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Physical Review A

Photon Statistics of Phase-Locked Coupled Light Beams

A.L. Kestenbaum, Eli A. Mishkin

Published May 1, 1972• 3 citations
Publisher

Abstract

A quantum statistical analysis of phase-locked coupled light beams is presented with the coupling mechanism between the individual modes provided by the time-modulated dielectric constant of the crystal inserted in the cavity. The intense electric field that drives the modulator is treated classically. A Hamiltonian including the modulator interaction is formulated and the Heisenberg equations of motion of the mode operators are solved utilizing a generating function of the mode operators. The evolution in time of the phase-space distribution functions for single and multiple modes is derived under various initial conditions. Initially coherent oscillating modes are shown to conserve their coherent character. Below threshold, the modes are assumed to be chaotic Gaussian. It is shown that when all modes have the same average number of photons, they remain uncorrelated Gaussian. When the average photon numbers are nonidentical, the field remains a pure noise field, but the various modes are correlated. The analysis of the resonant-case mode of operation, properly interpreted, leads directly to the case of nonresonant modulation.

DOI
10.1103/physreva.5.2267

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•Jan 1, 1980•1 citation
A quantum statistical analysis of phase-locked coupled light beams is presented with the coupling mechanism between the individual modes provided by the time-modulated dielectric constant of the crystal inserted in the cavity. The intense electric field that drives the modulator is treated classically. A Hamiltonian including the modulator interaction is formulated and the Heisenberg equations of motion of the mode operators are solved utilizing a generating function of the mode operators. The evolution in time of the phase-space distribution functions for single and multiple modes is derived under various initial conditions. Initially coherent oscillating modes are shown to conserve their coherent character. Below threshold, the modes are assumed to be chaotic Gaussian. It is shown that when all modes have the same average number of photons, they remain uncorrelated Gaussian. When the average photon numbers are nonidentical, the field remains a pure noise field, but the various modes are correlated. The analysis of the resonant-case mode of operation, properly interpreted, leads directly to the case of nonresonant modulation.
The statistical properties of systems of coupled quantum-mechanical harmonic oscillators are analyzed. The Hamiltonian for the system is assumed to be an inhomogeneous quadratic form in the creation and annihilation operators, and is allowed to have an explicit time dependence. The relationship to classical theory is emphasized by expressing pure states in terms of the coherent-state vectors, and density operators by means of the PPP representation and an analogous representation involving the Wigner function. The state which evolves from an initially coherent state of the system is found, and equations governing the time evolution of the Wigner function and the weight function for the PPP representation are derived, in differential and integral form, for arbitrary initial states of the system. The results remain valid for couplings which do not preserve the vacuum state, and for cases in which the time dependence of the coupling parameters gives rise to large-scale amplification of the initial field intensities. The analysis is performed by first treating general linear inhomogeneous canonical transformations on the oscillator variables, and then specializing to the case in which these transformations represent the solutions for the Heisenberg operators in terms of their initial values. The results are illustrated within the context of a model of parametric amplification.
The quantum theory of the stimulated Raman and Brillouin effects, the parametric amplifier, and the frequency converter is developed, starting with a simple Hamiltonian for a system of three coupled field modes. The intense incident beam is treated classically. The evolution in time of the statistical properties of the quantized Stokes and anti-Stokes waves are analyzed by means of appropriate time-dependent phasespace distributions under various assumptions for the initial fields. Expansions of the field variables in terms of complete sets of orthogonal operators are discussed. Several regimes of operation are considered. It is shown that an initially coherent state develops, in general, into a superposition of a coherent state and a chaotic state. The amplitudes of the coherent components follow the same equations of motion as the mode operators. The chaotic components stem from the amplification of the vacuum fluctuations.
In this paper, the authors investigate the coupling of the radiations generated by a non-degenerate three-level cascade laser and a parametric oscillator in a cavity coupled with a two-mode squeezed vacuum reservoir, using a 50 : 50 beam splitter. They analyze the quantum statistical properties of the light superposed via the standard beam splitter with phase shift ϕ . When the beams of light superpose in phase ( ϕ = 0 ) and operate above threshold, they demonstrate that the mean and variance of photon number increase initially and remain constant over time. They also show that these quantities significantly rise with the linear gain coefficient when more atoms are initially prepared in the top energy level, and, regardless of the initial atomic preparation, increase with the driving amplitude of the parametric oscillator. Furthermore, at steady state and above threshold, the authors reveal that the superposed radiation, despite exhibiting super-Poissonian statistics, displays non-classical photon number correlations due to strong cross-correlations between the cavity modes of the coupled systems.
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We consider a system consisting of a large individual quantum dot with excitonic resonance coupled to a single-mode photonic cavity in the nonlinear regime when exciton-exciton interaction becomes important. Quantum statistics of coupled exciton-photon modes is studied for two regimes of driving: a monochromatic input field and a field with periodically time-modulated amplitude. We show that sub-Poissonian statistics for both modes are realized in the case of monochromatic driving for transient and steady-state regimes in the presence of decoherence and cavity-induced feedback. We also demonstrate that variances of quantum fluctuations of photon and exciton numbers display oscillations in the case of modulated input. In this case, we show an improvement of the degree of sub-Poissonian statistics and antibunching for both modes at periodic sequence of definite time intervals in comparison with the case of the steady-state regime for monochromatic driving. We also observe the Wigner functions with negative values in phase space for a time-modulated exciton-photon system.
Two-photon absorption and emission for a collection of two level atoms from a single mode beam of light are considered and the statistical proper- ties of the field after the interaction are investigated without placing any restriction of the population of the atomic levels . The dynamical behaviour of the field is described by assuming the atomic levels maintained with a nearly fixed popula- tion distribution . By applying iteration methods the time development for the density operator of the field is written and an expression for the coherent component is obtained . This component appears to be present only when there is a coherent component in the field before the interaction . The statistical properties of the field after the interaction, when the mean photon number is large initially, are then discussed and explicit calculations are carried out for some special cases of initial fields . Finally, it is shown that two-photon spontaneous emission field can be expressed as a superposition of chaotic fields .