Download Why spontaneous emission

PG lectures 2004-05
Spontaneous emission
Outline
Lectures
1-2
Introduction
What is it?
Why does it happen?
Deriving the A coefficient.
Full quantum description
3-4
Modifying the ‘environment’.
Two atoms: superradiance
A mirror: Cavity QED.
Oscillating charge
2p(m=0) to 1s
2p(m=1) to 1s
2p(m=1) to 1s
Dipolar radiation
2p(m=0) to 1s
2p(m=1) to 1s
Spontaneous emission: what do the
greats have to say?
Dad
I understand that light is emitted when an atom decays from an excited state to a lower energy state.
Feynman
That’s right
Dad
And light consists of particles called photons
Feynman
Yes
Dad
So the photon ‘particle’ must be inside the atom when it is in the excited state.
Feynman
Well no.
Dad
Well how do you explain that the photon comes out of the atom when it was not there in the first place.
Pause
Feynman
I’m sorry. I don’t know I can’t explain it to you.
(Feynman, Physics Teacher 1969)
The light quanta has the peculiarity that it apparently ceases to exist when it is in one of its stationary states, namely
the zero state….When a light quanta is absorbed it is said to jump into this zero state and when one is emitted it can
be considered to jump from the zero state to one in which it is physically in evidence, so that it appears to have been
created. Since there is no limit to the number of light quanta that may be created in this way we must suppose that
there are an infinite number of light quanta in the zero state. (Dirac 1927)
What is this zero state? Vacuum energy.
Second quantisation of the electromagnetic field
see Loudon The quantum theory of light pp. 130-143
The idea of a photon is most easily expressed for an EM field inside a …perfectly reflecting cavity. Loudon p. 1
One mode of EM field
creation
1/ 2
  
ˆ
 ( aˆk  aˆk* ) ê
Ak  
 2 0V 
k
destruction
=1,2 polarisation
(n  12 )
a*
a
n photons
1
2

Vacuum state
University
of Durham
Quantum theory of spontaneous emission
Statement of the problem
1. Interpret spontaneous emission as:
transition induced by the vacuum field.
(a) only accounts for half the spontaneous decay rate
(b) vacuum fluctuations alone also lead to the
wrong sign of the electron spin anomaly g-2
2. Solution: vacuum fluctuations + self-reaction
(the interaction of an electron with its own field)
both contribute to observable processes.
3. However, their respective contributions cannot
be uniquely defined.
4. A matter of interpretation!
Outline
1. Semi-classical: Fermi’s golden rule
vacuum only accounts for half the decay rate.
2. QED: Radiative reaction or self reaction
(electromagentic mass)
3. Decay rate and level shifts (Lamb shift)
due to radiative reaction
4. Vacuum fluctuations + radiative reaction
excited states decay at a rate G
lowest energy state does not decay
P. W. Milonni, The quantum vacuum, (Acad. Press, 1994).
P. W. Milonni, Why spontaneous emission? Am. J. Phys. 52, 340 (1984).
2-level time dependent perturbation theory
  C1 1 e -iE t /   C2 2 e -iE t / 
1
2
i t   ( H 0  H ' )
where H '  er 0 cos t
electric dipole approximation
V12

C1 
C2 cos t e-i0t
i
2
V12 sin 2 (   0 )t / 2
2
C1  2

(   0 ) 2
Decay rate due to vacuum fluctuation
C1
2
2
sin 2 (   0 )t / 2

d ( )
2

2 2
1
(



)
0
2  0 
V12
0
1
 3
 ( )  g ( )   2 3
2
2 c
V12

2
 e 1 r cos 2
2
2
2
1 2
 e 1r 2
3
0
  C1 
e r12  3
2
G

3
60c
2
2
2
1 2 2
 e r12
3
Field produced by the atom – the source field
H  H atom  H int  H field
H int
e2

( p.A  A.p)
2m
H field   (a*a  12 )
H atom
p2

V
2m
1/ 2
  
 (a  a * ) ê
A  
 2 0V 
electric dipole approximation
1
ie
a  a, H   ia 
r.ê
1/ 2
i
(2 0V )
free field
source field
t
ie
it
i ( t t ')
a(t )  a(0)e 
d
t
'
e
ê .r(t' )
1/ 2 
(2 0V ) 0
Radiative reaction


Cut-off
sr
1/ 2
  
  (ak  ak* ) ê
 i
 2 0V  k
e
e
r m (0) 
r

3
3
30 c
60 c
 m (t ) 
m
-it
e
 d
 m
mc 2
 0   m 

Add source field to classical equation of motion
mr   02 r  e(   sr )  e 
e2
30c
3
r m (0) 
e2
60c
3
r
(m  m)r   02 r  e  F sr
e
Reactive reaction or
Electromagnetic m 

(
0
)
3 m
Self-reaction force
30 c
mass
Decay rate due to radiative reaction
W 
e2
60c
r.r 
3
 d   2 
(r.r )  r 
3 
60c  dt

e2
r  r12 cos 0t
r
Averaging over a cycle
2
 r12
W 
e
2
cos 0t  r12
2
60c 3
2
1
2
e r12  04
2
r2 
2
2
120c 3
Decay of the population
C1 
W
 0
e r12  03
2

2
120c 3
G Only half of the

missing half!
4
Fully quantized treatment
e2
H int  
( p.A  A.p)
2m
i
  Ck (   * )( ak  ak* )  (ak  ak* )(   * )
2 k
Symmetric ordering


e 0
Ck 
r .ê
1 / 2 12 
(2 0  kV )
Atomic lowering and raising operators
1 2
*  2 1
Population difference  z  2 2  1 1
H atom  12 0 z
u, v, w in optical
Bloch equations
Equations of motion
a k  i k ak  Ck (   * )

1
  i 0   Ck ( z (ak  ak* )  (ak  ak* ) z
2 k

 z   Ck (   * )( ak  ak* )  (ak  ak* )(   * )
k
t
aksr  Ck  dt '[ (t ' )   * (t ' )]ei k (t t ')
Markov approximation
 (t ' )   (t )e i
0
sr
*
C
a

(


i

)


i


 k k
1
1
k

1 
0
m
d
0   0

2 
0
m
d
0   0
Level shifts: sum over all levels gives Lamb shift!
0 (t 't )
Decay rate due to radiative reaction II
 zzsr  2
As
 z   22  11  2 22 1
sr
 22
 
Conclusion: both vacuum fluctuations and radiative reaction
contribute equally to the decay rate giving a total rate 2G.
Vacuum field contributed revisited
Integrate equation of motion for  and substitute in equation
for z and evaluate for the vacuum field
 (t )   (0)e i t
0
t

1
*
  Ck  dt ' z (t ' ) ak (t ' )  ak (t ' )  .....
2 k
0


 z   Ck (   * )( ak  ak* )  (ak  ak* )(   *
k
 (t ' )   (t )e i
Now
and
0 (t 't )
a(t )  a(0)e i k t
0 aa* z 0   z 0 a*a  1 0   z
0 ......a 0  0

)
0 a*.... 0  0
 zvf  2   z
Summing the vacuum and radiative
reaction contributions
 z  2 22 1  1  211
Radiative reaction
Vacuum fluctuations
Total
sr
 22
 
vf
 22
   (2 22  1)
 22  222
11sr  
11vf    (2 11  1)
11  2 ( 11  1)
Atom in excited state
 22  2 
Atom in ground state
11  0
Summary
1. Semi-classical or quantum vacuum fluctuations
only accounts for half the decay rate.
2. Symmetric ordering of the operators
self-reaction contributes the other half.
3. For the lowest energy state, the contribution of
vacuum field and self reaction cancel.
Spontaneous emission in multilevel
systems
Second order perturbation theory
E

E
1
i

t
j
1
j
1



Cj 
j H ' 1 C1e
j1

i
j V1 1 ei ( j1  ) t  1
C j (t ) 

( j1   )
2
1
1
-i2 j t

C2   2 H ' j C j e
i j
2 V2 j j V1 1 -i1t
1

C2  
(e - 1)e -i2t
1   
i j
Raman transitions
Example one electron atom with I=3/2 (e.g. Na or Rb-87)
2P
3/2
d
c
2P
1/2
b
a
eff
1 2


eff 
2S
1/2
2
1
c1a ca 2  c1bcb 2 c1c cc 2  c1d cd 2

12
1/ 2
3 / 2
1
1
1
1
 
0 
0 

6
3
3
6
1
1
1
1
1
1
2,1 
0    
2,1          
0
2
2
2
2
2
2
2,0 
1,1  
3
1
1
 
 
0
2
2 3
6
1,1 
1,1  
2
1
 
0
3
3
3
1
1
 
 
0
2
2 3
6
1,1 
2
1
 
0
3
3
1
1
1
1
 
0 
0 

3
6
6
3
1,0  
0,0  
1
1
1
1
 
0 
0 

3
6
6
3
2
1 1
1 1

3 1 / 2 3  3 / 2
1.5
2
3

1
6

1
3
2,2    
1
1
1
1
 
0 
0 

6
3
3
6
1,0  
Rabi frequency (MHz)
2,2    

1
3
1
0.5
0
760
1,1   
1,0 
1
  
2

0,0 
1
  
2

1,1   
770
780
790
800
Wavelength (nm)
810
Multilevel spontaneous emission
j
R
k
k
1
j
k V2 j j V1 1
1   
2
2
c2 a cak c2 bcbk  c1c cck 2 G
 spont  

3/ 2
4 9
k 1, 4 1 / 2
1H
c
b
 1
2  2 G
 spont   2  2 
 1/ 2  3 / 2  4 3
Spontaneous emission rate (MHz)
a
10
2
3
1
4
x 10
8
5
87Rb
6
4
2
0
760
770
780
790
800
Wavelength (nm)
810
Modification of spontaneous emission by
a cavity
wi f
2
 2

2
f V i  (0 )
R
Energy per unit frequency per unit volume

w0 

 ( ) 
 cavVcav
 ( ) 
 2 2
c
3
1
2

L( 2 R  L )
2

 cav
w02 L
 fsr

 fsr
c



2 L
Vcav 
L
 cav
4
Bad and good cavities
Atom-field coupling
(vacuum Rabi frequency)
1/ 2
Atom decay

dEv d   

g
 

  2 0V 
1/ 2
 d 

 
 2 0V 
2
Condition for enhanced spontaneous emission
k
Cavity decay
1. g >  : cavity induced decay faster than vacuum
2. Bad cavity k > g : photon escapes and does not reinteract
Strong coupling regime g > , k
Dressed state picture
 , n1
b,n
a , n1
  i
H  
 g
g
 , n1

Bad cavity k > g
 cav
,n
b , n-1
a,n

,n
g 

 ik 
k
g2
 
4k
Strong couling g > , k
 cav
g2
 
4k