Artificial Transmission Line Synthesis Tailored for Traveling-Wave Parametric Processes

TL;DR

The paper addresses the lack of systematic design tools for traveling-wave parametric amplifiers by developing a general synthesis framework for lossless artificial transmission lines (ATLs). It articulates two complementary design paths—periodic loading of frequency-independent components and filter-based synthesis of frequency responses in uniform lines—and derives a generalized dispersion relation and stopband/phase-matching strategies. The authors validate the framework by designing and simulating two novel TWPAs: a four-wave-mixing kinetic-inductance TWPA with a novel phase-matching architecture and an ambidextrous, backward-pumped TWPA based on a right-left-handed transmission line, achieving wideband gain and suppression of spurious processes. The work provides a systematic toolkit for ATL-based device concepts in superconducting circuits, potentially enabling optimized quantum-limited amplification and exotic photon-state generation.

Abstract

Artificial transmission lines built with lumped-element inductors and capacitors form the backbone of broadband, nearly quantum-limited traveling-wave parametric amplifiers (TWPAs). However, systematic design methods for TWPAs, and more generally artificial transmission lines, are lacking. Here, I develop a general synthesis framework for lossless artificial transmission lines by borrowing from periodic structure theory and passive network synthesis. These complementary approaches divide the design space: periodic loading synthesis employs spatial modulation of frequency-independent components, while filter synthesis employs frequency-dependent responses in spatially-uniform components. When tailoring transmission lines for parametric processes, nonlinear elements are added, typically nonlinear inductances in superconducting circuits, while ensuring energy and momentum conservation between interacting tones. Applying this framework, I design a kinetic inductance TWPA with a novel phase-matching architecture, and a backward-pumped Josephson TWPA exploiting an ambidextrous i.e., right-left-handed transmission line.

Figures (10)

• Figure 1: General topology of an artificial transmission line. (a) Cascaded unit cells, comprising series reactances (typically inductors) and shunt susceptances (typically capacitors), can be made antimetric (blue) or symmetric (orange). (b) The synthesis framework divides into two approaches: periodic loading (left) where the inductors and capacitors vary spatially, and filter synthesis (right) where the spatially-invariant reactances (X) and susceptances (B) are engineered to give a specific frequency response.
• Figure 2: Example of stopband synthesis via periodic modulation of the ATL shunt capacitance. (a) The band diagram shows the dispersion relation for the periodically loaded ATL with two synthesized stopbands at $8$ and $24$ GHz. Each line represents a different spatial harmonic $n$, with blue lines indicating modes where $k + nk_d > 0$ (forward propagation) and orange lines where $k + nk_d < 0$ (backward propagation). The stopbands appear at the edge of the Brillouin zones where counter-propagating modes couple, creating the bandgaps shown in gray. The inset shows the normalized capacitance modulation profile over one supercell ($d = 59$ cells), consisting of two overlapped cosine modulations. (b) Zoom of the first stopband at $8$ GHz with width $\Delta_1^+ = 0.2$ GHz. (c) Zoom of the third stopband at $24$ GHz with width $\Delta_3^- = 2$ GHz.
• Figure 3: Concept of a low-pass ATL prototype. (a) Both filters and ATLs can be characterized by their reflection coefficient $\Gamma$. For a filter, $\Gamma$ relates to the input impedance $Z_1$ computed from the ABCD matrix with a $1$ termination. For an ATL, $\Gamma$ relates to the Bloch impedance $Z_b$ computed from the unit cell ABCD matrix with the Bloch impedance as its own termination. (b) The transmission $1-|\Gamma|^2$ through a B2-ATL prototype (blue) presents a low-pass behavior comparable to that of a B2 filter (orange), enabling filter synthesis techniques for ATL design.
• Figure 4: Filter transformations of a low-pass ATL prototype. (a) Starting from a low-pass ATL whose unit cell comprises a series inductance and a shunt capacitance, systematic frequency transformations of the complex frequency $s=j\omega$ create different pass-stop responses. (b) The high-pass transformation inverts the component types, creating a left-handed ATL with $k < 0$ in the passband. (c) The band-pass transformation produces series and parallel LC resonators, yielding an ambidextrous (composite right-left-handed) ATL with the passband transitioning from $k < 0$ to $k > 0$ at the transmission zero $\omega_1$. (d) The low-pass and notch transformation creates either series or parallel filter configurations, introducing a transmission pole at $\omega_0$, and a zero at $\omega_1$. For each transformation, the transmission $1 - |\Gamma|^2$ and wavenumber $k$ demonstrate the distinct amplitude and phase characteristics of the resulting ATL.
• Figure 5: Linear response design of the 4WM KTWPA. (a) The supercell comprises a $9$-cell LC ladder with periodically modulated shunt capacitors, where one unit cell contains a series LC resonator (rpm filter) placed in parallel with the series inductor. (b) The power transmission $|S_{21}|$ (in decibel) through a $5004$-cell ATL shows the phase-matching notch at $10$ GHz created by the rpm filter, and the third-harmonic stopband at $27$ GHz, created by the periodic capacitor modulation. (c) The wavenumber $k$ of the corresponding infinite lattice demonstrates rapid phase accumulation near $10$ GHz essential for 4WM phase matching, and the bandgap at $27$ GHz that suppresses third-harmonic generation.