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TitleProceedings of the 7th International Symposium on Foundations of Quantum Mechanics in the Light of New Technology Isom-Tokyo '01: Advanced Research Laobratory ... Hatoyama, Saitama, Japan 27-30 August 2001
File Size22.2 MB
Total Pages349
Document Text Contents
Page 1

Proceedings of the 7th International Symposium on




Edited by Yoshimasa A. Ono • Kazuo Fujikawa

World Scientific

Page 2

Proceedings of the 7th International Symposium on





Page 174

Htr = Zft(oiait(t)ai(t) + V(t),

V(t)= ih-Z gijkl ait(t)ajt(t) ak(t)ai(t) ,

i ^ j , k ^ 1, (2)

Hext=/dkfcMkbkt(t)bk(t), (3)

Hint = -i^rl/2/dk{ko(k)bk(t)a0t(t)

- KO*(k)bkt(t)a0(t)}, (4)
where Htr is the Hamiltonian of the
harmonic trap. This consists of the
oscillation energy at frequency coj and the

interaction Hamiltonian V(t) which
describes the redistribution of the atoms by
elastic collisions. The operator aj is the

second quantized annihilation operator for
the mode i, and the coefficients g represent
the total transfer rates between levels by
Using the rotating wave approximation we

can reduce the interaction term V(t) to the
energy conservation terms. Some terms of
them can be ignored, because the level 2
decays very fast by the effect of the
evaporative cooling.
The term Hext is the Hamiltonian for the

out-put reservoir and Hint represents the

interaction between the laser mode ao and

the output free space field bk-

The coupling constant, K (k) = T^KoCk),
depends on k and its dispersion relation is
expressed by cok = fck2/2M (M is the mass

of the atoms) which differs from that for
the light. Hence, we treat this coupling to
the external free fields without the BMA,
and assume that the external field is empty
at the initial time, since initially there is no
matter. The term Hj is the Hamiltonian of

the reservoir j 0 = 1. 2), and Haj is the
interaction energy between the reservoir j
and the atoms in the trap level j , in which
the BMA is assumed.
3. Fundamental Equations in HP

3.1 Quantum Mechanical Langevin

First, eliminating the operators of the
reservoirs 1 and 2 from the Heisenberg
equations for each operator, we obtain the
coupled quantum mechanical Langevin
equations of the annihilation operators
a2,ai,ao an ( l Dk for the higher, pumping,
and lasing modes, and the external field,
respectively. In Fig.l, K^and KJ are the

decay rates of the higher mode &2 and of the

pumping mode ai, respectively.

In order to build up a condensate in the
ground state, it is necessary for the rates to
obey the inequality

K2»Kl»K0- (6)

Under the condition (6), we derive the
equation of the lasing mode ao(t) by

eliminating a2 and &\ adiabatically as


Stepl: let aj(t) = ai(t)e-imit, i =0,1,2 and

eliminate a2 adiabatically from both

equations of Si and ao.

Step 2: Eliminate the pumping mode Si

adiabatically from the equation of fin.

Having performed these steps, we finally
obtain the following Langevin equation of
the lasing mode SQ,

da0(t)/dt = -Yo'(t)+ (a+iQ 2)^0(1)

- (p+i£20)no(t) ao(t) - £(t)+F0t (t), (7)

YO'O) = Y0 (0 - KO ,
YO(t) = r/f(t-t')ao(t')dt, (8)

8kt, (9)

§(t) =/dkK0(k)e-i&k0-1 o) h>k(t0), (10)

5k = a)k-coo» (11)

a =A (K/KI)2 - KO,

P = 4A(K/KI)(KO/KI) ,

Qo=2gOOOO, Qi=(K/Ki)gi010,

A = |g021l|2/K2.
Here, a is the effective gain coefficient, (3

Page 175


is the self saturation coefficient of the gain,
A is the transition rate from the pumping
mode to the laser mode, and QQ Q^ present

the self energy shift, respectively , which
affect the spectra] property of the atom
laser. In the coefficients a and p, K is the
pumping rate from the reservoir 1 to the
level 1 in the trap , and it is introduced
phenomenologically in the equation of the
pumping mode aj.

Equation (7) is the fundamental equation of
the lasing mode, which contains time
convolution term YO(0 due to non Born-
Markovian coupling to the external field.
The terms YO 0) and §(t) correspond to the

damping factor and the fluctuation
operator, respectively, which appear in the
usual laser theory.

On the other hand, the equation of the
external field bk(t) is solved directly and the

solution is given by


+r1/2K0*(k) S dt'ao(t')eiok(t-t'). (12)

The Heisenberg equation of the expectation
value of arj(t) is obtained from Eq. (7),

d<arj(t)>/dt = -<y'o(t)>- [a- i (corj+Ql)
- (i Qo + (3n0)] <a0(t)>- T < §(t)> . (13)

When the BMA is valid, we obtain
d<a0(t)>/dt= [(a -ico0)- pn0]<a0(t)>, (14)

where QQ and £2j are ignored for simplicity

when we are interested only in the
occupation number of the ground state.
The Heisenberg equations (13) and (14) of

the lasing mode show
l)Equation (13) contains the effects of non

Born-Markovian output coupler.
2) Equations (13) and (14) show that a

CW atom laser behaves as a self-sustained
oscillator, since they have a third-order non-
linear saturation term. Similar discussions
from another approach have been reported
independently by Scully5).

3) It is worth noting that Eq.(14) is closely
analogous to the single mode laser equation.
3.2 Solution
For simplicity, we ignore the fluctuation

force operator Fot from the Markovian
reservoirs 1 and 2, and the nonlinear
saturation term. In order to solve the
Langevin equation, Laplace transformation

£[g(s)]=/g(t)e"stds and inverse Laplace

transformation, £~ [ ] are applied to the
final Eq. (7). The solutions for the laser
mode and output mode are given by


{s+(iQ ra)+r£[?(s)]n, (15)

bkO) = bk(t0) e-i<%t


- TK0*(k) e-i^V Jh(t-t')§(t')dt', (16)

h(t) = £-[(s+5k)-1{s+(iS2ra)

+r£[f(s)]}-l](t). (17)
The expectation value of the output inten-

sity <nk(t)> is obtained from Eq. (16) as


= r|k0(k)|2< a0t(0)a0(0) > |h(t)|2, (18)

where initially external reservoir is empty
i.e., <bk(0)> = 0.

For a pulsed atom laser, we can ignore

the pumping mechanism, so we put
a = 0 and p = 0. In addition we ignore Qi

for simplicity. Then the solution is given by

<ao(t)> = <ao(0)> e-too*

£-l[s+ £[f(s)]]-l(t). (19)
There is no further restrictions except

<%(0)> = 0, or <bk(0)> = 0.

In this stage, we do not assume the BMA.
4. Projection Operator Approach by the
Heisenberg Picture (HP)
In this section, We assume Born approxi-
mation with the non-Markovian process.
We treat the simplest model of the atom

Page 348


Ueno, K.
Ukena, A.
Ulam-Orgikh, D.
Utsumi, Y.
Vogels, J.M.
Watanabe, K.
Weihs, G.
Williams, D.A.
Wemsdorfer, W.
Wineland, D.J.
Yamaguchi, T.
Yamamoto, T.
Yamamoto, Y.
Yanagimachi, S.
Yoneda, T.
Zeilinger, A.
Zimmermann, M.








Page 349


This book discusses fundamental problems

in quantum physics, with emphasis on

quantum coherence and decoherence.

Papers covering the wide range of quantum

physics are included: atom optics, quantum

optics, quantum computing, quantum

information, cryptography, macroscopic

quantum phenomena, mesoscopic physics,

physics of precise measurements, and

fundamental problems in quantum physics.

The book will serve not only as a good

introduction to quantum coherence and

decoherence for newcomers in this field,

but also as a reference for experts.

www. worldscientific. com
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ISBN 981-238-130-9

9 "789812"381309"

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