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 different circuit designs for particular applications. We obtain formally exact amplitudes (`supercoefficients') of these parametric processes for driven SNAIL-based and SQUID-based circuits. 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 supercoefficients 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 multi-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 supercoefficients 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.

• The paper introduces a method using "supercoefficients" and normal-ordered expansion to calculate exact amplitudes of parametric processes in driven Josephson circuits, valid even under strong drives.
• The framework's utility is demonstrated through applications including estimating Kerr-cat qubit parameters, analyzing chaotic behavior, and extending to multi-mode circuits with weakly coupled linear modes.
• This robust theoretical model provides a powerful tool for designing and analyzing superconducting circuits, facilitating comparison across designs and advancing quantum computing and quantum information science.

Exact Amplitudes of Parametric Processes in Driven Josephson Circuits

This paper presents an in-depth analysis of parametric processes within Josephson circuits, specifically targeting applications that harness the unique properties of these superconducting systems. The authors introduce a comprehensive and systematic method for quantifying the amplitudes of arbitrary parametric processes using a normal-ordered expansion of the Hamiltonian governing Josephson circuits. This approach allows for precise modeling of a range of circuit designs, thereby facilitating the optimization of these systems for specific applications, such as amplifiers and quantum bits.

The core of the methodology is grounded in the introduction of "supercoefficients," which provide the exact amplitudes for parametric processes. These coefficients incorporate detailed information on circuit topologies, nonlinearity forms, and the influences of parametric drives. This approach proves particularly advantageous in scenarios involving strong parametric drives, where traditional weak-drive approximations are inadequate. The paper provides closed-form expressions for circuits devoid of stray inductors, as well as tractable formulations for those incorporating such elements.

Applications of the presented framework are illustrated through two significant examples: the estimation of Hamiltonian parameters for Kerr-cat qubits, and the exploration of chaotic behavior emergence criteria in Kerr-cat systems. Additionally, the framework is extended to circuits with multiple degrees of freedom, where linear modes are weakly coupled to a single nonlinear mode. This extension is exemplified by the activation of a beam-splitter interaction between two cavities connected through driven nonlinear elements.

The use of supercoefficients facilitates not only the encapsulation of the complexities of circuit structures and drives but also enables systematic comparison across different circuit designs and geometries, effectively aiding the development of superconducting quantum computing technologies. Moreover, the approach is adapted to tackle scenarios involving higher-harmonic models of Josephson junctions and circuits subject to multiple parametric drives, thereby encompassing a wide array of potential applications within quantum electronics and quantum information science.

In summary, this paper sets forth a robust theoretical model that broadens the capabilities of analyzing superconducting circuits under varying parametric conditions. The implications for both practical implementations and theoretical advancements in quantum information science are extensive, providing researchers with a powerful tool for circuit design and analysis in the rapidly evolving field of quantum technology. This work not only enhances the understanding of Josephson circuits under parametric driving but also lays the groundwork for future exploration into complex multi-mode circuit dynamics and their applications in quantum computing and beyond.