«So by observation of the atomic fluorescence I(t), say with the eye [...] one can directly monitor the quantum jumps on the weak transition.»

Possibility of Direct Observation of Quantum Jumps. R. Cook and H. Kimble in Phys. Rev. Lett. 54:1023 (1985).  What the paper says!?

This paper is the first theoretical description—although a simplified one (rate equations, incoherent pumping)—of the intermittency in an isolated V quantum system, showing that a single event (a quantum jump) can have macroscopic observable consequences (switch on and off of an intense emission).

a number of effects that can be observed in the fluorescence of a single atom (or ion) are totally masked when many atoms contribute to the fluorescence. The effect considered here is of this type.

They credit Dehmelt with the idea of the V-system, although his seminal input is unavailable while his publication on the topic came after the present work, but it did come with an experimental verification:

The idea we shall develop was first suggested by Dehmelt as a way to detect a weak transition in single-atom spectroscopy.

Here, «the purpose of this Letter is to present a first theoretical treatment of the above single-atom double-resonance effect. For simplicity we limit the analysis to the case of incoherent excitation» (they say the coherent case will be published elsewhere). As a result, the theory is simply rate equations:

With change of variable (their Eq. 3) for the probabilities that the weak transition is excited or not,

The description of Dehmelt's scheme is insightful:

Although single-atom fluorescence from a strong optical transition (~$10^8$ photons/sec) is readily detected either visually (using a microscope) or photoelectrically, the direct detection of a fluorescent or absorptive line profile of a very weak transition (say 1 photon/sec) is problematic. To circumvent this difficulty, Dehmelt proposed the double-resonance scheme illustrated in Fig. 1, in which the weak transition of interest, 0↔2, and a strong transition, 0↔1, have a common lower level.

The important point is that the fluorescence emanating from the strong transition is a direct indicator of the state of excitation of the weak transition; the fluorescence is off when level 2 is occupied and on when it is not.

The probabilities for the states do not say anything about the fluctuations, and the Authors turn to probabilities of the type described in their footnote 4:

⁴The mathematical method we shall employ is similar to that used by A. Burshtein (Zh. Eksp Teor. Fiz. 49, 1362 (1965) [Sov. Phys. JETP 22, 939 (1966)] } and by P. Zoller and F. Ehlotzky [J. Phys. B 10, 3023 (1977)] in the analysis of random jump processes. It is a generalization of the method used by S. O. Rice [in Selected Topics in Noise and Stochastic Processes, edited by N. Wax (Dover, New York, 1954), pp. 176—181] to study the bivalued random telegraph signal. An extensive list of references to the literature on jump processes is given by B. W. Shore, J. Opt. Soc. Am. 81, 176 (1984).

This gives rise to the beautiful (simple and elegant) mathematical structure for the problem:

From this, they derive the distribution of on and off times, which are exponentially distributed with different constants (their Eqs. (10-11)). They thus describe in this way the characteristic telegraphic signal profile of the signal:

in a single-atom double-resonance experiment with a strong and a weak transition, the atomic fluorescence flashes on and off as the atom undergoes weak transitions.

They also compute $g^{(2)}$ from such rate-equation arguments

At the time of writing, this effect has not been observed, but probably will be in the near future.

The effect was observed by Nagourney et al. the following year.

Probably because this is (or could be) covered in their Footnote 4, they do not solve Eqs. (8), which is do-able: taking the generating functions $G_\pm(s,T)=\sum_n s^n P_{n,\pm}(T)$, their Eqs. (8) become a linear system $$\dot G_+ = sR_+G_- - R_-G_+,\qquad \dot G_- = sR_-G_+ - R_+G_-,$$ with eigenvalues $$\lambda_\pm(s) = -\frac{\Sigma}{2} \pm \sqrt{\frac{(R_+-R_-)^2}{4} + s^2 R_+R_-},\qquad \Sigma \equiv R_+ + R_-.$$ Initial conditions being $P_{0,\pm}(0)=\mathscr{P}_\pm$, we have $G_\pm(s,T)$ in closed-form and expanding it in $s$ thus gives us $P_{n,\pm}(T)$: $$P_{2k,-}(T) = \frac{(R_+R_-)^k\,T^{2k}}{(2k)!}\,e^{-R_+T}\;{}_1F_1\!\big(k;\,2k{+}1;\,(R_+{-}R_-)T\big),$$ $$P_{2k+1,+}(T) = \frac{R_+^{\,k+1}R_-^{\,k}\,T^{2k+1}}{(2k+1)!}\,e^{-R_+T}\;{}_1F_1\!\big(k{+}1;\,2k{+}2;\,(R_+{-}R_-)T\big),$$ in terms of Kummer functions ${}_1F_1(k{+}1;2k{+}2;x)$, which will become simple sinh functions.

Because $\mathcal{P}_-R_+=\mathcal{P}_+R_-$ (in the Authors' notation), stationarity enforces that $$P_{2k+1,+} = P_{2k+1,−}$$ $$P_{2k,+}(R₊\leftrightarrow R₋) = P_{2k,−}(R₋\leftrightarrow R₊)$$

These are plots of the analytical expressions for the beginning of the chain:

where we can see that $P_{1,-}=P_{1,+}$ (different $y$-scale). Besides, each panel conserves its start-state probability: every curve in the left panel is seeded from + (start in the dark state), so the whole left chain—$P_{0,+}, P_{1,-}, P_{2,+}, P_{3,-}, \dots$—sums to $\mathcal{P}_+= 0.2$ at every $T$ while the right chain sums to $\mathcal{P}_-= 0.8$.

The Mathematica expressions are provided here:

Stot = Rp + Rm;
Pp = Rp/Stot;   (* stationary prob. of being in + (dark/shelved) *)
Pm = Rm/Stot;   (* stationary prob. of being in - (bright)       *)

(* even n = 2k *)
P2kPlus[k_, T_] :=
  Pp (Rp Rm)^k T^(2 k)/(2 k)! Exp[-Rm T] Hypergeometric1F1[k, 2 k + 1, (Rm - Rp) T];
P2kMinus[k_, T_] :=
  Pm (Rp Rm)^k T^(2 k)/(2 k)! Exp[-Rp T] Hypergeometric1F1[k, 2 k + 1, (Rp - Rm) T];

(* odd n = 2k+1 *)
P2k1Plus[k_, T_] :=
  Pm Rp^(k + 1) Rm^k T^(2 k + 1)/(2 k + 1)! Exp[-Rp T] Hypergeometric1F1[k + 1, 2 k + 2, (Rp - Rm) T];
P2k1Minus[k_, T_] :=
  Pp Rm^(k + 1) Rp^k T^(2 k + 1)/(2 k + 1)! Exp[-Rm T] Hypergeometric1F1[k + 1, 2 k + 2, (Rm - Rp) T];

Those are not quite needed for the results that the Authors focus on in this paper, which are the distributions of dwell times, and their $g^{(2)}$ which is however not provided in a very explicit manner. Here it is: $$g^{(2)}(T) = \frac{C(T)}{\langle I\rangle^2} = 1 + \frac{R_+}{R_-}\,e^{-(R_++R_-)T},\qquad g^{(2)}(0)=\frac{1}{\mathscr{P}_-} = 1+\frac{R_+}{R_-} > 1.$$

I'm not sure I like very much their $S(\omega)$ and should probably look into the proper frequency-resolved photon correlation of exactly their (incoherently-pumped) case.

Note that this is for the coarse-grain intensity, and a full quantum-optical $g^{(2)}(\tau)$ would display at least antibunching at $\tau=0$ and Rabi oscillations under coherent driving, of which such results would provide an envelope whenever applicable. This has been dealt with by others after them, e.g., Refs. ⋯ (see Ref. for a review).