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The coupling of an atom to an electromagnetic field can be described in a number of different ways. 2) where V(r) is the Coulomb interaction which binds the electron to the atomic nucleus. 5) with zero static potential = 0 . A= 0 . A(r) will be responsible for exciting the atom. The second term is quadratic in the vector potential and because it is independent of the atomic state will not excite transitions and will be ignored. The minimal substitution form of the interaction is widely used in quantum optics (see for example the paper by Mandel, Sudarshan and Wolf, Reprinted Paper 9 ) .

11) that V(t) excites transitions between all accessible states |k> to the state |&>. (0) = 1. Population is lost from |i> as time goes on so that a^(t) is diminished by couplings V^f(t) to excited states f ^ i. Population is increased in the initially unoccupied excited states |f> , so that the probability amplitudes for such excited states, af(t) increase. 13) P ^ f( t ) =a*(t)a f(t). 11) for the probability amplitudes are exactly soluble only for very simple cases. In^general we must make approximations concerning the strength of the coupling V(t).

25) Coherence Functions 49 which is the density matrix for the two-state system. the measurable information about our system. It contains all of The expectation value of an operator 6 can be written as <6> = or in our new notation as, <6> = p n o n + p 1 2o 2 1 + p 2 1o 1 2 . 26) that is, the sum of the diagonal elements. 22) as ( (+) W(t) = S Tr{pE ~^(t)E (t)} . Loudon 1973). 25) to that for a statistical mixture. ^For a particular realisation r of the ensemble, the expectation value is .