We give an asymptotic analytic solution for the generic atom-laser system with gain in a D-dimensional trap, and show that this has a non-Thomas-Fermi behavior. The effect is due to Bose-enhanced condensate growth, which creates a local-density maximum and a corresponding outward momentum component. In addition, the solution predicts amplified center-of-mass oscillations, leading to enhanced center-of-mass temperature

A.S. Parkins

B. Kneer

B.P. Anderson

C.W. Gardiner

E. Zaremba

E.P. Gross

E.W. Hagley

F. Dalfovo

G.M. Moy

H.-J. Miesner

H.T.C. Stoof

I. Bloch

K. V. Kheruntsyan

L.P. Pitaevskii

M.-O. Mewes

M.E. Fermann

M.H. Anderson

P. D. Drummond

P.D. Drummond

T. Nikuni

W. Kohn

Yu. Kagan

arXiv

Drummond, P. D.

Kheruntsyan, K. V.

University of Queensland eSpace

Asymptotic solutions to the Gross-Pitaevskii gain equation: Growth of a Bose-Einstein condensate

PHYSICAL REVIEW A, VOLUME 63, 013605Asymptotic solutions to the Gross-Pitaevskii gain equation: Growth of a Bose-Einstein condensate
P. D. Drummond and K. V. Kheruntsyan
Department of Physics, The University of Queensland, Brisbane, Queensland 4072, Australia
~Received 27 July 2000; published 11 December 2000!
We give an asymptotic analytic solution for the generic atom-laser system with gain in a D-dimensional trap,
and show that this has a non-Thomas-Fermi behavior. The effect is due to Bose-enhanced condensate growth,
which creates a local-density maximum and a corresponding outward momentum component. In addition, the
solution predicts amplified center-of-mass oscillations, leading to enhanced center-of-mass temperature.
DOI: 10.1103/PhysRevA.63.013605 PACS number~s!: 03.75.Fi, 03.65.Ge, 05.30.JpI. INTRODUCTION
The description of Bose-Einstein condensate ~BEC!
growth @1–4# has become important due to the need to un-
derstand the physics of atom lasers @5#. These recently de-
veloped devices that emit coherent wavelike beams of atoms
promise a new generation of precision measurements, appli-
cations in nanotechnology, and novel tests of fundamental
concepts in quantum theory. While it is possible to perform
first-principles numerical simulations of the relevant equa-
tions @6#, a great deal of physical insight can be obtained
from an analytic solution.
In this paper, we give an analytic asymptotic solution to
the Gross-Pitaevskii equation describing the early stages of
condensate growth in a trap. The physical insight we obtain
from this is that a growing nonequilibrium condensate has a
nonuniform momentum distribution across the condensed re-
gion. As a result, the observed density behaves as though the
trap frequency is increased, relative to the usual Thomas-
Fermi behavior @7# of an equilibrium BEC. In addition, our
solutions show center-of-mass oscillations whose kinetic en-
ergy is amplified with the condensate growth. This implies
that, while BEC’s formed through evaporative cooling may
have a low temperature for their internal degrees of freedom,
the temperature for the center-of-mass motion is higher,
leading to increased noise in the velocity and direction of
atom laser beams.
II. GROSS-PITAEVSKII GAIN MODEL
We start by considering a commonly used model of a
one-component trapped Bose-Einstein condensate—the
Gross-Pitaevskii ~GP! equation @8# modified by a linear gain
term g @9#, of the form
]C~x,t !
]t
5Fg2iS \2m 21V~x!1UuCu2D GC . ~1!
Here C(x,t) is the mean-field amplitude @so that uC(x,t)u2
is the particle number density#, m is the atomic mass, and U
is the effective interaction potential. In the treatment of D
equal to one, two, or three space dimensions, U is given by
U54p\aL32D/m , where a is the scattering length and L is
the confinement length. The potential term V(x) is due to an
optical or magnetic trap, which we assume is harmonic. In1050-2947/2000/63~1!/013605~5!/$15.00 63 0136the simplest rotationally symmetric case, the trap potential is
given by V(x)5mv2uxu2/(2\), where v is the trap oscilla-
tion frequency.
It is helpful to understand the physical implications of Eq.
~1! by an analysis of the relevant time scales. In the present
case, there are three different relevant time scales. These are
the time scale for the condensate growth, tg51/g; the trap
oscillation time scale, tv51/v; and the ‘‘healing’’ time
scale, th51/@UuC(x0 ,t)u2# , associated with the mean-field
interaction potential at the condensate center of mass x0.
Since current BEC’s typically have a relatively high den-
sity, the healing time is usually the smallest, which results in
a Thomas-Fermi ~TF! type equilibrium condensate. If the
gain time scale is long enough, then one may expect that
even during growth, the condensate will adiabatically follow
the TF solution. This is commonly assumed in analyzing
experimental data @1#. Another possibility is that the gain
time scale is shorter than the trap oscillation period, in which
case we should no longer expect adiabatic TF-like behavior.
A. Asymptotically growing solution
To show this distinction, we now proceed to give an ana-
lytic nonequilibrium solution of the GP equation with gain.
First, we expand the field in terms of the amplitude and
phase,
C~x,t !5A~x,t !e2if(x,t)/AU , ~2!
and obtain the following coupled equations:
]A
]t
5gA1
\
2m A2f1
\
m
fA , ~3!
]f
]t
5A21
mv2
2\ uxu
21
\
2m S ufu22 1A 2A D . ~4!
Next, we wish to investigate possible asymptotic solutions
for long times, i.e., steadily growing solutions, valid some
time after initial nucleation of the condensate, yet before any
gain saturation has occurred. We consider first the rotation-
ally symmetric case, where 25]r21@(D21)/r#]r , with r
5uxu. Following the construction successfully used recently
in optical fiber environments @10#—and extending these to
the current multidimensional case of a trapped BEC—we
suppose that the amplitude has a self-similar behavior at©2000 The American Physical Society05-1
P. D. DRUMMOND AND K. V. KHERUNTSYAN PHYSICAL REVIEW A 63 013605large time. The phase is assumed to depend on the atomic
density at the condensate center, and to have a uniform chirp
giving a radially dependent outward momentum. Thus, we
seek a solution in the form of
A5l~ t ! f r/l~ t !, ~5!
f5
l2~ t !
2g˜ t
2
mg˜
2\ r
2
. ~6!
Here l(t) is a scaling function, while t and g˜ are unknown
coefficients. Using this construction, we find, from Eqs. ~3!
and ~4!,
]l~ t !
]t S f 2 r f 8l D5gl f 2 f 8g˜ r2 D2 l f g˜ , ~7!
]l~ t !
]t S 1g˜ t l D 5~l f !21 \v˜ 22m r22 \2m S f 9f l2 1 D21f lr f 8D ,
~8!
where we have defined v˜ 2[v21g˜ 2.
From the first ~amplitude! equation, the solution requires
the twin conditions that
]l~ t !
]t
5S g2 Dg˜2 D l5g˜l . ~9!
This implies that the scaling function l grows exponentially
with time, having a solution of l(t)5exp(g˜t), where we can
immediately solve for the growth rate, since clearly g˜
52g/(D12). The physical interpretation of this equation is
rather straightforward; the growth of the amplitude at any
radial point is reduced below the single-mode growth rate g,
due to an outward flow of atoms, which transfers part of the
increased density to a larger radius. This effect increases
with dimensionality of the space.
The phase equation can now be simplified using the fact
that the last two terms involving derivatives have terms in
1/l , and hence become exponentially smaller than the earlier
terms, at long times, as the healing time becomes smaller.
This immediately gives the following solution for f:
f 5H A@12uxu2/R~ t !2#/t for uxu,R~ t !,
0 for uxu.R~ t !,
~10!
where R(t) is the maximum radius given by
R~ t !5eg˜ tA2\/~mtv˜ 2!. ~11!
The remaining unknown constant t is obtained by evalu-
ating the integral N(t)5* uC(x,t)u2dDx for the total number
of particles:
N~ t !5eg˜ (D12)tS 2\
mv˜ 2
D D/2 2CD t2(D12)/2UD~D12 ! . ~12!
01360Here CD is a dimension-dependent constant, with C1
52,C252p , and C354p . This gives the expected result
that the overall atom number grows exponentially with a
gain constant of 2g5g˜ (D12); that is, N(t)
5N(0)exp(2gt). The reason for this apparent difference in
gain constants is that the relatively slower growth in density
at the center of the condensate is exactly compensated for by
the increase in condensate radius with time.
The constant t is solved in terms of N(0), so that
t5F S 2\m D D/2 2CDUD~D12 !N~0 !v˜ DG
2/(D12)
. ~13!
This coincides with the healing time for a condensate with
atom number N(0), in TF equilibrium, but with an effective
trap oscillation frequency v˜ instead of v .
The final result for the particle number density
uC(x,t)u25l2 f 2/U is
uC~x,t !u25
e2g
˜ t
tU S 12 uxu2R~ t !2D . ~14!
While this gives an asymptotic solution in terms of the
initial atom number, the solution needs to be compared with
the usual TF solution to understand the physical implica-
tions. In particular, we can notice an interesting scaling be-
havior for the radius, as a function of the trap frequency. In
the present case, the radius scales as
R}v˜ 22/(D12)5~v21g˜ 2!21/(D12). ~15!
In equilibrium TF solutions, there is a similar behavior,
except that there is no outward momentum term. As a con-
sequence, the TF radius (RTF}v22/(D12)) is always larger at
a given total atom number, and the density is lower at the
center, than in these nonequilibrium solutions with gain
present. The physical reason for this is simply the rapid
Bose-enhanced increase in number density ~and hence pres-
sure! at the condensate center during the nucleation process.
This effect becomes appreciable when g@v or tg!1/v , so
that the time scale for condensate growth is faster than the
trap oscillation period.
B. Numerical results
We now turn to comparisons between the analytic
asymptotic solution and exact numerical results. In Fig. 1,
we plot the rms radius r¯(t) of a growing BEC versus t. The
result found from the above asymptotic solution in three di-
mensions is simply given by r¯(t)5A3/7R(t). This is repre-
sented by the dashed line, for g510v and N(0)510. For
comparison, the TF rms radius for the same values of N(t) is
represented by the dotted line. The full lines correspond to
the results of direct numerical simulations of the GP equation
with gain, with initial Gaussian wave functions of different
widths. The values of other parameters are chosen to corre-
spond to a 87Rb BEC (m51.44 kg,U355310217 m3/s) in
a trap with v/2p5100 Hz. As one can see from the graph,5-2
ASYMPTOTIC SOLUTIONS TO THE GROSS- . . . PHYSICAL REVIEW A 63 013605despite the initial differences, the mean radii approach the
same asymptotic value of ;A3/7R(t), which is different
from the TF result for the same total number of particles.
The outward momentum density p5upu where p5
2i\@C*(C)2H.c.#/2 is plotted in Fig. 2, showing that
in a growing BEC the flow of atoms from the trap center
increases initially with the distance, reaches a maximum, and
vanishes at r*R .
III. ASYMMETRIC CASE AND CENTER-OF-MASS
OSCILLATIONS
The above rotationally symmetric results can easily be
generalized to asymmetric cases, where the trap oscillation
frequencies v i in each space direction are different. In this
case, the trap potential term in Eqs. ~1! and ~4! is replaced by
V~x!5
m
2\ (i51
D
v i
2xi
2
. ~16!
Assuming again that the amplitude has a self-similar be-
havior at long times, one can use the previous construction.
In addition, we find that a more general solution can be ob-
FIG. 1. The rms radius r¯ ~in a logarithmic scale! of a growing
condensate versus t, for g510v . The full lines 1, 2, and 3 corre-
spond, respectively, to initial Gaussian wave functions having rms
radii that are twice as larger as, equal to, and twice as small as the
asymptotic rms radius r¯(0)5A3/7R(0). The dashed line is the
asymptotic result, while the dotted line corresponds to the equilib-
rium ~TF! solution.
FIG. 2. The momentum density p versus the radial distance
from the center of the trap r. The full line corresponds to the result-
ing distribution of case 1 in Fig. 1; the dashed line is the corre-
sponding asymptotic solution.01360tained. Following the approach of Kohn’s theorem @11#, gen-
eralized to include gain, we allow for center-of-mass oscil-
lation of the growing condensate, independent of the
interparticle interactions.
Thus, we seek a solution in the form of
A5l~ t ! f @Dx~ t !/l~ t !# ,
f5f0~ t !1
l~ t !2
2g˜ t
2
m
\
Fg˜Dx~ t !22 1Dx~ t !x˙0~ t !G ,
where Dx(t)[x2x0(t), x0(t) is the center of mass, and the
dot stands for the time derivative.
Using the same procedure as before, we first find—from
the amplitude equation—that the function l(t) is given by
l~ t !5exp~g˜ t !, ~17!
where g˜52g/(D12), as previously.
To treat the phase equation, in which we neglect the last
term }1/l that becomes exponentially small at long times,
we assume that
f 25F11F2~Dx2,
where the functions F1 and F2 are to be found by equating
the terms in powers of Dxi . From the terms in Dxi , we find
that each component of x0(t) satisfies the equation x¨ 0,i
1v ix0,i50, so that the condensate center of mass oscillates
according to
x0,i~ t !5a0,i cos~v it1d0,i!, ~18!
where a0,i and d0,i are the initial amplitude and phase.
By equating the terms in (Dxi2 and the zeroth-order
terms, respectively, we obtain an equation for f0(t),
f˙ 01
m
2h (i51
D
@~x˙ 0,i!
22v i
2x0,i#50,
as well as solutions to F1 and F2:
F151/t ,
F252
m
2\l~ t ! (i51
D
~v i
21g˜ 2!Dxi~ t !
@Dx~ t !#2
.
Combining these together and using the solutions for the
center-of-mass coordinates, we finally obtain that the solu-
tion for the function f0(t) is given by
f0~ t !52~m/2\!@x0~ t !]x˙0~ t !# , ~19!
while the solution for f is
f 5 1
At S 12(i51
D
@xi2x0,i~ t !#
2
Ri~ t !2
D 1/2, ~20!5-3
P. D. DRUMMOND AND K. V. KHERUNTSYAN PHYSICAL REVIEW A 63 013605in the region of space where the expression in large brackets
is positive, and f 50 outside that region.
The maximum radius Ri(t) in each space direction is
Ri~ t !5eg
˜ tA2\/~mtv˜ i2!, ~21!
where we have introduced v˜ i
2[v i
21g˜ 2. In addition, the con-
stant t is solved, as previously, in terms of N(0) by evalu-
ating the integral for the total number of particles N(t)
5N(0)exp(2gt). The resulting expression for t is given by
the same equation as before, Eq. ~13!, except that v˜ D is
replaced by ) j51
D v˜ j . This again corresponds to the healing
time in a TF condensate, with N(0) particles and effective
trap oscillation frequencies v˜ i .
The final result for the particle number density is now
uC~x,t !u25
e2g
˜ t
tU S 12(i51
D
@xi2x0,i~ t !#
2
Ri~ t !2
D . ~22!
An example showing growth of the BEC, while the conden-
sate center of mass oscillates, is shown in Fig. 3.
IV. SUMMARY
The physical interpretation of these results is that the
asymptotic solution with gain has a density distribution that
is similar to the TF solution, except that the trap oscillation
effective frequency is increased, with v i
2→v˜ i25v i21g˜ 2. For
a given gain constant, this has the strongest effect for weakly
trapped ~low-frequency! directions in an asymmetric trap.
The effective frequency v˜ i only modifies the density distri-
bution, since the center of mass still oscillates with the origi-
nal trap oscillation frequency v i. In addition, the solution
gives an outward momentum component due to the bosonic
FIG. 3. Growth of a BEC with a center-of-mass oscillation
present. Shown is the condensate density uC(x ,t)u2, as found from
the asymptotic solution in a symmetric trap, with a center-of-mass
oscillation in the x direction and N(0)510. The gain coefficient is
chosen as g50.1v , while a0,154.86 mm and d0,15p/2. Other pa-
rameter values are as previously.01360stimulation effect, which is stronger in the center of the trap.
The results obtained show that Bose stimulation can occur
to moving condensates as easily as to stationary ones. In
cases such as this, the amplitude of the center-of-mass oscil-
lation does not change during the condensate growth, while
the total mass of the condensate increases. This means that
the condensate center-of-mass kinetic energy, EK
5N(t)m( iv i2a0,i2 , can grow exponentially to large values,
even in the absence of technical noise. While the present
model of gain is simplified, similar types of center-of-mass
motion are found in first-principles simulations of conden-
sate formation @6#. This suggests that, while BEC’s formed
through stimulated emission may have a low temperature
relative to the center of mass, the center-of-mass motion may
itself have a higher temperature. Current experimental mea-
surements of single-atom velocity distributions @12# appear
insensitive to the center-of-mass temperature of the conden-
sate. The effect is analogous to the increased uncertainty in
the frequency and pointing stability of a multimode optical
laser, compared to a single-mode laser.
We emphasize that the asymptotic solutions given here
are only applicable for condensate nuclei that have already
formed, as the spontaneous-emission noise is omitted. The
gain medium is assumed to be unsaturated and to equilibrate
rapidly, giving a uniform gain constant across the growing
condensate. In addition, one can expect damping to occur
due to interactions with noncondensed atoms @13#, which are
not treated here. These interactions, however, can only
equilibrate the temperatures of condensed and noncondensed
center-of-mass motion, rather than providing additional cool-
ing of the condensate center-of-mass motion. Other damping
mechanisms that will intrinsically be present in a BEC are
two-body losses, which effectively change g to g2GuCu2,
where G is the two-body loss rate. While these affect the
form of the asymptotic solution, two-body ~or higher-order!
losses do not change the amplitude of the condensate center-
of-mass oscillation.
In summary, we have found an asymptotic solution to the
Gross-Pitaevskii equation with gain, which has the advan-
tage of yielding an explicit analytic result of great physical
transparency. The solution shows that the nonequilibrium be-
havior of a growing Bose-Einstein condensate generally in-
cludes an outward momentum component and spatial oscil-
lations. As a result, we suggest that the state of a trapped
BEC is not a canonical ensemble, and should be character-
ized by at least two distinct temperatures: one for the internal
degrees of freedom and one for the center-of-mass motion.
ACKNOWLEDGMENT
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Asymptotic solutions to the Gross-Pitaevskii gain equation: Growth of a Bose-Einstein condensate

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