The effect of relaxation on absorption lineshapes

A very important result of the theory of quantum dynamics is the connection between the time evolution in a given spectral region and the absorption lineshape into the same region. That such a cormection exists is to be expected, because the time evolution is detennined by the distribution of initial amplitudes among exact eigenstates according to Eq. (2.6), while the absorption process, in principle, prepares these initial amplitudes in the spectral region of interest. 

To see this connection in more detail we extend the model of Figs 9.1 and Eq. (9.2) to include two discrete states, the ground state g) and an excited state lx), and a continuum of states I/) that may represent the ground state dressed by environmental or radiation field states. We assume that l ) is the only excited state in the relevant spectral region that is radiatively coupled to the ground state g) so it can be initially prepared as explained in Section 9.2.4. In the subspace that encompasses the state [s ) and the continuum Z), the former plays the same role as state 11) in Fig. 9.1. We now focus on the excitation from g to s specifically we pose the question What is the corresponding absorption lineshape  

Coming back to the model of Fig. 9.2, we already know that the coupling V between the state and the manifold /) leads to the decay of 5 following an initial preparation of the system in this state. We are also given that the ground state Ig) is coupled radiatively only to s but not to /), that is, (5 /l g) 0 

The complete system under consideration now comprises both the molecule and the radiation field, and the corresponding Hamiltonian is 

As was indicated above, a state of the radiation field is defined by specifying population of each mode, and in particular single photon states (one photon in mode k of frequency a , no photons in other modes) will be denoted by Ik), k), or ty) as will be convenient. We will sometimes use vac) = 0. 0,0,0. ..) to denote the vacuum or ground state of the radiation field, that is, the state with no photons. 

However, for excited atoms in coUisionless conditions only the radiative relaxation channel is open. Here coUisionless means that the time between collisions is much longer than the radiative relaxation time. 

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We next consider the effect of thermal relaxation on the absorption lineshape. We start with the Bloch equations in the form (10.184),... 

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