# Quantum Probes for Ohmic Environments at Thermal Equilibrium

## TL;DR

This paper investigates how thermal fluctuations affect the ability of quantum probes, specifically a single qubit, to characterize Ohmic environments at thermal equilibrium, highlighting conditions where quantum probing remains effective.

## Contribution

It extends previous zero-temperature analysis to finite temperatures, analyzing the impact of thermal noise on quantum probing of Ohmic environments.

## Key findings

- Thermal fluctuations reduce precision for low cutoff frequencies ($\,\omega_c \lesssim T$).
- For higher cutoff frequencies, environment structure dominates decoherence.
- Single qubit probes remain effective for estimating larger cutoff frequencies despite thermal noise.

## Abstract

It is often the case that the environment of a quantum system may be described as a bath of oscillators with Ohmic density of states. In turn, the precise characterization of these classes of environments is a crucial tool to engineer decoherence or to tailor quantum information protocols. Recently, the use of quantum probes in characterizing Ohmic environments at zero-temperature has been discussed, showing that a single qubit provides precise estimation of the cutoff frequency. On the other hand, thermal noise often spoil quantum probing schemes, and for this reason we here extend the analysis to complex system at thermal equilibrium. In particular, we discuss the interplay between thermal fluctuations and time evolution in determining the precision {attainable by} quantum probes. Our results show that the presence of thermal fluctuations degrades the precision for low values of the cutoff frequency, i.e. values of the order $\omega_c \lesssim T$ (in natural units). For larger values of $\omega_c$ decoherence is mostly due to the structure of environment, rather than thermal fluctuations, such that quantum probing by a single qubit is still an effective estimation procedure.