##### Document Text Contents

Page 1

Proceedings of the 7th International Symposium on

OF

IN THE LIGHT OF

NEW TECHNOLOGY

Edited by Yoshimasa A. Ono • Kazuo Fujikawa

World Scientific

Page 2

Proceedings of the 7th International Symposium on

FOUNDATIONSOF

OUnNTUM MECHRNIC5

•

iirniElieiifoF

D

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

collisions.

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

Equations

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

follows:

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)

f(t)=/dk|k0(k)|2e-i

8kt, (9)

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

5k = a)k-coo» (11)

where

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

158

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

bk(t)=bk(t0)e-i&kt

+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

aVt)=a(o)£-[{a'0(0)-£[§(s)]}

{s+(iQ ra)+r£[?(s)]n, (15)

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

_ri/2ko*(k)ao(0)ei

wOth(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

<nk(t)>=<bkt(t)bk(t)>

= 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

331

Ueno, K.

Ukena, A.

Ulam-Orgikh, D.

Utsumi, Y.

Vogels, J.M.

Watanabe, K.

Weihs, G.

Williams, D.A.

Wemsdorfer, W.

Wineland, D.J.

Xu,K.

Yamaguchi, T.

Yamamoto, T.

Yamamoto, Y.

Yanagimachi, S.

Yoneda, T.

Zeilinger, A.

Zimmermann, M.

229

279

287

32

122

132

44

24

161

48

122

201

12

91

144

283

44

253

Page 349

FOUNDATIONS OF

OUHNTUM MECHANICS

IN THE LIGHT OF

NEW TECHNOLOGY

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

5076 he

ISBN 981-238-130-9

9 "789812"381309"

Proceedings of the 7th International Symposium on

OF

IN THE LIGHT OF

NEW TECHNOLOGY

Edited by Yoshimasa A. Ono • Kazuo Fujikawa

World Scientific

Page 2

Proceedings of the 7th International Symposium on

FOUNDATIONSOF

OUnNTUM MECHRNIC5

•

iirniElieiifoF

D

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

collisions.

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

Equations

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

follows:

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)

f(t)=/dk|k0(k)|2e-i

8kt, (9)

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

5k = a)k-coo» (11)

where

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

158

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

bk(t)=bk(t0)e-i&kt

+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

aVt)=a(o)£-[{a'0(0)-£[§(s)]}

{s+(iQ ra)+r£[?(s)]n, (15)

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

_ri/2ko*(k)ao(0)ei

wOth(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

<nk(t)>=<bkt(t)bk(t)>

= 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

331

Ueno, K.

Ukena, A.

Ulam-Orgikh, D.

Utsumi, Y.

Vogels, J.M.

Watanabe, K.

Weihs, G.

Williams, D.A.

Wemsdorfer, W.

Wineland, D.J.

Xu,K.

Yamaguchi, T.

Yamamoto, T.

Yamamoto, Y.

Yanagimachi, S.

Yoneda, T.

Zeilinger, A.

Zimmermann, M.

229

279

287

32

122

132

44

24

161

48

122

201

12

91

144

283

44

253

Page 349

FOUNDATIONS OF

OUHNTUM MECHANICS

IN THE LIGHT OF

NEW TECHNOLOGY

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

5076 he

ISBN 981-238-130-9

9 "789812"381309"