Cavity QED I: Simple Calculations

Abstract

Having seen that fluctuations outside the small-noise limit present the phasespace methods with fundamental difficulties, we look in the remaining four chapters at alternative ways of treating quantum fluctuations in this regime. The topic is not entirely new to us, since we handled the problem of resonance fluorescence in Chapter 2 without the need for either phase-space representations or a system size expansion. Of course this example is rather trivial, falling as it does within the class of solvable one-particle problems. Phase-space representations do an excellent job in the opposite limit, where the nonlinear physics builds upon the cooperation of many atoms and many photons, and the fluctuations may be viewed as small perturbations (“fuzz”) about classical nonlinear dynamics—the positive P representation even allows the “fuzz” to be squeezed. The real difficulties lie in the intermediate regime, where it may be hard, if not impossible, to solve density matrix equations exactly as we did for resonance fluorescence, and yet a small-noise approximation to a phase-space equation of motion may not be made either. We meet two new techniques in the remaining chapters: a systematic expansion of the density matrix equations for weak excitation (Sects. 16.1 and 16.3.4) and the quantum trajectory method (Chaps. 17–19). There is also more to be done with exact solutions of density matrix equations for systems of small or moderate size, analytically in the former case (Sect. 13.2) and in the latter with the help of a computer (Sect. 16.3.6).