By Rudolf Kingslake

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L Pjd3k g(k)l"'(k~ WA>. 62) is clear when we note that, there, a = 1/2 + iLl and (3 = (r /2)n +iLl'. 24) 34 2. 27) + 2L1' + L1. The symmetric grouping of terms we have adopted identifies a transition rate from 12)--+ II), described by the term proportional to b/2)(n+I), and a transition rate from II)--+ 12), described by the term proportional to ('y/2)n. The former contains a rate for spontaneous transitions, independent of n, and a rate for stimulated transitions induced by thermal photons, proportional to n; the latter gives a rate for absorptive transitions which take thermal photons from the equilibrium electromagnetic field.

98)] which are manifestly quantum mechanical expressions, respecting operator order and employing a quantum mechanical propagator. For those cases where a linearized treatment of fluctuations may be made, there is, it is true, a version of the formula much closer to Onsager, where one has a linear set of mean-value equations analogous to macroscopic transport equations [Eqs. 108)]. Even here, though, quantum mechanical features can turn up, such as antibunched or squeezed fluctuations. It is important to note, furthermore, that the quantum regression formula is not limited to the small fluctuation regime.

From this density operator we can obtain time-dependent expectation values for any operator acting in the Hilbert space of the system S. What, however, about products of operators evaluated at two different times? Of particular interest, for example, will be the first-order and second-order correlation functions of the electromagnetic field. For a single mode these are given by G(l)(t, t + T) ex (at(t)a(t + T)), Q( 2)(t, t + T) ex (at(t)at(t + T)a(t + T)a(t)). The first-order correlation function is required for calculating the spectrum of the field.

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Applied optics and optical engineering,Vol.IV by Rudolf Kingslake

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