Optimal initial states for quantum Fisher information in linearized cavity optomechanics
Abstract
We find the optimal initial states of a linearized cavity-optomechanical system for estimating the single-photon coupling by quantum Fisher information. In the red-detuned (beam-splitter) regime the interaction exchanges excitations between the two modes, and for pure inputs the Fisher information is four times the variance of the excitation-exchange operator, reducing the optimization to variance maximization with a closed-form solution. With the mechanical mode in a Fock state the probe is degenerate: any state saturating the photon-number constraint is optimal, and a Fock reference with many phonons enhances the per-photon sensitivity linearly. For an arbitrary fixed mechanical reference it becomes a phase-matched squeezed vacuum, from an exact bound on that coherence obtained by a Lagrange-dual argument. Jointly optimizing both modes at fixed total excitation number, the quantum Fisher information obeys a Heisenberg bound quadratic in the energy, saturated by equal superpositions of the two extremal eigenstates of the generator, equivalently a two-mode NOON state after a balanced beam splitter; balanced product Fock states reach only a linear enhancement. Exact diagonalization in truncated Fock spaces confirms all the results, the familiar single-photon input being the simplest saturating case. Three extensions are worked out: the blue sideband has an unbounded two-mode-squeezing spectrum, so the entangled optimum becomes impossible and the balanced product Fock state is exactly optimal at definite energy; under cavity loss the information accumulation turns linear on the cavity-lifetime timescale, at a renewal-optimal duration of about two and a half lifetimes; and for a state-of-the-art quantum-coherent coupling experiment the per-hour relative precision lies between one part in a million and ten parts in a million, number and thermal references being the most robust.
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BibTeX
@misc{lu2026optimal,
title = {{Optimal initial states for quantum Fisher information in linearized cavity optomechanics}},
author = {Wangjun Lu and Qing Yu and Ying Li and Cuilu Zhai and Rui Zhang and Zhao-Hui Peng and Shiqing Tang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.01417},
eprint = {2610.01417},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.01417},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Optimal initial states for quantum Fisher information in linearized cavity optomechanics
AU - Wangjun Lu
AU - Qing Yu
AU - Ying Li
AU - Cuilu Zhai
AU - Rui Zhang
AU - Zhao-Hui Peng
AU - Shiqing Tang
PY - 2026
DA - 2026/10/01
PB - arXiv
DO - 10.48550/arxiv.2610.01417
UR - https://arxiv.org/abs/2610.01417
AB - We find the optimal initial states of a linearized cavity-optomechanical system for estimating the single-photon coupling by quantum Fisher information. In the red-detuned (beam-splitter) regime the interaction exchanges excitations between the two modes, and for pure inputs the Fisher information is four times the variance of the excitation-exchange operator, reducing the optimization to variance maximization with a closed-form solution. With the mechanical mode in a Fock state the probe is degenerate: any state saturating the photon-number constraint is optimal, and a Fock reference with many phonons enhances the per-photon sensitivity linearly. For an arbitrary fixed mechanical reference it becomes a phase-matched squeezed vacuum, from an exact bound on that coherence obtained by a Lagrange-dual argument. Jointly optimizing both modes at fixed total excitation number, the quantum Fisher information obeys a Heisenberg bound quadratic in the energy, saturated by equal superpositions of the two extremal eigenstates of the generator, equivalently a two-mode NOON state after a balanced beam splitter; balanced product Fock states reach only a linear enhancement. Exact diagonalization in truncated Fock spaces confirms all the results, the familiar single-photon input being the simplest saturating case. Three extensions are worked out: the blue sideband has an unbounded two-mode-squeezing spectrum, so the entangled optimum becomes impossible and the balanced product Fock state is exactly optimal at definite energy; under cavity loss the information accumulation turns linear on the cavity-lifetime timescale, at a renewal-optimal duration of about two and a half lifetimes; and for a state-of-the-art quantum-coherent coupling experiment the per-hour relative precision lies between one part in a million and ten parts in a million, number and thermal references being the most robust.
ER -