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Some Exact Properties of the Nonequilibrium Response Function for Transient Photoabsorption

The physical interpretation of time-resolved photoabsorption experiments is not as straightforward as for the more conventional photoabsorption experiments conducted on equilibrium systems. In fact, the relation between the transient photoabsorption spectrum and the properties of the examined sample can be rather intricate since the former is a complicated functional of both the driving pump and the feeble probe fields. In this work we critically review the derivation of the time-resolved photoabsorption spectrum in terms of the nonequilibrium dipole response function $χ$ and assess its domain of validity. We then analyze $χ$ in detail and discuss a few exact properties useful to interpret the transient spectrum {\em during} (overlapping regime) and {\em after} (nonoverlapping regime) the action of the pump. The nonoverlapping regime is the simplest to address. The absorption energies are indeed independent of the delay between the pump and probe pulses and hence the transient spectrum can change only by a rearrangement of the spectral weights. We give a close expression of these spectral weights in two limiting cases (ultrashort and everlasting monochromatic probes) and highlight their strong dependence on coherence and probe-envelope. In the overlapping regime we obtain a Lehmann-like representation of $χ$ in terms of light-dressed states and provide a unifying framework of various well known effects in pump-driven systems. We also show the emergence of spectral sub-structures due to the finite duration of the pump pulse.