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Physical Review A

Quantum limits on noise in dual input-output linear optical amplifiers and attenuators

R. Loudon, Ottavia Jedrkiewicz, Stephen Mark Barnett, John Jeffers

Published Mar 13, 2003• 5 citations

Abstract

The input-output relations for linear amplifiers and attenuators that have two input and two output channels are used to derive inequalities that relate their gain profiles and output noise spectra. The results generalize earlier derivations, which mainly focus their attention on single-channel devices, to the two-ended amplifiers and attenuators often used in practical communications systems. The present inequalities are satisfied by the results of previous calculations for specific model systems. It is shown that; in contrast to single-channel devices, a two-ended system can act as an amplifier for some input signals and an attenuator for others, even when all the signal frequencies are the same. The output from the two-channel amplifier has a minimum noise determined by the sum of the gains for both input channels, even when only one input channel is used and the other is in its vacuum state. The conditions on device construction needed to achieve equal gains for signals that arrive at the two ends of the device are determined. The present results reduce to those of single-channel theory in special cases where the two output channels are each separately fed by only one of the two input channels.

DOI: 10.1103/physreva.67.033803

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How much noise does quantum mechanics require a linear amplifier to add to a signal it processes? An analysis of narrow-band amplifiers (single-mode input and output) yields a fundamental theorem for phase-insensitive linear amplifiers; it requires such an amplifier, in the limit of high gain, to add noise which, referred to the input, is at least as large as the half-quantum of zero-point fluctuations. For phase-sensitive linear amplifiers, which can respond differently to the two quadrature phases ("$cos\ensuremath{\omega}t$" and "$sin\ensuremath{\omega}t$"), the single-mode analysis yields an amplifier uncertainty principle---a lower limit on the product of the noises added to the two phases. A multimode treatment of linear amplifiers generalizes the single-mode analysis to amplifiers with nonzero bandwidth. The results for phase-insensitive amplifiers remain the same, but for phase-sensitive amplifiers there emerge bandwidth-dependent corrections to the single-mode results. Specifically, there is a bandwidth-dependent lower limit on the noise carried by one quadrature phase of a signal and a corresponding lower limit on the noise a high-gain linear amplifier must add to one quadrature phase. Particular attention is focused on developing a multimode description of signals with unequal noise in the two quadrature phases.

Conventional amplifiers add quantum noise to a signal. Both amplifiers and attenuators lead to a reduction of the signal-to-noise ratio. We find, however, that devices exhibiting squeezed fluctuations in the gain or loss medium lead to an improved signal-to-noise ratio, in one quadrature of the output, compared to the limits associated with conventional devices. We investigate amplifier and attenuator noise properties using a simple model in which the gain or loss medium is a broad-band reservoir in a specially prepared, or ‘rigged’, state.

The noise properties of the two main kinds of linear optical amplifier, phase-insensitive and phase-sensitive, are reviewed, with particular emphasis on their minimum-noise limits. An optical communications link is considered, in which identical sections of attenuating fiber alternate with identical linear amplifiers. The limits on the overall signal-to-noise ratios of such chains are derived. More attention is paid to the signal-to-noise ratio appropriate to phase-sensitive detection, but some results for direct detection are also given.

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.

In an ideal linear amplifier, the output signal is linearly related to the input signal with an additive noise that is independent of the input. The decoherence of a quantum-mechanical state as a result of optical amplification is usually assumed to be due to the addition of quantum noise. Here we show that entanglement between the input signal and the amplifying medium can produce an exponentially-large amount of decoherence in an ideal optical amplifier even when the gain is arbitrarily close to unity and the added noise is negligible. These effects occur for macroscopic superposition states, where even a small amount of gain can leave a significant amount of which-path information in the environment. Our results show that the usual input/output relation of a linear amplifier does not provide a complete description of the output state when post-selection is used.

This chapter begins by specializing the general definition of optical amplifiers to a special class of linear amplifiers that includes all those in common use. Amplifiers are divided into the categories of phase insensitive or phase sensitive, and each of these categories is subdivided into the categories of traveling-wave or regenerative. Illustrative examples are given for each of the four types. This is followed by a general description of single-input, single-output linear amplifiers that includes the input-output equation, the conditions for phase insensitivity, and the constraints imposed by unitarity. This general theory is used to determine the noise limits for linear amplifiers. The general amplifier uncertainty principle obtained in this way shows that there are noise free amplifiers.