Effective Study of Superconducting Quantum Circuits
Abstract
We apply the momentous quantum mechanics formalism to the non-perturbative study of superconducting circuits in the transmon regime, deriving effective equations of motion for the bare Josephson junction (JJ), the cavity--JJ system, and a dissipative resonator. A Gaussian closure on the hierarchy of quantum moments resums the full cosine nonlinearity into the closed-form effective Hamiltonian 1/2[V(ϕ+ϕs)+V(ϕ−ϕs)], which is non-perturbative and preserves the periodicity and boundedness of the Josephson potential at all phase amplitudes; the standard Kerr (Duffing) approximation is recovered as a special case. We derive the quantum-dressed frequency ωeff=Ωp√(cos(ϕzpf/ϕ0)), and benchmark the effective dynamics against exact Mathieu-function diagonalization, with the quantum width G2,0(t) providing the most sensitive diagnostic of the closure's validity and marks the boundary of the Gaussian approximation more sharply than ⟨ϕ̂⟩(t) does. We compare the Caldirola--Kanai, Bateman, and Lindblad descriptions: the Bateman dynamics, quantized with a switched symplectic structure, preserves the Heisenberg bound, admits exact closed-form moment solutions, and reproduces the Lindblad benchmark for weak damping, i.e., within the analytic error bound (λ/ω1)2 ϕzpf2.
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BibTeX
@misc{valdez2026effective,
title = {{Effective Study of Superconducting Quantum Circuits}},
author = {Carlos Raul Javier Valdez and Hector Hugo Hernandez Hernandez and Guillermo Chacon-Acosta},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.25296},
eprint = {2609.25296},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.25296},
}
RIS
TY - UNPB
TI - Effective Study of Superconducting Quantum Circuits
AU - Carlos Raul Javier Valdez
AU - Hector Hugo Hernandez Hernandez
AU - Guillermo Chacon-Acosta
PY - 2026
DA - 2026/09/21
PB - arXiv
DO - 10.48550/arxiv.2609.25296
UR - https://arxiv.org/abs/2609.25296
AB - We apply the momentous quantum mechanics formalism to the non-perturbative study of superconducting circuits in the transmon regime, deriving effective equations of motion for the bare Josephson junction (JJ), the cavity--JJ system, and a dissipative resonator. A Gaussian closure on the hierarchy of quantum moments resums the full cosine nonlinearity into the closed-form effective Hamiltonian 1/2[V(ϕ+ϕ_s)+V(ϕ−ϕ_s)], which is non-perturbative and preserves the periodicity and boundedness of the Josephson potential at all phase amplitudes; the standard Kerr (Duffing) approximation is recovered as a special case. We derive the quantum-dressed frequency ω_eff=Ω_p√(cos(ϕ_zpf/ϕ_0)), and benchmark the effective dynamics against exact Mathieu-function diagonalization, with the quantum width G^2,0(t) providing the most sensitive diagnostic of the closure's validity and marks the boundary of the Gaussian approximation more sharply than ⟨ϕ̂⟩(t) does. We compare the Caldirola--Kanai, Bateman, and Lindblad descriptions: the Bateman dynamics, quantized with a switched symplectic structure, preserves the Heisenberg bound, admits exact closed-form moment solutions, and reproduces the Lindblad benchmark for weak damping, i.e., within the analytic error bound (λ/ω_1)^2 ϕ_zpf^2.
ER -