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Journal of Optics B Quantum and Semiclassical Optics

Sub-Poissonian behaviour in the second harmonic generation

JiríBajer, Ondřej Haderka, Jan Peřina

Published Aug 13, 1999• 12 citations
Publisher

Abstract

A special class of quantum solutions of the second harmonic generation process exhibiting time-stable sub-Poissonian behaviour with Fano factor F 0.83 is numerically analysed. A theoretical prediction of that specific Fano number and an explanation of the extraordinary time stability of the sub-Poissonian behaviour is given using the semiclassical method of classical trajectories.

DOI
10.1088/1464-4266/1/5/306

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A special class of quantum solutions of the second harmonic generation process exhibiting time-stable sub-Poissonian behaviour with Fano factor F 0.83 is numerically analysed. A theoretical prediction of that specific Fano number and an explanation of the extraordinary time stability of the sub-Poissonian behaviour is given using the semiclassical method of classical trajectories.
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Second-harmonic generation in the no-energy-transfer regime can be a source of quasi-stationary sub-Poissonian light as was recently shown by Bajer et al ( Bajer J, Haderka O and Perina J 1999 J. Opt. B: Quantum Semiclass. Opt. 1 529 ). We generalize their results for higher-harmonic generation by applying the numerical method of Hamiltonian diagonalization and the analytical semiclassical description of classical trajectories. The quasi-stationary behaviour of the sub-Poissonian photocount noise in the no-energy-transfer regime is explained. An approximate formula for the Fano factor is derived for arbitrary harmonics. It is predicted that the deepest quasi-stationary reduction of photocount noise in the no-energy-transfer regime is achieved in the third-harmonic generation.
In this review, we compare different descriptions of photon-number statistics in harmonic generation processes within quantum, classical and semiclassical approaches. First, we study the exact quantum evolution of the harmonic generation by applying numerical methods including those of Hamiltonian diagonalization and global characteristics. We show explicitly that the harmonic generations can indeed serve as a source of nonclassical light. Then, we demonstrate that the quasi-stationary sub-Poissonian light can be generated in these quantum processes under conditions corresponding to the so-called no-energy-transfer regime known in classical nonlinear optics. By applying method of classical trajectories, we demonstrate that the analytical predictions of the Fano factors are in good agreement with the quantum results. On comparing second and higher harmonic generations in the no-energy-transfer regime, we show that the highest noise reduction is achieved in third-harmonic generation with the Fano-factor of the third harmonic equal to 13/16.
In this, the first of two papers on the subject, we present an analysis of a non-linear optical system with a mode coupled to its second harmonic, exhibiting a rich variety of non-equilibrium steady-state transitions. Here we use a semiclassical analysis to examine the steady states and their stability, and thus predict the existence of various instabilities. We find that this system exhibits both first- and second-order phase transition-like behaviour, as well as having a hard mode transition. The application of this system as a switching device is also considered.
The quantum statistical equations for the field produced in the process of second-harmonic generation (SHG) are expressed, based on a method proposed for four-wave mixing.1 We start from the following assumptions: 1) the field absorption in the nonlinear medium may be neglected; 2) the amplitude of the fundamental field is much greater than that of the SH field; 3) the Wigner function associated with the density operator of the field depends significantly only on the variables of the SH field. The coefficients of the Fokker–Planck equation thus obtained depend on the Laplace transforms of the correlation functions of the atomic polarization operator. The moments of the Wigner function2 are related to the moments of the SH field photon number; consequently, we impose the condition for sub-Poissonian photon-statistics and derive the corresponding relationships satisfied by the atomic correlation functions. A numerical interpretation is given.