Non-perturbative analytical diagonalization of Hamiltonians with application to coupling suppression and enhancement in cQED

TL;DR

This paper introduces two symbolic, non-perturbative methods for deriving effective Hamiltonians in quantum systems, enabling precise control of coupling in superconducting qubits without relying on perturbation assumptions.

Contribution

It presents the NPAD and RSWT methods, which are algebraic, automatable, and outperform traditional techniques in accuracy and efficiency for quantum Hamiltonian modeling.

Findings

NPAD works without perturbation assumptions and converges for wide models.

RSWT reduces computational complexity with linear growth in terms.

Methods accurately estimate coupling strengths in superconducting qubits.

Abstract

Deriving effective Hamiltonian models plays an essential role in quantum theory, with particular emphasis in recent years on control and engineering problems. In this work, we present two symbolic methods for computing effective Hamiltonian models: the Non-perturbative Analytical Diagonalization (NPAD) and the Recursive Schrieffer-Wolff Transformation (RSWT). NPAD makes use of the Jacobi iteration and works without the assumptions of perturbation theory while retaining convergence, allowing to treat a very wide range of models. In the perturbation regime, it reduces to RSWT, which takes advantage of an in-built recursive structure where remarkably the number of terms increases only linearly with perturbation order, exponentially decreasing the number of terms compared to the ubiquitous Schrieffer-Wolff method. In this regime, NPAD further gives an exponential reduction in terms, i.e. superexponential compared to Schrieffer-Wolff, relevant to high precision expansions. Both methods consist of algebraic expressions and can be easily automated for symbolic computation. To demonstrate the application of the methods, we study the ZZ and cross-resonance interactions of superconducting qubits systems. We investigate both suppressing and engineering the coupling in near-resonant and quasi-dispersive regimes. With the proposed methods, the coupling strength in the effective Hamiltonians can be estimated with high precision comparable to numerical results.