New optical equations for the interactio

Physica A 161 (1989) 525-538
North-Holland, Amsterdam

NEW OPTICAL EQUATIONS FOR THE INTERACTION
OF A TWO-LEVEL ATOM WITH A SINGLE MODE OF
THE ELECTROMAGNETIC FIELD

A. SANDULESCU
Central h~stitute of Physics, Bucharest, P.O. Box" MG-6, Romania
E. STEFANESCU
R & D Center for Electronic Components, R-72296 Bucharest. Romania

Received 17 January 1989
Revised manuscript received 12 June 1989

Based on the theory of open quantum systems we derive new optical equations, more
general than the conventional optical Bloch equations: new terms describing couplings of the
observables through the environment and an asymmetry of the dcphasing rates for the two
polarization observables are obtained. We show that, duc to the new coupling of the
population witl, the polarization through the environment, negative values of the absorption
coefficient are possible. We find experimental evidence of this new coupling by comparing the

bistability characteristic obtained from the new equations, for a nonlinear Fabrv-Pcrot
resonator, with the experimental data of Sandlc and Gallaghcr.

1. Introduction

In a previous paper [1], the damping of collective coordinates in deep
inelastic collisions has been analyzed on the basis of the Lindblad theory of
open quantum systems. It was shown that various master equations used in the
literature for the description of the damped collective modes are particular
cases of the Lindblad equation. Recently it was shown that this equation gives
a very good description of the charge equilibration in deep inelastic reactions
[2].
In this paper, we use the same method for the quantum system of an
ensemble of twe-level atoms interacting with a single mode of the
electromagnetic field, usually described by the Maxwell-Bloch equations [3-6].
In these equations the atom is described by the expectation values of the Pauli
spin observables tr,, o',., o'_ and its interaction with the environment by three
ph,~n,-,,~,~-cdc,oie-,.' parnmete,';" the spontaneous relaxation rate of the
0378-4371/89/$03.50 © Elsevier Science Publishers B.V.
(Nolth-Holland Physics Publishing Division)

526

A. Sandulescu and E. Stefanescu / New optical equations for atom-field interaction

polarization variables (tr x ) and ( % ) (the dephasing rate Yi ), the spontaneous
relaxation rate of the population variable ( ~ ) (the decay rate YlI) and the
zero-point value of the population variable trc~ (the equilibrium population
N o = No "° , where Ar is the atom density).
Introducing the dissipation in agreement with the Lindblad theorem, we
obtain a quantum master equation, where quantum friction and quantum
diffusion processes are included. This master equation leads to field equations,
which, for a rather large density of atoms, are in agreement with the classical
Maxwell equations with polarization for dissipative media. At the same time,
we obtain optical equations where the interaction with the environment is
described by nine phenomenological parameters: three spontaneous relaxation
rates, y ~, 3"~, "YlI;three coupling rates through the environment, s, 3'1, )2; and
three zero-point values, D t, DE, D 3.
We derive equations for the amplitudes of the polarization and of the
population without the "'rotating wave approximation". Consequently,

modified expressions for the absorption coefficient and for the dephasing of the
electromagnetic wave, due to its propagation through the atomic medium, are
obtained.
We find also fundamental constraints for the phenomenological parameters
and uncertainty relations for the field and atomic variables.
As an example, we analyze an optical bistable Fabry-Perot resonator with
the new equations for amplitudes and derive an analytical expression for the
bistability characteristic, including the coupling of the polarization with the
population and the asymmetric dephasing of the polarization variables. This
expression contains the analytical expressions previously derived by McCall
and Gibbs [7], Agrawal and Carmichael [8, 9], Carmichaei and Hermann
[10-121.
Finally, we compare the theoretical results with the experimental data of
Sandle and Gailagher [13] and show that the asymmetry of the peaktransmission characteristic as a function of the atomic detuning is well
described by the new equations.

2. Quantum master equation

We consider an injected single mode linearly polarized radiation field Et of
the frequency to and the wave vector kl, propagating through a medium of

two-level atoms with the transition frequency too, the electric dipole moment/x
and the density N. Due to the interaction between the radiation field and the
atoms, higher-order harmonics E, of frequency no and wave vector k,, are
gencratcd. For the electric dipole interaction, the Hamiltonian of the system

A. Sandulescu and E, Stefanescu / New optical equations for atom-fieM interaction

527

takes the form

H = - ~hto,, E cr i. + ~ ~ (n'-to'q, + p;)
j
n
+ tiY~cr,,/ ( n o q , , sin k,,- rj + p. cos k,, • rj),

(1)

in

where fi = ~/V'--i,V, V is the volume of quantization and p,, and q,, arc the
canonical variables of the harmonic n.
According to Lindblad's theorem [1], the dynamics of this physical system
with dissipative coupling can be described by the following equation of motion:

dt

~ [H. P] +

(2)

(IV.p. vii1 + [v,,. pv,*,]) .

where the operators V,, are linear combinations of the system observables with
complex coefficients characterizing the interaction with the environment.

V,, = E (a,,,,p,, + b,,,,q,,) + E (A,,io'.~. + B,,iot ~, + C,,o"_) .
v

(3)

j

With expressions (1) and (3) eq. (2) takes the explicit form
dp
= - ~ i Ill,,, p l - i ~fi" ~.,, [(noJq,, sin k,,. r, + p,, cos k,, . r ) t r i- P]
dt
A,
i~--h ( [ p , , . o q , , + q , , o ] - l q ~ . p p , , + p , p ] )

+~,

h ~_ lq..[q,,.oll +

([q..lp..oll+[p..Iq,,.oll)

i j
j
j
i

~
~D,.([~,.,p~,.+¢;O]-[¢~,

+~,.

D,,f, [P,,.[P,,Oll

p~J,. + o . i ,.Pl)

1

+

i

D

:(I

+

ip]

-

[~J pcr~+cr / p])

+-~D'i =,:([cr,po'J i + , ~ p ] - [o-~, po-,.~ +: p l ) '

'. ~" , [~J:, pll-A{.:[¢.{ , [¢~• p l l - ;t':.,.[~'[~',.,,., pl]
,, ,
i p ] ] + [ ¢ J ,., [~.~,p]])
+ F ~.,.,.qa,..[o~,..
i [ ~ i=, p l l + [ ~ : , ' [ o'j,., pll)
J
+ r,.~([~,..

(4)

528

A. Sandulescu and E. Stefanescu 1 New optical equations for atom-field interaction

where H o is the Hamiltonian of the system without interaction and the
phenomenologicai parameters have the expressions
h~
w

Dqqn

2

Dpq, =

,

- -h4Z

'g

A{,,. = 2h

h ~/b,,,,b*,,.
2 .

Dpp,,

. a.,,a,,,, ,

*
(a,,,,b,*,. + a,,,,b,,~)
C

,

A{.=

•

a,, = --2i

1 ~

J

2-"h

- a ,,,, b ,,,, ) ,

i*

A~.~ = ~

A~ A . .

~ B .J~ B ,,j* ,

(5)
F j~Y = -4"-h
1 ~ (A j~B,.j, + A,.j, B~,),
.
F,.. =

---

1 ~, ( a r jC , ,j , + a,,j , c ~ ) ,
1 Z (C~A~,* + C~*A~,)

2ih
|

J. =
D,,~

4h ~

F~.~ =-4---h ~

1 ~(A*
i*
i*
,
,.B,, - A , B{,),

O~y -

2ih

D~,-

'

E

( a , , cJ, ,

1 Z (C /

"

2ih ~

j*

i*

,,a,

-

J j* ) ,
B,,C,,

J*

i

- C~ a ~ ) .

A physical interpretation of these coefficients can be obtained from the
equations for the expectation values or variances of the observables. The
coefficients corresponding to the harmonic oscillator have already been
discussed in ref. [1]. At the same time. from the master equation (4) we obtain
the following field equation:

d2E,,
dE,,
" "
"
dt 2 +2a,, dt + ( n ' w ' + A T ' ) E "

X" [
mop.
"7'
nto(cr~}
EO g
.

t

(d(o-{>,, +A,,
dt

d
dt

+ 3"i ( ~ ) - ( o , , , - s)(~,,) + ~ , ( ~ ) - o ,
+ ( w o - s ) ( ~ . , . ) + 3''~ (or,,) + (3'2 - , ~ ) ( ~ )

=o,
- D2 = 0 ,

(8)

+ 3', ( ~ ) +(T2 + ,~)(or;.) + yll(O-: ) - D 3 = 0 ,

dt
where

3", = 4 ( a , . +

A.,)

s =4r~ ,. ,

Yl =4F:x ,

t

D i = 4D,.: ,

3'" = (Ax,. + A,.~)

D, = 4D:,. ,

ZI = 4(A:., + A,.:).

"/2 = 4 r ,.,. ,
D 3 = 4Dx ,. ,

and

.~: 2~ ~ TrI(nwq,, + p,,)pl
h

n

is the normalized field variable. In comparison with the optical Bloch
equations, these equations contain six additional parameters: the zero-point
values D, and D 2 for the polarization variables, the coupling coefficients s, Yi,
"/2 and the asymmetric dephasing rate (y 'l - 3' i ) / 2 .
Consequently, the master equation includes three kinds of processes owing
to the coupling with the environment: the friction, described by the parameters
A, A, the diffusion described by the parameters D and the coupling between
the observables discussed by the parameters F.

3. Equations for tile amplitudes

Following the model originally adopted by Feynman, Vernon and Hellwarth
[3], we consider the atomic observables expectation values as functions of the
"amplitudes" u, v, w as components of the Bloch vector in the "rotating
frame" of the frequency to,

530

A. Sandulescu and E. Stefanescu / New optical equations for atom-field interaction

(try) = u cos

tot -

(o':.) = - u sin
=

-

v

wt -

sin
v

wt,

cos

(9)

tot.

w

For a system of atoms with density Ac, from expressions (9) one obtains the
following expressions of the macroscopic polarization S = N(o-.~) and of the
population N = A/"( o': ):
S = ~(,ff e -i'°' + .Se* e i ' ° ' ) .

(m)

N = -,/¢w,
where
= N(u - iv).

(11)

With expressions (9), (10) a n d (11), eqs. (8) become
d5¢'
d t + (%. + iA)5¢ + y~Se*

e 2i'°' =

(ix - y e i°'' + ix* e " " ° ' ) N + D e ~'°' ,

dN
+ TII(N- N3) = ~[(ix* - T* ei°'t
dt

(12)
+

ix e-2"°')ow

- (ix - T e i'°' + ix* e2"°')Se*],
where
r

yz = ( y i + ~ / i ) / 2 .
y = yt + iy~.

D =

F

Ff

y,, = (y~ - y ~ ) / 2 ,
N(D~

+iD2),

a-~- o)o--S--

o) ,

N 3 = ~"Os/Yll .

From these equations, one can obtain the conventional optical Bloch equations
in the " r o t a t i n g wave a p p r o x i m a t i o n " by neglecting the rapidly varying terms.
In order to include the effects of these terms, we consider solutions of E, S and
N in the form of Fourier expansions,
E = ~;~, + ~(~,. e-,,o, + F . ~ ei,O,) + ~ ( ~~, e

-._i,o,

+ ~ : ~ e2i,,,, ) + - - - ,

S = ~ , + ~ (Se] e - ' " + 5e~: e"°') + ~t(5e2 e-2"°~ + 5~:~ e-'"°') + - ' - ,

(13)

N = A'~, + ~(?(, e -i~'' + jV'~ e ''°') + ½(~'~ e-2"°' + ~..~ e2,O,) + . . . .
where ~o, ~t, ~2 . . . . .
~ , get, ow:. . . . . . . ~,, ?,"~, ?,\ . . . .
functions and consequently we call them " a m p l i t u d e s " .

are slowly varying
For amplitudcs of

A. Sandulescu a~d E. Stefanescu I New optical equations for atom-fieM interaction

531

order zero and one, from (12) we obtain the following equations:
dCJ'~
dt

+ ( T '~ +

~aS¢, +

ia), 9°, = -T,W,,

% - i('/2 - 2 g o )

2

WI + ix, N o = 0 ,
(14)

dXo
m + ~,(x0 - N3)+ ,,,, ~ o = '~(Se,x ; - ~eTx,),
dt

dSro
dt

+

T:.~,

+ ~;(6) - N,) = o ,

where X0 = ( ~ l h ) ~ o , X , = ( l x l h ) ~
, N, = NDIly ~ .
From these equations we observe that the new parameter Yt # 0 is essential
for the existence of a coupling of the polarization variable b~l with the electric
field variable X~, i.e. the existence of an interaction of the atom with the
electric field.
In this case, one obtains the polarization equation

d.Y~

(T'~ - iT~(T~ - 2x.)fT.

d----~ +

)
2ix..,~,';.
+ iA ~, = 1 -- i(y: - 2X,)) IT, '

1 - i(Y2 - 2X())/TI

(15)

which can be compared with the conventional Bloch polarization equation

dm,
d--T + ( T l + ia)Sel = iX,~o -

(16)

We can easily see that the new polarization equation (15), in comparison
with eq. (16), describes new physical phenomena. For instance, in the steady
state, with the notations
-

,

Ti

~=
o

2 F,
Yi

TtI=Z!

1

--T ±TII

N3

'

Yl
2

1

X~=

2

(17)

lit

Nc:

,

/..1
'

8--

t,

X1
,

E = - - "

l

T l ,'YII

one obtains the polarization amplk.:de
~,=2N c

X.,

[8

~ +i(i + FS)le

T: (l+FS)~- + ( 8 - ~)" + ( 1 + r s ) l e l ~"

(18)

532

A. Sandulescu and E. Stefanescu I New optical equations for atom-field interaction

For the slowly varying amplitudes Xt+ and Xt- of the field variables, Xt =
Xt+ ei*~ + Xt- e-ik:, the Maxwell equations take the form [10]

dxt+

/xg
- i N .~,+,

dxt/xg 5Pl_
dz = - i -~,
X0 =

(19)

t~ G , ,
e0 h

where g = o#.L/2e,,, 5¢I + = (k/2"rr) .l'r,, Yt (kz) e -'k: dz, ~ _ = (k/2rr) J'~,,.fft(kz) ×
eik"dz. Eqs. (19) with expression (18) describe the propagation of an
electromagnetic plane wave, taking into account the self-reflection effect [14]
and they will be used in Section 4 for the derivation of the transmittivity
characteristic of an optical bistable device with a Fabry-Perot cavity. However.
neglecting the counter propagating wave due to the self-reflection, one obtains
the absorption coefficient

=

1 dlel
o~,,
l+Fa
I~1 dz = T (1 + ra): + ( a - ¢): + ( 1 + ra)l~l-"

(21))

and the corresponding dephasing
dO

t~-

1+1"6a ,
a o = 4#gN~/ehy ~.

(21)

dz

where
From these expressions one observes that the absorption coefficient a and
the dephasing dO/dz have an asymmetry as a function of the atomic detuning
which is not contained in the Bloch equations. For the condition 1 + / ' 6 < 0,
the absorption coefficient a becomes negative, i.e., due to the coupling of the
polarization with the population through the environment, the electromagnetic
wave is amplified.
From (14), (17)-(19) we also observe that the quantities F and ~ are
functions of I~!-',
tg

l"= 3'2 + _/z-.,~,
"/1
e.h',/.[
f

~ - ~'= F .

Y=
t?

D~

N %.',, "

1+

1)~ ra
(1 + r a

)

+ (a - ~)-~ I~l~
(22)

533

A. Sandulescu and E. Stefanescu / New optical equations ]'or atom-field interaction

With these expressions, the absorption coefficient a and the dephasing dO/dz
take an additional nonlinear dependence on the electric field connected with its
zero-point value )o, or with the zero-point polarization ~,. For rather small
values of the atomic density.
X ~

eoh y ~y,_

(23)

,

21.t2yz

this nonlinearity can be neglected.

4. Fundamental constraints and uncertainty relations

From the Schwartz inequality, one obtains the following inequalities for the
phenomenologicai parameters of the master equation (4):

t,",x'-.
D qq. >~ 0 ,

Dye . >i 0 ,

i 2

i"

A.I.:.A~.: - A.,:. i> ~D.,:.

,

A~.,~O,

A(.: ~ 0 ,

A i,, ~ 0 ,

DppnDqq,, - O ~ q n ~

,

(24)
•"

i2

J
i
/A.,.,.A:.,.
- r',.:
>~ ~D,._.

i2
A ,J. : A , .i , . - F:.,
1> ] D~.:.

Due to their generality we call these incqualities fundamental constraints.
From the inequ'dity
(25)

T r ( p V : V,,) >t Tr(pV,:) Tr(pV,,)

valid for every positive mapping V,,---,Tr(pV,) [1]. one obtains the following
uncertainty relations:
~l 21~n

D,~q.Am,,,(t ) + D~w Aqq~(t ) - 2Dpq. Apq,.(t)/>
Z l T. x l m ~ ~. ,¢ ~. g ]

,

a l* ~, ~

-.

"J

~.6~

Ax:.A=(t ) + A . , J y : . ( t )
J

, J

.'

,'
- 2 F y A-y~(t)
I> D,..' ( o r : ,' ( )t ) ,
J

.J , J
;
J
A~.:A~x(t) + Ax:,a::(t)
- 2r~x,a:~(t)
> / .o,

j

( o-',(t)),

where Apv,,, A q q n , . . • are the variances of the observables p, q . . . .
quantities Apqn(t ), . . . are of the form

and the

534

A. Sandulescu and E. Stefanescu / New optical equations for atom-field interaction

At,q,,(t) = Tr ~p(t)
/
Pnqn + qnPn)\ _ ( p ~ ( t ) ) ( q , , ( t ) )

~

°

•

~

•

From (24) and (26) we observe that we have obtained a number of inequalities
equal to the number of phenomenological coefficients•

5. Open quantum systems and optical bistability
We consider an optical bistable device [7-19] with a non-linear medium of
two-level atoms and a F a b r y - P e r o t resonator with mirrors of reflectivity R,
T = 1 - R (fig. 1). g'~ is the electric field amplitude of the incident wave, ~T is
the electric field amplitude of the transmitted wave and ~+, g'_ are the electric
field amplitudes of the waves propagating in the two directions of the z-axis.
We take the variable of the electric field in the cavity of the form X~ =
x ~ ( f e - i t k z + ° ) - b ei~kz+°)), where f and b are the normalized slowly varying
amplitudes of the electric field waves and 0 the dephasing due to the
interaction with the atomic medium.
We define the McCall coefficients a o, a t from the Fourier expansion [8, 9,
121

( l + F{~) 2 + (8 -- ~:)2 + (1 +

Fs)I I-"

= a~} + a 1 e i{2~z+2°) + a I e -i(2kz+2°) + . . . .

(27)

When thc condition (23) is satisfied, relation (27) yields
a o = [ ( 8 - ~)2 + (1 + / ' 3 ) ( 1 + F{5 + b 2 + f : -

2bf)] -t/z

x [ ( 3 - ~:)2 + ( 1 + FB)(1 + F6 + b 2 + f 2 + 2 b f ) ] - , / 2

1
a, = 2(1 +

£~)bf {[('~

- ~:)-' + ( 1 + F,~)(1 + £ ~ + b 2 + f 2 ) ] a 0 -

!
I
!

!
!

o

t

2

Fig. 1. Schematic d i a g r a m of an optical bistable F a b r y - P e r o t r e s o n a t o r .

(28)

1}.

A. Sandulescu and E. Stefanescu / New optical equations for atom-field interaction

535

Neglecting the phase deviation 0 on the length l of the resonator, from (18)
and (27) one obtains
2N ~
St~ : X~ ---z-,y±[6 - ~ + i(l + F6)](-aob + a~f),
(29)

2N e
Sl_ = X.~ _."2-57-,[6 - ~: + i(1 + F~$) ] ( a , , f - alb ) .

On the other hand, we consider the "mean-field approximation" [ 16], when the
electric field amplitudes can be written in the form

f(z)= x[1 + a(z)l,

b(z)=

X/-R xI1 -

a(z)],

(30)

with the conditions A ,~ 1, O "~ 1, where x = f(0).
Integrating the field equations (19) with expressions (29) and (30) one
obtains

o,[ 1- (1 + ( 1 + r a

a,=~(t)=~,

+(a-~)-"

'
(31)

O, = O ( i ) = 1 + 1-'6 A .

A much simpler form for the parameter 31 can be obtained, considering
f b ~ ( 1 + 1-'6)2 + ( 6 - ~5)2 + ( 1 + ra)(f: + b:),
(32)

%1/2

AI

(1 + r a ) -~+ ( a -

~:)-" + ( 3 + 2 r a ) x -~ "

From the boundary conditions for a Fabry-Perot cavity we obtain a
transmission characteristic as a function of the propagation parameters A~ and
01,
A_.2
lyl 2 = x:{ ! [1 + R -~ - 2R cos(20, + 2~)1 + 4
T2
T
+ 211 + cos(20, + 24~)] --A~}
T2

,

(33)

where y = XJ(X,~/T) with Xt = (l~/h)~1 is the input field variable and ~b is the
cavity detuning. The first term describes the dispersive optical bistability and
the last two terms the influence of the absorption. For a small detuning
2(0 + 40