Distributed Phase-Insensitive Displacement Sensing

TL;DR

This work addresses phase-insensitive amplitude sensing with distributed bosonic sensors that experience identical, randomly phased displacements. By deriving a phase-averaged quantum Fisher information bound, it shows a linear-in-total-excitation quantum advantage, $F_Q^{(\alpha)} \le 4M\big(2\langle\hat N\rangle +1\big)$, and identifies checkerboard multimode states that saturate the bound with joint-parity measurements. It also analyzes two practical sensing strategies—splitting a single-mode state across modes and embedding the same state in each mode—and shows they achieve $F_C/F_Q^{\text{SQL}} = 1 + 2\langle\hat a^\dagger\hat a\rangle_\psi$, with robustness that depends on the dominant decoherence channel. The decoherence study reveals distinct scalability and robustness patterns under loss, heating, dephasing, and phase jitter, guiding experimental choices across platforms such as trapped ions, optomechanical resonators, circuits-QED, and photonics. Overall, the results establish a distributed quantum advantage for phase-insensitive sensing and provide actionable measurement strategies under realistic noise.

Abstract

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.

Figures (3)

• Figure 1: Considered setup and main results. (a) The sensing system consisting of $M$ bosonic modes. In the $j^{\text{th}}$ experimental shot, all the modes experience a phase-space displacement with common amplitude $\alpha$ and $\varphi_j$. The phase is not fixed between the shots and becomes fully randomized after many experimental shots Grochowski2025. We show that the quantum Fisher information $F_\text{Q}$ for estimating $\alpha$ is bounded by the mode occupations $\langle \hat{n}_i\rangle$ and first-order bosonic correlations, totaling $F_\text{Q}^{\text{max}}$. (b) Pure states with definite joint parity, $\hat{\Pi} \hat{\rho} \hat{\Pi} = \hat{\rho}$, giving a checkerboard structure in Fock space, can saturate this bound in the small-$\alpha$ limit under joint-parity measurements $\hat{\Pi}$ (implemented via local parity, collective operations, or excitation-resolving detectors), with $F_\text{C} = F_\text{Q}^{\text{max}} - \mathcal{O} (\alpha^2)$, where $F_\text{C}$ is the classical Fisher information. (c) We analyze two sensitivity-optimal measurement strategies subject to single-mode state preparation with at most $n_\text{max}$ average excitations per mode. In the first, a single-mode state $\ket{\psi}$ of definite parity (e.g., squeezed vacuum, Fock, or cat) is split across many modes; in the second, the same state $\ket{\psi}$ is prepared independently in each mode. Both yield $F_\text{C}/F_\text{Q}^{\mathrm{SQL}} = 1 + 2\langle \hat{a}^\dagger \hat{a} \rangle_\psi$, with the standard quantum limit (SQL) given by $F_\text{Q}^{\mathrm{SQL}} = 4 M$. Notably, the first strategy achieves a metrological gain over the SQL that scales linearly with the total excitation number $\langle \hat{N} \rangle = \langle \sum_i \hat{n}_i\rangle = \langle \hat{a}^\dagger \hat{a} \rangle_\psi$. Which strategy is optimal depends on the dominant noise: the first is more robust to excitation loss and heating, while the second better suppresses dephasing and intermode phase fluctuations.
• Figure S1: The phase-sensitive and phase-insensitive measurement schemes considered in this comparison. If there is no phase mixing involved, it is assumed that the displacement direction is known and aligned optimally with respect to the probe state and the homodyne detection. $\hat{x}$ signifies homodyne detection, $\hat{\pi}$ single-mode parity measurement, while $\hat{\Pi} = \hat{\pi}_1 \hat{\pi}_2$ is the joint parity measurement. The box with the label displacement means equal phase-space displacement in the modes; the beam-splitter label means a balanced beam-splitter; however, if it is the second in the order, it is the reverse of the first one; the 'phase mixing' box signifies that the phase of the displacement varies from shot to shot. State $\ket{G}$ is the squeezed vacuum (ground state), and $\ket{\psi}$ is an arbitrary state with a definite parity (that also includes the squeezed vacuum). The top row (a-c) describes the usual homodyne scheme for the force sensing, while the bottom one (d-f) concerns the schemes described in the main text involving parity measurements. (a) Two modes are prepared in the same squeezed vacuum state and then, after the action of displacement, are independently measured via a homodyne or the parity measurement is made. Such a scheme gives twice the quantum Fisher information of a single-mode scheme, which in the large-$\langle \hat{N} \rangle$ and small-$\alpha$ limit gives $F_\text{C} \approx 16 \langle \hat{N} \rangle$ [cf. Eqs. \ref{['eqS1a']} and \ref{['GaussPar']}], where $\hat{N} = \hat{a}^\dagger _1 \hat{a} _1 + \hat{a}^\dagger _2 \hat{a} _2$. (b) The usual interferometric scheme, where a Gaussian state is prepared in one of the modes, then split into two modes via a beam splitter, recombined back into the first mode, and then a homodyne or parity measurement is made. Such a scheme gives the same Fisher information as a scheme in (a) if the same state $\ket{G}$ is used; however, the total number of excitations $\langle \hat{N} \rangle$ is lower, hence the saturation of the bound (24) from the main text at $F_\text{C} \approx 32 \langle \hat{N} \rangle$ [see Eqs. \ref{['eqS1b']} and \ref{['GaussPar']}]. This saturation is signified by the red star. (c) The same scheme as in (b), however, in this case, the phase is not fixed between the shots. The Fisher information tends to 0 as $\alpha \rightarrow 0$ [cf. Eq. \ref{['eqS1c']}], showing inadequacy of this usual scheme in the phase-scrambled case. (d) The 'single-mode' case, where a state $\ket{\psi}$ is prepared independently in both modes, and then each of the modes undergoes the parity measurement. In this case, the phase is also randomized. The Fisher information is lower than in the analogous case (a); however, it also scales with $\langle \hat{N} \rangle$. Homodyne measurement in this case would yield no advantage. (e) The same scheme as in (d), but this time, the total parity is measured [a binary observable, unlike in case (d)]. The Fisher information is the same as in the case (d). (f) The scheme analogous to (b)---a single mode is prepared in the state $\ket{\psi}$, and then split into two modes. As the measurement is the total parity, the recombination by the second beam-splitter is optional as passive operations preserve the parity. However, such a recombination allows for performing the total parity measurement by only a single local parity detector, bringing the amount of the needed resources to the same level as (b). Similarly to (b), this scheme saturates the theoretical bound given by Eq. (10) in the main text and scales with $\langle \hat{N} \rangle$ even with phase scrambling.
• Figure S2: Fisher information of parity measurement for different two-mode states as a function of measured displacement $\alpha$. Both plots provide a comparison between single-mode squeezed vacuum (blue, Gauss.), Fock (green), and cat (yellow) states that are prepared in both modes independently, and then a joint-parity measurement is taken (dotted, 'separable' case, sep.), are prepared in both modes independently, and then local parity measurements are taken without computing total parity (dashed, 'single-mode' case, sm.), are prepared in a single mode and then delocalized via a beam-splitter (solid, 'delocalized' case, del.). (a) Comparison of the states with the same average phonon number $n=5$. In every case, Gaussian states exhibit a smaller dynamical range (range of $\alpha$ such that $F>8 M$) than cat and Fock states, in that order. Single-mode strategy exhibits the largest dynamical range, followed by the delocalized and separable ones. This is contrary to the phase-fixed case, where both single-mode and delocalized strategies exhibit infinite dynamical ranges in the noiseless scenario. (b) Comparison of states with different average occupation numbers for the delocalized strategy. The larger the occupation number, the faster the metrological advantage decays as a function of $\alpha$.