Artificial discovery of lattice models for wave transport

Abstract

Wave transport devices, such as amplifiers, frequency converters, and nonreciprocal devices, are essential for modern communication, signal processing, and sensing applications. Of particular interest are traveling wave setups, which offer excellent gain and bandwidth properties. So far, the conceptual design of those devices has relied on human ingenuity. This makes it difficult and time-consuming to explore the full design space under a variety of constraints and target functionalities. In our work, we present a method which automates this challenge. By optimizing the discrete and continuous parameters of periodic coupled-mode lattices, our approach identifies the simplest lattices that achieve the target transport functionality, and we apply it to discover new schemes for directional amplifiers, isolators, and frequency demultiplexers. Leveraging automated symbolic regression tools, we find closed analytical expressions that facilitate the discovery of generalizable construction rules. Moreover, we utilize important conceptual connections between the device transport properties and non-Hermitian topology. The resulting structures can be implemented on a variety of platforms, including microwave, optical, and optomechanical systems. Our approach opens the door to extensions like the artificial discovery of lattice models with desired properties in higher dimensions or with nonlinear interactions.

Figures (6)

• Figure 1: Automated discovery of lattice models for optimized wave transport. (a) Possible target functionalities for wave transport. These include unidirectional amplifiers, frequency demultiplexers, and isolators (from top to bottom). (b) Target properties relevant for the design of wave transport devices. (c) Non-Hermitian lattice model. We consider an open one-dimensional lattice of identical unit cells, each of which consists of multiple modes (differently colored circles). The modes can be coupled via two-mode squeezing (blue edges), and real and complex-valued beamsplitter interactions (black and green edges). They can be detuned (black self-loops) and coupled to external waveguides or baths, leading to dissipation (represented by waves emanating from the mode). Such a lattice can amplify an input signal from one end to the other. The amplification is characterized by the gain rate, which is frequency dependent (see inset). (d) Illustrations of some of the many hardware platforms suitable for wave transport in periodic structures. These include superconducting optomechanical circuits youssefi2022topological, plasmonic waveguides Wetter2023Observation, photonic crystal nanobeams slim2024optomechanical, and superconducting circuits macklin2015near (from top to bottom). (e) Optimization protocol. Given a candidate lattice, the continuous optimization (blue background) optimizes the continuous parameters to achieve the target functionality defined over the loss function $\mathcal{L}$. If the loss can be minimized to zero, the lattice is added to the list of valid lattices (green book); otherwise, to the list of invalid lattices (red book). The discrete optimization (yellow background) suggests new lattices until all possible lattices have been sorted into the two lists.
• Figure 2: Connection between the gain properties and the transfer matrix. (a) Gain factor (blue) and reverse gain (orange) as a function of the input frequency. (b) Transfer matrix approach. The transfer matrix describes how excitations on the current and previous unit cell are passed on to the next unit cell. (c) Spectrum of the transfer matrix. The $M$th smallest eigenvalue is the gain rate per unit cell, the $(M+1)$th smallest eigenvalue is the inverse of the reverse gain rate. The distance between the $M$th smallest eigenvalue and its nearest neighbors determines the ripple. (d) Connection between the transfer matrix and non-Hermitian topology, shown for two exemplary points from (c). The curve shows the real and imaginary part of $\det(H(k)-\omega\mathds{1})$ as a function of $k$. In (i), $\abs{\lambda_M}$ is greater than 1.0 (green-shaded area in (c)), resulting in a topological winding number $w=-1$, see \ref{['eq:winding']}. In (ii), $\abs{\lambda_M}$ and $1/\abs{\lambda_{M+1}}$ are smaller than 1.0 resulting in a winding number $w=0$.
• Figure 3: Designing an amplifier with constant gain rate over a wide bandwidth. (a) Target behavior. The device is designed to amplify a signal from left to right, while attenuating in the reverse direction. The forward gain should remain as constant as possible across the frequency range of interest. (b) One of the discovered lattices, see Supplemental Material for an exhaustive list. (c) Transfer matrix spectrum of the lattice in (b). The green-shaded area is the frequency range in which $\abs{\lambda_M}$ is greater than 1.0. (d) Forward and reverse gain profiles (blue and orange) for the lattice in (b) with 10.0 unit cells. (e,f) Topology of the designed lattice. In (e), $\omega$ equals $0$ (within the green shaded area in (c)). In (f), $\omega$ equals $2.5\kappa_\mathrm{in}$ (outside of the green-shaded area).
• Figure 4: Automated discovery of symbolic expressions of the optimized lattice models. (a) Pathway to find symbolic expressions. Given a candidate lattice model (here \ref{['fig:amplifier']}(b)), we run the continuous optimization for different target gain rates $s_\mathrm{target}$, creating a dataset of gain rates and lattice model parameters. Using the symbolic regression tool AIFeynman 2, we find symbolic expressions that are Pareto-optimal, i.e., having the smallest fit error given a certain complexity. The fit error is defined as the mean error description length, and complexity as the description length of the symbolic expression, see AIFeynman2wu2019toward. (b) Comparison between the dataset values and \ref{['equ:gain_rate_equation']} found via symbolic regression. (c) Design of amplifiers with a target gain rate per unit cell and target bandwidth (dashed red lines) using the discovered symbolic expressions in \ref{['equ:gain_rate_equation', 'equ:bandwidth_equation', 'equ:sqeezing_rates_1', 'equ:sqeezing_rates_2']}. We engineer an amplifier with a large bandwidth and small gain rate per unit cell (left), and with a large gain rate and a small bandwidth (right).
• Figure 5: Automated design of an isolator. (a) Target behavior. The device transmits a signal from left to right with unity transmission, while suppressing the reverse direction. (b) Previously proposed few-mode isolator schemes. (LI), proposed in lecocq_nonreciprocal_2017sliwa2015reconfigurablehabraken_continuous_2012, couples input and output (orange) to one auxiliary mode (gray). In the following, we consider the case where the auxiliary mode is overdamped and acts as a bath, see metelmann_nonreciprocal_2015. In (LII), the input and output are microwave modes (orange) coupled via two strongly detuned mechanical modes (gray), see bernier2017nonreciprocal. (c) Two of the discovered lattice models, see Supplemental Material for an exhaustive list. (I) only uses beamsplitter couplings, (II) also incorporates squeezing. The gray modes in (I) are lossless. (d) Transmission profile of the isolator schemes from (b) and (c). To ensure a fair comparison, the in- and out-coupling rates of all schemes are set to $\kappa_\mathrm{in}$. The schemes (LI), (LII), and (I) share the same transmission spectrum (blue curve). The reverse transmission (not shown) vanishes for (LI) and (LII). For the lattices (I) and (II), it is exponentially suppressed in the thermodynamic limit. The length of these lattices is set to $N=40$. (e) Spectrum of the transfer matrix for lattice (II).