Download Atom-radiation Interaction

Atom-radiation Interaction
• Einstein’s coefficients represent a phenomenological
description of the matter-radiation interaction
• Prescription for computing the values of the A and B
coefficients
• No definition of the validity limits
• Need for a more general theory for transition
probabilities
F. De Matteis
Quantum Optics
L2/1
Atom-radiation Interaction
Ηˆ   i d
dt
Atom with no radiation. No explicit time-dependency
Atomic
Hamiltonian
n r , t   exp  iEnt /  y n r  Hˆ E y n r   En y n r 
yn stationary states Average value of any observable is time
independent.
Let’s look at a two level system
1 r , t   exp  iE1t /  y 1 r 
2 r , t   exp  iE 2t /  y 2 r 
F. De Matteis
Quantum Optics
0  E2  E1
2
Atom-radiation Interaction
Atom in presence of an em radiation field.
ˆ  Η
ˆ Η
ˆ t   Η
ˆ
Η
E
I
R
HE Atomic
Hamiltonian
HI Interaction
Hamiltonian
HR Radiation
Hamiltonian
HR describes the em field energy
HI describes the interaction of the em field with the two level atomic
system
yn no more stationary states Linear combination of stationary states
with time dependent coefficients Ci(t)
r , t   C1 t  1 r,t   C2 t  2 r,t 
 r, t 
F. De Matteis
2
dV  C1 t   C2 t   1
2
Quantum Optics
2
3
Interaction Hamiltonian
Atom in a radiation field
Ηˆ
E

d2
dC
dC2 
 d
 Ηˆ I C11  C2 2   i C1 1  C2
 1 1  2

dt
dt
dt
dt 

r , t   C1 t  1 r,t   C2 t  2 r,t 
 r, t 
2
dV  C1 t   C2 t   1
2
2
Let’s left-multiply by the atomic problem solutions i and
integrate
dCi
* ˆ
  Η I C11  C2 2 dV  i dt
i
C1P11  C2 exp  i0t P12  i dC1 dt

 C1 exp i0t P21  C2 P22  i dC2 dt
F. De Matteis
Quantum Optics
Pij  y i r H Iy j r dV
*
4
Atom-radiation Interaction
40  2
11
a0 

5

10
m
2
me
10  4  ka0  1
Dipole Approximation
Eo cos t
N
d   e rj
j 1
k
Hˆ I  d  Eo cos t
Pij  y i r H Iy j r dV
*
Bo cos t
C2 exp  i0t P12  i dC1 dt

 C1 exp i0t P21  i dC2 dt
 r, t 
F. De Matteis
2
P12  P21
*
P11  P22  0
P12  E0 d12  cos t  V12 cos t
dV  C1 t   C2 t   1
2
Quantum Optics
2
5
Transition Rate
 V cos t exp  i0t C2  i dC1 dt C1 0  1
 *
 V cos t exp i0t C1  i dC2 dt C2 0  0
C2 t 
t
2
 B12W  
V  V12  E0 d12  
12 Transition Rate
for absorption
V  0  E2  E1  
Small perturbation
Solve by successive approximations
Ci≈Ci(0)
C1 t   1


*
1  exp i0   t  1  exp i 0   t 
V
C t  



2

2 
0  
0  


Let’s substitute in the first member of equation
F. De Matteis
Quantum Optics
6
Transition Rate
 V cos t exp  i0t C2  i dC1 dt C1 0  1
 *
 V cos t exp i0t C1  i dC2 dt C2 0  0
Small perturbation
Second order approx.
V  0  E2  E1  
2

C1 t   1  V  f t 

V * 1  exp i 0   t  1  exp i 0   t  



C2 t   2 
0  
0  



C2 t 
t
2
 B12W  
2
2




W

d


E

V
0

Einstein’s theory of absorption and emission apply to
not too-strong fields
F. De Matteis
Quantum Optics
7
Rotating wave Approximation


*
V
C t  
 2
2

C1 t   1
1  exp i0   t  1  exp i 0   t 



0  
0  


Second term in C2(t) is much larger than first
C2 t 
2
sin 2 0   t 2
1 2 2
2
0   t 2
V

V
t
sinc
2
4
0   
1 2 2



V t
0
4
2
Increasing t 
Max increases
Nodes approach the
origin of axes
F. De Matteis
Quantum Optics
8
Rotating wave Approximation
Transition Frequency 0 ±
1
2


W

d



E
0
0

2

 
V12  E0  D12 
 E0eX 12  

C2 t 
2 e X12

 0 2
2
2
1
2 0  2 

1
2
0  
sin 2 0   t 2
W  
d
2
0   
Einstein’s theory rely on the assumption of broad-band illumination of the
atomic system (the transition line is totally covered) W()~W(0) in the
interval 
t
C2 t  
2
2e 2 X12 W 0  t
2
 0 2
4
2  t
4
sin 2 x
dx
2
x 
2e 2 X12 W 0  t 2
2
C2 t  
 t  1
2
 0
4
2
2
2
e
X
2
12 W 0   t
C2 t  
t  1
2
 0
2
2
F. De Matteis
Quantum Optics
9
Einstein’s coefficients
System of N identical two-level atoms
eX 12  y 1*ε  Dy 2 d 3r  y 1*ε  ery 2 d 3r
Electric Polarization
Vector
X12
 ε  D12
2
 D12 cos 2  
2
C2 t 
2
t
 B12W  
 D12
3 0 2
2
B12 
2

1
2
D12
3
Spatial orientation of dipole moment is
random in an atomic or molecular gas.
Let’s take an average over the possible
orientations
t  1
A semiclassic theory
doesn’t include
spontaneous emission
C2 t  
2
A21 
2
/ D12 W 0   t
2
 0 2
2/
g103 D12
2
 g
B12 
 cg
30 g 2 c 3
3
0 1
2 3
2
03 D12
me10

 8
 6 108 s 1
3
5 6 3
90c
3 40  c
2
Hydrogen
atom
F. De Matteis
Aps
A
1

  1.6 ns
Quantum Optics
10
Optical Bloch Equations
Monochromatic em field at frequency 
 V cos t exp  i0t C2  i dC1 dt C1 0  1
 *
 V cos t exp i0t C1  i dC2 dt C2 0  0
Quadratic terms of coefficients Ci
 11  C1 2  N1 N

2
  22  C2  N 2 N

  21  C1C2*

 12  C1*C2
 d 22 dt   d11 dt 

  i cos t V * exp i0t 12  V exp  i0t  21
d dt   d * dt  iV cos t exp  i t    
21
0
11
22
 12


d
d
d
ij  Ci C *j  C *j Ci
dt
dt
dt
11   22  1
According to the rotating-wave approximation
 21  12*
 d 22 dt   d11 dt 

  i 1 V * exp i0   t 12  V exp  i0   t  21

2

1
*
d

dt


d

dt

i
V exp  i 0   t 11   22 
 12
21
2


F. De Matteis

Quantum Optics
Optical
Bloch
Equations
11
Rabi Oscillations
Let’s take the particular case
2200 1200
The solution of Bloch eq is:
 2  0     V
2

2


2
2
2  t 


V
/

sin
 
 22
 2 


  exp i   t V /  2 sin  t       sin  t   i cos t 


12
0
0

2
2
2











Rabi Oscillation where |V| is Rabi frequency
The solutions are derived assuming strictly monochromatic light (frequency ) i.e.
A width of the incident frequency distribution smaller than the atomic transition
linewidth.
The broadening of the transition line modifies the Bloch equations.
|V|<<(0) limit yields the Einstein coefficient treatment
F. De Matteis
Quantum Optics
12
Radiative broadening
The theory of absorption and emission contains an intrinsic line broadening
mechanism linked to spontaneous emission.
Effect of spontaneous emission in the derivation of the susceptibility of a gas of
2-level atoms


E (t )  E0 cos t 
1 
 E0 exp it   exp  it 
2


P(t )   0 E t  
1 
  0 E0     exp it      exp  it 
2
 Z


d t     t  e  ex j  t dV
 j 1

*
X

N 
P(t )  d t 
V
Take a two level system


d t   e C1*C2 d12 exp  i0t   C2*C1d 21 exp i0t 
 X 22  X 11  0

*
 X 12  X 21
F. De Matteis
Dipole moment d is an observable  real quantity
Quantum Optics
13
Radiative Broadening
The rate equations lead to steady-state wavefunction  time-independent in the
absence of any applied em field.
If initial state is the excited state, a way must exist to relax to fundamental state.
Let’s introduce a term to represent the spontaneous emission
V cos t exp  i0t C2  i dC1 dt
V  E0 d12 
V * cos t exp i0t C1  iC2  i dC2 dt
With no-incident radiation we easily get the solution
C2 t   C2 0exp   t 
N 2 t   N C2 t   N 2 0 exp  2 t 
2
2  A21  1
R
The susceptibility is found by substitution of the rate eq. solutions
Correct to first perturbative order in V  C1=1 e C2(t)
1  expi0   t expi0   t
C2 t    V * 

2  0    i
0    i 
F. De Matteis
Quantum Optics
C1 t   1 o ( V )
2
14
Radiative Broadening
Substitute the C2(t) expression in the atomic dipole moment
 exp it 
1 d 21E0
exp  it  exp  it 
exp it  
d t   
d12 




2 




i





i





i





i

0
0
0
 0

Pt  
N
1
d t    0 E0    exp  it     exp it
V
2
2
N d12 

1
1

       *  
   

3 0 V  0    i 0    i 
K

 ' ' ( )
c
Making again the rotating wave approximation, we end up with
N d12 0
 
K
3 0 cV 0   2   2
2
F. De Matteis
Lorentzian Lineshape.
Natural linewidth of the spectral line
Quantum Optics
15
Lorentzian lineshape
 
FL   
0   2   2
Lorentzian lineshape around the
angular frequecy 0.
Unit Area
FWHM  2  A21  
1
FL 0  

Hydrogen Atom
03 D12
me10

 8
 6 108 s 1
3
5 6 3
90 c
3 40  c
2
Ap  s
A
1
r
F. De Matteis
 r  1.6 ns
108
     
3 1015
Minimal transition linewidth.
The linear time dependency of the
transition probability ( expression for B)
is valid for times sustantially longer than r
Quantum Optics
16
Power Broadening
2
N d12 
1
1

   


3 0 V  0    i 0    i

       *  

Correct result to the second order in the dipole matrix element D12 for the linear
response of the atomic gas to the incident beam.
Generalized maintaining the rotating wave approx and including the effects of
spontaneous emission
d 22 dt 

 i 1 V * exp i0   t12  V exp  i0   t 21  222

2

1
d

dt

i
V exp  i0   t11   22   12

12
2



Only rate eq. for C2(t) has been modified.
The spontaneous emission introduces a damping term in the solutions which
are no longer purely obscillatory. After a sufficiently long time the system
reaches a stedy state.
F. De Matteis
Quantum Optics
17
Power Broadening
d 22 dt 

 i 1 V * ~12  V~21  222

2
 ~
1
~
~
d12 dt  i V 11   22   12  i0   12
2

Bloch equations become in
virtue of the substitution:
~12  expi0   t12
~   *
21

12
and complex conjugate for 21
Setting all the rate to zero, the
equilibrium steady-state solutions
d t  
  21d12 exp  i0t   12d 21 exp i0t 
Pt  

N
d t  
V
2

V 4
  22 
0   2   2  V 2 2


   exp  i    t 12 V 0    i 
0
 12
0   2   2  V 2 2

1
  0 E0    exp  it     exp it
2
2
No more linear because of |V|2

N d12 
0     i
       *   in the denominator
   
2
2
2
3 0V  0       V 2 
2
  0       V
2
F. De Matteis
2
2
2

FWHM  2  2  V
2
Quantum Optics

2
18
Collision Broadening
The collisions between atoms in a gas during the interaction with the em radiation
produce further line-broadening
p( )d  1  0  exp   /  0 d
4a 2 N

0
V
1
k BT
Probability of free flight time  between collisions
0 free flight mean time 4a2 collision cross-section
M
Instantaneous Collision (<< 0)
Collisions influence optical processes only indirectly by changing the state of the system
before and after the collision.
Inelastic Collisions Additional decay terms for the atomic level populations besides to
the radiative one 
Elastic Collisions  the atom remains in the initial state with a phase change of the
wavefunction
An additional decay rate in the off-diagonal Bloch equations
d12 dt 
iV
exp  i 0   t11   22    ' 12
2

 '  V 2 4
  22 
0   2   '2  V 2  '   2


0    i 'V 2
   exp  i   t
0
2
2
2
 12


 '   2





'

V
0

F. De Matteis
Quantum Optics
 '     coll
 coll  1  0

FWHM  2  '2  V  '   2
2

 0  3 10 11 s   R 100
19
Doppler Broadening
During the absorption (and emission) of a photon by means of an atomic gas, the
total momentum must be conserved too
 E  1 Mv 2    E  1 Mv 2
E2  E1
    v1z c    2 / 2 Mc 2

5
 v1z   v1z c  10
   0 1   
c   / 2 Mc 2  10 9


0 
1
1

Mv1  k  Mv 2

2
2
2
2
The distribution of the absorption frequencies
mirrors the Maxwell distribution


exp  M v z  / 2k BT dvz 
2


 exp      / 2 c  d
 exp  Mc 2   0  / 202 k BT c 0 d 
2
2
2
0
0
Gaussiana lineshape
FWHM  2  2 2 ln 2 
FG   
F. De Matteis
1
2
2
20
c

k BT / M 2 ln 2
exp    0  / 2 2
2

Quantum Optics
20
Composite Broadening Mechanism
When different lineshape broadening
mechanismes are simultaneously
present, the lineshapes combine by
convolution of the functions
F  ; 0    F1  ; 0 F2  ; 0   d
The combination of two lines of the same type gives a lineshape
with a width equal to the sum of the widths
  1   2
2  21  22
The combination of two lines of different type gives a composite (Voigt)
lineshape
Lorentzian lineshape  Homogeneous broadening mechanism
Gaussian lineshape  Inhomogeneous broadening mechanism
For homogeneous broadening the uncertainty principle applies
(consequence of the finite time during which the atom can emit/absorb
undisturbed)   property of the Fourier transform
  t  1
F. De Matteis
Quantum Optics
21
Optical BlochEq. / Rate eq.

d
i *~
d






V 12  V~21  222
11
 dt 22
dt
2
d
i
 ~12  V 11   22   i 0      '~12
2
 dt

 '     coll
2  1 
 22 0  0
12 0  0
2   ' 
' 





exp

2

t

    2   '2    2  2   '2

0
 0

2
2
V  2 0      ' 2   ' cos0   t   40    ' sin 0   t 
 22 


4 
0   2   '2 0   2  2   '2

 exp   ' t






12 




iV 2
exp   ' t   exp  i0   t  12 0exp   ' t 
i 0      '
d
d
N1   N 2  N 2 A21  N1 B12W    N 2 B21W  
dt
dt
N2 0  0
F. De Matteis
N 2 t   NBW A1  exp  At 
Quantum Optics
Weak incident
beam
22
Optical Bloch Eq./ Rate eq.
Two distinct regimes in which the solutions of Bloch eq. Resembles the time dependence of the
excited-state population obtained by the rate equations (Einstein coefficients )
2   ' 
' 


    2   '2     2  2   '2 exp  2t  

0
 0

2
2

V  2 0      ' 2   ' cos0   t   40    ' sin 0   t 
 22 


4 
0   2   '2 0   2  2   '2

 exp   ' t










2. Collision broadening much greater than radiative one
’
t  1   1  '
 22 
V
2
' 
1  exp  2t 
2
2
4 0      '
N 2 t   NBW A1  exp  2t 
1. Broad band incident light
2’


 22d   V / 4 1  exp  2t 
2
 '   
The rate equations are valid in general when:
• The bandwidth of the incident light exceeds the
atomic transition linewidth
• The dephasing broadening (collision + Doppler)
greatly exceeds the radiative linewidth of the
transition
F. De Matteis
Quantum Optics
23