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Optics Letters

Quantum noise limits on frequency and timing for optical pulses in a linear high-gain amplifier

Victor V. Kozlov

Published Oct 1, 2002• 0 citations
Publisher

Abstract

Frequency and timing noise added in a linear optical amplifier of gain G grows as (G-1)/G. Provided that the pulse is initially in a coherent state, the output frequency and timing noise cannot exceed twice the input value. It is also shown that, in a transmission line with zero net gain and loss, the frequency and timing noise added in the lossy medium is equal to or larger than the noise added in the amplifier.

DOI
10.1364/ol.27.001723

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Frequency and timing noise added in a linear optical amplifier of gain G grows as (G-1)/G. Provided that the pulse is initially in a coherent state, the output frequency and timing noise cannot exceed twice the input value. It is also shown that, in a transmission line with zero net gain and loss, the frequency and timing noise added in the lossy medium is equal to or larger than the noise added in the amplifier.
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.
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.
The purpose of a phase-preserving linear amplifier is to make a small signal larger, regardless of its phase, so that it can be perceived by instruments incapable of resolving the original signal, while sacrificing as little as possible in signal-to-noise ratio. Quantum mechanics limits how well this can be done: A high-gain linear amplifier must degrade the signal-to-noise ratio; the noise added by the amplifier, when referred to the input, must be at least half a quantum at the operating frequency. This well-known quantum limit only constrains the second moments of the added noise. Here we derive the quantum constraints on the entire distribution of added noise: We show that any phase-preserving linear amplifier is equivalent to a parametric amplifier with a physical state for the ancillary mode; the noise added to the amplified field mode is distributed according to the Wigner function of the ancilla state.
In a linear amplifier, the output signal is linearly related to the input signal with an additive noise that is independent of the input. Here we show that entanglement between the input signal and the amplifying medium can produce exponentially-large decoherence in an ideal optical amplifier even when the gain is arbitrarily close to unity and the added noise is arbitrarily small. These nonlinear effects show that the usual input/output relationship of a 'linear' amplifier does not provide a complete description of its performance.
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.
We consider the fundamental noise properties of propagation through slow- and fast-light optical media based on gain or loss processes. For purely quantum mechanical reasons, any gain or loss process will add noise to a transmitted light field. We derive a relation between the noise figure describing the decreased signal-to-noise ratio of the transmitted laser pulse and the fractional delay or advancement of the pulse. We apply these results explicitly to the situation of operation on the line center of a gain or loss line. We find that for an ideal gain medium the noise figure never exceeds a factor of two. For a loss medium, there is no limit as to how large the noise figure can become. The increased noise in this case is the result of the random loss of photons from the optical field.