Metrology of weak quantum perturbations

TL;DR

This paper investigates methods for precisely estimating weak quantum perturbations in Hamiltonians, comparing stationary and dynamical measurement schemes, and identifying optimal strategies for different quantum models.

Contribution

It introduces a framework for quantum metrology of weak perturbations, deriving optimal measurement strategies and precision bounds for systems with multiple couplings.

Findings

Dynamical schemes can outperform stationary measurements in estimating couplings.

Optimal initial states and interaction times enhance estimation precision.

Explicit results for qubit models demonstrate practical applicability.

Abstract

We consider quantum systems with a Hamiltonian containing a weak perturbation i.e. $\boldsymbol{H=H_0} + \boldsymbol{\lambda} \cdot \boldsymbol{\tilde{H}}$, $\boldsymbol{\lambda}= \{\lambda_1, \lambda_2,...\}$, $\boldsymbol{\tilde{H}}$ $= \{H_1, H_2,...\}$, $\left|\boldsymbol{\lambda}\right| \ll 1$, and address situations where $\boldsymbol{\tilde{H}}$ is known but the values of the couplings $\boldsymbol{\lambda}$ are unknown, and should be determined by performing measurements on the system. We consider two scenarios: in the first one we assume that measurements are performed on a given stationary state of the system, e.g., the ground state, whereas in the second one an initial state is prepared and then measured after evolution. In both cases, we look for the optimal measurements to estimate the couplings and evaluate the ultimate limits to precision. In particular, we derive general results for one and two couplings, and analyze in details some specific qubit models. Our results indicates that dynamical estimation schemes may provide enhanced precision upon a suitable choice of the initial preparation and the interaction time.