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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-i0t 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 C1 e r12 3 2 G 3 60c 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 ia r.ê 1/ 2 i (2 0V ) free field source field t ie it 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 30 c 60 c m (t ) m -it e d m mc 2 0 m Add source field to classical equation of motion mr 02 r e( sr ) e e2 30c 3 r m (0) e2 60c 3 r (m m)r 02 r e F sr e Reactive reaction or Electromagnetic m ( 0 ) 3 m Self-reaction force 30 c mass Decay rate due to radiative reaction W e2 60c r.r 3 d 2 (r.r ) r 3 60c dt e2 r r12 cos 0t r Averaging over a cycle 2 r12 W e 2 cos 0t r12 2 60c 3 2 1 2 e r12 04 2 r2 2 2 120c 3 Decay of the population C1 W 0 e r12 03 2 2 120c 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 2G. 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 1 211 Radiative reaction Vacuum fluctuations Total sr 22 vf 22 (2 22 1) 22 222 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 -i2 j t C2 2 H ' j C j e i j 2 V2 j j V1 1 -i1t 1 C2 (e - 1)e -i2t 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 12 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 , n1 b,n a , n1 i H g g , n1 Bad cavity k > g cav ,n b , n-1 a,n ,n g ik k g2 4k Strong couling g > , k cav g2 4k