Semi-classical introduction
Atomic Hamiltonian:
(ground state is 0 of energy)
Dipole operator:
H A  0 b b
0  Eb  Ea
 
a
D  d ab ( a b  b a )




d ab  b D a  a D b  e a r b
Semi-classical laser field:
Interaction Hamiltonian:
b
a
b
 
 
E(r , t )  E0 (r ) cos( Lt )
VAL
 
  D  E (0, t )
b
Stimulated
absorption
a
atom at
origin, dipole
approximation
Stimulated
emission
Schrödinger equation
General state of atom:
ca (0)  1
Initially:
Schrödinger Eq:
(1)
   L   0 (2)
(t )  ca (t ) a  cb (t ) b ei0t
cb (0)  0
 
1
ica (t )  a ( D   ) E0 b cb (t )(eit  e i (  20 )t )
2
 
1
icb (t )  b ( D   ) E0 a ca (t )(e it  ei (  20 )t )
2
Perturbation theory
Solution in
atomic basis
Dressed state
Perturbation theory
b VAL a   0
ca (t )  1
 
1  (d ab   ) E0 / 
i1t
I (t )  I (t  2 0t )
cb (t )  
2
1 t
2 2
cb (t ) 
2
4
I (t)
I (t ) 

2  i t / 2
e
sin( t / 2)
t

 I (t  2 0t )  2 Re I (t ) I (t  2 0t )
2
2

1
0.8
0.7
|I(t)|
Finite time
“delta” function
0.9
Fast oscillating overlap term
0.6
0.5
0.4
0.3
0.2
0.1
0
-40
-30
-20
-10
0
t
10
20
30
1 t
2
cb (t ) 
2
t 
40
2
 ( )   (  20 ) Fermi golden rule
Perturbation theory (2)
cb (t ) 
2
1    0
1
2
2
sin 2 (t / 2)
|cb(1t)|2
0.01
0.009
perturbation theory
0.008
0.007
2
|cb(1t)|
" exact" theory
0.006
0.005
0.004
0.003
0.002
0.001
0
0
0.5
1
1.5
2
2.5
1t
3
3.5
4
4.5
5

1

1
 10
 10
“Exact” solution in atomic basis
 ca (t )   
0
  
i
i (  2 0 ) t
it

c
(
t
)
2

e

e
 b 
 1




1 eit  e i (  20 )t  ca (t ) 


 c (t ) 
0
 b 
Rotating wave approximation
 ca (t )    0
  
i
it

c
(
t
)
2

e
 b 
 1
ca (t )  e
  12   2
it / 2
1eit  ca (t ) 



0  cb (t ) 



cos(

t
/
2
)

i
sin(

t
/
2
)





1 it / 2
cb (t )  i e
sin( t / 2)

Atomic basis occupations
|c (t)| 2
 0
b
1
0.9
  0.21
0.8
  21
|c (t)|2
b
0.7
  2.21
0.6
0.5
0.4
0.3
0.2
0.1
0
0
5
10
15
t
20
25
30
Dressed state
Complete Hamiltonian:
H  H A  VAL
Diagonalize
New basis, in which initial condition (all the population
in the ground state) is not an eigenstate
Rabi oscillations are naturally understood
Dressed state (2)
Comfortable ansatz:
Rotating wave frame (not an
approximation!)
(t )  ca (t ) a  cb (t )eit i0t b
Schrödinger Eq:
 ca (t )    0
  
i
 cb (t )  2  1
1  ca (t ) 



 2  cb (t ) 
Time independent matrix!
Dressed state (3)

Eigenvalues:
1
   
2 2
 u 
1
  
12  (   ) 2
 v 
   


 1 
  1
 0
 
1
b
a
 ca (t )    0
  
i
 cb (t )  2  1
1  ca (t ) 



 2  cb (t ) 
1   2
2
 u 
1
  
12  (  ) 2
 v 
  


 1 
  1
12
Eb  
4
12
Ea 
4
1
 
 0
  1