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arXiv:2109.11628 · PRX Quantum

Synthesis of Parametrically Coupled Networks

Ofer Naaman, José Aumentado

Published May 16, 2022• 49 citations
arXivHTMLPDF

Abstract

We show that a common language can be used to unify the description of parametrically-coupled circuits--parametric amplifiers, frequency converters, and parametric nonreciprocal devices--with that of band-pass filter and impedance matching networks. This enables one to readily adapt network synthesis methods from microwave engineering in the design of parametrically-coupled devices having prescribed transfer characteristics, e.g. gain, bandwidth, return loss, and isolation. We review basic practical aspects of coupled mode theory and filter synthesis, and then show how to apply both, on an equal footing, to the design of multi-pole, broadband parametric and non-reciprocal networks. We supplement the discussion with a range of examples and reference designs.

DOI
10.1103/prxquantum.3.020201

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We propose and realize a multi-resonator coupled impedance-matched Josephson parametric amplifier. Parallel plate capacitors serve as couplers to achieve coupling between the nonlinear resonator containing the Josephson loop and the half-wavelength resonator, and between the half-wavelength resonators themselves. By precisely designing each coupling strength, we effectively extend the amplifier's bandwidth. Through a combination of circuit-level and electromagnetic field simulations, we optimize the structural parameters of the resonators and capacitors. The two measured devices have 15 dB gain with a bandwidth of 1.3 GHz and 20 dB gain with a bandwidth of 1.1 GHz, respectively, with near-quantum-limited noise temperatures.
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Josephson parametric devices are widely used in superconducting quantum computing research but suffer from an inherent gain-bandwidth trade-off. This limitation is partly overcome by coupling the device to its input/output transmission line via a bandpass filter, leading to wider bandwidth at undiminished gain. Here we perform a non-perturbative circuit analysis in terms of dressed transmission-line modes for representative resonant coupling circuits, going beyond the weak-coupling treatment. The strong frequency dependence of the resulting coupling coefficients implies that the Markov approximation commonly employed in cQED analysis is inadequate. By retaining the full frequency dependence of the coupling, we arrive at a non-Markovian form of the quantum Langevin equation with the frequency-dependent complex-valued self-energy of the coupling in place of a single damping parameter. We also consistently generalize the input-output relations and unitarity conditions. Using the exact self-energies of elementary filter networks -- a series- and parallel-LC circuit and a simple representative bandpass filter consisting of their combination -- we calculate the generalized parametric gain factors. Compared with their Markovian counterpart, these gain profiles are strongly modified. We find bandwidth broadening not only in the established parameter regime, where the self-energy of the coupling is in resonance with the device and its real part has unity slope, but also within off-resonant parameter regimes where the real part of the self-energy is large. Our results offer insight for the bandwidth engineering of Josephson parametric devices using simple coupling networks.
Superconducting parametric amplifiers have great promise for quantum-limited readout of superconducting qubits and detectors. Until recently, most superconducting parametric amplifiers had been based on resonant structures, limiting their bandwidth and dynamic range. Broadband traveling-wave parametric amplifiers based both on the nonlinear kinetic inductance of superconducting thin films and on Josephson junctions are in development. By modifying the dispersion property of the amplifier circuit, referred to as dispersion engineering, the gain can be greatly enhanced and the size can be reduced. We present two theoretical frameworks for analyzing and understanding such parametric amplifiers: 1) generalized coupled-mode equations and 2) a finite difference time domain (FDTD) model combined with a small signal analysis. We show how these analytical and numerical tools may be used to understand device performance.
In this thesis we propose new methods in multi-qubit multi-mode circuit quantum electrodynamics (circuit-QED) architectures. In Chapter 2 we describe a direct parity measurement method for three qubits, which can be realized in 2D circuit-QED with a possible extension to four qubits in a 3D circuit-QED setup for the implementation of the surface code. In Chapter 3 we show how to derive Hamiltonians and compute relaxation rates of the multi-mode superconducting microwave circuits consisting of single Josephson junctions using an exact impedance synthesis technique (the Brune synthesis) and applying previous formalisms for lumped element circuit quantization. In the rest of the thesis we extend our method in Chapter 3 to multi-junction (multi-qubit) multi-mode circuits through the use of state-space descriptions which allows us to quantize any multiport microwave superconducting circuit with a reciprocal lossy impedance response.