Quantum metrology through spectral measurements in quantum optics

TL;DR

This work provides a general, physically grounded framework for quantum metrology using frequency-resolved measurements of radiation from continuously driven quantum emitters. By modeling frequency-filtered modes as ancillary sensors in a cascaded network, it defines classical and quantum Fisher information for single- and multi-mode detection, and demonstrates how spectral filtering, detector linewidth, and mean-field displacement can optimize parameter estimation. The authors show that two-mode photon correlations can yield orders-of-magnitude metrological gains near leapfrog transitions and that mean-field engineering can saturate the quantum Fisher information, effectively achieving optimal measurements. The framework is illustrated with a driven two-level system and extended to a transmon and optomechanical systems, underscoring its applicability to diverse quantum-optical platforms and its potential to guide practical sensing and spectroscopy designs.

Abstract

Continuously monitored quantum systems are emerging as promising platforms for quantum metrology, where a central challenge is to identify measurement strategies that optimally extract information about unknown parameters encoded in the complex quantum state of emitted radiation. Different measurement strategies effectively access distinct temporal modes of the emitted field, and the resulting choice of mode can strongly impact the information available for parameter estimation. While a ubiquitous approach in quantum optics is to select frequency modes through spectral filtering, the metrological potential of this technique has not yet been systematically quantified. We develop a theoretical framework to assess this potential by modeling spectral detection as a cascaded quantum system, allowing us to reconstruct the full density matrix of frequency-filtered photonic modes and to compute their associated Fisher information. This framework provides a minimal yet general method to benchmark the performance of spectral measurements in quantum optics, allowing to identify optimal filtering strategies in terms of frequency selection, detector linewidth, and metrological gain accessible through higher-order frequency-resolved correlations and mean-field engineering. These results lay the groundwork for identifying and designing optimal sensing strategies in practical quantum-optical platforms.

Figures (12)

• Figure 1: (a) Sketch of the metrological setup. A coherently driven emitter---characterized by the parameter to be estimated, $\theta\in\{\Delta,\Omega,\gamma\}$---emits radiation that is split by a beam splitter and routed into two frequency-selective sensors, each centered at $\Delta_i$ and featuring a spectral resolution $\Gamma_i$. (b) Metrological protocol. A sensor collects the radiation emitted by the source, giving access to the filtered density matrix of the output field in the frequency domain. This process allows the extraction of the $\theta$-dependent photon-number distribution, $p(n|\theta) = \langle n | \hat{\rho}_\theta | n \rangle$ (top panel). An estimation strategy can then be employed to infer $\theta$ from $p(n|\theta)$. The precision of any unbiased estimator is bounded by the inverse of the Fisher information (bottom panel): $I_\Delta$ (solid black), $F_\Delta$ (solid blue), and $\text{SNR}_\Delta$ (dashed blue), which gives the precision of an estimation based only on the mean population of the sensor $\langle \hat{\xi}^\dagger \hat{\xi} \rangle$ (black-dashed curve in the top panel) instead of the full photon-counting distribution. All quantities correspond to the estimation of the qubit–laser detuning parameter $\Delta$. (c) Joint $\theta$-dependent probability distribution, $p(n_1, n_2 | \theta)$ for simultaneous detection with two sensors. The top and right panels display the marginal distributions for each detector, $p(n_i|\theta)$, highlighting their sensitivity to a small perturbation in the parameter, $\theta + \delta\theta$ (a non-infinitesimal perturbation $\delta\theta = -0.65\gamma$ was chosen to enhance the discernibility of the curves). Parameters: (b) $\Omega=10\gamma$, $\Delta=10^{-5}\gamma$, $\Gamma=10^{-1}\gamma$, $\varepsilon=0.9$. (c) $\Omega=2\gamma$, $\Delta=0$, $\Gamma=10^{-2}\gamma$, $\varepsilon=0.75$, $\Delta_1=0$, $\Delta_2=5\cdot10^{-3}\gamma$;
• Figure 2: Frequency-resolved Fisher information for one-sensor for the estimation of $\theta=\Delta$. (a) $F_\Delta$ (black solid curve) and $\langle \hat{\xi}^{\dagger k} \hat{\xi}^k \rangle$ (red dashed curves, $k=1,2,3,4$) in the frequency domain for a fixed value of $\Omega=3\gamma$. (b) $\langle \hat{\xi}^\dagger \hat{\xi} \rangle$ (fluorescence spectrum of the source) and (c) $F_\Delta$ in terms of the driving strength $\Omega$ and the detection frequency $\Delta_\xi$. The Mollow ladder is illustrated in the inset of panel (b). Parameters: (a) $\Omega=3\gamma$; (a-c) $\Delta=10^{-6}\gamma$, $\Gamma=10^{-1}\gamma$, $\varepsilon=0.1$.
• Figure 3: Impact of sensor linewidth $\Gamma$ on the performance of the estimation protocol for a single sensor. (a) $I_\Delta$ (grey solid), $F_\Delta$ (blue solid), and $\text{SNR}_\Delta$ (red solid) as a function of $\Gamma$, shown for $\Omega=3\gamma$. Additionally, $F_\Delta^{\alpha_{\text{opt}}}$ (blue dashed) and $F_\Delta^{P}$ (yellow dot-dashed) are shown. (b) $F_\Delta$ in terms of the driving strength $\Omega$ and the sensor linewidth $\Gamma$. The optimal regime is approximately bounded by $\Gamma \approx (10^{-2}\Omega,10\Omega)$, i.e., where the sensor preserves spectral resolution and metrological performance is enhanced. Parameters: $\Delta=10^{-6}\gamma,\ \Delta_\xi=\Omega,\ \varepsilon=1$.
• Figure 4: Optimizing quantum parameter estimation via mean-field engineering for a single sensor. (a) Mean-field engineering scheme: the multi-mode radiation is mixed with a coherent field via an unbalanced beam splitter in the limit of perfect transmittance, yielding an engineered signal that is frequency-filtered. (b) $F^\alpha_\Delta$ over the phase space of the local oscillator displacement. Two points are highlighted: the displacement $\alpha_\text{fluct}\approx-1.58 i$, and the optimal displacement $\alpha_{\text{opt}}\approx-4.3-1.4i$. (c) $I_\Delta$ (grey solid), $F_\Delta$ (blue solid), $F_\Delta^{\alpha_{\text{fluct}}}$ (blue dot-dashed), and $F_\Delta^{\alpha_{\text{opt}}}$ (blue dashed) in the frequency domain ($\Delta_\xi$). (d) Same quantities as in (c), now plotted against the emitter-laser detuning ($\Delta$), with $\Delta_\xi=\Omega$. Parameters: (b-d) $\Omega=\gamma/2\sqrt{\gamma}$, $\Gamma=10^{-1}\gamma$, $\varepsilon=0.5$; (b,c) $\Delta=10^{-6}\gamma$; (b) $\Delta_\xi=0$; (d) $\Delta_\xi=\Omega$.
• Figure 5: Frequency-resolved Fisher information for two sensors to estimate $\theta=\Delta$. Panels (a-c) correspond to different sensor linewidths $\Gamma=\{10^{-1},1 ,10\}\gamma$. Each column shares the same structure. Top row: $I_\Delta$ (grey solid), $F_\Delta$ (blue solid), $F_\Delta^I$ (blue dot-dashed), $F_\Delta^{\vec{\alpha}_{\text{opt}}}$ (blue dashed), and $g^{(2)}_\Gamma$ (red solid) in terms of one sensor-laser detuning $\Delta_1$, with the second fixed at $\Delta_2=2\Omega$. Middle row: $g^{(2)}_\Gamma$ in the frequency domain ($\Delta_1,\Delta_2$). Bottom row: $F_\Delta$ (upper-half) and $F_\Delta/F_\Delta^I$ (lower-half) in the frequency domain ($\Delta_1,\Delta_2$). Parameters: (a-c) $\Omega=10\gamma,\ \Delta=0,\ \varepsilon=0.5.$