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Physical Review A

Quantum Theory of the Threshold and Growth of Optical Parametric Oscillation

P. N. Keating

Published Jan 1, 1971• 10 citations
Publisher

Abstract

The problem of the threshold and growth of parametric oscillation in a lossy optical cavity is treated quantum theoretically. It is shown that during the growth of oscillation each of the generated modes is statistically as chaotic as those observed previously in the lossless case, and is so chaotic that the additional uncertainty introduced by the loss processes has no further effect on the single-mode photon distributions. However, there is considerable cross correlation between the idler and signal modes. The cross correlation in the numbers of idler and signal quanta is essentially complete for large numbers, but is shown to be reduced when the numbers are smaller, because of noise due to loss. Correlations in phase are also investigated and it is shown that, under certain statistical conditions necessary for phase to be a relevant concept, the phase correlation is also essentially complete.

DOI
10.1103/physreva.3.180

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Journal of Modern Optics•Jul 1, 1988•1 citation
The problem of the threshold and growth of parametric oscillation in a lossy optical cavity is treated quantum theoretically. It is shown that during the growth of oscillation each of the generated modes is statistically as chaotic as those observed previously in the lossless case, and is so chaotic that the additional uncertainty introduced by the loss processes has no further effect on the single-mode photon distributions. However, there is considerable cross correlation between the idler and signal modes. The cross correlation in the numbers of idler and signal quanta is essentially complete for large numbers, but is shown to be reduced when the numbers are smaller, because of noise due to loss. Correlations in phase are also investigated and it is shown that, under certain statistical conditions necessary for phase to be a relevant concept, the phase correlation is also essentially complete.
A general approach for describing complex quantum-mechanical systems in terms of ccc-number variables is discussed on the basis of a formalism developed by Lax. We apply this formalism to the particular problem of noise fluctuations in parametric oscillators. From a microscopic Hamiltonian describing light interacting with optically active atoms in a cavity with loss, we arrive at an equation for the motion in terms of ccc-number variables. We then solve this system for the noise properties of the signal mode near threshold and above, when the pump and idler modes may be treated adiabatically, and compare our work with the results of Graham, who treated the signal and idler symmetrically.
We present a quantum-statistical analysis of the steady-state light fields in driven parametric oscillation in a cavity. Using the solution of a Fokker-Planck equation for the generalized PPP representation of the signal and idler modes, we calculate the mean photon numbers, second-order correlation functions, and intermode correlation function as functions of the driving field. The generalized PPP distribution describes the signal and idler modes' statistics over the whole range of driving-field strengths except in the region far below the oscillation threshold. The second-order correlation functions are found to violate the Cauchy-Schwartz inequality, a violation allowed because the PPP distribution is complex. Squeezing is also found in a linear combination of the signal and idler fields.
We present a quantum-theoretical treatment of biphoton generation in single-resonant type-II parametric down-conversion. The nonlinear medium is continuously pumped and is placed inside a cavity which is resonant for the signal field, but nonresonant for the idler deflected by an intracavity polarizing beam splitter. The intensity of the classical pump is assumed to be sufficiently low in order to yield a biphoton production rate that is small compared to the cavity loss rate. Explicit expressions are derived for the rate of biphoton generation and for the biphoton wave function. The output spectra of the signal and idler field are determined, as well as the second-order signal-idler cross-correlation function which is shown to be asymmetric with respect to the time delay. Due to frequency entanglement in the signal-idler photon pair, the idler spectrum is found to reveal the longitudinal mode structure of the cavity, even though the idler field is not resonant.
SUMMARY Parametric processes in the interaction of electromagnetic field with a medium in an optical cavity are well known to produce states of light with nonclassical properties. The quantum correlations between these states can be employed for a realization of an EinsteinPodolsky-Rosen (EPR) paradox [l], in the original formulation using continuous variables, as proposed [2] and experimentally demonstrated [3] in the case of Nondegenerate Parametric Amplification below threshold. In this paper we study EPR-like behaviour between the signal and idler fields emitted by a degenerate optical parametric oscillator (OPO) above threshold; these fields, generated by conversion of a pump photon of frequency 2w0 in two photons of frequency w. , are spatially separated in the far field, showing non-local quantum correlations. The system is described by the full Hamiltonian for the operators associated with the pump and signal fields; the optical cavity introduces proper damping constants and corresponding vacuum field operators [4,5]. Above the threshold for the generation of the signal field, we consider the fluctuations in the intensity difference and in the phase sum of the signal beams, which are related to their degree of mutual correlation. By linearizing around the stationary state, we calculate the spectral variances of these quantities (i.e. the Fourier transform of the two time correlation functions). The Heisenberg uncertainty principle sets a lower limit to the product of the spectral variances for the fluctuations in the intensity and in the phase of each individual signal field; by using the idler beam to infer results of measurements on the correlated signal beam, we have an EPR paradox if the product of the spectral variances for the fluctuations in the intensity difference and phase sum of the fields, is below the uncertainty limit. This feature is apparent for a wide interval around zero frequency in the far field of the active OPO system, depending on the parameter space; in particular there is maximum correlation between signal beams, and therefore maximum breaking of the uncertainty limit, if the damping constant of the signal field is larger than that of the pump field.
Using the quantum theory of a non-degenerate parametric amplifier, we discuss the phase correlations between signal and idler modes well above threshold. Particular attention is paid to the influence which fluctuation of the driving laser has on these correlations. We point out that the signal and idler mode phase correlation may be tuned by control of the driving laser field intensity and linewidth. For coherent driving, the mode phases are negatively cross-correlated. We stress the qualitative importance of these mode correlations for a class of two-photon resonant excitation experiments in atoms. In particular, the non-degenerate parametric oscillator represents a concrete model of a source with negatively cross-correlated phase fluctuations.