Correlation functions and the AC Stark effect ?

We present a Heisenberg picture QED treatment of the resonance fluorescence of a two-level atom. It is shown that the equations of motion for first-order correlation functions have an asymmetry with respect to the two time arguments. This asymmetry arises in a correct and careful evaluation of the unequal-time commutator between free-field and atomic operators. General solutions of these equations are found without going to the stationary limit. A deceptively simple-sounding problem has recently received much attention. That problem is the behaviour of a two-level atom interacting with a constant amplitude monochromatic field tuned near resonance. The calculation of the spectral density of the fluorescent light has received particular attention (see Kimble and Mandel 1975a, b and references therein), especially since the report of recent experiments observing the characteristic three-peaked spectrum of the AC Stark effect by Schuda et a1 (1974). This spectrum is of some fundamental interest as it demonstrates the existence of quantum fluctuations in the atomic radiation. This interest makes it desirable to examine carefully the assumptions and approximations made in the theoretical derivations. Mollow (1969) has carried out a semiclassical treatment in which he assumed that the interaction is a stationary Markovian process. Oliver et a1 (1971), Agarwal (1974), Milonni (1974), Smithers and Freedhoff (1974), Carmichael and Walls (1975), Hassan and Bullough (1975) and Mollow (1975a) have approached the problem using fully quantized formalisms and end up with equivalent results. If one carries out such a calculation in the Heisenberg picture, one inevitably encounters atomic correlation functions which must be evaluated to determine the spectrum of the scattered light. Some previous attempts (Oliver 1971, Agarwal 1974) to solve this problem have avoided the difficult task of explicitly calculating these correlation functions and the associated two-time operator products by appealing to the fluctuation-regression theorem (Lax 1963, Haake 1973). Another approach sometimes used (Milonni 1974, Hassan and Bullough 1975) is to assume that the freefield operators commute with the atomic variables at all times, which is similar to assuming a c-number external field (Mollow 1969, Herrmann et a1 1973). Other techniques are used, but none of these approaches has been justified from first principles, so that if one wishes to be sure of the exact predictions to be compared with experiment, one must determine the correlation functions directly. (This point has t This work was supported in part by the ONR through contract "14-68-A-0091.