Gardiner's Phonon for Bose-Einstein Condensation: a Physical Realization of The

Home , Bose gas

Gardiner's phonon for Bose-Einstein Condensation: A physical realization of the

q -deformed Boson

Chang-Pu Sun , Sixia Yu

Institute of theoretical Physics, Academia Sinica, Beijing 100080, P.R. China

Yi-Bao Gao

Department of Physics, Beijing Polytechnic University, Beijing 100022, P.R. China

The Gardiner's phonon presented for a particle-numb er conserving approximation metho d to de-

scrib e the dynamics of Bose-Einstein Condesation (BEC) (C.W. Gardiner, Phys. Rev. A 56, 1414

(1997)) is shown to b e a physical realization of the q -deformed b oson, whichwas abstractly devel-

op ed in quantum group theory. On this observation, the coherent output of BEC atoms driven bya

radio frequency (r.f ) eld is analyzed in the viewp ointofa q-deformed Fock space. It is illustrated

that the q -deformation of b osonic commutation relation corresp onds to the non-ideal BEC with the

nite particle number N of condensated atoms. Up to order 1=N , the coherent output state of the

untrapp ed atoms minimizes the uncertainty relation like a coherent state do es in the ideal case of

BEC that N approaches in nityor q =12=N approaches one.

tions for the Gardiner's phonon op erators will no longer I. INTRODUCTION

ob ey the commutation relation of the Heisenb erg-Weyl

algebra but the q -deformed b osonic commutation rela-

Since the intro duction of the concept of q -deformed b o-

tion

son [1] by di erent authors indep endently in the develop-

y y y

ment of the quantum group theory [2{4], there have b een

b; b bb qb b =1; (1)

many e orts of nding its physical realizations [5,6], e.g.,

where the deformation constant q dep ends on the total

ttings the deformed sp ectra of rotation and oscillation

atomic particle numb er. By using this q -deformed bo-

for molecules and nuclei [7,8]. In our opinion, those in-

son algebra we will discuss the e ects of the niteness of

vestigations can b e regarded as merely phenomenological

the particle numb er on the output of MIT exp erimentof

b ecause a q -deformed structure is p ostulated in advance

BEC.

without giving it a microscopic mechanism. In this pap er

This pap er is organized as follows. In sec.I I after

it will b e shown that a physical and natural realization

the reformulation of the Gardiner's mo di ed Bogoliub ov

of the q -deformed b oson is provided by the phonon op er-

metho d for the two-level atoms, the q -deformed commu-

ators, whichwas presented recently by Gardiner [10] for

tation relation for the Gardiner's phonon op erator is de-

the description of Bose-Einstein condensation (BEC).

rived in case of large but nite particle numb er. The

As well known, to deal with the dynamics of Bose-

excitation of the Gardiner's phonons is also considered

Einstein condensated gas the Bogoliub ov approximation

for small N . In sec.I I I, the Hamiltonian describing the

in quantum many-b o dy theory is an ecient approach,

coherent output in MIT exp eriment for atomic laser [13]

in which the creation and annihilation op erators for con-

is discussed in terms of the q -deformed algebra. When

densated atoms are substituted by c-numb ers. One short-

q = 1 or N ! 1 it describ es the coherent output of

coming of this metho d is that the total atomic particle-

BEC from a vacuum to a coherent states. For general q ,

number may not b e conserved after the approximation.

or nite but large condensated atomic particle numb er,

Or a symmetry may b e broken. To remedy this default,

it rusults in a nonlinear dynamic Heisenb erg equation

Gardiner suggested a mo di ed Bogoliub ov approxima-

for the q -deformed b osonic op erators. Its solution at the

tion byintro ducing phonon op erators which conserve the

rst order of 1=N is analyzed. In sec.IV dressed phonon,

total atomic particle number N and ob ey the b osonic

a generalization of the Gardiner's phonon in case of quan-

commutation relations in case of N !1. In this sense

tized radiation eld, is disscussed and it satis es also a

this phonon op erator approach gives an elegant in nite

q -deformed b osonic algebra for a large phonton number

atomic particle number approximation theory for BEC

and atomic particle numb er. Finally there are some con-

taking into account the conservation of the total atomic

cluding remarks in the last section.

numb er.

What will be investigated here is the case that the

total atomic particle number N is very large but not

II. Q-DEFORMED BOSONIC ALGEBRA FOR

in nite. That is, we shall consider the e ects of order

THE GARDINER'S PHONON

o(1=N ). And we shall fo cus on an algebraic metho d of

treating the e ects of nite condensated particle number

Gardiner's starting p ointtointro duce the phonon op-

in atomic BEC. As it turns out, the commutation rela-

erators is to consider a system of the weakly interacting 1

Bose gas. Without losing generality,we consider for the with q = 1 2 . This is exactly a typical q -deformed

momenta classical radiation eld interacting with two- commutation relation. As N !1 or q ! 1, the usual

level atoms with b and b denoting the creation and an- commutation relation of Heisenb erg-Weyl algebra is re-

nihilation op erators for the atoms in the untrapp ed state gained.

and b and b for the creation and annihilation op erators Since the Gardiner's phonon excitation for BEC can

of the atoms in the trapp ed ground state. The simpli ed b e mathematically understo o d in terms of q -deformed b o-

Hamiltonian for the MIT exp eriment reads son system, the algebraic metho d previously develop ed in

quantum group theory can b e of use for the construction

y y

H =h! b b +hg (t)b b +h:c:; (2)

e e g

e e

of a q -deformed Fock space. The existence of a vacuum

state j0i such that bj0i = 0 is guaranteed by the funda-

where g (t) is a time-dep endent coupling co ecient for the

mental structure of the q -deformed b oson algebra [14].

classical sweeping r.f. eld coupled to those two states

Starting with the vacuum, from relation

with level di erence h! . Note that the total atomic

n n

y y

y n y yn1

particle number N = b b + b b is conserved. For con-

e g

bb = q b b + hnib ; (6)

e g

venience we de ne =1=N for large particle numb er.

where the q -deformed c-number hni = n n(n 1) )

In the thermo dynamical limit N ! 1, the Bogoli-

is up to the rst order of and approaches to n when

ub ov approximation is usually applied, in which the lad-

N ! 1, the q -deformed Fock space for the Gardiner's

der op erators b ;b of the ground state are replaced by

phonon excitation of BEC is spanned by bases

a c-number N , where N is the average number of

c c

the ininital condensated atoms. As a result Hamiltonian

jni = b j0i (7)

Eq.(2) b ecomes a forced harmonic oscillator with the ex-

hni!

ternal force term h N g (t)b +h:c. Obviously, it will

drive the atom in the untrapp ed excited state from a with hni! = hnihn 1ih1i (n = 0;1;:::;N). The

vacuum state to a coherent state and thus lead to a co-

action of op erators b and b are then represented on the

herent output of BEC atoms in the propagating mo de . deformed Fock space as:

p p

In this case the condensated atoms are initially describ ed

b jni = hn +1ijn +1i; bjni = hnijn 1i (8)

by a coherent state [13].

However, this app oroximation destroyes a symmetry

the actions of the creation and annihilation op erators b

of the Hamiltonian Eq.(2), i.e., the conservation of the

and b coincide with their direct actions on the subspace

total particle numb er is violated b ecause of [N; H ] 6=0.

V of the Fock space of a two-mo de b oson on which the

According to Gardiner, the phonon op erators are de ned

total particle numb er is conserved.

as:

Actually, the q -deformed vacuum state is physically a

two-mo de b oson state jN ;0i and it is trivially annihi-

1 1

y y y

p p

b = b b b b ; b = : (3)

g e

lated by the phonon op erator b. So the vacuum state

N N

can be regarded as a case that all b osons condensed in

These op erators act invariantly on the subspace V

the ground state and tends to form a BEC. By map-

spanned by bases jN ; nijN n; ni (n =0;1;:::;N),

ping backtotwo-mo de b oson space, the q -deformed Fock

where Fock sates

state jni jN ; ni implies a phonon excitation that n

b osons are created in the propagating mo de by destroy-

ym yn

b jm; ni = b j0i

ing n b osons in trapp ed mo de. This kind of fundamental

e g

m!n!

pro cess of phonon excitation essentially guaranteed the

particle numb er conservation.

with m; n =0;1;2;::: spann the Fock space H of a two

In the ab ove discussion ab out the Gardiner's phonon

mo de b oson.

excitation, wehave linearized commutator h f (b b; )

A straightforward calculation leads to the following

so that a q -deformed commutation rule was obtained.

commutation relation b etween the phonon op eraotr and

Essentially this linearization establishes a physical real-

it Hermitian conjugate:

ization of the q -deformed algebra. However, if the total

particle number N is not large enough, then h can not y y y

=1 b =f(b b;); (4) b; b b

b e approximated by a linear function. From the commu-

tation relations b etween h and b; b

with f (x; )= 1 + 2(1 2x) + . Keeping only

2 2

the lowest order of for a very large total atomic particle

y y

h; b = b ; [h; b]= b; (9)

numb er, the commutator ab ove b ecomes

N N

we see that the algebra of Gardiner's op erators is a rescal-

y y

=12b b (5a) b; b

ing of algebra su(2) with factor N . When N ! 1, it

approaches the Heisenb erg-Weyl algebra b ecause h ap-

proaches identity. Its representation space is a nite di-

b; b =1; (5b)

mensional space is found to b e spanned by bases given in

q 2

Eq.(7) with hni = n n(n 1) b eing exact. It is inter- evaluate the variances of the Hermitian quadrature phase

esting to notice that hni coincide with the q -deformed op erators [16]

number in the rst order of . The action of those

y y

b(t)b (t) b(t)+b (t)

creation and annihlation op erators have the same form.

p p

; X (t)= : (14) X (t)=

2 1

On this nite dimensional space we have bj0i = 0 and

2 i 2

hj0i = j0i together with b jN i =0. It is worthy to p oint

One can easily nd that

out that the last equality can not be comp osed to the

usual Heisenb erg-Weyl algebra b ecause such a constrain

2 2 2

(15) X (t)=h0jX j0ih0jX j0i =

1;2

will make it is imp ossible for the creation op erator to

1;2 1;2

b e the Hermitian conjugate to the annihilation op erator.

are time-indep endent constants that minimize the uncer-

Physically the usual b oson algebra and its q -deformation

2 2

tainty pro duct X (t) X (t)=1=4. It re ects that

with q b eing not a ro ot of unity describ e systems with-

1 2

the outputted atoms are just in a coherent state. This

out particle numb er conservation such as the black body

judgements for the case with in nite N can b e similarly

radiation, but this nite dimensional representation pro-

used to analyses the in uence of q -deformation with nite

vide us with a suitable algebraic framework to treat the

atomic numb er on the dynamics of outputting BEC.

dynamic pro cess where the particle numb er is conserved.

In practice condensated atomic number N cannot b e

in nite and therefore the Bogoliub ov approximation can

not work well and so an approximation metho d with

I I I. THE COHERENT OUTPUT OF BEC AS A

Q-DEFORMED BOSONIC EXCITATION

higher accuracy that for the generic Bogoliub ov ap-

proximation or its Gardiners mo di cation. And the q -

deformed b oson algebra disscussed in the previous sec-

In this section we shall apply the concept of the q -

tion provides us with a algebric structure of treating this

deformed b oson algebra for the Gardiner's phonon exci-

correction of nite atomic particle numb er.

tation to the theoretical analysis of the MIT exp eriment

Up to the approximation of order , as discussed previ-

for the coherent output of the BEC atoms from a trap.

ously, the dynamics of Gardiner's phonon excitation for

By denoting N = b b , in terms of the deformed b oson

e e

coherent output of BEC is determined by the q -deformed

op erators the Hamiltonian can b e rewritten as:

b oson commutation relation. The corresp onding Heisen-

b erg equation

H =h! N +h N g(t)b +h:c: : (10)

q e e

b(t)=i! b(t) i(t)+2i (t)b (t)b(t) (16)

For an ideal BEC with in nite atoms condensated in the

trapp ed state, the q -deformed b osonic commutation rela-

is then a nonlinear one. It is should be remarked here

tion b ecomes the usual commutation relation [b; b ]=1

that that the e ective coupling (t) = g (t) N is nite

of HW algebra. Together with commutator [N ;b]= b,

in the thermo dynamical limit of an in nite number of

a linear Heisenb erg equation

atoms in an in nite volume but with the density xed,

i.e., n = lim (N=V ) is nite. This is b ecause

N;V !1

b = i! b i(t) (11)

the coupling constant g (t) between the r.f. pulse and

atoms must b e prop ortional to the inverse of the square

is acquired where (t)=g(t) N is nite as N tends to

ro ot of the the e ective mo de volume V . So we have

in nity. Its solution can b e easily found to b e

(t) / N=V , the square ro ot of the density of the con-

i! t

densated atoms in the thermo dynamical limit. For this

b(t)=b(0)e + (t) (12)

reason the nonlinear terms in Eq.(16) can be handled

as a p erturbation in comparison with other terms. Oth-

with a non-vanishing exp ectation value in the vacuum

erwise, if (t) were not xed as N approaches in nity,

state:

the non-liner term should not b e regarded as a p erturba-

tion due to its square ampli cation of p erturbation. The

i! (tt ) 0 0

(t )dt : (13) (t)=h0jb(t)j0i=i e

thermo dynamical limit is our key p oint to deal with the

dynamics of q-deformation for the coherent output of the

Therefore when the BEC is initially at a Fock state, one

BEC atoms with a p erturbation metho d.

obtains a coherent state output of the untrapp ed atoms.

Becasue of the nature of the deformed b onon, in gen-

This is just a direct manifestation of the quantum coher-

eral, it is only neccissary to solve this nonlinear Heisen-

y y

ence through a factorization hb (t)b(t)i = hb (t)ihb(t)i. In

b erg equation Eq.(16) according to the rst order of by

fact for in nite N , the average phonon number hb (t)b(t)i

p erturbation. In fact if the annihilation op erator at time

approach the average atomic number b b of outputting

t is expanded as

in the untrapp ed mo de.

i! t

Further, to consider the quantum coherence of the out-

b(t)=b e + (t)+b (t) (17)

0 1

putting atoms represented by phonon op erators b(t), we

with b = b(0), one obtains immediately

0 3

y y

2 i! t

b (t)=(t)b + (t)e b 2 (t)b b + (t); (18a) where N = a a is the photon numb er op erator and g is a

1 0 0 0

0 0

real coupling constant. Obviously there is a conservative

i! (tt ) 0 0 2 0

(t)=2i e (t )j (t )j dt ; (18b)

quantity=N +N in addition to N . Wenow de ne

0 e

the dressed phonon op erators as

i! t 0 0 0

(t)=2i e (t ) (t )dt : (18c)

1 1

y y y y

p p

a b b ; B = ab b : (25) B =

e g

g e

N N

As a result the average outputted atomic number

For convenience we denote = 1= for large . The

hN (t)i in the untrapp ed excited state with nite con-

dressed phonon op erators satisfy the following commuta-

denstated BEC particle numb er correction is given by

tion relation:

hN (t)i = j (t)j + j (t)j + 2Re ( (t) (t) ) : (19)

2 y

=12( + )N + (3N N ): (26) B; B

0 e 0 e

The quantum uctuation of the quadrature phase op-

In terms of those dressed phonon op erators the Hamilto-

erators X (t) and X (t) in the nal state can also be

1 2

nian Eq.(24) can b e rewritten as

explicitly determined as:

H = ! N + ! +! N + (B +B ): (27)

d g 0 e d

2 2 2

X = j (t)j + (t)+ (t) ; (20a)

where = g N and ! = ! ! ! .

d e g 0

2 2 2

X = j (t)j (t) (t) : (20b)

As N (or ) tends to in nity the dressed phonon op-

erators B; B and N will give a representation of al-

Keeping the rst order of , one obains immediately the

gebra su(2). As b oth N and tends in nity one ob-

follwing relation

tains Heisenb erg-Weyl algebra [B; B ] = 1. When we

keep the rst order in o( ) and o( ), and considering

2 2

B B = N + o( + ), we obtain a q -deformed b oson

e 0 X X = j (t)j : (21)

1 2

algebra

The term of re ects the devotion from the thermo dy-

B; B =1 (28)

q namical limit as the e ect of the q -deformation with nite

condensated particle numb er.

with the deforming constant given by q =12( + ).

d 0

Since the commutation relation of quadrature phase

Within this kind of approximation, the Heisenb erg

op erators X (t) and X (t) is deformed into

1 2

equation of motion satis ed by the dressed phonon is

given by:

[X (t);X (t)] = i 1 2b (t)b(t) ; (22)

1 2

B = i! B i +2i( +) B B: (29)

the corresp onding Heisenb erg's uncertainty relationship

d 0 d

of those quadratures in the output taking accountof -

Here the coupling constant g is prop ortional to the in-

nite atomic numb er should b e:

verse of the square ro ot of volume V which makes

2 2 y

nite as N and approaches in nity. Because the evolu-

X X h0jb (t)b(t)j0i: (23)

1 2

tion equation Eq.(28) is the same with Heisenb erg equa-

tion Eq.(16), all the results discussed in the previous sec-

Considering h0jb (t)b(t)j0i = j (t)j + o( ) the coherent

tion should also b e valid in case of quantized r.f. eld.

output state of the untrapp ed atoms minimizes the un-

certainty relation. Because, as we known, the squeezed

state and coherent state can be de ned to be states

V. REMARKS

minimize the uncertainty relation between the p osition

and momentum, the output state of the atomic lase can

To conclude this pap er, we treat the same problem

be likened to a kind of q -deformed coherent state or

in the Schrodinger picture. Without the e ect of q -

even a squeezed state b ecause uctuations Eq.(26a) and

deformation caused by nite N , the dynamic pro cess

Eq.(26b) are inequal.

can be describ ed as a factorized evolution of the wave

function [16]. Supp ose the initial state is j(0)i = j =

N i j0i i.e., all atoms o ccupied the trapp ed groud

IV. DRESSED PHONON OPERATOR g e

state with a coherent state while there is no atom in

the propagating mo de. Driven by a r.f. eld g (t) =

When the r.f eld is a quantized one which is describ ed

1=V ) with a xed frequency ! , the g exp(i! t)(g /

f f

by a single-mo de b oson whose annihilation op erator is

system will have exactly a factorized evolution wave func-

denoted as a,we consider the following Hamiltonian:

tion of the form j(t)i = j i j (t)i even for a nite

g e

y y y

b b + ab b ); (24) H = ! N + ! N + ! N + g (a N . Such a factorization structure of wave function rst

e g d e e g g 0 0

g e 4

p ointed out by Mewes et al is very crucial to realize their [13] M.Mews, M.R. Andrews, D.S. Durfee, D.M. Kurn, C.G.

Townsend and W. Ketterle, Phys. Rev. Lett. 78, 583

exp eriment, and its dynamical origins mainly dep end on

(1997)

the linearity of the original coupling system of b and

[14] C.P. Sun, IL. Nuuvo. Cimento B 34, 499 (1993)

b . The coherent state of the second comp onent j (t)i

g e

[15] C.P. Sun, Y.B. Gao, H.F. Dong and S.R. Zhao Phys. Rev.

of j(t)i means a coherent output of atoms in propa-

E 57 (1998) 3900

gating mo de in the MIT exp eriment. The in uence of

[16] C.P. Sun, H. Zhan, Y.X. Miao, and J.M. Li Commun.

nite N on Schro edinger wave function is manifested by

Theoretical Phys. 161, 29 (1998)

the outputting amplitude j (t)j/ N=V j sin( t)j where

2 2

(! ! ) =4+g . Because the factor N=V =

is nite and tends to j! ! j in the thermo dynam-

ics limit, the results from the Bogoliub ov approximation

mentioned in Sec.I I is just recovered by this exact solu-

tion.

There is another nonlinear e ect di erent from that re-

sulting from the q -deformation with nite N . In presence

of interatomic interaction in the trapp ed or untrapp ed

y y

states, which roughly is of the form b b b b , the corre-

k l

i j

sp onding Heisenb erg equations are no longer linear and

thus the factorization structure will b e broken down. It

is noticed that such a non-linearity is essentially di erent

from that resulting from the q-deformation with nite N

in this pap er. The later originates from the subsystem

isolation of a large system.

ACKNOWLEDGMENT

This work is partly supp orted by the NFS of China.

Electronic address: [email protected]

Internet www site: http:// www.itp.ac.cn/~suncp

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