Exact amplitudes of parametric processes in driven Josephson circuits

TL;DR

The paper develops a general, algebraic framework for exact amplitudes of parametric processes in driven Josephson circuits using a normal-ordered Hamiltonian expansion and the concept of supercoefficients $C_{nl,p}$. It derives closed-form $C_{nl,p}$ for symmetric circuits and extends to an eigenstate basis and multi-DOF configurations, so the amplitudes encode circuit topology, nonlinearity, and drive in the strong-drive regime. Applications to Kerr-cat dynamics, chaos onset, and beam-splitter interactions show how to estimate Hamiltonian parameters (e.g., Kerr nonlinearity $K$, squeezing $\epsilon_2$, and $g_{BS}$) and to benchmark circuit designs for robustness and performance. Overall, the SC framework offers a universal, tractable alternative to standard $n$-based expansions, enabling pre-fabrication circuit optimization under strong drives.$

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') 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.

Figures (6)

• Figure 1: (Color online) Maximum Kerr-cat size for different (a) SNAIL-based and (b) SQUID-based circuit designs (see Table \ref{['table:cEQD_schemes']}) with minimal Kerr nonlinearity $K_\mathrm{lim}=1MHz$, effective drive amplitude $\Pi=0.5$, and combined index $\mathrm{S}\leq8$ for $L\neq\infty$ (see Eq. (\ref{['eq:supercoefficiennt_as_sum']})). For (a), (b) parameters are derived from Grimm2020frattini2022squeezed. More details on circuit optimization and simulation parameters are presented in Appendix \ref{['app:kerr_cat_scs']}.
• Figure 2: (Color online) Parametric curves $|\epsilon_2(\Pi)|/\omega_q(\Pi)$ vs. $|\alpha|^2(\Pi)$ probes criteria for chaos onset for different SNAIL-based ($N=3$) circuit designs: A ($M=1$, $K=1.15MHz$ for an undriven circuit), B ($M=2$, $K=-2.58MHz$), C ($M=1$, $K=6.76MHz$), D ($M=2$, $K=0.72MHz$). Solid lines: SC approach with $\mathrm{S}\leq13$, dashed line: $\mathrm{S}\leq4$ (used in Ref. chávezcarlos2024). More details on the simulation parameters are presented in Table \ref{['tab:kerr_chaos_params']} and Appendix \ref{['app:KC_chaos']}.
• Figure 3: (Color online) Beam-splitter interaction versus effective drive strength, showing sharp downturns due to multiphoton resonances for different flux biases with $\mathrm{S} \leq 13$ (see Eq. \ref{['eq:supercoefficiennt_as_sum']}). As a result of the Stark shift, the resonance condition $\tilde{\delta} = \delta + \Delta_a \approx 0$ is met at a particular drive strength (see Eq. \ref{['eq:g_bs']}), where $\delta = \omega_a^\prime + \omega_b^\prime - 2\omega_d$ and the circuit is driven at $\omega_d = \omega_c^\prime - \omega_b^\prime$. The presented curves demonstrate that multiphoton resonance manifests differently across various ranges of external flux: a sharp downturn shifting toward weaker drives for $\varphi_e \lesssim 0.37$ (dashed curves), no visible resonance within an intermediate range exemplified by $\varphi_e = 0.38$ (dash-dotted curve), and mild downturns for $\varphi_e \gtrsim 0.4$ (solid curves). The criteria for the values of $\Pi$ for each curve are derived from the constraints $g_{ab}/\tilde{\delta},\, g_{ac}/\tilde{\delta} \leq 0.25$, $\Tilde{\Pi}=\Pi/n_\mathrm{zpf}\lesssim2$ (see Eqs. \ref{['eq:g_bs']}, \ref{['eq_app:eff_ham_3modes']}, and step 6 in Appendix \ref{['app:circuit_opt']}). The circuit parameters match the architecture described in Ref. ChapmanStijn2023 (see details in Appendix \ref{['app:beam_splitter']}).
• Figure 4: (Color online) Maximum on-off ratio between the beam-splitter interaction and the cavity-cavity cross-Kerr interaction, $\chi_{bc}$, for different (a) SNAIL-based and (b) SQUID-based circuit designs with $\chi_{bc} \geq 30Hz$, effective drive amplitude $\Pi = 1$, and combined index $\mathrm{S} \leq 9$. Panels (c) and (d) show the corresponding beam-splitter interaction strengths. Crossed cells in (a) and (c) indicate circuits that fall outside the scope of the desired parameter range. Other simulation parameters follow Ref. ChapmanStijn2023 and are detailed in Appendix \ref{['app:beam_splitter']}.
• Figure 5: Superconducting nonlinear asymmetric inductive elements (SNAIL) with $N$ large Josephson junctions.