Correspondence between quantum and classical stochastic processes for a laser

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.

B. Shore

Joseph V. Pulé

Abstract It is found that the correlation functions of quantum stochastic processes are very similar to the corresponding functions for classical stochastic processes. It is shown also that if the state for the process satisfies the KMS boundary condition then this is reflected in the analyticity of the continuation of the correlation.

C. V. Heer

M. Mirzaee, B. Askari, Ali Motazedifard +1 more

Inspired by the Green's function (GF) approach in quantum field theory (QFT) and many body physics, we have developed a mathematical formalism to investigate classical multiplicative stochastic processes. Based on this approach, the interacting GF of any dynamical system subjected to classical stochastic noises, which enter into the system equations of motion in a multiplicative way, can be obtained from the noninteracting (free of noise) GF through an infinite perturbative series which may converge to an exact closed form under special conditions. Using this formalism, we have studied the effects of classical noises of the driving laser on the coherent population trapping (CPT) which have a crucial role in the performance of CPT-based atomic clocks. We have shown that if the bandwidth of the colored noise is sufficiently larger than the system damping rate, the infinite series corresponding to the interacting GF can be approximated by the closed form. The presented formalism enables us to investigate all kinds of homogeneous and inhomogeneous broadening mechanisms on the CPT transmission resonance lineshape, including dephasing due to atomic collisions, power/Doppler broadenings, as well as the broadening mechanisms due to phase and amplitude fluctuations of driving laser and compared their destructive effect with each other. It should be emphasized that the presented formalism is applicable to any dynamical system with multiplicative stochastic noises and the CPT phenomenon is just a prototype for the application of the presented formalism in practice.

Sergio A Albeverio, Giulio Casati, D. Merlini