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Physical Review A

Generation and amplification of number states by nondegenerate parametric oscillators with idler-measurement feedback

Gunnar G. Bjork, Y. Yamamoto

Published Jan 1, 1988• 51 citations
Publisher

Abstract

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.

DOI
10.1103/physreva.37.125

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Annual Meeting Optical Society of America•Jan 1, 1992•0 citations
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.
If the idler photon flux is continuously measured in a photon-counting detector and the photocurrent fluctuations are fed back to control the pump intensity, the output signal state will be an amplitude-squeezed state (near-number state) at low frequencies and high feedback loop gain. Electromagnetic field in number-phase minimum-uncertainty states (amplitude- squeezed states), and its limit (photon-number states) have potential applications in optical communication, precision measurement, and atomic spectroscopy, and gravitational wave detection. These states have been the focus of considerable attention.
A quantum analysis of the above-threshold intensity fluctuations in a nondegenerate parametric oscillator with direct-detection feedback onto the pump amplitude is presented. We derive a master equation for the signal (in-loop) and idler (out-of-loop) modes by adiabatically eliminating the pump mode and incorporating a feedback term, using the Wiseman-Milburn quantum feedback theroy [Phys. Rev. Lett. 70, 548 (1993)]. In the absence of feedback and far above threshold, we find that both beams are 50% intensity squeezed. For small negative (positive) feedback, the intensity fluctuations in the out-of-loop (in-loop) beam are reduced further. For larger values of feedback, the fluctuations grow, the fields eventually becoming unsqueezed.
The phase-fluctuation correlation between the signal and idler fields in nondegenerate parametric optical amplifiers and oscillators is analyzed. It is shown that close to the threshold pump power, the correlation is sufficient to reduce the phase fluctuations below those of a coherent state. The phase-fluctuation correlation is negative due to the simultaneous phase matching and energy conservation. This restricts the usefulness of the correlation. The pump source's phase noise will furthermore set a limit to the minimum achievable phase spectral density. The conditions for surpassing the standard quantum limit are elaborated. Finally, applications to coherent optical communication and gravitational wave detection are discussed.
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.
Signal and idler waves produced by a nondegenerate parametric amplifier feature strong positive correlation between their photon numbers, or in-phase amplitudes. They feature equally strong negative correlation between their phases, or quadrature-phase amplitudes. These photon twins can produce arbitrarily squeezed states via state reduction by appropriate measurements of an idler wave. When the quadrature amplitude of an idler wave is measured, the signal wave is collapsed to a quadrature-amplitude squeezed state. When the two-quadrature amplitudes of an idler wave are measured simultaneously, the signal wave becomes a coherent state.
Quantum fluctuations in an optical amplifier are of fundamental as well as technological significance and lead to basic limitations on noise performance. It is well-known that the signal-to-noise ratio (SNR) of an injected signal is degraded in the amplification process due to added noise from thermal or (at best) vacuum fluctuations of the internal modes of the amplifier.[1] However, if the internal modes are coupled to a squeezed vacuum, the added noise in amplification can be reduced.[2] With this motivation, we have constructed a nondegenerate parametric amplifier whose idler mode is itself coupled to a squeezed vacuum, which is generated by a second parametric amplifier. Phase-sensitive noise amplification of a vacuum input to the signal mode is observed with the minimum amplified noise level 0.5dB below the usual phase-insensitive amplified noise level for the amplified signal of the same amplifier but with the idler mode coupled to a vacuum state. Prospects for improvement by increasing the squeezing that is coupled into the idler mode of the amplifier beyond the current level of 2dB as well as surprising results from our theoretical analysis which includes losses will be discussed.