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Journal of Physics B Atomic Molecular and Optical Physics

Optimal quantum estimation of the displacement in optical imaging

Yang Gao, Le‐Man Kuang

Published Sep 9, 2019• 2 citations
Publisher

Abstract

Abstract We present a general method to find the ultimate limits on the resolution of optical image on a surface perpendicular to the light beam propagation. We prove that the optimal sensitivity always obeys the standard quantum limit for a single mode light. In the absence of photon losses, we find the possibility to achieve the Heisenberg limit by using two-mode or multi-mode lights. We also use the variational approach to obtain the lower bound on the sensitivity considering the effects of photon losses.

DOI
10.1088/1361-6455/ab42b7

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Abstract We present a general method to find the ultimate limits on the resolution of optical image on a surface perpendicular to the light beam propagation. We prove that the optimal sensitivity always obeys the standard quantum limit for a single mode light. In the absence of photon losses, we find the possibility to achieve the Heisenberg limit by using two-mode or multi-mode lights. We also use the variational approach to obtain the lower bound on the sensitivity considering the effects of photon losses.
Tracking small transverse displacements of an optical beam with ultra-high accuracy is a fundamental problem underlying numerous important applications ranging from pointing, acquisition and tracking for establishing a lasercom link, to atomic force microscopy for imaging with atomic-scale resolution. Determining what is the optimal quantum-optical probe and the best achievable sensitivity of measuring a small transverse optical beam displacement, is the fundamental question central to these sensing schemes. By mapping this problem to an array of nested Mach-Zehnder interferometers, we explicitly construct the optimal probe state. It is entangled across the spatial modes allowed within the Fresnel number product of the propagation geometry, and entangled across the temporal modes within the time-bandwidth product of the optical probe. We show that the optimal sensitivity of measuring the beam displacement achieves a Heisenberg limited scaling over both the number of temporal modes and the average number of photons transmitted per mode. Surprisingly, we discover a sub-Heisenberg limited scaling over the number of available spatial modes. To qualify the quantum enhancement, we also establish the optimal sensitivity of a classical-light probe, which gives shot-noise limit over both the number of temporal modes and the number of photons per mode, and Heisenberg limited scaling over the number of spatial modes. Finally, we construct an explicit design for quantum-enhanced beam tracking, which uses a Gaussian (multi-mode-entangled squeezed-state) probe and a Gaussian (multi-mode homodyne) receiver.
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We consider estimating a small transverse displacement of an optical beam over a line-of-sight propagation path: a problem that has numerous important applications ranging from establishing a lasercom link, single-molecule tracking, guided munition, to atomic force microscopy. We establish the ultimate quantum limit of the accuracy of sensing a beam displacement, and quantify the classical-quantum gap. Further, using normal-mode decomposition of the Fresnel propagation kernel, and insights from recent work on entanglement-assisted sensing, we find a near-term realizable multi-spatio-temporal-mode continuous-variable entangled-state probe and a receiver design, which attains the quantum precision limit. We find a Heisenberg-limited sensitivity enhancement in terms of the number of entangled temporal modes, and a curious super-Heisenberg quantum enhanced scaling in terms of the number of entangled spatial modes permitted by the diffraction-limited beam propagation geometry.
Estimation of an optical beam's transverse displacement is a canonical imaging problem fundamental to numerous optical imaging and sensing tasks. Quantum enhancements to the measurement precision in this problem have been studied extensively. However, previous studies have neither accounted for diffraction loss in full generality, nor have they addressed how to jointly optimize the spatial mode and the balance between squeezing and coherent amplitude. Here we show that, in the small-displacement limit, the seemingly intractable infinite-spatial-mode problem can be reduced to a compact three-mode interaction framework. We quantify the improvement afforded by an optimized single-spatial-mode Gaussian-state probe over the optimal classical laser probe, and show that a two-spatial-mode homodyne receiver is asymptotically optimal for the former in the limit of high probe energy. Our findings reveal a strategy for identifying quantum-optimal probes in the presence of generic multimode linear probe-target interaction and photon loss.