The decay of an initially prepared level

4 Golden-rule transition rates 12.4.1 The decay of an initially prepared level 

Let us now return to the two model Hamiltonians introduced in Section 12.2, and drop from now on the subscripts S and SB from the coupling operators. Using the polaron transformation we can describe both models (12.1), (12.2) and (12.4)-(12.6) in a similar language, where the difference enters in the form ofthe coupling to the boson bath 

In one coupling model we use (12.2b) where, for simplicity, terms with j = j are disregarded 

Sometimes an additional approximation is invoked by disregarding in (12.28a) terms that caimot conserve energy in the lowest order treatment. Under this so-called rotating wave approximation the coupling (12.28a) is replaced by (for E2 E ) 

The RWA keeps only terms for which 1 — 2 tkoa can be small, the argument being that strongly oscillating terms make only a small contribution to the transition rate. 

The rationale for this approximation can be seen in the interaction picture in which V becomes Vpt) = exp -(i/K)Hot) 

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Another relaxation process encountered in isolated molecules is the phenomenon of intramolecular vibrational relaxation. Following excitation of a high-lying vibrational level associated with a particular molecular mode, the excitation energy can rapidly spread to other nuclear modes. This is again a case of an initially prepared single state decaying into an effective continuum. 

A nonperturbative treatment of the exponential decay of an excited two-level system has already been given by Weisskopf and Wigner [7, 8] in their seminal work. If the initially prepared quantum state of the matter-field system is of the form e) 0 0) the time evolution of the resulting quantum state can be decomposed according to... 

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