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Physical Review A

Noise Calculations for Optical Parametric Oscillators

David R. White, W. H. Louisell

Published May 1, 1970• 16 citations
Publisher

Abstract

A general approach for describing complex quantum-mechanical systems in terms of $c$-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 $c$-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.

DOI
10.1103/physreva.1.1347

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Physical Review A•Jan 1, 1988•51 citations
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.
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 quantum analysis of the intensity fluctuations is given for the nondegenerate parametric oscillator both above and below threshold. Theoretical analysis and experiments by Heidmann, Horowicz, Reynaud, Giacobino, and Fabre [Phys. Rev. Lett. 59, 2555 (1987)] have shown that a noise suppression below the vacuum level is possible in the difference of the intensities of the signal and idler modes. The effect of nonequal cavity decay rates and intracavity losses is shown to modify significantly the fluctuation spectra for the difference in the intensity of the signal and idler beams. The results of exact nonlinear solutions for the internal cavity modes are discussed. We demonstrate that a very sensitive absorption detector may be made by operating the oscillator near threshold.
Most treatments of noise characteristics of optical parametric oscillators (OPOs) have been based on linearization of the equations of motion. This approach, valid far below and far above threshold, breaks down in the threshold region as it ignores the nonlinearity of interaction. We present a treatment that takes into account both the nonlinearity and the quantum mechanical nature of noise in these oscillators. Analytical expressions for the steady-state amplitude distribution functions are derived. Both the degenerate and the nondegenerate OPOs are considered. These steady-state distributions are used to study the coherence properties of the OPOs in the threshold region. These studies bridge the noise characteristics of the OPOs below and above threshold. Results are compared with classical oscillators and differences and similarities are discussed. Special attention is given to quantum mechanical features that survive even as the oscillator amplitude grows and reaches macroscopic proportions.
We derive the fundamental noise properties of a parametric amplifier (paramp) which is driven by a monochromatic pump but which has many lossy signal and idler modes. We use a quantum-mechanical approach similar to that employed by Louisell and co-workers in their treatment of paramps and lasers with lossless modes. Our results are not directly deducible from the lossless mode analysis, but they do give the same limiting noise temperature as that analysis. This minimum noise temperature is the same as that of an ideal laser. In fact, we find that one can always design a laser whose field correlation functions would be the same as those of any given paramp. Many results for laser noise in various configurations can be carried over directly and applied to parametric amplifiers by using correspondence substitutions which we set down. For example, in analogy to Pound's description of the maser (in terms of a Nyquist theorem extended to negative temperatures), the calculation of paramp noise is found to be equivalent to applying Nyquist's theorem to both the active and passive elements in the (classically computed) equivalent circuit of a signal mode of interest. From this one obtains the spectra of effective noise generators which, when amplified in the circuit, duplicate the "quantum" as well as thermal noise that is present in the output. The Nyquist theorem is extended to apply to each active element by simply taking the element temperatures to be minus the real temperature of the corresponding idler mode (whose parametric coupling into the signal circuit gives rise to that element) multiplied by the ratio of the pump minus the idler to the idler frequencies.
Noise spectra of the in-phase and quadrature-phase components of the signal and idler fields of a nondegenerate optical parametric amplifier are calculated. It is shown that, as is suggested by the operator Manley-Rowe relations, the external in-phase (amplitude) fluctuations of the signal and idler fields are correlated in the frequency region lower than the signal and idler cavity bandwidths. If the idler photon flux is continuously measured and the photocurrent fluctuations are fed back to control the pump intensity, the signal output will feature an amplitude noise spectral density reduced to below the standard quantum limit (amplitude squeezing) in the frequency region below the feedback loop and signal and idler cavity bandwidths. An all-optical realization of such a negative-feedback parametric oscillator is presented. It is also shown that a nondegenerate parametric amplifier with idler measurement feedback can add a fixed number of photons to a photon-number state.