Quantum Noise. IV. Quantum Theory of Noise Sources

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.