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Journal of Optics B Quantum and Semiclassical Optics

Sub-Poissonian photon statistics of higher harmonics: quantum predictions via classical trajectories

J. Bajer, Adam Miranowicz

Published May 11, 2000• 13 citations
Publisher

Abstract

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.

DOI
10.1088/1464-4266/2/3/102

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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.
Sub-Poissonian light generation in non-degenerate three-wave mixing is studied numerically and analytically within quantum and classical approaches. Husimi Q -functions and their classical trajectory simulations are analysed to reveal a special regime corresponding to the time-stable sub-Poissonian photocount statistics of the sum-frequency mode. Conditions for the observation of this regime are discussed. Theoretical predictions of the Fano factor and explanation of the extraordinary stabilization of the sub-Poissonian photocount behaviour are obtained analytically by applying the classical trajectories. Scaling laws for the maximum sub-Poissonian behaviour are found. Noise suppression levels in non-degenerate versus degenerate three-wave mixing are discussed on different time scales compared to the revival times. It is shown that the non-degenerate conversion offers much better stabilization of the suppressed noise in comparison to the degenerate process.
The squeezing and sub-Poissonian eects of light in third harmonic generation are investigated based on the fully quantum mechanical approach up to the rst order Hamiltonian interaction in gt, where g is the coupling constant between the modes per second and t is the interaction time between the waves during the process in a nonlinear medium.The coupled Heisenberg equations of motion involving real and imaginary parts of the quadrature operators are established.The occurrence of amplitude squeezing eects in both the quadratures of the radiation eld in the fundamental mode is investigated and found to be dependent on the selective phase values of the eld amplitude.The photon statistics of the pump mode in this process have also been investigated and found to be sub-Poissonian in nature.It is shown that for particular phase values the amplitude squeezing and sub-Poissonian photon statistics of light occur simultaneously.It is observed that there is no possibility to produce squeezed light in the harmonic mode up to rst-order interaction in gt.Further, it is found that the normal squeezing in the harmonic mode directly depends upon the amplitude-cubed squeezing of the initial pump eld to the case of second-order interaction in gt.This gives a method of converting higher-order (amplitude-cubed) squeezing of the fundamental mode into normal squeezing of the harmonic mode and vice versa.
We report the generation of sub-Poissonian pulses of light by means of traveling-wave second-harmonic generation (SHG) in a type II phase-matched nonlinear crystal. On direct detection, the quantum noise on the output light, which is polarized orthogonally to the input fundamental field, is measured to be below the shotnoise limit by as much as 0.3 +/- 0.2 dB (6 +/- 4%). We show that the input fundamental-power dependence of the measured Fano factor, direct-detection noise as a fraction of the shot-noise limit, is in qualitative agreement with the quantum theory of SHG.