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IEEE Journal of Quantum Electronics

The effect of pump coherence on frequency conversion and parametric amplification

B. Crosignani, Paolo Di Porto, U. Ganiel, S. Solimeno, Avigail Yariv

Published Sep 1, 1972• 8 citations
Publisher

Abstract

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.

DOI
10.1109/jqe.1972.1077283

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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.
A quantum-statistical treatment of degenerate (single-mode) parametric amplification is given in which the pump is allowed to have time-dependent amplitude and phase. This relaxes earlier restrictions of a constant pump in exact resonance with the signal. Solutions are obtained for the case where the ratio of the time derivative of the pump phase to the coupling strength between pump and signal is constant. This includes the special case of simple offresonance, where the pump has a constant frequency differing from twice the signal frequency by a given amount (and has constant amplitude). The Wigner distribution for the signal mode is obtained (the PPP distribution does not in general exist as a well-defined function beyond a finite time interval for this process) and is used to determine the time development of the expectation values of various field operators of interest, such as field strengths and fluctuations. A critical point exists in the solution for the photon annihilation operator such that the behavior of the solution and the resulting quantities which it determines is divided into an exponential region and an oscillatory region. The critical point defines, from the expectation value for the photon number, the condition for amplification. For the case of simple off-resonance, this gives the frequency range of amplification. A calculation without using the rotating-wave approximation is given in the Appendix which shows that, to first order, the same range is obtained as when the approximation is used. The effect of damping is given, showing, e.g., how it slows the growth rate of the signal (in the region of amplification) and how it narrows the frequency range of amplification. The effect of a time-dependent pump (and damping) on the breakdown of the PPP distribution is also described.
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.
A quantum theory of parametric amplification and frequency conversion of an optical image in coupled nonlinear optical processes that include one parametric amplification process at high-frequency pumping and two up-conversion processes in the same pump field is developed. The field momentum operator that takes into account the diffraction and group velocities of the waves is used to derive the quantum equations related to the spatial dynamics of the images during the interaction. An optical scheme for the amplification and conversion of a close image is considered. The mean photon number density and signal-to-noise ratio are calculated in the fixed-pump-field approximation for images at various frequencies. It has been established that the signal-to-noise ratio decreases with increasing interaction length in the amplified image and increases in the images at the generated frequencies, tending to asymptotic values for all interacting waves. The variance of the difference of the numbers of photons is calculated for various pairs of frequencies. The quantum entanglement of the optical images formed in a high-frequency pump field is shown to be converted to higher frequencies during the generation of sum frequencies. Thus, two pairs of entangled optical images are produced in the process considered.
A unified treatment of the description of the statistical dynamical properties of two-mode quadratic optical parametric processes with intense coherent or stochastic pumping (parametric amplification, frequency conversion, second subharmonic generation, etc. as special cases are involved) including the lossy mechanism is developed. The Heisenberg-Langevin equations are derived and their solutions are used to obtain the quantum characteristic functions and the corresponding quasidistributions. The Fokker-Planck equation approach is adopted as well and the equivalence of both the approaches is explicitly found.
Pumping a nonlinear crystal by an intense radiation results in the optical parametric generation of photons in two modes (the signal and the idler). The quantized electromagnetic field in these modes is described by a continuous-variable quantum state, which is entangled if the pump is a coherent state produced by a laser. The signal and the idler modes remain populated by photons even if the pump becomes incoherent (dephased by a medium, superposed with a thermal state, or produced by an alternative source such as the superluminescent diode). However, the incoherent pump does effect the entanglement and purity of the signal and the idler modes, which is of vital importance for quantum information applications and interferometry. Here we develop an approach to infer the signal-idler entanglement and purity for a general quantum incoherent pump with the given Glauber-Sudarshan function. We show that the signal-idler entanglement is extremely sensitive to the phase distribution of the pump and illustrate our findings by physically relevant examples of the incoherent pump: the noisy coherent state, slightly dephased and phase-averaged coherent states, the thermal state, and states modulated by the Kerr medium. The effect of an incoherent pump on the combined quadratures is discussed as well.