Phase diffusion and the small-noise approximation in linear amplifiers: Limitations and beyond

Andy Chia1, Michal Hajdušek1, Rosario Fazio2,3, Leong-Chuan Kwek1,4,5, and Vlatko Vedral1,6

1Centre for Quantum Technologies, National University of Singapore, 3 Science Drive 2, Singapore 117543
2ICTP, Strada Costiera 11, I-34151 Trieste, Italy
3Dipartemento di Fisica, Università di Napoli "Federico II", Monte S. Angelo, I-80126, Italy
4MajuLab, CNRS-UNS-NUS-NTU International Joint Research Unit, UMI 3654, Singapore
5National Institute of Education, Nanyang Technological University, 1 Nanyang Walk, Singapore 637616
6Department of Physics, University of Oxford, Parks Road, Oxford, OX1 3PU, UK

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Abstract

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.

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