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Quantum Information and Measurement (QIM)

Phase uncertainty in quantum linear amplifiers beyond the small-noise approximation

A. Chia, Michal Hajdušek, Rosario Fazio, L. C. Kwek, Vlatko Vedral

Published Jan 1, 2019• 0 citations
Publisher

Abstract

We estimate the output phase uncertainty of linear amplifiers in a practical regime (weak input, ideal amplification, and large gain) without the small-noise assumption. Furthermore, the small-noise assumption is shown to fail in this regime.
DOI
10.1364/qim.2019.t5a.47

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The phase of an optical field inside a linear amplifier is widely known to diffuse with a diffusion coefficient that is inversely proportional to the photon number. The same process occurs in lasers which limits its intrinsic linewidth and makes the phase uncertainty difficult to calculate. The most commonly used simplification is to assume a narrow photon-number distribution for the optical field (which we call the small-noise approximation). For coherent light, this condition is determined by the average photon number. The small-noise approximation relies on (i) the input to have a good signal-to-noise ratio, and (ii) that such a signal-to-noise ratio can be maintained throughout the amplification process. Here we ask: For a coherent input, how many photons must be present in the input to a quantum linear amplifier for the phase noise at the output to be amenable to a small-noise analysis? We address these questions by showing how the phase uncertainty can be obtained without recourse to the small-noise approximation. It is shown that for an ideal linear amplifier (i.e. an amplifier most favourable to the small-noise approximation), the small-noise approximation breaks down with only a few photons on average. Interestingly, when the input strength is increased to tens of photons, the small-noise approximation can be seen to perform much better and the process of phase diffusion permits a small-noise analysis. This demarcates the limit of the small-noise assumption in linear amplifiers as such an assumption is less true for a nonideal amplifier.
The phase of an optical field inside a linear amplifier is widely known to diffuse with a diffusion coefficient that is inversely proportional to the photon number. The same process occurs in lasers which limits its intrinsic linewidth and makes the phase uncertainty difficult to calculate. The most commonly used simplification is to assume a narrow photon-number distribution for the optical field (which we call the small-noise approximation). This relies on (i) the input to have a good signal-to-noise ratio, and (ii) that such a signal-to-noise ratio can be maintained throughout the amplification process. Here we ask: How many photons must be present in the input to a quantum linear amplifier for the phase noise to be amenable to a small-noise analysis? We show that for a linear amplifier which is signal-to-noise optimised (i.e. an amplifier most favourable to the small-noise approximation), the small-noise approximation breaks down for a coherent-state input with only a few photons on average. Interestingly, when the input strength is increased to tens of photons, the small-noise approximation can be seen to perform much better and the process of phase diffusion permits a small-noise analysis. This demarcates the limit of the small-noise assumption in linear amplifiers as such an assumption is less true for an amplifier which does not have its signal-to-noise performance optimised. In order to explore this limit we propose a description of phase diffusion that goes beyond the small-noise approximation which we call the inverse-number expansion. The small-noise approximation can then be understood as the lowest-order contribution to the inverse-number expansion while phase diffusion with only a few-photon input can be captured by the inclusion of higher-order terms.
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.
Amplification of quantum states is inevitably accompanied with the introduction of noise at the output. For protocols that are probabilistic with heralded success, noiseless linear amplification in theory may still be possible. When the protocol is successful, it can lead to an output that is a noiselessly amplified copy of the input. When the protocol is unsuccessful, the output state is degraded and is usually discarded. Probabilistic protocols may improve the performance of some quantum information protocols, but not for metrology if the whole statistics is taken into consideration. We calculate the precision limits on estimating the phase of coherent states using a noiseless linear amplifier by computing its quantum Fisher information and we show that on average, the noiseless linear amplifier does not improve the phase estimate. We also discuss the case where abstention from measurement can reduce the cost for estimation.
According to the fundamental laws of quantum optics, noise is necessarily added to the system when one tries to clone or amplify a quantum state. However, it has recently been shown that the quantum noise related to the operation of a linear phase-insensitive amplifier can be avoided when the requirement of a deterministic operation is relaxed. Nondeterministic noiseless linear amplifiers are therefore realizable. Usually nondeterministic amplifiers rely on using single photon sources. We have, in contrast, recently proposed an amplification scheme in which no external energy is added to the signal, but the energy required to amplify the signal originates from the stochastic fluctuations in the field itself. Applying our amplification scheme, we examine the amplifier gain and the success rate as well as the properties of the output states after successful and failed amplification processes. We also optimize the setup to find the maximum success rates in terms of the reflectivities of the beam splitters used in the setup. In addition, we discuss the nonidealities related to the operation of our setup and the relation of our setup with the previous setups.
The principal problem of light amplifiers is to find a means to amplify a coherent amplitude without decreasing the SNR. Consider a signal with amplitude squeezed quantum fluctuations, that is, less noise in the amplitude quadrature than the phase quadrature. The objective is to find a phase sensitive amplifier, which, while amplifying the coherent amplitude, adds quantum noise preferentially to the phase quadrature. We take as a concrete model for the amplifier an intracavity parametric amplifier. We go beyond the usual linear analysis and include the nonlinear regime where pump depletion is important. We find that in the linear region the quantum noise added by the amplification process may be reduced by squeezing the idler mode, while in the nonlinear regime the quantum fluctuations from the pump become more important. These results are of direct application to practical light amplifiers.