Distributed Phase-Insensitive Displacement Sensing

Distributed quantum sensing leverages quantum correlations among multiple sensors to enhance the precision of parameter estimation beyond classical limits. Most existing approaches target phase estimation and rely on a shared phase reference between the signal and the probe, yet many relevant scenarios deal with regimes where such a reference is absent, making the estimation of force or field amplitudes the main task. We study this phase-insensitive regime for bosonic sensors that undergo identical displacements with common phases randomly varying between experimental runs. We derive analytical bounds on the achievable precision and show that it is determined by first-order normal correlations between modes in the probe state, constrained by their average excitations. These correlations yield a collective sensitivity enhancement over the standard quantum limit, with a gain that grows linearly in the total excitation number, revealing a distributed quantum advantage even without a global phase reference. We identify families of multimode states with definite joint parity that saturate this limit and can be probed efficiently via local parity measurements already demonstrated or emerging in several quantum platforms. We further demonstrate that experimentally relevant decoherence channels favor two distinct sensing strategies: splitting of a single-mode nonclassical state among the modes, which is robust to loss and heating, and separable probes, which are instead resilient to dephasing and phase jitter. Our results are relevant to multimode continuous platforms, including trapped-ion, solid-state mechanical, optomechanical, superconducting, and photonic systems.

💡 Research Summary

This paper introduces a new framework for distributed quantum sensing that operates without a global phase reference, focusing on the estimation of the amplitude of a displacement applied identically to many bosonic sensors. The authors consider M bosonic modes that each experience a phase‑space displacement ˆD(α,φ)=∏i exp(iαÂi) with a fixed amplitude α but a random global phase φ that is uniformly distributed from shot to shot. After averaging over φ, the effective channel is ρ_α=∫0^{2π}dφ/(2π) ˆD(α,φ) ρ ˆD†(α,φ).

Using the quantum Fisher information (QFI) as the figure of merit, they derive an analytical bound for any probe state ρ. The bound separates into a self‑contribution that depends only on the average photon numbers ⟨n_i⟩ of each mode and a cross‑mode contribution that depends on first‑order normal correlations ⟨a_i†a_j⟩ between different modes. Explicitly,

F_Q ≤ 4∑i(2⟨n_i⟩+1) + 4∑{i≠j}⟨a_i†a_j + a_i a_j†⟩.

Applying the Cauchy‑Schwarz inequality to the correlations yields a simple scaling law

F_Q ≤ 4M + 8M⟨N⟩, ⟨N⟩=∑_i⟨n_i⟩,

which shows that the standard quantum limit (SQL) of 4M can be surpassed by a factor that grows linearly with the total mean excitation number. This linear Heisenberg‑type scaling is achieved even though the phase of the displacement is completely unknown.

To saturate the bound, the authors identify a class of probe states that possess a definite joint parity, ˆΠ ρ = ±ρ, often called “checkerboard” states because their Fock‑space populations form an alternating pattern. For such states, a binary measurement of the global parity, M_±=