Nonlinear optical spectroscopy of single, few, and many molecules: Nonequilibrium Green’s function QED approach

Nonlinear optical signals from an assembly of $N$ noninteracting particles consist of an incoherent and a coherent component, whose magnitudes scale $\sim N$ and $\sim N(N-1)$, respectively. A unified microscopic description of both types of signals is developed using a quantum electrodynamical (QED) treatment of the optical fields. Closed nonequilibrium Green’s function expressions are derived that incorporate both stimulated and spontaneous processes. General $(n+1)$-wave mixing experiments are discussed as an example of spontaneously generated signals. When performed on a single particle, such signals cannot be expressed in terms of the $n$th order polarization, as predicted by the semiclassical theory. Stimulated processes are shown to be purely incoherent in nature. Within the QED framework, heterodyne-detected wave mixing signals are simply viewed as incoherent stimulated emission, whereas homodyne signals are generated by coherent spontaneous emission.

Figure 1

Loop diagram for SLE. ${\mathbf{k}}_1$ is the incoming field and ${\mathbf{k}}_2$ is the signal field. Note that the interactions are time ordered within each strand, but not between strands. $s_1,s_2,s_3$ are the delay times between the interactions along the loop.

Figure 2

Time-ordered Feynman diagrams of SLE, generated by shifting the arrows of Fig. 1 along each strand thus changing their relative time ordering. Each of the possible three relative positions then gives one fully time ordered diagram (double sided Feynman diagram).

Figure 3

Three level systems with sequential dipole couplings used in the derivation of the Kramers Heisenberg relation for [(a) Eq. (46)] the SLE and (b) the pump-probe signal.

Figure 4

Eight loop diagrams for the pump-probe signal [Eq. (52)]. Note that (a) coincides with the SLE (Fig. 1).

Figure 5

Double sided Feynman diagrams resulting from Fig. 4 upon breaking the loops into time ordered contributions. (a)–(d) Time ordered diagrams corresponding to the loops shown in (a)–(d) of Fig. 4, respectively. Since the four loops with three interactions on the bra [Figs. 4e, 4f, 4g, 4h] are already fully time ordered, each gives a single double sided diagram. These are not repeated here. Overall, the eight loop diagrams of Fig. 4 yield 16 Feynman diagrams.