Temperature and non-Markovian parameter estimation in quantum Brownian motion

TL;DR

This work addresses metrology in a non-Markovian quantum Brownian motion setting using PM-correlated Gaussian probes. It combines an exact, non-Born-Markov QBM description with a Gaussian-state QFI framework to quantify precision in estimating both the bath temperature $T$ and the non-Markovian witness parameter $x=rac{ extomega_c}{ extomega_0}$, while incorporating initial PM correlations through $oldsymbol{oldsymbol{ ho}}$ (or $oldsymbol{oldsymbol{ m abla}}$). The results show that memory effects and PM correlations can synergistically enhance estimation accuracy beyond the shot-noise limit in many regimes, including low temperatures and strongly non-Markovian dynamics, and they provide a methodology to infer the non-Markovianity witness from QFI. The findings highlight PM correlations as practical resources for quantum sensing in noisy, structured environments and suggest extensions to other spectral densities and experimental platforms such as optomechanics and superconducting devices.

Abstract

We investigate a quantum metrological protocol operating in a non-Markovian environment by employing the quantum Brownian motion (QBM) model, in which the system is linearly coupled to a reservoir of harmonic oscillators. Specifically, we use a position-momentum (PM) correlated Gaussian state as a probe to examine how memory effects influence the evolution of the system's covariance matrix in the weak coupling regime under both high- and low-temperature conditions. To confirm the presence of non-Markovian behavior, we apply two well-established non-Markovianity quantifiers. Furthermore, we estimate both the channel's sample temperature and its non-Markovianity witness parameter. Our results demonstrate that non-Markovianity and PM correlations can jointly be valuable resources to enhance metrological performance.

Figures (7)

• Figure 1: Estimation protocol scheme: An initial probe state with frequency $\omega_0$ is prepared, exhibiting a non-zero position-momentum correlation $(\gamma \neq 0)$. The probe undergoes non-Markovian dynamics during evolution through a bosonic quantum Brownian motion channel at temperature $T$. In this stage, the environmental parameters of the system are encoded into the probe state via non-unitary evolution. Subsequently, measurements are made to estimate the unknown channel parameters. The non-Markovian regime is controlled by adjusting the ratio $x=\omega_c/\omega_0$ , which compares the cut-off frequency $\omega_c$ of the bath to the frequency $\omega_0$ of the probe state.
• Figure 2: Temporal behavior of the non-Markovianity quantifier $\mathcal{N}$ and purity $\mu$ (with different initial PM correlations $\gamma$) under high-temperature ($k_{B}T/\hbar\omega_{c}=1000$, left panels) and low-temperature ($k_{B}T/\hbar\omega_{c}=10$, right panels) regimes. Even in an almost Markovian regime ($x=5.0$), the non-Markovian characteristic persists only in the $\mathcal{N}$ plots. This contrasts with the revival signatures, which are typically observed only in the short-time dynamics in most cases, particularly when state-dependent quantifiers of non-Markovianity, such as trace distance or purity, are used. Additionally, we observe that the initial PM correlation $\gamma$ enhances the oscillations and revival features of purity compared to the uncorrelated case $\gamma=0$, emphasizing its non-Markovian nature.
• Figure 3: The temporal evolution of the Fisher information $T^2\mathcal{F}_T$ (varying the initial PM correlations $\gamma$) is displayed under both high-temperature ($k_{B}T/\hbar\omega_{c}=1000$, left panels) and low-temperature regimes ($k_{B}T/\hbar\omega_{c}=10$, right panels) and for different non-Markovianity witness parameter $x$. The Fisher information $\mathcal{I}_0 = \mathcal{F}_T(\tau, x, T, \gamma = 0)$ for the uncorrelated probe ($\gamma = 0$) represents the shot-noise limit.
• Figure 4: The temporal behavior of the Quantum Fisher Information (QFI) $\mathcal{F}_x$ (varying the initial PM correlations $\gamma$) is displayed under both high-temperature ($k_{B}T/\hbar\omega_{c}=1000$, left panels) and low-temperature regimes ($k_{B}T/\hbar\omega_{c}=10$, right panels). Additionally, we analyze the QFI $\mathcal{F}_x$ under a high (low) non-Markovian environment in the top (bottom) panels. The QFI $\mathcal{F}_x$ of an initially uncorrelated Gaussian state $(\gamma = 0)$ corresponds to the SNL. Observe a meteorological enhancement (beating the SNL) as the PM correlation $\gamma$ and the non-Markovianity quantifier (small values of $x$) increase.
• Figure 5: Dynamic of the QFI, $\mathcal{F}^{\gamma=0}_x$ (solid red line), and non-Markovianity quantifier, $\mathcal{N}$ (black dashed line), for both high-temperature ($k_{B}T/\hbar\omega_{c}=1000$) and low-temperature conditions ($k_{B}T/\hbar\omega_{c}=10$) considering different non-Markovianity witness parameter $x$.