Fundamental limits to frequency estimation: A comprehensive microscopic perspective

TL;DR

This paper investigates the fundamental limits of frequency estimation using qubit probes under noise, deriving how different dynamical regimes affect the ultimate precision scaling, including non-Markovian effects and non-phase-covariant dynamics.

Contribution

It provides a microscopic derivation of the probe dynamics and characterizes the ultimate precision limits across various dissipative regimes, extending beyond the phase-covariant assumption.

Findings

Ultimate precision scaling can be $1/N^{3/2}$ or $1/N^{7/4}$ depending on the dynamics.

Non-Markovian and non-phase-covariant effects significantly influence estimation limits.

The study offers a comprehensive microscopic framework covering all dissipative regimes.

Abstract

We consider a scenario in which qubit-like probes are used to sense an external field that linearly affects their energy splitting. Following the frequency estimation approach in which one optimizes the state and sensing time of the probes to maximize the sensitivity, we provide a systematic study of the attainable precision under the impact of noise originating from independent bosonic baths. We invoke an explicit microscopic derivation of the probe dynamics using the spin-boson model with weak coupling of arbitrary geometry and clarify how the secular approximation leads to a phase-covariant dynamics, where the noise terms commute with the field Hamiltonian, while the inclusion of non-secular terms breaks the phase-covariance. Moreover, unless one restricts to a particular (i.e., Ohmic) spectral density of the bath modes, the noise terms may contain relevant information about the frequency to be estimated. Thus, by considering general evolutions of a single probe, we study regimes in which these two effects have a non-negligible impact on the achievable precision. We then consider baths of Ohmic spectral density yet fully accounting for the lack of phase-covariance, in order to characterize the ultimate attainable scaling of precision when $N$ probes are used in parallel. Crucially, we show that beyond the semigroup (Lindbladian) regime the Zeno limit imposing the $1/N^{3/2}$ scaling of the mean squared error, derived assuming phase-covariance, generalises to any dynamics of the probes, unless the latter are coupled to the baths in the direction perfectly transversal to the encoding---when a novel scaling of $1/N^{7/4}$ arises. As our microscopic approach covers all classes of dissipative dynamics, from semigroup to non-Markovian ones (potentially non-phase-covariant), it provides an exhaustive picture, in which all the different asymptotic scalings of precision naturally emerge.