Recently, a one-to-one correspondence between non-trivial non-Hermitian topology and directional amplification has been demonstrated, theoretically and experimentally, for the case of one complex band. Here, we extend our framework to multiple bands and higher spatial dimension. This proves to be far from trivial. Building on the singular value decomposition, we introduce a new quantity that we dub generalised singular spectrum (GSS). The GSS allows us to define physically meaningful bands related to the system's scattering behaviour and to define invariants for novel notions of point gaps (non-Hermitian topology) and line gaps (Hermitian-like topology), respectively. For both invariants, we prove a bulk-boundary correspondence and show that they give rise to two different kinds of topological edge modes. We illustrate our results with a 1D non-Hermitian Su-Schrieffer-Heeger (SSH) model and a 2D non-Hermitian model that features corner-to-corner amplification. Our work is relevant for many state-of-the-art experimental platforms and it sets the stage for applications such as novel directional amplifiers and non-reciprocal sensors.
Training of physical neural networks
Ali Momeni,
Babak Rahmani,
Benjamin Scellier,
Logan G. Wright,
Peter L. McMahon,
Clara C. Wanjura,
Yuhang Li,
Anas Skalli,
Natalia G. Berloff, et al.
Physical neural networks (PNNs) are a class of neural-like networks that make use of analogue physical systems to perform computations. Although at present confined to small-scale laboratory demonstrations, PNNs could one day transform how artificial intelligence (AI) calculations are performed. Could we train AI models many orders of magnitude larger than present ones? Could we perform model inference locally and privately on edge devices? Research over the past few years has shown that the answer to these questions is probably “yes, with enough research”. Because PNNs can make use of analogue physical computations more directly, flexibly and opportunistically than traditional computing hardware, they could change what is possible and practical for AI systems. To do this, however, will require notable progress, rethinking both how AI models work and how they are trained—primarily by considering the problems through the constraints of the underlying hardware physics. To train PNNs, backpropagation-based and backpropagation-free approaches are now being explored. These methods have various trade-offs and, so far, no method has been shown to scale to large models with the same performance as the backpropagation algorithm widely used in deep learning today. However, this challenge has been rapidly changing and a diverse ecosystem of training techniques provides clues for how PNNs may one day be used to create both more efficient and larger-scale realizations of present-scale AI models.
Training nonlinear optical neural networks with Scattering Backpropagation
Nicola Dal Cin,
Florian Marquardt,
Clara C. Wanjura
As deep learning applications continue to deploy increasingly large artificial neural networks, the associated high energy demands are creating a need for alternative neuromorphic approaches. Optics and photonics are particularly compelling platforms as they offer high speeds and energy efficiency. Neuromorphic systems based on nonlinear optics promise high expressivity with a minimal number of parameters. However, so far, there is no efficient and generic physics-based training method allowing us to extract gradients for the most general class of nonlinear optical systems. In this work, we present Scattering Backpropagation, an efficient method for experimentally measuring approximated gradients for nonlinear optical neural networks. Remarkably, our approach does not require a mathematical model of the physical nonlinearity, and only involves two scattering experiments to extract all gradient approximations. The estimation precision depends on the deviation from reciprocity. We successfully apply our method to well-known benchmarks such as XOR and MNIST. Scattering Backpropagation is widely applicable to existing state-of-the-art, scalable platforms, such as optics, microwave, and also extends to other physical platforms such as electrical circuits.
Artificial discovery of lattice models for wave transport
Jonas Landgraf,
Clara C. Wanjura,
Vittorio Peano,
Florian Marquardt
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.
Magnetic tunnel junctions driven by hybrid optical-electrical signals as a flexible neuromorphic computing platform
Felix Oberbauer,
Tristan Joachim Winkel,
Tim Böhnert,
Clara C. Wanjura,
Marcel S. Claro,
Luana Benetti,
Ihsan Çaha,
Francis Leonard Deepak,
Farshad Moradi, et al.
Communications Physics
8
329
(2025)
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Magnetic tunnel junctions (MTJs) offer a promising pathway toward energy-efficient neuromorphic computing due to their nanoscale footprint, nonvolatile switching, and intrinsic nonlinear dynamics that emulate synaptic behavior. However, generating large thermoelectric voltages with bias-tunable nonlinearities for neuromorphic use remains largely unexplored. Here, we introduce a hybrid opto-electrical excitation scheme—combining pulsed laser heating with DC bias—to drive MTJs into the nonlinear bias-enhanced tunnel magneto-Seebeck regime. This regime yields thermoelectric voltages in the tens of millivolts with a strong contrast between magnetic states, while also revealing spiking and double-switching behavior linked to vortex dynamics and fixed-layer depinning. The thermovoltage exhibits cubic dependence on bias current, enabling tunable synaptic weights. We simulate a single-layer neuromorphic network using optically encoded inputs and achieve 93.7% classification accuracy on handwritten digits. These results establish hybrid-driven MTJs as a compact, CMOS-compatible platform for neuromorphic computing, integrating optical input with spintronic functionality.
Quantum equilibrium propagation for efficient training of quantum systems based on Onsager reciprocity
Clara C. Wanjura,
Florian Marquardt
Nature Communications
16
6595
(2025)
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The widespread adoption of machine learning and artificial intelligence in all branches of science and technology creates a need for energy-efficient, alternative hardware. While such neuromorphic systems have been demonstrated in a wide range of platforms, it remains an open challenge to find efficient and general physics-based training approaches. Equilibrium propagation (EP), the most widely studied approach, has been introduced for classical energy-based models relaxing to an equilibrium. Here, we show a direct connection between EP and Onsager reciprocity and exploit this to derive a quantum version of EP. For an arbitrary quantum system, this can now be used to extract training gradients with respect to all tuneable parameters via a single linear response experiment. We illustrate this new concept in examples in which the input or the task is of quantum-mechanical nature, e.g., the recognition of many-body ground states, phase discovery, sensing, and phase boundary exploration. Quantum EP may be used to solve challenges such as quantum phase discovery for Hamiltonians which are classically hard to simulate or even partially unknown. Our scheme is relevant for a variety of quantum simulation platforms such as ion chains, superconducting circuits, Rydberg atom tweezer arrays and ultracold atoms in optical lattices.
Massive quantum systems as interfaces of quantum mechanics and gravity
Sougato Bose,
Ivette Fuentes,
Andrew A. Geraci,
Saba Mehsar Khan,
Sofia Qvarfort,
Markus Rademacher,
Muddassar Rashid,
Marko Toroš,
Hendrik Ulbricht, et al.
Reviews of Modern Physics
97
015003
(2025)
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The traditional view from particle physics is that quantum-gravity effects should become detectable only at extremely high energies and small length scales. Owing to the significant technological challenges involved, there has been limited progress in identifying experimentally detectable effects that can be accessed in the foreseeable future. However, in recent decades, the size and mass of quantum systems that can be controlled in the laboratory have reached unprecedented scales, enabled by advances in ground-state cooling and quantum-control techniques. Preparations of massive systems in quantum states pave the way for the exploration of a low-energy regime in which gravity can be both sourced and probed by quantum systems. Such approaches constitute an increasingly viable alternative to accelerator-based, laser-interferometric, torsion-balance, and cosmological tests of gravity. This review provides an overview of proposals where massive quantum systems act as interfaces between quantum mechanics and gravity. Conceptual difficulties in the theoretical description of quantum systems in the presence of gravity are discussed, tools for modeling massive quantum systems in the laboratory are reviewed, and an overview of the current state-of-the-art experimental landscape is provided. Proposals covered in this review include precision tests of gravity, tests of gravitationally induced wave-function collapse and decoherence, and gravity-mediated entanglement. The review concludes with an outlook and a summary of the key questions raised.
Kontakt
ForschungsgruppeClara Wanjura
Max-Planck-Institut für die Physik des Lichts Staudtstr. 2 91058 Erlangen
