Nonperturbative Analytical Diagonalization of Hamiltonians with Application to Circuit QED

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 nonperturbative 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 us 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 the 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 the Schrieffer-Wolff transformation, 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 qubit systems. We investigate both suppressing and engineering the coupling in near-resonant and quasidispersive regimes. With the proposed methods, the coupling strength in the effective Hamiltonians can be estimated with high precision comparable to numerical results.