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Physical Review A

Quantum optics of traveling-wave attenuators and amplifiers

John Jeffers, N. Imoto, R. Loudon

Published Apr 1, 1993• 118 citations
Publisher

Abstract

We use a continuous-mode quantization scheme to derive relations between the output- and input-field operators for traveling-wave propagation along attenuating and amplifying optical fibers. These relations provide complete information on the temporal and longitudinal spatial developments of the signal field. They are used here to obtain the effects of propagation on the first and second moments of the photocount in direct detection and of the signal field measured in balanced homodyne detection. Some of the results are similar to those obtained for attenuation or amplification of standing waves in cavities, and, for example, the survival of any input squeezing still limits the maximum gain to twofold. There are, however, additional propagation effects for the traveling-wave system. Thus, in direct detection, it is necessary to take account of the changes in gain profile with propagation distance, and in homodyne detection there are fundamental quantum-mechanical restrictions on the minimum field uncertainties that can be achieved in measurements at separated space-time points. These uncertainty properties are derived in detail and illustrated by the example of a squeezed input signal.

DOI
10.1103/physreva.47.3346

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We use a continuous-mode quantization scheme to derive relations between the output- and input-field operators for traveling-wave propagation along attenuating and amplifying optical fibers. These relations provide complete information on the temporal and longitudinal spatial developments of the signal field. They are used here to obtain the effects of propagation on the first and second moments of the photocount in direct detection and of the signal field measured in balanced homodyne detection. Some of the results are similar to those obtained for attenuation or amplification of standing waves in cavities, and, for example, the survival of any input squeezing still limits the maximum gain to twofold. There are, however, additional propagation effects for the traveling-wave system. Thus, in direct detection, it is necessary to take account of the changes in gain profile with propagation distance, and in homodyne detection there are fundamental quantum-mechanical restrictions on the minimum field uncertainties that can be achieved in measurements at separated space-time points. These uncertainty properties are derived in detail and illustrated by the example of a squeezed input signal.
The properties of a linear optical amplifier or attenuator in which the input light is coupled to a collection of non-saturable atoms are considered. The photon-number factorial moments and probability distribution of the amplifier output are derived for arbitrary input statistics. Relations are obtained between the output and input variances in photon number, phase-angle cosine, and electric-field magnitude. The effects of amplification and attenuation on signal-to-noise ratio are obtained for both direct and homodyne detection. Particular attention is paid to the extents to which the non-classical properties of photon antibunching and squeezing are preserved by amplification and attenuation. It is found that both properties can at best survive only twofold intensity amplification.
Abstract Linear quantum input–output relation based models are widely applied to describe the light propagation in a lossy medium. The details of the interaction and the associated added noise depend on whether the device is configured to operate as an amplifier or an attenuator. Using the traveling wave (TW) approach, we generalize the linear material model to simultaneously account for both the emission and absorption processes and to have point-wise defined noise field statistics and intensity dependent interaction strengths. Thus, our approach describes the quantum input–output relations of linear media with net attenuation, amplification or transparency without pre-selection of the operation point. The TW approach is then applied to investigate materials at thermal equilibrium, inverted materials, the transparency limit where losses are compensated, and the saturating amplifiers. We also apply the approach to investigate media in nonuniform states which can be e.g. consequences of a temperature gradient over the medium or a position dependent inversion of the amplifier. Furthermore, by using the generalized model we investigate devices with intensity dependent interactions and show how an initial thermal field transforms to a field having coherent statistics due to gain saturation.
The theory of balanced homodyne and heterodyne detection is developed for inputs in which the signal field is in an arbitrary quantum state and the local-oscillator field is in a highly excited coherent state. Exact expressions are derived for the photocount moment-generating functions in the special case of a coherent signal. For more general signals, the first two moments of the photocount probability distribution are determined. The moments are evaluated for the examples of a coherent signal with a chaotic noise component, and for squeezed light derived from a degenerate and from a non-degenerate parametric amplifier. The corresponding moments for direct detection are obtained so that comparisons can be made. The Kelley-Kleiner photocount distribution formula is adapted to balanced detection schemes. Light beams are characterized throughout by their energy fluxes, and the theory accordingly describes steady-state experiments.
Starting with the canonical quantization procedure for the electromagnetic field inside an effective (linear or nonlinear) medium, we present a direct-space formulation of the theory of quantum optics. This approach does not use the conventional modal decomposition of the field, but relies on the electromagnetic momentum operator defined in terms of local electric- and magnetic-field operators. The momentum operator contains all the information on the spatial characteristics of the field, and can describe the translation of a short light pulse in a nonlinear medium, without a modal analysis of the pulse. Propagation is described through an operatorial wave equation that relates the temporal evolution of an electromagnetic pulse to its spatial progression. Through this equation, the direct-space approach to quantum optics can treat traveling-wave nonlinear-optical phenomena and, at the same time, account for their quantum statistics. The theory is applied to squeezed-light generation by the parametric down-conversion of a short laser pulse, as an illustration.
Introduction In previous chapters we studied wave propagation in fiber OPAs from a classical standpoint, i.e. starting from Maxwell's equations. That approach provides only a partial view of the rich nature of the electromagnetic field, namely its wave nature. However, a complete description of the electromagnetic field requires quantum mechanics. This quantum description of the electromagnetic field (called quantum electrodynamics, or QED) predicts fluctuations in measurable quantities due to the Heisenberg uncertainty principle and sets ultimate limits on the performance of devices such as optical amplifiers. A remarkable result of this theory is that the statistics and thus the noise performance of a system are affected by the quantum statistics of the source and by the physical measurement that is made. Thus a different analysis is needed depending on whether homodyne, heterodyne, or direct detection (where the field simply impinges on a photodiode) is used. Direct detection reveals the particle nature of the electromagnetic field. More specifically, energy arrives in discrete units, called photons, each of which may create an electron–hole pair in a semiconductor photodetector. In particular, the number of photons in an optical pulse can be thought of as a random variable having a mean and a variance. By deriving the mean and variance of the photocurrent that would be present if one performed direct detection of ideal laser light at the input of an optical amplifier and then a similar measurement at its exit, one can quantify the degradation of the signal-to-noise ratio (SNR) caused by the amplifier.