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Distinct from closed quantum systems, non-Hermitian system can have exceptional points (EPs) where both eigenvalues and eigenvectors coalesce. Recently, it has been proposed and demonstrated that EPs can enhance the performance of sensors in terms of amplification of detected signal. Meanwhile, the noise might also be amplified at EPs and it is not obvious whether exceptional points will still improve the performance of sensors when both signal and noise are amplified. We develop quantum noise theory to systematically calculate the signal and noise associated with the EP sensors. We then compute quantum Fisher information to extract a lower bound of the sensitivity of EP sensors. Finally, we explicitly construct an EP sensing scheme based on heterodyne detection to achieve the same scaling of the ultimate sensitivity with enhanced performance. Our results can be generalized to higher order EPs for any bosonic non-Hermitian system with linear interactions.
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Recently, sensors with resonances at exceptional points (EPs) have been suggested to have a vastly improved sensitivity due to the extraordinary scaling of the complex frequency splitting of the $n$ initially degenerate modes with the $n$-th root of the perturbation. We show here that the resulting quantum-limited signal to noise at EPs is proportional to the perturbation, and comparable to other sensors, thus providing the same precision. The complex frequency splitting close to EPs is therefore not suited to estimate the precision of EP sensors. The underlying reason of this counter-intuitive result is that the mode fields, described by the eigenvectors, are equal for all modes at the EP, and are strongly changing with the perturbation.
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