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Physical Review

Quantum Noise. IV. Quantum Theory of Noise Sources

M. Lax

Published May 6, 1966• 451 citations
Publisher

Abstract

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.

DOI
10.1103/physrev.145.110

Related papers

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

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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 $b$, ${b}^{\ifmmode\dagger\else\textdagger\fi{}}$ and the population variables ${N}_{1}$ and ${N}_{2}$ 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 $P$ is replaced by Langevin equations for $\ensuremath{\beta}$, ${\ensuremath{\beta}}^{*}$, ${N}_{1}$, and ${N}_{2}$ 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 $P(I)$ valid at all operating levels, and (b) to the phase linewidth. The transformation to new variables $N=\frac{1}{2}(m+n)$, $k=m\ensuremath{-}n$, 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 $\ensuremath{\rho}(N)$.

Physical Review•Sep 10, 1969•99 citations

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

M. Lax

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Physical Review•May 10, 1967•214 citations

NOISE AND DISSIPATION IN QUANTUM THEORY

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Reviews in Mathematical Physics•Jan 1, 1990•40 citations

Quantum Noise. XI. Multitime Correspondence between Quantum and Classical Stochastic Processes

M. Lax

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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We derive and evaluate equations of motion for the mean values and variances of the components of spins collectively coupled to a broadband squeezed radiation reservoir. Our formalism bridges between a single two-level emitter, represented by spin components that relax at different rates, depending on the degree of squeezing, to an ensemble of emitters represented by a large collective spin, whose components relax independently of the squeezing. For a single spin, the steady-state fluctuations in the transverse components are independent of the squeezing, while the steady state of a large spin ensemble reflects the statistics of the squeezed reservoir. This follows from an analysis of the Langevin noise contributions to the equations of motion and their consequences for the first and second moments of the spin operators. We argue that the difference between a single and many spins is related to whether vacuum fluctuations or radiation reaction dominate the coupling of the spin system to the radiation environment.

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How Good Is Langevin

N. G. van Kampen

UNLABELLED: The Langevin equation is universally applied to describe noise in physical systems of all kinds. To derive it, rather than postulate it, one has to introduce a noise source such as an environment or "reservoir". The solution requires that at some initial time the reservoir is in equilibrium and uncorrelated with the system. This is unphysical and, moreover, in the quantum mechanical case leads to inconsistencies. The way to overcome this anomaly is to consider the equilibrium of system and noise source combined. Then the resulting time-dependent equations for the correlation of various operators can be given. To find out whether they can be reproduced by a Langevin equation, the result is applied to the explicitly solvable system of a harmonic oscillator in a harmonic bath. CONCLUSION: Langevin cannot describe quantum noise but belongs to the realm of mesoscopic physics.

The Journal of Physical Chemistry B•Sep 28, 2005•7 citations
For a quantum system describable in a Markoffian way, via a set of variables a=a1,a2,\ensuremath⋯\mathrm{a}={{a}_{1}, {a}_{2}, \ensuremath{\cdots}}a=a1​,a2​,\ensuremath⋯, we show that the Langevin noise sources F\ensuremathμ{F}_{\ensuremath{\mu}}F\ensuremathμ​ in the operator equations of motion da\ensuremathμdt=A\ensuremathμ(a)+F\ensuremathμ\frac{d{a}_{\ensuremath{\mu}}}{\mathrm{dt}}={A}_{\ensuremath{\mu}}(\mathrm{a})+{F}_{\ensuremath{\mu}}dtda\ensuremathμ​​=A\ensuremathμ​(a)+F\ensuremathμ​ possess second moments 〈F\ensuremathμ(t)F\ensuremathν(u)〉=2〈D\ensuremathμ\ensuremathν(a,t)〉\ensuremathδ(t\ensuremath−u)〈{F}_{\ensuremath{\mu}}(t){F}_{\ensuremath{\nu}}(u)〉=2〈{D}_{\ensuremath{\mu}\ensuremath{\nu}}(\mathrm{a}, t)〉\ensuremath{\delta}(t\ensuremath{-}u)〈F\ensuremathμ​(t)F\ensuremathν​(u)〉=2〈D\ensuremathμ\ensuremathν​(a,t)〉\ensuremathδ(t\ensuremath−u). The diffusion coefficients D\ensuremathμ\ensuremathν{D}_{\ensuremath{\mu}\ensuremath{\nu}}D\ensuremathμ\ensuremathν​ can be determined from a knowledge of the mean equations of motion via the (exact) time-dependent Einstein relation 2〈D\ensuremathμ\ensuremathν〉=\ensuremath−〈A\ensuremathμa\ensuremathν〉\ensuremath−〈a\ensuremathμA\ensuremathν〉+d〈a\ensuremathμ(t)a\ensuremathν(t)〉dt2〈{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}}2〈D\ensuremathμ\ensuremathν​〉=\ensuremath−〈A\ensuremathμ​a\ensuremathν​〉\ensuremath−〈a\ensuremathμ​A\ensuremathν​〉+dtd〈a\ensuremathμ​(t)a\ensuremathν​(t)〉​, where 〈〉〈 〉〈〉 represents a reservoir average. The sources F\ensuremathμ{F}_{\ensuremath{\mu}}F\ensuremathμ​, F\ensuremathν{F}_{\ensuremath{\nu}}F\ensuremathν​ do not commute with one another, and as a result the commutation rules of the a\ensuremathμ{a}_{\ensuremath{\mu}}a\ensuremathμ​ 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 \ensuremathΔ\ensuremathω12+\ensuremathΔ\ensuremathω23+\ensuremathΔ\ensuremathω31\ensuremath{\Delta}{\ensuremath{\omega}}_{12}+\ensuremath{\Delta}{\ensuremath{\omega}}_{23}+\ensuremath{\Delta}{\ensuremath{\omega}}_{31}\ensuremathΔ\ensuremathω12​+\ensuremathΔ\ensuremathω23​+\ensuremathΔ\ensuremathω31​ need not vanish.
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 $\ensuremath{\rho}(N)$.
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 bbb, ${b}^{\ifmmode\dagger\else\textdagger\fi{}}$ and the population difference DDD. The corresponding equation obeyed by the density matrix ${\ensuremath{\rho}}_{\mathrm{mn}}(D, t)$ in the "nnn 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 bbb 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′sb'\mathrm{s}b′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 III, $\ensuremath{\varphi}$, DDD, 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 DDD. 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 DDD. The equation for $\ensuremath{\rho}(b, {b}^{\ifmmode\dagger\else\textdagger\fi{}}, t)$, rewritten in the nnn representation agrees with corresponding results of Scully and Lamb.
A model independent generalization of quantum mechanics, including the usual as well as the dissipative quantum systems, is proposed. The theory is developed deductively from the basic principles of the standard quantum theory, the only new qualitative assumption being that we allow the wave operator at time t of a quantum system to be non-differentiable (in t) in the usual sense, but only in an appropriately defined (Sec. 5) stochastic sense. The resulting theory is shown to lead to a natural generalization of the usual quantum equations of motion, both in the form of the Schrödinger equation in interaction representation (Sec. 6) and of the Heisenberg equation (Sec. 8). The former equation leads in particular to a quantum fluctuation-dissipation relation of Einstein’s type. The latter equation is a generalized Langevin equation, from which the known form of the generalized master equation can be deduced via the quantum Feynmann-Kac technique (Secs. 9 and 10). For quantum noises with increments commuting with the past the quantum Langevin equation defines a closed system of (usually nonlinear) stochastic differential equations for the observables defining the coefficients of the noises. Such systems are parametrized by certain Lie algebras of observables of the system (Sec. 10). With appropriate choices of these Lie algebras one can deduce generalizations and corrections of several phenomenological equations previously introduced at different times to explain different phenomena. Two examples are considered: the Lie algebra [q, p]=i (Sec. 12), which is shown to lead to the equations of the damped harmonic oscillator; and the Lie algebra of SO(3) (Sec. 13) which is shown to lead to the Bloch equations. In both cases the equations obtained are independent of the model of noise. Moreover, in the former case, it is proved that the only possible noises which preserve the commutation relations of p, q are the quantum Brownian motions, commonly used in laser theory and solid state physics.
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.
We derive and evaluate equations of motion for the mean values and variances of the components of spins collectively coupled to a broadband squeezed radiation reservoir. Our formalism bridges between a single two-level emitter, represented by spin components that relax at different rates, depending on the degree of squeezing, to an ensemble of emitters represented by a large collective spin, whose components relax independently of the squeezing. For a single spin, the steady-state fluctuations in the transverse components are independent of the squeezing, while the steady state of a large spin ensemble reflects the statistics of the squeezed reservoir. This follows from an analysis of the Langevin noise contributions to the equations of motion and their consequences for the first and second moments of the spin operators. We argue that the difference between a single and many spins is related to whether vacuum fluctuations or radiation reaction dominate the coupling of the spin system to the radiation environment.
UNLABELLED: The Langevin equation is universally applied to describe noise in physical systems of all kinds. To derive it, rather than postulate it, one has to introduce a noise source such as an environment or "reservoir". The solution requires that at some initial time the reservoir is in equilibrium and uncorrelated with the system. This is unphysical and, moreover, in the quantum mechanical case leads to inconsistencies. The way to overcome this anomaly is to consider the equilibrium of system and noise source combined. Then the resulting time-dependent equations for the correlation of various operators can be given. To find out whether they can be reproduced by a Langevin equation, the result is applied to the explicitly solvable system of a harmonic oscillator in a harmonic bath. CONCLUSION: Langevin cannot describe quantum noise but belongs to the realm of mesoscopic physics.