Designing open quantum systems for enabling quantum enhanced sensing through classical measurements

TL;DR

The work addresses how quantum-enhanced metrology in open, many-body systems can be realized using only classical measurements of the environment. By deriving explicit saturation criteria for photon counting and homodyne detection within the input-output framework and applying them to spin-boson models (including the Tavis–Cummings and generalized Dicke cases), the authors show that the classical Fisher information from continuous environmental monitoring can match the ultimate system–environment QFI in the long-time limit. They identify saturating classes of states, analyze the impact of detuning and measurement phase, and demonstrate practical sensing improvements achievable without joint system–environment measurements. These results provide design principles for open-quantum-sensor architectures, clarifying when classical measurements suffice to harness many-body quantum enhancement and how to tailor dynamics to maximize metrological gain. Overall, the paper bridges fundamental QFI limits with experimentally accessible sensing protocols in nonequilibrium many-body quantum systems.

Abstract

Quantum systems in nonequilibrium conditions, where coherent many-body interactions compete with dissipative effects, can feature rich phase diagrams and emergent critical behavior. Associated collective effects, together with the continuous observation of quanta dissipated into the environment -- typically photons -- allow to achieve quantum enhanced parameter estimation. However, protocols for tapping this enhancement typically involve intricate measurements on the combined system-environment state. Here we show that many-body quantum enhancement can in fact be obtained through classical measurements, such as photon counting and homodyne detection. We illustrate this in detail for a class of open spin-boson models which can be realized in trapped-ion or cavity QED setups. Our findings highlight a route towards the design of systems that enable a practical implementation of quantum enhanced metrology through continuous classical measurements.

Figures (6)

• Figure 1: Parameter estimation with open many-body system via continuous monitoring. (a) To illustrate our ideas we consider spin-boson models, which consist of $N$ two-level atoms, with ground state $\ket{g}$ and excited state $\ket{e}$, driven by a laser (Rabi frequency $\Omega$ and detuning $\Delta$) and collectively interacting with a bosonic mode (detuned from the laser by $\delta=\omega_{\mathrm{B}} - \omega_{\mathrm{L}}$). Decay of bosonic excitations into the environment (rate $\kappa$) are continuously monitored. Typical photocounting emission record in the stationary (b) $\Omega/\kappa =0.1$ and in the time-crystal regime (c) $\Omega/\kappa=2$ of the Tavis-Cummings model [cf. Eq. \ref{['eq:TC_Hamiltonian']}], with $\lambda/\kappa=0.5$. QFI of system-environment state (dashed) and QFI for photocounting trajectories (solid), for sensing the Rabi frequency $\Omega$, averaged over $10^4$ trajectories, for system sizes of $N=5,10,15$. The initial state features all spins in the excited state and the bosonic mode in the vacuum. In panel (b) the dashed-dotted line represents the analytical value of the system-environment QFI in the stationary regime SM.
• Figure 2: Saturation with homodyne measurements. System-environment QFI (dashed) and QFI for the homodyne unravelling (solid) averaged over $2\times10^4$ trajectories with $\Phi=0,\pi/8,\pi/4,3\pi/8,\pi/2$. (a) Sensing $\Omega$ in the time-crystal regime of the Tavis-Cummings model with $N=5$, $\Omega/\kappa=2$ and $\lambda/\kappa=0.5$. (b) Sensing $\lambda$ in the critical regime of the generalized Dicke model with $N=5$, $\Omega/\kappa=1$, $\lambda/\kappa=1$ and $\delta=0$. The initial state is $\ket{\psi(0)} = \ket{S,S}\otimes\ket{0}$. The saturation condition [cf. Eq. \ref{['eq:conditions_saturation']}] is illustrated in the insets for a trajectory with $N=5$.
• Figure 3: Saturation in the generalized Dicke model with photon counting. System-environment QFI (dashed) and QFI for the photocounting unravelling (solid) averaged over $10^4$ trajectories for (a) $\delta/\kappa=1$ and (b) $\delta=0$, with $\Omega/\kappa=1$, $\lambda/\kappa=1$, $N=5,7,9$ and sensing $\lambda$. The initial state is $\ket{\psi(0)} = \ket{S,S}\otimes\ket{0}$. The condition for saturation [cf. Eq. \ref{['eq:conditions_saturation']}] is illustrated in the insets for a trajectory with $N=5$.
• Figure S1: Many-body enhanced sensitivity in the generalized Dicke model. Time-evolution of the system-environment QFI for $\Delta=0$, $\delta/\kappa=1$, $N=5,7,9,11$ and considering the interaction given in Eq. \ref{['eq:GD_Hamiltonian']}. (a) Sensitivity for estimating $\lambda/\kappa=1$ in the superradiant $\Omega/\kappa=0.4$ (blue), critical $\Omega/\kappa=1$ (green) and normal $\Omega/\kappa=2$ (red) regime. (b) Sensitivity for estimating $\Omega/\kappa=1$ in the normal $\lambda/\kappa=0.7$ (red), critical $\lambda/\kappa=1$ (green) and superradiant $\lambda/\kappa=1.3$ (blue) regime. The system is initially in the state $\ket{\psi(0)} = \ket{S,S}\otimes\ket{0}$.
• Figure S2: Saturation with photon counting for initial states outside the saturating class. (a)-(c) System-environment QFI (dashed) and QFI of the photocounting trajectories (solid), averaged over $10^4$ trajectories, associated to sensing $\Omega$ for the model given in Eq. \ref{['eq:TC_Hamiltonian']}, with $\Omega/\kappa=0.1$ in the stationary (red) and $\Omega/\kappa=2$ in the time-crystal regime (blue) for $\lambda/\kappa=0.5$ and $N=5$. The dashed-dotted line is the analytical value in the stationary regime. The inset shows $\mathrm{Im}(\braket{\psi|\phi})$ for a single trajectory in the stationary (red) and the time-crystal (blue) regime. The initial state is hereby $\ket{S,\theta,\phi}$carmichael_analytical_1980, with (a) $(\theta,\phi) = (\pi/2,0)$, (b) $(\theta,\phi) = (\pi/4,\pi/8)$ and (c) $(\theta,\phi) = (\pi,0)$. (d) System-environment QFI (dashed) and QFI of the photocounting trajectories (solid), averaged over $10^4$ trajectories, associated to sensing $\lambda$ for the model given in Eq. \ref{['eq:GD_Hamiltonian']}, with $\Omega/\kappa=1$, $\lambda/\kappa=1$, $\delta/\kappa=0$, $N=5$ and the initial state $(\theta,\phi) = (\pi/2,0)$. The inset shows again the evolution of $\mathrm{Im}(\braket{\psi|\phi})$ for a single trajectory and latter parameters.