Equivalent Circuit Model of JPAs

• Equivalent Circuit Models of JPAs are representations that capture the quantum-limited dynamics of superconducting amplifiers using a combination of linear LC resonators and nonlinear Josephson junction elements.
• The model rigorously linearizes complex interactions, enabling prediction of phenomena such as vacuum Rabi splitting, qubit Lamb shift, and parametric amplification gain through detailed circuit simulation.
• Advanced circuit constructions guide practical enhancements in impedance matching, bandwidth expansion, and noise minimization, offering actionable insights for scalable quantum readout architectures.

Josephson Parametric Amplifiers (JPAs) are superconducting nonlinear microwave amplifiers that leverage the parametric modulation of a Josephson inductance to achieve quantum-limited amplification. The equivalent circuit model of JPAs distills their essential physical and dynamical properties into a representation composed of passive linear and nonlinear circuit elements, facilitating both rigorous analysis and efficient simulation. This modeling framework forms the basis for quantitative prediction of key amplifier characteristics and enables practical optimization for quantum measurement and information processing tasks.

1. Foundational Hamiltonian and Linearized Equivalent Circuits

The core physical description begins with a combined system Hamiltonian encompassing the energy stored in the resonator, the interaction with the quantum system, and the quantum system itself:

$$H_\text{tot} = \frac{1}{2}(C_r \hat{V}_\text{res}^2 + L_r \hat{I}_\text{res}^2) + \frac{\hbar g}{2V_\text{RMS}} \sigma_1 \hat{V}_\text{res} + \frac{\hbar\omega_q}{2} \sigma_3$$

as developed in (Mátyás et al., 2011). Here, the first two terms encode a quantum LC-resonator (with voltage and current operators), the third describes resonator–qubit coupling, and the final term is the bare qubit energy splitting.

In the dispersive regime (detuning $\Delta = \omega_q - \omega_r \gg g$), the system can be linearized. The resulting coupled equations—the resonator–Bloch equations (RBEs)—reduce to a form mathematically equivalent to linear passive circuits, amenable to conventional circuit simulation. The resonator is modeled as an LC, and the qubit is effectively represented as an RLC, with interactions realized by either a coupling capacitor or a mutual inductance, depending on whether the JPA is based on charge or flux qubits.

Circuit parameters are derived from the RBEs:

$$C_r = \frac{1}{Z_0\omega_r},\quad L_r = \frac{Z_0}{\omega_r}$$

with qubit decoherence times $T_1$, $T_2$ and resonator decay rate $\gamma$ dictating effective resistances.

This linear equivalent model accurately captures observed phenomena, such as transmission anticrossing (vacuum Rabi splitting) and qubit Lamb shift, directly reflecting experimental measurements.

2. Nonlinear Inductance: Josephson Junctions and Arrays

JPAs employ the nonlinearity of Josephson junctions, whose current–phase relation is $I=I_c\sin\phi$, and energy $E(\phi) = -E_J\cos\phi$ (with $E_J = \hbar I_c/2e$) (Salmanogli et al., 17 Jul 2025). For small phase fluctuation, this is expanded to quartic order, yielding a "Kerr oscillatory" Hamiltonian for a single junction:

$H = Q^2/(2C) + (E_J/2) \phi^2 - (E_J/24) \phi^4$

where the quadratic term sets the resonance and the $\phi^4$ term provides the Kerr nonlinearity critical for parametric amplification.

Array architectures distribute the overall phase drop across $N$ junctions, leading to substantial dilution of nonlinearity: $K_\text{eff} \approx K_\text{single}/N$. This increases power handling and dynamic range, with direct mapping to a single nonlinear LC circuit (Planat et al., 2018).

Arrays are described by Lagrangians summing up capacitive and Josephson terms. Effective linear parameters are extracted by diagonalization:

$$C_\text{eff,n} = \vec{\phi}_n^\mathrm{T} \hat{C} \vec{\phi}_n,\quad L_\text{eff,n}^{-1} = \vec{\phi}_n^\mathrm{T} \hat{L}^{-1} \vec{\phi}_n$$

for mode $n$.

3. Advanced Equivalent Circuit Construction and Nonlinearities

Full circuit-level modeling involves incorporating detailed multi-junction nonlinear energy expansions, as in Josephson ring modulator (JRM)-based JPAs (Liu et al., 2019):

$$E_\text{JRM} \approx -4E_J\sin(\varphi_\text{ext}/4)\varphi_a \varphi_b \varphi_c - \frac{E_J}{96}\cos(\varphi_\text{ext}/4)(\varphi_a^4 + \varphi_b^4 + 16\varphi_c^4) + \cdots$$

Here, the cubic term drives parametric amplification, and quartic/quintic terms limit dynamic range.

Key circuit parameters:

• Shunt inductance ratio $\beta=L_J/L_\text{in}$ controls nonlinearity
• Participation ratio $p=L_\text{JRM}/(2L_\text{out}+L_\text{JRM})$ governs how strongly the JRM couples to the modes

Tuning external flux and shunt/series inductances strategically suppresses undesired nonlinearities, enhancing saturation power.

4. Impedance Engineering, Matching Networks, and Bandwidth Enhancement

Bandwidth and impedance matching are critical in JPA optimization. Integrated superconducting transmission line transformers, especially in Ruthroff topologies, extend operational bandwidth by reducing input impedance and resonance $Q$ (Ranzani et al., 2022, Patel et al., 12 Jul 2025). The transformer model is expressed as

$$Z_\text{ext} = 2Z_\text{oo}\frac{Z_0\cos\vartheta - 2jZ_\text{oo}\sin\vartheta}{4Z_\text{oo}(\cos\vartheta+1) - jZ_0\sin\vartheta}$$

where $Z_\text{oo}$ is odd mode impedance, $\vartheta$ the electrical length.

Matching networks based on Legendre polynomials can be systematically synthesized via continued fraction expansion from power loss functions $P_{IL}(\omega) = A[1 + k^2 P_N^2(\omega)]$, improving gain flatness and passband ripple (Kaufman et al., 2023). The resulting network prototype coefficients guide the synthesis of direct coupled-resonator implementations.

Impedance engineering via lumped-element LC networks, including Josephson arrays, achieves wide gain-bandwidth products (e.g., 18 dB over 400 MHz at 5.3 GHz for state-of-the-art devices fabricated in single-step lithography (Patel et al., 12 Jul 2025)).

5. Quantum-Limited Amplification and Dynamical Response

The dynamical response of JPAs is rigorously described in terms of Kerr Hamiltonians and quantum Langevin equations (Bhoite et al., 30 Jul 2025):

$$H_r = \hbar\omega_0 A^\dagger A + \frac{\hbar K}{2}A^\dagger A^\dagger AA$$

For a strong classical pump, steady-state operation is determined by a cubic equation for intracavity field energy, with solutions indicating possible bistability.

Linearization around this operating point enables calculation of parametric and intermodulation gain:

$$G_S = \frac{|\Gamma(\omega)|^2}{(\omega^2 + \lambda_0^2)(\omega^2 + \lambda_1^2)}$$

$$G_I = \frac{4\gamma^2 |V|^2}{(\omega^2 + \lambda_0^2)(\omega^2 + \lambda_1^2)}$$

where eigenvalues $\lambda_{0,1}$ define resonance properties.

Operating near critical points (e.g., $\omega_0 - \omega_p + K B^2 = 0$) allows optimization of gain and noise, but may approach the onset of bistability.

6. Practical Simulation, Optimization, and Multiphysics Approaches

Equivalent circuit models enable time-domain simulation using classical circuit tools and open-source simulators like JoSIM (Küçükyılmaz et al., 25 Aug 2025). Lumped element representations—Norton equivalents, LC resonators, nonlinear inductors—allow for calculation of frequency-domain reflection coefficients ($S_{11}$), bandwidth, and power handling.

Simulation frameworks based on “quantum-adapted” X-parameters interface harmonic balance solutions with quantum operator bases, supporting inclusion of real-world impairments and parasitics (Peng et al., 2022). Validation against analytical models and experiments is essential for accurate device design.

Multiphysics 1D and prospective 3D numerical modeling (using finite element and leapfrog time marching) incorporates arbitrary component variations, essential for understanding performance variability and tolerance sensitivity (Elkin et al., 22 Mar 2024).

7. Performance Metrics, Efficiency, and Applications

Key figures of merit include gain, bandwidth, compression point, quantum efficiency (QE), and power added efficiency (PAE). Polynomial current–phase relation modeling enables the tailoring and optimization of nonlinearity, with high PAE values attainable via staged or chained amplifiers (e.g., up to 95% for cascaded architectures versus <0.1% in uncompensated traditional JPAs) (Hougland et al., 19 Feb 2024). Saturation power is strongly influenced by the circuit’s nonlinearity distribution and parameter tuning.

JPAs are central to quantum system readout chains, qubit measurement, quantum-limited RF amplification, and precision microwave quantum optics. Advanced architectures incorporating Blochnium elements, comb-like gain networks, and tailored matching further enhance selectivity, dynamic range, and integration for scalable quantum computing deployments (Salmanogli et al., 17 Jul 2025).

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The equivalent circuit model of JPAs provides a quantitative, versatile framework connecting fundamental quantum nonlinear oscillator dynamics to practical amplifier construction, performance optimization, and scalable simulation. These models underpin the analysis, synthesis, and implementation of JPAs in quantum-limited amplification scenarios and enable further innovation through rigorous integration of nonlinear circuit theory, engineered impedance environments, and multiphysics simulation methodologies.