Exact amplitudes of parametric processes in driven Josephson circuits

Abstract

We present a general approach for analyzing arbitrary parametric processes in Josephson circuits within a single-degree-of-freedom approximation. Introducing a systematic normal-ordered expansion for the Hamiltonian of parametrically driven superconducting circuits, we present a flexible procedure to describe parametric processes and to compare and optimize different circuit designs for particular applications. We obtain formally exact amplitudes [“supercoefficients” (SCs)] of these parametric processes for driven circuits based on superconducting nonlinear asymmetric inductive elements and superconducting quantum interference devices. The corresponding amplitudes contain complete information about the circuit topology, the form of the nonlinearity, and the parametric drive, making them, in particular, well suited for the study of the strong-drive regime. We present a closed-form expression for SCs describing circuits without stray inductors and a tractable formulation for those with it. We demonstrate the versatility of the approach by applying it to the estimation of Kerr-cat qubit Hamiltonian parameters and by examining the criterion for the emergence of chaos in Kerr-cat qubits. Additionally, we extend the approach to multiple-degree-of-freedom circuits comprising multiple linear modes weakly coupled to a single nonlinear mode. We apply this generalized framework to study the activation of a beam-splitter interaction between two cavities coupled via driven nonlinear elements. Finally, utilizing the flexibility of the proposed approach, we separately derive SCs for the higher-harmonics model of Josephson junctions, circuits with multiple drives, and the expansion of the Hamiltonian in the exact eigenstate basis for Josephson circuits with specific symmetries.