Coarse-grained effective Hamiltonian via the Magnus Expansion for a three-level system

TL;DR

This paper introduces a systematic method using the Magnus expansion to derive effective Hamiltonians for three-level quantum systems, improving accuracy and reducing ambiguities in low-energy subspace approximations.

Contribution

It presents a novel approach employing the Magnus expansion for ambiguity-free effective Hamiltonians, emphasizing proper time coarse-graining for accuracy.

Findings

Magnus expansion provides a systematic way to derive effective Hamiltonians.

Proper time coarse-graining is crucial for the validity of approximations.

Validated effective Hamiltonians with tailored fidelities of quantum operations.

Abstract

Quantum state processing is one of the main tools of quantum technologies. While real systems are complicated and/or may be driven by non-ideal control they may nevertheless exhibit simple dynamics approximately confined to a low-energy Hilbert subspace. Adiabatic elimination is the simplest approximation scheme allowing us to derive in certain cases an effective Hamiltonian operating in a low-dimensional Hilbert subspace. However, these approximations may present ambiguities and difficulties hindering a systematic improvement of their accuracy in larger and larger systems. Here we use the Magnus expansion as a systematic tool to derive ambiguity-free effective Hamiltonians. We show that the validity of the approximations ultimately leverages only on a properly done coarse-graining in time of the exact dynamics. We validate the accuracy of the obtained effective Hamiltonians with suitably tailored fidelities of quantum operations.