Quantum Fluctuations in a Laser Soliton

Abstract

The Heisenberg–Langevin equation for a spatial laser soliton is constructed within consistent quantum electrodynamics. Canonical variables for the generation field and for a two-level material medium consisting of a medium that generates a laser and a medium that realizes saturable absorption are discussed in detail. It is assumed that the laser generation evolves in time much more slowly than the atomic medium. This makes it possible to apply the adiabatic approximation and construct a closed equation for the amplitude of the laser field. The equation has been derived paying special attention to the definition of Langevin sources, which play a decisive role in the formation of quantum statistical features of solitons. To ensure the observation procedure of the quantum squeezing of a soliton, the laser generation has been synchronized by applying an external weak electromagnetic field.

Appendices

SUB-POISSONIAN LASER GENERATION WITH SATURABLE ABSORPTION

1. INTRODUCTION

Equation (97) that was obtained in the main body of the text can be used to discuss a rather wide range of physical conditions. First, these are laser problems on the effect of saturated absorption on laser generation. In this case, the spatial aspect can be or cannot be taken into account. Second, these are problems of generation of a laser soliton with or without an external driving force that synchronizes the field in the cavity. All these problems are methodologically close to each other; therefore, it is reasonable to discuss them whenever possible simultaneously.

The simplest of these problems is the laser problem in the plane wave approximation in the presence of saturated absorption in the system. We have already discussed it in [18]. Two possible measuring procedures were considered in that work, which are related to the observation of the spectrum of fluctuations in the number of photons in laser generation and the observation of quadrature squeezing. It was assumed that the excitation of the laser active medium can be either a completely random Poissonian process or a regular sub-Poissonian process.

The solution of the problem of a laser with saturable absorption has a double meaning. First, this is a physical situation that is interesting in itself. Second, methodologically, it substantially intersects with the problem of generation of a laser soliton and, therefore, can be useful purely technically, since the laser system is formally much simpler than the soliton system, if only because the latter system contains a spatial aspect. At present, however, a situation has arisen that makes it difficult to perform potentially useful comparative procedures between the laser problem of [18] and the problem solved here and taking into account the spatial aspect.

The point is that there and here the problems were solved in terms of collective variables for both active and passive media. Furthermore, here, we introduce the collective coherence operator for the two media in the same way, while, in [18], it was introduced in different ways. This led to the fact that comparative procedures proved to be difficult. This, of course, is inconvenient, and, in this appendix, we will try to compensate for this inconvenience by reformulating the definition of collective coherences.

2. MAIN EQUATION FOR LASER GENERATION

We will solve the problem in the plane-wave approximation, i.e., completely ignoring the spatial aspect. Assuming that ν_in = 0 in Eq. (97), we rewrite it in the form

$$\begin{gathered} \dot {a} = - \frac{1}{2}\kappa (a - {{a}_{{{\text{in}}}}}) \\ + \;\frac{1}{2}\left( {\frac{A}{{1 + \beta {{{\left| a \right|}}^{2}}}} - \frac{{{{A}_{p}}}}{{1 + {{\beta }_{p}}{{{\left| a \right|}}^{2}}}}} \right)a + \Phi + {{\Phi }_{p}}. \\ \end{gathered} $$

(A.1)

All other frequency detunings are also zero: ${{\omega }_{0}}$ = ${{\omega }_{a}}$ = ${{\omega }_{p}}$.

Recall that this equation is a consequence of the operator equation

$$\dot {\hat {a}} = - \frac{1}{2}\kappa (\hat {a} - {{\hat {a}}_{{{\text{in}}}}}) + g\hat {\sigma } + {{g}_{p}}\hat {\pi } + {{\hat {F}}_{a}}.$$

(A.2)

Here, $\hat {\sigma }$ and $\hat {\pi }$ are the collective coherences of the active and passive media, respectively. These variables are defined in this paper as linear superpositions of individual coherences ${{\hat {\sigma }}^{j}}$ and ${{\hat {\pi }}^{j}}$,

$$\hat {\sigma } = - i\sum\limits_j \,{{\hat {\sigma }}^{j}}\Theta (t - {{t}_{j}}),\quad {{\hat {\sigma }}^{j}} = {{\left| 1 \right\rangle }_{{jj}}}\left\langle 2 \right|,$$

(A.3)

$$\hat {\pi } = - i\sum\limits_j \,{{\hat {\pi }}^{j}}\Theta (t - {{t}_{j}}),\quad {{\hat {\pi }}^{j}} = {{\left| 1 \right\rangle }_{{jj}}}\left\langle 2 \right|.$$

(A.4)

In [18], operator $\hat {\pi }$ was defined somewhat differently; namely, it was presented as a linear superposition of individual operators $\mathop {\hat {\pi }}\nolimits^{j\dag }$.

As far as Langevin sources are concerned, in accordance with expressions (105)–(109), we can write the following nonzero correlation relations for the active medium:

$$\left\langle {{{\Phi }^{2}}} \right\rangle = - \frac{1}{2}{{\left( {\frac{{\beta a}}{{1 + I}}} \right)}^{2}}{{R}_{a}}(1 + {{s}_{a}}{\text{/2}}),$$

(A.5)

$$\left\langle {{{{\left| \Phi \right|}}^{2}}} \right\rangle = \frac{\beta }{{1 + I}}{{R}_{a}} - \frac{1}{2}{{\left( {\frac{{\beta \left| a \right|}}{{1 + I}}} \right)}^{2}}{{R}_{a}}(1 + {{s}_{a}}{\text{/2}}).$$

(A.6)

For the passive medium, we have

$$\left\langle {\Phi _{p}^{2}} \right\rangle = \frac{1}{2}{{\left( {\frac{{{{\beta }_{p}}a}}{{1 + {{I}_{p}}}}} \right)}^{2}}{{R}_{p}}(1 - {{s}_{p}}{\text{/}}2),$$

(A.7)

$$\left\langle {{{{\left| {{{\Phi }_{p}}} \right|}}^{2}}} \right\rangle = \frac{1}{2}{{\left( {\frac{{{{\beta }_{p}}\left| a \right|}}{{1 + {{I}_{p}}}}} \right)}^{2}}{{R}_{p}}(1 - {{s}_{p}}{\text{/}}2).$$

(A.8)

We note that the last two formulas in [18] look different. This is a direct consequence of the fact that we defined the collective coherence of the passive medium differently.

3. FLUCTUATIONS IN THE NUMBER OF PHOTONS

Initially, we will assume that the external synchronizing field is absent (a_in = 0) and write the field amplitude in the form $a = \sqrt n \exp(i\varphi )$. Then, instead of original equation (A.1), we obtain two equations of the form

$$\dot {n} = - \kappa n + \left( {\frac{A}{{1 + \beta n}} - \frac{{{{A}_{p}}}}{{1 + {{\beta }_{p}}n}}} \right)n + {{\Phi }_{n}},$$

(A.9)

$$\dot {\varphi } = {{\Phi }_{\varphi }}.$$

(A.10)

Here, the Langevin sources are expressed via the initial sources as follows:

$${{\Phi }_{n}} = [a{\text{*}}\Phi + a\Phi {\text{*}}] + [a{\text{*}}{{\Phi }_{p}} + a\Phi _{p}^{{\text{*}}}],$$

(A.11)

$${{\Phi }_{\varphi }} = - \frac{i}{n}[a{\text{*}}\Phi - a\Phi {\text{*}}] - \frac{i}{n}[a{\text{*}}{{\Phi }_{p}} - a\Phi _{p}^{{\text{*}}}].$$

(A.12)

We will seek the solution to Eq. (A.9) in the form

$$n(t) = \bar {n} + \varepsilon (t),\quad \bar {n} \gg \varepsilon (t).$$

(A.13)

Here, $\bar {n}$ is the stationary classical solution, for which the following equality holds:

$$\kappa = \frac{A}{{1 + \beta \bar {n}}} - \frac{{{{A}_{p}}}}{{1 + {{\beta }_{p}}\bar {n}}}.$$

(A.14)

Linearizing Eq. (A.9) with respect to fluctuation ε, we obtain a simple equation of the form

$$\dot {\varepsilon } = - D\varepsilon + {{\Phi }_{n}},$$

(A.15)

where coefficient D is expressed as follows through laser parameters taking into account equality (A.14):

$$D = \kappa \frac{I}{{1 + I}}\left[ {1 + \frac{{({{A}_{p}}{\text{/}}\kappa )(1 - {v})}}{{{{{(1 + {v}I)}}^{2}}}}} \right],\quad {v} = {{\beta }_{p}}{\text{/}}\beta .$$

(A.16)

As is seen, at ${v}$ < 1, D is always positive; i.e., the stationary solution is always stable. At the same time, at ${v}$ > 1, the stationary solution proves to be stable only if ${v}$ is not too large: ${v}$ < ${{{v}}_{0}}$, were

$${{{v}}_{0}} = \frac{{{{\kappa }_{p}}{\text{/}}\kappa + 1}}{{{{\kappa }_{p}}{\text{/}}\kappa - I}},\quad {{\kappa }_{p}} = \frac{{{{A}_{p}}}}{{1 + {v}I}}.$$

(A.17)

Let us rewrite Eq. (A.15) in the spectral representation,

$${{\varepsilon }_{\omega }} = \frac{{{{\Phi }_{{n,\omega }}}}}{{D - i\omega }}.$$

(A.18)

To obtain the spectrum of Langevin noise, we will take into account definition (A.11); then,

$$\left\langle {{{{\left| {{{\Phi }_{{n,\omega }}}} \right|}}^{2}}} \right\rangle = 2\tilde {n}{{D}_{1}},$$

(A.19)

where

$$\begin{gathered} {{D}_{1}} = \kappa \frac{{1 - ({{s}_{a}}{\text{/}}2)I}}{{1 + I}} \\ + \;\frac{{{{A}_{p}}}}{{1 + {v}I}}\left( {\frac{{1 - ({{s}_{a}}{\text{/}}2)I}}{{1 + I}} + \frac{{(1 - {{s}_{p}}{\text{/}}2){v}I}}{{1 + {v}I}}} \right). \\ \end{gathered} $$

(A.20)

In the main, the explicit expression for coefficient D₁ coincides with that obtained in [18]. Only two refinements were made. First, for generality, we took into account the possibility of regular excitation of not only active, but also passive medium. As a result, an additional factor ($0 < {{s}_{p}} < 1$) appeared in the formula. Second, in the previous formula, the parenthesized multiplier (${v}I$) in the numerator of the second term was erroneously missed.

Applying the standard measuring procedure, we can obtain the spectrum of the photocurrent noise in the form

$$\left\langle {{{{\left| {\delta {{i}_{\omega }}} \right|}}^{2}}} \right\rangle {\text{/}}\left\langle i \right\rangle = 1 + \frac{{2\kappa {{D}_{1}}}}{{{{D}^{2}} + {{\omega }^{2}}}},\quad \left\langle i \right\rangle = \kappa {{\left| {\tilde {a}} \right|}^{2}}.$$

(A.21)

Recall that this formula is valid only if stability condition (A.17) for stationary generation is satisfied.

4. PHASE DIFFUSION IN A LASER FIELD WITHOUT SYNCHRONIZATION

Now we will follow the behavior of the phase of the generation field. Recall that, according to semiclassical theory, the phase turns out to be a random variable and, thus, in no way can be definite. To understand how quantum theory relates to this, we need to return to Eq. (A.7). We will integrate it in the time interval from t₁ to t₂, and, as a result, we obtain the equality

$$\varphi ({{t}_{2}}) - \varphi ({{t}_{1}}) = \int\limits_{{{t}_{1}}}^{{{t}_{2}}} \,dt{{\Phi }_{\varphi }}.$$

(A.22)

Since the average value of the Langevin source is zero, $\left\langle {{{\Phi }_{\varphi }}} \right\rangle$ = 0, this means that the average value of the generation phase remains unchanged with time, $\left\langle {\varphi ({{t}_{2}})} \right\rangle$ = $\left\langle {\varphi ({{t}_{1}})} \right\rangle$. Moreover, we cannot predict the result of the phase measurement. If we, nevertheless, succeeded in determining it, due to its fluctuations, we still will lose this information after some characteristic time. Indeed, the rms phase fluctuations are defined by the quantity

$$\begin{gathered} \left\langle {{{{[\varphi ({{t}_{2}}) - \varphi ({{t}_{1}})]}}^{2}}} \right\rangle = \int\limits_{{{t}_{1}}}^{{{t}_{2}}} {\int {dtdt{\kern 1pt} '\left\langle {{{\Phi }_{\varphi }}(t){{\Phi }_{\varphi }}(t{\kern 1pt} ')} \right\rangle } } \\ = \left\langle {\Phi _{\varphi }^{2}} \right\rangle ({{t}_{2}} - {{t}_{1}}). \\ \end{gathered} $$

(A.23)

In accordance with (A.10), we can find that

$$\left\langle {\Phi _{\varphi }^{2}} \right\rangle = \frac{{2\kappa }}{{\bar {n}}}\left( {1 + \frac{{{{A}_{p}}{\text{/}}\kappa }}{{1 + {v}I}}} \right).$$

(A.24)

We will interpret expression (A.23) as phase diffusion, which destroys information about the phase within a characteristic diffusion time ${{\left\langle {\Phi _{\varphi }^{2}} \right\rangle }^{{ - 1}}}$. As we can see, the saturated cell reduces the diffusion time. In addition, we can conclude that, in this case, if there is no synchronization of the laser field, phase diffusion does not depend on the photon statistics. Indeed, the result is independent of statistical parameters ${{s}_{a}}$ and ${{s}_{p}}$ of the medium excitation.

Comparing the latter formula with the corresponding formula obtained in [18], we can see that, here, coefficient 2 is common, while, in [18], it refers only to the second term in the parentheses.

5. QUADRATURE SQUEEZING OF A LASER FIELD WITH A SATURATED ABSORPTION UNDER THE PHASE SYNCHRONIZATION CONDITIONS

In the preceding section, we discussed laser generation with saturated absorption in the absence of synchronization. There, the main operating variables were fluctuations in the number of photons and phase fluctuations. In that section, we could see that, upon regular pumping of active medium atoms, the photon statistics turned out to be sub-Poissonian with complete or partial suppression of shot fluctuations in the saturation regime at I, ${v}I \gg 1$. At the same time, saturated absorption could markedly enhance phase diffusion provided that ${{A}_{p}}{\text{/}}\kappa \gg 1$.

To understand the role played by synchronization, we will return to Eq. (A.1). If an external field with amplitude ${{a}_{{{\text{in}}}}}$ = $\sqrt {{{n}_{{{\text{in}}}}}} \exp(i{{\varphi }_{{{\text{in}}}}})$ is in a coherent state, then, as is known, the phase of the generation field is “captured” by the external field. Clearly, phase fluctuations do occur, but they now turn out to be small. In connection with this, we assume that the amplitude of the generation field can be written as

$$a(t) = \tilde {a} + \delta a(t),\quad \tilde {a} \gg \delta a(t).$$

(A.25)

Here, stationary amplitude $\tilde {a}$ is defined by the equality

$$\begin{gathered} \kappa (1 - \mu ) = \frac{A}{{1 + \beta {{{\left| {\tilde {a}} \right|}}^{2}}}} - \frac{{{{A}_{p}}}}{{1 + {{\beta }_{p}}{{{\left| {\tilde {a}} \right|}}^{2}}}}, \\ \mu = \sqrt {{{n}_{{{\text{in}}}}}{\text{/}}\tilde {n}.} \\ \end{gathered} $$

(A.26)

Here, we are interested mostly not in the complex amplitude itself, but in its quadratures. Fluctuations of quadratures in a convenient basis are written as

$$\delta x = \frac{1}{2}({{e}^{{ - i{{\varphi }_{{{\text{in}}}}}}}}\delta a + {{e}^{{i{{\varphi }_{{{\text{in}}}}}}}}\delta a{\text{*}}),$$

(A.27)

$$\delta y = \frac{1}{{2i}}({{e}^{{ - i{{\varphi }_{{{\text{in}}}}}}}}\delta a - {{e}^{{i{{\varphi }_{{{\text{in}}}}}}}}\delta a{\text{*}}).$$

(A.28)

Because Eq. (A.1) is linearized in fluctuations of quadratures, we obtain two independent equations for these quadratures,

$$\delta x = - (\kappa \mu {\text{/2}} + {{D}_{\mu }})\delta x + {{\Phi }_{x}},$$

(A.29)

$$\delta y = - \kappa \mu {\text{/2}}\delta y + {{\Phi }_{y}}.$$

(A.30)

Now, coefficient ${{D}_{\mu }}$ turns out to be dependent of synchronization parameter µ and be defined by the expression

$${{D}_{\mu }} = \kappa \frac{I}{{1 + I}}\left[ {1 - \mu + \frac{{({{A}_{p}}{\text{/}}\kappa )(1 - {v})}}{{{{{(1 + {v}I)}}^{2}}}}} \right].$$

(A.31)

As compared to (A.16), a dependence on µ appeared here. The expressions completely coincide at µ = 0.

Concerning the Langevin sources in Eqs. (A.29) and (A.30), they can be expressed via initial sources Φ and Φ_p as follows:

$$\begin{gathered} {{\Phi }_{x}} = \frac{1}{2}[{{e}^{{ - i{{\varphi }_{{{\text{in}}}}}}}}\Phi + {{e}^{{i{{\varphi }_{{{\text{in}}}}}}}}\Phi {\text{*}}] \\ + \;\frac{1}{2}[{{e}^{{ - i{{\varphi }_{{{\text{in}}}}}}}}{{\Phi }_{p}} + {{e}^{{i{{\varphi }_{{{\text{in}}}}}}}}\Phi _{p}^{{\text{*}}}], \\ \end{gathered} $$

(A.32)

$$\begin{gathered} {{\Phi }_{y}} = \frac{1}{{2i}}[{{e}^{{ - i{{\varphi }_{{{\text{in}}}}}}}}\Phi - {{e}^{{i{{\varphi }_{{{\text{in}}}}}}}}\Phi {\text{*}}] \\ + \;\frac{1}{{2i}}[{{e}^{{ - i{{\varphi }_{{{\text{in}}}}}}}}{{\Phi }_{p}} - {{e}^{{i{{\varphi }_{{{\text{in}}}}}}}}\Phi _{p}^{ * }]. \\ \end{gathered} $$

(A.33)

In addition, we will discuss the situation in terms of Fourier components. Equations (A.29) and (A.30) then turn into equalities

$$ - i\omega \delta {{x}_{\omega }} = - (\kappa \mu {\text{/}}2 + {{D}_{\mu }})\delta {{x}_{\omega }} + {{\Phi }_{{x\omega }}},$$

(A.34)

$$ - i\omega \delta {{y}_{\omega }} = - \kappa \mu {\text{/}}2\delta {{y}_{\omega }} + {{\Phi }_{{y\omega }}}.$$

(A.35)

Now, taking into account correlation relations (24), (25) for Φ and Φ_p, we can obtain

$$\left\langle {{{{\left| {{{\Phi }_{{x\omega }}}} \right|}}^{2}}} \right\rangle = {{D}_{{1,\mu }}}{\text{/}}2,$$

(A.36)

$$\left\langle {{{{\left| {{{\Phi }_{{y\omega }}}} \right|}}^{2}}} \right\rangle = \kappa {\text{/}}2(1 - \mu ) + \frac{{{{A}_{p}}}}{{1 + {{{\tilde {I}}}_{p}}}},$$

(A.37)

where coefficient ${{D}_{{1,\mu }}}$ has the form

$$\begin{gathered} {{D}_{{1,\mu }}} = \kappa (1 - \mu )\frac{{1 - ({{s}_{a}}{\text{/}}2)I}}{{1 + I}} + \frac{{{{A}_{p}}}}{{1 + {v}I}} \\ \times \;\left( {\frac{{1 - ({{s}_{a}}{\text{/}}2)I}}{{1 + I}} + \frac{{(1 - ({{s}_{p}}{\text{/}}2)){v}I}}{{1 + {v}I}}} \right). \\ \end{gathered} $$

(A.38)

From knowing the spectral powers of noise (36) and (37), we can explicitly write the spectral powers of the corresponding quadrature components inside the cavity field, and using the well-known input–output re-lation, attribute them to the observables outside the cavity.

To observe quadratures, the so-called “balanced homodyne detection” is commonly used. In this approach, a signal with amplitude ${{\hat {a}}_{{{\text{out}}}}}(t)$ that has emerged from a cavity, prior to reaching a photodetector, is mixed with classical field L(t) of a local oscillator. We are interested most in precisely the x-quadrature of the field. Bearing in mind the preceding formulas of this section, we obtain the spectrum of noise of the photocurrent in the form

$$\left\langle {{{{\left| {\delta {{i}_{\omega }}} \right|}}^{2}}} \right\rangle {\text{/}}{{\left| L \right|}^{2}} = 1 + \frac{{2\kappa {{D}_{{1,\mu }}}}}{{{{{({{D}_{\mu }} + \kappa \mu {\text{/}}2)}}^{2}} + {{\omega }^{2}}}}.$$

(A.39)

In the saturation regime of the two media, ${{D}_{\mu }} = \kappa (1 - \mu )$ and ${{D}_{{1,\mu }}}$ = $- {{s}_{a}}{\text{/}}2\kappa (1 - \mu )$. It can be easily verified that, in this case,

$$\left\langle {{{{\left| {\delta {{i}_{{\omega = 0}}}} \right|}}^{2}}} \right\rangle {\text{/}}{{\left| L \right|}^{2}} = {{\mu }^{2}}{\text{/}}4 \ll 1.$$

(A.40)

We can infer that an additional cell introduced into a laser cavity does not destroy quantum particular features of a sub-Poissonian laser irrespective of whether it is synchronized with an external field or not.

LANGEVIN SOURCES IN EQUATIONS FOR A TWO-LEVEL MEDIUM

Calculations in this work were performed using simplest models of the field and matter. The laser field was assumed to be a quasi-plane and quasi-monochromatic wave, while the material medium was considered consisting of two-level atoms. Under these conditions, we can use the results obtained previously in [17] based on the fluctuation–dissipation theorem. Summarizing these results for the case with a transverse spatial distribution of the field, we can write the following correlation relations for active medium sources

$$\left\langle {\hat {F}_{i}^{\dag }({\boldsymbol{\rho }},t){{{\hat {F}}}_{j}}({\boldsymbol{\rho }}{\kern 1pt} ',t{\kern 1pt} ')} \right\rangle = \left\langle {{{{\hat {F}}}_{i}}{{{\hat {F}}}_{j}}} \right\rangle \delta (t - t{\kern 1pt} '){{\delta }^{2}}({\boldsymbol{\rho }} - {\boldsymbol{\rho }}{\kern 1pt} '),$$

(B.1)

$$\left\langle {{{{\hat {F}}}_{2}}{{{\hat {F}}}_{2}}} \right\rangle = {{\Gamma }_{2}}{{\bar {\sigma }}_{2}} + {{R}_{a}}(1 - {{s}_{a}}),$$

(B.2)

$$\left\langle {{{{\hat {F}}}^{\dag }}\hat {F}} \right\rangle = {{\Gamma }_{1}}{{\bar {\sigma }}_{2}} + {{R}_{a}},$$

(B.3)

$$\left\langle {{{{\hat {F}}}_{1}}{{{\hat {F}}}_{1}}} \right\rangle = {{\Gamma }_{1}}{{\bar {\sigma }}_{1}},$$

(B.4)

$$\left\langle {{{{\hat {F}}}_{1}}\hat {F}} \right\rangle = {{\Gamma }_{1}}\bar {\sigma },$$

(B.5)

$$\left\langle {\hat {F}{{{\hat {F}}}_{2}}} \right\rangle = {{\Gamma }_{2}}\bar {\sigma },$$

(B.6)

$$\left\langle {\hat {F}{{{\hat {F}}}^{\dag }}} \right\rangle = {{\Gamma }_{2}}{{\bar {\sigma }}_{1}}.$$

(B.7)

Here, Γ₁ is the decay rate of the population of the lower atomic level, Γ₂ is the decay rate of the population of the upper atomic level, R_a is the pumping rate of the upper atomic level, and s_a is the pumping statistical parameter. We assumed that the coherence decay rate of the laser transition is Γ = (Γ₁ + Γ₂)/2. Here, attention should be drawn to the parameter $0 \leqslant {{s}_{a}} \leqslant 1$, which provides the possibility of taking into account not only the average excitation rate of the upper laser level, R_a, but also to monitor excitation fluctuations. In the limiting case of s_a = 0, atoms are excited independently of each other and randomly in time (Poissonian pumping). At the same time, at s_a = 1, there is strictly regular pumping without noise (sub-Poissonian pumping).

Using the same approaches that were developed in [17], we can write the actual correlation functions for the Langevin sources of the passive medium in the form

$$\left\langle {\hat {G}_{i}^{\dag }({\boldsymbol{\rho }},t){{{\hat {G}}}_{j}}({\boldsymbol{\rho }}{\kern 1pt} ',t{\kern 1pt} ')} \right\rangle = \left\langle {{{{\hat {G}}}_{i}}{{{\hat {G}}}_{j}}} \right\rangle \delta (t - t{\kern 1pt} '){{\delta }^{2}}({\boldsymbol{\rho }} - {\boldsymbol{\rho }}{\kern 1pt} '),$$

(B.8)

$$\left\langle {{{{\hat {G}}}_{1}}{{{\hat {G}}}_{1}}} \right\rangle = {{\gamma }_{1}}{{\bar {\pi }}_{1}} + {{R}_{p}}(1 - {{s}_{p}}),$$

(B.9)

$$\left\langle {{{{\hat {G}}}^{\dag }}\hat {G}} \right\rangle = {{\gamma }_{1}}{{\bar {\pi }}_{2}},$$

(B.10)

$$\left\langle {{{{\hat {G}}}_{2}}{{{\hat {G}}}_{2}}} \right\rangle = {{\gamma }_{2}}{{\bar {\pi }}_{2}},$$

(B.11)

$$\left\langle {\hat {G}{{{\hat {G}}}_{2}}} \right\rangle = {{\gamma }_{2}}\bar {\pi },$$

(B.12)

$$\left\langle {{{{\hat {G}}}_{1}}\hat {G}} \right\rangle = {{\gamma }_{1}}\bar {\pi },$$

(B.13)

$$\left\langle {\hat {G}{{{\hat {G}}}^{\dag }}} \right\rangle = {{\gamma }_{2}}{{\bar {\pi }}_{1}} + {{R}_{p}},$$

(B.14)

Here, γ is the coherence decay rate of the transition with saturated absorption, γ₁ is the population decay rate of the lower atomic level, γ₂ is the population decay rate of the upper atomic level, R_p is the pumping rate of the lower atomic level, and s_p is the pumping statistical parameter. The meaning of parameter s_p is the same as that of s_a but with respect to the lower level of the passive medium.

The correlation relations in the c-number representation for the active medium are given by

$$\begin{gathered} \langle F_{i}^{{\text{*}}}({\boldsymbol{\rho }},t){{F}_{j}}({\boldsymbol{\rho }}{\kern 1pt} ',t{\kern 1pt} ')\rangle \\ = \langle F_{i}^{{\text{*}}}({\boldsymbol{\rho }},t){{F}_{j}}({\boldsymbol{\rho }},t)\rangle \delta (t - t{\kern 1pt} '){{\delta }^{2}}({\boldsymbol{\rho }} - {\boldsymbol{\rho }}{\kern 1pt} '), \\ \end{gathered} $$

(B.15)

$$\left\langle {{{{\left| F \right|}}^{2}}} \right\rangle = {{\Gamma }_{1}}{{\bar {\sigma }}_{2}} + {{R}_{a}},$$

(B.16)

$$\left\langle {{{F}^{2}}} \right\rangle = 2g\overline {\sigma a} ,$$

(B.17)

$$\left\langle {{{F}_{1}}F} \right\rangle = {{\Gamma }_{1}}\bar {\sigma },$$

(B.18)

$$\left\langle {{{F}_{2}}{{F}_{2}}} \right\rangle = {{\Gamma }_{2}}{{\bar {\sigma }}_{2}} + {{R}_{a}}(1 - {{s}_{a}}) - g(\overline {a{\text{*}}\sigma } + \overline {a\sigma {\text{*}}} ),$$

(B.19)

$$\left\langle {{{F}_{1}}{{F}_{1}}} \right\rangle = {{\Gamma }_{1}}{{\bar {\sigma }}_{1}} - g(\overline {a{\text{*}}\sigma } + \overline {a\sigma {\text{*}}} ),$$

(B.20)

$$\left\langle {{{F}_{2}}{{F}_{1}}} \right\rangle = g(\overline {a{\text{*}}\sigma } + \overline {a\sigma {\text{*}}} ).$$

(B.21)

The correlation relations for the passive medium are as follows:

$$\langle G_{i}^{{\text{*}}}({\boldsymbol{\rho }},t){{G}_{j}}({\boldsymbol{\rho }}{\kern 1pt} ',t{\kern 1pt} ')\rangle = \langle G_{i}^{{\text{*}}}{{G}_{j}}\rangle \delta (t - t{\kern 1pt} '){{\delta }^{2}}({\boldsymbol{\rho }} - {\boldsymbol{\rho }}{\kern 1pt} '),$$

(B.22)

$$\begin{gathered} \left\langle {{{{\left| G \right|}}^{2}}} \right\rangle = {{R}_{p}} + {{\gamma }_{1}}{{\pi }_{2}} - {{\gamma }_{1}}{{\pi }_{1}} \\ + \;{{\gamma }_{2}}{{\pi }_{2}} + 2{{g}_{p}}(\overline {a{\text{*}}\pi } + \overline {a\pi {\text{*}}} ), \\ \end{gathered} $$

(B.23)

$$\left\langle {{{G}^{2}}} \right\rangle = 2{{g}_{p}}\overline {\pi a} ,$$

(B.24)

$$\left\langle {{{G}_{2}}G} \right\rangle = {{\gamma }_{2}}\bar {\pi },$$

(B.25)

$$\left\langle {G_{1}^{2}} \right\rangle = {{\gamma }_{1}}{{\bar {\pi }}_{1}} + {{R}_{p}}(1 - {{s}_{p}}) - {{g}_{p}}(\overline {a{\text{*}}\pi } + \overline {a\pi {\text{*}}} ),$$

(B.26)

$$\left\langle {G_{2}^{2}} \right\rangle = {{\gamma }_{2}}{{\bar {\pi }}_{2}} - {{g}_{p}}(\overline {a{\text{*}}\pi } + \overline {a\pi {\text{*}}} ),$$

(B.27)

$$\left\langle {{{G}_{1}}{{G}_{2}}} \right\rangle = {{g}_{p}}(\overline {a{\text{*}}\pi } + \overline {a\pi {\text{*}}} ).$$

(B.28)