Stability of Atomic Bose-Einstein Condensate with Negative Scattering Length

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Stability of Atomic Bose-Einstein Condensate with Negative Scattering Length

Weiping Zhang, B. Sanders, A. Mann

To cite this version:

Weiping Zhang, B. Sanders, A. Mann. Stability of Atomic Bose-Einstein Condensate with Neg- ative Scattering Length. Journal de Physique I, EDP Sciences, 1996, 6 (11), pp.1411-1415.

�10.1051/jp1:1996154�. �jpa-00247254�

Stability of Atomic Bose-Einstein Condensate with Negative Scattering Length

Weiping Zhang (~),

B-C- Sanders

(~)

^{and A. Mann} (~~~,*)

(~) ^{School of} Mathematics,

Physics, Computing

and Electronics,

and Centre for Lasers and

Applications, Macquarie University,

North

Ryde,

New South Wales 2109, Australia

(~)

Department

of

Physics,

Technion-Israel Institute of

Technology,

Haifa 32 o00, Israel

(Received

26

February1996,

revised 24 June1996,

accepted

1

August 1996)

PACS.42.50.Vk Mechanical effects of

light

_on atoms, molecules, electrons and ions PACS.32.80.-t Photon interactions with atoms

Abstract. We discuss the

stability

of

a small-scale Bose-Einstein condensate with nega- tive scattering

length.

The stable state is found

by obtaining

^the appropriate ^basis using self-

consistent field theory ^for an attractive one-dimensional

potential.

Bose-Einstein Condensation in dilute atomic gases has been reaJized for

8~Rb Ill,

for

~Li

_{[2j and} for

~~Na [3j.

^{In the latter} case a condensate has formed

despite

the presence of _an attractive

potentiaJ. Although suprising

at

first,

Bose-Einstein Condensation is

possible

in attractive

potentials.

Here _we determine the

appropnate

basis states for _an attractive one-dimensional

potential

and _use this basis set to show that _a stable Bose-Einstein Condensate with

negative scattering length

_can indeed form.

A nonlinear Hamiltonian for _an atomic beam in _an attractive à-function

potential

^is _given

by

_[4j

É

=

-j /

_dz

)t(xj$il(xj

+

/

_dz

_dx' ôt(xjilt _{(xl jV(x} x')il(x'jil(xj, (ij

where

/(x)

_{is the atomic} field annihilation

operator,

^and

V(x

-x' _is the nonlinear

dipole-dipole

interaction

arising

in the

study

_on nonlinear atom

optics [4, 5j.

^The

potential

is assumed to extend _{over a range} of the order of the

optical wavelength;

as this scale _is much smaller than the interatomic

separation

^for a dilute _{gas, we can}

employ

the

pseudopotential approximation

_[6]

vjx xl

t

-xôjx ^xl j. j2j

The

pseudopotential approximation

^is

justified

for _a dilute _gas where the

density

^{of the} _gas n

and the _s-wave

scattering length

a

satisfy

the condition

Ra~

_< 1.

(3)

(*)

Author for

correspondence (e-mail: ady@physics.technion.ac.il)

@

Les

Éditions

_de

Physique

1996

1412 JOURNAL DE

PHYSIQUE

I N°11

For N atoms in the

condensate,

the field

operators

can be

decomposed

_as

Î~(X)

"

~7b(X)àb

+

à~Ç7~(X)â~

(4)

where the functions

qJb(x)

and

(qJ~(x))

_are

eigenfunctions

of the self-consistent nonlinear

Schrôdinger equation

Î~ _~~~

~~~~~~~~~~~~~

~~

_~~~

obtained

by applying

the

time-independent

Hartree-Fock

approximation

to

equation il)

with the

pseudopotential (2).

The function

I?i

qJb(x)

=

fisech ~

(N -1)xxi ⁽⁶⁾

2 2

is an

eigenstate

of the nonlinear

equation

with

corresponding eigenvalue

Eb

=

ilN _{1)xi~ /8,} j7)

and trie _set of _states

~~~~~

fi~~~~ _~~~

l

)XX /2j

i~

i ₊ i~

e~~~'/-iix~/2

_~

are

eigenstates

of trie modified _version of

(5),

with

corresponding eigenvalues E~

=

(N

1)~x~~~/8,

where trie effective

density

_)qJbÎ~

replaces

trie exact

density

_)qJ)~

iii.

Trie

eigenfunc-

tion

qJb(x)

is trie

unique

bound _state for trie

system

with _a

negative

_energy value.

Conversely,

trie

eigenfunctions (qJ~(x)), corresponding

to trie continuum index ~, possess

nonnegative

_en-

ergy values. Trie _{energy gap} betweenlhe bound state and trie continuum mdicates that _a stable state is

possible,

in contradistinction to trie _momentum

eigenstate

_expansion of

/(x)

where trie zero-momentum

eigenstate

_is unstable due _to trie presence of trie attractive

poten-

tial.

(This instability

is manifested

by

trie _{appearance of}

an

imaginary

excitation spectrum in the

Bogoliubov approximation

for this

case.)

The

decomposition (4)

aJlows the Hamiltonian

É

_{to be rewritten in terms of the} annihilation operators

âb

and

â~

and their

conjugates à)

and

âÎ.

_The

_Bogoliubov approximation

_[8] involves

replacing âÎ

and

âb by @, _replacing

_N-1

_{by N,}

_and

_discarding

lower-order _{terms with}

respect

to N for

sufficiently large

N.

Thus,

the Hamiltonian

(1)

reduces to

Éred

=

NEb

+

/ _d~E~âlâ~

+

xN~ )qJb(x))~dx

2

~~

d~

dl

(E$~, ^â~â~,

2

+(E$~, )"âÎâÎ,

₊

_2E[~,âlâ~, ), (9j

"~~~~~

Bj~,

=

/

_dz

_jç7llx)l~

v7~lx)v7~,lx) _~~°~

and

E[~,

"

/ dz1i7b(xji~ ç7i(xjç7~,ixj. Ill)

are the collision

coefficients,

which _can be

written, using equations (6, 8),

_as

~~~' ÎÎ /

~~~~~~Î~Î~

^~~~

^~~~Î~ÎÎ~'

^~~~~

~ÎI~~ÎÎÎI'

~~~~

Integrating (12) produces

trie

expression

_I?1

~~~' ~Î ÎÎ/~Î~ÎÎ/Î~ÎÎ' _sinh[(Î Î _Î)r

/2]

^~~~~

which is

sharply peaked

at ~ + ~'. Trie

major

contribution to trie Hamiltonian

(9)

is from terms for which _~

= +~'.

Trie reduced Hamiltonian is still not amenable to an exact

analytic

solution.

However,

trie Hamiltonian _can be

separated

into two

components: (a)

trie

approximate

^{reduced Hamiitoman} which retains

only

trie dominant

diagonal

terms ~ = ~' and trie dominant

ofl-diagonal

terms ~ =

-~' and

(b)

trie residual terms which _are treated later. Trie

approximate

^reduced Hamiltonian is thus written _as

ÉS~~~

"

Êb

₊

/

_d~

_{lE~ 91~)1àlà~} /

_d~

_91~)

(à~à-~

⁺

àlà~

~j li~)

Î Î

~°~

E~

=

NE~

+

1~N2 / jç~~j~)j4dx. (15)

2

~~~~

~~~~

Î~Î~ ÎÎÎ

~~~~

With

respect

to the

approximate

^reduced Hamiltonian

(14),

the

integrals

can be restricted to

nonnegative

_~

by rewriting

^the

expression

_as

É]fl~~°~

=

Êb

₊ ^~ d~

[E~ g(~)] âlâ~

₊

âÎ _~â-~

^~ _d~

_{g(~) â~â-~}

₊

âÎâÎ

~

(17)

Using

the

Bogoliubov

transformation [8],

Î~

=

u~â~ ~~âÎ

~

(18)

and its

conjugate

expression, we obtain the final _energy

spectrum

E]$~~°~

=

~/[E~ ^{Eb g(x)]~ [g(~)]~}

ç~

~~X~ ~/(~~ ~)

~

~j~ ~)~2

₊ 3~~4_

(ig)

4/à

The reduced _energy

(19)

is a

positive

^{real number} in the limit that _{~ -} 0. The excitation

spectrum

^contains a finite _gap which _anses from the bound state nature of the

condensate;

therefore,

at

sulliciently

low

temperatures,

^{the atoms will}

stay

in the condensate until the

temperature

increases to the order of the gap. This _is similar to the situation in the BCS

theory

^of

superconductivity

where the finite gap

prevents

energy

dissipation.

Of _course trie

approximate

^reduced Hamiltonian

(14) neglects

^{the residual terms in the} re-

duced Hamiltonian

(9).

The _reason for

neglecting

^{these terms is that} a resonance is evident

1414 JOURNAL DE

PHYSIQUE

I N°11

in the collision coefficients

(10)

and

(11).

The dominant contribution _to the collision coeffi- cient

(13)

arises for _~

= +~'. The residual terms will

modify

trie size of trie _gap.

However,

trie

Bogoliubov approach provides

_a

good approximation

for _trie

ground

and low-excited

states, and,

_as trie exact Hamiltonian bas _{a gap,} trie _terms

neglected

in going from

equations il, ₂₎

_to

equation (9)

do not

substantially

alter trie value of trie gap. Trie modification _to trie size of trie gap caused

by

trie

neglected

residual terms is

certainly

of interest but not amenable _{to ana-}

lytical

calculation.

Therefore,

trie final _energy

spectrum (19)

_represents trie closest calculation achievable

by analytic

_means.

At low

temperatures,

the condensate formed

by putting

the atoms in the bourra state

(6)

is indeed

stable,

and _an attractive

potential

does allow the existence of _a stable Bose conden-

sate.

However,

this bound state condensate _can be very diflerent from the condensate for the

field in the absence of _{an attractive}

potential

_or in the _presence of _a

repulsive potential.

In the attractive interaction à-function

potential

the zero-momentum state is not _a stable Bose

condensate;

instead the stable _state is

given by equation (6). Creating

_a stable Bose-Einstein condensate with the attractive interaction would require

'cooling'

the atoms into such _{a state.}

Once the

temperature drops

below the _{gap energy, a} swift condensation into this bound _state condensate may be

expected.

It is

interesting

to contrast this

stability

for _any number N of atoms in the condensate with trie result of

Baym

and Pethick [9]. ^Their solution is stable for N restricted between 0 and _a

specific

_upper bound. Trie diflerent result is _a consequence of trie diflerent

approaches adopted

here and in reference [9]. ^In reference _{[9] the}

three-dimensional trap potential

is assumed to be the dominant

potential

and is used to determine the nature of the

macroscopic

wavefunction of the condensate. In contrast _our

approach,

which

employs

_a one-dimensional self-consistent field theoretic

technique, yields

exact solutions of the

Schrôdinger equation:

the resultant

wavefunction _narrows in space and the energy gap increases with

N~.

_This

ensures that the

wavefunction chosen here is stable for all N.

Extending

this

analysis

to the three-dimensional version of the self-consistent nonlinear

Schrôdinger equation (5)

would be

interesting,

but the

three-dimensional _{case is} much _more

complicated

where _even issues such _as the existence of _a stable sohton solution itself _must be considered

[loi.

Whereas the above results _are

obviously

tied to the

specific

interaction

being considered,

the

qualitative

results should holà for

general

attractive one-dimensional

potentials.

The restriction here _{to one} dimension _is motivated

by

the treatment of _a confined Bose _{gas as} treated in

reference [4]. ^For an attractive interaction the self-consistent Hartree-Fock

equations

will

yield

a lowest

single-partiale state,

and the Hartree state obtained

by putting

all the _atoms into this

single partiale

state will form _{a stable} condensate with _a real excitation spectrum

containing

a finite energy gap. This will support the condensate

against breakup

at

sufficiently

low

temperatures

and may cause a swift condensation when the

temperature

_is lowered below the gap energy.

Acknowledgments

This research has been

supported by

_a

Macquarie University

Research

Grant, by

the Fund for

Promotion of Research at the

Technion, by

the Technion VPR

Fund,

and

by

GIF German-

Israeh Foundation for Research and

Development.

WZ is

supported by

_an Austrahan Research Council Research

Fellowship.

ILeferences

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M.H.,

Ensher

J-R-,

Matthews

M.R.,

Wieman C.E. and Cornell

E-A-,

Science 269

(1995)

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Bradley C.C.,

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C.A.,

Tollett J-J- and Hulet

R-G-, Phys.

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1687.

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K-B-,

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M.-O.,

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M.R.,

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D.S.,

Kurn D.M. and Ketterle

W., Phys.

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[4]

Zhang W., Phys.

Lett. A 176

(1993) 225; Zhang W.,

Walls D.F, and Sanders

B.C., Phys.

Reu. Lett. 72

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60.

[5] Lenz

G., Meystre

P, and

Wright E-M-, Phys.

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Huang K.,

Statistical Mechanics

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New

York, 1963).

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G, and Pethick

C.J., Phys.

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Rubenchik A.M., and Zakharov

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103:

particularly equation (149)

and discussion.