Optimal adaptive control for quantum metrology with time-dependent Hamiltonians

TL;DR

This paper develops an optimal adaptive control strategy for quantum metrology involving time-dependent Hamiltonians, achieving enhanced precision scaling beyond traditional limits, with practical measurement schemes demonstrated on a qubit system.

Contribution

It derives the optimal quantum Fisher information and control schemes for time-dependent Hamiltonians, revealing improved scaling and addressing complex dynamics in quantum metrology.

Findings

Optimal control enhances Fisher information in time-dependent systems.

Achieves $T^{4}$ scaling in a qubit magnetic field estimation.

Additional control needed for level crossings in Hamiltonians.

Abstract

Quantum metrology has been studied for a wide range of systems with time-independent Hamiltonians. For systems with time-dependent Hamiltonians, however, due to the complexity of dynamics, little has been known about quantum metrology. Here we investigate quantum metrology with time-dependent Hamiltonians to bridge this gap. We obtain the optimal quantum Fisher information for parameters in time-dependent Hamiltonians, and show proper Hamiltonian control is necessary to optimize the Fisher information. We derive the optimal Hamiltonian control, which is generally adaptive, and the measurement scheme to attain the optimal Fisher information. In a minimal example of a qubit in a rotating magnetic field, we find a surprising result that the fundamental limit of $T^{2}$ time scaling of quantum Fisher information can be broken with time-dependent Hamiltonians, which reaches $T^{4}$ in estimating the rotation frequency of the field. We conclude by considering level crossings in the derivatives of the Hamiltonians, and point out additional control is necessary for that case.