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 $\tfrac{1}{2}[V(\phi+\phi_s)+V(\phi-\phi_s)]$, 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 $\omega_{\rm eff}=\Omega_p\sqrt{\cos(\phi_{\rm zpf}/\phi_0)}$, 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 $\langle\hat\phi\rangle(t)$ 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 $(\lambda/\omega_1)^2\,\phi_{\rm zpf}^2$.

arXiv:2609.25296 [quant-ph]

(or arXiv:2609.25296v1 [quant-ph] for this version)

https://doi.org/10.48550/arXiv.2609.25296 Focus to learn more arXiv-issued DOI via DataCite (pending registration)

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