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Quantum Optics Journal of the European Optical Society Part B

Number and phase quantum fluctuations in second harmonic generation

R. Tanaś, Tserensodnom Gantsog, Roman Zawodny

Published Aug 1, 1991• 20 citations
Publisher

Abstract

Photon number and phase fluctuations and correlations in the second harmonic generation are discussed. The new Pegg-Barnett Hermitian phase formalism is used to deal with the phase properties of the field. The method of numerical diagonalization of the interaction Hamiltonian is applied to find the state evolution and, consequently, the number and phase quantum fluctuations. It is shown that the joint phase probability distribution evolves into a multi-peak structure indicating clearly the transition from the second harmonic to the down-conversion regime, and back. The evolution of all relevant quantities is illustrated graphically, and their dependence on the mean number of photons of the initial field is shown explicitly. The number-phase uncertainty product for the fundamental mode is plotted making evident the abrupt transition from the low to the high level of quantum fluctuations.

DOI
10.1088/0954-8998/3/4/004

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Photon number and phase fluctuations and correlations in the second harmonic generation are discussed. The new Pegg-Barnett Hermitian phase formalism is used to deal with the phase properties of the field. The method of numerical diagonalization of the interaction Hamiltonian is applied to find the state evolution and, consequently, the number and phase quantum fluctuations. It is shown that the joint phase probability distribution evolves into a multi-peak structure indicating clearly the transition from the second harmonic to the down-conversion regime, and back. The evolution of all relevant quantities is illustrated graphically, and their dependence on the mean number of photons of the initial field is shown explicitly. The number-phase uncertainty product for the fundamental mode is plotted making evident the abrupt transition from the low to the high level of quantum fluctuations.
For the process of intracavity second-harmonic generation in the region of unstable dynamics, the joint fluctuations of numbers of photons and phases of the fundamental mode and second harmonic are studied in positive P representation. It is shown that above the bifurcation point of the optical system the distribution functions of the phases of the fundamental mode and second harmonic have a two-component structure. The quantum trajectories of the phases of these modes are not localized in the components of their states, which demonstrates the presence of quantum-mechanical interference between these components. In both components of each state the same values of numbers of photons are realized. It is shown that in the case of coherent initial states of both modes the joint distribution functions of the numbers of photons and phases are in the oscillatory regime. With time, the modes pass from a two-component delocalized state to a one-component one and then back to the two-component state. The time of oscillation of the distribution function is of the order of the photon's lifetime in the cavity volume. Self-matching of phases of interacting modes is investigated above the bifurcation point.
We present a comprehensive and self-consistent theory of the relative phase between fundamental and harmonic modes in the nonlinear process of second-harmonic generation. We introduce an adequate polar decomposition of the field amplitudes that leads to a truly Hermitian relative-phase operator, whose eigenstates correctly describe the phase properties of the fields. We find the probability distribution for the relative phase and, by using a numerical diagonalization of the interaction Hamiltonian, we study its time evolution. This evolution shows quite different asymptotic behaviours for long times depending on the total number of quanta.
The dynamics of correlation function of quantum fluctuations of quadrature amplitudes, as well as the dynamics of correlation function of quantum fluctuations of photon-numbers of fundamental and the third-harmonics mode are investigated for the process of intracavity generation of the third-harmonic. In the case of the evolution of system from initial vacuum state of both modes into the region of great times of interaction, both correlation functions tend to the stationary value. It is shown that these correlations depend strongly on the non-linear coupling coefficient between these modes. In the case of small coupling coefficients the correlation of fluctuations of the quadrature amplitudes as well as the correlation of fluctuations of photon-number are weak. As the coupling coefficient is rising both correlations are also rising (the state of the sub-systems becomes more entangled by the corresponding variables). By further rise of the coupling coefficient the correlation of quadrature amplitudes of fluctuations vanishes (the entanglement of subsystem states over the quadrature amplitudes is also vanishing) but the correlation of fluctuations of the photon-numbers is not vanishing (the subsystems remain entangled over the photon-number variables).
Using Pegg-Barnett formalism the author has studied the phase properties of displaced number states. It has been found that the phase probability distribution exhibits the same oscillations as the photon number distribution. Using the Hermitian phase operator the fluctuations of the phase of the displaced number states have been studied. The dependence of the fluctuations on the value of the reference phase have been analysed in detail. It has been found that phase of displaced number states exhibit squeezing in the high-intensity limit of the driving field. These states are approximately equivalent to phase states with precisely defined phase. It has been shown that displaced number states are not minimum uncertainty states with respect to the phase and the number operators.