Single-Molecule Photon Counting Statistics via Generalized Optical Bloch Equations

We derive the generating function for single molecule photon emission events within the context of the stochastically modulated optical Bloch equations. Statistical properties of single molecule photon counting experiments are deduced from a set of coupled differential equations only slightly more complicated than the Bloch equations themselves. This formulation allows for the study of photon bunching and antibunching within a single theoretical framework and provides a description of these behaviors that emphasizes the connection with single molecule experiments. Application is made to the spectroscopy of a chromophore coupled to a single two level system.

Figure 1

The average number of emitted photons as a function of time for the two state jump model with physical parameters as in the text. Once the coherence associated with the initial condition disappears we see linear growth in time. Inset: Long time behavior of the photon count. The solid line is our numerical result and the dotted line represents the limiting case of perturbative expansion in $\Omega$ in the long time limit, first derived by Barkai et al. [15]. The slight difference in slope reflects the approximate nature of the perturbative solution. Reduction of $\Omega$ by an order of magnitude results in plots that are indistinguishable.

Figure 2

Mandel’s $Q$ parameter as a function of time for the model discussed in the text. Inset: The long time limit with comparison to perturbative analytical results (dotted line) of Ref. 15. As in Fig. 1, the perturbative result becomes indistinguishable from the exact result when $\Omega$ is reduced by an order of magnitude.

Figure 3

Manifestation of Rabi oscillations observed under high field conditions. The dotted line is ${\rho }_{bb}(t)$. The solid line, which represents average cumulative photon emission, grows only when ${\rho }_{bb}(t)$ is occupied. The behavior dies out as coherence is lost.