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arXiv:2509.11457

Optimal single-mode squeezing for beam displacement sensing

Wenhua He, Christos N. Gagatsos, Dalziel Joseph Wilson, Saikat Guha

Published Sep 14, 2025• 1 citation
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Abstract

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.

DOI
10.48550/arxiv.2509.11457

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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.
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.
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.
This Letter addresses a canonical imaging problem: how to estimate a small transverse displacement of a laser beam as precisely as possible. This problem is central to many applications, including scanning-probe atomic force microscopy, free-space optical communication, and optomechanics. Based on the geometry of the imaging system, the center wavelength of the laser probe, and the return-path propagation length, the authors identify the optimal spatial mode in which to excite a laser probe, when paired with the most sensitive receiver allowed by quantum estimation theory. Several phase-sensitive, mode-resolving receivers that achieve the mentioned quantum precision limit are presented.
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.
Bounds on the ultimate precision attainable in the estimation of a parameter in Gaussian quantum metrology are obtained when the average number of bosonic probes is fixed.We identify the optimal input probe state among generic (mixed in general) Gaussian states with a fixed average number of probe photons for the estimation of a parameter contained in a generic multimode interferometric optical circuit, namely, a passive linear circuit preserving the total number of photons.The optimal Gaussian input state is essentially a single-mode squeezed vacuum, and the ultimate precision is achieved by a homodyne measurement on the single mode.We also reveal the best strategy for the estimation when we are given L identical target circuits and are allowed to apply passive linear controls in between with an arbitrary number of ancilla modes introduced.