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Physical Review

Quantum Noise. X. Density-Matrix Treatment of Field and Population-Difference Fluctuations

M. Lax

Published May 10, 1967• 214 citations
Publisher

Abstract

Starting with the quantum-noise-source model of a maser developed in the fourth paper in this series and specializing to the case in which all atomic parameters except the population difference vary rapidly compared with the field, an equation is developed for the density operator $\ensuremath{\rho}(b, {b}^{\ifmmode\dagger\else\textdagger\fi{}}, D, t)$ that depends on the electromagnetic field variables $b$, ${b}^{\ifmmode\dagger\else\textdagger\fi{}}$ and the population difference $D$. The corresponding equation obeyed by the density matrix ${\ensuremath{\rho}}_{\mathrm{mn}}(D, t)$ in the "$n$ representation" is obtained. The antinormal ordering correspondence $\ensuremath{\rho}(b, {b}^{\ifmmode\dagger\else\textdagger\fi{}}, D, t)=\mathcal{a}{\overline{\ensuremath{\rho}}}^{(a)}(\ensuremath{\beta}, {\ensuremath{\beta}}^{*}, D, t)$ [in which $\ensuremath{\beta}$ is replaced by $b$ and ${\ensuremath{\beta}}^{*}$ by ${b}^{\ifmmode\dagger\else\textdagger\fi{}}$ with all ${b}^{\ifmmode\dagger\else\textdagger\fi{}}'\mathrm{s}$ to the right of all $b'\mathrm{s}$] defines an associated classical function ${\overline{\ensuremath{\rho}}}^{(a)}$. The equation for $\frac{\ensuremath{\partial}\ensuremath{\rho}}{\ensuremath{\partial}t}$ corresponds to a Fokker-Planck equation for $\frac{\ensuremath{\partial}{\overline{\ensuremath{\rho}}}^{(a)}}{\ensuremath{\partial}t}$. Under the Markoffian assumption, an associated classical random process is uniquely defined. The quantum-mechanical average of normal-ordered time-ordered operators is given by simple integrals over the corresponding classical distribution functions. For one-time operators this result is well known. For two-time operators, a proof is given using the quantum-regression theorem of the second paper in this series. Amplitude and phase fluctuations are discussed by rewriting the Fokker-Planck equation in the variables $I$, $\ensuremath{\varphi}$, $D$, where $\ensuremath{\beta}={I}^{\frac{1}{2}}\mathrm{exp}(\ensuremath{-}i\ensuremath{\varphi})$. The Langevin description of the classical random process associated with ${\overline{\ensuremath{\rho}}}^{(a)}(\ensuremath{\beta}, {\ensuremath{\beta}}^{*}, D, t)$ is used to make a classical adiabatic elimination of $D$. The resulting equations for ${\overline{\ensuremath{\rho}}}^{(a)}(\ensuremath{\beta}, {\ensuremath{\beta}}^{*}, t)$ and $\ensuremath{\rho}(b, {b}^{\ifmmode\dagger\else\textdagger\fi{}}, t)$ are shown to agree with the corresponding results of the ninth paper in this series, obtained by a more difficult quantum adiabatic elimination of $D$. The equation for $\ensuremath{\rho}(b, {b}^{\ifmmode\dagger\else\textdagger\fi{}}, t)$, rewritten in the $n$ representation agrees with corresponding results of Scully and Lamb.

DOI
10.1103/physrev.157.213

Related papers

Quantum Noise. XII. Density-Operator Treatment of Field and Population Fluctuations

M. Lax, W. H. Louisell

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A correspondence ${a}_{j}\ensuremath{\leftrightarrow}{\ensuremath{\alpha}}_{j}$ between operators $\mathrm{a}=[{a}_{1}, {a}_{2}, \ensuremath{\cdots}{a}_{f}]$ and $c$ numbers $\ensuremath{\alpha}=[{\ensuremath{\alpha}}_{1}, {\ensuremath{\alpha}}_{2}, \ensuremath{\cdots}{\ensuremath{\alpha}}_{f}]$ together with an arbitrary ordering rule $\mathcal{C}$ (e.g., in sequence from 1 to $f$) permit an association $M(\mathrm{a})=\mathcal{C}{M}^{(c)}(\ensuremath{\alpha})$ between a general operator $M(\mathrm{a})$ and an associated $c$ number function ${M}^{(c)}(\ensuremath{\alpha})$. A quasiprobability $P(\ensuremath{\alpha}, t)$ is then defined so that a general ensemble average can be written as an ordinary integration: $〈M(\mathrm{a}(t))〉=\ensuremath{\int}{M}^{(c)}(\ensuremath{\alpha})d\ensuremath{\alpha}P(\ensuremath{\alpha}, t)$. The equation for $\frac{\ensuremath{\partial}P(\ensuremath{\alpha}, t)}{\ensuremath{\partial}t}$ suggests that the $\ensuremath{\alpha}$ obeys a classical Markoff process. If this classical Markoff process is taken literally, multitime classical averages can be computed. Do these correspond to appropriate quantum averages? For the case of field operators such that $[b, {b}^{\ifmmode\dagger\else\textdagger\fi{}}]=1$, important in discussing laser statistics, we show that with ${a}_{1}={b}^{\ifmmode\dagger\else\textdagger\fi{}}$ and ${a}_{f}=b$, the classical multitime average is equivalent to the average of the corresponding quantum operators written in time-ordered, normal-ordered sequence. For the atomic operators in a laser problem, we obtain the desired correspondence, but find that the more complicated commutation rules necessarily lead to derivative correction terms when multitime averages are taken. Our derivation of multitime averages is based on the quantum regression theorem. We show that this theorem is equivalent to assuming the quantum system to be Markoffian, by showing that it leads to an appropriate factorization of a multitime density matrix and to a Chapman-Kolmogoroff-like condition on the conditional density matrix.

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The Fokker-Planck equation for quantum noise of theN-level system

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The European Physical Journal A•Aug 1, 1966•13 citations
Starting with the quantum-noise-source model of a maser developed in the fourth paper in this series and specializing to the case in which all atomic parameters except the population difference vary rapidly compared with the field, an equation is developed for the density operator \ensuremathρ(b,b\ifmmode†\else\textdagger\fi,D,t)\ensuremath{\rho}(b, {b}^{\ifmmode\dagger\else\textdagger\fi{}}, D, t)\ensuremathρ(b,b\ifmmode†\else\textdagger\fi,D,t) that depends on the electromagnetic field variables bbb, b\ifmmode†\else\textdagger\fi{b}^{\ifmmode\dagger\else\textdagger\fi{}}b\ifmmode†\else\textdagger\fi and the population difference DDD. The corresponding equation obeyed by the density matrix \ensuremathρmn(D,t){\ensuremath{\rho}}_{\mathrm{mn}}(D, t)\ensuremathρmn​(D,t) in the "nnn representation" is obtained. The antinormal ordering correspondence \ensuremathρ(b,b\ifmmode†\else\textdagger\fi,D,t)=a\ensuremathρ‾(a)(\ensuremathβ,\ensuremathβ∗,D,t)\ensuremath{\rho}(b, {b}^{\ifmmode\dagger\else\textdagger\fi{}}, D, t)=\mathcal{a}{\overline{\ensuremath{\rho}}}^{(a)}(\ensuremath{\beta}, {\ensuremath{\beta}}^{*}, D, t)\ensuremathρ(b,b\ifmmode†\else\textdagger\fi,D,t)=a\ensuremathρ​(a)(\ensuremathβ,\ensuremathβ∗,D,t) [in which \ensuremathβ\ensuremath{\beta}\ensuremathβ is replaced by bbb and \ensuremathβ∗{\ensuremath{\beta}}^{*}\ensuremathβ∗ by b\ifmmode†\else\textdagger\fi{b}^{\ifmmode\dagger\else\textdagger\fi{}}b\ifmmode†\else\textdagger\fi with all {b}^{\ifmmode\dagger\else\textdagger\fi{}}'\mathrm{s} to the right of all b′sb'\mathrm{s}b′s] defines an associated classical function \ensuremathρ‾(a){\overline{\ensuremath{\rho}}}^{(a)}\ensuremathρ​(a). The equation for \ensuremath∂\ensuremathρ\ensuremath∂t\frac{\ensuremath{\partial}\ensuremath{\rho}}{\ensuremath{\partial}t}\ensuremath∂t\ensuremath∂\ensuremathρ​ corresponds to a Fokker-Planck equation for \ensuremath∂\ensuremathρ‾(a)\ensuremath∂t\frac{\ensuremath{\partial}{\overline{\ensuremath{\rho}}}^{(a)}}{\ensuremath{\partial}t}\ensuremath∂t\ensuremath∂\ensuremathρ​(a)​. Under the Markoffian assumption, an associated classical random process is uniquely defined. The quantum-mechanical average of normal-ordered time-ordered operators is given by simple integrals over the corresponding classical distribution functions. For one-time operators this result is well known. For two-time operators, a proof is given using the quantum-regression theorem of the second paper in this series. Amplitude and phase fluctuations are discussed by rewriting the Fokker-Planck equation in the variables III, \ensuremathφ\ensuremath{\varphi}\ensuremathφ, DDD, where \ensuremathβ=I12exp(\ensuremath−i\ensuremathφ)\ensuremath{\beta}={I}^{\frac{1}{2}}\mathrm{exp}(\ensuremath{-}i\ensuremath{\varphi})\ensuremathβ=I21​exp(\ensuremath−i\ensuremathφ). The Langevin description of the classical random process associated with \ensuremathρ‾(a)(\ensuremathβ,\ensuremathβ∗,D,t){\overline{\ensuremath{\rho}}}^{(a)}(\ensuremath{\beta}, {\ensuremath{\beta}}^{*}, D, t)\ensuremathρ​(a)(\ensuremathβ,\ensuremathβ∗,D,t) is used to make a classical adiabatic elimination of DDD. The resulting equations for \ensuremathρ‾(a)(\ensuremathβ,\ensuremathβ∗,t){\overline{\ensuremath{\rho}}}^{(a)}(\ensuremath{\beta}, {\ensuremath{\beta}}^{*}, t)\ensuremathρ​(a)(\ensuremathβ,\ensuremathβ∗,t) and \ensuremathρ(b,b\ifmmode†\else\textdagger\fi,t)\ensuremath{\rho}(b, {b}^{\ifmmode\dagger\else\textdagger\fi{}}, t)\ensuremathρ(b,b\ifmmode†\else\textdagger\fi,t) are shown to agree with the corresponding results of the ninth paper in this series, obtained by a more difficult quantum adiabatic elimination of DDD. The equation for \ensuremathρ(b,b\ifmmode†\else\textdagger\fi,t)\ensuremath{\rho}(b, {b}^{\ifmmode\dagger\else\textdagger\fi{}}, t)\ensuremathρ(b,b\ifmmode†\else\textdagger\fi,t), rewritten in the nnn representation agrees with corresponding results of Scully and Lamb.
A fully quantum-mechanical treatment is given of field and population fluctuations in a homogeneously broadened laser. The field and population reservoirs have been replaced by dissipation coefficients (or transition probabilities) and noncommuting noise sources. The atomic polarization variables are adiabatically eliminated. The mean motion of an arbitrary operator function $M(b, {b}^{\ifmmode\dagger\else\textdagger\fi{}}, {N}_{1}, {N}_{2})$ of the field variables bbb, ${b}^{\ifmmode\dagger\else\textdagger\fi{}}$ and the population variables N1{N}_{1}N1​ and N2{N}_{2}N2​ is calculated. The equation of motion for the density matrix $\ensuremath{\rho}(b, {b}^{\ifmmode\dagger\else\textdagger\fi{}}, {N}_{1}, {N}_{2}, t)$ is deduced, as well as the Fokker-Planck equation for the associated classical distribution function $P(\ensuremath{\beta}, {\ensuremath{\beta}}^{*}, {N}_{1}, {N}_{2}, t)$. The diffusion coefficients in the latter equation demonstrate the shot-noise origin of the fluctuations. The Fokker-Planck equation for PPP is replaced by Langevin equations for $\ensuremath{\beta}$, ${\ensuremath{\beta}}^{*}$, N1{N}_{1}N1​, and N2{N}_{2}N2​ to facilitate adiabatic elimination of the population variables (when valid). The resulting Langevin equations for $\ensuremath{\beta}$, ${\ensuremath{\beta}}^{*}$ are converted to a Fokker-Planck equation for $P(\ensuremath{\beta}, {\ensuremath{\beta}}^{*}, t)$, to an operator equation for $\ensuremath{\rho}(b, {b}^{\ifmmode\dagger\else\textdagger\fi{}}, t)$, and to an equation for ${\ensuremath{\rho}}_{\mathrm{mn}}$, the field density matrix in the photon-number representation. The transformation $\ensuremath{\beta}={I}^{\frac{1}{2}}{e}^{\ensuremath{-}i\ensuremath{\phi}}$ to an ($I, \ensuremath{\phi}$) representation leads (a) to the steady-state solution P(I)P(I)P(I) valid at all operating levels, and (b) to the phase linewidth. The transformation to new variables N=12(m+n)N=\frac{1}{2}(m+n)N=21​(m+n), $k=m\ensuremath{-}n$, permits the equation for ${\ensuremath{\rho}}_{\mathrm{mn}}$ to be replaced by a Fokker-Planck equation for $\ensuremath{\rho}(N, k, t)$. The latter permits a verification of the previously obtained phase linewidth and a solution for the steady-state $\ensuremath{\rho}(N)$.
A dynamic correspondence is set up between the density matrix of the electromagnetic field and an associated classical function. Means of one and two time quantum operators are shown to be obtainable from the corresponding averages of the associated classical random problem. When all atomic response rates are fast compared to photon rates, the equation of motion is obtained for the field density matrix. The associated classical problem is shown to obey a Fokker-Planck equation. Over a broad region that includes the threshold region, the Fokker-Planck equation is reduced to that for the rotating-wave van der Pol oscillator for which exact numerical solutions have been previously obtained by Hempstead and Lax.
A correspondence ${a}_{j}\ensuremath{\leftrightarrow}{\ensuremath{\alpha}}_{j}$ between operators $\mathrm{a}=[{a}_{1}, {a}_{2}, \ensuremath{\cdots}{a}_{f}]$ and ccc numbers $\ensuremath{\alpha}=[{\ensuremath{\alpha}}_{1}, {\ensuremath{\alpha}}_{2}, \ensuremath{\cdots}{\ensuremath{\alpha}}_{f}]$ together with an arbitrary ordering rule C\mathcal{C}C (e.g., in sequence from 1 to fff) permit an association $M(\mathrm{a})=\mathcal{C}{M}^{(c)}(\ensuremath{\alpha})$ between a general operator M(a)M(\mathrm{a})M(a) and an associated ccc number function ${M}^{(c)}(\ensuremath{\alpha})$. A quasiprobability $P(\ensuremath{\alpha}, t)$ is then defined so that a general ensemble average can be written as an ordinary integration: $〈M(\mathrm{a}(t))〉=\ensuremath{\int}{M}^{(c)}(\ensuremath{\alpha})d\ensuremath{\alpha}P(\ensuremath{\alpha}, t)$. The equation for $\frac{\ensuremath{\partial}P(\ensuremath{\alpha}, t)}{\ensuremath{\partial}t}$ suggests that the $\ensuremath{\alpha}$ obeys a classical Markoff process. If this classical Markoff process is taken literally, multitime classical averages can be computed. Do these correspond to appropriate quantum averages? For the case of field operators such that $[b, {b}^{\ifmmode\dagger\else\textdagger\fi{}}]=1$, important in discussing laser statistics, we show that with ${a}_{1}={b}^{\ifmmode\dagger\else\textdagger\fi{}}$ and af=b{a}_{f}=baf​=b, the classical multitime average is equivalent to the average of the corresponding quantum operators written in time-ordered, normal-ordered sequence. For the atomic operators in a laser problem, we obtain the desired correspondence, but find that the more complicated commutation rules necessarily lead to derivative correction terms when multitime averages are taken. Our derivation of multitime averages is based on the quantum regression theorem. We show that this theorem is equivalent to assuming the quantum system to be Markoffian, by showing that it leads to an appropriate factorization of a multitime density matrix and to a Chapman-Kolmogoroff-like condition on the conditional density matrix.
For a quantum system describable in a Markoffian way, via a set of variables $\mathrm{a}={{a}_{1}, {a}_{2}, \ensuremath{\cdots}}$, we show that the Langevin noise sources ${F}_{\ensuremath{\mu}}$ in the operator equations of motion $\frac{d{a}_{\ensuremath{\mu}}}{\mathrm{dt}}={A}_{\ensuremath{\mu}}(\mathrm{a})+{F}_{\ensuremath{\mu}}$ possess second moments $〈{F}_{\ensuremath{\mu}}(t){F}_{\ensuremath{\nu}}(u)〉=2〈{D}_{\ensuremath{\mu}\ensuremath{\nu}}(\mathrm{a}, t)〉\ensuremath{\delta}(t\ensuremath{-}u)$. The diffusion coefficients ${D}_{\ensuremath{\mu}\ensuremath{\nu}}$ can be determined from a knowledge of the mean equations of motion via the (exact) time-dependent Einstein relation $2〈{D}_{\ensuremath{\mu}\ensuremath{\nu}}〉=\ensuremath{-}〈{A}_{\ensuremath{\mu}}{a}_{\ensuremath{\nu}}〉\ensuremath{-}〈{a}_{\ensuremath{\mu}}{A}_{\ensuremath{\nu}}〉+\frac{d〈{a}_{\ensuremath{\mu}}(t){a}_{\ensuremath{\nu}}(t)〉}{\mathrm{dt}}$, where 〈〉〈 〉〈〉 represents a reservoir average. The sources ${F}_{\ensuremath{\mu}}$, ${F}_{\ensuremath{\nu}}$ do not commute with one another, and as a result the commutation rules of the ${a}_{\ensuremath{\mu}}$ are shown to be preserved in time. The mean motion and diffusion coefficients are calculated for a harmonic oscillator, and for a set of atomic levels. We prove that two dynamically coupled systems (e.g., field and atoms) have uncorrelated Langevin forces if they are coupled to independent reservoirs. Radiation-field-atom coupling adds no new noise sources. We thus obtain simply the maser model including noise sources used in Quantum Noise V. Direct calculations of the mean motion and fluctuations in a system coupled to a reservoir yield relationships in agreement with the Einstein relation. For reservoirs violating time reversal, anomalous frequency shifts are found possible that violate the Ritz combination principle since $\ensuremath{\Delta}{\ensuremath{\omega}}_{12}+\ensuremath{\Delta}{\ensuremath{\omega}}_{23}+\ensuremath{\Delta}{\ensuremath{\omega}}_{31}$ need not vanish.
An exact ccc-number generalized Fokker-Planck equation is derived for six noncommuting variables: the field variables, atomic populations, and atomic polarization of a laser model treated extensively by Lax. Langevin procedures yield the correct Fokker-Planck terms. Adiabatic eliminations are performed on the classical Markoff process given by the Fokker-Planck equation to compare with previous results. It is found that higher-order diffusion coefficients influence the second-order ones slightly during adiabatic elimination.