2k 2 2 ˆ =兺 H + ⍀k zk + . 2 ¯ 1¯a2¯a3 2a k=1 2
Pn1n2n3共兲 = Z共兲−1exp共− En1n2n3兲,
共55兲
⌿n1n2n3共z1,z2,z3兲 = 具z1,z2,z3兩n1,n2,n3典,
共56兲
ˆ 兩n ,n ,n 典 = E H 1 2 3 n1n2n3兩n1,n2,n3典.
共57兲
Calculating the sum in Eq. 共54兲 with the results found in the previous section allows us to write the probability density in the following form:
再
3
w共z1,z2,z3兲 = C exp − 兺
k=1
冋
冉
1 ⍀k tanh ប⍀k 2 ប
where the normalization constant is 3
C=兿
k=1
冋
冉
1 ⍀k tanh ប⍀k 2 ប
冊册
冊册 冎
z2k , 共58兲
1/2
,
共59兲
and the characteristic frequencies ⍀k are given by Eq. 共37兲. Taking into account that as = ¯as + y s, we find that 具as典 = ¯as and that the dispersions of the ellipsoidal parameters are
IV. QUANTUM FLUCTUATIONS AT LOW TEMPERATURES
3
D共as兲 =
The quantum fluctuations of the ellipsoidal parameters as of trapped Bose-Einstein condensates in the case of thermal equilibrium can be calculated directly from the Hamiltonian 共25兲, where ak and k in the quantum case are the generalized coordinates and corresponding canonical momenta. We note that in this paper we do not take into account collisional interactions of the BEC with the thermal cloud. As long as the deviations, y k = ak − ¯ak, of the ellipsoidal parameters are small 共兩y k兩 / ¯ak Ⰶ 1兲, which holds at low temperatures, we may use the results of the previous section to write the Hamiltonian operator in quadratic form:
具as2典
− 具as典 = 2
具y s2典
= 兺 共us共k兲兲2具z2k 典.
共60兲
k=1
Note that the notation 具¯典 signifies averages over the probability density w defined in Eq. 共58兲: 具F典 = 兰兰兰Fwdz1dz2dz3. Using Eqs. 共58兲 and 共59兲 one may find the standard deviation ⌬zk = 冑具z2k 典 as ⌬zk =
冑
冉
冊
1 ប coth ប⍀k . 2 2⍀k
共61兲
Hence the dispersions of the ellipsoidal parameters can be written:
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KRUGLOV, OLSEN, AND COLLETT 3
D共as兲 = 兺
k=1
冉
冊
ប共us共k兲兲2 1 coth ប⍀k . 2 2⍀k
共62兲
check that all normalization conditions are fulfilled for the weight factors wsk ⬅ 共us共k兲兲2: 3
共us共k兲兲2 = 1 兺 s=1
The solution of the linear homogeneous system Eqs. 共33兲 for eigenvectors u共k兲 j in the axially symmetric trap is
at k = 1,2,3, 共69兲
3
u共k兲 1
=
u共k兲 2
=
共⍀2k − 22兲0 共⍀2k
− 3 2兲 2 −
u共k兲 . 4 3
共63兲
共⍀2k − 32兲2 − 4
=
320 − ⍀2k 2202共22 − ⍀2k 兲
,
共64兲
共⌬a⬜兲2 =
which follows from Eq. 共17兲 and taking into account the 3 normalization condition 兺s=1 共us共k兲兲2 = 1 we find u共k兲 1
=
u共k兲 2
u共k兲 3
=
=
±共⍀2k − 320兲
冑2共⍀2k − 320兲2 + 4202 ,
冑2
,
u共3兲 2 =−
1
冑2 ,
wsk ⬅
冢
冉冊
冉 冊
1 1 ⌫1 coth共 2 ប⍀1兲 ⌫2 coth共 2 ប⍀2兲 + 2 2 ⍀1 ⍀2
册
1 1 coth共 2 ប⍀3兲 , 2 ⍀3
冋
共70兲
册
coth共 21 ប⍀1兲 coth共 21 ប⍀2兲 ប ⌫2 + ⌫1 , 2 ⍀1 ⍀2 共71兲
共66兲
u共3兲 3 = 0.
共67兲
Using these equations it is easy to verify that ⌳u共3兲 = ⍀23u共3兲. Taking the equations ⍀23 = 22 and ⍀21⍀22 = 10220 into account 关which follow from Eq. 共17兲兴 and using an explicit solution 共42兲 and 共43兲 for three eigenfrequencies ⍀k and explicit solutions 共63兲–共67兲 for eigenvectors u共k兲 j one can write, after some algebra, the weight matrix wsk as
共us共k兲兲2
共⌬a3兲2 =
冋冉 冊
where
±20
冑2共⍀2k − 320兲2 + 4202 .
1
ប 2 +
共65兲
In the axially symmetric case all three eigenfrequencies ⍀k are different. However, as follows from Eqs. 共65兲 and 共66兲, the determinant of the matrix usk 共=us共k兲兲 is zero. The reason for this degeneracy is connected with the fact that the denominator of Eq. 共63兲 is zero at ⍀3 = 冑2. To avoid this degeneracy we will use Eqs. 共65兲 and 共66兲 when k = 1,2 and for k = 3 we find the eigenvector u共3兲 as the vector product u共3兲 = u共1兲 ⫻ u共2兲. This leads directly to the equations u共3兲 1 =
at s = 1,2,3.
These normalization conditions directly follow from Eq. 共68兲 and the identity ⌫1 + ⌫2 = 1. Using Eqs. 共62兲 and 共68兲 we can write the variances 共⌬as兲2 = D共as兲 in the axially symmetric case in the form:
Using 1
共us共k兲兲2 = 1 兺 k=1
⌫1/2 ⌫2/2 1/2
冣
= ⌫1/2 ⌫2/2 1/2 , 0 ⌫1 ⌫2
⌫1 =
⍀21 − ⍀22
,
⌫2 =
320 − ⍀22 ⍀21 − ⍀22
= 1 − ⌫1 .
共72兲
Here ⌬a⬜ = ⌬a1 = ⌬a2 due to the axial symmetry. We will present expressions for the general case where all three trapping frequencies are different in Appendix B. We note that the effective mass = Ncm / 7 is a function of the temperature T in Eqs. 共70兲 and 共71兲 because Nc = Nc共T兲. One can fit the temperature variation of the function Nc共T兲 to the functional form Nc共T兲 = N关1 − 共T / Tc兲␣兴 共for T ⬍ Tc兲, where Tc and ␣ are fitting parameters. Obviously, this fitting formula is a simple generalization of the ideal Bose gas result where Tc is the critical temperature for Bose-Einstein condensation. For example, in the case of the ideal Bose gas we have ␣ = 3 / 2, for liquid helium ␣ ⯝ 11/ 2, and ␣ ⯝ 2.3 for weakly interacting Bose gas in the isotropic harmonic potential trap 关12兴. These results for the variances simplify for high temperatures 共the classical limit兲, when the conditions k BT Ⰷ 1, ប⍀k
共73兲
k = 1,2,3,
are fulfilled. In this case, using the approximation coth共ប⍀k / 2兲 ⯝ 2 / 共ប⍀k兲, one can find from Eqs. 共70兲–共72兲 asymptotically exact equations:
共68兲
where ⌫1 = 共⍀21 − 320兲 / 共⍀21 − ⍀22兲 and ⌫2 = 共320 − ⍀22兲 / 共⍀21 − ⍀22兲. We note that the weight matrix wsk is degenerate in the axially symmetric case since D共a1兲 = D共a2兲, but the matrix usk = us共k兲 is not degenerate 关see Eqs. 共63兲–共67兲兴. One can
⍀21 − 320
⌬a⬜ ⯝
1
冑
2kBT , 5
⌬a3 ⯝
1 0
冑
2kBT . 5
共74兲
These equations are connected with the classical limit when the temperature is sufficiently high 关Eq. 共73兲兴 because in this case the standard deviations do not depend on the constant ប. Equations 共62兲 also yield a simple formula for the average standard deviation ⌬a:
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QUANTUM AND THERMAL FLUCTUATIONS OF TRAPPED…
冉兺 3
⌬a ⬅ 冑共⌬a1兲2 + 共⌬a2兲2 + 共⌬a3兲2 =
ប coth共 21 ប⍀k兲 2⍀k
k=1
冊
1/2
共75兲
兲 → 1, In the limiting case T → 0 we find that coth共 which simplifies the above expressions. For example, for zero temperature the above formula has the form 1 2 ប⍀k
3
共⌬a兲2 =
=
ប 兺 ⍀−1 2 k=1 k ប 2
冉冑 冑 1
2
+
2 520
+
冊
冊 冉冑
冉
1 1 ប 1 + = 共⌬a⬜兲 = 4 4 ⍀2 ⍀3
共78兲
冢冣 冢冣 ±
u共1兲 =
± ±
冑3 1
冑3
,
u共2兲 =
1
⫿
冑6
⫿
1
±
冑3
1
冑6
冢冣
2
−
1
冑2
.
0
冑6
共79兲
It is readily verified that, despite the degeneracy, ⍀2 = ⍀3 = 冑2, the three vectors are orthogonal. Hence the degeneracy does not cause any difficulty and, using Eq. 共62兲, we find 共⌬a1兲2 = 共⌬a2兲2 = 共⌬a3兲2 =
ប 6
冋 冉 冊 1
冑5
coth
冑5 2
冉 冑 冊册
ប + 冑2coth
1
2
ប
2
+
冑 冊
ប . 2 5 1
共81兲
冊 冉
冉
冊
共82兲
1 ប ប = . 2⍀2 冑10 0
共83兲
Finally we note that Eqs. 共52兲 and 共53兲 yield the expression for the standard deviations of the energies Ek for the modes k = 1,2,3 as
⌬Ek =
ប⍀k 2
冑 冉 coth2
冊
1 ប⍀k − 1, 2
共84兲
which coincides with the classical formula for a quantum harmonic oscillator with the frequency ⍀k. Hence in the classical limit given by Eq. 共73兲, in contrast to Eqs. 共74兲 we have a linear dependence of the deviation ⌬Ek on the temperature: ⌬Ek ⯝ kBT.
1
u共3兲 =
1
V. SCALING OF THERMAL FLUCTUATIONS OF TRAPPED BEC
冑2
,
冉冑
1 1 1 1 ប ប 1 + + = , 4 2 冑2 4 ⍀1 ⍀3
共⌬a3兲2 =
冊
Case 2 共Spherical form兲 0 = . In the spherically symmetric case ⍀1 = 冑5 , ⍀2 = ⍀3 = 冑2, the expressions for the basis vectors take the form 1
共⌬a⬜兲2 =
1 ប 3 , 共77兲 + 10 冑2
1 ប ប 共⌬a3兲2 = = . 2 ⍀ 1 2 冑3 0
1 3
We note here that the above expressions for the spherically symmetric case also follow from Eqs. 共70兲 and 共71兲 with ⌫1 = 2 / 3 and ⌫2 = 1 / 3. Case 3 共Cigar form兲 Ⰷ 0. In the limiting case when 0 / → 0 we find that ⌫1 → 1 and ⌫2 → 0 which again leads to significant simplification of the formulas 共70兲 and 共71兲. For quantum fluctuations 共T = 0兲 we find
冑2 3 . 共76兲 2 + 冑 10 50
The formulas for ⌬a given by Eqs. 共75兲 and 共76兲 call to mind the equations for average standard deviations of a threedimensional quantum oscillator at temperatures T and T = 0, respectively, with characteristic frequencies ⍀k and mass . To conclude this section we consider the limiting cases for the formulas 共70兲 and 共71兲 which connected with different trapping geometries Eqs. 共44兲–共46兲. Case 1 共Disc form兲 0 Ⰷ . In the limiting case when / 0 → 0 we find that ⌫1 → 0 and ⌫2 → 1, which leads to significant simplification of the formulas 共70兲 and 共71兲. For quantum fluctuations 共T = 0兲, one can find 2
共⌬a1兲2 = 共⌬a2兲2 = 共⌬a3兲2 =
.
We have considered that the condensate is at thermal equilibrium at a temperature T which may be zero. Beginning with a mean-field, Gross-Pitaevskii theory, and developing equations which lend themselves to quantization, we have been able to predict dynamical effects which are not calculable in the GPE approach. Our approach gives values for the excitation frequencies equal to those found using other methods, but also predicts a quantum mechanical uncertainty in the physical dimensions of the condensate. We note that the collective modes of the condensate are weakly temperature dependent in a range of temperatures below Tc, even when the condensate is strongly depleted. All our calculations were based on the effective Hamiltonian
,
3
共80兲 where the three variances are equal because of the spherical symmetry. In particular for quantum fluctuations 共T = 0兲 this equation takes the form
3
2 2 G ˆ = − ប 兺 + 兺 2a 2 + , H k k 2 2 k=1 ak 2 k=1 a 1a 2a 3
共85兲
which, for small deviations of the ellipsoidal parameters, has the form 共52兲. We can also introduce the bosonic annihilation
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KRUGLOV, OLSEN, AND COLLETT
and creation operators for the diagonalized fluctuations of the ellipsoidal parameters bˆk =
bˆ†k =
冑 冑
⍀k zk + i 2ប ⍀k zk − i 2ប
冑 冑
1 ˜ k, 2 ⍀ kប 共86兲 1 ˜ k, 2 ⍀ kប
which allows us to write a linearized Hamiltonian as
冉
3
冊
5G ˆ = 兺 ប⍀ bˆ†bˆ + 1 + . H k k k 2 ¯ 1¯a2¯a3 2a k=1
共87兲
Finally, we give the explicit expressions for the fluctuations of the ellipsoidal parameters in dimensionless form. For transverse fluctuations we introduce the characteristic length l, the dimensionless temperature ⌰ and the dimensionless frequency as l=
冑
ប ,
k BT , ប
⌰=
=
0 ,
共88兲
then the dimensionless transverse dispersion given by Eq. 共70兲 is
冉 冊 冉 ⌬a⬜ l
2
=
+
+
冊
FIG. 1. The dimensionless transverse dispersion 共⌬a⬜ / l兲2 as a function of the dimensionless temperature ⌰ for different values of = 0.1, 1, and 10. A greater parameter gives the curve with a greater dispersion. There is very little dependence on , with the upper 共chain兲 line, the middle 共broken兲 line, and the lower 共full兲 line being for = 10, 1, and 0.1, respectively. The subfigure shows the same curves for 0 艋 ⌰ 艋 0.6.
1 3 2 − 23 2 + f共兲 coth共 2⌰ 冑2 + 2 2 + f共兲兲
冉
8f共兲
−2+
3 2 2
+ f共兲
8f共兲
1 coth共 冑2⌰ 兲
4 冑2
冊
冑2 +
3 2 2
f 0共 0兲 =
+ f共兲
coth共 冑2 + 23 2 − f共兲兲 冑2 + 23 2 − f共兲 1 2⌰
共89兲
,
where function f共兲 has the form f共兲 =
冑
5 共2 − 2兲2 + 4 . 4
冑
5 共1 − 220兲2 + . 4
共93兲
We note that in all these equations the effective mass = Ncm / 7 is a function of the temperature T 共see previous section兲. Considering Eqs. 共89兲 and 共92兲 for fixed parameters , 0 and ⌰ , ⌰0 we can find the dimensionless transverse and longitudinal dispersions as functions of ⌰ , ⌰0 and , 0, respectively. These curves are represented by Figs. 1–4 for different values of the parameters. We can see that the curve in Fig. 1 is similar to that in Fig. 2 共they practically coincide兲
共90兲
For longitudinal fluctuations one may introduce the characteristic length l0, the dimensionless temperature ⌰0 and the dimensionless frequency 0 as l0 =
冑
ប , 0
⌰0 =
k BT , ប0
0 =
1 = , 0
共91兲
then the dimensionless longitudinal dispersion given by Eq. 共71兲 is
冉 冊 冉 ⌬a3 l0
2
=
+
3 2
冉
冊
共 2⌰ 冑 2 + 220 + f 0共0兲兲
− 220 + f 0共0兲 coth 4f 0共0兲
− 23 + 220 + f 0共0兲 4f 0共0兲
冊
1
3
0
冑 23 + 220 + f 0共0兲 coth共 2⌰1 冑 23 + 220 − f 0共0兲兲 冑 23 + 220 − f 0共0兲 , 0
共92兲 where the function f 0共0兲 is given as
FIG. 2. The dimensionless longitudinal dispersion 共⌬a3 / l0兲2 as a function of the dimensionless temperature ⌰0. Here the upper 共chain兲 line, the middle 共broken兲 line, and the lower 共full兲 line are for 0 = 10, 1, and 0.1, respectively. The subfigure shows the same curves for 0 艋 ⌰0 艋 0.6.
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an accuracy of about 20%; at ⌰ = 2 and ⌰0 = 2 they both have an accuracy about 5% and at higher temperatures become practically exact. Hence one may consider the intervals of temperatures ⌰ ⲏ 2 and ⌰0 ⲏ 2 as the regions of classical fluctuations for transverse and longitudinal components, respectively. Thus, from Figs. 1–4 it follows that the dimensionless transverse and longitudinal dispersions with high accuracy do not depend on the frequency parameters and 0 of the trap and they have the same dependence on dimensionless temperatures. This scaling is an approximate but nevertheless very accurate law and it was found in this section by direct numerical calculations of Eqs. 共70兲 and 共71兲. We can express this scaling of thermal fluctuations of trapped BEC by the approximate, but very accurate formulas: FIG. 3. The dimensionless transverse dispersion 共⌬a⬜ / l兲 as a function of the dimensionless parameter = 0 / for different values of dimensionless temperatures ⌰ = 0,0.1,…,0.6. A greater temperature gives a curve with a greater value of dispersion 共⌬a⬜ / l兲2.
冉 冊 ⌬a⬜ l
2
when ⌰ and ⌰0 are greater than 0.6. Figures 3 and 4 also demonstrate a very weak dependence of the dimensionless transverse and longitudinal dispersions on the frequency parameters and 0 when ⌰ and ⌰0 are greater than 0.6. This is the result of our choice of the scaling parameters and the nature of the transverse and longitudinal fluctuations. Hence the dimensionless transverse and longitudinal dispersions have the same behavior considered as functions of dimensionless temperature, except in a small region near zero. For these dimensionless variables the asymptotically exact Eqs. 共74兲 have the form
冉 冊 ⌬a⬜ l
2
2 ⯝ ⌰, 5
冉 冊 ⌬a3 l0
2
2 ⯝ ⌰0 , 5
共94兲
when ⌰ Ⰷ 1 and ⌰0 Ⰷ 1 respectively. However, Fig. 1 and Fig. 2 show that these formulas at ⌰ = 1 and ⌰0 = 1 both have
2
= ⌶共⌰兲,
冉 冊 ⌬a3 l0
2
= ⌶共⌰0兲,
共95兲
where ⌰ and ⌰0 are greater than 0.6. One can find the scaling function ⌶共x兲 by making the choice that and 0 in Eqs. 共89兲 and 共92兲 are equal to 1 共 = 0兲 that corresponds to the middle 共broken兲 curves of Figs. 1 and 2: ⌶共x兲 =
1
6 冑5
冉冑 冊
coth
冉冑 冊
5 1 coth + 2x 3 冑2
1
2x
.
共96兲
At x ⲏ 2 this equation yields ⌶共x兲 ⯝ 共2 / 5兲x, in accordance with the asymptotic limit 关see Eqs. 共94兲兴. Using, for example, a characteristic frequency of the trap = / 2 = 100 Hz one can find that the characteristic temperature connected with the dimensionless temperature ⌰ = 0.6 to be T ⯝ 2.9 nK. VI. CONCLUDING REMARKS
In this paper we have studied the dynamics of the collective modes of a trapped dilute Bose gas in an approach which goes beyond the Gross-Pitaevskii theory. Our treatment is essentially different from previous ones 共see for example the papers 关12–16兴 mentioned above兲. The approach developed in this paper was first used to find the roton structure of the excitation spectrum in liquid helium II 关27兴. We found in the previous section a scaling of the thermal fluctuations of trapped BEC which takes place when the dimensionless temperatures ⌰ and ⌰0 are greater than 0.6. This scaling means that the dimensionless dispersions do not depend on the frequencies k of the trap and the transverse and longitudinal ones both have the same dependence on the dimensionless temperatures. In the general case when the three frequencies k, 共k = 1,2,3兲 of the trap are different one can define the scaling parameters: lk =
冑
ប , k
⌰k =
k BT , បk
k = 1,2,3.
共97兲
Equations 共95兲 can then be generalized as FIG. 4. The dimensionless longitudinal dispersion 共⌬a3 / l0兲 as a function of the dimensionless parameter 0 = / 0 for different values of dimensionless temperatures ⌰0 = 0,0.1,…,0.6. A greater temperature gives a curve with a greater value of dispersion 共⌬a3 / l0兲2.
冉 冊 ⌬ak lk
2
2
= ⌶共⌰k兲,
k = 1,2,3,
共98兲
where the scaling function ⌶共x兲 is given by Eq. 共96兲 and ⌰k 艌 0.6. This generalization is proved by the results ob-
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tained in Appendix B. We note again that in all these equations the effective mass = Ncm / 7 is a function of the temperature T. Finally, we discuss a simple estimation of the damping effect for low-energy excitations at temperatures kBT ⱗ 0, where 0 is the chemical potential of the trap and we suppose that Nex / Nc Ⰶ 1, where Nex is the number of excited atoms and Nc is the number of condensed atoms. We also assume here that col⍀k Ⰷ 1, where col is collision time and hence in this case collective excitations have hydrodynamic nature. The solution of the classical Eq. 共14兲 has the form given by Eq. 共35兲 where zk = Akcos共⍀kt + ⌽k兲 and Ak are the arbitrary constants. However, when the damping is provided by the interaction of the low-energy excitations of a trapped BEC with the thermal excitations, the amplitudes Ak 共k = 1, 2, 3兲 depend slowly on time: Ak = Aoke−kt , k / ⍀k Ⰶ 1. Hence the damping parameters can be defined as k = −␦Ak / 共Ak␦t兲. When the conditions mentioned above are fulfilled using the ergodic hypothesis one may assume that ␦Ak / ␦t ⬃ ⌬zk / ⌬t, where ⌬Ek⌬t ⬃ ប. Thus the damping parameters can be estimated as k ⬃ ⌬zk⌬Ek, where the standard deviations ⌬zk and ⌬Ek are given by Eqs. 共61兲 and 共84兲. Hence for the collisionless regime col⍀k Ⰷ 1 and Nex / Nc Ⰶ 1 in the region of classical fluctuation ⌰ , ⌰0 ⲏ 2 and kBT ⱗ 0, the temperature dependence of the damping parameters has the form k ⬃ T3/2, which is in agreement with the result reported in 关38兴. We consider this discussion of the damping effect for low-energy excitations to emphasize that the approach developed in this paper is relevant to this problem. However, the full treatment of it would require consideration of the interaction of the BEC with the noncondensed thermal cloud.
␥=
ប , 2m
Nc共t兲 =
A g共t兲 = − 2␥共ⵜA兲共ⵜ兲 − ␥Aⵜ2 + A, 2 t
冋
− ␥共ⵜ兲2 +
册
共A1兲
3
A = − ␥ⵜ2A + 兺 kx2k A + kA3 , t k=1 共A2兲
where
4បa0 , m
k=
共A3兲
冕
A2共x,t兲dx.
共A4兲
Equations 共A1兲 and 共2兲–共4兲 then lead to a set of equations 共with k = 1,2,3兲 1 df k = − 2␥ck f k + gk共t兲f k , 2 dt
共A5兲
where 3
˙ 共t兲 N
c . gk共t兲 = g共t兲 ⬅ 兺 N 共t兲 c k=1
共A6兲
We now introduce the characteristic dimensionless parameter 2 E c = 8 n ca 0l m ,
共A7兲
where lm is the minimum of the lengths along the three axes of the trapped BEC and nc is the average condensate density. For the conditions a0 ⬎ 0 共positive s-wave scattering length兲 and Ec Ⰷ 1, we can neglect the term −␥ⵜ2A on the right-hand side of Eq. 共A2兲. Taking into account Eqs. 共2兲–共4兲, we can now write Eq. 共A2兲 in the form 3
兺 k=1
冉
4␥c2k +
dck + k dt
+ 共 f 1 f 2 f 3兲−2
冊共
冉冕
f 1 f 2 f 3 f 2k 兲−2exp 2
冊
t
gk共t⬘兲dt⬘ 2k
0
d0 = − kF2共1, 2, 3兲 . dt
共A8兲
Due to the fact that the right-hand side of Eq. 共A8兲 depends only on the k, we can write a system of ordinary differential equations 共for k = 1,2,3兲,
冉 冕
dck + 4␥c2k + k = k2k f 4k 共 f 1 f 2 f 3兲2exp − 2 dt
APPENDIX A: SOLUTION OF GROSS-PITAEVSKII GAIN EQUATION
The modeling of an atom laser which includes a semiclassical treatment of pumping and losses for a trapped BEC leads to a generalized GPE with a linear gain term 关23兴. Here we consider in detail the self-similar solution of this model because it can be generalized to describe a number of different experimental situations 关30–37兴. Introducing the real amplitude A共x , t兲 and phase 共x , t兲, so that = A exp共i兲 we may write Eq. 共1兲 as the system of equations
m2k , 2ប
with the normalization condition
ACKNOWLEDGMENTS
This research was supported by the Marsden Fund of the Royal Society of New Zealand and the New Zealand Foundation for Research, Science and Technology 共Grant No. UFRJ0001兲.
k =
冊
t
gk共t⬘兲dt⬘ .
0
共A9兲 We also find that d0 = − k 共 f 1 f 2 f 3兲 2 , dt and
冋
3
F共1, 2, 3兲 = 1 − 兺 k2k k=1
册 冉 1/2
共A10兲
3
冊
S 1 − 兺 k2k , k=1
共A11兲 where the k are positive constants and S共x兲 is the Heaviside step function. Note here that by writing the equation for the phase in the form of Eq. 共A10兲, we are fixing a normalization condition for the function F共1 , 2 , 3兲. This does not alter the generality of the solution because Eq. 共2兲 allows for a scaling factor in the definitions of f 1 , f 2 , f 3 and F. In accord with Eq. 共A11兲, we fix this scaling factor by the condition F共0 , 0 , 0兲 = 1.
033604-10
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QUANTUM AND THERMAL FLUCTUATIONS OF TRAPPED…
Let us now consider the normalization condition, Eq. 共A4兲, where the amplitude is given by Eqs. 共A1兲 and 共A11兲. Calculating the integral on the right-hand side of Eq. 共A4兲, we can write the number of atoms in the trapped BEC as
冉冕
冉 冊
8 Nc共t兲 = 共123兲−1/2exp 15 Introducing the definition
冉冕
g共t⬘兲dt⬘ .
0
冊
t
Nk共t兲 = Nk共0兲exp
gk共t⬘兲dt⬘ ,
0
where
冉 冊
8 Nk共0兲 = 15
冊
t
⌬k = 共⍀2k − 321兲共⍀2k − 322兲 − 2122 ,
and u共k兲 3 may be found from the normalization condition
兺j 共u共k兲j 兲2 = 1.
共A12兲
共A13兲
1/3
共⍀2k − 222兲13
冑Q k
共A14兲 u共k兲 2 = ±
we can rewrite Eq. 共A12兲 in the form
共⍀2k − 221兲23
3
Nc共t兲 = 兿 Nk共t兲.
共A15兲
k=1
u共k兲 3 = ±
Hence the self-similar variables k are defined as
k =
Nk共0兲f 2k 共t兲 xk , Nk共t兲
共A16兲
and the ellipsoidal parameters have the form
冉 冊
15 ak共t兲 = 8
1/3
Nk共t兲 . f 2k 共t兲
共A17兲
2 共k兲 2 共k兲 12u共k兲 1 + 共32 − ⍀k 兲u2 + 23u3 = 0,
共B1兲
u共k兲 1 = u共k兲 2 =
−
2 2 2 2 u共3兲 1 = 关 共⍀1 − 21兲⌬2 − 共⍀2 − 21兲⌬1兴
2 2 2 2 u共3兲 2 = 关 共⍀2 − 22兲⌬1 − 共⍀1 − 22兲⌬2兴
2 3
冑Q 1 Q 2 , 1 3
共B7兲
冑Q 1 Q 2 ,
2 2 2 2 u共3兲 3 = 关 共⍀1 − 22兲共⍀2 − 21兲
1223
冑Q 1 Q 2 .
We now find that the matrix usk 共=us共k兲兲, defined by Eq. 共B5兲 for k = 1,2 and by Eq. 共B7兲 for k = 3, is not degenerate 共the determinant being equal to 1兲, when Q1 and Q2 are not equal to zero. Using Eqs. 共62兲 and 共B5兲–共B7兲, we may now find the standard deviations in the general case with all three frequencies different: ⌬a1 =
222兲13 共k兲 u3 , ⌬k
共⍀2k − 221兲23 共k兲 u3 , ⌬k
共B5兲
冑Q k ,
Using the orthonormality of the eigenvectors, we can give a general definition for u共3兲 via the product u共1兲 ⫻ u共2兲:
The solution to these equations is 共⍀2k
⌬k
− 共⍀22 − 222兲共⍀21 − 221兲兴
共k兲 共k兲 共321 − ⍀2k 兲u共k兲 1 + 12u2 + 13u3 = 0,
,
Qk = 共⍀2k − 222兲22123 + 共⍀2k − 221兲22223 + ⌬2k . 共B6兲
APPENDIX B: GENERAL ASYMMETRIC CASE
2 共k兲 2 共k兲 13u共k兲 1 + 32u2 + 共33 − ⍀k 兲u3 = 0.
冑Q k
,
where
Combining Eqs. 共13兲 and 共A11兲 with the definitions of Eqs. 共A13兲, 共A16兲, and 共A17兲, we can represent the amplitude A共x , t兲 in the form given by Eqs. 共7兲 and 共8兲. Equations 共A5兲, 共A9兲, and 共A10兲 and the definition of Eq. 共A17兲 yield Eqs. 共9兲–共11兲 and the initial conditions of Eq. 共12兲. Hence, by using the variables defined in Eq. 共A17兲 we are able to exclude the indeterminate parameters k in our self-similar solution.
In the general case where all three trapping frequencies are different, the equations for the eigenvectors, u共k兲, have the form
共B4兲
We note here that these expressions are not valid in the axially symmetric case where 1 = 2 = and 3 = 0, because ⍀23 then becomes equal to 22, so that ⌬3 = 0 and the solutions are undefined. However, for k = 1,2, we find the normalized solutions u共k兲 1 = ±
−1/2 , k
共B3兲
共B2兲
冋
冉 冉
ប共⍀21 − 222兲22123 1 coth ប⍀1 2 2 ⍀ 1Q 1
冊 冊
+
ប共⍀22 − 222兲22123 1 coth ប⍀2 2 2 ⍀ 2Q 2
+
ប2223关共⍀21 − 221兲⌬2 − 共⍀22 − 221兲⌬1兴2 2 ⍀ 3Q 1Q 2
册
1/2
, 共B8兲
where 033604-11
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KRUGLOV, OLSEN, AND COLLETT
⌬a2 =
冋
冉
冊
冉
冊
ប共⍀21 − 221兲22223 ប共⍀22 − 221兲22223 ប2123关共⍀22 − 222兲⌬1 − 共⍀21 − 222兲⌬2兴2 1 1 coth ប⍀1 + coth ប⍀2 + 2 2 2 ⍀ 1Q 1 2 ⍀ 2Q 2 2 ⍀ 3Q 1Q 2
册
1/2
,
共B9兲 ⌬a3 =
冋 +
冉
冊
冉
ប⌬21 ប⌬22 1 1 coth ប⍀1 + coth ប⍀2 2 2 2 ⍀ 1Q 1 2 ⍀ 2Q 2
冊
冉
ប212243关共⍀21 − 222兲共⍀22 − 221兲 − 共⍀22 − 222兲共⍀21 − 221兲兴2 1 coth ប⍀3 2 2 ⍀ 3Q 1Q 2
冊册
1/2
.
共B10兲
2
To conclude, we note that these expressions for the standard deviations are valid only when Q1 and Q2 are not equal to zero. For example in the spherically symmetric case Q2 = 0 and hence in this degenerate case we should use Eq. 共80兲. Thus, despite the generality of the results given here, in the axially and spherically symmetric cases we should use the equations derived in Sec. IV. Actually the above general equations can be used even in the spherically symmetric case as long as we use an accurate limiting procedure. In the spherically symmetric case Eq. 共B3兲 can be written as ⌬k = 共⍀2k − 42兲共⍀2k − 22兲; hence the normalized Eq. 共B2兲 for k = 1,2 yields
Defining u共3兲 via the product u共1兲 ⫻ u共2兲, one may verify that these eigenvectors lead to Eq. 共80兲. We also note that the three expressions given above can be combined to give the same result as in Eq. 共75兲 for the average standard deviation 3 共⌬ai兲2兴1/2. ⌬a = 关兺i=1
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共k兲 u共k兲 1 = u2 = ±
u共k兲 3 = ±
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冑共⍀2k − 42兲2 + 24 , 共⍀2k − 42兲
共B11兲
冑共⍀2k − 42兲2 + 24 .
PHYSICAL REVIEW A 72, 033604 共2005兲
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