Systematic Construction of Time-Dependent Hamiltonians for Microwave-Driven Josephson Circuits

TL;DR

The paper tackles the challenge of deriving accurate time-dependent Hamiltonians for microwave-driven Josephson circuits with complex geometries. It introduces three complementary, geometry-aware methods—displaced-frame (DF), irrotational-gauge (IG), and overlap—to extract drive parameters from classical electromagnetic simulations, yielding Hamiltonians that accurately describe coherent drive dynamics in lumped and distributed circuits. It further develops PVNR, a port-noise susceptibility framework, to compute drive-induced decoherence rates via Fermi’s golden rule and Floquet–Markov theory, fully incorporating realistic noise spectra and port correlations. The approach is validated through case studies involving Purcell decay and drive-induced decoherence in transmon and SQUID-based circuits, showing agreement with HFSS eigenmode and admittance-based analyses while enabling efficient design optimization. This geometry-aware workflow enables fast, reliable design iterations for high-fidelity operations in superconducting quantum computing, and lays groundwork for extensions to pulsed drives, non-Markovian noise, nonreciprocal elements, and hybrid electromechanical systems.

Abstract

Time-dependent electromagnetic drives are fundamental for controlling complex quantum systems, including superconducting Josephson circuits. In these devices, accurate time-dependent Hamiltonian models are imperative for predicting their dynamics and designing high-fidelity quantum operations. Existing numerical methods, such as black-box quantization (BBQ) and energy-participation ratio (EPR), excel at modeling the static Hamiltonians of Josephson circuits. However, these techniques do not fully capture the behavior of driven circuits stimulated by external microwave drives, nor do they include a generalized approach to account for the inevitable noise and dissipation that enter through microwave ports. Here, we introduce novel numerical techniques that leverage classical microwave simulations that can be efficiently executed in finite element solvers, to obtain the time-dependent Hamiltonian of a microwave-driven superconducting circuit with arbitrary geometries. Importantly, our techniques do not rely on a lumped-element description of the superconducting circuit, in contrast to previous approaches to tackling this problem. We demonstrate the versatility of our approach by characterizing the driven properties of realistic circuit devices in complex electromagnetic environments, including coherent dynamics due to charge and flux modulation, as well as drive-induced relaxation and dephasing. Our techniques offer a powerful toolbox for optimizing circuit designs and advancing practical applications in superconducting quantum computing.

Figures (14)

• Figure 1: Flow diagram for numerically modeling a general Josephson circuit.a Two types of driven Josephson circuits: (i) lumped circuits with negligible distributed linear inductance, and (ii) distributed circuits. Each type is amenable to different methods developed in this work. b The time-dependent Hamiltonians in different reference frames are derived using three methods: (i) the displaced frame (DF) method, (ii) the irrotational-gauge (IG) method, and (iii) the overlap method. The first two methods are junction-centered approaches, focusing on simulating flux across junctions under excitation (or equivalently the impedance matrix $\mathbf{Z}(\omega)$) to extract the junction phase displacements $A_k(t)$ or effective phase modulations $F_{\mathrm{J}_{k}}(t)$ for each junction $k$. Conversely, the overlap method focuses on three-dimensional electric fields of the driven system. By computing the overlap between the displacement field of the driven circuit $\mathbf{D}_d(\mathbf{r}, t)$ and the electric field mode profile of each eigenmode $\mathbf{f}_i(\mathbf{r})$, we can extract the effective charge and residual phase displacement, denoted as $g_i(t)$ and $A_{\mathrm{res}, k}(t)$, respectively. While all three methods apply to lumped circuits, only the DF and overlap methods are suitable for distributed circuits. c Hamiltonians obtained with these methods describe the coherent dynamics of a driven circuit, enabling the extraction of relevant properties such as coherent process rates and AC Stark shifts. d The drive parameters are also useful for obtaining the circuit susceptibility function with respect to the drive port. This function allows for Fermi's golden rule or Floquet-Markov calculations of the circuit decoherence due to the thermal bath at the drive port.
• Figure 2: A lumped-element flux-tunable circuit under voltage and flux modulation. Circuit quantization that properly accounts for the EMF effect can produce a gauge-dependent circuit Hamiltonian, Eq. (\ref{['eq:H_SQUID']}). However, translating a physical circuit layout into such a lumped-element model is challenging due to the complexities of the circuit's geometry and the external field profile.
• Figure 3: Illustration of using the junction-centered approaches.a Black-box representation of Josephson circuit with $M$ linearized Josephson junctions driven by a voltage source $V_{\text{S}}$ with internal impedance $Z_0$. The impedance matrix $\mathbf{Z}(\omega)$ is used to obtained the modulation parameters in the displaced frame (junction phase displacements) or the irrotational gauge Hamiltonians (effective phase modulations) using Eqs. (\ref{['eq:A-from-Z']}) and (\ref{['eq:f-from-Z']}), respectively. b The irrotational gauge effective phase modulations can also be obtained using the impedance matrix of the same circuit, but with the junctions opened, using Eq. (\ref{['eq:f-from-Z-open']}). c Illustration of a SQUID-based flux-tunable transmon circuit that is modulated by an on-chip flux line (green). The two arrows indicate the directed lines $\mathrm{DL}_1$ and $\mathrm{DL}_2$, along which the phase displacements of the two junctions are evaluated in Eq. \ref{['eq:A_line_int']}. The inductances of the two junctions are $L_{\mathrm{J}_{1}} = L_{\mathrm{J}_{2}} = 15$ nH, and the (linearized) mode frequency is 7.673 GHz. The input power is set to be -50 dBm. d Phase displacements $\bar{A}_1$ and $\bar{A}_2$ of the two junctions in circuit c obtained with the displaced-frame method. Magnitudes and phases are shown separately. e Effective phase modulation parameters $\bar{F}_{\mathrm{J}_{1}}$ and $\bar{F}_{\mathrm{J}_{2}}$ of the same circuit obtained with the irrotational-gauge method, where filled markers refer to the circuit with closed junctions (a) and empty markers to opened junctions (b). Magnitudes and phases are shown separately. The shaded region indicates the frequency range where numerical convergence is difficult to reach for the closed junction approach due to the mode resonance, whereas the opened junction approach mitigates this issue and provides correct results over the entire frequency range.
• Figure 4: Illustration of the overlap method.a An example circuit where a flux-tunable transmon device is inductively coupled to a stub cavity Lu2023. b The overlap method involves two finite-element simulations: the eigenmode simulation of the electric field mode profile $\mathbf{f}_i$ of the $i$-th normal mode, and the driven simulation of the displacement field $\mathbf{D}_d$ at frequency $\omega_d$. By computing the overlap integral $O_i$ [Eq. (\ref{['eq:electric_overlap']})] between $\mathbf{f}_i$ and $\mathbf{D}_d$, we can obtain the effective charge modulation $\bar{g}_i$ for mode $i$ through Eq. (\ref{['eq:gi_overlap']}). c The amplitude of the $k$-th junction phase displacement contributed by the first $N$ modes $\lvert\sum_{i=1}^N\bar{A}_k^{(i)}\rvert$ (empty markers), compared against the amplitude of the total junction phase displacement $\lvert A_k\rvert$ (filled markers). The former is calculated from $\bar{g}_i$ obtained via the overlap method, while the latter is extracted using the displaced-frame method. Both simulations are performed under a drive power of -50 dBm and a drive frequency range of $2-14$ GHz. The contribution from the lowest $N$ modes progressively reproduces the total junction phase displacement as $N$ increases.
• Figure 5: The two pictures of the drive port voltage noise.a Josephson circuit driven by a sinusoidal voltage source $V_{\mathrm{S}_{}}$ with fluctuations, loaded by a resistance $Z_0$ at temperature $T$. b The voltage noise at the lossy resistor and the voltage source can be modeled as an ideal resistor in series with a voltage noise source $\delta V$ that incoherently drives the Josephson circuit, in addition to the coherent voltage drive.