Fundamental Limits of Continuous Gaussian Quantum Metrology

TL;DR

This work establishes a comprehensive framework for quantum metrology with multimode free bosons under continuous Gaussian measurements, deriving analytical expressions for the asymptotic growth rates of the global QFI $I_G$ and environmental QFI $I_E$ and identifying fundamental bounds in terms of mode number $M$ and bosonic excitation resources. The authors distinguish two coupling regimes—dissipative (contractive toward a steady state) and zero-damping (oscillatory and energy-growing)—and show that, with local couplings, both QFIs scale linearly in time and with $M$, while globally coupled or engineered nonreciprocal (non-Hermitian skin) setups can achieve quadratic or exponential gains in $I_G$, though $I_E$ may scale differently. In dissipative scenarios, $I_G$ and $I_E$ share the same long-time growth rate, saturating the bound with optimal quadratic scaling in $M$ under global coupling, whereas in zero-damping regimes, $I_G$ can outpace $I_E$, with nonreciprocal designs yielding exponential boosts in $I_G$ at energy cost. The results illuminate resource trade-offs and scaling laws for continuous bosonic sensing and point to experimental platforms like coupled cavity arrays and trapped-particle arrays as viable testbeds for these metrological advantages.

Abstract

Continuous quantum metrology holds promise for realizing high-precision sensing by harnessing information progressively carried away by the radiation quanta emitted into the environment. Despite recent progress, a comprehensive understanding of the fundamental precision limits of continuous metrology with bosonic systems is currently lacking. We develop a general theoretical framework for quantum metrology with multimode free bosons under continuous Gaussian measurements. We derive analytical expressions for the asymptotic growth rates of the global quantum Fisher information (QFI) and the environmental QFI, which quantify the total information encoded in the joint system-environment state and the information accessible from the emitted radiation, respectively. We derive fundamental bounds on these quantities, showing that while Heisenberg-type scaling with the number of modes is attainable, the precision scales at most linearly with time and a meaningful energy resource. To illustrate our findings, we analyze several concrete setups, including coupled cavity arrays and trapped particle arrays. While a local setup yields a standard linear scaling with resources, a globally coupled setup can achieve the optimal quadratic scaling in terms of the mode number. Furthermore, we demonstrate that a nonreciprocal setup can leverage the non-Hermitian skin effect to realize an exponentially enhanced global QFI. Notably, however, this enhancement cannot be reflected in the environmental QFI, highlighting a fundamental distinction between the information stored within the joint state and the information radiated into the environment. These findings establish an understanding of the resource trade-offs and scaling behaviors in continuous bosonic sensing.

Figures (9)

• Figure 1: Schematic illustrations of continuous quantum sensing protocols with (a) dissipative or (b) zero-damping system-environment couplings. In both cases, the environment consists of the electromagnetic fields. A linear external field $E$ to be estimated is encoded in the array dynamics. (a) Coupled cavity array, where $M$ coupled cavity modes are connected to waveguides. Photons leaking from the cavities are collected by the waveguides. (b) Trapped particle array, where $M$ trapped particles are arranged periodically.
• Figure 2: Local setup. (a) Time evolution of diagonal elements of the covariance matrix $\Gamma_\phi$. The variances of the quadrature operators converge to steady-state values, indicating that the system reaches a unique steady state. (b) Mode-number scaling of the asymptotic rate of the global QFI, normalized by the total bosonic excitation number $\bar{n}$. The dots are obtained from numerical simulations of $\dot{I}_{\rm G}$, while the solid curve represents the analytical result obtained from the steady-state expression $\dot{I}_{\rm G,st}$ (cf. Eq. (\ref{['eq4B2']})). The parameters are set to be $M=60$ for (a), with $\Delta=0.5,\zeta=0.3$, and $E=0.1$ for (b).
• Figure 3: Hybrid local-and-global setup. (a) Time evolution of diagonal elements of the covariance matrix $\Gamma_\phi$. As in the local-only setup, the system converges to a unique steady state. (b) Mode-number scaling of the asymptotic rate of the global QFI, normalized by the bosonic excitation number. The dots are obtained from numerical simulations of $\dot{I}_{\rm G}/\bar{n}$, while the solid curve represents the analytical result from Eq. (\ref{['eq4B2']}). The parameters are set to be $M=60$ for (a), with $\Delta=0.5,\zeta=0.1,\gamma=0.3$, and $E=0.1$ for (b).
• Figure 4: Local setup. (a) Time evolution of diagonal elements of the covariance matrix $\Gamma_\phi$. (b) Time evolution of the global and environmental QFIs. (c) Mode-number scaling of the asymptotic rate of the global QFI, normalized by the bosonic excitation rate $\dot{I}_{\rm G}/\dot{\bar{n}}$. (d) Mode-number scaling of the asymptotic rate of the environmental QFI $\dot{I}_{\rm E}$. In panels (c) and (d), the dots are obtained from numerical simulations, while the solid lines represent the analytical results for the steady values $\dot{I}_{\rm G,st},\dot{\bar{n}}_{\rm st}$, and $\dot{I}_{\rm E,st}$ (cf. Eqs. (\ref{['eq5C3']}), (\ref{['eq5D1']}), and (\ref{['eq5E1']})). The parameters are set to be $M=60$ for (a) and (b), with $\gamma=0.1$ and $E=0.1$ for all the panels.
• Figure 5: Global setup. (a) Array of $M$ trapped particles that is placed inside an optical cavity. A collective observable of all particles is monitored by detecting the light leaking from the cavity. (b) Time evolution of the covariance matrix $\Gamma_\phi$ in a log-log plot. (c) Time evolution of the global and environmental QFIs normalized by evolved time $t$ in a log-log plot. (d) Asymptotic rate of the global QFI normalized by the excitation rate $\dot{I}_{\rm G}/\dot{\bar{n}}$. (e) Asymptotic rate of the environmental QFI $\dot{I}_{\rm E}$. In panels (d) and (e), the dots are extracted from numerical results, while the solid curves represent the analytical results for the stationary regime. The parameters are $M=60$ for (b) and (c), with $\gamma=0.1$ and $E=0.1$ for all the panels.