PG lectures 200405 Spontaneous emission

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PG lectures 2004-05 Spontaneous emission
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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.
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Oscillating charge
2p(m1) to 1s
2p(m1) to 1s
2p(m0) to 1s
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Dipolar radiation
2p(m0) to 1s
2p(m1) to 1s
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Spontaneous emission what do the greats have to
say?
I understand that light is emitted when an atom
decays from an excited state to a lower energy
state. Thats right And light consists of
particles called photons Yes So the photon
particle must be inside the atom when it is in
the excited state. Well no. Well how do you
explain that the photon comes out of the atom
when it was not there in the first place. Im
sorry. I dont know I cant explain it to you.
Dad Feynman Dad Feynman Dad Feynman Dad Pau
se Feynman
(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)
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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
creation
One mode of EM field
k
destruction
l1,2 polarisation
a
a
n photons
Vacuum state
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Quantum theory of spontaneous emission
University of Durham
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Statement of the problem

• 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
• Solution vacuum fluctuations self-reaction
• (the interaction of an electron with its own
  field)
• both contribute to observable processes.
• However, their respective contributions cannot
• be uniquely defined.
• 4. A matter of interpretation!

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Outline

• Semi-classical Fermis golden rule
• vacuum only accounts for half the decay rate.
• QED Radiative reaction or self reaction
• (electromagentic mass)
• Decay rate and level shifts (Lamb shift)
• due to radiative reaction
• 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).
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2-level time dependent perturbation theory
where
electric dipole approximation
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Decay rate due to vacuum fluctuation
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Field produced by the atom the source field
electric dipole approximation
free field
source field
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Radiative reaction
Cut-off
Add source field to classical equation of motion
Reactive reaction or Self-reaction force
Electromagnetic mass
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Decay rate due to radiative reaction
Averaging over a cycle
Decay of the population
Only half of the missing half!
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Fully quantized treatment
Symmetric ordering
Atomic lowering and raising operators
Population difference
u, v, w in optical Bloch equations
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Equations of motion
Markov approximation
Level shifts sum over all levels gives Lamb
shift!
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Decay rate due to radiative reaction II
As
Conclusion both vacuum fluctuations and
radiative reaction contribute equally to the
decay rate giving a total rate 2bG.
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Vacuum field contributed revisited
Integrate equation of motion for s and substitute
in equation for sz and evaluate for the vacuum
field
Now
and
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Summing the vacuum and radiative reaction
contributions
Radiative reaction
Vacuum fluctuations
Total
Atom in excited state
Atom in ground state
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Summary

• Semi-classical or quantum vacuum fluctuations
• only accounts for half the decay rate.
• Symmetric ordering of the operators
• self-reaction contributes the other half.
• For the lowest energy state, the contribution of
• vacuum field and self reaction cancel.

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Spontaneous emission in multilevel systems
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Second order perturbation theory
2
1
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Raman transitions
Example one electron atom with I3/2 (e.g. Na or
Rb-87)
2P3/2
d
c
b
2P1/2
a
2
2S1/2
1
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(No Transcript)
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Multilevel spontaneous emission
j
k
1
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1H
c
b
a
87Rb
3
4
2
1
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Modification of spontaneous emission by a cavity
R
Energy per unit frequency per unit volume
L
r(w)
w
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Bad and good cavities
Atom-field coupling (vacuum Rabi frequency)
Atom decay
g
Condition for enhanced spontaneous emission 1. g
gt g cavity induced decay faster than vacuum 2.
Bad cavity k gt g photon escapes and does not
reinteract
k
Cavity decay
Strong coupling regime g gt g, k
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Dressed state picture
, n1
b , n
g
a , n1
- , n1
Bad cavity k gt g
w
, n
b , n-1
g
Strong couling g gt g, k
a , n
k
- , n