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International Conference and School on Quantum Electronics: Laser Physics and Applications

Parametric four-photon mixing: exact analytical solutions in Jacobi functions

Anelia Dakova, Diana Dakova, Valeri I. Slavchev, L. M. Kovachev, Sofia Milenkova, Zara Kasapeteva

Published Jan 29, 2019• 2 citations
Publisher

Abstract

The problem with generating of new frequencies (signal and idler waves) in parametric four-photon processes without including the Raman scatter was solved only in approximation for a constant electric field or a constant intensity of the pump wave. The overall task which includes the change in intensity and the phase of the pump wave has not been studied in detail in the literature up to now. The main idea of present work is to solve the more general problem in which we have energy exchange between pump wave, signal and idler waves.

DOI
10.1117/12.2518961

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The problem with generating of new frequencies (signal and idler waves) in parametric four-photon processes without including the Raman scatter was solved only in approximation for a constant electric field or a constant intensity of the pump wave. The overall task which includes the change in intensity and the phase of the pump wave has not been studied in detail in the literature up to now. The main idea of present work is to solve the more general problem in which we have energy exchange between pump wave, signal and idler waves.
The equations for plane-wave steady-state propagation of n fields interacting via four wave processes in a Raman medium have been known since at least 1962. Complete analytic solutions have only been found for the process of harmonic generation. However, solutions have been obtained in the gain regime assuming zero pump depletion. Assuming all fields satisfy the phase matching condition we have found general analytic solutions to these equations. The solutions are complete in that they describe both pump deletion and saturation. Some example problems will be studied.
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.
A general theory of transient stimulated Raman scattering and the phase-matching diagrams are developed based on an interaction Hamiltonian which includes all the possible competing interaction channels. In a Raman system pumped by multiple crossing beams with multiple Stokes and anti-Stokes components, these channels are defined by the interactions among fields at different frequencies and/or k vectors of the pump pairs and Stokes and anti-Stokes fields. An ensemble-averaged equation of motion for the active phonon mode is derived within the Markovian approximation in which the phonon-phonon coupling (pressure effect) is reduced to an energy relaxation and a phase fluctuation. A complete set of transient coupled equations including phase modulation, Stark shift, medium (anharmonic) saturation, and all the possible four-wave-mixing channels is presented. In the transient regime, it is found that the effects of the Stark shift on the Raman efficiency are mainly governed by the input pump energy in contrast to the steady-state theory where the pump intensity plays the main role. Both analytic and numerical methods are developed in describing the nonlinear effects due to the energy feedback mechanism in the strong pump-depletion regime. Twelve possible four-wave-mixing channels are analyzed for nonlinear phase diagrams characterized by the angles among pump pairs, and Stokes and anti-Stokes fields. Finally, numerical examples are shown for the effects of the coherence lengths of high-order Stokes fields and pump pairs on the generation of low-order Stokes fields.
We present a full quantum analysis of resonant forward four-wave mixing based on electromagnetically induced transparency (EIT). In particular, we study the regime of efficient nonlinear conversion with low-intensity fields that has been predicted from a semiclassical analysis. We derive an effective nonlinear interaction Hamiltonian in the adiabatic limit. In contrast to conventional nonlinear optics, this Hamiltonian does not have a power expansion in the fields and the conversion length decreases with decreasing input power. We analyze the stationary wave-mixing process in the forward scattering configuration using an exact numerical analysis for up to 103{10}^{3}103 input photons and compare the results with a mean-field approach. Due to quantum effects, complete conversion from the two pump fields into the signal and idler modes is achieved only asymptotically for large coherent pump intensities or for pump fields in few-photon Fock states. The signal and idler fields are perfectly quantum correlated which has potential applications in quantum communication schemes. We also discuss the implementation of a single-photon phase gate for continuous quantum computation.