Superbunching in cathodoluminescence: A master equation approach

We propose a theoretical model of a master equation for cathodoluminescence (CL). The master equation describes simultaneous excitation of multiple emitters by an incoming electron and radiative decay of individual emitters. We investigate the normalized second-order correlation function, $g^{\left(2\right)}\left(\tau \right)$, of this model. We derive the exact formula for the zero-time delay correlation, $g^{\left(2\right)}\left(0\right)$, and show that the model successfully describes giant bunching (superbunching) in the CL. We also derive an approximate form of $g^{\left(2\right)}\left(\tau \right)$, which is valid for small excitation rate. Furthermore, we discuss the state of the radiation field of the CL. We reveal that the superbunching results from a mixture of an excited photon state and the vacuum state and that this type of state is realized in the CL.

Figure 1

Zero-time delay correlation $g^{\left(2\right)}\left(0\right)$ [Eq. (16)] as a function of the excitation rate $\gamma$ (normalized by the radiative lifetime ${\tau }_{\mathrm{rad}}$). The curves from bottom to top correspond to $N=2$, 3, 4, 10, and 20, respectively. The dotted line is a visual guide proportional to $1/\gamma$.

Figure 2

Normalized second-order correlation function $g^{\left(2\right)}\left(\tau \right)$ for (a) $N=2$ and (b) $N=10$. Semilogarithmic plots of $g^{\left(2\right)}\left(\tau \right)-1$ are shown for excitation rates ranging from ${\tau }_{\mathrm{rad}}\gamma =0.001$ to 0.1. The symbols, solid lines, and dotted lines represent the numerical results, approximate formula (17), and decay curves proportional to $e^{-\tau /{\tau }_{\mathrm{rad}}}$ [Eq. (20)], respectively. For ${\tau }_{\mathrm{rad}}\gamma =0.1$ and $N=10$ [solid circles in (b)], data points for $\tau /{\tau }_{\mathrm{rad}}\gtrsim 2.5$ are not shown because $g^{\left(2\right)}\left(\tau \right)-1$ is extremely small $[g^{\left(2\right)}\left(\tau \right)\approx 1]$ in this region.

Figure 3

A sequence of random pulses of radiation intensity $I_{\mathrm{rad}}$. The intensity profile of each pulse is $I_{\mathrm{rad}}\approx N{e}^{-\tau /{\tau }_{\mathrm{rad}}}$ on average, so the mean duration of the pulse is roughly equal to ${\tau }_{\mathrm{rad}}(logN+\delta )$.