## Abstract

In this tutorial, Langevin stochastic equations and Markov probability equations are used to model electron- and photon-number fluctuations in three- and four-level lasers. Equations are derived for the moments of the electron and photon numbers (means, variances and correlations). Both approaches produce the same moment equations. In the Langevin approach, the moments of the noise terms must be specified by other means, whereas in the Markov approach, they are determined self-consistently and satisfy the shot-noise rule: For each process that is modeled by the rate equations, the driving terms in the variance equations equal the moduli of the associated terms in the mean equations. The driving terms in the correlation equations have the same magnitudes as the variance terms, but can be positive or negative, depending on whether the changes in the electron and photon numbers are correlated or anti-correlated, respectively. Formulas are derived for the relative intensity noise and its spectrum. The consequences of these results for three- and four-level lasers are discussed.

© 2020 Optical Society of America

## 1. INTRODUCTION

In this tutorial, the noise properties of three- and four-level atom lasers are reviewed in detail. Four-level lasers involve levels 0 (ground), 1 (lower), 2 (upper), and 3 (excited). Type A three-level lasers involve levels 0, 1, and 2, whereas type B lasers involve levels 1, 2, and 3. Let $S$ be the number of signal (laser) photons in a cavity. Then the laser performance is characterized by the equilibrium number ${S_0}$, which is proportional to the average photon flux (output power), and the number deviation (fluctuation) ${S_1}$, which varies randomly in time. Two important metrics are the relative intensity noise (RIN) ${\left\langle {S_1^2(t)} \right\rangle _t}/S_0^2$, where ${\left\langle \right\rangle _t}$ denotes a time average, and the RIN spectrum ${\left\langle {|{S_1}(\omega {{)|}^2}} \right\rangle _e}/S_0^2T$, where ${\left\langle \right\rangle _e}$ denotes an ensemble average and $T$ is the integration time. Both quantities can be measured.

Laser evolution can be modeled by rate equations for the electron and photon numbers [1,2]. These equations, which model absorption, emission, loss, and pumping, are stated in Section 2. In ideal lasers, electrons decay quickly from level 3 to level 2 or from level 1 to level 0, so only the populations of two levels need to be retained: 0 and 2 for a four-level or type A laser, and 1 and 2 for a type B laser. These simplifications are used to derive a set of three general rate equations, which apply to all of the aforementioned lasers.

The general rate equations are deterministic, so they can describe the growth and saturation of the signal-photon number, but cannot describe photon-number fluctuations. One can mimic the effects of quantum fluctuations by adding random source terms to the rate equations, with one source term for each deterministic process. According to the shot-noise rule (which remains to be justified), the source terms are statistically independent, and the variance of each source term is the mean of the deterministic term with which it associated. The stochastic rate equations that result are called Langevin equations [3,4] and are stated in Section 3. They depend nonlinearly on the electron and photon numbers, so in most circumstances one has to solve them numerically. However, one can develop physical insight into laser noise by solving them approximately. By linearizing the Langevin equations about their equilibrium solutions, one can derive stochastic equations for the number deviations, and by treating the noise terms in these equations carefully, one can derive deterministic equations for their means, variances, and correlations. Although the linearized Langevin equations and their associated moment equations are useful, I decided to delay discussions of their consequences until after the source strengths have been determined.

Let $F(m,n,s)$ be the probability that there are $m$ lowest-level (0 or 1) electrons, $n$ upper-level (2) electrons, and $s$ photons in the cavity, where $m$, $n$, and $s$ are nonnegative integers. The Markov equation  for this probability distribution function is stated in Section 4. It is based on the assumption that the electron and photon numbers change randomly, but at the average rates associated with the general rate equations. By calculating the moments of the Markov equation, one can derive equations for the number means, variances, and correlations from first principles. In the Markov approach, the driving terms in the variance and correlation equations appear naturally, and validate the shot-noise formulas upon which the Langevin approach is based. Although the Markov moment equations are exact, they do not form a closed set, because they involve higher-order deviation moments. One can close them by neglecting these deviation moments, in which case they reduce to the linearized Langevin moment equations.

With the validity of the linearized Langevin equations and their source terms established, their consequences are determined in Section 5. By solving the linearized Langevin equations in the frequency domain, one can derive a formula for the RIN spectrum. By inverting this spectrum, one obtains a formula for the autocorrelation function ${\langle {S_1}(t){S_1}(t + \tau )\rangle _t}/S_0^2$, where $\tau$ is the time delay, and by letting $\tau \to 0$, one obtains a formula for the RIN. One can also derive a formula for the RIN by solving the Langevin moment equations in steady state. These metrics (and the formulas for them) are standard [5,6]. They allow one to quantify the noise performance of a particular laser and compare the properties of different types of laser. The main conclusions of this article are summarized in Section 6.

In a previous tutorial , I reviewed the mathematical methods one requires to analyze a laser-like equation for the photon number, and the physical insights one develops by doing so. These methods and insights are also required for this tutorial. Only those generalizations that are required for multiple-variable systems are described herein. My previous article included a quantum-optical analysis of number fluctuations. Because of length limitations, this article includes only classical analyses (deterministic, stochastic, and probabilistic).

For convenience, this article includes five appendices. In Appendix A, some properties and characterizations of stochastic systems are described. In Appendix B, the rules of stochastic calculus are reviewed for an arbitrary system of differential equations, then used to derive general formulas for the Langevin moment equations. In Appendix C, these formulas (which are not linearized) are applied to the Langevin equations for three- and four-level lasers. Moment equations are stated for the electron and photon numbers that are consistent with the Markov moment equations of Section 4. In Appendix D, some generalizations of the Markov moment equations are derived. These generalized equations are consistent with the formulas of Appendix B. Finally, in Appendix E, contour integration is used to inverse transform the RIN spectrum, in order to determine the autocorrelation function and the RIN.

## 2. RATE EQUATIONS

In this section, the electron and photon rate equations for three- and four-level lasers are stated and simplified, as described in the introduction. Readers who are familiar with this material [1,2] can proceed directly to Section 2.D.

#### A. Four-Level Laser

Consider a laser with four electron levels: 0 (ground), 1 (lower), 2 (upper), and 3 (excited), as illustrated in Fig. 1. Examples of such lasers include helium–neon and neodinium–YAG. Let ${N_j}$ be the number of electrons in level $j$. Then the rate equations for the electron numbers are

$${d_t}{N_3}\def\LDeqtab{}={a_{30}}P{N_0} - {a_{30}}(P + 1){N_3} - {a_{32}}{N_3},$$
$${d_t}{N_2}\def\LDeqtab{}={a_{32}}{N_3} + {a_{21}}S{N_1} - {a_{21}}(S + 1){N_2},$$
$${d_t}{N_1}\def\LDeqtab{}=- {a_{21}}S{N_1} + {a_{21}}(S + 1){N_2} - {a_{10}}{N_1},$$
$${d_t}{N_0}\def\LDeqtab{}=- {a_{30}}P{N_0} + {a_{30}}(P + 1){N_3} + {a_{10}}{N_1},$$
where $P$ is the number of pump photons, $S$ is the number of signal (laser) photons, and ${a_{jk}}$ is the rate of spontaneous emission between levels $j$ and $k$. The transition $3 \leftrightarrow 0$ is stimulated by the pump and the transition $2 \leftrightarrow 1$ is stimulated by the signal, whereas the transitions (decays) $3 \to 2$ and $1 \to 0$ are purely spontaneous. For simplicity, the transition $2 \to 0$ is neglected, as are transitions from the aforementioned levels to other levels. It follows from Eqs. (1)–(4) that
$${d_t}({N_0} + {N_1} + {N_2} + {N_3}) = 0.$$
The total electron number is conserved. Fig. 1. Levels and transitions for a four-level laser.

If ${a_{32}} \gg {a_{30}}P$, then ${N_3}$ tends quickly to its quasi-steady-state value

$${N_3} = {a_{30}}P{N_0}/[{a_{32}} + {a_{30}}(P + 1)] \approx {a_{30}}P{N_0}/{a_{32}},$$
which is much less than ${N_0}$. Likewise, if ${a_{10}} \gg {a_{21}}S$, then ${N_1}$ tends quickly to its quasi-steady-state value
$${N_1} = {a_{21}}(S + 1){N_2}/({a_{10}} + {a_{21}}S) \approx {a_{21}}(S + 1){N_2}/{a_{10}},$$
which is much less than ${N_2}$. Electrons in level 3 drop quickly to level 2, and electrons in level 1 drop quickly to level 0. In this case, Eqs. (1)–(4) can be replaced by the reduced set of equations
$${d_t}{N_2} \def\LDeqtab{}\approx {a_{30}}P{N_0} - {a_{21}}(S + 1){N_2},$$
$${d_t}{N_0} \def\LDeqtab{}\approx - {a_{30}}P{N_0} + {a_{21}}(S + 1){N_2}.$$
Equations (8) and (9) are what one would obtain by assuming that electrons transition directly between level 0 and level 2. The first terms on the right sides of these equations represent stimulated pump-photon absorption, whereas the second terms represent stimulated and spontaneous signal-photon emission. It follows from Eqs. (8) and (9) that
$${d_t}({N_0} + {N_2}) \approx 0.$$
The reduced equations conserve the “total” electron number ${N_0} + {N_2}$.

The rate equation for the signal-photon number is

$$\begin{split}{d_t}S &= {a_{21}}{N_2}(S + 1) - {a_{21}}{N_1}S - bS \\ &\approx {a_{21}}{N_2}(S + 1) - bS,\end{split}$$
where $b$ is the transmission-loss rate. Stimulated and spontaneous emission increase the signal-photon number, whereas transmission loss decreases the number and stimulated absorption is unimportant. Spontaneous emission seeds the signal growth, but otherwise is unimportant.

The emission terms in Eqs. (8), (9), and (11) pertain to one lasing mode. Although the stimulated terms are correct, the spontaneous terms in Eqs. (8) and (9) should be augmented by $\mp {a_{nl}}{N_2}$, respectively, where ${a_{nl}} = {a_{21}}I$ is the spontaneous emission rate multiplied by the number of non-lasing (inert) modes into which photons are also emitted. Electrons decay from level 2 to level 1 at the specified rate, then decay immediately to level 0. These additional decays increase the laser threshold (the pump power required for population inversion and signal net gain) but do not affect the well-above-threshold laser evolution significantly (because $S \gg I$).

#### B. Type A Three-Level Laser

Now consider a laser with three electron levels: 0 (ground), 1 (lower), and 2 (upper), as illustrated in Fig. 2. In the notation of Section 2.A, the rate equations for the electron numbers are

$$\begin{split}{d_t}{N_2} &= {a_{20}}P{N_0} - {a_{20}}(P + 1){N_2} + {a_{21}}S{N_1}\\&\quad - {a_{21}}(S + 1){N_2} - {a_{nl}}{N_2},\end{split}$$
$${d_t}{N_1}\def\LDeqtab{} = - {a_{21}}S{N_1} + {a_{21}}(S + 1){N_2} + {a_{nl}}{N_2} - {a_{10}}{N_1},$$
$${d_t}{N_0}\def\LDeqtab{} = - {a_{20}}P{N_0} + {a_{20}}(P + 1){N_2} + {a_{10}}{N_1}.$$
The transition $2 \leftrightarrow 0$ is stimulated by the pump and the transition $2 \leftrightarrow 1$ is stimulated by the signal, whereas the transition (decay) $1 \to 0$ is purely spontaneous. It follows from Eqs. (12)–(14) that
$${d_t}({N_0} + {N_1} + {N_2}) = 0.$$
The total electron number is conserved. The main difference between this laser and a four-level laser is that pump-photon emission limits the upper-level population. Fig. 2. Levels and transitions for a type A three-level laser.

If ${a_{10}} \gg {a_{21}}S$, electrons in level 1 drop quickly to level 0, so ${N_1} \ll {N_2},{N_0}$. In this case, Eqs. (12)–(14) can be replaced by the reduced set of equations

$$\begin{split}{d_t}{N_2} &\approx {a_{20}}P{N_0} - {a_{20}}(P + 1){N_2}\\&\quad - {a_{21}}(S + 1){N_2} - {a_{nl}}{N_2},\end{split}$$
$$\begin{split}{d_t}{N_0} &\approx - {a_{20}}P{N_0} + {a_{20}}(P + 1){N_2}\\&\quad + {a_{21}}(S + 1){N_2} + {a_{nl}}{N_2}.\end{split}$$
Equations (16) and (17) are what one would obtain by assuming that electrons transition directly from level 2 to level 0. The first terms on the right sides represent pump-photon absorption, the second terms represent pump-photon emission, the third terms represent signal-photon emission and the fourth terms represent spontaneous emission into non-lasing modes. It follows from Eqs. (16) and (17) that
$${d_t}({N_0} + {N_2}) = 0.$$
Once again, the reduced equations conserve the “total” electron number ${N_0} + {N_2}$.

The rate equation for the signal-photon number is

$$\begin{split}{d_t}S& = {a_{21}}{N_2}(S + 1) - {a_{21}}{N_1}S - bS, \\ &\approx {a_{21}}{N_2}(S + 1) - bS.\end{split}$$
Once again, stimulated and spontaneous emission increase the signal-photon number, whereas transmission loss decreases the number and stimulated absorption is unimportant.

#### C. Type B Three-Level Laser

Finally, consider a laser with three different electron levels: 1 (lower), 2 (upper), and 3 (excited), as illustrated in Fig. 3. Examples of such lasers include erbium and ruby. In the notation of Section 2.A, the rate equations for the electron numbers are

$${d_t}{N_3} \def\LDeqtab{}= {a_{31}}P{N_1} - {a_{31}}(P + 1){N_3} - {a_{32}}{N_3},$$
$${d_t}{N_2}\def\LDeqtab{} = {a_{32}}{N_3} + {a_{21}}S{N_1} - {a_{21}}(S + 1){N_2} - {a_{nl}}{N_2},$$
$$\begin{split}{d_t}{N_1} &= - {a_{31}}P{N_1} + {a_{31}}(P + 1){N_3} - {a_{21}}S{N_1}\\&\quad + {a_{21}}(S + 1){N_2} + {a_{nl}}{N_2}.\end{split}$$
The transition $3 \leftrightarrow 1$ is stimulated by the pump and the transition $2 \leftrightarrow 1$ is stimulated by the signal, whereas the transition (decay) $3 \to 2$ is purely spontaneous. It follows from Eqs. (20)–(22) that
$${d_t}({N_1} + {N_2} + {N_3}) = 0.$$
The total electron number is conserved.

If ${a_{32}} \gg {a_{31}}P$, then ${N_3} \ll {N_1}$, ${N_2}$. In this case, Eqs. (20)–(22) can be replaced by the reduced set of equations

$${d_t}{N_2} \def\LDeqtab{}\approx {a_{31}}P{N_1} + {a_{21}}S{N_1} - {a_{21}}(S + 1){N_2} - {a_{nl}}{N_2},$$
$${d_t}{N_1} \def\LDeqtab{}\approx - {a_{31}}P{N_1} - {a_{21}}S{N_1} + {a_{21}}(S + 1){N_2} + {a_{nl}}{N_2}.$$
Equations (24) and (25) are what one would obtain by assuming that electrons transition directly from level 1 to level 2. The first and second terms on the right sides represent pump- and signal-photon absorption, respectively, whereas the third terms represent signal-photon emission. It follows from Eqs. (24) and (25) that
$${d_t}({N_1} + {N_2}) \approx 0.$$
The reduced equations conserve the “total” electron number ${N_t} = {N_1} + {N_2}$. Fig. 3. Levels and transitions for a type B three-level laser. Table 1. Rate Coefficients for Three Types of Lasera Fig. 4. Normalized electron and photon numbers $N/{N_t}$ (blue) and $S/{N_t}$ (red) plotted as functions of time ($bt$) for a four-level laser with the gain parameter $a{N_t}/b = 10$. The pump parameter $c/b = 0.01$ (left) and 0.10 (right). Fig. 5. Normalized electron and photon numbers $N/{N_t}$ (blue) and $S/{N_t}$ (red) plotted as functions of time ($bt$) for a type B three-level laser with the gain parameter $a{N_t}/b = 10$. The pump parameter $c/b = 0.01$ (left) and 0.10 (right).

The rate equation for the signal-photon number is

$${d_t}S = {a_{21}}{N_2}(S + 1) - {a_{21}}{N_1}S - bS.$$
Emission increases the signal-photon number, whereas absorption and loss decrease it. The main difference between this laser and a four-level laser is that ${N_2} \gt {N_t}/2$ is required for net gain (so the laser threshold is higher).

#### D. General Rate Equations

For all three types of laser, the rate equations can be written in the forms

$${d_t}S \def\LDeqtab{}= - {a^\prime}MS + aN(S + 1) - bS,$$
$${d_t}N \def\LDeqtab{}= {a^\prime}MS - aN(S + 1) + cM - {c^\prime}N - dN,$$
$${d_t}M \def\LDeqtab{}= - {a^\prime}MS + aN(S + 1) - cM + {c^\prime}N + dN,$$
where $N = {N_2}$ is the number of upper-level electrons. For ideal four-level and type A three-level lasers, $M = {N_0}$ is the number of ground-level electrons, whereas for type B three-level lasers, $M = {N_1}$ is the number of lower-level electrons. The rate coefficients are specified in Table 1. For four-level and type A lasers, ${a^\prime} = 0$ (no stimulated signal-photon absorption), and for four-level and type B lasers, ${c^\prime} = 0$ (no stimulated pump-photon emission). Only for type A lasers is ${c^\prime} \ne 0$ (stimulated pump-photon emission).

Equations (28)–(30) can be solved numerically. It is convenient to measure the electron and photon numbers in units of ${N_t}$, and time in units of $1/b$, which is of the order of the transit time (of light across the cavity). The dimensionless rate equations involve three dimensionless parameters: the gain parameter $a{N_t}/b$, the pumping parameter $c/b$, and the damping parameter $d/b$. To simplify my discussions of laser evolution, I will consider only the weak-damping (moderate-to-strong pumping) regime, in which $d/b$ is much smaller than the other parameters and can be neglected. If $a{N_t}/b = 10$, then the inversion $N$ (or $N - M$) required to counteract loss is much smaller than ${N_t}$, which is typical. It only remains to determine how the laser evolution depends on the pump strength.

The evolution of a four-level laser is illustrated in Fig. 4. For short times, the photon number is small and the upper-level-electron number increases linearly with time. If this growth were to continue, all the electrons would be in the upper level after $b/c$ transit times. When $aN \gt b$, the photons experience net gain, their number increases rapidly (spikes) and the electron number decreases concomitantly. When $aN \lt b$, the photons experience net loss and their number decreases rapidly. For moderate pumping (left image), this process repeats itself several times before a steady state is attained, in which the gain rate $aN$ is clamped at its threshold value $b$ and the photon flux $bS$ equals the pumping rate $cM$. The diminishing oscillations are called relaxation oscillations (ROs) [8,9]. For strong pumping (right image), the photon number spikes, then tends rapidly and almost monotonically to its steady-state value. The evolution of a type A laser is similar. Compared to the previous results, the peak values of $N/{N_t}$ and $S/{N_t}$ are slightly lower, because stimulated pump-photon emission limits the number of electrons.

The evolution of a type B laser is illustrated in Fig. 5. Initially, the photon number is small and the upper-level-electron number grows linearly with time. Subsequently, the growth of the upper-electron number slows because the lower-electron number decreases. Like a four-level laser, for moderate pumping (left image), the photon number spikes, then exhibits ROs, whereas for strong pumping (right image), the photon number spikes, then tends almost monotonically to its steady-state value. Compared to the four-level results, the onset of photon generation occurs later, and the peak values of $N/{N_t}$ and $S/{N_t}$ are higher, because $N$ must exceed ${N_t}/2$ to produce net gain. In steady state, the net-gain rate $a(N - M)$ is clamped at its threshold value $b$. The photon flux $bS$ still equals the pumping rate $cM$, but the steady-state value of $M$ is lower by a factor of 2.

## 3. LANGEVIN EQUATIONS

The rate equations (28)–(30) are deterministic, so they can describe the growth and saturation of the signal-photon number, but cannot describe photon-number fluctuations. One can mimic the effects of quantum fluctuations by adding random source terms to the rate equations, one source term for each deterministic process. In Section 3.A, the stochastic (Langevin) rate equations are stated and the properties of their source terms are discussed briefly. In Section 3.B, the Langevin equations are perturbed about an above-threshold equilibrium. The result is a set of linear stochastic equations for the electron- and photon-number deviations. The upper- and lowest-level electron deviations are anti-correlated, so the lowest-electron deviation can be eliminated. In Section 3.C, the simplified Langevin equations are used to model ROs, and in Section 3.D, they are used to derive deterministic moment equations for the upper-electron and photon-number variances and correlation.

#### A. Langevin Equations

The Langevin equations associated with the rate equations are

$${d_t}S \def\LDeqtab{}= - {a^\prime}MS + aN(S + 1) - bS - R_a^\prime + {R_a} - {R_b},$$
$$\begin{split}{d_t}N &= {a^\prime}MS - aN(S + 1) + cM - {c^\prime}N - dN\\&\quad + R_a^\prime - {R_a}+ {R_c} - R_c^\prime - {R_d},\end{split}$$
$$\begin{split}{d_t}M &= - {a^\prime}MS + aN(S + 1) - cM + {c^\prime}N + dN\\&\quad - R_a^\prime + {R_a} - {R_c}+ R_c^\prime + {R_d}.\end{split}$$
On the right side of these equations, each source term ${R_j}$ is the product of $S_j^{1/2}$, which is a deterministic function of the electron and photon numbers, and ${r_j}$, which is a random function of time with the properties
$$\langle {r_j}(t)\rangle = 0,\quad \langle {r_j}(t){r_k}({t^\prime})\rangle = {\delta _{jk}}\delta (t - {t^\prime}),$$
where $\langle \rangle$ denotes an ensemble average. The first part of Eq. (34) states that positive and negative impulses are equally likely, whereas the second part states that different impulses are statistically independent. Notice that Eqs. (32) and (33) conserve the total electron number, even in the presence of noise.

Formulas for the source strengths ${S_j}$ will be derived in Section 4. In the meantime, one can specify the source strengths tentatively by analogy with the known properties of shot noise [10,11]. In this phenomenon, electrons enter a detector at random times, but at a specified mean rate ($\gamma$). It was shown in  [7,12] that the number of electrons in the detector has Poisson statistics, in which the number mean and variance both equal $\gamma t$. The variance increases (is driven) at the same rate as the mean. These results only apply to a constant driving process (for example, pumping of the upper-level electrons with constant $M$), but it is reasonable to suppose that similar results apply to the other processes, which are assumed to be independent. With these assumptions, the source strengths

$$\begin{split}S_a^\prime &= {a^\prime}\langle MS\rangle ,\quad {S_a} = a\langle N(S + 1)\rangle , \quad {S_b} = b\langle S\rangle , \\ {S_c} &= c\langle M\rangle ,\quad S_c^\prime = {c^\prime}\langle N\rangle ,\quad {S_d} = d\langle N\rangle \end{split}$$
equal the corresponding rates in the mean equations (provided that $\langle {R_j}\rangle = 0$).

In Eqs. (32) and (33), the deterministic ${c^\prime}$ and $d$ terms appear in the combination ${c^\prime}N + dN = ({c^\prime} + d)N$, and the random terms appear in the combination $R_c^\prime + {R_d}$. The random terms both decrease (increase) $N$ and increase (decrease) $M$, and neither affects $S$. Hence, one can replace the deterministic terms by ${d^\prime}N$, where ${d^\prime} = {c^\prime} + d$, and the random terms by $R_d^\prime$, where $S_d^\prime = {c^\prime}\langle N\rangle + d\langle N\rangle = {d^\prime}\langle N\rangle$.

To be precise, random source terms should be added to the fundamental rate equations (1)–(4) and (11), (12)–(14) and (19), and (20)–(22) and (27). According to Eqs. (35), the source strengths equal the corresponding deterministic rates, so the same approximations that allow one to neglect some of the deterministic terms also allow one to neglect the corresponding source terms. Hence, one need only add to the rate equations source terms that correspond to the retained deterministic terms.

Equations (31)–(33) are stochastic and depend nonlinearly on the electron and photon numbers, so in most circumstances one has to solve them numerically (and repeatedly, to accumulate reliable statistics). However, one can develop physical insight into laser noise by solving them approximately.

#### B. Linearized Langevin Equations

According to Eq. (28), the growth of the signal-photon number is initiated by spontaneous emission. Below threshold, the output photon flux is proportional to the spontaneous-emission rate. However, above threshold, the photon flux depends only weakly on this rate, so spontaneous emission can be neglected. In this regime, the Langevin equations (31)–(33) can be analyzed perturbatively. Let $M = {M_0} + {M_1}$, $N = {N_0} + {N_1}$, and $S = {S_0} + {S_1}$, where the subscript 0 denotes a (large) zeroth-order quantity and the subscript 1 denotes a (small) first-order quantity (deviation). Then, in zeroth-order equilibrium and without noise, Eqs. (31)–(33) imply that

$$a{N_0} - {a^\prime}{M_0} = b,$$
$$(a{N_0} - {a^\prime}{M_0}){S_0} = b{S_0} = c{M_0} - {d^\prime}{N_0}.$$
Net gain ($a{N_0} - {a^\prime}{M_0}$) compensates loss ($b$), and the output photon flux ($b{S_0}$) equals the difference between the rates of stimulated pump-photon absorption ($c{M_0}$) and pump- and non-lasing-photon emission (${d^\prime}{N_0}$). In the strong-pumping regime, $c{M_0} \gg {d^\prime}{N_0}$.

The first-order equations, with noise, are

$$\begin{split}{d_t}{S_1} &= (a{N_0} - {a^\prime}{M_0} - b){S_1} + (a{S_0}){N_1} - ({a^\prime}{S_0}){M_1}\\&\quad - R_a^\prime + {R_a} - {R_b},\end{split}$$
$$\begin{split}{d_t}{N_1} &= ({a^\prime}{M_0} - a{N_0}){S_1} - (a{S_0} + {d^\prime}){N_1}\\&\quad + ({a^\prime}{S_0} + c){M_1}+ R_a^\prime - {R_a} + {R_c} - R_d^\prime,\end{split}$$
$$\begin{split}{d_t}{M_1} &= (a{N_0} - {a^\prime}{M_0}){S_1} + (a{S_0} + {d^\prime}){N_1}\\&\quad - ({a^\prime}{S_0} + c){M_1} - R_a^\prime + {R_a} - {R_c} + R_d^\prime,\end{split}$$
where the source terms only depend on zeroth-order quantities. One can use Eqs. (38)–(40) to study the evolution of number deviations. However,
$${d_t}({M_1} + {N_1}) = 0,$$
so one can conclude that ${M_1} = - {N_1}$ and eliminate the ${M_1}$ equation.

The remaining rate equations are

$$\begin{split}{d_t}{S_1} &= (a{N_0} - {a^\prime}{M_0} - b){S_1} + (a{S_0} + {a^\prime}{S_0}){N_1}\\&\quad - R_a^\prime + {R_a} - {R_b},\end{split}$$
$$\begin{split}{d_t}{N_1} &= ({a^\prime}{M_0} - a{N_0}){S_1} - (a{S_0} + {a^\prime}{S_0} + c + {d^\prime}){N_1}\\&\quad + R_a^\prime - {R_a} + {R_c} - R_d^\prime.\end{split}$$
These linearized Langevin equations can be rewritten in the compact forms 
$${d_t}{S_1} \def\LDeqtab{}= {\gamma _{\textit ss}}{S_1} + {\gamma _{\textit sn}}{N_1} + {R_s},$$
$${d_t}{N_1}\def\LDeqtab{} = - {\gamma _{\textit ns}}{S_1} - {\gamma _{\textit nn}}{N_1} + {R_n}.$$
On the right side of these equations, the compound sources have the properties
$$\langle {R_j}(t)\rangle = 0,\quad\langle {R_j}(t){R_k}({t^\prime})\rangle = {R_{jk}}\delta (t - {t^\prime}),$$
where the source strengths (variances and correlation) are
$$\begin{split}{R_{\textit ss}} &= S_a^\prime + {S_a} + {S_b},\quad {R_{\textit ns}} = - S_a^\prime - {S_a}, \\ {R_{\textit nn}} &= S_a^\prime + {S_a} + {S_c} + S_d^\prime.\end{split}$$
For convenience, the coupling coefficients are specified in Table 2 and the source strengths are specified in Table 3. In the analyses that follow, the ${\gamma _{\textit ss}}$ terms are retained so that the results remain valid for more complicated models in which ${\gamma _{\textit ss}} \ne 0$ [5,6]. Table 2. Coupling Coefficients for Three Types of Lasera Table 3. Source Strengths for Three Types of Lasera

#### C. Relaxation Oscillations

The characteristic equation associated with Eqs. (44) and (45) can be written in the form

$${\gamma ^2} + 2{\nu _0}\gamma + \omega _0^2 = 0,$$
where the squared frequency and damping parameters are
$$\omega _0^2 = {\gamma _{\textit ns}}{\gamma _{\textit sn}} - {\gamma _{\textit nn}}{\gamma _{\textit ss}}, \quad {\nu _0} = ({\gamma _{\textit nn}} - {\gamma _{\textit ss}})/2,$$
respectively [9,13]. If ${\nu _0}$ is less than, equal to, or greater than ${\omega _0}$, the laser is termed under-damped, critically damped, or over-damped, respectively. For laser models in which ${\gamma _{\textit ss}} = 0$ (which include all the models discussed herein), ${\omega _0} = ({\gamma _{\textit ns}}{\gamma _{\textit sn}}{)^{1/2}}$ and ${\nu _0} = {\gamma _{\textit nn}}/2$. It follows from Eq. (48) that the characteristic exponent $\gamma = - {\nu _0} \pm i{\omega _r}$, where the resonance frequency ${\omega _r} = (\omega _0^2 - \nu _0^2{)^{1/2}}$. In terms of the rate coefficients, the exponent
$$\gamma = ({\gamma _{\textit ss}} - {\gamma _{\textit nn}})/2 \pm i{[{\gamma _{\textit ns}}{\gamma _{\textit sn}} - {({\gamma _{\textit nn}} + {\gamma _{\textit ss}})^2}/4]^{1/2}}.$$

In the absence of noise (${R_s}$, ${R_n} = 0$), the solutions of Eqs. (44) and (45) can be written in the forms

$${S_1}(t) \def\LDeqtab{}= {G_{\textit ss}}(t){S_1}(0) + {G_{\textit sn}}(t){N_1}(0),$$
$${N_1}(t) \def\LDeqtab{}= {G_{\textit ns}}(t){S_1}(0) + {G_{\textit nn}}(t){N_1}(0),$$
where the transfer (Green) functions are
$${G_{\textit ss}}(t) \def\LDeqtab{}= [\cos ({\omega _r}t) + {\gamma _a}\sin ({\omega _r}t)/{\omega _r}]\exp ( - {\nu _0}t),$$
$${G_{\textit sn}}(t)\def\LDeqtab{} = [{\gamma _{\textit sn}}\sin ({\omega _r}t)/{\omega _r}]\exp ( - {\nu _0}t),$$
$${G_{\textit ns}}(t)\def\LDeqtab{}= - [{\gamma _{\textit ns}}\sin ({\omega _r}t)/{\omega _r}]\exp ( - {\nu _0}t),$$
$${G_{\textit nn}}(t) \def\LDeqtab{}= [\cos ({\omega _r}t) - {\gamma _a}\sin ({\omega _r}t)/{\omega _r}]\exp ( - {\nu _0}t),$$
and the rate coefficient ${\gamma _a} = ({\gamma _{\textit nn}} + {\gamma _{\textit ss}})/2$. The Green function ${G_{jk}}(t)$ describes the effect on deviation $j$ at time $t$ of a unit initial value of deviation $k$. [More generally, the Green function ${G_{jk}}(t - {t^\prime})$ describes the effect on deviation $j$ at time $t$ of a unit impulse applied to deviation $k$ at the earlier time ${t^\prime}$]. Although solutions (51) and (52) cannot describe the first photon-number spike (which is a nonlinear phenomenon), they should describe the subsequent evolutions of the electron and photon numbers. They predict that the electron- and photon-number deviations exhibit ROs if the laser is under-damped and decrease monotonically if the laser is critically damped or over-damped.

Consider the four-level laser of Fig. 4. For moderate pumping ($c/b = 0.01$), the normalized frequencies predicted by Eqs. (49) are ${\omega _0}/b = 0.30$, ${\nu _0}/b = 0.05$, and ${\omega _r}/b = 0.30$, and the normalized period $b{\tau _r} = 2\pi b/{\omega _r} = 21$. For strong pumping ($c/b = 0.10$), the frequencies are ${\omega _0}/b = 0.95$, ${\nu _0}/b = 0.50$, and ${\omega _r}/b = 0.81$, and the period $b{\tau _r} = 7.8$. The predicted periods are consistent with the numerical results displayed in Fig. 4. The predicted frequencies and periods of a type A laser are similar.

Now consider the type B laser of Fig. 5. For moderate pumping ($c/b = 0.01$), the predicted frequencies are ${\omega _0}/b = 0.30$, ${\nu _0}/b = 0.05$, and ${\omega _r}/b = 0.30$, and the predicted period $b{\tau _r} = 21$. For strong pumping ($c/b = 0.10$), the frequencies are ${\omega _0}/b = 0.95$, ${\nu _0}/b = 0.50$, and ${\omega _r}/b = 0.81$, and the period $b{\tau _r} = 7.8$. The predicted periods are consistent with the numerical results displayed in Fig. 5. Thus, perturbation theory describes the approach to equilibrium accurately.

In these examples, the type B frequencies and damping rates are equal to the four-level values, because the examples involve the same pump parameters. The factors of 2 in the coupling coefficients listed in Table 2 (which are due to signal-photon absorption) are compensated by reduced values of ${S_0} = c{M_0}/b$. In a four-level laser, ${M_0} = {N_t} - {N_i}$, where ${N_i} = b/a$ is the inversion, whereas in a type B laser, ${M_0} = ({N_t} - {N_i})/2$.

Equations (44) and (45) can also be solved analytically in the presence of noise (${R_s}$, ${R_n} \ne 0$). However, I will refrain from doing so until the formulas for the source strengths have been validated.

#### D. Linearized Langevin Moment Equations

One can use Eqs. (44) and (45) to derive equations for the second-order deviation moments $\langle S_1^2\rangle$, $\langle {N_1}{S_1}\rangle$, and $\langle N_1^2\rangle$. Before doing so, a brief discussion of stochastic calculus is required. Let $\delta t$ be a short time interval and consider the noise increment $\int _0^{\delta t}{r_j}(t){\rm d}t$. Then it follows from the second of Eqs. (34) that

$$\left\langle {\left[\int_0^{\delta t}{r_j}(t){\rm d}t\right]^2}\right\rangle ={\int _0^{\delta t}} {{\int_0^{\delta t}}}\langle {r_j}(t){r_j}({t^\prime})\rangle {\rm d}t{\rm d}{t^\prime} = \delta t.$$
In a crude sense, the noise increment is of order $\delta {t^{1/2}}$. This scaling differentiates stochastic calculus from deterministic calculus. Stochastic calculus is reviewed in Appendix B. Ito calculus is based on the approximation that the source functions have zero correlation time, whereas Stratonovich calculus is based on the assumption that they have very short, but nonzero, correlation times. Ito calculus is an idealization, because no noise source has a zero correlation time (infinite frequency bandwidth), and no admitting system has infinite frequency bandwidth (zero response time). Nonetheless, I chose to use it in this article because it is intuitive and its predictions are consistent with the results of Section 4, which are based on the similar assumption of instantaneous changes in photon number.

In Ito calculus, the differential equations (44) and (45) are equivalent to the difference equations

$${S_1}(\delta t) \def\LDeqtab{}\approx (1 + {\gamma _{\textit ss}}\delta t){S_1} + ({\gamma _{\textit sn}}\delta t){N_1} + \int _0^{\delta t}{R_s}(t){\rm d}t,$$
$${N_1}(\delta t) \def\LDeqtab{}\approx - ({\gamma _{\textit ns}}\delta t){S_1} + (1 - {\gamma _{\textit nn}}\delta t){N_1} + \int _0^{\delta t}{R_n}(t){\rm d}t.$$
By combining Eqs. (58) and (59), averaging the equations that result and letting $\delta t \to 0$, one obtains the variance and correlation equations
$${d_t}\langle S_1^2\rangle\def\LDeqtab{} = 2{\gamma _{\textit ss}}\langle S_1^2\rangle + 2{\gamma _{\textit sn}}\langle {N_1}{S_1}\rangle + {R_{\textit ss}},$$
$$\begin{split}{d_t}\langle {N_1}{S_1}\rangle &= - {\gamma _{\textit ns}}\left\langle {S_1^2} \right\rangle + ({\gamma _{\textit ss}} - {\gamma _{\textit nn}})\left\langle {{N_1}{S_1}} \right\rangle\\&\quad + {\gamma _{\textit sn}}\langle N_1^2\rangle + {R_{\textit ns}},\end{split}$$
$${d_t}\langle N_1^2\rangle\def\LDeqtab{} = - 2{\gamma _{\textit ns}}\langle {N_1}{S_1}\rangle - 2{\gamma _{\textit nn}}\langle N_1^2\rangle + {R_{\textit nn}}.$$
Equations (60)–(62) are deterministic, so it is straightforward to solve them analytically. However, I will refrain from doing so until the formulas for the source strengths have been validated.

## 4. MARKOV EQUATIONS

Although the Langevin equations (31)–(33) and (38)–(40) model electron- and photon-number fluctuations, they are based on the premise that these quantities are real numbers that change continuously, whereas they are actually nonnegative integers that change discontinuously (instantaneously). Following Shimoda et al.  and McCumber , one can replace the Langevin equations by the family of Markov equations

$$\begin{split} {d_t}F(m,n,s) &= - {a^\prime}msF(m,n,s) + {a^\prime}(m + 1)(s + 1) F(m + 1,n - 1,s + 1)\\[-5pt] &\quad - an(s + 1)F(m,n,s) + a(n + 1)sF(m - 1,n + 1,s - 1) \\[-5pt] &\quad- bsF(m,n,s) + b(s + 1)F(m,n,s + 1) \\[-5pt]&\quad - cmF(m,n,s) + c(m + 1)F(m + 1,n - 1,s) \\[-5pt]&\quad- {c^\prime}nF(m,n,s) + {c^\prime}(n + 1)F(m - 1,n + 1,s) \\[-5pt]&\quad- dnF(m,n,s) + d(n + 1)F(m - 1,n + 1,s),\end{split}$$
where $F(m,n,s)$ is the probability that $m$ lowest-level electrons, $n$ upper-level electrons and $s$ signal photons are in the cavity. (For $m = 0$, the second $a$ term and ${c^\prime}$ term are omitted, for $n = 0$, the second ${a^\prime}$ and $c$ terms are omitted, and for $s = 0$, the second $a$ term is omitted.) These equations are based on the assumption that electrons and photons are created and destroyed randomly, but at specified mean rates. The signal-absorption rate is proportional to $ms$, the signal-emission rate is proportional to $n(s + 1)$, and the signal-loss rate is proportional to $s$. (In the absence of spontaneous emission, the signal-emission rate is proportional to $ns$.) The pump-absorption rate is proportional to $mp$, the pump-emission rate is proportional to $n(p + 1)$, and both processes change the number of pump photons. However, if the pump is strong ($p,p + 1 \approx P \gg 1$), then both rates are proportional to $P$, which remains constant and is included in the formulas for $c$ and ${c^\prime}$. In this approximation, the effects on the electrons of pump- and non-lasing-photon emission are indistinguishable, so one can replace the ${c^\prime}$ and $d$ terms in Eq. (63) by a pair of ${d^\prime}$ terms (${d^\prime} = {c^\prime} + d$).

The Markov equation (63) describes the evolution of an ensemble of members. Let $(m,n,s)$ denote the member with $m$ lowest-level electrons, $n$ upper-level electrons, and $s$ signal photons, and consider the effects of signal absorption. If $(m,n,s) \to (m - 1,n + 1,s - 1)$, then $F(m,n,s)$ decreases, whereas if $(m + 1,n - 1,s + 1) \to (m,n,s)$, then $F(m,n,s)$ increases. For signal emission, if $(m,n,s) \to (m + 1,n - 1,s + 1)$, then $F(m,n,s)$ decreases, whereas if $(m - 1,n + 1,s - 1) \to (m,n,s)$, then $F(m,n,s)$ increases. For signal loss, which does not involve the electrons, if $(m,n,s) \to (m,n,s - 1)$, then $F(m,n,s)$ decreases, whereas if $(m,n,s + 1) \to (m,n,s)$, then $F(m,n,s)$ increases. The descriptions of the electron decay and pumping transitions, which do not involve the signal photons, are similar.

The total probability $T = {\sum _0^\infty}{\sum _0^\infty}{\sum _0^\infty}F(m,n,s)$, where the summations are over $m$, $n$ and $s$, respectively. It follows from Eq. (63) that

$$\begin{split}{d_t}T &= - {a^\prime}\sum\limits_0 \sum\limits_0 \sum\limits_0 msF(m,n,s) + {a^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 (m + 1)(s + 1)F(m + 1,n - 1,s + 1) \\[-5pt]&\quad - a\sum\limits_0 \sum\limits_0 \sum\limits_0 n(s + 1)F(m,n,s)+ a\sum\limits_1 \sum\limits_0 \sum\limits_1 (n + 1)sF(m - 1,n + 1,s - 1) \\[-5pt]&\quad - b\sum\limits_0 \sum\limits_0 \sum\limits_0 sF(m,n,s) + b\sum\limits_0 \sum\limits_0 \sum\limits_0 (s + 1)F(m,n,s + 1) \\[-5pt]&\quad - c\sum\limits_0 \sum\limits_0 \sum\limits_0 mF(m,n,s) + c\sum\limits_0 \sum\limits_1 \sum\limits_0 (m + 1)F(m + 1,n - 1,s) \\[-5pt]&\quad - {d^\prime}\sum\limits_0 \sum\limits_0 \sum\limits_0 nF(m,n,s) + {d^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_0 (n + 1)F(m - 1,n + 1,s) \\[-5pt]& = - {a^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_1 msF(m,n,s) + {a^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_1 msF(m,n,s) \\[-5pt]&\quad - a\sum\limits_0 \sum\limits_1 \sum\limits_0 n(s + 1)F(m,n,s) + a\sum\limits_0 \sum\limits_1 \sum\limits_0 n(s + 1)F(m,n,s) \\[-5pt] &\quad- b\sum\limits_0 \sum\limits_0 \sum\limits_1 sF(m,n,s) + b\sum\limits_0 \sum\limits_0 \sum\limits_1 sF(m,n,s) \\[-5pt]&\quad - c\sum\limits_1 \sum\limits_0 \sum\limits_0 mF(m,n,s) + c\sum\limits_1 \sum\limits_0 \sum\limits_0 mF(m,n,s) \\[-5pt]&\quad - {d^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 nF(m,n,s) + {d^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 nF(m,n,s) \\[-5pt]& = 0.\end{split}$$
Thus, the Markov equation conserves the total probability, which is normalized to 1. Notice that the manipulation of the $b$ terms in Eq. (64) did not change $m$ and $n$, so one can multiply these terms by any polynomial of $m$ and $n$ and they would still cancel. Likewise, the manipulation of the $c$ and ${d^\prime}$ terms did not change $s$, so one can multiply these terms by any polynomial of $s$ and they would still cancel: Signal loss does not affect the electrons directly, and pump-induced electron transitions do not affect the signal photons directly.

#### A. Markov Moment Equations

The number moments $\langle {m}^{\alpha} {n^\beta }{s^\gamma }\rangle = \sum _0^\infty \sum _0^\infty \sum _0^\infty {m^\alpha }{n^\beta }{s^\gamma }F(m,n,s)$. Three first-order moments and six second-order moments are required to model number fluctuations. While writing this article, I had to decide between doing nine shorter calculations (${a^\prime} = 0$ or ${c^\prime} = 0$) three times and doing nine longer calculations (${a^\prime} \ne 0$ and ${c^\prime} \ne 0$) once. I chose the latter option, which is the lesser evil. The first signal-photon moment obeys the equation

$$\begin{split}{d_t}\langle s\rangle &= - {a^\prime}\sum\limits_0 \sum\limits_0 \sum\limits_0 m{s^2}F(m,n,s) + {a^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 (m + 1)s(s + 1)F(m + 1,n - 1,s + 1) \\[-2pt] &\quad - a\sum\limits_0 \sum\limits_0 \sum\limits_0 ns(s + 1)F(m,n,s) + a\sum\limits_1 \sum\limits_0 \sum\limits_1 (n + 1){s^2}F(m - 1,n + 1,s - 1) \\[-2pt] &\quad - b\sum\limits_0 \sum\limits_0 \sum\limits_0 {s^2}F(m,n,s) + b\sum\limits_0 \sum\limits_0 \sum\limits_0 s(s + 1)F(m,n,s + 1) \\[-2pt] & = - {a^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_1 m{s^2}F(m,n,s) + {a^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_1 ms(s - 1)F(m,n,s) \\[-2pt] &\quad - a\sum\limits_0 \sum\limits_1 \sum\limits_0 ns(s + 1)F(m,n,s) + a\sum\limits_0 \sum\limits_1 \sum\limits_0 n{(s + 1)^2}F(m,n,s) \\[-2pt] &\quad - b\sum\limits_0 \sum\limits_0 \sum\limits_1 {s^2}F(m,n,s) + b\sum\limits_0 \sum\limits_0 \sum\limits_1 s(s - 1)F(m,n,s).\end{split}$$
The $c$ and ${d^\prime}$ terms were omitted from Eq. (65) because they do not affect $\langle s\rangle$ directly, as explained above. By combining the terms on the right side, one obtains the first-moment equation
$${d_t}\langle s\rangle = - {a^\prime}\langle ms\rangle + a\langle n(s + 1)\rangle - b\langle s\rangle ,$$
where $a\langle ms\rangle$ is the mean (stimulated) absorption rate, $a\langle {n(s + 1)} \rangle$ is the mean (stimulated and spontaneous) emission rate, and $b$ is the mean loss rate.

The first upper-level-electron moment obeys the equation

$$\begin{split}{d_t}\langle n\rangle &= - {a^\prime}\sum\limits_0 \sum\limits_0 \sum\limits_0 mnsF(m,n,s) + {a^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 (m + 1)n(s + 1)F(m + 1,n - 1,s + 1) \\[-2pt] &\quad - a\sum\limits_0 \sum\limits_0 \sum\limits_0 {n^2}(s + 1)F(m,n,s) + a\sum\limits_1 \sum\limits_0 \sum\limits_1 n(n + 1)sF(m - 1,n + 1,s - 1) \\[-2pt] &\quad - c\sum\limits_0 \sum\limits_0 \sum\limits_0 mnF(m,n,s) + c\sum\limits_0 \sum\limits_1 \sum\limits_0 (m + 1)nF(m + 1,n - 1,s) \\[-2pt] &\quad - {d^\prime}\sum\limits_0 \sum\limits_0 \sum\limits_0 {n^2}F(m,n,s) + {d^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_0 n(n + 1)F(m - 1,n + 1,s) \\[-2pt] & = - {a^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_1 mnsF(m,n,s) + {a^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_1 m(n + 1)sF(m,n,s) \\[-2pt] &\quad - a\sum\limits_0 \sum\limits_1 \sum\limits_0 {n^2}(s + 1)F(m,n,s) + a\sum\limits_0 \sum\limits_1 \sum\limits_0 n(n - 1)(s + 1)F(m,n,s) \\[-2pt] &\quad - c\sum\limits_1 \sum\limits_0 \sum\limits_0 mnF(m,n,s) + c\sum\limits_1 \sum\limits_0 \sum\limits_0 m(n + 1)F(m,n,s) \\[-2pt] &\quad - {d^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 {n^2}F(m,n,s) + {d^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 n(n - 1)F(m,n,s).\end{split}$$
The $b$ terms were omitted from Eq. (67) because they do not affect $\langle n \rangle$ directly. By combining the terms on the right side, one obtains the first-moment equation
$${d_t}\langle n \rangle = {a^\prime}\langle {ms}\rangle - a\langle {n(s + 1)} \rangle + c\langle m \rangle - {d^\prime}\langle n \rangle ,$$
where $c\langle m \rangle$ is the mean rate of pump-stimulated absorption and ${d^\prime}\langle n \rangle$ is the mean rate of pump-stimulated emission.

The first lowest-level-electron moment obeys the equation

$$\begin{split}{d_t}\langle m \rangle &= - {a^\prime}\sum\limits_0 \sum\limits_0 \sum\limits_0 {m^2}sF(m,n,s) + {a^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 m(m + 1)(s + 1)F(m + 1,n - 1,s + 1) \\[-2pt] &\quad - a\sum\limits_0 \sum\limits_0 \sum\limits_0 mn(s + 1)F(m,n,s) + a\sum\limits_1 \sum\limits_0 \sum\limits_1 m(n + 1)sF(m - 1,n + 1,s - 1) \\[-2pt] &\quad - c\sum\limits_0 \sum\limits_0 \sum\limits_0 {m^2}F(m,n,s) + c\sum\limits_0 \sum\limits_1 \sum\limits_0 m(m + 1)F(m + 1,n - 1,s) \\[-2pt] &\quad - {d^\prime}\sum\limits_0 \sum\limits_0 \sum\limits_0 mnF(m,n,s) + {d^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_0 m(n + 1)F(m - 1,n + 1,s) \\[-2pt] & = - {a^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_1 {m^2}sF(m,n,s) + {a^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_1 m(m - 1)sF(m,n,s) \\[-2pt] &\quad - a\sum\limits_0 \sum\limits_1 \sum\limits_0 mn(s + 1)F(m,n,s) + a\sum\limits_0 \sum\limits_1 \sum\limits_0 (m + 1)n(s + 1)F(m,n,s) \\[-2pt] &\quad - c\sum\limits_1 \sum\limits_0 \sum\limits_0 {m^2}F(m,n,s) + c\sum\limits_1 \sum\limits_0 \sum\limits_0 m(m - 1)F(m,n,s) \\[-2pt] &\quad - {d^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 mnF(m,n,s) + {d^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 (m + 1)nF(m,n,s).\end{split}$$
By combining the terms on the right side of Eq. (69), one obtains the first-moment equation
$${d_t}\langle m \rangle = - {a^\prime}\langle {ms} \rangle + a\langle {n(s + 1)} \rangle - c\langle m \rangle + {d^\prime}\langle n \rangle .$$
Equations (66), (68) and (70) show that the electron- and photon-number means change at the mean rates associated with the Langevin equations (28)–(30). This commonsense result makes the Markov approach seem reasonable. Furthermore, signal-photon absorption and emission conserve the sums $\langle m \rangle + \langle n \rangle$ and $\langle n \rangle + \langle s \rangle$, and the difference $\langle m \rangle - \langle s \rangle$, and pump-photon absorption and emission conserve the sum $\langle m \rangle + \langle n \rangle$. These conservation laws were built into Eq. (63), so their appearances serve as algebra checks. Notice that the first moments are coupled to the second moments $\langle {ms} \rangle$ and $\langle {ns} \rangle$, which are not yet known. For reference, the $\pm a\langle n \rangle$ terms in Eqs. (66), (68) and (70) are associated with spontaneous emission.

Now consider the second number moments. The variances will be determined first, because the calculations are similar to the first-moment calculations. The second photon moment obeys the equation

$$\begin{split}{d_t}\left\langle {{s^2}} \right\rangle &= - {a^\prime}\sum\limits_0 \sum\limits_0 \sum\limits_0 m{s^3}F(m,n,s) + {a^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 (m + 1){s^2}(s + 1)F(m + 1,n - 1,s + 1) \\[-2pt] &\quad - a\sum\limits_0 \sum\limits_0 \sum\limits_0 n{s^2}(s + 1)F(m,n,s) + a\sum\limits_1 \sum\limits_0 \sum\limits_1 (n + 1){s^3}F(m - 1,n + 1,s - 1) \\[-2pt] &\quad - b\sum\limits_0 \sum\limits_0 \sum\limits_0 {s^3}F(m,n,s) + b\sum\limits_0 \sum\limits_0 \sum\limits_0 {s^2}(s + 1)F(m,n,s + 1) \\[-2pt] & = - {a^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_1 m{s^3}F(m,n,s) + {a^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_1 ms{(s - 1)^2}F(m,n,s) \\[-2pt] &\quad - a\sum\limits_0 \sum\limits_1 \sum\limits_0 n{s^2}(s + 1)F(m,n,s) + a\sum\limits_0 \sum\limits_1 \sum\limits_0 n{(s + 1)^3}F(m,n,s) \\[-2pt] &\quad - b\sum\limits_0 \sum\limits_0 \sum\limits_1 {s^3}F(m,n,s) + b\sum\limits_0 \sum\limits_0 \sum\limits_1 s{(s - 1)^2}F(m,n,s).\end{split}$$
The $c$ and ${d^\prime}$ terms were omitted from Eq. (71) because they do not affect $\left\langle {{s^2}} \right\rangle$ directly. By combining the terms on the right side, one obtains the second-moment equation
$${d_t}\langle {s^2}\rangle = {a^\prime}( - 2\langle m{s^2}\rangle \, + \,\langle ms\rangle ) + a(2\langle n{s^2}\rangle + 3\langle ns\rangle + \langle n\rangle ) + b( - 2\langle {s^2}\rangle + \langle s\rangle ).$$
The second upper-electron moment obeys the equation
$$\begin{split}{d_t}\left\langle {{n^2}} \right\rangle &= - {a^\prime}\sum\limits_0 \sum\limits_0 \sum\limits_0 m{n^2}sF(m,n,s) + {a^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 (m + 1){n^2}(s + 1)F(m + 1,n - 1,s + 1) \\[-3pt] &\quad - a\sum\limits_0 \sum\limits_0 \sum\limits_0 {n^3}(s + 1)F(m,n,s) + a\sum\limits_1 \sum\limits_0 \sum\limits_1 {n^2}(n + 1)sF(m - 1,n + 1,s - 1) \\[-3pt] &\quad - c\sum\limits_0 \sum\limits_0 \sum\limits_0 m{n^2}F(m,n,s) + c\sum\limits_0 \sum\limits_1 \sum\limits_0 (m + 1){n^2}F(m + 1,n - 1,s) \\[-3pt] &\quad - {d^\prime}\sum\limits_0 \sum\limits_0 \sum\limits_0 {n^3}F(m,n,s) + {d^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_0 {n^2}(n + 1)F(m - 1,n + 1,s) \\[-3pt] & = - {a^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_1 m{n^2}sF(m,n,s) + {a^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_1 m{(n + 1)^2}sF(m,n,s) \\[-3pt] &\quad - a\sum\limits_0 \sum\limits_1 \sum\limits_0 {n^3}(s + 1)F(m,n,s) + a\sum\limits_0 \sum\limits_1 \sum\limits_0 n{(n - 1)^2}(s + 1)F(m,n,s) \\[-3pt] &\quad - c\sum\limits_1 \sum\limits_0 \sum\limits_0 m{n^2}F(m,n,s) + c\sum\limits_1 \sum\limits_0 \sum\limits_0 m{(n + 1)^2}F(m,n,s) \\[-3pt] &\quad - {d^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 {n^3}F(m,n,s) + {d^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 n{(n - 1)^2}F(m,n,s).\end{split}$$
The $b$ terms were omitted from Eq. (73) because they do not affect $\left\langle {{n^2}} \right\rangle$ directly. By combining the terms on the right side, one obtains the second-moment equation
$$\begin{split}{d_t}\left\langle {{n^2}} \right\rangle& = {a^\prime}(2\langle {mns} \rangle + \langle {ms} \rangle ) + a( - 2\langle {{n^2}s} \rangle - 2\langle {{n^2}} \rangle + \langle {ns} \rangle + \langle n \rangle ) \\[-3pt] &\quad + c(2\langle {mn} \rangle + \langle m \rangle ) + {d^\prime}( - 2\langle {{n^2}} \rangle + \langle n\rangle ).\end{split}$$
The second lowest-electron moment obeys the equation
$$\begin{split}{d_t}\left\langle {{m^2}} \right\rangle &= - {a^\prime}\sum\limits_0 \sum\limits_0 \sum\limits_0 {m^3}sF(m,n,s) + {a^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 {m^2}(m + 1)(s + 1)F(m + 1,n - 1,s + 1) \\[-3pt] &\quad - a\sum\limits_0 \sum\limits_0 \sum\limits_0 {m^2}n(s + 1)F(m,n,s) + a\sum\limits_1 \sum\limits_0 \sum\limits_1 {m^2}(n + 1)sF(m - 1,n + 1,s - 1) \\[-3pt] &\quad - c\sum\limits_0 \sum\limits_0 \sum\limits_0 {m^3}F(m,n,s) + c\sum\limits_0 \sum\limits_1 \sum\limits_0 {m^2}(m + 1)F(m + 1,n - 1,s) \\[-3pt] &\quad - {d^\prime}\sum\limits_0 \sum\limits_0 \sum\limits_0 {m^2}nF(m,n,s) + {d^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_0 {m^2}(n + 1)F(m - 1,n + 1,s) \\[-3pt] & = - {a^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_1 {m^3}sF(m,n,s) + {a^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_1 m{(m - 1)^2}sF(m,n,s) \\[-3pt] &\quad - a\sum\limits_0 \sum\limits_1 \sum\limits_0 {m^2}n(s + 1)F(m,n,s) + a\sum\limits_0 \sum\limits_1 \sum\limits_0 {(m + 1)^2}n(s + 1)F(m,n,s) \\[-3pt] &\quad - c\sum\limits_1 \sum\limits_0 \sum\limits_0 {m^3}F(m,n,s) + c\sum\limits_1 \sum\limits_0 \sum\limits_0 m{(m - 1)^2}F(m,n,s) \\[-3pt] &\quad - {d^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 {m^2}nF(m,n,s) + {d^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 {(m + 1)^2}nF(m,n,s).\end{split}$$
By combining the terms on the right side of Eq. (75), one obtains the second-moment equation
$$\begin{split}{d_t}\left\langle {{m^2}} \right\rangle &= {a^\prime}( - 2\left\langle {{m^2}s} \right\rangle + \langle {ms}\rangle ) + a(2\langle {mns} \rangle + 2\langle {mn} \rangle + \langle {ns} \rangle + \langle n \rangle ) \\ &\quad + c( - 2\left\langle {{m^2}} \right\rangle + \langle m \rangle ) + {d^\prime}(2\langle {mn} \rangle + \langle n \rangle ).\end{split}$$
Notice that the second individual moments are coupled to third moments, which are unknown.

The number deviations $\delta s = s - \langle s \rangle$, $\delta n = n - \langle n \rangle$ and $\delta m = m - \langle m \rangle$, so the second deviation moments $\left\langle {\delta {s^2}} \right\rangle = \left\langle {{s^2}} \right\rangle - {\left\langle s \right\rangle ^2}$ and $\langle {\delta n\delta s} \rangle = \langle {ns} \rangle - \langle n \rangle \langle s \rangle$, and the other deviation moments are defined similarly. By combining Eqs. (66), (68) and (70) with Eqs. (72), (74) and (76), one obtains the variance equations

$$\begin{split}{d_t}\left\langle {\delta {s^2}} \right\rangle &= - 2{a^\prime}\langle {\delta ms\delta s} \rangle + 2a\langle {\delta ns\delta s} \rangle + 2a\langle {\delta n\delta s} \rangle - 2b\left\langle {\delta {s^2}} \right\rangle \\[-2pt] &\quad + {a^\prime}\langle {ms} \rangle + a\langle {n(s + 1)} \rangle + b\langle s \rangle , \end{split}$$
$$\begin{split}{d_t}\left\langle {\delta {n^2}} \right\rangle &= 2{a^\prime}\langle {\delta ms\delta n} \rangle - 2a\langle {\delta ns\delta n} \rangle - 2a\left\langle {\delta {n^2}} \right\rangle + 2c\langle {\delta m\delta n} \rangle - 2{d^\prime}\left\langle {\delta {n^2}} \right\rangle\\&\quad+ {a^\prime}\langle {ms} \rangle + a\langle {n(s + 1)} \rangle + c\langle m \rangle + {d^\prime}\langle n\rangle , \end{split}$$
$$\begin{split}{d_t}\left\langle {\delta {m^2}} \right\rangle &= - 2{a^\prime}\langle {\delta ms\delta m} \rangle + 2a\langle {\delta ns\delta m} \rangle + 2a\langle {\delta m\delta n} \rangle - 2c\left\langle {\delta {m^2}} \right\rangle + 2{d^\prime}\langle {\delta m\delta n}\rangle\\&\quad+ {a^\prime}\langle {ms} \rangle + a\langle {n(s + 1)} \rangle + c\langle m \rangle + {d^\prime}\langle n\rangle .\end{split}$$
Notice that the driving terms in Eqs. (77)–(79) are consistent with the shot-noise rule. For reference, the terms $2a\langle {\delta n\delta s} \rangle + a\langle n\rangle$, $- 2a\left\langle {\delta {n^2}} \right\rangle + a\langle n\rangle$, and $2a\langle {\delta m\delta n} \rangle + a\langle n \rangle$ are associated with spontaneous emission.

The second photon and upper-electron moment obeys the equation

$$\begin{split}{d_t}\langle {ns} \rangle& = - {a^\prime}\sum\limits_0 \sum\limits_0 \sum\limits_0 mn{s^2}F(m,n,s) + {a^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 (m + 1)ns(s + 1)F(m + 1,n - 1,s + 1) \\[-2pt] &\quad - a\sum\limits_0 \sum\limits_0 \sum\limits_0 {n^2}s(s + 1)F(m,n,s) + a\sum\limits_1 \sum\limits_0 \sum\limits_1 n(n + 1){s^2}F(m - 1,n + 1,s - 1) \\[-2pt] &\quad - b\sum\limits_0 \sum\limits_0 \sum\limits_0 n{s^2}F(m,n,s) + b\sum\limits_0 \sum\limits_0 \sum\limits_0 ns(s + 1)F(m,n,s + 1) \\[-2pt] &\quad - c\sum\limits_0 \sum\limits_0 \sum\limits_0 mnsF(m,n,s) + c\sum\limits_0 \sum\limits_1 \sum\limits_0 (m + 1)nsF(m + 1,n - 1,s) \\[-2pt] &\quad - {d^\prime}\sum\limits_0 \sum\limits_0 \sum\limits_0 {n^2}sF(m,n,s) + {d^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_0 n(n + 1)sF(m - 1,n + 1,s) \\[-2pt] & = - {a^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_1 mn{s^2}F(m,n,s) + {a^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_1 m(n + 1)s(s - 1)F(m,n,s) \\[-2pt] &\quad - a\sum\limits_0 \sum\limits_1 \sum\limits_0 {n^2}s(s + 1)F(m,n,s) + a\sum\limits_0 \sum\limits_1 \sum\limits_0 n(n - 1)(s + {1)^2}F(m,n,s) \\[-2pt] &\quad - b\sum\limits_0 \sum\limits_0 \sum\limits_1 n{s^2}F(m,n,s) + b\sum\limits_0 \sum\limits_0 \sum\limits_1 ns(s - 1)F(m,n,s) \\[-2pt] &\quad - c\sum\limits_1 \sum\limits_0 \sum\limits_0 mnsF(m,n,s) + c\sum\limits_1 \sum\limits_0 \sum\limits_0 m(n + 1)sF(m,n,s) \\[-2pt] &\quad - {d^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 {n^2}sF(m,n,s) + {d^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 n(n - 1)sF(m,n,s).\end{split}$$
By combining the terms on the right side of Eq. (80), one obtains the second-moment equation
$$\begin{split}{d_t}\langle {ns} \rangle &= {a^\prime}( - \langle {mns}\rangle + \left\langle {m{s^2}} \right\rangle - \langle {ms} \rangle ) + a(\left\langle {{n^2}s} \right\rangle - \left\langle {n{s^2}} \right\rangle + \left\langle {{n^2}} \right\rangle - 2\langle {ns}\rangle -\langle n \rangle ) \\[-2pt] &\quad - b \langle {ns}\rangle + c \langle {ms} \rangle - {d^\prime}\langle {ns} \rangle .\end{split}$$
The second upper- and lowest-level electron moment obeys the equation
$$\begin{split}{d_t}\langle {mn} \rangle &= - {a^\prime}\sum\limits_0 \sum\limits_0 \sum\limits_0 {m^2}nsF(m,n,s) + {a^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 m(m + 1)n(s + 1)F(m + 1,n - 1,s + 1) \\[-3pt] &\quad - a\sum\limits_0 \sum\limits_0 \sum\limits_0 m{n^2}(s + 1)F(m,n,s) + a\sum\limits_1 \sum\limits_0 \sum\limits_1 mn(n + 1)sF(m - 1,n + 1,s - 1) \\[-3pt] &\quad - c\sum\limits_0 \sum\limits_0 \sum\limits_0 {m^2}nF(m,n,s) + c\sum\limits_0 \sum\limits_1 \sum\limits_0 m(m + 1)nF(m + 1,n - 1,s) \\[-3pt] &\quad - {d^\prime}\sum\limits_0 \sum\limits_0 \sum\limits_0 m{n^2}F(m,n,s) + {d^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_0 mn(n + 1)F(m - 1,n + 1,s) \\[-3pt] & = - {a^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_1 {m^2}nsF(m,n,s) + {a^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_1 m(m - 1)(n + 1)sF(m,n,s) \\[-3pt] &\quad - a\sum\limits_0 \sum\limits_1 \sum\limits_0 m{n^2}(s + 1)F(m,n,s) + a\sum\limits_0 \sum\limits_1 \sum\limits_0 (m + 1)n(n - 1)(s + 1)F(m,n,s) \\[-3pt] &\quad - c\sum\limits_1 \sum\limits_0 \sum\limits_0 {m^2}nF(m,n,s) + c\sum\limits_1 \sum\limits_0 \sum\limits_0 m(m - 1)(n + 1)F(m,n,s) \\[-3pt] &\quad - {d^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 m{n^2}F(m,n,s) + {d^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 (m + 1)n(n - 1)F(m,n,s).\end{split}$$
The $b$ terms were omitted from Eq. (82) because they do not affect $\langle {mn} \rangle$ directly. By combining the terms on the right side, one obtains the second-moment equation
$$\begin{split}{d_t}\langle {mn} \rangle& = {a^\prime}(\left\langle {{m^2}s} \right\rangle - \langle {mns} \rangle - \langle {ms} \rangle ) + a( - \langle {mns} \rangle + \left\langle {{n^2}s} \right\rangle - \langle {mn} \rangle + \left\langle {{n^2}} \right\rangle - \langle {ns}\rangle - \langle n \rangle ) \\[-3pt] &\quad + c(\left\langle {{m^2}} \right\rangle - \langle {mn} \rangle - \langle m \rangle ) + {d^\prime}( - \langle {mn} \rangle + \left\langle {{n^2}} \right\rangle - \langle n \rangle ).\end{split}$$
The second photon and lowest-level electron moment obeys the equation
$$\begin{split}{d_t}\langle {ms}\rangle& = - {a^\prime}\sum\limits_0 \sum\limits_0 \sum\limits_0 {m^2}{s^2}F(m,n,s) + {a^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 m(m + 1)s(s + 1)F(m + 1,n - 1,s + 1) \\[-3pt] &\quad - a\sum\limits_0 \sum\limits_0 \sum\limits_0 mns(s + 1)F(m,n,s) + a\sum\limits_1 \sum\limits_0 \sum\limits_1 m(n + 1){s^2}F(m - 1,n + 1,s - 1) \\[-3pt] &\quad - b\sum\limits_0 \sum\limits_0 \sum\limits_0 m{s^2}F(m,n,s) + b\sum\limits_0 \sum\limits_0 \sum\limits_0 ms(s + 1)F(m,n,s + 1) \\[-3pt] &\quad - c\sum\limits_0 \sum\limits_0 \sum\limits_0 {m^2}sF(m,n,s) + c\sum\limits_0 \sum\limits_1 \sum\limits_0 m(m + 1)sF(m + 1,n - 1,s) \\[-3pt] &\quad - {d^\prime}\sum\limits_0 \sum\limits_0 \sum\limits_0 mnsF(m,n,s) + {d^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_0 m(n + 1)sF(m - 1,n + 1,s) \\[-3pt] & = - {a^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_1 {m^2}{s^2}F(m,n,s) + {a^\prime}\sum\limits_1 \sum\limits_0 \sum\limits_1 m(m - 1)s(s - 1)F(m,n,s) \\[-3pt] &\quad - a\sum\limits_0 \sum\limits_1 \sum\limits_0 mns(s + 1)F(m,n,s) + a\sum\limits_0 \sum\limits_1 \sum\limits_0 (m + 1)n{(s + 1)^2}F(m,n,s) \\[-3pt] &\quad - b\sum\limits_0 \sum\limits_0 \sum\limits_1 m{s^2}F(m,n,s) + b\sum\limits_0 \sum\limits_0 \sum\limits_1 ms(s - 1)F(m,n,s) \\[-3pt] &\quad - c\sum\limits_1 \sum\limits_0 \sum\limits_0 {m^2}sF(m,n,s) + c\sum\limits_1 \sum\limits_0 \sum\limits_0 m(m - 1)sF(m,n,s) \\[-3pt] &\quad - {d^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 mnsF(m,n,s) + {d^\prime}\sum\limits_0 \sum\limits_1 \sum\limits_0 (m + 1)nsF(m,n,s).\end{split}$$
By combining the terms on the right side of Eq. (84), one obtains the second-moment equation
$$\begin{split}{d_t}\langle {ms} \rangle& = {a^\prime}( - \left\langle {{m^2}s} \right\rangle - \left\langle {m{s^2}} \right\rangle + \langle {ms} \rangle )\\ &\quad + a(\langle {mns} \rangle + \left\langle {n{s^2}} \right\rangle + \langle {mn} \rangle + 2\langle {ns} \rangle + \langle n \rangle ) \\ &\quad - b\langle {ms} \rangle - c\langle {ms} \rangle + {d^\prime}\langle {ns} \rangle .\end{split}$$
The second mixed moments are also coupled to third moments, which are unknown.

By combining Eqs. (66), (68) and (70) with Eqs. (81), (83) and (85), one obtains the correlation equations

$$\begin{split}{d_t}\langle {\delta n\delta s} \rangle& = {a^\prime}(\langle {\delta ms\delta s} \rangle - \langle {\delta ms\delta n} \rangle )\\ &\quad + a(\langle {\delta ns\delta n}\rangle - \langle {\delta ns\delta s} \rangle ) + a\left\langle {\delta {n^2}} \right\rangle \\ &\quad - (a + b + {d^\prime})\langle {\delta n\delta s} \rangle + c\langle {\delta m\delta s} \rangle\\ &\quad - {a^\prime}\langle {ms} \rangle - a\langle {n(s + 1)} \rangle , \end{split}$$
$$\begin{split}{d_t}\langle {\delta m\delta n} \rangle &= {a^\prime}(\langle {\delta ms\delta m} \rangle - \langle {\delta ms\delta n} \rangle )\\ &\quad + a(\langle {\delta ns\delta n} \rangle - \langle {\delta ns\delta m} \rangle ) \\ &\quad - (a + c + {d^\prime})\langle {\delta m\delta n} \rangle + (a + {d^\prime})\left\langle {\delta {n^2}} \right\rangle \\ &\quad + c\left\langle {\delta {m^2}} \right\rangle - {a^\prime}\langle {ms}\rangle - a\langle {n(s + 1)} \rangle\\ &\quad - c\langle m \rangle - {d^\prime}\langle n \rangle , \end{split}$$
$$\begin{split}{d_t}\langle {\delta m\delta s} \rangle &= - {a^\prime}(\langle {\delta ms\delta m} \rangle + \langle {\delta ms\delta s} \rangle )\\ &\quad + a(\langle {\delta ns\delta m} \rangle + \langle {\delta ns\delta s} \rangle ) + a\langle {\delta m\delta n} \rangle \\&\quad+ a\langle {\delta n\delta s} \rangle - (b + c)\langle {\delta m\delta s } \rangle \\ &\quad+ {d^\prime}\langle {\delta n\delta s} \rangle + {a^\prime}\langle {ms}\rangle + a\langle {n(s + 1)} \rangle .\end{split}$$
Notice that the driving terms in Eqs. (86)–(88) are consistent with the shot-noise rule. In the first and second of these equations, the driving terms are negative, because the deviations involved are anti-correlated. For reference, the terms $a\left\langle {\delta {n^2}} \right\rangle - a\langle {\delta n\delta s} \rangle - a\langle n \rangle$, $- a\langle {\delta m\delta n} \rangle + a\left\langle {\delta {n^2}} \right\rangle - a\langle n \rangle$, and $a\langle {\delta m\delta n} \rangle + a\langle {\delta n\delta s} \rangle + a\langle n \rangle$ are associated with spontaneous emission.

By combining Eqs. (78), (79) and (87), one can show that

$${d_t}\left\langle {{{(\delta m + \delta n)}^2}} \right\rangle = 0.$$
The upper-electron and lowest-electron deviations are completely anti-correlated. If one retains only the absorption and emission terms in the variance and correlation equations, one can show that
$${d_t}\left\langle {{{(\delta n + \delta s)}^2}} \right\rangle = 0 = {d_t}\left\langle {{{(\delta m - \delta s)}^2}} \right\rangle .$$
In the absence of loss and pumping, the photon and upper-electron deviations are completely anti-correlated, whereas the photon and lowest-electron deviations are completely correlated. These correlations were built into Eq. (63), so their appearances serve as algebra checks.

#### B. Closed Moment Equations

Within the Markov framework, the mean equations (66), (68) and (70), the variance equations (77)–(79), and the correlation equations (86)–(88) are exact. However, the Markov moment equations do not form a closed set. Fortunately, for a practical laser the first moments (means) are much larger than the deviations and their moments. By making the approximations $\langle {ms} \rangle \approx \langle m \rangle \langle s \rangle$ and $\langle {ns} \rangle \approx\langle n \rangle \langle s \rangle$ in Eqs. (66), (68) and (70), which amounts to neglecting the correlations $\delta m\delta s$ and $\delta n\delta s$, respectively, one obtains the approximate mean equations

$${d_t}\langle s \rangle \def\LDeqtab{}\approx - {a^\prime}\langle m \rangle \langle s \rangle + a\langle n \rangle (\langle s \rangle + 1) - b\langle s\rangle ,$$
$${d_t}\langle n \rangle \def\LDeqtab{}\approx {a^\prime}\langle m \rangle \langle s \rangle - a\langle n \rangle (\langle s \rangle + 1) + c\langle m \rangle - {d^\prime}\langle n \rangle ,$$
$${d_t}\langle m \rangle \def\LDeqtab{}\approx - {a^\prime}\langle m\rangle \langle s \rangle + a\langle n \rangle (\langle s \rangle + 1) - c\langle m \rangle + {d^\prime}\langle n \rangle .$$
Equations (91)–(93) are equivalent to the rate equations (28)–(30), which is a reasonable result. The spontaneous emission terms were retained only to allow the signal number to grow from zero. For all other purposes, they can be neglected, because $1 \lt |\delta s| \ll \langle s \rangle$. Notice that the approximate mean equations are closed. Because Eqs. (92) and (93) conserve the total electron number $\langle m \rangle + \langle n \rangle$, one can omit the latter equation.

At this stage, one could use the approximations $\delta ms \approx \langle m \rangle \delta s + \langle s \rangle \delta m$ and $\delta ns \approx \langle n \rangle \delta s + \langle s \rangle \delta n$ to simplify all six variance and correlation equations. However, Eq. (89) shows that the electron-number fluctuations are anti-correlated. By making the substitution $\delta m = - \delta n$ in Eqs. (87) and (88), one obtains the negatives of Eqs. (78) and (86), respectively. Hence, it is sufficient to retain and simplify Eqs. (77), (78) and (86). By making the aforementioned substitution and omitting the spontaneous-emission terms, which are smaller than the other terms by factors of $\langle m \rangle$, $\langle n \rangle$ and $\langle s \rangle$, one obtains the intermediate equations

$$\begin{split}{d_t}\left\langle {\delta {s^2}} \right\rangle &= - 2{a^\prime}\langle {\delta ms\delta s} \rangle + 2a\langle {\delta ns\delta s} \rangle - 2b\left\langle {\delta {s^2}} \right\rangle \\ &\quad + {a^\prime}\langle {ms} \rangle + a\langle {ns} \rangle + b\langle s \rangle , \end{split}$$
$$\begin{split} {d_t}\langle {\delta n\delta s} \rangle &= - {a^\prime}\langle {\delta ms\delta n} \rangle + {a^\prime}\langle {\delta ms\delta s} \rangle + a\langle {\delta ns\delta n} \rangle \\&\quad - a\langle {\delta ns\delta s} \rangle - (b + c + {d^\prime})\langle {\delta n\delta s} \rangle\\&\quad - {a^\prime}\langle {ms} \rangle - a\langle {ns} \rangle ,\end{split}$$
$$\begin{split} {d_t}\left\langle {\delta {n^2}} \right\rangle &= 2{a^\prime}\langle {\delta ms\delta n} \rangle - 2a\langle {\delta ns\delta n} \rangle - 2(c + {d^\prime})\left\langle {\delta {n^2}} \right\rangle \\&\quad+ {a^\prime}\langle {ms} \rangle + a\langle {ns} \rangle + c\langle m \rangle + {d^\prime}\langle n \rangle .\end{split}$$
These equations still satisfy the first of Eqs. (90). By making the aforementioned approximations for $\langle {ms} \rangle$, $\langle {ns} \rangle$, $\delta ms$ and $\delta ns$, one obtains the approximate variance and correlation equations
$$\begin{split}{d_t}\left\langle {\delta {s^2}} \right\rangle &= 2({a^\prime}\langle s \rangle + a\langle s \rangle )\langle {\delta n\delta s} \rangle\\ &\quad + 2(a\langle n \rangle - {a^\prime}\langle m \rangle - b)\left\langle {\delta {s^2}} \right\rangle \\ &\quad + {a^\prime}\langle m \rangle \langle s \rangle + a\langle n \rangle \langle s \rangle + b\langle s \rangle , \end{split}$$
$$\begin{split}{d_t}\langle {\delta n\delta s} \rangle& = ({a^\prime}\langle s \rangle + a\langle s \rangle )\left\langle {\delta {n^2}} \right\rangle\\ &\quad + (a\langle n \rangle - {a^\prime}\langle m \rangle - b)\langle {\delta n\delta s} \rangle \\ &\quad - ({a^\prime}\langle s \rangle + a\langle s \rangle + c + {d^\prime})\langle {\delta n\delta s} \rangle\\ &\quad - (a\langle n \rangle - {a^\prime}\langle m \rangle )\left\langle {\delta {s^2}} \right\rangle\\ &\quad - {a^\prime}\langle m \rangle \langle s \rangle - a\langle n \rangle \langle s \rangle , \end{split}$$
$$\begin{split}{d_t}\left\langle {\delta {n^2}} \right\rangle &= - 2({a^\prime}\langle s \rangle + a\langle s \rangle + c + {d^\prime})\left\langle {\delta {n^2}} \right\rangle\\ &\quad - 2(a \langle n \rangle - {a^\prime}\langle m \rangle )\langle {\delta n\delta s} \rangle\\&\quad+ {a^\prime}\langle m \rangle \langle s \rangle + a \langle n \rangle \langle s \rangle + c \langle m \rangle + {d^\prime}\langle n \rangle .\end{split}$$
Equations (91), (92) and (97)–(99) form a complete set. The driving terms in Eqs. (97)–(99) justify the shot-noise rule: The driving terms in the variance equations are the sums of the moduli of the associated terms in the mean equations, whereas those in the correlation equation have the same magnitudes but opposite signs, because the changes in the signal-photon and upper-electron numbers are anti-correlated. Furthermore, the coefficients of the variances and correlation on the right sides of Eqs. (97)–(99) are consistent with the coefficients in Eqs. (60)–(62), which were based on the linearized Langevin equations, so the closure approximations are sound.

As a bonus, the shot-noise rule derived in this section is consistent with the laws of quantum optics. The Heisenberg-picture calculations  are based on quantum Langevin equations for the number operators. The operator source terms in these equations have properties that are consistent with the shot-noise rule. The Schrödinger-picture calculations  are based on an equation for the signal density operator in the number-state representation. The diagonal entries of this equation are consistent with the Markov equation [Eq. (63)].

#### C. Source Strengths for Different Lasers

It is instructive to consider the source strengths associated with different lasers. In the notation of Table 3, for a four-level laser the compound source strengths ${R_{\textit ss}}$, ${R_{\textit sn}}$, and ${R_{\textit nn}}$ are $a{N_0}{S_0} + b{S_0}$, $- a{N_0}{S_0}$, and $a{N_0}{S_0} + c{M_0} + d{N_0}$, respectively, where the equilibrium conditions are $a{N_0} = b$ and $b{S_0} = c{M_0} - d{N_0}$. In terms of the photon flux $b{S_0}$, the source strengths are $2b{S_0}$, $- b{S_0}$, and $2b{S_0} + 2d{N_0}$. In the well-above-threshold regime ($b{S_0} \gg d{N_0}$), these strengths depend on only the photon flux.

For a type A laser, the compound source strengths are $a{N_0}{S_0} + b{S_0}$, $- a{N_0}{S_0}$, and $a{N_0}{S_0} + c{M_0} + (c + d){N_0}$, where the operating conditions are $a{N_0} = b$ and $b{S_0} = c{M_0} - (c + d){N_0}$. In terms of the photon flux, the source strengths are $2b{S_0}$, $- b{S_0}$, and $2b{S_0} + 2(c + d){N_0}$. Typically, the electron inversion (${N_0}$) is much smaller than the total electron number (${N_t} = {M_0} + {N_0}$), so $c{N_0} \ll c{M_0} \approx b{S_0}$. Type A lasers are only slightly noisier than four-level lasers (and the extra noise is restricted to the electron-number fluctuations).

### Tables (3) Table 1. Rate Coefficients for Three Types of Lasera Table 2. Coupling Coefficients for Three Types of Lasera Table 3. Source Strengths for Three Types of Lasera

### Equations (192)

$d t N 3 = a 30 P N 0 − a 30 ( P + 1 ) N 3 − a 32 N 3 ,$
$d t N 2 = a 32 N 3 + a 21 S N 1 − a 21 ( S + 1 ) N 2 ,$
$d t N 1 = − a 21 S N 1 + a 21 ( S + 1 ) N 2 − a 10 N 1 ,$
$d t N 0 = − a 30 P N 0 + a 30 ( P + 1 ) N 3 + a 10 N 1 ,$
$d t ( N 0 + N 1 + N 2 + N 3 ) = 0.$
$N 3 = a 30 P N 0 / [ a 32 + a 30 ( P + 1 ) ] ≈ a 30 P N 0 / a 32 ,$
$N 1 = a 21 ( S + 1 ) N 2 / ( a 10 + a 21 S ) ≈ a 21 ( S + 1 ) N 2 / a 10 ,$
$d t N 2 ≈ a 30 P N 0 − a 21 ( S + 1 ) N 2 ,$
$d t N 0 ≈ − a 30 P N 0 + a 21 ( S + 1 ) N 2 .$
$d t ( N 0 + N 2 ) ≈ 0.$
$d t S = a 21 N 2 ( S + 1 ) − a 21 N 1 S − b S ≈ a 21 N 2 ( S + 1 ) − b S ,$
$d t N 2 = a 20 P N 0 − a 20 ( P + 1 ) N 2 + a 21 S N 1 − a 21 ( S + 1 ) N 2 − a n l N 2 ,$
$d t N 1 = − a 21 S N 1 + a 21 ( S + 1 ) N 2 + a n l N 2 − a 10 N 1 ,$
$d t N 0 = − a 20 P N 0 + a 20 ( P + 1 ) N 2 + a 10 N 1 .$
$d t ( N 0 + N 1 + N 2 ) = 0.$
$d t N 2 ≈ a 20 P N 0 − a 20 ( P + 1 ) N 2 − a 21 ( S + 1 ) N 2 − a n l N 2 ,$
$d t N 0 ≈ − a 20 P N 0 + a 20 ( P + 1 ) N 2 + a 21 ( S + 1 ) N 2 + a n l N 2 .$
$d t ( N 0 + N 2 ) = 0.$
$d t S = a 21 N 2 ( S + 1 ) − a 21 N 1 S − b S , ≈ a 21 N 2 ( S + 1 ) − b S .$
$d t N 3 = a 31 P N 1 − a 31 ( P + 1 ) N 3 − a 32 N 3 ,$
$d t N 2 = a 32 N 3 + a 21 S N 1 − a 21 ( S + 1 ) N 2 − a n l N 2 ,$
$d t N 1 = − a 31 P N 1 + a 31 ( P + 1 ) N 3 − a 21 S N 1 + a 21 ( S + 1 ) N 2 + a n l N 2 .$
$d t ( N 1 + N 2 + N 3 ) = 0.$
$d t N 2 ≈ a 31 P N 1 + a 21 S N 1 − a 21 ( S + 1 ) N 2 − a n l N 2 ,$
$d t N 1 ≈ − a 31 P N 1 − a 21 S N 1 + a 21 ( S + 1 ) N 2 + a n l N 2 .$
$d t ( N 1 + N 2 ) ≈ 0.$
$d t S = a 21 N 2 ( S + 1 ) − a 21 N 1 S − b S .$
$d t S = − a ′ M S + a N ( S + 1 ) − b S ,$
$d t N = a ′ M S − a N ( S + 1 ) + c M − c ′ N − d N ,$
$d t M = − a ′ M S + a N ( S + 1 ) − c M + c ′ N + d N ,$
$d t S = − a ′ M S + a N ( S + 1 ) − b S − R a ′ + R a − R b ,$
$d t N = a ′ M S − a N ( S + 1 ) + c M − c ′ N − d N + R a ′ − R a + R c − R c ′ − R d ,$
$d t M = − a ′ M S + a N ( S + 1 ) − c M + c ′ N + d N − R a ′ + R a − R c + R c ′ + R d .$
$⟨ r j ( t ) ⟩ = 0 , ⟨ r j ( t ) r k ( t ′ ) ⟩ = δ j k δ ( t − t ′ ) ,$
$S a ′ = a ′ ⟨ M S ⟩ , S a = a ⟨ N ( S + 1 ) ⟩ , S b = b ⟨ S ⟩ , S c = c ⟨ M ⟩ , S c ′ = c ′ ⟨ N ⟩ , S d = d ⟨ N ⟩$
$a N 0 − a ′ M 0 = b ,$
$( a N 0 − a ′ M 0 ) S 0 = b S 0 = c M 0 − d ′ N 0 .$
$d t S 1 = ( a N 0 − a ′ M 0 − b ) S 1 + ( a S 0 ) N 1 − ( a ′ S 0 ) M 1 − R a ′ + R a − R b ,$
$d t N 1 = ( a ′ M 0 − a N 0 ) S 1 − ( a S 0 + d ′ ) N 1 + ( a ′ S 0 + c ) M 1 + R a ′ − R a + R c − R d ′ ,$
$d t M 1 = ( a N 0 − a ′ M 0 ) S 1 + ( a S 0 + d ′ ) N 1 − ( a ′ S 0 + c ) M 1 − R a ′ + R a − R c + R d ′ ,$
$d t ( M 1 + N 1 ) = 0 ,$
$d t S 1 = ( a N 0 − a ′ M 0 − b ) S 1 + ( a S 0 + a ′ S 0 ) N 1 − R a ′ + R a − R b ,$
$d t N 1 = ( a ′ M 0 − a N 0 ) S 1 − ( a S 0 + a ′ S 0 + c + d ′ ) N 1 + R a ′ − R a + R c − R d ′ .$
$d t S 1 = γ s s S 1 + γ s n N 1 + R s ,$
$d t N 1 = − γ n s S 1 − γ n n N 1 + R n .$
$⟨ R j ( t ) ⟩ = 0 , ⟨ R j ( t ) R k ( t ′ ) ⟩ = R j k δ ( t − t ′ ) ,$
$R s s = S a ′ + S a + S b , R n s = − S a ′ − S a , R n n = S a ′ + S a + S c + S d ′ .$
$γ 2 + 2 ν 0 γ + ω 0 2 = 0 ,$
$ω 0 2 = γ n s γ s n − γ n n γ s s , ν 0 = ( γ n n − γ s s ) / 2 ,$
$γ = ( γ s s − γ n n ) / 2 ± i [ γ n s γ s n − ( γ n n + γ s s ) 2 / 4 ] 1 / 2 .$
$S 1 ( t ) = G s s ( t ) S 1 ( 0 ) + G s n ( t ) N 1 ( 0 ) ,$
$N 1 ( t ) = G n s ( t ) S 1 ( 0 ) + G n n ( t ) N 1 ( 0 ) ,$
$G s s ( t ) = [ cos ⁡ ( ω r t ) + γ a sin ⁡ ( ω r t ) / ω r ] exp ⁡ ( − ν 0 t ) ,$
$G s n ( t ) = [ γ s n sin ⁡ ( ω r t ) / ω r ] exp ⁡ ( − ν 0 t ) ,$
$G n s ( t ) = − [ γ n s sin ⁡ ( ω r t ) / ω r ] exp ⁡ ( − ν 0 t ) ,$
$G n n ( t ) = [ cos ⁡ ( ω r t ) − γ a sin ⁡ ( ω r t ) / ω r ] exp ⁡ ( − ν 0 t ) ,$
$⟨ [ ∫ 0 δ t r j ( t ) d t ] 2 ⟩ = ∫ 0 δ t ∫ 0 δ t ⟨ r j ( t ) r j ( t ′ ) ⟩ d t d t ′ = δ t .$
$S 1 ( δ t ) ≈ ( 1 + γ s s δ t ) S 1 + ( γ s n δ t ) N 1 + ∫ 0 δ t R s ( t ) d t ,$
$N 1 ( δ t ) ≈ − ( γ n s δ t ) S 1 + ( 1 − γ n n δ t ) N 1 + ∫ 0 δ t R n ( t ) d t .$
$d t ⟨ S 1 2 ⟩ = 2 γ s s ⟨ S 1 2 ⟩ + 2 γ s n ⟨ N 1 S 1 ⟩ + R s s ,$
$d t ⟨ N 1 S 1 ⟩ = − γ n s ⟨ S 1 2 ⟩ + ( γ s s − γ n n ) ⟨ N 1 S 1 ⟩ + γ s n ⟨ N 1 2 ⟩ + R n s ,$
$d t ⟨ N 1 2 ⟩ = − 2 γ n s ⟨ N 1 S 1 ⟩ − 2 γ n n ⟨ N 1 2 ⟩ + R n n .$
$d t F ( m , n , s ) = − a ′ m s F ( m , n , s ) + a ′ ( m + 1 ) ( s + 1 ) F ( m + 1 , n − 1 , s + 1 ) − a n ( s + 1 ) F ( m , n , s ) + a ( n + 1 ) s F ( m − 1 , n + 1 , s − 1 ) − b s F ( m , n , s ) + b ( s + 1 ) F ( m , n , s + 1 ) − c m F ( m , n , s ) + c ( m + 1 ) F ( m + 1 , n − 1 , s ) − c ′ n F ( m , n , s ) + c ′ ( n + 1 ) F ( m − 1 , n + 1 , s ) − d n F ( m , n , s ) + d ( n + 1 ) F ( m − 1 , n + 1 , s ) ,$
$d t T = − a ′ ∑ 0 ∑ 0 ∑ 0 m s F ( m , n , s ) + a ′ ∑ 0 ∑ 1 ∑ 0 ( m + 1 ) ( s + 1 ) F ( m + 1 , n − 1 , s + 1 ) − a ∑ 0 ∑ 0 ∑ 0 n ( s + 1 ) F ( m , n , s ) + a ∑ 1 ∑ 0 ∑ 1 ( n + 1 ) s F ( m − 1 , n + 1 , s − 1 ) − b ∑ 0 ∑ 0 ∑ 0 s F ( m , n , s ) + b ∑ 0 ∑ 0 ∑ 0 ( s + 1 ) F ( m , n , s + 1 ) − c ∑ 0 ∑ 0 ∑ 0 m F ( m , n , s ) + c ∑ 0 ∑ 1 ∑ 0 ( m + 1 ) F ( m + 1 , n − 1 , s ) − d ′ ∑ 0 ∑ 0 ∑ 0 n F ( m , n , s ) + d ′ ∑ 1 ∑ 0 ∑ 0 ( n + 1 ) F ( m − 1 , n + 1 , s ) = − a ′ ∑ 1 ∑ 0 ∑ 1 m s F ( m , n , s ) + a ′ ∑ 1 ∑ 0 ∑ 1 m s F ( m , n , s ) − a ∑ 0 ∑ 1 ∑ 0 n ( s + 1 ) F ( m , n , s ) + a ∑ 0 ∑ 1 ∑ 0 n ( s + 1 ) F ( m , n , s ) − b ∑ 0 ∑ 0 ∑ 1 s F ( m , n , s ) + b ∑ 0 ∑ 0 ∑ 1 s F ( m , n , s ) − c ∑ 1 ∑ 0 ∑ 0 m F ( m , n , s ) + c ∑ 1 ∑ 0 ∑ 0 m F ( m , n , s ) − d ′ ∑ 0 ∑ 1 ∑ 0 n F ( m , n , s ) + d ′ ∑ 0 ∑ 1 ∑ 0 n F ( m , n , s ) = 0.$
$d t ⟨ s ⟩ = − a ′ ∑ 0 ∑ 0 ∑ 0 m s 2 F ( m , n , s ) + a ′ ∑ 0 ∑ 1 ∑ 0 ( m + 1 ) s ( s + 1 ) F ( m + 1 , n − 1 , s + 1 ) − a ∑ 0 ∑ 0 ∑ 0 n s ( s + 1 ) F ( m , n , s ) + a ∑ 1 ∑ 0 ∑ 1 ( n + 1 ) s 2 F ( m − 1 , n + 1 , s − 1 ) − b ∑ 0 ∑ 0 ∑ 0 s 2 F ( m , n , s ) + b ∑ 0 ∑ 0 ∑ 0 s ( s + 1 ) F ( m , n , s + 1 ) = − a ′ ∑ 1 ∑ 0 ∑ 1 m s 2 F ( m , n , s ) + a ′ ∑ 1 ∑ 0 ∑ 1 m s ( s − 1 ) F ( m , n , s ) − a ∑ 0 ∑ 1 ∑ 0 n s ( s + 1 ) F ( m , n , s ) + a ∑ 0 ∑ 1 ∑ 0 n ( s + 1 ) 2 F ( m , n , s ) − b ∑ 0 ∑ 0 ∑ 1 s 2 F ( m , n , s ) + b ∑ 0 ∑ 0 ∑ 1 s ( s − 1 ) F ( m , n , s ) .$
$d t ⟨ s ⟩ = − a ′ ⟨ m s ⟩ + a ⟨ n ( s + 1 ) ⟩ − b ⟨ s ⟩ ,$
$d t ⟨ n ⟩ = − a ′ ∑ 0 ∑ 0 ∑ 0 m n s F ( m , n , s ) + a ′ ∑ 0 ∑ 1 ∑ 0 ( m + 1 ) n ( s + 1 ) F ( m + 1 , n − 1 , s + 1 ) − a ∑ 0 ∑ 0 ∑ 0 n 2 ( s + 1 ) F ( m , n , s ) + a ∑ 1 ∑ 0 ∑ 1 n ( n + 1 ) s F ( m − 1 , n + 1 , s − 1 ) − c ∑ 0 ∑ 0 ∑ 0 m n F ( m , n , s ) + c ∑ 0 ∑ 1 ∑ 0 ( m + 1 ) n F ( m + 1 , n − 1 , s ) − d ′ ∑ 0 ∑ 0 ∑ 0 n 2 F ( m , n , s ) + d ′ ∑ 1 ∑ 0 ∑ 0 n ( n + 1 ) F ( m − 1 , n + 1 , s ) = − a ′ ∑ 1 ∑ 0 ∑ 1 m n s F ( m , n , s ) + a ′ ∑ 1 ∑ 0 ∑ 1 m ( n + 1 ) s F ( m , n , s ) − a ∑ 0 ∑ 1 ∑ 0 n 2 ( s + 1 ) F ( m , n , s ) + a ∑ 0 ∑ 1 ∑ 0 n ( n − 1 ) ( s + 1 ) F ( m , n , s ) − c ∑ 1 ∑ 0 ∑ 0 m n F ( m , n , s ) + c ∑ 1 ∑ 0 ∑ 0 m ( n + 1 ) F ( m , n , s ) − d ′ ∑ 0 ∑ 1 ∑ 0 n 2 F ( m , n , s ) + d ′ ∑ 0 ∑ 1 ∑ 0 n ( n − 1 ) F ( m , n , s ) .$
$d t ⟨ n ⟩ = a ′ ⟨ m s ⟩ − a ⟨ n ( s + 1 ) ⟩ + c ⟨ m ⟩ − d ′ ⟨ n ⟩ ,$
$d t ⟨ m ⟩ = − a ′ ∑ 0 ∑ 0 ∑ 0 m 2 s F ( m , n , s ) + a ′ ∑ 0 ∑ 1 ∑ 0 m ( m + 1 ) ( s + 1 ) F ( m + 1 , n − 1 , s + 1 ) − a ∑ 0 ∑ 0 ∑ 0 m n ( s + 1 ) F ( m , n , s ) + a ∑ 1 ∑ 0 ∑ 1 m ( n + 1 ) s F ( m − 1 , n + 1 , s − 1 ) − c ∑ 0 ∑ 0 ∑ 0 m 2 F ( m , n , s ) + c ∑ 0 ∑ 1 ∑ 0 m ( m + 1 ) F ( m + 1 , n − 1 , s ) − d ′ ∑ 0 ∑ 0 ∑ 0 m n F ( m , n , s ) + d ′ ∑ 1 ∑ 0 ∑ 0 m ( n + 1 ) F ( m − 1 , n + 1 , s ) = − a ′ ∑ 1 ∑ 0 ∑ 1 m 2 s F ( m , n , s ) + a ′ ∑ 1 ∑ 0 ∑ 1 m ( m − 1 ) s F ( m , n , s ) − a ∑ 0 ∑ 1 ∑ 0 m n ( s + 1 ) F ( m , n , s ) + a ∑ 0 ∑ 1 ∑ 0 ( m + 1 ) n ( s + 1 ) F ( m , n , s ) − c ∑ 1 ∑ 0 ∑ 0 m 2 F ( m , n , s ) + c ∑ 1 ∑ 0 ∑ 0 m ( m − 1 ) F ( m , n , s ) − d ′ ∑ 0 ∑ 1 ∑ 0 m n F ( m , n , s ) + d ′ ∑ 0 ∑ 1 ∑ 0 ( m + 1 ) n F ( m , n , s ) .$
$d t ⟨ m ⟩ = − a ′ ⟨ m s ⟩ + a ⟨ n ( s + 1 ) ⟩ − c ⟨ m ⟩ + d ′ ⟨ n ⟩ .$
$d t ⟨ s 2 ⟩ = − a ′ ∑ 0 ∑ 0 ∑ 0 m s 3 F ( m , n , s ) + a ′ ∑ 0 ∑ 1 ∑ 0 ( m + 1 ) s 2 ( s + 1 ) F ( m + 1 , n − 1 , s + 1 ) − a ∑ 0 ∑ 0 ∑ 0 n s 2 ( s + 1 ) F ( m , n , s ) + a ∑ 1 ∑ 0 ∑ 1 ( n + 1 ) s 3 F ( m − 1 , n + 1 , s − 1 ) − b ∑ 0 ∑ 0 ∑ 0 s 3 F ( m , n , s ) + b ∑ 0 ∑ 0 ∑ 0 s 2 ( s + 1 ) F ( m , n , s + 1 ) = − a ′ ∑ 1 ∑ 0 ∑ 1 m s 3 F ( m , n , s ) + a ′ ∑ 1 ∑ 0 ∑ 1 m s ( s − 1 ) 2 F ( m , n , s ) − a ∑ 0 ∑ 1 ∑ 0 n s 2 ( s + 1 ) F ( m , n , s ) + a ∑ 0 ∑ 1 ∑ 0 n ( s + 1 ) 3 F ( m , n , s ) − b ∑ 0 ∑ 0 ∑ 1 s 3 F ( m , n , s ) + b ∑ 0 ∑ 0 ∑ 1 s ( s − 1 ) 2 F ( m , n , s ) .$
$d t ⟨ s 2 ⟩ = a ′ ( − 2 ⟨ m s 2 ⟩ + ⟨ m s ⟩ ) + a ( 2 ⟨ n s 2 ⟩ + 3 ⟨ n s ⟩ + ⟨ n ⟩ ) + b ( − 2 ⟨ s 2 ⟩ + ⟨ s ⟩ ) .$
$d t ⟨ n 2 ⟩ = − a ′ ∑ 0 ∑ 0 ∑ 0 m n 2 s F ( m , n , s ) + a ′ ∑ 0 ∑ 1 ∑ 0 ( m + 1 ) n 2 ( s + 1 ) F ( m + 1 , n − 1 , s + 1 ) − a ∑ 0 ∑ 0 ∑ 0 n 3 ( s + 1 ) F ( m , n , s ) + a ∑ 1 ∑ 0 ∑ 1 n 2 ( n + 1 ) s F ( m − 1 , n + 1 , s − 1 ) − c ∑ 0 ∑ 0 ∑ 0 m n 2 F ( m , n , s ) + c ∑ 0 ∑ 1 ∑ 0 ( m + 1 ) n 2 F ( m + 1 , n − 1 , s ) − d ′ ∑ 0 ∑ 0 ∑ 0 n 3 F ( m , n , s ) + d ′ ∑ 1 ∑ 0 ∑ 0 n 2 ( n + 1 ) F ( m − 1 , n + 1 , s ) = − a ′ ∑ 1 ∑ 0 ∑ 1 m n 2 s F ( m , n , s ) + a ′ ∑ 1 ∑ 0 ∑ 1 m ( n + 1 ) 2 s F ( m , n , s ) − a ∑ 0 ∑ 1 ∑ 0 n 3 ( s + 1 ) F ( m , n , s ) + a ∑ 0 ∑ 1 ∑ 0 n ( n − 1 ) 2 ( s + 1 ) F ( m , n , s ) − c ∑ 1 ∑ 0 ∑ 0 m n 2 F ( m , n , s ) + c ∑ 1 ∑ 0 ∑ 0 m ( n + 1 ) 2 F ( m , n , s ) − d ′ ∑ 0 ∑ 1 ∑ 0 n 3 F ( m , n , s ) + d ′ ∑ 0 ∑ 1 ∑ 0 n ( n − 1 ) 2 F ( m , n , s ) .$
$d t ⟨ n 2 ⟩ = a ′ ( 2 ⟨ m n s ⟩ + ⟨ m s ⟩ ) + a ( − 2 ⟨ n 2 s ⟩ − 2 ⟨ n 2 ⟩ + ⟨ n s ⟩ + ⟨ n ⟩ ) + c ( 2 ⟨ m n ⟩ + ⟨ m ⟩ ) + d ′ ( − 2 ⟨ n 2 ⟩ + ⟨ n ⟩ ) .$
$d t ⟨ m 2 ⟩ = − a ′ ∑ 0 ∑ 0 ∑ 0 m 3 s F ( m , n , s ) + a ′ ∑ 0 ∑ 1 ∑ 0 m 2 ( m + 1 ) ( s + 1 ) F ( m + 1 , n − 1 , s + 1 ) − a ∑ 0 ∑ 0 ∑ 0 m 2 n ( s + 1 ) F ( m , n , s ) + a ∑ 1 ∑ 0 ∑ 1 m 2 ( n + 1 ) s F ( m − 1 , n + 1 , s − 1 ) − c ∑ 0 ∑ 0 ∑ 0 m 3 F ( m , n , s ) + c ∑ 0 ∑ 1 ∑ 0 m 2 ( m + 1 ) F ( m + 1 , n − 1 , s ) − d ′ ∑ 0 ∑ 0 ∑ 0 m 2 n F ( m , n , s ) + d ′ ∑ 1 ∑ 0 ∑ 0 m 2 ( n + 1 ) F ( m − 1 , n + 1 , s ) = − a ′ ∑ 1 ∑ 0 ∑ 1 m 3 s F ( m , n , s ) + a ′ ∑ 1 ∑ 0 ∑ 1 m ( m − 1 ) 2 s F ( m , n , s ) − a ∑ 0 ∑ 1 ∑ 0 m 2 n ( s + 1 ) F ( m , n , s ) + a ∑ 0 ∑ 1 ∑ 0 ( m + 1 ) 2 n ( s + 1 ) F ( m , n , s ) − c ∑ 1 ∑ 0 ∑ 0 m 3 F ( m , n , s ) + c ∑ 1 ∑ 0 ∑ 0 m ( m − 1 ) 2 F ( m , n , s ) − d ′ ∑ 0 ∑ 1 ∑ 0 m 2 n F ( m , n , s ) + d ′ ∑ 0 ∑ 1 ∑ 0 ( m + 1 ) 2 n F ( m , n , s ) .$
$d t ⟨ m 2 ⟩ = a ′ ( − 2 ⟨ m 2 s ⟩ + ⟨ m s ⟩ ) + a ( 2 ⟨ m n s ⟩ + 2 ⟨ m n ⟩ + ⟨ n s ⟩ + ⟨ n ⟩ ) + c ( − 2 ⟨ m 2 ⟩ + ⟨ m ⟩ ) + d ′ ( 2 ⟨ m n ⟩ + ⟨ n ⟩ ) .$
$d t ⟨ δ s 2 ⟩ = − 2 a ′ ⟨ δ m s δ s ⟩ + 2 a ⟨ δ n s δ s ⟩ + 2 a ⟨ δ n δ s ⟩ − 2 b ⟨ δ s 2 ⟩ + a ′ ⟨ m s ⟩ + a ⟨ n ( s + 1 ) ⟩ + b ⟨ s ⟩ ,$
$d t ⟨ δ n 2 ⟩ = 2 a ′ ⟨ δ m s δ n ⟩ − 2 a ⟨ δ n s δ n ⟩ − 2 a ⟨ δ n 2 ⟩ + 2 c ⟨ δ m δ n ⟩ − 2 d ′ ⟨ δ n 2 ⟩ + a ′ ⟨ m s ⟩ + a ⟨ n ( s + 1 ) ⟩ + c ⟨ m ⟩ + d ′ ⟨ n ⟩ ,$
$d t ⟨ δ m 2 ⟩ = − 2 a ′ ⟨ δ m s δ m ⟩ + 2 a ⟨ δ n s δ m ⟩ + 2 a ⟨ δ m δ n ⟩ − 2 c ⟨ δ m 2 ⟩ + 2 d ′ ⟨ δ m δ n ⟩ + a ′ ⟨ m s ⟩ + a ⟨ n ( s + 1 ) ⟩ + c ⟨ m ⟩ + d ′ ⟨ n ⟩ .$
$d t ⟨ n s ⟩ = − a ′ ∑ 0 ∑ 0 ∑ 0 m n s 2 F ( m , n , s ) + a ′ ∑ 0 ∑ 1 ∑ 0 ( m + 1 ) n s ( s + 1 ) F ( m + 1 , n − 1 , s + 1 ) − a ∑ 0 ∑ 0 ∑ 0 n 2 s ( s + 1 ) F ( m , n , s ) + a ∑ 1 ∑ 0 ∑ 1 n ( n + 1 ) s 2 F ( m − 1 , n + 1 , s − 1 ) − b ∑ 0 ∑ 0 ∑ 0 n s 2 F ( m , n , s ) + b ∑ 0 ∑ 0 ∑ 0 n s ( s + 1 ) F ( m , n , s + 1 ) − c ∑ 0 ∑ 0 ∑ 0 m n s F ( m , n , s ) + c ∑ 0 ∑ 1 ∑ 0 ( m + 1 ) n s F ( m + 1 , n − 1 , s ) − d ′ ∑ 0 ∑ 0 ∑ 0 n 2 s F ( m , n , s ) + d ′ ∑ 1 ∑ 0 ∑ 0 n ( n + 1 ) s F ( m − 1 , n + 1 , s ) = − a ′ ∑ 1 ∑ 0 ∑ 1 m n s 2 F ( m , n , s ) + a ′ ∑ 1 ∑ 0 ∑ 1 m ( n + 1 ) s ( s − 1 ) F ( m , n , s ) − a ∑ 0 ∑ 1 ∑ 0 n 2 s ( s + 1 ) F ( m , n , s ) + a ∑ 0 ∑ 1 ∑ 0 n ( n − 1 ) ( s + 1 ) 2 F ( m , n , s ) − b ∑ 0 ∑ 0 ∑ 1 n s 2 F ( m , n , s ) + b ∑ 0 ∑ 0 ∑ 1 n s ( s − 1 ) F ( m , n , s ) − c ∑ 1 ∑ 0 ∑ 0 m n s F ( m , n , s ) + c ∑ 1 ∑ 0 ∑ 0 m ( n + 1 ) s F ( m , n , s ) − d ′ ∑ 0 ∑ 1 ∑ 0 n 2 s F ( m , n , s ) + d ′ ∑ 0 ∑ 1 ∑ 0 n ( n − 1 ) s F ( m , n , s ) .$
$d t ⟨ n s ⟩ = a ′ ( − ⟨ m n s ⟩ + ⟨ m s 2 ⟩ − ⟨ m s ⟩ ) + a ( ⟨ n 2 s ⟩ − ⟨ n s 2 ⟩ + ⟨ n 2 ⟩ − 2 ⟨ n s ⟩ − ⟨ n ⟩ ) − b ⟨ n s ⟩ + c ⟨ m s ⟩ − d ′ ⟨ n s ⟩ .$
$d t ⟨ m n ⟩ = − a ′ ∑ 0 ∑ 0 ∑ 0 m 2 n s F ( m , n , s ) + a ′ ∑ 0 ∑ 1 ∑ 0 m ( m + 1 ) n ( s + 1 ) F ( m + 1 , n − 1 , s + 1 ) − a ∑ 0 ∑ 0 ∑ 0 m n 2 ( s + 1 ) F ( m , n , s ) + a ∑ 1 ∑ 0 ∑ 1 m n ( n + 1 ) s F ( m − 1 , n + 1 , s − 1 ) − c ∑ 0 ∑ 0 ∑ 0 m 2 n F ( m , n , s ) + c ∑ 0 ∑ 1 ∑ 0 m ( m + 1 ) n F ( m + 1 , n − 1 , s ) − d ′ ∑ 0 ∑ 0 ∑ 0 m n 2 F ( m , n , s ) + d ′ ∑ 1 ∑ 0 ∑ 0 m n ( n + 1 ) F ( m − 1 , n + 1 , s ) = − a ′ ∑ 1 ∑ 0 ∑ 1 m 2 n s F ( m , n , s ) + a ′ ∑ 1 ∑ 0 ∑ 1 m ( m − 1 ) ( n + 1 ) s F ( m , n , s ) − a ∑ 0 ∑ 1 ∑ 0 m n 2 ( s + 1 ) F ( m , n , s ) + a ∑ 0 ∑ 1 ∑ 0 ( m + 1 ) n ( n − 1 ) ( s + 1 ) F ( m , n , s ) − c ∑ 1 ∑ 0 ∑ 0 m 2 n F ( m , n , s ) + c ∑ 1 ∑ 0 ∑ 0 m ( m − 1 ) ( n + 1 ) F ( m , n , s ) − d ′ ∑ 0 ∑ 1 ∑ 0 m n 2 F ( m , n , s ) + d ′ ∑ 0 ∑ 1 ∑ 0 ( m + 1 ) n ( n − 1 ) F ( m , n , s ) .$
$d t ⟨ m n ⟩ = a ′ ( ⟨ m 2 s ⟩ − ⟨ m n s ⟩ − ⟨ m s ⟩ ) + a ( − ⟨ m n s ⟩ + ⟨ n 2 s ⟩ − ⟨ m n ⟩ + ⟨ n 2 ⟩ − ⟨ n s ⟩ − ⟨ n ⟩ ) + c ( ⟨ m 2 ⟩ − ⟨ m n ⟩ − ⟨ m ⟩ ) + d ′ ( − ⟨ m n ⟩ + ⟨ n 2 ⟩ − ⟨ n ⟩ ) .$
$d t ⟨ m s ⟩ = − a ′ ∑ 0 ∑ 0 ∑ 0 m 2 s 2 F ( m , n , s ) + a ′ ∑ 0 ∑ 1 ∑ 0 m ( m + 1 ) s ( s + 1 ) F ( m + 1 , n − 1 , s + 1 ) − a ∑ 0 ∑ 0 ∑ 0 m n s ( s + 1 ) F ( m , n , s ) + a ∑ 1 ∑ 0 ∑ 1 m ( n + 1 ) s 2 F ( m − 1 , n + 1 , s − 1 ) − b ∑ 0 ∑ 0 ∑ 0 m s 2 F ( m , n , s ) + b ∑ 0 ∑ 0 ∑ 0 m s ( s + 1 ) F ( m , n , s + 1 ) − c ∑ 0 ∑ 0 ∑ 0 m 2 s F ( m , n , s ) + c ∑ 0 ∑ 1 ∑ 0 m ( m + 1 ) s F ( m + 1 , n − 1 , s ) − d ′ ∑ 0 ∑ 0 ∑ 0 m n s F ( m , n , s ) + d ′ ∑ 1 ∑ 0 ∑ 0 m ( n + 1 ) s F ( m − 1 , n + 1 , s ) = − a ′ ∑ 1 ∑ 0 ∑ 1 m 2 s 2 F ( m , n , s ) + a ′ ∑ 1 ∑ 0 ∑ 1 m ( m − 1 ) s ( s − 1 ) F ( m , n , s ) − a ∑ 0 ∑ 1 ∑ 0 m n s ( s + 1 ) F ( m , n , s ) + a ∑ 0 ∑ 1 ∑ 0 ( m + 1 ) n ( s + 1 ) 2 F ( m , n , s ) − b ∑ 0 ∑ 0 ∑ 1 m s 2 F ( m , n , s ) + b ∑ 0 ∑ 0 ∑ 1 m s ( s − 1 ) F ( m , n , s ) − c ∑ 1 ∑ 0 ∑ 0 m 2 s F ( m , n , s ) + c ∑ 1 ∑ 0 ∑ 0 m ( m − 1 ) s F ( m , n , s ) − d ′ ∑ 0 ∑ 1 ∑ 0 m n s F ( m , n , s ) + d ′ ∑ 0 ∑ 1 ∑ 0 ( m + 1 ) n s F ( m , n , s ) .$
$d t ⟨ m s ⟩ = a ′ ( − ⟨ m 2 s ⟩ − ⟨ m s 2 ⟩ + ⟨ m s ⟩ ) + a ( ⟨ m n s ⟩ + ⟨ n s 2 ⟩ + ⟨ m n ⟩ + 2 ⟨ n s ⟩ + ⟨ n ⟩ ) − b ⟨ m s ⟩ − c ⟨ m s ⟩ + d ′ ⟨ n s ⟩ .$
$d t ⟨ δ n δ s ⟩ = a ′ ( ⟨ δ m s δ s ⟩ − ⟨ δ m s δ n ⟩ ) + a ( ⟨ δ n s δ n ⟩ − ⟨ δ n s δ s ⟩ ) + a ⟨ δ n 2 ⟩ − ( a + b + d ′ ) ⟨ δ n δ s ⟩ + c ⟨ δ m δ s ⟩ − a ′ ⟨ m s ⟩ − a ⟨ n ( s + 1 ) ⟩ ,$
$d t ⟨ δ m δ n ⟩ = a ′ ( ⟨ δ m s δ m ⟩ − ⟨ δ m s δ n ⟩ ) + a ( ⟨ δ n s δ n ⟩ − ⟨ δ n s δ m ⟩ ) − ( a + c + d ′ ) ⟨ δ m δ n ⟩ + ( a + d ′ ) ⟨ δ n 2 ⟩ + c ⟨ δ m 2 ⟩ − a ′ ⟨ m s ⟩ − a ⟨ n ( s + 1 ) ⟩ − c ⟨ m ⟩ − d ′ ⟨ n ⟩ ,$
$d t ⟨ δ m δ s ⟩ = − a ′ ( ⟨ δ m s δ m ⟩ + ⟨ δ m s δ s ⟩ ) + a ( ⟨ δ n s δ m ⟩ + ⟨ δ n s δ s ⟩ ) + a ⟨ δ m δ n ⟩ + a ⟨ δ n δ s ⟩ − ( b + c ) ⟨ δ m δ s ⟩ + d ′ ⟨ δ n δ s ⟩ + a ′ ⟨ m s ⟩ + a ⟨ n ( s + 1 ) ⟩ .$
$d t ⟨ ( δ m + δ n ) 2 ⟩ = 0.$
$d t ⟨ ( δ n + δ s ) 2 ⟩ = 0 = d t ⟨ ( δ m − δ s ) 2 ⟩ .$
$d t ⟨ s ⟩ ≈ − a ′ ⟨ m ⟩ ⟨ s ⟩ + a ⟨ n ⟩ ( ⟨ s ⟩ + 1 ) − b ⟨ s ⟩ ,$
$d t ⟨ n ⟩ ≈ a ′ ⟨ m ⟩ ⟨ s ⟩ − a ⟨ n ⟩ ( ⟨ s ⟩ + 1 ) + c ⟨ m ⟩ − d ′ ⟨ n ⟩ ,$
$d t ⟨ m ⟩ ≈ − a ′ ⟨ m ⟩ ⟨ s ⟩ + a ⟨ n ⟩ ( ⟨ s ⟩ + 1 ) − c ⟨ m ⟩ + d ′ ⟨ n ⟩ .$
$d t ⟨ δ s 2 ⟩ = − 2 a ′ ⟨ δ m s δ s ⟩ + 2 a ⟨ δ n s δ s ⟩ − 2 b ⟨ δ s 2 ⟩ + a ′ ⟨ m s ⟩ + a ⟨ n s ⟩ + b ⟨ s ⟩ ,$
$d t ⟨ δ n δ s ⟩ = − a ′ ⟨ δ m s δ n ⟩ + a ′ ⟨ δ m s δ s ⟩ + a ⟨ δ n s δ n ⟩ − a ⟨ δ n s δ s ⟩ − ( b + c + d ′ ) ⟨ δ n δ s ⟩ − a ′ ⟨ m s ⟩ − a ⟨ n s ⟩ ,$
$d t ⟨ δ n 2 ⟩ = 2 a ′ ⟨ δ m s δ n ⟩ − 2 a ⟨ δ n s δ n ⟩ − 2 ( c + d ′ ) ⟨ δ n 2 ⟩ + a ′ ⟨ m s ⟩ + a ⟨ n s ⟩ + c ⟨ m ⟩ + d ′ ⟨ n ⟩ .$
$d t ⟨ δ s 2 ⟩ = 2 ( a ′ ⟨ s ⟩ + a ⟨ s ⟩ ) ⟨ δ n δ s ⟩ + 2 ( a ⟨ n ⟩ − a ′ ⟨ m ⟩ − b ) ⟨ δ s 2 ⟩ + a ′ ⟨ m ⟩ ⟨ s ⟩ + a ⟨ n ⟩ ⟨ s ⟩ + b ⟨ s ⟩ ,$
$d t ⟨ δ n δ s ⟩ = ( a ′ ⟨ s ⟩ + a ⟨ s ⟩ ) ⟨ δ n 2 ⟩ + ( a ⟨ n ⟩ − a ′ ⟨ m ⟩ − b ) ⟨ δ n δ s ⟩ − ( a ′ ⟨ s ⟩ + a ⟨ s ⟩ + c + d ′ ) ⟨ δ n δ s ⟩ − ( a ⟨ n ⟩ − a ′ ⟨ m ⟩ ) ⟨ δ s 2 ⟩ − a ′ ⟨ m ⟩ ⟨ s ⟩ − a ⟨ n ⟩ ⟨ s ⟩ ,$
$d t ⟨ δ n 2 ⟩ = − 2 ( a ′ ⟨ s ⟩ + a ⟨ s ⟩ + c + d ′ ) ⟨ δ n 2 ⟩ − 2 ( a ⟨ n ⟩ − a ′ ⟨ m ⟩ ) ⟨ δ n δ s ⟩ + a ′ ⟨ m ⟩ ⟨ s ⟩ + a ⟨ n ⟩ ⟨ s ⟩ + c ⟨ m ⟩ + d ′ ⟨ n ⟩ .$
$F ( ω ) = ∫ − T / 2 T / 2 F ( t ) exp ⁡ ( i ω t ) d t , F ( t ) = ∫ ∞ ∞ F ( ω ) exp ⁡ ( − i ω t ) d ω / 2 π ,$
$S 1 ( ω ) = ( γ n n − i ω ) R s ( ω ) + γ s n R n ( ω ) ω 0 2 − 2 i ν 0 ω − ω 2 ,$
$N 1 ( ω ) = − γ n s R s ( ω ) − ( γ s s + i ω ) R n ( ω ) ω 0 2 − 2 i ν 0 ω − ω 2 ,$
$⟨ R j ( ω ) ⟩ = 0 , ⟨ R j ( ω ) R k ( ω ) ⟩ = R j k T .$
$Q ( ω ) = ⟨ | S 1 ( ω ) | 2 ⟩ S 0 2 T = A + B ω 2 S 0 2 [ ( ω 0 2 − ω 2 ) 2 + 4 ν 0 2 ω 2 ] ,$
$A = γ n n 2 R s s + 2 γ n n γ s n R n s + γ s n 2 R n n , B = R s s .$
$( b S 0 ) Q ( 0 ) = ( c ′ ) 2 + ( a ′ S 0 ′ ) 2 + ( a ′ S 0 ′ + c ′ ) 2 ( a ′ S 0 ′ ) 2 ,$
$( b S 0 ) Q ( 0 ) = a ′ ( c ′ ) 2 + ( 2 a ′ S 0 ′ ) 2 + ( 2 a ′ S 0 ′ + c ′ ) 2 ( 2 a ′ S 0 ′ ) 2 ,$
$( b S 0 ) Q ( ω 0 ) = ( c ′ ) 2 + ( a ′ S 0 ′ ) 2 + ( a ′ S 0 ′ + c ′ ) 2 + 2 ( a ′ S 0 ′ ) ( a ′ S 0 ′ ) ( a ′ S 0 ′ + c ′ ) 2 ,$
$( b S 0 ) Q ( ω 0 ) = a ′ ( c ′ ) 2 + ( 2 a ′ S 0 ′ ) 2 + ( 2 a ′ S 0 ′ + c ′ ) 2 + ( a ′ + 1 ) ( 2 a ′ S 0 ′ ) ( 2 a ′ S 0 ′ ) ( 2 a ′ S 0 ′ + c ′ ) 2 .$
$C ( τ ) = exp ⁡ ( − ν 0 τ ) 4 ν 0 ω r S 0 2 R e [ A + B ( ω r − i ν 0 ) 2 ω r − i ν 0 exp ⁡ ( − i ω r τ ) ] .$
$C ( 0 ) = A + B ω 0 2 4 ν 0 ω 0 2 S 0 2 .$
$[ − 2 γ s s − 2 γ s n 0 γ n s γ n n − γ s s − γ s n 0 2 γ n s 2 γ n n ] [ ⟨ S 1 2 ⟩ ⟨ N 1 S 1 ⟩ ⟨ N 1 2 ⟩ ] = [ R s s R n s R n n ] .$
$[ ⟨ S 1 2 ⟩ ⟨ N 1 S 1 ⟩ ⟨ N 1 2 ⟩ ] = 1 Δ [ γ n s γ s n + γ n n ( γ n n − γ s s ) 2 γ n n γ s n γ s n 2 − γ n n γ n s − 2 γ n n γ s s − γ s n γ s s γ n s 2 2 γ n s γ s s γ n s γ s n − γ s s ( γ n n − γ s s ) ] [ R s s R n s R n n ] ,$
$Δ = 2 ( γ n n − γ s s ) ( γ n s γ s n − γ n n γ s s ) = 4 ν 0 ω 0 2 .$
$⟨ S 1 2 ⟩ S 0 2 = ( γ n n 2 + γ n s γ s n − γ n n γ s s ) R s s + 2 γ n n γ s n R n s + γ s n 2 R n n 2 ( γ n n − γ s s ) ( γ n s γ s n − γ n n γ s s ) S 0 2 .$
$F ( ω ) = ∫ − T / 2 T / 2 F ( t ) exp ⁡ ( i ω t ) d t , F ( t ) = ∫ ∞ ∞ F ( ω ) exp ⁡ ( − i ω t ) d ω / 2 π ,$
$∫ − T / 2 T / 2 | F ( t ) | 2 d t = ∫ ∞ ∞ | F ( ω ) | 2 d ω / 2 π .$
$⟨ F ( t ) ⟩ e = ⟨ F ( t ) ⟩ t = ∫ − T / 2 T / 2 F ( t ) d t / T ,$
$C j k ( τ ) = ⟨ F j ( t ) F k ( t + τ ) ⟩ e = ⟨ F j ( t ) F k ( t + τ ) ⟩ t .$
$C j k ( ω ) = ⟨ F j ( t ) exp ⁡ ( − i ω t ) F k ( ω ) ⟩ t = F j ∗ ( ω ) F k ( ω ) / T .$
$⟨ F j ( t ) ⟩ e = 0 , ⟨ F j ( t ) F k ( t ′ ) ⟩ e = S j k δ ( t − t ′ ) ,$
$⟨ F j ( ω ) F k ∗ ( ω ) ⟩ e = ⟨ ∬ F j ( t ) F k ( t ′ ) exp ⁡ [ i ω ( t − t ′ ) ] d t d t ′ ⟩ e = ∬ S j k δ ( t − t ′ ) exp ⁡ [ i ω ( t − t ′ ) ] d t d t ′ = S j k T .$
$⟨ F j ( ω ) F k ∗ ( ω ′ ) ⟩ e = S j k T s i n c [ ( ω − ω ′ ) T / 2 ] ≈ 2 π S j k δ ( ω − ω ′ ) .$
$⟨ F ( ω ) F ∗ ( ω ′ ) ⟩ e = S f ( ω ) δ ( ω − ω ′ )$
$∫ − T / 2 T / 2 F ˙ ( t ) exp ⁡ ( i ω t ) d t = F ( t ) exp ⁡ ( i ω t ) | − T / 2 T / 2 − i ω ∫ − T / 2 T / 2 F ( t ) exp ⁡ ( i ω t ) d t .$
$d t X i = a i ( X ) + ∑ k b i k ( X ) r k ( t ) ,$
$⟨ r k ( t ) ⟩ = 0 , ⟨ r k ( t ) r l ( t ′ ) ⟩ = δ k l δ ( t − t ′ ) .$
$⟨ δ w k ⟩ = 0 , ⟨ δ w k 2 ⟩ = δ t .$
$δ X i = ∫ 0 δ t a i [ X ( t ) ] d t + ∑ k ∫ 0 δ t b i k [ X ( t ) ] r k ( t ) d t ≈ a i ( X ) δ t + ∑ k ∫ 0 δ t [ b i k ( X ) + ∑ j ∂ j b i k ( X ) ∑ l b j l ( X ) ∫ 0 t r l ( t ′ ) d t ′ ] r k ( t ) d t = a i ( X ) δ t + ∑ k b i k ( X ) ∫ 0 δ t r k ( t ) d t + ∑ k ∑ j ∑ l ∂ j b i k ( X ) b j l ( X ) ∫ 0 δ t ∫ 0 t r k ( t ) r l ( t ′ ) d t ′ d t ,$
$∑ j ∑ k b j k ( X ) ∂ j b i k ( X ) ∫ 0 δ t ∫ 0 t ⟨ r k ( t ) r k ( t ′ ) ⟩ d t ′ d t .$
$⟨ δ X i ⟩ ≈ a i ( X ) δ t .$
$⟨ δ X i ⟩ ≈ a i ( X ) δ t + ∑ j ∑ k b j k ( X ) ∂ j b i k ( X ) δ t / 2.$
$⟨ δ X i 2 ⟩ ≈ ∑ k b i k 2 ( X ) δ t .$
$X i ( 0 ) r l ( 0 ) ≈ { X i ( − δ t ) + ∑ k b i k [ X ( − δ t ) ] ∫ − δ t 0 r k ( t ) d t } r l ( 0 ) .$
$⟨ X i r l ⟩ = 0.$
$⟨ X i r l ⟩ = b i l ( X ) / 2.$
$⟨ R i k ( X , t ) ⟩ = 0 , ⟨ R i k ( X , t ) R j l ( X , t ′ ) ⟩ = b i k ( X ) b j k ( X ) δ k l δ ( t − t ′ ) .$
$δ X i ≈ a i ( X ) δ t + ∑ k b i k ( X ) ∫ 0 δ t r k ( t ) d t .$
$d t ⟨ X i ⟩ = ⟨ a i ( X ) ⟩ .$
$δ f ( X ) ≈ ∑ i f i ( X ) δ X i + ∑ i ∑ j f i j ( X ) δ X i δ X j / 2.$
$⟨ δ X i δ X j ⟩ ≈ ∑ k ∑ l b i k ( X ) b j l ( X ) ∫ 0 δ t ∫ 0 δ t ⟨ r k ( t ) r l ( t ′ ) ⟩ d t d t ′ = ∑ k b i k ( X ) b j k ( X ) δ t .$
$d t f ( X ) = ∑ i f i ( X ) d t X i + ∑ i ∑ j ∑ k f i j ( X ) b i k ( X ) b j k ( X ) / 2.$
$d t ⟨ X i 2 ⟩ = 2 ⟨ X i a i ( X ) ⟩ + ∑ k ⟨ b i k 2 ( X ) ⟩ ,$
$d t ⟨ X i X j ⟩ = ⟨ X j a i ( X ) ⟩ + ⟨ X i a j ( X ) ⟩ + ∑ k ⟨ b i k ( X ) b j k ( X ) ⟩ .$
$d t ⟨ δ X i 2 ⟩ = 2 ⟨ δ X i δ a i ( X ) ⟩ + ∑ k ⟨ b i k 2 ( X ) ⟩ ,$
$d t ⟨ δ X i δ X j ⟩ = ⟨ δ X j δ a i ( X ) ⟩ + ⟨ δ X i δ a j ( X ) ⟩ + ∑ k ⟨ b i k ( X ) b j k ( X ) ⟩ ,$
$d t X = A X + B R ( t ) ,$
$⟨ R ( t ) ⟩ = 0 , ⟨ R ( t ) R T ( t ′ ) ⟩ = I δ ( t − t ′ ) ,$
$d t Y = e − A t B R ( t ) .$
$X ( t ) = e A t X ( 0 ) + ∫ 0 t e A ( t − t ′ ) B R ( t ′ ) d t ′ .$
$⟨ X ( t ) ⟩ = e A t ⟨ X ( 0 ) ⟩ .$
$d t ⟨ X ⟩ = A ⟨ X ⟩ .$
$X ( t ) X T ( t ) = [ e A t X + ∫ 0 t e A ( t − t ′ ) B R ( t ′ ) d t ′ ] × [ X T e A T t + ∫ 0 t R T ( t ′ ′ ) B T e A T ( t − t ′ ′ ) d t ′ ′ ] ,$
$⟨ X ( t ) X T ( t ) ⟩ = e A t ⟨ X X T ⟩ e A T t + ∫ 0 t e A ( t − t ′ ) B B T e A T ( t − t ′ ) d t ′ .$
$d t ⟨ X X T ⟩ = A ⟨ X X T ⟩ + ⟨ X X T ⟩ A T + B B T .$
$d t ⟨ X i 2 ⟩ = 2 ∑ k a i k ⟨ X i X k ⟩ + ∑ k b i k 2 ,$
$d t ⟨ X i X j ⟩ = ∑ k a i k ⟨ X j X k ⟩ + ∑ k a j k ⟨ X i X k ⟩ + ∑ k b i k b j k ,$
$d t ⟨ M ⟩ = − a ′ ⟨ M S ⟩ + a ⟨ N ( S + 1 ) ⟩ − c ⟨ M ⟩ + d ′ ⟨ N ⟩ ,$
$d t ⟨ N ⟩ = a ′ ⟨ M S ⟩ − a ⟨ N ( S + 1 ) ⟩ + c ⟨ M ⟩ − d ′ ⟨ N ⟩ ,$
$d t ⟨ S ⟩ = − a ′ ⟨ M S ⟩ + a ⟨ N ( S + 1 ) ⟩ − b ⟨ S ⟩ .$
$d t ⟨ δ M 2 ⟩ = − 2 a ′ ⟨ δ M S δ M ⟩ + 2 a ⟨ δ N S δ M ⟩ + 2 a ⟨ δ M δ N ⟩ − 2 c ⟨ δ M 2 ⟩ + 2 d ′ ⟨ δ M δ N ⟩ + a ′ ⟨ M S ⟩ + a ⟨ N ( S + 1 ) ⟩ + c ⟨ M ⟩ + d ′ ⟨ N ⟩ ,$
$d t ⟨ δ N 2 ⟩ = 2 a ′ ⟨ δ M S δ N ⟩ − 2 a ⟨ δ N S δ N ⟩ − 2 a ⟨ δ N 2 ⟩ + 2 c ⟨ δ M δ N ⟩ − 2 d ′ ⟨ δ N 2 ⟩ + a ′ ⟨ M S ⟩ + a ⟨ N ( S + 1 ) ⟩ + c ⟨ M ⟩ + d ′ ⟨ N ⟩ ,$
$d t ⟨ δ S 2 ⟩ = − 2 a ′ ⟨ δ M S δ S ⟩ + 2 a ⟨ δ N S δ S ⟩ + 2 a ⟨ δ N δ S ⟩ − 2 b ⟨ δ S 2 ⟩ + a ′ ⟨ M S ⟩ + a ⟨ N ( S + 1 ) ⟩ + b ⟨ S ⟩ ,$
$d t ⟨ δ M δ N ⟩ = − a ′ ⟨ δ M S δ N ⟩ + a ⟨ δ N S δ N ⟩ + a ⟨ δ N 2 ⟩ − c ⟨ δ M δ N ⟩ + d ′ ⟨ δ N 2 ⟩ + a ′ ⟨ δ M S δ M ⟩ − a ⟨ δ N S δ M ⟩ − a ⟨ δ M δ N ⟩ + c ⟨ δ M 2 ⟩ − d ′ ⟨ δ M δ N ⟩ − a ′ ⟨ M S ⟩ − a ⟨ N ( S + 1 ) ⟩ − c ⟨ M ⟩ − d ′ ⟨ N ⟩ ,$
$d t ⟨ δ M δ S ⟩ = − a ′ ⟨ δ M S δ S ⟩ + a ⟨ δ N S δ S ⟩ + a ⟨ δ N δ S ⟩ − c ⟨ δ M δ S ⟩ + d ′ ⟨ δ N δ S ⟩ − a ′ ⟨ δ M S δ M ⟩ + a ⟨ δ N S δ M ⟩ + a ⟨ δ M δ N ⟩ − b ⟨ δ M δ S ⟩ + a ′ ⟨ M S ⟩ + a ⟨ N ( S + 1 ) ⟩ ,$
$d t ⟨ δ N δ S ⟩ = a ′ ⟨ δ M S δ S ⟩ − a ⟨ δ N S δ S ⟩ − a ⟨ δ N δ S ⟩ + c ⟨ δ M δ S ⟩ − d ′ ⟨ δ N δ S ⟩ − a ′ ⟨ δ M S δ N ⟩ + a ⟨ δ N S δ N ⟩ + a ⟨ δ N 2 ⟩ − b ⟨ δ N δ S ⟩ − a ′ ⟨ M S ⟩ − a ⟨ N ( S + 1 ) ⟩ .$
$d t ⟨ ( δ M + δ N ) 2 ⟩ = 2 ⟨ δ M ( δ A m + δ A n ) ⟩ + 2 ⟨ δ N ( δ A m + δ A n ) ⟩ = 0 ,$
$d t N = − a ( N , S ) + c ( N ) ,$
$d t S = a ( N , S ) − b ( S ) ,$
$d t F ( n , s ) = − a ( n , s ) F ( n , s ) + a ( n + 1 , s − 1 ) F ( n + 1 , s − 1 ) − b ( s ) F ( n , s ) + b ( s + 1 ) F ( n , s + 1 ) − c ( n ) F ( n , s ) + c ( n − 1 ) F ( n − 1 , s ) ,$
$d t T = − ∑ 0 ∑ 0 a ( n , s ) F ( n , s ) + ∑ 0 ∑ 1 a ( n + 1 , s − 1 ) F ( n + 1 , s − 1 ) − ∑ 0 ∑ 0 b ( s ) F ( n , s ) + ∑ 0 ∑ 0 b ( s + 1 ) F ( n , s + 1 ) − ∑ 0 ∑ 0 c ( n ) F ( n , s ) + ∑ 1 ∑ 0 c ( n − 1 ) F ( n − 1 , s ) = − ∑ 1 ∑ 0 a ( n , s ) F ( n , s ) + ∑ 1 ∑ 0 a ( n , s ) F ( n , s ) − ∑ 0 ∑ 1 b ( s ) F ( n , s ) + ∑ 0 ∑ 1 b ( s ) F ( n , s ) − ∑ 0 ∑ 0 c ( n ) F ( n , s ) + ∑ 0 ∑ 0 c ( n ) F ( n , s ) = 0 ,$
$d t ⟨ n ⟩ = − ∑ 0 ∑ 0 n a ( n , s ) F ( n , s ) + ∑ 0 ∑ 1 n a ( n + 1 , s − 1 ) F ( n + 1 , s − 1 ) − ∑ 0 ∑ 0 n c ( n ) F ( n , s ) + ∑ 1 ∑ 0 n c ( n − 1 ) F ( n − 1 , s ) = − ∑ 1 ∑ 0 n a ( n , s ) F ( n , s ) + ∑ 1 ∑ 0 ( n − 1 ) a ( n , s ) F ( n , s ) − ∑ 0 ∑ 0 n c ( n ) F ( n , s ) + ∑ 0 ∑ 0 ( n + 1 ) c ( n ) F ( n , s ) .$
$d t ⟨ n ⟩ = − ⟨ a ( n , s ) ⟩ + ⟨ c ( n ) ⟩ ,$
$d t ⟨ s ⟩ = − ∑ 0 ∑ 0 s a ( n , s ) F ( n , s ) + ∑ 0 ∑ 1 s a ( n + 1 , s − 1 ) F ( n + 1 , s − 1 ) − ∑ 0 ∑ 0 s b ( s ) F ( n , s ) + ∑ 0 ∑ 0 s b ( s + 1 ) F ( n , s + 1 ) = − ∑ 1 ∑ 0 s a ( n , s ) F ( n , s ) + ∑ 1 ∑ 0 ( s + 1 ) a ( n , s ) F ( n , s ) − ∑ 0 ∑ 1 s b ( s ) F ( n , s ) + ∑ 0 ∑ 1 ( s − 1 ) b ( s ) F ( n , s ) .$
$d t ⟨ s ⟩ = ⟨ a ( n , s ) ⟩ − ⟨ b ( s ) ⟩ ,$
$d t ⟨ n 2 ⟩ = − ∑ 0 ∑ 0 n 2 a ( n , s ) F ( n , s ) + ∑ 0 ∑ 1 n 2 a ( n + 1 , s − 1 ) F ( n + 1 , s − 1 ) − ∑ 0 ∑ 0 n 2 c ( n ) F ( n , s ) + ∑ 1 ∑ 0 n 2 c ( n − 1 ) F ( n − 1 , s ) = − ∑ 1 ∑ 0 n 2 a ( n , s ) F ( n , s ) + ∑ 1 ∑ 0 ( n − 1 ) 2 a ( n , s ) F ( n , s ) − ∑ 0 ∑ 0 n 2 c ( n ) F ( n , s ) + ∑ 0 ∑ 0 ( n + 1 ) 2 c ( n ) F ( n , s ) = − 2 ⟨ n a ( n , s ) ⟩ + ⟨ a ( n , s ) ⟩ + 2 ⟨ n c ( n ) ⟩ + ⟨ c ( n ) ⟩ ,$
$d t ⟨ n s ⟩ = − ∑ 0 ∑ 0 n s a ( n , s ) F ( n , s ) + ∑ 0 ∑ 1 n s a ( n + 1 , s − 1 ) F ( n + 1 , s − 1 ) − ∑ 0 ∑ 0 n s b ( s ) F ( n , s ) + ∑ 0 ∑ 0 n s b ( s + 1 ) F ( n , s + 1 ) − ∑ 0 ∑ 0 n s c ( n ) F ( n , s ) + ∑ 1 ∑ 0 n s c ( n − 1 ) F ( n − 1 , s ) = − ∑ 1 ∑ 0 n s a ( n , s ) F ( n , s ) + ∑ 1 ∑ 0 ( n − 1 ) ( s + 1 ) a ( n , s ) F ( n , s ) − ∑ 0 ∑ 1 n s b ( s ) F ( n , s ) + ∑ 0 ∑ 1 n ( s − 1 ) b ( s ) F ( n , s ) − ∑ 0 ∑ 0 n s c ( n ) F ( n , s ) + ∑ 0 ∑ 0 ( n + 1 ) s c ( n ) F ( n , s ) = ⟨ n a ( n , s ) ⟩ − ⟨ s a ( n , s ) ⟩ − ⟨ a ( n , s ) ⟩ − ⟨ n b ( s ) ⟩ + ⟨ s c ( n ) ⟩ ,$
$d t ⟨ s 2 ⟩ = − ∑ 0 ∑ 0 s 2 a ( n , s ) F ( n , s ) + ∑ 0 ∑ 1 s 2 a ( n + 1 , s − 1 ) F ( n + 1 , s − 1 ) − ∑ 0 ∑ 0 s 2 b ( s ) F ( n , s ) + ∑ 0 ∑ 0 s 2 b ( s + 1 ) F ( n , s + 1 ) = − ∑ 1 ∑ 0 s 2 a ( n , s ) F ( n , s ) + ∑ 1 ∑ 0 ( s + 1 ) 2 a ( n , s ) F ( n , s ) − ∑ 0 ∑ 1 s 2 b ( s ) F ( n , s ) + ∑ 0 ∑ 1 ( s − 1 ) 2 b ( s ) F ( n , s ) = 2 ⟨ s a ( n , s ) ⟩ + ⟨ a ( n , s ) ⟩ − 2 ⟨ s b ( s ) ⟩ + ⟨ b ( s ) ⟩ .$
$d t ⟨ δ n 2 ⟩ = − 2 ⟨ δ n δ a ( n , s ) ⟩ + 2 ⟨ δ n δ c ( n ) ⟩ + ⟨ a ( n , s ) ⟩ + ⟨ c ( n ) ⟩ ,$
$d t ⟨ δ n δ s ⟩ = ⟨ δ n δ a ( n , s ) ⟩ − ⟨ δ s δ a ( n , s ) ⟩ − ⟨ δ n δ b ( s ) ⟩ + ⟨ δ s δ c ( n ) ⟩ − ⟨ a ( n , s ) ⟩ ,$
$d t ⟨ δ s 2 ⟩ = 2 ⟨ δ s δ a ( n , s ) ⟩ − 2 ⟨ δ s δ b ( s ) ⟩ + ⟨ a ( n , s ) ⟩ + ⟨ b ( s ) ⟩ ,$
$C ( τ ) = ∫ − ∞ ∞ ( A + B ω 2 ) exp ⁡ ( − i ω τ ) [ ( ω 0 2 − ω 2 ) 2 + 4 ν 0 2 ω 2 ] d ω 2 π ,$
$∫ Γ F ( z ) d z = 2 π i ∑ j r e s j ,$
$( ω − ω r − i ν 0 ) ( ω − ω r + i ν 0 ) ( ω + ω r − i ν 0 ) ( ω + ω r + i ν 0 ) .$
$( − 2 i ν 0 ) 2 ( ω r − i ν 0 ) ( 2 ω r ) .$
$A + B ( ω r − i ν 0 ) 2 8 ν 0 ω r ( ω r − i ν 0 ) exp ⁡ ( − i ω r τ − ν 0 τ ) .$
$A + B ( ω r + i ν 0 ) 2 8 ν 0 ω r ( ω r + i ν 0 ) exp ⁡ ( i ω r τ − ν 0 τ ) .$
$C ( τ ) = exp ⁡ ( − ν 0 τ ) 4 ν 0 ω r R e [ A + B ( ω r − i ν 0 ) 2 ω r − i ν 0 exp ⁡ ( − i ω r τ ) ] .$
$I 1 ( t ) = ∫ − ∞ ∞ exp ⁡ ( − i ω t ) d ω ω 0 2 − 2 i ν 0 ω − ω 2 , I 2 ( t ) = ∫ − ∞ ∞ − i ω exp ⁡ ( − i ω t ) d ω ω 0 2 − 2 i ν 0 ω − ω 2 .$
$− ( ω − ω r + i ν 0 ) ( ω + ω r + i ν 0 ) .$
$I 1 ( t ) = − exp ⁡ ( − i ω r t − ν 0 t ) / 2 i ω r + exp ⁡ ( i ω r t − ν 0 t ) / 2 i ω r = sin ⁡ ( ω r t ) exp ⁡ ( − ν 0 t ) / ω r ,$
$I 2 ( t ) = ( i ω r + ν 0 ) exp ⁡ ( − i ω r t − ν 0 t ) / 2 i ω r + ( i ω r − ν 0 ) exp ⁡ ( i ω r t − ν 0 t ) / 2 i ω r = [ cos ⁡ ( ω r t ) − ν 0 sin ⁡ ( ω r t ) / ω r ] exp ⁡ ( − ν 0 t ) .$