PG lectures 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.

2p(m=0) to 1s 2p(m=1) to 1s Oscillating charge

2p(m=0) to 1s 2p(m=1) to 1s Dipolar radiation

I understand that light is emitted when an atom decays from an excited state to a lower energy state. That’s 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. I’m sorry. I don’t know I can’t explain it to you. Dad Feynman Dad Feynman Dad Feynman Dad Pause Feynman 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) Spontaneous emission: what do the greats have to say? (Feynman, Physics Teacher 1969)

What is this zero state? Vacuum energy. Second quantisation of the electromagnetic field see Loudon The quantum theory of light pp a* a n photons destruction creation k One mode of EM field =1,2 polarisation Vacuum state The idea of a photon is most easily expressed for an EM field inside a …perfectly reflecting cavity. Loudon p. 1

Quantum theory of spontaneous emission University of Durham

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  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).

where 2-level time dependent perturbation theory electric dipole approximation

Decay rate due to vacuum fluctuation

free fieldsource field Field produced by the atom – the source field electric dipole approximation

Radiative reaction Add source field to classical equation of motion Electromagnetic mass Reactive reaction or Self-reaction force Cut-off

Decay rate due to radiative reaction Averaging over a cycle Decay of the population Only half of the missing half!

Fully quantized treatment Symmetric ordering Atomic lowering and raising operators Population differenceu, v, w in optical Bloch equations

Markov approximation Equations of motion Level shifts: sum over all levels gives Lamb shift!

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 

Vacuum field contributed revisited Integrate equation of motion for  and substitute in equation for  z and evaluate for the vacuum field Now and

Summing the vacuum and radiative reaction contributions Radiative reactionVacuum fluctuationsTotal Atom in excited state Atom in ground state

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

1 2 Second order perturbation theory

2 P 3/2 2 P 1/2 2 S 1/2 Raman transitions 1 2 a b c d Example one electron atom with I=3/2 (e.g. Na or Rb-87)

1 k Multilevel spontaneous emission j

1 342 a b c 87 Rb 1H1H

Modification of spontaneous emission by a cavity   Energy per unit frequency per unit volume L R

Bad and good cavities   Atom decay Cavity decay Atom-field coupling (vacuum Rabi frequency) Strong coupling regime g >  Condition for enhanced spontaneous emission 1. g >  : cavity induced decay faster than vacuum 2. Bad cavity  > g : photon escapes and does not reinteract

Dressed state picture , n , n , n  1 , n  1 b , n a , n  1 g   b , n-1 a , n  Bad cavity  > g Strong couling g > 