Multiparameter estimation with position-momentum correlated Gaussian probes

Abstract

Gaussian quantum probes have been widely used in quantum metrology and thermometry, where the goal is to estimate the temperature of an environment with which the probe interacts. It was recently shown that introducing initial position-momentum (PM) correlations in such probes can enhance the estimation precision compared to standard, uncorrelated Gaussian states. Motivated by these findings, we investigate whether PM correlations can also be advantageous in a simultaneous estimation setting, specifically, when estimating both the PM correlations themselves and the effective environment temperature that interacts with the probe. Using the Quantum Fisher Information Matrix, we derive new precision bounds for this joint estimation task. Additionally, we demonstrate that such correlations can serve as a resource to improve temperature estimation within this multiparameter context. Finally, we analyze the compatibility between the two parameters, establishing conditions under which the derived bounds can be saturated.

Figures (7)

• Figure 1: Protocol to describe the simultaneous estimation of the parameters $\Lambda$ (effective environmental coupling constant) and $\gamma$ (position–momentum correlations). Fullerene wave packets with momentum $\vec{k}$ and an initial coherent length $\ell_0$ are produced during the initialization procedure. The focalization stage then takes place, producing initial position–momentum correlations, represented by $\gamma$. After then, there is interaction with a Markovian thermal reservoir, and the effective environmental coupling is quantified by $\Lambda$. Multiple parameters are estimated simultaneously during the readout process.
• Figure 2: Temporal behavior of the performance ratio $\mathcal{R}$ for increasing values of the environmental effective coupling constant (effective temperature): (a) $\Lambda = 3 \times 10^{15}$ m$^{-2}$s$^{-1}$ ($T=16.9$ mK), (b) $\Lambda = 3 \times 10^{20}$ m$^{-2}$s$^{-1}$ ($T=36.5$ K), and (c) $\Lambda = 3 \times 10^{22}$ m$^{-2}$s$^{-1}$ ($T=786$ K), demonstrating the mitigation of decoherence in simultaneous estimation via initial correlations $\gamma$ within specific time windows.
• Figure 3: Performance ratio $\mathcal{R}$ as a function of time and initial correlation $\gamma$ for increasing values of the environmental effective constant: (a) $\Lambda = 3 \times 10^{15}$ m$^{-2}$s$^{-1}$ ($T=16.9$ mK), (b) $\Lambda = 3 \times 10^{20}$ m$^{-2}$s$^{-1}$ ($T=36.5$ K), and (c) $\Lambda = 3 \times 10^{22}$ m$^{-2}$s$^{-1}$ ($T=786$ K), identifying a clear parameter window where simultaneous estimation outperforms the individual strategy, even under conditions of strong decoherence.
• Figure 4: The behavior of the Quantum Fisher Information Matrix (QFIM) elements is shown. The solid lines represent the newly derived bounds, denoted as $\widetilde{\mathcal{F}}_{ij}$, corresponding to the framework of simultaneous multiparameter estimation. In contrast, the dashed lines indicate the bounds $\mathcal{F}_{ij}$ associated with the individual estimation of each parameter. The plots from left to right correspond to three distinct regimes: a weak environmental effect regime with $\Lambda = 3 \times 10^{15}\mathrm{m}^{-2}\mathrm{s}^{-1}$ ($T=16.9$ mK - left), an intermediate regime with $\Lambda = 3 \times 10^{20}\mathrm{m}^{-2}\mathrm{s}^{-1}$ ($T=36.5$ K - center), and a strong environmental effect regime with $\Lambda = 3 \times 10^{22}\mathrm{m}^{-2}\mathrm{s}^{-1}$ ($T=786$ K - right). In all cases, the interaction time is fixed at $t = 40\mu\mathrm{s}$. Our analysis of the newly derived bounds confirms that, given appropriate initial correlations $\gamma$, the simultaneous precision can attain the optimal individual bound.
• Figure 5: Temporal behavior of the newly derived bound $\widetilde{\mathcal{F}}_{\Lambda \Lambda}$ for effective temperature estimation is analyzed within the framework of simultaneous multiparameter estimation for different values of the initial correlation parameter $\gamma$. Panels (a) correspond to a weak environmental effect, characterized by $\Lambda = 3 \times 10^{15}\mathrm{m}^{-2}\mathrm{s}^{-1}$ ($T=16.9$ mK), while (b) represent a strong environmental effect with $\Lambda = 3 \times 10^{22}\mathrm{m}^{-2}\mathrm{s}^{-1}$ ($T=786$ K). It is important to emphasize that, in the regime of effective weak coupling (a), the newly derived precision bounds are practically indistinguishable from those obtained using the individual estimation scheme analyzed in Ref. Porto2024, with both curves being superposed and, for this reason, only one is shown in the figure.