# Statistical Methods in Quantum Optics 1: Master Equations by Howard J. Carmichael

0'+· The derivation of the master equation for a two-level atom then follows in complete analogy to the derivation of the master equation for the harmonic oscillator, aside from two minor differences: (1) The explicit evaluation of the summation over reservoir oscillators now involves a summation over wavevector directions and polarization states.

HsR k,>. with ' >. · · ik·rA~k _ -ze - - ek "'k >. , and corresponding frequencies Wk and unit polarization vectors ek,>... The atom is positioned at r A, and Vis the quantization volume. "'k,>. is the dipole coupling constant for the electromagnetic field mode with wavevector k and polarization >.. The general formalism from Sect. t "'k,>.. k,>.. 17b) k,>. 18a) k,>.. 18b) k,>. 19a) O"+eiWAt. 19), consider the Heisenberg equation of motion §1 = i~WAei(wAo-z/2)t(O'zO'- _ (T_O'z)e-i(WAO"z/2)t = -iWABI· This is trivially solved to give BI(t) = si(O)e-iWAt = (T_e-iwAt.

60b) - 8jkfi 2 j). 34). Since the reservoir subsystems are statistically independent and all reservoir operators have zero mean, all of the cross terms involving correlation functions for products of operators from different reservoir subsystems will vanish. Thus, the spontaneous emission terms arising from the interaction with f 1 and f 2 are obtained exactly as in Sect. 1. 61); the others follow in a similar form. 59a), (F3(t)F3(t'))R 1 = tr[Rw L L Kljk Klj'k' (rLrlkei(w 1rwlk)t- 8jkfilj) j,k j',k' x ( r 1t j,r1k'e i(w 13-t-wlk')t' = tr[ Rw ( LL - 8 )] j'k'nlj' Kljk Klj'k' djrlk dj'rlk' ei(wlj-wlk)tei(wv-wlk' )t' j,k j',k' -L L j -L L j,k j' +L L j Kljj Klj'k' fi1j rL,rlk' ei(w 13'-w 1k')t' j',k' j' Kljk Klj'j' rLrlk filj' ei(w 13 -w 1k)t) Kljj Ktj' j' filj filj'' J 2.