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 $\chi$ and assess its domain of validity. We then analyze $\chi$ in detail and discuss a few exact properties useful to interpret the transient spectrum during (overlapping regime) and 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-type representation of $\chi$ 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 substructures due to the finite duration of the pump pulse.

Figure 1

Illustration of a PA experiment.

Figure 2

Illustration of a TR-PA experiment.

Figure 3

Absorption energies as a function of the pump intensity $P\in [0,0.2]$ for ${\omega }_P=0$ [dc (gray) region] and as a function of ${\omega }_P\in [0,0.9]$ for $P=0.2$ [ac region] of a four-level quantum system consisting of the states $1,a\in A$ with energy $E_1=0$, $E_a=21.2$, and states $b,2\in B$ with energies $E_b=20.6,E_2=23.1$ (energies in eV). The absorption at energy ${\Omega }_{12}$ is independent of $P$ and not shown. The color scale indicates the oscillator strength of the transition [$\langle {\Psi }_{\alpha A}^{\left(0\right)}\left|\widehat{d}\right|{\Psi }_{\gamma B}^{\left(1\right)}\rangle$ for ${\Omega }_{\alpha \gamma }+{\omega }_P$, $\langle {\Psi }_{\alpha B}^{\left(1\right)}\left|\widehat{d}\right|{\Psi }_{\gamma A}^{\left(0\right)}\rangle$ for ${\Omega }_{\alpha \gamma }-{\omega }_P$, and the sum of the two for ${\omega }_P=0$, see Eq. (54)] normalized to $\mathfrak{D}\equiv \langle {\Psi }_{\alpha }\left|\widehat{d}\right|{\Psi }_{\beta }\rangle$ with $\alpha =1,a$ and $\beta =b,2$. In the ac region we also display the absorption energy ${\omega }_P$ with oscillator strength ${\sum }_{\alpha }\langle {\Psi }_{\alpha A}^{\left(0\right)}|\widehat{d}|{\Psi }_{\gamma B}^{\left(1\right)}\rangle$, and $-{\omega }_P$ with oscillator strength ${\sum }_{\alpha }\langle {\Psi }_{\alpha B}^{\left(1\right)}|\widehat{d}|{\Psi }_{\gamma A}^{\left(0\right)}\rangle$.

Figure 4

TR-PA spectrum (normalized to the maximum) for the two-level system described in the main text. The parameters are ${\Omega }_{ab}=E_a-E_b=7$, $P=0.7$, ${\gamma }_a={\gamma }_b=\gamma =0.2$, ${\gamma }_P=0.02$, and ${\omega }_P={\Omega }_{ab}$. Energies in eV and times in fs.

Figure 5

TR-PA spectrum (normalized to the maximum) for the two-level system described in the main text at a delay $\tau =\pi /4{\omega }_P$. The parameters are ${\Omega }_{ab}=E_a-E_b=7$, ${\gamma }_a={\gamma }_b={\gamma }_P=0.04$. Different panels refer to different pump intensity $P=0.2,0.4,0.6,0.8$ and the solid (dashed) line refers to the resonant (off-resonant) pump frequency ${\omega }_P={\Omega }_{ab}+\Delta \omega$. Energies in eV.

Figure 6

TR-PA spectrum (normalized to the maximum) for the two-level system described in the main text. The parameters are ${\Omega }_{ab}=E_a-E_b=7$, ${\gamma }_a={\gamma }_b={\gamma }_P=0.04$, $P=0.4$, and ${\omega }_P={\Omega }_{ab}$. Energies in eV and times in fs.

Figure 7

TR-PA spectrum (normalized to the maximum) for the three-level system described in the main text. The parameters are $E_1=0$, $E_b=20.6$, $E_a=21.2$, $P=0.7$, $\gamma =0.2$, ${\gamma }_P=0.04$, and ${\omega }_P={\Omega }_{ab}$ (top panel) and ${\omega }_P={\Omega }_{ab}-0.5$ (bottom panel). Energies in eV and time in fs.

Figure 8

Linear term in $d_p$ of the TR-PA spectrum (normalized to the maximum) of Eq. (34) for the three-level system described in the main text. The probe Hamiltonian is given in Eq. (67) with $\lambda /e_0=0.01$. The rest of the parameters are the same as in Fig. 7. The pump frequency is ${\omega }_P={\Omega }_{ab}$ (top panel) and ${\omega }_P={\Omega }_{ab}-0.5$ (bottom panel). Energies in eV and time in fs.

Figure 9

Plot of $|\omega {\overline{d}}_p{\left(\omega \right)|}^2$ (normalized to the maximum) for the three-level system described in the main text. Same parameters as in the top panel of Fig. 8. Energies in eV and time in fs.