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A Linear Quantum Dynamic Theory for Coherent Output of Bose-Einstain Condensation

A Linear Quantum Dynamic Theory for Coherent Output of Bose-Einstain Condensation
A Linear Quantum Dynamic Theory for Coherent Output of Bose-Einstain Condensation

a r X i v :q u a n t -p h /9706041v 1 18 J u n 1997A Linear Quantum Dynamic Theory for Coherent Output of

Bose-Einstain Condensation

C.P.Sun a,e ,J.M.Li b ,H.Zhan c ,Y.X.Miao c ,S.R.Zhao d and G.Xu a

a Institute of Theoretical Physics,Academia Sinica,Beijing 100080,China

b Department of Physics,Peking University,Beijing 100871,China

c Institute of Theoretical Physics,Northeast Normal University,Changchun 130024,China

d Department of Applied Mathematics,University of Western Ontario,London,Ontario,Canada,N6A,5B7

e Department o

f Physics,the Chinese University of Hon

g Kong,Shatin,NT,Hong Kong

ABSTRACT A model for the coherent output coupler of the Bose-Einstein condensed atoms from a trap in the recent MIT experiment (Phys.Rev.Lett.,78(1997)582)is established with a simple many-boson system of two states with linear coupling.Its exact solution for the many-body problem shows a factorization of dynamical evolution process,i.e.,the wave function initially prepared in a direct product of a vacuum state and a coherent state remains in a direct product of two coherent states at any instance in the evolution of the total system.This conclusion always holds even for a system with a ?nite average particle number in the initial state.Its thermodynamical limit can be directly dealt with in the Bogoliubov approximation and manifests that an ideal condensate in the trap will remain in a coherent state after the r.f.interaction while the output-coupler pulse of atoms is also in a coherent state,which means a coherent output of atomic beam to form a macroscopic quantum state in a propagating mode.PACS numbers:03.65,03.75,05.30

1.Introduction

According to de Broglie matter wave theory,all matter in general can behave itself like a light that spreads out in space and combines with another one to form interference pattern.In principle,it is logical to image that a“matter wave laser”can be realized,in analog to a light laser,to produce the output coherent matter wave.Based on such fundamental consideration and stimulated by the experiments observing atomic Boson-Einstein condensate(BEC)in last two years[1-3],a number of e?orts coming from di?erent research groups have been placed on both its experimental setup and theoretical possibilities[4-7].Until the end of last year,a milestone experiment for the matter wave laser of bosonic atoms(also called atom laser)was accomplished by Mewes et al in MIT[8].

In their experiment,they coupled the toms trapped in a state|1 with that untrapped in another state|2 by a sweeping radiation frequency(r.f.)?eld.The r.f.?eld coherently turns the BEC atoms constrained in|1 into a propagating mode of atoms in|2 .The experiment shows that“the condensate which is initially in a coherent state remains in such a state after the r.f.interaction while the output-coupler pulse of atom is also in a coherent state”[8].In fact,a laser of photons above threshold can be understood in terms of the coherent state for its well-de?ned classical analogy with minimum quantum?uctuation in amplitude[9].Similarly,it is reasonable to believe that such a output of atoms in a single coherent state is something like the photon laser.Roughly,we can regard the dynamic process of wave function evolution driven by the r.f.?eld as a“stimulated”process of(massive)matter wave.

In this paper,we?rst use the Bogoliubov approximation[10,11]in thermodynamic limit to give a theoretical analysis for the dynamic lasering process of BEC atoms in the MIT experi-ment.We will show that such an approximation is not actually needed to reach the conclusion about coherent output in the requirement of very large N https://www.doczj.com/doc/c918722100.html,ly,even for the case that the number N c of atoms condensed in the ground state is not very large,thereby the Bogoliubov approximation is violated,the output of?nite atoms is still coherent so long as the system is interatomic interaction free and the atoms in the trap is initially in a coherent state.Theoreti-cally,this is the factorization structure for the evolution of the many-boson model of two state with linear coupling,i.e.,the wave function at any instance remains in a direct product of two coherent states if the system is initially prepared in a direct product of a vacuum state and a coherent state.This fact does not depend on the average number of condensed atoms in the initial state and it implies that the experiment result by Mewes et al is quite steady even for rather?nite N c.

2.A Dynamic Model of Output Coupler via Bogoliubov Approximation

The second quantization of model Hamiltonian for the MIT experiment can be written as

H=ˉhωa b?2b2+ˉhωR b?2b1exp ?i t0ω(τ)dτ +h.c (1)

in terms of the creation and annihilation operators,b?1,b?2,b1and b2,of bosonic atoms for the magnetically trapped state|1 and the untrapped state|2 with level di?erenceˉhωa.The r.f. pulse of varying frequencyω(t)is a sweeping classical electromagnetic?eld,which couples|1 and|2 through the dipole matrix elementˉhωR= V.V is the e?ective mode

volume and ε0is the vacuum permittivity.This many atom system without interatomic interac-tion can be modeled as a linear coupling system of two-oscillators.The main simpli?cation is to ignore the quantized motion of atomic center of mass in the trapped state by an inhomogeneous magnetic ?eld.As pointed out by Mewes et al,this inhomogeneous feature can be neglected for a su?ciently short r.f.pulse.

The MIT experiment is theoretically described as an initial value problem for Sch¨o rdinger equation governed by H with the initial state |ψ(0) =|α=√N c 1is a Glauber coherent state of the operator b 1characterizing N c atoms condensed in the trapped state |1 .No atoms occupy the untrapped state |2 initially.Why does the coherent state |α represent the BEC is still an open question,but there are some reasons enable one to believe it is right,such as the phase locking implied by BEC [12]and the correct average atomic number N c .The additional reason we believe is due to the connection with the Bogoliubov approximation [10,11],which replaces operators b 1and b ?1with a c -number

√N c b ?2exp ?i t

0ω(τ)dτ +h.c

.For the case of very large N c in BEC,the coupling term is very small in comparison with the term of ˉh ωR

√N c b ?2exp ?i t

0ω(τ)dτ +h.c .(2)

This is just the Bogoliubov approximation of the original Hamiltonian H .

Now let’s consider the thermodynamical limit of an in?nite number of atoms in an in?nite volume but with the density ?xed,i.e.,n c

=lim N,V →∞(N c /V )→constant ,the coupling term ˉh ωR ∝1/

√N c =ˉh g √

method calculating wave function developed in[14,15]will be used in this paper to obtain the exact solution for factorized wave function in section4.

3.Coherent Output via Factorization

The next step is to prove|Φ(t) is still a coherent state evolving from its vacuum state|0 2. It is quite clear that the Bogoliubov approximation Hamiltonian(2)is just a forced harmonic oscillator(FHO).It is well-known that the displacement property of a quantized FHO results dynamically in a coherent state for the initial vacuum state.By considering the case ofω(τ) independent of time,i.e.,ω(τ)=ω,the solution of the Hamiltonian(2)can be found in some standard textbook[16]|Φ(t) =|?α(t) with

?α(t)=ωR

? e i?t?1 e?iωa t,(4)

Namely,the total wave function in the BEC with large N c can be factorized in the direct product of two coherent states|ψ(t) =|α ?|?α(t) .The second components implies the output-couple pulse of the atomic beam is in the coherent state,which“can be regarded as a pulsed atomic laser”.

Notice again that the above theoretical discussion about the output-coupler for atomic laser is only with the help of a highly-simpli?ed many-boson model of two states with linear coupling rather than the complicated nonlinear one,such as the system of Gross-Petaevskii equation. Because the atomic number of the coherent output in the untrapped state

ψ(t)|b?2b2|ψ(t) =|?α|2=2ω2R

n c ∞0exp iπn c

very large N c is not necessary so long as the atoms are prepared in a coherent state|α even with rather?nite average atom number N c= α|b?2b2|α .The coherent state is only a special

Poisson superposition of in?nite number states with marched phasesΘn=nθ,n=1,2,....

The second reason to consider the exact solution concerns the analogy in the photon case.In the theory of light laser,a su?cient strong light?eld can be treated as a classical electromagnetic ?eld and it can be regarded as a coherent state above the threshold[9].In this sense,the quantum e?ect mainly results from the quantum?uctuation,which is demonstrated by a weak light?eld. Therefore,for the matter wave of massive particles,the in?nite number case only corresponds to the classical analogy of photon?eld.Then,we need to study the case corresponding to the quantum?uctuation of the weak light?eld,for which the Bogoliubov approximation is broken down.

According to[14,15],the exact solution of the wave function of the linear system in the Schr¨o dinger picture can be easily obtained from that of the canonical operators,such as b1(t) and b2(t),in Heisenberg picture.In the case ofω(τ)independent of time,by solving a system of one-order di?erential equations resulting from the Heisenberg equations of b1(t)and b2(t),the exact expressions of these solutions are

b1(t)=α1(t)b1(0)+α2(t)b2(0),(7)

b2(t)=β1(t)b1(0)+β2(t)b2(0)(8)

where

β1(t)=?ωR

?2+4ω2R e iω+t?e iω?t e?iωt,(9)

β2(t)=

1

?2+4ω2R ω+e iω+t?ω?e iω?t

e?iωt,(10)

α1(t)=

1

?2+4ω2R ω+e iω?t?ω?e iω+t

,(11)

α2(t)=?ωR

?2+4ω2R e iω+t?e iω?t ,(12)

ω±=1

?2+4ω2R .(13)

From the property of the evolution operator,U(t),of the total system,i.e.,bα(t)=U(t)?bα(0)U(t), (α=1,2),the initial state|ψ(0) =|α=

2|α|2

∞ n=0αn

2|α|2

∞ n=0αn

That is just a direct product of two coherent states

|ψ(t) =|αα1(?t) ?|αα2(?t)

=| 2?t(cos(?t)+i sin(?t)cosθ) ?|?i 2?t sinθsin(?t)e?iωt (15) where tgθ=2ωR/?,where?=

2ˉh n c

?t?ωt (16)

2

of the bosonic?eld,?φ(x)= k

?φ(x)2 ? ?φ(x) 2do not depend on N c.This means the relative uncertainty of the amplitude,?φ/ ?φ(x) ,will approach zero only in the thermodynamics

limit:N c,V→∞,but N c/V→constant.Therefore,for?nite N c,the quantum?uctuation √

?φ∝

,but N c sinθ∝g√

2

N c 1and no atoms occupy the untrapped state|2 ,the spin-like dynamics of Rabi oscillation will create a factorized wave function|ψ(t) =|α(t) 1?|β(t) 2,which

is a product of two coherent states.Even for the case without ideal Bose-Einstein condensation

that N c is not su?ciently large and thus|α=

N c 1is coherent states.However,it seems very di?cult to prepare?nite numbers of atoms in a coherent state duo to the massive feature of atoms.The spin-like dynamics of Rabi oscillation will only create an entanglement evolution state|ψ(t) = i,j|αj(t) 1?|βi(t) 2for an arbitrary initial components|ψ(0) 1other than a coherent state.Only in BEC case that the?xed phase and amplitude of macroscopic wave function are de?ned and thus to form a Glauber coherent state, this entanglement state can be completely factorized.

Such a factorized structure of wave function?rst pointed out by Mewes et al is very crucial to realize their experiment,but it dynamical origins mainly depend on the linearity of coupling. For the case with an interatomic interaction in the trapped or untrapped states,which roughly is of the form b?i b?i b i b i,the corresponding Heisenberg equations are no longer linear and thus this factorized structure will be broken down.Therefore,the su?ciently weak interaction should be required for a coherent output in practical system.In fact,Ballagh et al[18]have modeled a theory for the evolution of two component BEC by generalizing the Gross-Pitaevskii equation.

Their theory introduces a non-linear element resulting from the interatomic interaction and shows a rich variety of phenomena of nonlinear e?ect such as the localization of atom in the untrapped state due to the nonlinear coherent coupling.

Notice that an interference experiment was also carried out by Andrews et al in the same research group at MIT[19]to examine the coherence of atoms in BEC by overlapping two expanding condensates?ying from the double well magnetically-trapped potential when this trap is adiabatically switched o?.The next step following this experiment was immediately accomplished to test the coherence of the output couplers rather than a pair of expanding atomic sources.They also have shown that a similar high-contrast regular pattern of interference.Based on an intuitional understanding of the MIT“atomic laser”in terms of coherent states described in this paper,a many–body quantum theory for the interference of output couplers can be present in a forecoming paper as a space-dependent generalization of present results.The non-vanishing order parameter of quantum system,concerning a symmetry broken involving the atomic number conservation,can be obtained for the description of regular pattern of stripes in these interference experiments.

One(CPS)of t he authors wish to express his sincere thanks to K.Young for inviting him to visit the Chinese University of Hong Kong as a C.N.Yang’s Fellow.The work is supported in part by the NSF of China.

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