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Physical Review A

Incoherent effects in the optical up-conversion process

W. J. Mielniczuk, Jacek Chrostowski

Published Mar 1, 1981• 10 citations
Publisher

Abstract

The theoretical treatment of the optical frequency up-conversion process with stochastic pumping is presented. This process consists of exchange of photons between two modes of different frequencies. It is assumed that the two modes are coupled by a short memory stochastic function. Statistically averaged occupation numbers of both modes were calculated exactly by means of the theory of multiplicative stochastic processes. The obtained nonperturbative solution sums secular terms and is valid for arbitrary times. The oscillatory-damped nature of the exchange of energy between two modes and the role of partial incoherence of the pump is discussed. It is shown that stochastic coupling causes the equipartition of energy between two modes for sufficiently long times. Analogies between an up-converter and a two-level atom driven by a single mode electromagnetic field are indicated. Optical Bloch equations for an up-conversion process are derived.

DOI
10.1103/physreva.23.1382

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The theoretical treatment of the optical frequency up-conversion process with stochastic pumping is presented. This process consists of exchange of photons between two modes of different frequencies. It is assumed that the two modes are coupled by a short memory stochastic function. Statistically averaged occupation numbers of both modes were calculated exactly by means of the theory of multiplicative stochastic processes. The obtained nonperturbative solution sums secular terms and is valid for arbitrary times. The oscillatory-damped nature of the exchange of energy between two modes and the role of partial incoherence of the pump is discussed. It is shown that stochastic coupling causes the equipartition of energy between two modes for sufficiently long times. Analogies between an up-converter and a two-level atom driven by a single mode electromagnetic field are indicated. Optical Bloch equations for an up-conversion process are derived.
The consequences of the deviation from ideal coherence of the driving (pump) field in nonlinear processes are examined. A detailed treatment of the processes of parametric amplification and frequency upconversion is presented. The treatment utilizes two different approximation methods to suit the two different physical situations considered. For slow phase variations of the pump field an adiabatic approach is useful, whereas in the treatment of fast phase variations (the nonadiabatic case) an equation describing the time evolution of thePrepresentation is found to be a good starting point. In this latter case, a closed hierarchy of equations for the moments of the photon number distribution in the signal (or idler) mode is found. In particular, we find that the efficiencies of both the amplification and the upconversion processes are reduced as a result of pump incoherence, the reduction becoming more severe in the highly incoherent case. For the frequency converter, the very nature of the process is modified as compared to the ideal case, and this modification is again of particular significance for the highly incoherent pump. Second-order moments are explicitly calculated and some of the statistical properties of the output signals are examined.
The finite coherence time of the driving field plays an important role in the rates of energy exchange taking place in nonlinear processes. As a typical example, the problems of the quantum parametric amplifier and frequency converter driven by a fluctuating pump are considered. A closed hierarchy of equations for the moments of the fields is given, which in turn is used for evaluating the average number of photons per mode and the relative variance as functions of time. Significant deviations from the case where the pump field is coherent are found.
The behavior of a simple theoretical model of a quantum-frequency converter with a time-dependent pump amplitude and phase is analyzed quantum mechanically. A sufficient condition ensuring periodic frequency conversion between the modes is found and exact solutions are given in these cases. The complete time-dependent density matrices which give the most complete statistical description of the system are presented for a variety of initial states of the system. In the general case where the sufficient condition is not satisfied, time-dependent-perturbation-theory results give corrections to the amplitude and the period of the energy exchange between the modes. Comparisons are made between these results and those of other authors.
The theory of optical Bloch equations with stochastic coefficients is discussed in detail. It is shown that Bloch equations with stochastic fields lead to multiplicative stochastic processes for which exact solutions can be obtained. The steady state and the transient regimes of the phase-diffusion model of the laser bandwidth are treated, and exact solutions of Bloch equations are obtained. The case of fluctuation due to collisions is also discussed. A simple stochastic model of radiation-induced collisions is presented. For each stochastic model of incoherences formulas for the longitudinal and transverse lifetimes are derived exactly. It is shown that the finite bandwidths of light or the collision incoherences do not, in general, have a simple additive effect in atomic equations of motion.