Photoelectron counting in quantum optics

Mandel formula: many countsHow to obtain probability of n transitions p n (t, t + T )Short-time probability p 1 (t, t + ∆t) = ηI (r)∆t for single electrontransition (ηI (r) transition rate).Long-time probability of n transitions p n (t, t + T ) ↔ n electrons.Markovian master equation for probabilities. p n (t) ≡ p n (0, t),p n (t + dt) = p n (t) × [1 − ηI (r)dt] + p n−1 (t) × ηI (r)dt (2)ddt p n(t) = ηI (r)[p n−1 (t) − p n (t)]. (3)Generating function G(s, t) ≡ ∑ ∞n=0 sn p n (t),∂ t G(s, t) = ηI (r)(s − 1)G(s, t).Solve with p 0 (0) = 1, p n (0) = 0, n > 0, G(s, 0) = 1.Thus G(s, t) = exp[ηI (r)t(s − 1)] = ∑ ∞ ¯nnn=0snn! e−¯n , ¯n = ηI (r)t.