Journal of Experimental and Theoretical Physics, Vol. 98, No. 5, 2004, pp. 908–917.Translated from Zhurnal Éksperimental’no

œ

i Teoretichesko

œ

Fiziki, Vol. 125, No. 5, 2004, pp. 1041–1051.Original Russian Text Copyright © 2004 by Kamchatnov.

ATOMS, SPECTRA, RADIATION

Expansion of Bose–Einstein Condensates Confinedin Quasi-One-Dimensional or Quasi-Two-Dimensional Traps

A. M. KamchatnovInstitute of Spectroscopy, Russian Academy of Sciences, Troitsk, Moscow oblast, 142190 Russia

e-mail: kamch@isan.troitsk.ruReceived October 31, 2003

Abstract—Solutions to the Gross–Pitaevskii equations are obtained in the hydrodynamic approximation for arepulsive Bose gas that expands after a quasi-one-dimensional or quasi-two-dimensional trap is removed. Theresults are expressed in terms of measurable parameters, such as the initial condensate size and the oscillationfrequencies of trapped particles. Three-dimensional effects are calculated by a variational method. The analyt-ical results are in good agreement with available experimental data. © 2004 MAIK “Nauka/Interperiodica”.

1. INTRODUCTION

The properties of a Bose–Einstein condensate inwhich particle motion is “frozen” or reduced to zero-point oscillations in one or two directions are the sub-ject of intensive studies [1–14]. In experiments, cigar-shaped quasi-one-dimensional condensates are createdby using optical dipole traps [1]. A quasi-two-dimen-sional condensate was created in an array of disc-shaped traps provided by the periodic potential of alaser beam [2]. When the traps are sufficiently deep, themotion along the array is frozen and the condensatesplits into several independent condensates confined inseparate potential wells.

Important experimental information about the prop-erties of a Bose–Einstein condensate confined in athree-dimensional trap can be extracted by measuringthe time-dependent density of the expanding atomiccloud after the trapping potential is switched off. In themean-field approximation, the dynamics of a dilutecondensate is described by the Gross–Pitaevskii equa-tion [14]

(1)

where

is the trapping potential,

(2)

is the nonlinear coupling constant associated with anatom–atom scattering length as , and the condensatewave function ψ is normalized to the number of atoms

(3)

i"∂ψ∂t------- "

2

2m-------∆ψ– Vext r( )ψ g ψ 2ψ,+ +=

V ext r( )12---m ωx

2x2 ωy2y2 ωz

2z2+ +( )=

g 4π"2as/m=

ψ 2 rd∫ N .=

1063-7761/04/9805- $26.00 © 20908

If the number of atoms is sufficiently large, then theGross–Pitaevskii equation can be transformed intohydrodynamic equations that admit simple self-similarsolutions describing both oscillations of a gas in a par-abolic trapping potential and its free three-dimensionalexpansion after the potential is switched off [15–18].This theory is perfectly consistent with experiment.

A different situation arises when some degrees offreedom of the expanding condensate remain frozen.Recently, condensate expansion was investigated inquasi-one-dimensional waveguides [1] and in systemsof two-dimensional discs [2]. This promising line ofresearch was pursued in several studies. In [19], quasi-one-dimensional condensate expansion was analyzedwithout taking into account the transverse “quantumpressure.” In [20], the effects due to quantum pressurewere taken into account for steady states, in which caseonly the two transverse modes contribute to the pres-sure. In [13], the ground states of condensates confinedin cigar- and disc-shaped traps were calculated by avariational method, but no analysis of the dynamics ofcondensate expansion was presented.

In this paper, an analytical study of quasi-one-dimensional and quasi-two-dimensional condensateexpansion is presented. Conditions are formulatedunder which the three-dimensional Gross–Pitaevskiiequation can be reduced to analogous equations infewer coordinates. These equations are solved in thehydrodynamic approximation under initial conditionscorresponding to a trapped condensate in equilibriumbefore the trap is switched off. The condensate expandseither along the axis of a quasi-one-dimensionalwaveguide or in the plane of a quasi-two-dimensionaltrap. However, if the conditions for reduction to Gross–Pitaevskii equations of lower dimension are violated,then the gas flow is three-dimensional. Three-dimen-sional effects in the flow are calculated by a variational

004 MAIK “Nauka/Interperiodica”

EXPANSION OF BOSE–EINSTEIN CONDENSATES 909

method. Finally, it is shown that the theoretical resultsagree with experiment.

2. QUASI-ONE-DIMENSIONAL AND QUASI-TWO-DIMENSIONAL

CONDENSATE EXPANSIONWITHOUT THREE-DIMENSIONAL EFFECTS

It is well known that the Gross–Pitaevskii equationcan be formulated as a principle of least action with theaction functional

(4)

where the Lagrangian density is

(5)

In the case of a cigar- or disc-shaped trap, one canreadily find conditions under which the tightlyrestrained degrees of freedom are frozen and theGross–Pitaevskii equation reduces to a one- or two-dimensional equation, respectively. Even though thisproblem has been considered more than once, webriefly review here the basic points of the derivation inorder to identify the essential parameters of the theoryand formulate conditions for its applicability.

2.1. One-Dimensional Expansion

If the longitudinal frequency ωz for an axially sym-metric trap is much less than the transverse trap fre-quency ω⊥ ,

(6)

and the transverse zero-point energy is much higherthan the nonlinear interaction energy per atom, then thetransverse motion reduces to the ground state of parti-cle oscillation, with the amplitude

Denoting by Z0 the characteristic size of the condensatealong the axis of a cigar-shaped trap, one can use theestimate

(see (3)) to write the corresponding condition as fol-lows (e.g., see [11]):

(7)

If this condition is satisfied, then the condensate wavefunction can be factorized:

(8)

S L t, Ld∫ + r,d∫= =

+i"2----- ψt*ψ ψtψ∗–( ) "

2

2m------- ∇ψ 2+=

+ V ext ψ 2 12---g ψ 4.+

λ ωz/ω⊥ ! 1,=

a⊥ "/mω⊥( )1/2.=

N ψ 2a⊥2 Z0∼

Nas/Z0 ! 1.

ψ r t,( ) φ x y,( )Ψ z t,( ),=

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHY

where

(9)

is the wave function of the ground state of transversemotion. Substituting (8) and (9) into (4) and (5) andintegrating the result over the condensate’s cross sec-tion, one obtains the action expressed in terms of theone-dimensional Lagrangian density

(10)

Then, the evolution of Ψ(z, t) obeys the one-dimen-sional Gross–Pitaevskii equation

(11)

where

(12)

is an effective coupling constant and Ψ is normalized as

(13)

Equation (11) determines the longitudinal dynamics ofa condensate in a cigar-shaped trap.

By the well-known substitution

, (14)

Eq. (11) is transformed into the system

(15)

(16)

In Eq. (16), the last term (“quantum pressure”) can beneglected if it is much smaller than the nonlinear term,i.e., if

(17)

Then, Eq. (16) reduces to

. (18)

φ x y,( )1

πa⊥

------------- x2 y2+

2a⊥2

----------------–

exp=

+1Di"2----- Ψt*Ψ ΨtΨ∗–( ) "

2

2m------- Ψz

2+=

+12---mωz

2z2 Ψ 2 g

4πa⊥2

------------ Ψ 4.+

i"Ψt"

2

2m-------Ψzz

12---mωz

2z2Ψ g1D Ψ 2Ψ,+ +–=

g1Dg

2πa⊥2

------------2"

2as

ma⊥2

-------------= =

Ψ 2 zd∫ N .=

Ψ z t,( ) ρ z t,( )im"

------ v z' t,( ) z'd

z

∫

exp=

ρt ρv( )z+ 0,=

v t vv z

g1D

m--------ρz ωz

2z+ + +

+"

2

2m2---------

ρz2

4ρ2--------

ρzz

2ρ------–

z

0.=

a⊥

Z0----- !

Nas

a⊥---------.

v t vv z

g1D

m--------ρz ωz

2z+ + + 0=

SICS Vol. 98 No. 5 2004

910 KAMCHATNOV

Combined with Eq. (15), it constitutes the hydrody-namic approximation describing the evolution of a con-densate.

The time-independent solution of the hydrodynamicequations is the well-known Thomas–Fermi distribu-tion of a one-dimensional condensate:

(19)

where the integration constant Z0 (longitudinal half-length of the condensate) can be expressed in terms ofthe number of atoms N as

(20)

Applicability conditions (7) and (17) for the one-dimensional hydrodynamic approximation can berewritten by substituting (20) as follows (see [11]):

(21)

Now, assume that the longitudinal trapping potentialis switched off and the condensate can freely expandalong the longitudinal axis. At the same time, it remainstransversely confined, and its transverse motionremains frozen in ground state (9). Accordingly, theexpansion can be described by hydrodynamic equa-tions (15) and (18) subject to initial conditions (19). Ananalogous problem in nonlinear optics was solved longago [22], with a “pressure” ρz in (18) having the oppositesign, and its solution was recently applied to describethree-dimensional condensate expansion [15–18]. Thisapproach is used here to analyze the case when the con-densate expands into a “waveguide.” A solution toEqs. (15) and (18) is sought in the form

(22)

where bz(t) and αz(t) satisfy the conditions

(23)

Substituting (22) into (15) and (18) yields

(24)

and the equation

(25)

for bz(t). The latter equation can easily be integrated toobtain an implicit formula for bz as a function of t:

(26)

ρ z( )3N4Z0--------- 1 z2

Z02

-----–

, v 0,= =

Z0 3Nasa⊥2 λ 2–( )1/3

.=

λ ! Nas/a⊥ ! 1/λ .

ρ z t,( )3N4Z0--------- 1

bz t( )---------- 1 z2

Z02bz

2 t( )----------------–

,=

v z t,( ) zα z t( ),=

bz 0( ) 1, α z 0( ) 0.= =

az t( ) bz t( )/bz t( ),=

bz ωz2/bz

2=

2ωzt bz bz 1–( )=

+12--- 2bz 1– 2 bz bz 1–( )+[ ]ln

JOURNAL OF EXPERIMENTAL

(see [23], where this solution was applied to describethe quasi-one-dimensional initial stage of the conden-sate expansion that follows after a disc-shaped trap isswitched off). The expression for α(t) in terms of bz(t)yields the velocity field:

(27)

The leading edge of the density distribution moves as

(28)

with the maximum velocity

(29)

At ,

(30)

the density and velocity distributions simplify to

(31)

and the maximum velocity tends to the constant value

(32)

These formulas describe inertial motion when the den-sity is so small that the nonlinear pressure does notaccelerate the gas any longer. Formula (32) is suitablefor comparison with experiment, because the asymp-totic value of the maximum velocity is expressed interms of measurable parameters: the longitudinal trapfrequency ωz and the initial half-width Z0 of the longi-tudinal Thomas–Fermi profile.

Expression (31) yields the asymptotic velocity dis-tribution

(33)

The mean kinetic energy is

(34)

2.2. Two-Dimensional Expansion

Two-dimensional condensate dynamics areobserved when the longitudinal trap frequency ωz is

v z t,( )2ωzzbz t( )

---------------- 1 1bz t( )----------– .=

zmax t( ) Z0bz t( )=

v max t( )dzmax

dt------------ 2Z0ωz 1 1

bz t( )----------– .= =

t @ ωz1–

bz t( ) 2ωzt, t @ ωz1– ,≈

ρ z t,( )3N

4v maxt---------------- 1 z2

v maxt( )2-------------------–

,≈

v z t,( )zt--, t @ ωz

1– ,≈

v max 2Z0ωz, t @ ωz1– .≈

ρ v( )dv3N

4v max-------------- 1 v 2

v max2

----------–

dv , v v max.≤=

Em

2N------- v 2ρ v( ) vd∫ 1

5---Emax,= =

Emax12---mv max

2 .=

AND THEORETICAL PHYSICS Vol. 98 No. 5 2004

EXPANSION OF BOSE–EINSTEIN CONDENSATES 911

much higher than the radial trap frequency ω⊥ , i.e.,when inequality (6) is replaced with the reverse one:

(35)

Now, assume that the motion along the z axis is frozen,i.e., the zero-point energy associated with the oscilla-tion amplitude az = ("/mωz)1/2 is much higher than thenonlinear energy. By virtue of the estimate

where R0 is the radius of the density distribution in theplane (x, y) of the trap, this condition leads to the ine-quality

(36)

If it holds, the condensate wave function can again befactorized:

(37)

where

(38)

is the longitudinal ground-state wave function. By sub-stituting (37) and (38) into (4) and (5) and integratingthe result over the longitudinal coordinate, the action isexpressed in terms of the effective two-dimensionalLagrangian density

(39)

The corresponding Euler–Lagrange equation is thetwo-dimensional Gross–Pitaevskii equation

(40)

where ∆⊥ = + is the transverse Laplace operator,g2D is an effective coupling constant expressed as

(41)

and Ψ is normalized as

(42)

Equation (40) describes the two-dimensional transversedynamics of a condensate in a disc-shaped trap.

λ ωz/ω⊥ @ 1.=

N ψ 2R02az,∼

Nas

az

--------- ! R0

az

-----

2

.

ψ r t,( ) φ z( )Ψ x y t, ,( ),=

φ z( )1

π1/4az1/2

---------------- z2

2az2

--------–

exp=

+2Di"2----- Ψt*Ψ ΨtΨ∗–( ) "

2

2m------- Ψx

2 Ψy2+( )+=

+12---mω⊥

2 x2 y2+( ) Ψ 2 g

2 2πaz

------------------- Ψ 4.+

i"Ψt"

2

2m-------∆⊥ Ψ–=

+12---mω⊥

2 x2 y2+( )Ψ g2D Ψ 2Ψ,+

∂x2 ∂y

2

g2Dg

2πaz

---------------2 2π"

2as

maz

-------------------------,= =

Ψ 2 xd yd∫ N .=

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHY

By the substitution

(43)

where r⊥ = (x, y) and v = (v x, v y), Eq. (40) is trans-formed into the system

(44)

(45)

where ∇ ⊥ = (∂x, ∂y) is the transverse gradient operator.The quantum pressure can be neglected if it is muchlower than the nonlinear pressure, i.e.,

(46)

Then, Eq. (45) reduces to the hydrodynamic equation

(47)

The time-independent solution of the hydrodynamicequations is the Thomas–Fermi profile

(48)

where r2 = x2 + y2 and the radius R0 of the density dis-tribution is determined by the number of atoms N:

(49)

This inequality can be used to rewrite inequality (36) ina more convenient form, and the applicability conditionfor the two-dimensional Thomas–Fermi approximationbecomes

(50)

After the transverse trapping potential is switchedoff, the condensate begins to expand radially, remain-ing bounded longitudinally. The radial expansion isdescribed by hydrodynamic equations (44) and (47)subject to initial conditions (48). Now, the solution issought in the form

(51)

where v is the radial velocity component, and b⊥ (t) and

Ψ r⊥ t,( ) ρ r⊥ t,( )im"

------ v r⊥' t,( ) r'd

r⊥

∫

exp ,=

ρt ∇ ⊥ ρv( )+ 0,=

vt v∇ ⊥( )vg2D

m--------∇ ⊥ ρ ω⊥

2 r+ + +

+"

2

2m2---------∇ ⊥

∇ ⊥ ρ( )2

4ρ2-----------------

∆⊥ ρ2ρ

----------– 0,=

1 ! Nas/az.

vt v∇ ⊥( )vg2D

m--------∇ ⊥ ρ ω⊥

2 r+ + + 0.=

ρ r( )2N

πR02

--------- 1 r2

R02

-----–

, v 0,= =

R016

2π----------Nasaz

3λ2

1/4

.=

1 ! Nas/az ! λ2.

ρ r t,( )2N

πR02

--------- 1

b⊥2 t( )

----------- 1 r2

R02b⊥

2 t( )-----------------–

,=

v r t,( ) rα⊥ t( ),=

SICS Vol. 98 No. 5 2004

912 KAMCHATNOV

α⊥ (t) satisfy the initial conditions

(52)

The substitution of (51) into (44) and (47) yields anequation relating b⊥ (t) to α⊥ (t),

and a differential equation for b⊥ (t),

(53)

The last equation is solved under the initial conditions

to obtain

(54)

and hence

Thus, simple expressions are obtained for the radialdensity and velocity distributions:

(55)

The leading edge of the radial density distributionmoves as

(56)

with the maximum velocity

(57)

At t @ ,

(58)

where

(59)

As in the one-dimensional case, these formulasdescribe inertial motion. Again, the maximum velocityis expressed in terms of measurable parameters, theradial frequency ω⊥ of the trap before it was switchedoff and the initial Thomas–Fermi radius R0.

b⊥ 0( ) 1, α⊥ 0( ) 0.= =

α⊥ b⊥ /b⊥ ,=

b⊥ ω⊥2 /b⊥

3 .=

b⊥ 0( ) 1, b⊥ 0( ) b⊥ 0( )α⊥ 0( ) 0= = =

b⊥ t( ) 1 ω⊥2 t2+ ,=

α⊥ t( ) b⊥ /b⊥ ω⊥2 t/ 1 ω⊥

2 t2+( ).= =

ρ r t,( )2N

πR02

--------- 1

1 ω⊥2 t2+

-------------------- 1 r2

R02 1 ω⊥

2 t2+( )-------------------------------–

,=

v r t,( )ω⊥

2 rt

1 ω⊥2 t2+

--------------------.=

rmax t( ) R0 1 ω⊥2 t2+=

v max t( )drmax

dt------------

R0ω⊥2 t

1 ω⊥2 t2+

------------------------.= =

ω⊥1–

ρ r t,( )2N

πv maxt2------------------ 1 r2

v maxt( )2-------------------–

,≈

v r t,( )rt--,≈

v max R0ω⊥ , t @ ω⊥1– .≈

JOURNAL OF EXPERIMENTAL

The asymptotic velocity distribution is

(60)

and the mean energy is

(61)

3. THREE-DIMENSIONAL EFFECTS: A VARIATIONAL APPROACH

If condition (7) or (36) is violated, then motionalong the smaller dimension of the condensate is notfrozen and must be taken into account. As shownin [24], this can be done by applying a simple varia-tional method. However, when the trap is highly aniso-tropic and the number of atoms in the condensate is suf-ficiently large, the trial distribution along the largerdimension should be the equilibrium Thomas–Fermiprofile, which differs substantially from the Gaussiandistribution assumed in [24]. This method was appliedto calculate the ground states of condensates in [13].Here, it is applied to condensate dynamics.

3.1. Cigar-Shaped Trap

In the case of a cigar-shaped trap with Thomas–Fermi axial density distribution, the variational conden-sate wave function has the form

(62)

where the parameters A, w⊥ , wz, α⊥ , and αz are functionsof time. It is assumed that Nas/Z0 ~ 1, i.e., the Thomas–Fermi limit radial profile is not reached and the radialwave function can be well approximated by a Gaussiandistribution. The parameter A is related to the widths w⊥and wz by normalization condition (3), which yields

(63)

Substituting (62) into (4) and (5) and integrating theresult, one obtains the averaged Lagrangian

(64)

ρ v( )dv4N

v max2

---------- 1 v 2

v max2

----------–

v dv ,=

E13---Emax, Emax

12---mv max

2 .= =

ψ Ar2

2w⊥2

----------–

1 z2

wz2

------–exp=

× i2--- α⊥

2 r2 α z2z2+( ) ,exp

A3N

4πw⊥2 wz

------------------- 1/2

.=

LN----

"2

2-----

dα⊥

dt---------- "

2

2m------- 1

w⊥4

------ α⊥2+

12---mω⊥

2+ + w⊥2=

+"

2

2-----

dα z

dt-------- "

2

2m-------α z

2 12---mω⊥

2 λ2+ + wz

2

5------

3Nas"2

5m------------------ 1

w⊥2 wz

------------,+

AND THEORETICAL PHYSICS Vol. 98 No. 5 2004

EXPANSION OF BOSE–EINSTEIN CONDENSATES 913

where the 1/ term is neglected. Indeed, this term is

much less than , because the estimates

imply that the condition 1/ ! αz is equivalent to thecondition λ1/2 ! Nas/a⊥ for the applicability of the Tho-mas–Fermi approximation in the longitudinal direction(see (21)).

Lagrangian (64) entails the well-known expressions

(65)

and the equations of motion for the widths

(66)

(67)

These equations differ from those derived in [24] bynumerical factors and by the absence of the term corre-sponding to longitudinal quantum pressure (it would beincorrect to retain this term in the approximationemployed here). In the dimensionless variables

(68)

Eqs. (66) and (67) are rewritten as

(69)

(70)

where the parameter

(71)

characterizes the radial nonlinear pressure. If q ! 1,then the second term on the right-hand side of (69) canbe neglected to obtain the time-independent solutionb⊥ 0 = bz0 = 1, which corresponds to the one-dimensionalapproximation (see Section 2). In this case, Eq. (70)describing free longitudinal expansion obviouslyreduces to Eq. (25).

The equilibrium values of b⊥ and bz are determinedby the equations

(72)

which differ from analogous equations obtained in [21]only by notation. They describe the state of the conden-sate before expansion.

wz4

α z2

wz Z0, α z m/"( )wz/wz mωz/" λ /a⊥2∼∼∼∼

wz2

α⊥m"---- 1

w⊥------

dw⊥

dt----------, α z

m"---- 1

wz

-----dwz

dt---------= =

w⊥ ω⊥2 w⊥+ "

2

m2w⊥3

-------------6Nas"

2

5m2------------------ 1

w⊥3 wz

------------,+=

wz λ2ω⊥2 wz+

3Nas"2

m2------------------ 1

w⊥2 wz

2-------------.=

b⊥ w⊥ /a⊥ , bz wz/Z0, τ ω⊥ t= = =

d2b⊥

dτ2----------- b⊥+ 1

b⊥3

-----2q5

------ 1

b⊥3 bz

----------,+=

d2bz

dτ2---------- λ2bz+

λ2

b⊥2 bz

2-----------,=

q λZ0/a⊥( )2=

b⊥ 0 bz0–3/2,

1

bz06

------ 12q5

------ 1bz0------,+= =

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHY

The condensate expansion that follows after the lon-gitudinal potential is switched off is described by theequations

(73)

They can readily be solved numerically under the initialconditions

(74)

where b⊥ 0 and bz0 are determined by (72). Figure 1shows the functions of time

,

where Z = Z0bz0 is the initial half-width of the conden-sate, for several values of N. When

,

we have the analytical solution obtained in Section 2. If

then the expansion starts from a radius larger than thezero-point oscillation amplitude a⊥ , and b⊥ 1 asτ ∞. At τ @ 1, the longitudinal expansionapproaches an inertial motion characterized by constantvelocities of atoms. The maximum velocity is readilyfound by using the conservation law corresponding toEq. (73):

(75)

At τ = 0, it yields the initial values in (74). As τ ∞,

and therefore

(76)

Then, Eqs. (72) are used to find

(77)

When q ! 1, i.e., b⊥ 0 = 1 and Z = Z0, this expression

d2b⊥

dτ2----------- b⊥+ 1

b⊥3

-----2q5

------ 1

b⊥3 bz

----------,d2bz

dτ2----------+

λ2

b⊥2 bz

2-----------.= =

b⊥ 0( ) b⊥ 0, b⊥ 0( ) 0,= =

bz 0( ) bz0, bz 0( ) 0,= =

b⊥ w⊥ /a⊥ , zmax Z0bz t( ) Zbz t( )/bz0= = =

λa⊥ /as ! N ! a⊥ / λas( )

N a⊥ / λas( ),>

12---

dbz

dτ--------

2 5λ2

2q--------

db⊥

dτ---------

2

b⊥2 1

b⊥2

-----+ ++

+λ2

b⊥2 bz

---------- const.=

dbz/dτ bz max, , b⊥ 1, bz ∞,

bz max, λ 2

b⊥ 02 bz0

---------------5q--- b⊥ 0

1b⊥ 0-------–

2

+ .=

v maxdzmax

dt------------

t ∞→

2Zωzb⊥ 0

1 b⊥ 02+

---------------------.= =

SICS Vol. 98 No. 5 2004

914 KAMCHATNOV

0 1 2 3 4 5 6tωz

1.1

1.2

1.3

1.4

b⊥

(a) (b)

1

2

3

0 1 2 3 4 5 6tωz

0.25

0.50

0.75

1.00

1.25

1.50

zmax, mm

1

2

3

Fig. 1. Radius b⊥ = w⊥ /a⊥ (a) and longitudinal size zmax (b) vs. time τ = tωz for a one-dimensional expanding condensate in a cigar-

shaped trap at a⊥ = 5 µm, as = 5 nm, and λ = 0.05. Curves 1, 2, and 3 correspond to N = 103, 104, and 105, respectively.

yields the one-dimensional velocity given by (32).When q @ 1, i.e., b⊥ 0 @ 1, it reduces to

Thus, not only the longitudinal condensate sizeincreases, but also the ratio of vmax to ωzR increases

from to 2 as the number of atoms in the condensateincrease.

Both asymptotic velocity distribution and expres-sion for the mean energy retain their form given by (33)and (34), respectively, where vmax is now given by (77).Figure 2 illustrates the dependence of the mean energyon N.

3.2. Disc-Shaped Trap

In the case of a disc-shaped trap with Thomas–Fermi radial density distribution, the variational wavefunction has the form

(78)

where the time-dependent parameters A, w⊥ , wz , α⊥ ,and αz are related by normalization condition (3):

(79)

Substituting (78) into (4) and (5) and integrating the

v max 2ωzZ .≈

2

ψ A 1 r2

w⊥2

------– z2

2wz2

---------–

exp=

× i2--- α⊥

2 r2 α z2z2+( ) ,exp

A2N

π3/2-------- 1

w⊥2 wz

------------ 1/2

.=

JOURNAL OF EXPERIMENTAL A

result, one obtains the Lagrangian

(80)

where the 1/ term is neglected, because this term is

LN----

"2

2-----

dα⊥

dt---------- "

2

2m-------α⊥

2 12---mω⊥

2+ + w⊥

2

3------=

+"

2

2-----

dα z

dt-------- "

2

2m------- 1

wz4

------ α z2+

12---mω⊥

2 λ2+ +wz

2

2------

+8

3 2π--------------

Nas"2

m--------------- 1

w⊥2 wz

------------,

w⊥4

2 × 1040 4 × 104 6 × 104 8 × 104 105

N

5

10

15

20

E/"ωz–

Fig. 2. Mean energy of atoms in a condensate (measured inenergy quanta "ωz of longitudinal oscillations) after one-dimensional expansion in a cigar-shaped trap vs. number ofatoms.

ND THEORETICAL PHYSICS Vol. 98 No. 5 2004

EXPANSION OF BOSE–EINSTEIN CONDENSATES 915

0 1 2 3 4 5 6tω⊥

(a)

0.1

0.2

0.3

0.4

0.5

0.6

Rmax, mm

1

2

3

0 1 2 3 4 5 6tω⊥

1.02

1.04

1.06

1.08

1.10(b)

bz

1

2

3

Fig. 3. Radius Rmax (a) and longitudinal size bz = wz/az (b) vs. time τ = tω⊥ for a two-dimensional expanding condensate in a disc-

shaped trap at az = 5 µm, as = 5 nm, and λ = 20. Curves 1, 2, and 3 correspond to N = 103, 104, and 105, respectively.

much less than by virtue of the applicability condi-tion Nas/az @ 1 for the radial Thomas–Fermi approxi-mation.

Lagrangian (80) entails expressions (65) and theequations of motion for the widths

(81)

(82)

which are rewritten in the dimensionless variables

(83)

where az = a⊥ / , as

(84)

(85)

The parameter

(86)

characterizes the longitudinal nonlinear pressure. Ifq ! 1, then the second term on the right-hand sideof (85) can be neglected to obtain the time-independentsolution

α⊥2

w⊥ ω⊥2 w⊥+

16

2π----------

Nas"2

m2--------------- 1

w⊥3 wz

------------,=

wz λ2ω⊥2 wz+ "

2

m2wz3

------------16

2π----------

Nas"2

m2--------------- 1

w⊥2 wz

2-------------,+=

b⊥ w⊥ /R0, bz wz/az, τ ω⊥ t,= = =

λ

d2b⊥

dτ2----------- b⊥+

1

b⊥3 bz

----------,=

d2bz

dτ2---------- λ2bz+ λ2

bz3

-----λ2q3

-------- 1

b⊥2 bz

2-----------.+=

q R0/λaz( )2=

b⊥ bz 1,= =

JOURNAL OF EXPERIMENTAL AND THEORETICAL PHY

which corresponds to the two-dimensional approxima-tion considered in Section 2. In this case, Eq. (84) obvi-ously reduces to (53).

The equilibrium values of b⊥ and bz are determinedby the equations

(87)

which again are identical to the equations obtainedin [21], except for notation. They describe the state ofthe condensate before expansion. The expansion in theplane of the trap is described by the equations

(88)

They can readily be solved numerically under initialconditions (74), where b⊥ 0 and bz0 are determinedby (87). Figure 3 shows the functions of time

where R = R0b⊥ 0 is the initial radius of the condensate,for several values of N. When

,

we have the analytical solution obtained in Section 2. If

then the expansion starts from a longitudinal size wz0larger than the zero-point oscillation amplitude az (i.e.,bz0 > 1), and bz 1 as τ ∞. At τ @ 1, the radialexpansion approaches an inertial motion characterizedby constant velocities of atoms. The maximum velocity

bz0 b⊥ 04– ,

1

b⊥ 016

------- 1q3--- 1

b⊥ 06

-------,+= =

d2b⊥

dτ2-----------

1

b⊥3 bz

----------,d2bz

dτ2---------- λ2bz+ λ2

bz3

-----λ2q3

-------- 1

b⊥2 bz

2-----------.+= =

Rmax Rb⊥ t( )/b⊥ 0, bz wz/az,= =

az/as ! N ! azλ2/as

N azλ2/as>

SICS Vol. 98 No. 5 2004

916 KAMCHATNOV

is found by using the conservation law corresponding toEq. (88):

(89)

which yields

(90)

Accordingly,

(91)

When q ! 1, i.e., bz0 = 1 and R = R0, this expressionyields the two-dimensional velocity given by (59).When q @ 1, i.e., bz0 @ 1, it reduces to

Both the asymptotic velocity distribution and expres-sion for the mean energy retain their form given by (60)and (61), respectively, where vmax is now given by (91).Figure 4 illustrates the dependence of the mean energyon N.

db⊥

dτ---------

2 3

λ2q-------- 1

2---

dbz

dτ--------

2 λ2

2----- bz

2 1

bz2

-----+ ++

+1

b⊥2 bz

---------- const,=

b⊥ max,1

b⊥ 02 bz0

---------------3

2q------ bz0

1bz0------–

2

+ .=

v maxdRmax

dt--------------

t ∞→

ω⊥ R1 3bz0

2+

2 1 bz02+( )

------------------------.= =

v max 3/2ω⊥ R.≈

2 × 1040 4 × 104 6 × 104 8 × 104

1

2

3

4

N

E/"ω⊥–

Fig. 4. Mean energy of atoms in a condensate (measured inenergy quanta "ω⊥ of longitudinal oscillations) after two-dimensional expansion in a disc-shaped trap vs. number ofatoms.

JOURNAL OF EXPERIMENTAL A

4. DISCUSSION

Let us compare the theoretical results with availableexperimental data.

An experimental study of condensate expansion in aquasi-one-dimensional trap was reported in [1], wherethe longitudinal half-length of the condensate was Z ≈100 µm and the time of transition to inertial motion wasapproximately 20 ms, which corresponds to the trapfrequency wz ≈ 50 s–1. Under these conditions, expres-sion (32) yields

In view of the assumptions made, good agreementbetween this estimate and the experimental valuevmax ≈ 5.9 mm/s is achieved without introducing anyadjustable parameters.

Expansion of a Bose–Einstein condensate in a sys-tem of two-dimensional disc-shaped traps was investi-gated in [2]. The results of that study showed that nolongitudinal expansion took place, and the condensatewas effectively divided into separate condensate “pan-cakes” confined in separate potential wells. Therefore,the two-dimensional theory can be applied to describethe radial expansion of each particular condensate.According to [2], the maximum radial velocity wasvmax ≈ 1.5–1.7 mm/s when the initial radius was R ≈13 µm and the radial trap frequency was ω⊥ ≈ 132 s–1.Expression (59) predicts

which is in good agreement with the experimentalvalue. Thus, the mean-field theory provides a gooddescription of condensate expansion in fewer coordi-nates as well. Deviations from the theoretical predic-tions would point to the existence of a condensate thatcould not be described by the mean-field theory.

ACKNOWLEDGMENTS

I thank F.Kh. Abdullaev, V.A. Brazhnyi, A. Gammal,V.V. Konotop, R.A. Kraenkel, and L. Tomio for valu-able discussions. This work was supported by theRussian Foundation for Basic Research, projectno. 01-01-00696.

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Translated by A. Betev

ICS Vol. 98 No. 5 2004