ABSTRACT
The statistical properties of photon sequences originating from single-molecule emitters differ from those produced by generic light sources. Let us start with the latter. A typical light source (such as a lightbulb) is composed of numerous independent lightemitting atoms or molecules. A sufficiently sensitive photodetector may be used to record the arrival time of each individual photon. The simplest assumption one could adopt to describe this kind of measurement is that the arrival times for different photons are completely uncorrelated (see, however, the discussion at the end of this section). If the properties of the light source (such as, e.g., its brightness) do not
depend on time then we should expect that the probability of detecting a photon during an infinitesimal time interval dt is proportional to dt , with the proportionality constant being independent of time or the arrival times of other photons. Let us call this probability λdt , where λ is the probability of detecting a photon per unit time (which will be referred to as the photon count rate). To estimate λ we could measure the number of photons n detected over a certain period of time t and divide it by t :
λ ≈ n/t. (7.1) But, since photon arrival times are random, the exact number of photons found during any finite time interval t will fluctuate around its mean value λt , as illustrated in Fig.7.1. A complete description of this process thus requires the probability wn(t) that a specified number of photons, n, is observed during the time interval t . To obtain this probability, it is helpful to think of the number of detected photons, n, as the “state” of the system at time t . At t = 0 we have n = 0. Each time a new photon is detected, the system undergoes a first-order irreversible “reaction” n → n + 1. The overall process thus can be described by the kinetic scheme:
0 λ→ 1 λ→ 2 λ→ 3 λ→ 4 → · · · . Accordingly, wn(t) is described by a system of differential equations of the form
dwn(t)/dt = λwn−1(t) − λwn(t), n > 0 (7.2) and
dw0(t)/dt = −λw0(t), (7.3) with the initial conditions
wn(0) = 0, n > 0 and
w0(0) = 1. The solution to Eq. 7.3 is
w0(t) = e−λt . (7.4) Substituting this into the differential equation for w1 (Eq. 7.2) we can find the time dependence of w1(t), then substitute the result into the equation for w2(t) and so on. Suppose we have found wn−1(t) and now wish to solve for wn(t). It is convenient to use the ansatz
wn(t) = ρn(t)e−λt .
