Theory of Frequency-Filtered and Time-Resolved $N$-Photon Correlations

A theory of correlations between $N$ photons of given frequencies and detected at given time delays is presented. These correlation functions are usually too cumbersome to be computed explicitly. We show that they are obtained exactly through intensity correlations between two-level sensors in the limit of their vanishing coupling to the system. This allows the computation of correlation functions hitherto unreachable. The uncertainties in time and frequency of the detection, which are necessary variables to describe the system, are intrinsic to the theory. We illustrate the power of our formalism with the example of the Jaynes-Cummings model, by showing how higher order photon correlations can bring new insights into the dynamics of open quantum systems.

Figure 1

(a) Scheme of our proposal to compute $N$-photon correlations between photons emitted at different times and frequencies from an open quantum open system $Q$. $N$ two-level systems of ascribed frequencies are weakly coupled to $Q$ and serve as correlation sensors at these frequencies, with their decay rate providing the detector linewidth. (b) Dissipative Jaynes-Cummings ladder up to the third rung with two of the cascades probed in panels (d) [with two sensors] and (e) [with three sensors]. Solid arrows show the fixed frequencies. Curved arrows show the scanning frequency ${\omega }_1$, at the transitions where the joint emission is strongly enhanced (dashed) or, on the other hand, suppressed (dotted). (c) Power spectra of emission probed by weak incoherent excitation ($P_{\sigma }={\gamma }_{\sigma }=0.01g$) for three cavities of decreasing quality ${\gamma }_a=0.01$ (solid), 0.1 (dashed) and $0.5g$ (dotted). (d) Two- and (e) three-photon correlations at zero delay for the three cavities, with sensor linewidths $\Gamma ={\gamma }_2$ (solid) and ${\gamma }_2/2$ (dashed and dotted).

Figure 2

(a) Two- and three-photon correlations at zero delay as a function of the sensor linewidth $\Gamma$, with frequencies of detection as shown in the insets i–iv. (b)–(c) $\tau$ dynamics of the correlation functions with $\Gamma ={\gamma }_2$ for, (b), two photons in the configurations of insets ii and iv and, (c), three photons in the configurations i and iii. Positive $\tau$ corresponds to detection in the order from top to bottom of the ladder. Parameters: $P_{\sigma }={\gamma }_{\sigma }=0.01g$, ${\gamma }_a=0.1g$.