Steady state response of the three-level system

In document The dynamic Stark effect in a J equals 0 to 1 to 0 three-level system (Page 51-55)

Probe laser frequency

Chapter 4: Theory of the weak probe experiment

4.1 Steady state response of the three-level system

The atomic density operator pA is defined as follows in terms of the upper (|a>), middle (|b>) and ground (|c>) states of the 3LS shown in fig. 4.1;

C h a p t e r 4 45

c c

P

a

=

E

E

PAij I i><31

(4.1)

i=a j=a

w h ere

pA i j

r e p r e s e n t s a m a t r i x e l e m e n t of

p A .

The c o m p l e t e d e n s i t y o p e r a t o r

p s

for the s y s t e m c o n t a i n s a f i e l d c o m p o n e n t

p F

. For systems w h e r e the atoms have

n e g l i g i b l e effect on the a p p l i e d fields ('no b a c k reaction'

assumption),

p s

m a y be written:

The tw o r a d i a t i o n f ield modes shall be d i s t i n g u i s h e d b y the

s u b s c r i p t

X,

w h e r e

X=1

refers to the p r obe field w h i c h is t u n e d close to r e s o n a n c e w i t h the |c> — > |b> t ransition, and

X=2

refers to the s t r o n g field, w h i c h is t u n e d close to r e s o n a n c e w i t h the |b> — > |a> transition.

The r a d i a t i o n fields, b e i n g p r o d u c e d b y lasers p u m p e d well

above threshold, are a s s u m e d to be in c o h e r e n t states | a^>

(Loudon 1973, c h . 7). The two fields are a s s u m e d to be

u n c o r r e l a t e d a n d m o n o c h r o m a t i c . The a b s e n c e of b a c k r e a ction

m e ans that the states of the tw o r a d i a t i o n fields are not

a l t e r e d by i n t e r a c t i o n s wi t h the atoms. This implies that

I 0C^|2 » 1. Therefore; Ps “ Pa Pf (4.2) 2

P

f

= n lak><akl

X=i

(4.3)

C h a p t e r 4 46

or, in terms of the h a r m o n i c o s c i l l a t o r n u m b e r states |n^>;

2 PF = n exp (- I <XjJ 2 ) X=l

E E

|gx l exp{icp^(n^-mx) } In^XmjJ

K

!

n

^ !)1/2 (4.4) w h e r e is a c o m p l e x n u m b e r w i t h p h a s e (p^ d e s c r i b i n g th e

state of the f i e l d m o d e

X,

an d the o c c u p a t i o n n u m b e r s of the h a r m o n i c o s c i l l a t o r states are n^ an d m^.

The i n t e n s i t y of the f l u o r e s c e n c e e m i t t e d fr o m a s y s t e m whose

d e n s i t y o p e r a t o r is

ps

is g i v e n by the e x p e c t a t i o n v a l u e of the f l u o r e s c e n c e m o n i t o r i n g o p e r a t o r L F (Corney 1977, p. 503) :

LF = ]£ a-JA lixil a*-li

(4.5)

The s u m m a t i o n is c a r r i e d out over the final (atomic) states of

the s p o n t a n e o u s t r a n s i t i o n s c a u s i n g the f l u o r e s c e n c e . In

e q u a t i o n 4.5 a co n s t a n t of p r o p o r t i o n a l i t y has b e e n d r o p p e d

for simplicity, e is the p o l a r i z a t i o n v e c t o r of the

f l u o r e s c e n t light and Jl is the e l e c t r i c d i p o l e o p erator. The

e x p e c t a t i o n v a l u e < L F> of the f l u o r e s c e n c e m o n i t o r i n g o p e r a t o r

is g i v e n by;

< L p> = T r ( p s Lf ) (4.6)

where the t r a c e is t a k e n over b o t h the a t o m and field

C h a p t e r 4 47

In the 3LS shown in fig. 4.1, < L F> is s i m p l y p r o p o r t i o n a l to the d i a g o n a l a t o m i c d e n s i t y m a t r i x e l e m e n t s (populations)

pAaa

and of the u p p e r a n d m i d d l e l e v e l s . The p r e s e n c e of

c o h e r e n c e s b e t w e e n the h y p e r f i n e c o m p o n e n t s of the m i d d l e levels of the od d i s o topes p r o d u c e s a d d i t i o n a l t e r m s in < L F> i n v o l v i n g off d i a g o n a l d e n s i t y m a t r i x elements.

The e q u a t i o n of m o t i o n (Liouville equation) for the d e n s i t y o p e r a t o r

p

m a y be written;

w h ere the dot d e n o t e s d i f f e r e n t i a t i o n w i t h re s p e c t to time, the square b r a c k e t s r e p r e s e n t the c o m m u t a t o r product, E is an o p e r a t o r g o v e r n i n g r a d i a t i v e d a m p i n g of b o t h d i a g o n a l

(populations) an d off d i a g o n a l (coherences) a t o m i c d e n s i t y m a t r i x e l e m e n t s a nd TT is P l a n c k ' s c o n s t a n t d i v i d e d b y 2k. In an i n t e r a c t i o n p i c t u r e w i t h r e spect to the r a d i a t i o n fields, the H a m i l t o n i a n H(t) c o n s i s t s of an ato m i c part and a time d e p e n d e n t i n t e r a c t i o n part:

The c o m p o n e n t s of the H a m i l t o n i a n in the r o t a t i n g wave and e l e c t r i c d i p o l e a p p r o x i m a t i o n s are d e f i n e d as follows;

p s

= (i TT)"1 [H

(t

) ,

p s ] - X p s

(4.7)

Chapter 4 48

HAtom li> = Ei 'i;> i=a,b,c

2

In document The dynamic Stark effect in a J equals 0 to 1 to 0 three-level system (Page 51-55)