Coarse-Grained Effective Hamiltonian via the Magnus Expansion for a Three-Level System

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 proper coarse-graining in time of the exact dynamics. We validate the accuracy of the obtained effective Hamiltonians with suitably tailored fidelities of quantum operations.

Automated summary

Quantum state processing is a crucial tool in quantum technologies, and effective Hamiltonians derived through adiabatic elimination play a significant role in simplifying complex systems. However, these approximations often face ambiguities and difficulties, limiting their accuracy for larger systems. In this study, we propose using the Magnus expansion as a systematic tool to derive unambiguous effective Hamiltonians, which only depend on proper time coarse-graining of the exact dynamics. The accuracy of the obtained effective Hamiltonians is validated through tailored fidelities of quantum operations.