Diese Seite zeigt alle Veröffentlichungen der MPL Theorieabteilung, beginnend 2016, inklusive aller zugehöriger Forschungsgruppen. Die individuellen Veröffentlichungslisten finden Sie auf den Seiten der Gruppen Marquardt, Krenn und Kunst.
Machine learning has shown significant breakthroughs in quantum science, where in particular deep neural networks exhibited remarkable power in modeling quantum manybody systems. Here, we explore how the capacity of datadriven deep neural networks in learning the dynamics of physical observables is correlated with the scrambling of quantum information. We train a neural network to find a mapping from the parameters of a model to the evolution of observables in random quantum circuits for various regimes of quantum<br>scrambling and test its \textit{generalization} and \textit{extrapolation} capabilities in applying it to unseen circuits. Our results show that a particular type of recurrent neural network is extremely powerful in generalizing its predictions within the system size and time window that it has been trained on for both, localized and scrambled regimes. These include<br>regimes where classical learning approaches are known to fail in sampling from a representation of the full wave function. Moreover, the considered neural network succeeds in \textit{extrapolating} its predictions beyond the time window and system size that it has been trained on for models that show localization, but not in scrambled regimes.
Precision Quantum Parameter Inference with Continuous Observation
Quantum Parameter Estimation (QPE) is important from the perspective of both fundamental quantum research and various practical applications of quantum technologies such as for developing optimal quantum control strategies. Standard and traditional methods for QPE involve projective measurements on thousands of identically prepared quantum systems. However, these methods face limitations, particularly in terms of the required number of samples and the associated experimental resources. In this work, we present a novel method for precise QPE that diverges from conventional techniques, employs continuous measurements, and enables accurate QPE with a single quantum trajectory. In an application, we demonstrate the use of the method for the task of parameter estimation and force sensing of a levitated nanoparticle.
Fully NonLinear Neuromorphic Computing with Linear Wave Scattering
The increasing complexity of neural networks and the energy consumption associated with training and inference create a need for alternative neuromorphic approaches, e.g. using optics. Current proposals and implementations rely on physical nonlinearities or optoelectronic conversion to realise the required nonlinear activation function. However, there are significant challenges with these approaches related to power levels, control, energyefficiency, and delays. Here, we present a scheme for a neuromorphic system that relies on linear wave scattering and yet achieves nonlinear processing with a high expressivity. The key idea is to inject the input via physical parameters that affect the scattering processes. Moreover, we show that gradients needed for training can be directly measured in scattering experiments. We predict classification accuracies on par with results obtained by standard artificial neural networks. Our proposal can be readily implemented with existing stateoftheart, scalable platforms, e.g. in optics, microwave and electrical circuits, and we propose an integratedphotonics implementation based on racetrack resonators that achieves high connectivity with a minimal number of waveguide crossings.
Discovering Local HiddenVariable Models for Arbitrary Multipartite Entangled States and Arbitrary Measurements
Measurement correlations in quantum systems can exhibit nonlocal behavior, a fundamental aspect of quantum mechanics with applications such as deviceindependent quantum information processing. However, the explicit construction of local hiddenvariable (LHV) models remains an outstanding challenge in the general setting. To address this, we develop an approach that employs gradientdescent algorithms from machine learning to find LHV models which reproduce the statis tics of arbitrary measurements for quantum manybody states. In contrast to previous approaches, our method employs a general ansatz, enabling it to discover an LHV model in all cases where the state is local. Therefore, it provides actual estimates for the critical noise levels at which twoqubit Werner states and threequbit GHZ and W states become nonlocal. Furthermore, we find evidence suggesting that twospin subsystems in the ground states of translationally invariant Hamiltonians are local, while bigger subsystems are in general not. Our method now offers a quantitative tool for determining the regimes of nonlocality in any given physical context, including scenarios involving nonequilibrium and decoherence.
Theory of symmetryresolved quenchdrive spectroscopy: Nonlinear response of phasefluctuating superconductors
Matteo Puviani
Physical Review B (109)
214515
(2024)

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Recent experiments on cuprates have shown the possibility of opening a gap above the superconducting critical temperature, in the socalled phasefluctuating state, by enhancing the phase coherence of preformed Cooper pairs. Quenchdrive spectroscopy, an implementation of 2D coherent spectroscopy, has emerged as a powerful tool for investigating outofequilibrium superconductors and their collective modes. In this paper, we enrich the quenchdrive scheme by developing a systematic generalization to study the nonlinear response of dwave incoherent Cooper pairs in a symmetryresolved manner. In particular, we not only show that it is possible to obtain a thirdharmonic signal from fully incoherent pairs with an equilibrium vanishing order parameter, but we also characterize the full flourishing 2D spectrum of the generated nonlinear response. The results provide a deeper theoretical insight on recent experimental results, opening the door to a symmetrydriven design of future experiments on unconventional and enhanced superconductors.
Merging automatic differentiation and the adjoint method for photonic inverse design
Alexander Luce, Rasoul Alaee, Fabian Knorr, Florian Marquardt
Machine Learning: Science and Technology
5(2)
025076
(2024)

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Optimizing the shapes and topology of physical devices is crucial for both scientific and technological advancements, given their wideranging implications across numerous industries and research areas. Innovations in shape and topology optimization have been observed across a wide range of fields, notably structural mechanics, fluid mechanics, and more recently, photonics. Gradientbased inverse design techniques have been particularly successful for photonic and optical problems, resulting in integrated, miniaturized hardware that has set new standards in device performance. To calculate the gradients, there are typically two approaches: namely, either by implementing specialized solvers using automatic differentiation (AD) or by deriving analytical solutions for gradient calculation and adjoint sources by hand. In this work, we propose a middle ground and present a hybrid approach that leverages and enables the benefits of AD for handling gradient derivation while using existing, proven but blackbox photonic solvers for numerical solutions. Utilizing the adjoint method, we make existing numerical solvers differentiable and seamlessly integrate them into an AD framework. Further, this enables users to integrate the optimization environment seamlessly with other autodifferentiable components such as machine learning, geometry generation, or intricate postprocessing which could lead to better photonic design workflows. We illustrate the approach through two distinct photonic optimization problems: optimizing the Purcell factor of a magnetic dipole in the vicinity of an optical nanocavity and enhancing the light extraction efficiency of a µLED.
Predicting atmospheric turbulence for secure quantum communications in free space
Tareq Jaouni, Lukas Scarfe, Frédéric Bouchard, Mario Krenn, Khabat Heshami, Francesco Di Colandrea, Ebrahim Karimi
Atmospheric turbulence is the main barrier to largescale freespace quantum communication networks. Aberrations distort optical information carriers, thus limiting or preventing the possibility of establishing a secure link between two parties. For this reason, forecasting the turbulence strength within an optical channel is highly desirable, as it allows for knowing the optimal timing to establish a secure link in advance. Here, we train a Recurrent Neural Network, TAROCCO, to predict the turbulence strength within a freespace channel. The training is based on weather and turbulence data collected over 9 months for a 5.4 km intracity freespace link across the City of Ottawa. The implications of accurate predictions from our network are demonstrated in a simulated high dimensional Quantum Key Distribution protocol based on orbital angular momentum states of light across different turbulence regimes. TAROCCO will be crucial in validating a freespace channel to optimally route the key exchange for secure communications in real experimental scenarios.
Coherent pair injection as a route towards the enhancement of supersolid order in manybody bosonic models
Emmanouil Grigoriou, Zhiyao Ning, Hang Su, Benjamin Löckler, Ming Li, Yoshitomo Kamiya, Carlos NavarreteBenlloch
Physical Review A (109)
063324
(2024)

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Over the last couple of decades, quantum simulators have been probing quantum manybody physics with un precedented levels of control. So far, the main focus has been on the access to novel observables and dynamical conditions related to condensedmatter models. However, the potential of quantum simulators goes beyond the traditional scope of condensedmatter physics: Being based on drivendissipative quantum optical platforms, quantum simulators allow for processes that are typically not considered in condensedmatter physics. These processes can enrich in unexplored ways the phase diagram of wellestablished models. Taking the extended BoseHubbard model as the guiding example, in this work we examine the impact of coherent pair injection, a process readily available in, for example, superconducting circuit arrays. The interest behind this process is that, in contrast to the standard injection of single excitations, it can be configured to preserve the U(1) symmetry underlying the model. We prove that this process favors both superfluid and densitywave order, as opposed to insulation or homogeneous states, thereby providing a novel route towards the access of lattice supersolidity.
Revisiting N2 with NeuralNetworkSupported CI
Yorick L. A. Schmerwitz, Louis Thirion, Gianluca Levi, Elvar Ö. Jónsson, Pavlo Bilous, Hannes Jónsson, Philipp Hansmann
We apply a recently proposed computational protocol for a neuralnetworksupported configura tion interaction (NN CI) calculation to the paradigmatic N2 molecule. By comparison of correlation energy, binding energy, and the full dissociation curve to experimental and full CI benchmarks, we demonstrate the applicability and robustness of our approach for the first time in the context of molecular systems, and offer thereby a new complementary tool in the family of machinelearning based computation methods. The main advantage of the method lies in the efficiency of the neural networkselected manybody basis set. Specifically, we approximate full CI results obtained on bases of ≈ 1010 Slater Determinants with only ≈ 105 determinants with good accuracy. The high effi ciency of the NN CI approach underlines its potential for broader applications such as structural optimizations and even computation of spectroscopic observables in systems for which computational resources are a limiting factor.
Protected gap closing and reopening in topologicalinsulator Josephson junctions
Jakob Schluck, Ella Nikodem, Anton Montag, Alexander Ziesen, Mahasweta Bagchi, Fabian Hassler, Yoichi Ando
In the seminal proposal by Fu and Kane, the superconducting proximity effect was predicted to transform the surface state of a topological insulator (TI) into a topological superconduc tor, forming a nonchiral 1D Majorana state within a linear Josephson junction on the TI surface. The hallmark of this 1D Majorana state is a robust gap closing as a function of the superconducting phase difference φ across the junction, which alternates in and out of the topological phase. These topological phasetransitions occur at φ = (2n + 1)π with integer n, leading to a 4πperiodicity of the ground state. While the 4πperiodicity has been indirectly inferred in the AC Josephson effect, the direct observation of the 1D Majorana state in a TI Josephson junction has remained contentious. Here, we report the direct observation of topological phasetransitions in a TI Josephson junction, where the local density of states is probed via tunnel contacts and φ is controlled by a flux loop. The observed transitions are independent of the chemical potential, reinforcing their topological origin. Under an applied perpendicular magnetic field, Josephson vortices form, making φ positiondependent. In this case, the gap closing occurs locally at the Josephson vortex cores where φ = (2n + 1)π, which we also observe. Our findings provide direct confirmation of the FuKane proposal and ro bust evidence for the emergence of topological superconductivity in a TI Josephson junction.
Quantum Equilibrium Propagation for efficient training of quantum systems based on Onsager reciprocity
The widespread adoption of machine learning and artificial intelligence in all branches of science and technology has created a need for energyefficient, alternative hardware platforms. While such neuromorphic approaches have been proposed and realised for a wide range of platforms, physically extracting the gradients required for training remains challenging as generic approaches only exist in certain cases. Equilibrium propagation (EP) is such a procedure that has been introduced and applied to classical energybased models which relax to an equilibrium. Here, we show a direct connection between EP and Onsager reciprocity and exploit this to derive a quantum version of EP. This can be used to optimize loss functions that depend on the expectation values of observables of an arbitrary quantum system. Specifically, we illustrate this new concept with supervised and unsupervised learning examples in which the input or the solvable task is of quantum mechanical nature, e.g., the recognition of quantum manybody ground states, quantum phase exploration, sensing and phase boundary exploration. We propose that in the future quantum EP may be used to solve tasks such as quantum phase discovery with a quantum simulator even for Hamiltonians which are numerically hard to simulate or even partially unknown. Our scheme is relevant for a variety of quantum simulation platforms such as ion chains, superconducting qubit arrays, neutral atom Rydberg tweezer arrays and strongly interacting atoms in optical lattices.
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 neurallike networks that leverage the properties of physical systems to perform computation. While PNNs are so far a niche research area with smallscale laboratory demonstrations, they are arguably one of the most underappreciated important opportunities in modern AI. Could we train AI models 1000x larger than current ones? Could we do this and also have them perform inference locally and privately on edge devices, such as smartphones or sensors? Research over the past few years has shown that the answer to all these questions is likely "yes, with enough research": PNNs could one day radically change what is possible and practical for AI systems. To do this will however require 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 at large scale, many methods including backpropagationbased and backpropagationfree approaches are now being explored. These methods have various tradeoffs, and so far no method has been shown to scale to the same scale and performance as the backpropagation algorithm widely used in deep learning today. However, this is rapidly changing, and a diverse ecosystem of training techniques provides clues for how PNNs may one day be utilized to create both more efficient realizations of currentscale AI models, and to enable unprecedentedscale models.
MetaDesigning Quantum Experiments with Language Models
Sören Arlt, Haonan Duan, Felix Li, Sang Michael Xie, Yuhuai Wu, Mario Krenn
Artificial Intelligence (AI) has the potential to sig nificantly advance scientific discovery by finding solutions beyond human capabilities. However, these superhuman solutions are often unintuitive and require considerable effort to uncover under lying principles, if possible at all. Here, we show how a codegenerating language model trained on synthetic data can not only find solutions to specific problems but can create metasolutions, which solve an entire class of problems in one shot and simultaneously offer insight into the underlying design principles. Specifically, for the design of new quantum physics experiments, our sequencetosequence transformer architec ture generates interpretable Python code that de scribes experimental blueprints for a whole class of quantum systems. We discover general and pre viously unknown design rules for infinitely large classes of quantum states. The ability to automat ically generate generalized patterns in readable computer code is a crucial step toward machines that help discover new scientific understanding – one of the central aims of physics.
Neuralnetworksupported basis optimizer for the configuration interaction problem in quantum manybody clusters: Feasibility study and numerical proof
Pavlo Bilous, Louis Thirion, Henri Menke, Maurits W. Haverkort, Adriana Pálffy, Philipp Hansmann
A deeplearning approach to optimize the selection of Slater determinants in configuration interaction calculations for condensedmatter quantum manybody systems is developed. We exemplify our algorithm on the discrete version of the singleimpurity Anderson model with up to 299 bath sites. Employing a neural network classifier and active learning, our algorithm enhances computational efficiency by iteratively identifying the most relevant Slater determinants for the groundstate wave function. We benchmark our results against established methods and investigate the efficiency of our approach as compared to other basis truncation schemes. Our algorithm demonstrates a substantial improvement in the efficiency of determinant selection, yielding a more compact and computationally manageable basis without compromising accuracy. Given the straightforward application of our neural networksupported selection scheme to other model Hamiltonians of quantum manybody clusters, our algorithm can significantly advance selective configuration interaction calculations in the context of correlated condensed matter.
Generation and humanexpert evaluation of interesting research ideas using knowledge graphs and large language models
Advanced artificial intelligence (AI) systems with access to millions of research papers could inspire new research ideas that may not be conceived by humans alone. However, how interesting are these AIgenerated ideas, and how can we improve their quality? Here, we introduce SciMuse, a system that uses an evolving knowledge graph built from more than 58 million scientific papers to generate personalized research ideas via an interface to GPT4. We conducted a largescale human evaluation with over 100 research group leaders from the Max Planck Society, who ranked more than 4,000 personalized research ideas based on their level of interest. This evaluation allows us to understand the relationships between scientific interest and the core properties of the knowledge graph. We find that dataefficient machine learning can predict research interest with high precision, allowing us to optimize the interestlevel of generated research ideas. This work represents a step towards an artificial scientific muse that could catalyze unforeseen collaborations and suggest interesting avenues for scientists.
Transfer learning in predicting quantum manybody dynamics: from physical observables to entanglement entropy
Philipp Schmidt, Florian Marquardt, Naeimeh Mohseni
Deep neural networks have demonstrated remarkable efficacy in extracting meaningful representations from complex datasets. This has propelled representation learning as a compelling area of research across diverse fields. One interesting open question is how beneficial representation learning can be for quantum manybody physics, with its notouriosly highdimensional state space. In this work, we showcase the capacity of a neural network that was trained on a subset of physical observables of a manybody system to partially acquire an implicit representation of the wave function. We illustrate this by demonstrating the effectiveness of reusing the representation learned by the neural network to enhance the learning process of another quantity derived from the quantum state. In particular, we focus on how the pretrained neural network can enhance the learning of entanglement entropy. This is of particular interest as directly measuring the entanglement in a manybody system is very challenging, while a subset of physical observables can be easily measured in experiments. We show the pretrained neural network learns the dynamics of entropy with fewer resources and higher precision in comparison with direct training on the entanglement entropy.
Symmetryinduced higherorder exceptional points in two dimensions
Anton Montag, Flore K. Kunst
Physical Review Research
6
023205
(2024)

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Exceptional points of order n (EPns) appear in nonHermitian systems as points where the eigen values and eigenvectors coalesce. Whereas EP2s generically appear in two dimensions (2D), higher order EPs require a higherdimensional parameter space to emerge. In this work, we provide a complete characterization the appearance of symmetryinduced higherorder EPs in 2D parameter space. We find that besides EP2s only EP3s, EP4s, and EP5s can be stabilized in 2D. Moreover, these higherorder EPs must always appear in pairs with their dispersion determined by the sym metries. Upon studying the complex spectral structure around these EPs, we find that depending on the symmetry, EP3s are accompanied by EP2 arcs, and 2 and 3level open Fermi structures. Similarly, EP4s and closely related EP5s, which arise due to multiple symmetries, are accompanied by exotic EP arcs and open Fermi structures. For each case, we provide an explicit example. We also comment on the topological charge of these EPs, and discuss similarities and differences between symmetryprotected higherorder EPs and EP2s.
Tackling Decision Processes with NonCumulative Objectives using Reinforcement Learning
Maximilian Nägele, Jan Olle, Thomas Fösel, Remmy Zen, Florian Marquardt
Markov decision processes (MDPs) are used to model a wide variety of applications ranging from game playing over robotics to finance. Their optimal policy typically maximizes the expected sum of rewards given at each step of the decision process. However, a large class of problems does not fit straightforwardly into this framework: Noncumulative Markov decision processes (NCMDPs), where instead of the expected sum of rewards, the expected value of an arbitrary function of the rewards is maximized. Example functions include the maximum of the rewards or their mean divided by their standard deviation. In this work, we introduce a general mapping of NCMDPs to standard MDPs. This allows all techniques developed to find optimal policies for MDPs, such as reinforcement learning or dynamic programming, to be directly applied to the larger class of NCMDPs. Focusing on reinforcement learning, we show applications in a diverse set of tasks, including classical control, portfolio optimization in finance, and discrete optimization problems. Given our approach, we can improve both final performance and training time compared to relying on standard MDPs.
Discovering Quantum Circuit Components with Program Synthesis
Leopoldo Sarra, Kevin Ellis, Florian Marquardt
Machine Learning: Science and Technology
5
025029
(2024)

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Despite rapid progress in the field, it is still challenging to discover new ways to leverage quantum computation: all quantum algorithms must be designed by hand, and quantum mechanics is notoriously counterintuitive. In this paper, we study how artificial intelligence, in the form of program synthesis, may help overcome some of these difficulties, by showing how a computer can incrementally learn concepts relevant to quantum circuit synthesis with experience, and reuse them in unseen tasks. In particular, we focus on the decomposition of unitary matrices into quantum circuits, and show how, starting from a set of elementary gates, we can automatically discover a library of useful new composite gates and use them to decompose increasingly complicated unitaries.
In optics and photonics, a small number of building blocks, like resonators, waveguides, arbitrary couplings, and parametric interactions, allow the design of a broad variety of devices and func tionalities, distinguished by their scattering properties. These include transducers, amplifiers, and nonreciprocal devices, like isolators or circulators. Usually, the design of such a system is hand crafted by an experienced scientist in a timeconsuming process where it remains uncertain whether the simplest possibility has indeed been found. In our work, we develop a discovery algorithm that automates this challenge. By optimizing the continuous and discrete system properties our auto mated search identifies the minimal resources required to realize the requested scattering behavior. In the spirit of artificial scientific discovery, it produces a complete list of interpretable solutions and leads to generalizable insights, as we illustrate in several examples. This now opens the door to rapid design in areas like photonic and microwave architectures or optomechanics.
Topologically Protected Transport in Engineered Mechanical Systems
Tirth Shah, Christian Brendel, Vittorio Peano, Florian Marquardt
Reviews of Modern Physics
96
021002
(2024)

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Mechanical vibrations are being harnessed for a variety of purposes and at many length scales, from the macroscopic world down to the nanoscale. The considerable design freedom in mechanical structures allows to engineer new<br>functionalities. In recent years, this has been exploited to generate setups that offer topologically protected transport of vibrational waves, both in the solid state and in fluids. Borrowing concepts from electronic physics and being crossfertilized by concurrent studies for cold atoms and electromagnetic waves, this field of topological transport in engineered mechanical systems offers a rich variety of phenomena and platforms. In this review, we provide a unifying overview of the various ideas employed in this area, summarize the different approaches and experimental implementations, and comment on the challenges as well as the prospects.
Detailed balance in nonequilibrium dynamics of granular matter: derivation and implications
Clara C. Wanjura, Amelie Mayländer, Othmar Marti, Raphael Blumenfeld
Discovering fundamental principles governing the dynamics of granular media has been a longstanding challenge. Recent predictions of detailed balance steady states (DBSS), supported by experimental observations in cyclic shear experiments of planar granular systems, called into question the common belief that the detailed balance principle is only a feature of equilibrium. Here, we first show analytically that DBSS in planar granular dynamics arise when a certain conditional cell order distribution is independent of the condition. We then demonstrate that this condition is met in rotational shear experiments, which indeed also give rise to robust DBSS. This suggests that DBSS not only exist but are also quite common. We also show that, when the unconditional cell order distribution maximises the entropy, as has been found recently, then this distribution is determined by a single parameter  the ratio of splitting and merging rates of cells of any arbitrary order. These results simplify the modelling of the complex dynamics of planar granular systems to the solution of recently proposed evolution equations, demonstrating their predictive power.<br>
Three perspectives on entropy dynamics in a nonHermitian twostate system
Alexander Felski, Alireza Beygi, Christos Karapoulitidis, S. P. Klevansky
A comparative study of entropy dynamics as an indicator of physical behavior in an open two state system with balanced gain and loss is presented. We distinguish the perspective taken in utilizing the conventional framework of Hermitianadjoint states from an approach that is based on biorthogonaladjoint states and a third case based on an isospectral mapping. In this it is demonstrated that their differences are rooted in the treatment of the environmental coupling mode. For unbroken PT symmetry of the system, a notable characteristic feature of the perspective taken is the presence or absence of purity oscillations, with an associated entropy revival. The description of the system is then continued from its PT symmetric pseudoHermitian phase into the regime of spontaneously broken symmetry, in the latter two approaches through a nonanalytic operator based continuation, yielding a Lindblad master equation based on the PT charge operator C. This phase transition indicates a general connection between the pseudoHermitian closedsystem and the Lindbladian opensystem formalism through a spontaneous breakdown of the underlying physical reflection symmetry.
Reservoir Engineering for Classical Nonlinear Fields
Benedikt Tissot, Hugo Ribeiro, Florian Marquardt
Physical Review Research
6
023015
(2024)

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Reservoir engineering has become a prominent tool to control quantum systems. Recently, there have been first experiments applying it to manybody systems, especially with a view to engineer particleconserving dissipation for quantum simulations using bosons. In this work, we explore the dissipative dynamics of these systems in the classical limit. We derive a general equation of motion capturing the effective nonlinear dissipation introduced by the bath and apply it to the special case of a BoseHubbard model, where it leads to an unconventional type of dissipative nonlinear Schr ̈odinger equation. Building on that, we study the dynamics of one and two solitons in such a dissipative classical field theory.
Optomechanical realization of the bosonic Kitaev chain
Jesse J. Slim, Clara C. Wanjura, Matteo Brunelli, Javier del Pino, Andreas Nunnenkamp, Ewold Verhagen
The fermionic Kitaev chain is a canonical model featuring topological Majorana zero modes. We report the experimental realization of its bosonic analogue in a nanooptomechanical network, in which the parametric interactions induce beamsplitter coupling and twomode squeezing among the nanomechanical modes, analogous to hopping and pwave pairing in the fermionic case, respectively. This specific structure gives rise to a set of extraordinary phenomena in the bosonic dynamics and transport. We observe quadraturedependent chiral amplification, exponential scaling of the gain with system size and strong sensitivity to boundary conditions. All these are linked to the unique nonHermitian topological nature of the bosonic Kitaev chain.<br>We probe the topological phase transition and uncover a rich dynamical phase diagram by controlling interaction phases and amplitudes. Finally, we present an experimental demonstration of an exponentially enhanced response to a small perturbation. These results represent the demonstration of a new synthetic phase of matter whose bosonic dynamics do not have fermionic parallels, and we have established a powerful system for studying nonHermitian topology and its applications for signal manipulation and sensing.
Supervised Training of NeuralNetwork Quantum States for the Next Nearest Neighbor Ising model
Zheyu Wu, Remmy Augusta Menzata Zen, Heitor P. Casagrande, Stéphane Bressan, Dario Poletti
Different neural network architectures can be unsupervisedly or supervis edly trained to represent quantum states. We explore and compare different strategies for the supervised training of feed forward neural network quan tum states. We empirically and comparatively evaluate the performance of feed forward neural network quantum states in different phases of matter for variants of the architecture, for different hyperparameters, and for two different loss functions, to which we refer as meansquared error and over lap, respectively. We consider the nextnearest neighbor Ising model for the diversity of its phases and focus on its paramagnetic, ferromagnetic, and pairantiferromagnetic phases. We observe that the overlap loss function al lows better training of the model across all phases, provided a rescaling of the neural network.
Quantum interference between distant creation processes
Johannes Pseiner, Manuel Erhard, Mario Krenn
Physical Review Research
6
013294
(2024)

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The search for macroscopic quantum phenomena is a fundamental pursuit in quantum mechanics. It allows us to test the limits of quantum physics and provides new avenues for exploring the interplay between quantum mechanics and relativity. In this work, we introduce a novel approach to generate macroscopic quantum systems by demonstrating that the creation process of a quantum system can span a macroscopic distance. Specifically, we generate photon pairs in a coherent superposition of two origins separated by up to 70 meters. This new approach not only provides an exciting opportunity for foundational experiments in quantum physics, but also has practical applications for highprecision measurements of distributed properties such as pressure and humidity of air or gases.
Exceptional points of any order in a generalized HatanoNelson model
Exceptional points (EPs) are truly nonHermitian (NH) degeneracies where matrices become defective. The order of such an EP is given by the number of coalescing eigenvectors. On the one hand, most work focusses on studying Nthorder EPs in N≤4dimensional NH Bloch Hamiltonians. On the other hand, some works have remarked on the existence of EPs of orders scaling with systems size in models exhibiting the NH skin effect. In this letter, we introduce a new type of EP and provide a recipe on how to realize EPs of arbitrary order not scaling with system size. We introduce a generalized version of the paradigmatic HatanoNelson model with longerrange hoppings. The EPs existing in this system show remarkable physical features: Their associated eigenstates are localized on a subset of sites and are exhibiting the NH skin effect. Furthermore, the EPs are robust against generic perturbations in the hopping strengths as well as against a specific form of onsite disorder.
Essential implications of similarities in nonHermitian systems
In this paper, we show that three different generalized similarities enclose all unitary and antiunitary symmetries that induce exceptional points in lowerdimensional nonHermitian systems. We prove that the generalized similarity conditions result in a larger class of systems than any class defined by a unitary or antiunitary symmetry. Further we highlight that the similarities enforce spectral symmetry on the Hamiltonian resulting in a reduction of the codimension of exceptional points. As a consequence we show that the similarities drive the emergence of exceptional points in lower dimensions without the more restrictive need for a unitary and/or antiunitary symmetry.
Quantum Circuit Discovery for FaultTolerant Logical State Preparation with Reinforcement Learning
Remmy Zen, Jan Olle, Luis Colmenarez, Matteo Puviani, Markus Müller, Florian Marquardt
One of the key aspects in the realization of largescale faulttolerant quantum computers is quan tum error correction (QEC). The first essential step of QEC is to encode the logical state into physical qubits in a faulttolerant manner. Recently, flagbased protocols have been introduced that use ancillary qubits to flag harmful errors. However, there is no clear recipe for finding a compact quantum circuit with flagbased protocols for faulttolerant logical state preparation. It is even more difficult when we consider the hardware constraints, such as qubit connectivity and gate set. In this work, we propose and explore reinforcement learning (RL) to automatically discover compact and hardwareadapted quantum circuits that faulttolerantly prepare the logical state of a QEC code. We show that RL discovers circuits with fewer gates and ancillary qubits than published results without and with hardware constraints of up to 15 physical qubits. Furthermore, RL allows for straightforward exploration of different qubit connectivities and the use of transfer learning to accelerate the discovery. More generally, our work opens the door towards the use of RL for the discovery of faulttolerant quantum circuits for addressing tasks beyond state preparation, including magic state preparation, logical gate synthesis, or syndrome measurement.
Virtual Reality for Understanding ArtificialIntelligencedriven Scientific Discovery with an Application in Quantum Optics
Philipp Schmidt, Sören Arlt, Carlos RuizGonzalez, Xuemei Gu, Carla Rodríguez, Mario Krenn
Generative Artificial Intelligence (AI) models can propose solutions to scientific problems beyond human capability. To truly make conceptual contributions, researchers need to be capable of understanding the AIgenerated structures and extracting the underlying concepts and ideas. When algorithms provide little explanatory reasoning alongside the output, scientists have to reverseengineer the fundamental insights behind proposals based solely on examples. This task can be challenging as the output is often highly complex and thus not immediately accessible to humans. In this work we show how transferring part of the analysis process into an immersive Virtual Reality (VR) environment can assist researchers in developing an understanding of AIgenerated solutions. We demonstrate the usefulness of VR in finding interpretable configurations of abstract graphs, representing Quantum Optics experiments. Thereby, we can manually discover new generalizations of AIdiscoveries as well as new understanding in experimental quantum optics. Furthermore, it allows us to customize the search space in an informed way  as a humanintheloop  to achieve significantly faster subsequent discovery iterations. As concrete examples, with this technology, we discover a new resourceefficient 3dimensional entanglement swapping scheme, as well as a 3dimensional 4particle GreenbergerHorneZeilingerstate analyzer. Our results show the potential of VR for increasing a human researcher's ability to derive knowledge from graphbased generative AI that, which is a common abstract data representation used in diverse fields of science.
NonHermitian chiral anomalies in interacting systems
Sharareh Sayyad
Physical Review Research (6)
L012028
(2024)

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The emergence of chiral anomaly entails various fascinating phenomena such as anomalous quantum Hall effect and chiral magnetic effect in different branches of (non)Hermitian physics. While in the singleparticle picture, anomalous currents merely appear due to the coupling of massless particles with background fields, manybody interactions can also be responsible for anomalous transport in interacting systems. In this Letter, we study anomalous chiral currents in systems where interacting massless fermions with complex Fermi velocities are coupled to complex gauge fields. Our results reveal that incorporating nonHermiticity and manybody interactions gives rise to additional terms in anomalous relations beyond their Hermitian counterparts. We further present that manybody corrections in the subsequent nonHermitian chiral magnetic field or anomalous Hall effect are nonvanishing in nonequilibrium or inhomogeneous systems. Our findings advance efforts in understanding anomalous transport in interacting nonHermitian systems.<br>
Deep Quantum Graph Dreaming: Deciphering Neural Network Insights into Quantum Experiments
Tareq Jaouni, Sören Arlt, Carlos RuizGonzalez, Ebrahim Karimi, Xuemei Gu, Mario Krenn
Machine Learning: Science and Technology (5)
015029
(2024)

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Despite their promise to facilitate new scientific discoveries, the opaqueness of neural networks presents a challenge in interpreting the logic behind their findings. Here, we use a eXplainableAI (XAI) technique called inception or deep dreaming, which has been invented in machine learning for computer vision. We use this techniques to explore what neural networks learn about quantum optics experiments. Our story begins by training a deep neural networks on the properties of quantum systems. Once trained, we "invert" the neural network – effectively asking how it imagines a quantum system with a specific property, and how it would continuously modify the quantum system to change a property. We find that the network can shift the initial distribution of properties of the quantum system, and we can conceptualize the learned strategies of the neural network. Interestingly, we find that, in the first layers, the neural network identifies simple properties, while in the deeper ones, it can identify complex quantum structures and even quantum entanglement. This is in reminiscence of longunderstood properties known in computer vision, which we now identify in a complex natural science task. Our approach could be useful in a more interpretable way to develop new advanced AIbased scientific discovery techniques in quantum physics.
Forecasting highimpact research topics via machine learning on evolving knowledge graphs
The exponential growth in scientific publications poses a severe challenge for human researchers. It forces attention to more narrow subfields, which makes it challenging to discover new impactful research ideas and collaborations outside one’s own field. While there are ways to predict a scientific paper’s future citation counts, they need the research to be finished and the paper written, usually assessing impact long after the idea was conceived. Here we show how to predict the impact of onsets of ideas that have never been published by researchers. For that, we developed a large evolving knowledge graph built from more than 21 million scientific papers. It combines a semantic network created from the content of the papers and an impact network created from the historic citations of papers. Using machine learning, we can predict the dynamic of the evolving network into the future with high accuracy, and thereby the impact of new research directions. We envision that the ability to predict the impact of new ideas will be a crucial component of future artificial muses that can inspire new impactful and interesting scientific ideas.
Training Coupled Phase Oscillators as a Neuromorphic Platform using Equilibrium Propagation
Qingshan Wang, Clara C. Wanjura, Florian Marquardt
Given the rapidly growing scale and resource requirements of machine learning applications, the idea of building more efficient learning machines much closer to the laws of physics is an attractive proposition. One central question for identifying promising candidates for such neuromorphic platforms is whether not only infer ence but also training can exploit the physical dynamics. In this work, we show that it is possible to successfully train a system of coupled phase oscillators—one of the most widely investigated nonlinear dynamical systems with a multitude of physical implementations, comprising laser arrays, coupled mechanical limit cycles, super fluids, and excitonpolaritons. To this end, we apply the approach of equilibrium propagation, which permits to extract training gradients via a physical realization of backpropagation, based only on local interactions. The complex energy landscape of the XY/ Kuramoto model leads to multistability, and we show how to address this challenge. Our study identifies coupled phase oscillators as a new generalpurpose neuromorphic platform and opens the door towards future experimental implementations.
Transfer learning from Hermitian to nonHermitian quantum manybody physics
Sharareh Sayyad, Jose L. Lado
Journal of Physics: Condensed Matter
36(185603)
(2024)

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Identifying phase boundaries of interacting systems is one of the key steps to understanding quantum manybody models. The development of various numerical and analytical methods has allowed exploring the phase diagrams of many Hermitian interacting systems. However, numerical challenges and scarcity of analytical solutions hinder obtaining phase boundaries in nonHermitian manybody models. Recent machine learning methods have emerged as a potential strategy to learn phase boundaries from various observables without having access to the full manybody wavefunc tion. Here, we show that a machine learning methodology trained solely on Hermitian correlation functions allows identifying phase boundaries of nonHermitian interacting models. These results demonstrate that Hermitian machine learning algorithms can be redeployed to nonHermitian mod els without requiring further training to reveal nonHermitian phase diagrams. Our findings es tablish transfer learning as a versatile strategy to leverage Hermitian physics to machine learning nonHermitian phenomena.
Catch and release of propagating bosonic field with nonMarkovian giant atom
Luting Xu, Lingzhen Guo
New Journal of Physics (26)
013025
(2024)

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The nonMarkovianity of physical systems is considered to be a valuable resource that has potential applications to quantum information processing. The control of traveling quantum fields encoded with information (flying qubit) is crucial for quantum networks. In this work, we propose to catch and release the propagating photon/phonon with a nonMarkovian giant atom, which is coupled to the environment via multiple coupling points. Based on the Heisenberg equation of motion for the giant atom and field operators, we calculate the time dependent scattering coefficients from the linear response theory and define the criteria for the nonMarkovian giant atom. We analyze and numerically verify that the field bound states due to nonMarkovianity can be harnessed to catch and release the propagating bosonic field on demand by tuning the parameters of giant atom.
Engineering Arbitrary Hamiltonians in Phase Space
Lingzhen Guo, Vittorio Peano
Physical Review Letters
132
023602
(2024)

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We introduce a general method to engineer arbitrary Hamiltonians in the Floquet phase space of a periodically driven oscillator, based on the noncommutative Fourier transformation technique. We establish the relationship between an arbitrary target Floquet Hamiltonian in phase space and the periodic driving potential in real space. We obtain analytical expressions for the driving potentials in real space that can generate novel Hamiltonians in phase space, e.g., rotational lattices and sharpboundary wells. Our protocol can be realized in a range of experimental platforms for nonclassical state generation and bosonic quantum computation.
2023
Merging automatic differentiation and the adjoint method for photonic inverse design
Alexander Luce, Rasoul Alaee, Fabian Knorr, Florian Marquardt
Machine Learning: Science and Technology
5(2)
025076
(2024)

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Optimizing the shapes and topology of physical devices is crucial for both scientific and technological advancements, given their wideranging implications across numerous industries and research areas. Innovations in shape and topology optimization have been observed across a wide range of fields, notably structural mechanics, fluid mechanics, and more recently, photonics. Gradientbased inverse design techniques have been particularly successful for photonic and optical problems, resulting in integrated, miniaturized hardware that has set new standards in device performance. To calculate the gradients, there are typically two approaches: namely, either by implementing specialized solvers using automatic differentiation (AD) or by deriving analytical solutions for gradient calculation and adjoint sources by hand. In this work, we propose a middle ground and present a hybrid approach that leverages and enables the benefits of AD for handling gradient derivation while using existing, proven but blackbox photonic solvers for numerical solutions. Utilizing the adjoint method, we make existing numerical solvers differentiable and seamlessly integrate them into an AD framework. Further, this enables users to integrate the optimization environment seamlessly with other autodifferentiable components such as machine learning, geometry generation, or intricate postprocessing which could lead to better photonic design workflows. We illustrate the approach through two distinct photonic optimization problems: optimizing the Purcell factor of a magnetic dipole in the vicinity of an optical nanocavity and enhancing the light extraction efficiency of a µLED.
Discovering Quantum Circuit Components with Program Synthesis
Leopoldo Sarra, Kevin Ellis, Florian Marquardt
Machine Learning: Science and Technology
5
025029
(2024)

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Despite rapid progress in the field, it is still challenging to discover new ways to leverage quantum computation: all quantum algorithms must be designed by hand, and quantum mechanics is notoriously counterintuitive. In this paper, we study how artificial intelligence, in the form of program synthesis, may help overcome some of these difficulties, by showing how a computer can incrementally learn concepts relevant to quantum circuit synthesis with experience, and reuse them in unseen tasks. In particular, we focus on the decomposition of unitary matrices into quantum circuits, and show how, starting from a set of elementary gates, we can automatically discover a library of useful new composite gates and use them to decompose increasingly complicated unitaries.
Modelaware reinforcement learning for highperformance Bayesian experimental design in quantum metrology
Quantum sensors offer control flexibility during estimation by allowing manipulation by the experimenter across various parameters. For each sensing platform, pinpointing the optimal controls to enhance the sensor’s precision remains a challenging task. While an analytical solution might be out of reach, machine learning offers a promising avenue for many systems of interest, especially given the capabilities of contemporary hardware. We have introduced a versatile procedure capable of optimizing a wide range of problems in quantum metrology, estimation, and hypothesis testing by combining modelaware reinforcement learning (RL) with Bayesian estimation based on particle filtering. To achieve this, we had to address the challenge of incorporating the many nondifferentiable steps of the estimation in the training process, such as measurements and the resampling of the particle filter. Modelaware RL is a gradientbased method, where the derivatives of the sensor’s precision are obtained through automatic differentiation (AD) in the simulation of the experiment. Our approach is suitable for optimizing both nonadaptive and adaptive strategies, using neural networks or other agents. We provide an implementation of this technique in the form of a Python library called qsensoropt, alongside several premade applications for relevant physical platforms, namely NV centers, photonic circuits, and optical cavities. This library will be released soon on PyPI. Leveraging our method, we’ve achieved results for many examples that surpass the current stateoftheart in experimental design. In addition to Bayesian estimation, leveraging modelaware RL, it is also possible to find optimal controls for the minimization of the Cram ́erRao bound, based on Fisher information.
Experimental Solutions to the HighDimensional Mean King's Problem
Tareq Jaouni, Xiaoqin Gao, Sören Arlt, Mario Krenn, Ebrahim Karimi
Vaidman, Aharanov, and Albert [Phys. Rev. Lett. 58(14), 1385 (1987)] put forward a puzzle called the mean king’s problem (MKP) that can be solved only by harnessing quantum entanglement. Primepowered solutions to the problem have been shown to exist, but they have not yet been experimentally realized for any dimension beyond two. We propose a general firstofitskind experimental scheme for solving the MKP in prime dimensions (D). Our search is guided by the digital discovery framework Pytheus, which finds highly interpretable graphbased representations of quantum optical experimental setups; using it, we find specific solutions and generalize to higher dimensions through human insight. As proof of principle, we present a detailed investigation of our solution for the three, five, and sevendimensional cases. We obtain maximum success probabilities of 82.3%, 56.2%, and 35.5%, respectively. We therefore posit that our computerinspired scheme yields solutions that implement Alice’s strategy with quantum advantage, demonstrating its promise for experimental implementation in quantum communication tasks.
Digital Discovery of 100 diverse Quantum Experiments with PyTheus
Carlos RuizGonzalez, Sören Arlt, Jan Petermann, Sharareh Sayyad, Tareq Jaouni, Ebrahim Karimi, Nora Tischler, Xuemei Gu, Mario Krenn
Photons are the physical system of choice for performing experimental tests of the foundations of quantum mechanics. Furthermore, photonic quantum technology is a main player in the second quantum revolution, promising the development of better sensors, secure communications, and quantumenhanced computation. These endeavors require generating specific quantum states or efficiently performing quantum tasks. The design of the corresponding optical experiments was historically powered by human creativity but is recently being automated with advanced computer algorithms and artificial intelligence. While several computerdesigned experiments have been experimentally realized, this approach has not yet been widely adopted by the broader photonic quantum optics community. The main roadblocks consist of most systems being closedsource, inefficient, or targeted to very specific usecases that are difficult to generalize. Here, we overcome these problems with a highlyefficient, opensource digital discovery framework PyTheus, which can employ a wide range of experimental devices from modern quantum labs to solve various tasks. This includes the discovery of highly entangled quantum states, quantum measurement schemes, quantum communication protocols, multiparticle quantum gates, as well as the optimization of continuous and discrete properties of quantum experiments or quantum states. PyTheus produces interpretable designs for complex experimental problems which human researchers can often readily conceptualize. PyTheus is an example of a powerful framework that can lead to scientific discoveries  one of the core goals of artificial intelligence in science. We hope it will help accelerate the development of quantum optics and provide new ideas in quantum hardware and technology.
Boosting the GottesmanKitaevPreskill quantum error correction with nonMarkovian feedback
Matteo Puviani, Sangkha Borah, Remmy Zen, Jan Olle, Florian Marquardt
Bosonic codes allow the encoding of a logical qubit in a single component device, utilizing the infinitely large Hilbert space of a harmonic oscillator. In particular, the GottesmanKitaevPreskill code has recently been demonstrated to be correctable well beyond the breakeven point of the best passive encoding in the same system. Current approaches to quantum error correction (QEC) for this system are based on protocols that use feedback, but the response is based only on the latest measurement outcome. In our work, we use the recently proposed FeedbackGRAPE (Gra dient Ascent Pulse Engineering with Feedback) method to train a recurrent neural network that provides a QEC scheme based on memory, responding in a nonMarkovian way to the full history of previous measurement outcomes, optimizing all subsequent unitary operations. This approach sig nificantly outperforms current strategies and paves the way for more powerful measurementbased QEC protocols.
Digital Discovery of interferometric Gravitational Wave Detectors
Gravitational waves, detected a century after they were first theorized, are spacetime distortions caused by some of the most cataclysmic events in the universe, including black hole mergers and supernovae. The successful detection of these waves has been made possible by ingenious detectors designed by human experts. Beyond these successful designs, the vast space of experimental config urations remains largely unexplored, offering an exciting territory potentially rich in innovative and unconventional detection strategies. Here, we demonstrate the application of artificial intelligence (AI) to systematically explore this enormous space, revealing novel topologies for gravitational wave (GW) detectors that outperform current nextgeneration designs under realistic experimental con straints. Our results span a broad range of astrophysical targets, such as black hole and neutron star mergers, supernovae, and primordial GW sources. Moreover, we are able to conceptualize the initially unorthodox discovered designs, emphasizing the potential of using AI algorithms not only in discovering but also in understanding these novel topologies. We’ve assembled more than 50 superior solutions in a publicly available Gravitational Wave Detector Zoo which could lead to many new surprising techniques. At a bigger picture, our approach is not limited to gravitational wave detectors and can be extended to AIdriven design of experiments across diverse domains of fundamental physics.
Topological Properties of a NonHermitian Quasi1D Chainwith a Flat Band
C. MartínezStrasser, M. A. J. Herrera, G. Palumbo, Flore K. Kunst, D. Bercioux
Advanced Quantum Technologies
7(2)
2300225
(2023)

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The spectral properties of a nonHermitian quasi1D lattice in two of the possible dimerization configurations are investigated. Specifically, it focuses on a nonHermitian diamond chain that presents a zeroenergy flat band. The flat band originates from wave interference and results in eigenstates with a finite contribution only on two sites of the unit cell. To achieve the nonHermitian characteristics, the system under study presents nonreciprocal hopping terms in the chain. This leads to the accumulation of eigenstates on the boundary of the system, known as the nonHermitian skin effect. Despite this accumulation of eigenstates, for one of the two considered configurations, it is possible to characterize the presence of nontrivial edge states at zero energy by a realspace topological invariant known as the biorthogonal polarization. This work shows that this invariant, evaluated using the destructive interference method, characterizes the nontrivial phase of the nonHermitian diamond chain. For the second nonHermitian configuration, there is a finite quantum metric associated with the flat band. Additionally, the system presents the skin effect despite the system having a purely real or imaginary spectrum. The two nonHermitian diamond chains can be mapped into two models of the SuSchriefferHeeger chains, either nonHermitian, and Hermitian, both in the presence of a flat band. This mapping allows to draw valuable insights into the behavior and properties of these systems.<br><br>
Optimizing ZXDiagrams with Deep Reinforcement Learning
ZXdiagrams are a powerful graphical language for the description of quantum processes with applications in fundamental quantum mechanics, quantum circuit optimization, tensor network simulation, and many more. The utility of ZXdiagrams relies on a set of local transformation rules that can be applied to them without changing the underlying quantum process they describe. These rules can be exploited to optimize the structure of ZXdiagrams for a range of applications. However, finding an optimal sequence of transformation rules is generally an open problem. In this work, we bring together ZXdiagrams with reinforcement learning, a machine learning technique designed to discover an optimal sequence of actions in a decisionmaking problem and show that a trained reinforcement learning agent can significantly outperform other optimization techniques like a greedy strategy or simulated annealing. The use of graph neural networks to encode the policy of the agent enables generalization to diagrams much bigger than seen during the training phase.
Symmetryprotected exceptional and nodal points in nonHermitian systems
Sharareh Sayyad, Marcus Stålhammar, Lukas Rødland, Flore K. Kunst
One of the unique features of nonHermitian (NH) systems is the appearance of nonHermitian degeneracies known as exceptional points~(EPs). The occurrence of EPs in NH systems requires satisfying constraints whose number can be reduced in the presence of some symmetries. This results in stabilizing the appearance of EPs. Even though two different types of EPs, namely defective and nondefective EPs, may emerge in NH systems, exploring the possibilities of stabilizing EPs has been only addressed for defective EPs, at which the Hamiltonian becomes nondiagonalizable. In this letter, we show that certain discrete symmetries, namely paritytime, parityparticlehole, and pseudoHermitian symmetry, may guarantee the occurrence of both defective and nondefective EPs. We extend this list of symmetries by including the nonHermitian timereversal symmetry in the twoband systems. <br>We further show that the nondefective EPs manifest themselves by i) the diagonalizability of nonHermitian Hamiltonian at these points and ii) the nondiagonalizability of the Hamiltonian along certain intersections of nondefective EPs. Twoband and fourband models exemplify our findings. Through an example, we further reveal that ordinary (Hermitian) nodal points may coexist with defective EPs in nonHermitian models when the above symmetries are relaxed.
NoCollapse Accurate Quantum Feedback Control via Conditional State Tomography
Sangkha Borah, Bijita Sarma
Physical Review Letters
131
210803
(2023)

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The effectiveness of measurementbased feedback control (MBFC) protocols is hindered by the presence of measurement noise, which impairs the ability to accurately infer the underlying dynamics of a quantum system from noisy continuous measurement records. To circumvent this limitation, a realtime stochastic state estimation approach is proposed in this work, that enables noisefree monitoring of the conditional dynamics, including the full density matrix of the quantum system, despite using noisy measurement data. This, in turn, enables the development of precise MBFC strategies that leads to effective control of quantum systems by essentially mitigating the constraints imposed by measurement noise, and has potential applications in various feedback quantum control scenarios. This approach is particularly important for machine learningbased control, where the AI controller can be trained with arbitrary conditional averages of observables, including the full density matrix, to quickly and accurately learn control strategies.
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.
The traditional view from particle physics is that quantum gravity effects should only become detectable at extremely high energies and small length scales. Due to the sig nificant technological challenges involved, there has been limited progress in identifying experimentally detectable effects that can be accessed in the foreseeable future. How ever, 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 groundstate cooling and quantumcontrol techniques. Preparations of massive systems in quantum states paves the way for the explorations of a lowenergy regime in which gravity can be both sourced and probed by quantum systems. Such approaches constitute an in creasingly viable alternative to acceleratorbased, laserinterferometric, torsionbalance, and cosmological tests of gravity. In this review, we provide an overview of propos als where massive quantum systems act as interfaces between quantum mechanics and gravity. We discuss conceptual difficulties in the theoretical description of quantum systems in the presence of gravity, review tools for modeling massive quantum systems in the laboratory, and provide an overview of the current stateoftheart experimen tal landscape. Proposals covered in this review include, among others, precision tests of gravity, tests of gravitationallyinduced wavefunction collapse and decoherence, as well as gravitymediated entanglement. We conclude the review with an outlook and discussion of future questions.
Simultaneous Discovery of Quantum Error Correction Codes and Encoders with a NoiseAware Reinforcement Learning Agent
Jan Olle, Remmy Zen, Matteo Puviani, Florian Marquardt
Finding optimal ways to protect quantum states from noise remains an outstanding challenge across all quantum technologies, and quantum error correction (QEC) is the most promising strategy to address this issue. Constructing QEC codes is a complex task that has historically been powered by human creativity with the discovery of a large zoo of families of codes. However, in the context of realworld scenarios there are two challenges: these codes have typically been categorized only for their performance under an idealized noise model and the implementationspecific optimal encoding circuit is not known. In this work, we train a Deep Reinforcement Learning agent that automatically discovers both QEC codes and their encoding circuits for a given gate set, qubit connectivity, and error model. We introduce the concept of a noiseaware metaagent, which learns to produce encoding strategies simultaneously for a range of noise models, thus leveraging transfer of insights between different situations. Moreover, thanks to the use of the stabilizer formalism and a vectorized Clifford simulator, our RL implementation is extremely efficient, allowing us to produce many codes and their encoders from scratch within seconds, with code distances varying from 3 to 5 and with up to 20 physical qubits. Our approach opens the door towards hardwareadapted accelerated discovery of QEC approaches across the full spectrum of quantum hardware platforms of interest.
Realizing a deep reinforcement learning agent discovering realtime feedback control strategies for a quantum system
Kevin Reuer, Jonas Landgraf, Thomas Fösel, James O'Sullivan, Liberto Beltrán, Abdulkadir Akin, Graham J. Norris, Ants Remm, Michael Kerschbaum, et al.
Nature Communications
14
7138
(2023)

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Realizing the full potential of quantum technologies requires precise realtime control on time scales much shorter than the coherence time. Modelfree reinforcement learning promises to discover efficient feedback strategies from scratch without relying on a description of the quantum system. However, developing and training a reinforcement learning agent able to operate in realtime using feedback has been an open challenge. Here, we have implemented such an agent for a single qubit as a submicrosecondlatency neural network on a fieldprogrammable gate array (FPGA). We demonstrate its use to efficiently initialize a superconducting qubit and train the agent based solely on measurements. Our work is a first step towards adoption of reinforcement learning for the control of quantum devices and more generally any physical device requiring lowlatency feedback.<br>
Deep Bayesian Experimental Design for Quantum ManyBody Systems
Leopoldo Sarra, Florian Marquardt
Machine Learning: Science and Technology (4)
045022
(2023)

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Bayesian experimental design is a technique that allows to efficiently select measurements to characterize a physical system by maximizing the expected information gain. Recent developments in deep neural networks and normalizing flows allow for a more efficient approximation of the posterior and thus the extension of this technique to complex highdimensional situations. In this paper, we show how this approach holds promise for adaptive measurement strategies to characterize presentday quantum technology platforms. In particular, we focus on arrays of coupled cavities and qubit arrays. Both represent model systems of high relevance for modern applications, like quantum simulations and computing, and both have been realized in platforms where measurement and control can be exploited to characterize and counteract unavoidable disorder. Thus, they represent ideal targets for applications of Bayesian experimental design.
Restoration of the nonHermitian bulkboundary correspondence viatopological amplification
Matteo Brunelli, Clara C. Wanjura, Andreas Nunnenkamp
NonHermitian (NH) lattice Hamiltonians display a unique kind of energy gap and ex treme sensitivity to boundary conditions. Due to the NH skin effect, the separation between edge and bulk states is blurred and the (conventional) bulkboundary corre spondence is lost. Here, we restore the bulkboundary correspondence for the most paradigmatic class of NH Hamiltonians, namely those with one complex band and with out symmetries. We obtain the desired NH Hamiltonian from the meanfield evolution of drivendissipative cavity arrays, in which NH terms—in the form of nonreciprocal hopping amplitudes, gain and loss—are explicitly modeled via coupling to (engineered and nonengineered) reservoirs. This approach removes the arbitrariness in the defini tion of the topological invariant, as pointgapped spectra differing by a complexenergy shift are not treated as equivalent; the origin of the complex plane provides a common reference (base point) for the evaluation of the topological invariant. This implies that topologically nontrivial Hamiltonians are only a strict subset of those with a point gap and that the NH skin effect does not have a topological origin. We analyze the NH Hamil tonians so obtained via the singular value decomposition, which allows to express the NH bulkboundary correspondence in the following simple form: an integer value ν of the topological invariant defined in the bulk corresponds to ν singular vectors exponen tially localized at the system edge under open boundary conditions, in which the sign of ν determines which edge. Nontrivial topology manifests as directional amplification of a coherent input with gain exponential in system size. Our work solves an outstanding problem in the theory of NH topological phases and opens up new avenues in topological photonics.
Forecasting the future of artificial intelligence with machine learningbased link prediction in an exponentially growing knowledge network
Mario Krenn, Lorenzo Buffoni, Bruno Coutinho, Sagi Eppel, Jacob Gates Foster, Andrew Gritsevskiy, Harlin Lee, Yichao Lu, Joao P. Moutinho, et al.
Nature Machine Intelligence
13261335
(2023)

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A tool that could suggest new personalized research directions and ideas by taking insights from the scientific literature could profoundly accelerate the progress of science. A field that might benefit from such an approach is artificial intelligence (AI) research, where the number of scientific publications has been growing exponentially over recent years, making it challenging for human researchers to keep track of the progress. Here we use AI techniques to predict the future research directions of AI itself. We introduce a graphbased benchmark based on realworld data—the Science4Cast benchmark, which aims to predict the future state of an evolving semantic network of AI. For that, we use more than 143,000 research papers and build up a knowledge network with more than 64,000 concept nodes. We then present ten diverse methods to tackle this task, ranging from pure statistical to pure learning methods. Surprisingly, the most powerful methods use a carefully curated set of network features, rather than an endtoend AI approach. These results indicate a great potential that can be unleashed for purely ML approaches without human knowledge. Ultimately, better predictions of new future research directions will be a crucial component of more advanced research suggestion tools.
Fast quantum control of cavities using an improved protocol without coherent errors
Jonas Landgraf, Christa Flühmann, Thomas Fösel, Florian Marquardt, Robert J. Schoelkopf
The selective numberdependent arbitrary phase (SNAP) gates form a powerful class of quantum gates, imparting arbitrarily chosen phases to the Fock modes of a cavity. However, for short pulses, coherent errors limit the performance. Here we demonstrate in theory and experiment that such errors can be completely suppressed, provided that the pulse times exceed a specific limit. The resulting shorter gate times also reduce incoherent errors. Our approach needs only a small number of frequency components, the resulting pulses can be interpreted easily, and it is compatible with faulttolerant schemes.
XLuminA: An Autodifferentiating Discovery Framework for SuperResolution Microscopy
Carla Rodríguez Mangues, Sören Arlt, Leonhard Möckl, Mario Krenn
In this work we introduce XLuminA, an original computational framework designed for the discovery of novel optical hardware in superresolution microscopy. Our framework offers autodifferentiation capabilities, allowing for the fast and efficient simulation and automated design of entirely new optical setups from scratch. We showcase its potential by rediscovering three foundational experiments, each one covering different areas in optics: an optical telescope, STED microscopy and the focusing beyond the diffraction limit of a radially polarized light beam. Intriguingly, for this last experiment, the machine found an alternative solution following the same physical principle exploited for breaking the diffraction limit. With XLuminA, we can go beyond simple optimization and calibration of known experimental setups, opening the door to potentially uncovering new microscopy concepts within the vast landscape of experimental possibilities.
DeepLearning Approach for the Atomic Configuration Interaction Problem on Large Basis Sets
Pavlo Bilous, Adriana Pálffy, Florian Marquardt
Physical Review Letters
131(13)
(2023)

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Highprecision atomic structure calculations require accurate modeling of electronic correlations typically addressed via the configuration interaction (CI) problem on a multiconfiguration wave function expansion. The latter can easily become challenging or infeasibly large even for advanced supercomputers. Here, we develop a deeplearning approach which allows us to preselect the most relevant configurations out of large CI basis sets until the targeted energy precision is achieved. The large CI computation is thereby replaced by a series of smaller ones performed on an iteratively expanding basis subset managed by a neural network. While dense architectures as used in quantum chemistry fail, we show that a convolutional neural network naturally accounts for the physical structure of the basis set and allows for robust and accurate CI calculations. The method was benchmarked on basis sets of moderate size allowing for the direct CI calculation, and further demonstrated on prohibitively large sets where the direct computation is not possible.
Cooling microwave fields into general multimode Gaussian states
Nahid Yazdi, Juan José GarcíaRipoll, Diego Porras, Carlos NavarreteBenlloch
New Journal of Physics
25
083052
(2023)

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We show that a collection of lossy multichromatically modulated qubits can be used to dissipa tively engineer arbitrary Gaussian states of a set of bosonic modes. Our ideas are especially suited to superconductingcircuit architectures, where all the required ingredients are experimentally avail able. The generation of such multimode Gaussian states is necessary for many applications, most notably measurementbased quantum computation. We build upon some of our previous proposals, where we showed how to generate singlemode and twomode squeezed states through cooling and lasing. Special care must be taken when extending these ideas to many bosonic modes, and we discuss here how to overcome all the limitations and hurdles that naturally appear. We illustrate our ideas with a fully worked out example consisting of GHZ states, but have also tested several other examples such as cluster states. All these examples allow us to show that it is possible to use a set of N lossy qubits to cool down a bosonic chain of N modes to any desired Gaussian state.
Roadmap on structured waves
Konstantin Y Bliokh, Ebrahim Karimi, Miles J Padgett, Miguel A Alonso, Mark R Dennis, Angela Dudley, Andrew Forbes, Sina Zahedpour, Scott W Hancock, et al.
Structured waves are ubiquitous for all areas of wave physics, both classical and quantum, where the wavefields are inhomogeneous and cannot be approximated by a single plane wave. Even the interference of two plane waves, or of a single inhomogeneous (evanescent) wave, provides a number of nontrivial phenomena and additional functionalities as compared to a single plane wave. Complex wavefields with inhomogeneities in the amplitude, phase, and polarization, including topological structures and singularities, underpin modern nanooptics and photonics, yet they are equally important, e.g. for quantum matter waves, acoustics, water waves, etc. Structured waves are crucial in optical and electron microscopy, wave propagation and scattering, imaging, communications, quantum optics, topological and nonHermitian wave systems, quantum condensedmatter systems, optomechanics, plasmonics and metamaterials, optical and acoustic manipulation, and so forth. This Roadmap is written collectively by prominent researchers and aims to survey the role of structured waves in various areas of wave physics. Providing background, current research, and anticipating future developments, it will be of interest to a wide crossdisciplinary audience.
Selflearning Machines based on Hamiltonian Echo Backpropagation
Victor LopezPastor, Florian Marquardt
Physical Review X
13(3)
031020
(2023)

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A physical selflearning machine can be defined as a nonlinear dynamical system that can be trained on data (similar to artificial neural networks), but where the update of the internal degrees of freedom that serve as learnable parameters happens autonomously. In this way, neither external processing and feedback nor knowledge of (and control of) these internal degrees of freedom is required. We introduce a general scheme for selflearning in any timereversible Hamiltonian system. We illustrate the training of such a selflearning machine numerically for the case of coupled nonlinear wave fields.
Efficient approaches to quantum control and feedback are essential for quantum technologies, from sensing to quantum computation. Openloop control tasks have been successfully solved using optimization techniques, including methods such as gradientascent pulse engineering (GRAPE) , relying on a differentiable model of the quantum dynamics. For feedback tasks, such methods are not directly applicable, since the aim is to discover strategies conditioned on measurement outcomes. In this work, we introduce feedback GRAPE, which borrows some concepts from modelfree reinforcement learning to incorporate the response to strong stochastic (discrete or continuous) measurements, while still performing direct gradient ascent through the quantum dynamics. We illustrate its power considering various scenarios based on cavityQED setups. Our method yields interpretable feedback strategies for state preparation and stabilization in the presence of noise. Our approach could be employed for discovering strategies in a wide range of feedback tasks, from calibration of multiqubit devices to linearoptics quantum computation strategies, quantum enhanced sensing with adaptive measurements, and quantum error correction.
Quadrature nonreciprocity in bosonic networks without breaking timereversal symmetry
Clara C. Wanjura, Jesse J. Slim, Javier del Pino, Matteo Brunelli, Ewold Verhagen, Andreas Nunnenkamp
Nature Physics
19
14291436
(2023)

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Nonreciprocity means that the transmission of a signal depends on its direction of propagation. Despite vastly different platforms and underlying working principles, the realizations of nonreciprocal transport in linear, timeindependent systems rely on Aharonov–Bohm interference among several pathways and require breaking timereversal symmetry. Here we extend the notion of nonreciprocity to unidirectional bosonic transport in systems with a timereversal symmetric Hamiltonian by exploiting interference between beamsplitter (excitationpreserving) and twomodesqueezing (excitation nonpreserving) interactions. In contrast to standard nonreciprocity, this unidirectional transport manifests when the mode quadratures are resolved with respect to an external reference phase. Accordingly, we dub this phenomenon ‘quadrature nonreciprocity’. We experimentally demonstrate it in the minimal system of two coupled nanomechanical modes orchestrated by optomechanical interactions. Next, we develop a theoretical framework to characterize the class of networks exhibiting quadrature nonreciprocity based on features of their particle–hole graphs. In addition to unidirectionality, these networks can exhibit an even–odd pairing between collective quadratures, which we confirm experimentally in a fourmode system, and an exponential endtoend gain in the case of arrays of cavities.
Topological phase diagrams of exactly solvable nonHermitian interacting Kitaev chains
Sharareh Sayyad, Jose L. Lado
Physical Review Research
5(2)
L022046
(2023)

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Manybody interactions give rise to the appearance of exotic phases in Hermitian physics. Despite their importance, manybody effects remain an open problem in nonHermitian physics due to the complexity of treating manybody interactions. Here, we present a family of exact and numerical phase diagrams for nonHermitian interacting Kitaev chains. In particular, we establish the exact phase boundaries for the dimerized KitaevHubbard chain with complexvalued Hubbard interactions. Our results reveal that some of the Hermitian phases disappear as nonHermiticity is enhanced. Based on our analytical findings, we explore the regime of the model that goes beyond the solvable regime, revealing regimes where nonHermitian topological degeneracy remains. The combination of our exact and numerical phase diagrams provides an extensive description of a family of nonHermitian interacting models. Our results provide a stepping stone toward characterizing nonHermitian topology in realistic interacting quantum manybody systems.<br><br>
Recent advances in the SelfReferencing Embedding Strings (SELFIES) library
Alston Lo, Robert Pollice, AkshatKumar Nigam, Andrew D. White, Mario Krenn, Alán AspuruGuzik
Digital Discovery
10.1039/d3dd00044c
(2023)

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Stringbased molecular representations play a crucial role in cheminformatics applications, and with the growing success of deep learning in chemistry, have been readily adopted into machine learning pipelines. However, traditional stringbased representations such as SMILES are often prone to syntactic and semantic errors when produced by generative models. To address these problems, a novel representation, SELFreferencing embedded strings (SELFIES), was proposed that is inherently 100% robust, alongside an accompanying opensource implementation called selfies. Since then, we have generalized SELFIES to support a wider range of molecules and semantic constraints, and streamlined its underlying grammar. We have implemented this updated representation in subsequent versions of selfies, where we have also made major advances with respect to design, efficiency, and supported features. Hence, we present the current status of selfies (version 2.1.1) in this manuscript. Our library, selfies, is available at GitHub (https://github.com/aspuruguzikgroup/selfies).<br>
Deep recurrent networks predicting the gap evolution in adiabatic quantum computing
Naeimeh Mohseni, Carlos NavarreteBenlloch, Tim Byrnes, Florian Marquardt
One of the main challenges in quantum physics is predicting efficiently the dynamics of observables in manybody problems out of equilibrium. A particular example occurs in adiabatic quantum computing, where finding the structure of the instantaneous gap of the Hamiltonian is crucial in order to optimize the speed of the computation. Inspired by this challenge, in this work we explore the potential of deep learning for discovering a mapping from the parameters that fully identify a problem Hamiltonian to the full evolution of the gap during an adiabatic sweep applying different network architectures. Through this example, we find that a limiting factor for the learnability of the dynamics is the size of the input, that is, how the number of parameters needed to identify the Hamiltonian scales with the system size. We demonstrate that a long shortterm memory network succeeds in predicting the gap when the parameter space scales linearly with system size. Remarkably, we show that once this architecture is combined with a convolutional neural network to deal with the spatial structure of the model, the gap evolution can even be predicted for system sizes larger than the ones seen by the neural network during training. This provides a significant speedup in comparison with the existing exact and approximate algorithms in calculating the gap.
Classical Phase Space Crystals in Open Environment
Ali Emami Kopaei, Krzysztof Sacha, Lingzhen Guo
Physical Review B
107
214302
(2023)

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It was recently discovered that a crystalline manybody state can exist in the phase space of a closed dynamical system. A phase space crystal can be an anomalous Chern insulator that supports chiral topological transport without breaking physical timereversal symmetry [L. Guo et al., Phys. Rev. B 105, 094301 (2022)]. In this work, we further study the effects of an open dissipative environment with thermal noise and identify the existence condition of classical phase space crystals in realistic scenarios. By defining a crystal order parameter, we plot the phase diagram in the parameter space of dissipation rate, interaction, and temperature. Our present work paves the way to realize phase space crystals and explore anomalous chiral transport in experiments.
Multiphoton nonlocal quantum interference controlled by an undetected photon
Kaiyi Qian, Kai Wang, Leizhen Chen, Hou Zhaohua, Mario Krenn, Shining Zhu, XiaoSong Ma
Nature Communications
14
1480 (2023)
(2023)

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The interference of quanta lies at the heart of quantum physics. The multipartite generalization<br>of singlequanta interference creates entanglement, the coherent superposition of states shared by several quanta. Entanglement allows nonlocal correlations between many quanta and hence is a key resource for quantum information technology. Entanglement is typically considered to be essential for creating nonlocal correlations, manifested by multipartite interference. Here, we show that this is not the case and demonstrate multiphoton nonlocal quantum interference without entanglement of any intrinsic properties of the photons. We harness the superposition of the physical origin of a fourphoton product state, which leads to constructive and destructive interference of the photons’ mere existence. With the intrinsic indistinguishability in the generation process of photons, we realize fourphoton frustrated quantum interference. We furthermore establish nonlocal control of multipartite quantum interference, in which we tune the phase of one undetected photon and observe the interference of the other three photons. Our work paves the way for fundamental studies of nonlocality and potential applications in quantum technologies.
PT symmetryprotected exceptional cones and analogue Hawking radiation
Marcus Stålhammar, Jorge LaranaAragon, Lucas Rødland, Flore K. Kunst
New Journal of Physics
25
043012
(2023)

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NonHermitian Hamiltonians, which effectively describe dissipative systems, and analogue gravity models, which simulate properties of gravitational objects, comprise seemingly different areas of current research. Here, we investigate the interplay between the two by relating paritytime symmetric dissipative Weyltype Hamiltonians to analogue Schwarzschild black holes emitting Hawking radiation. We show that the exceptional points of these Hamiltonians form tilted cones mimicking the behavior of the light cone of a radially infalling observer approaching a black hole horizon. We further investigate the presence of tunneling processes, reminiscent of those happening in black holes, in a concrete example model. We interpret the nontrivial result as the purely thermal contribution to analogue Hawking radiation in a Schwarzschild black hole. Assuming that our particular Hamiltonian models a photonic crystal, we discuss the concrete nature of the analogue Hawking radiation in this particular setup.
Quenchdrive spectroscopy and highharmonic generation in BCS superconductors
Matteo Puviani, Rafael Haenel, Dirk Manske
Physical Review B (107)
094501
(2023)

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In pumpprobe spectroscopies, THz pulses are used to quench a system, which is subsequently probed by either a THz or optical pulse. In contrast, thirdharmonic generation experiments employ a single multicycle driving pulse and measure the induced third harmonic. In this work, we analyze a spectroscopy setup where both a quench and a drive are applied and twodimensional spectra as a function of time and quenchdrive delay are recorded. We calculate the time evolution of the nonlinear current generated in the superconductor within an Andersonpseudospin framework and characterize all experimental signatures using a quasiequilibrium approach. We analyze the superconducting response in Fourier space with respect to both the frequencies corresponding to the real time and the quenchdrive delay time. In particular, we show the presence of a transient modulation of higher harmonics, induced by a wave mixing process of the drive with the quench pulse, which probes both quasiparticle and collective excitations of the superconducting condensate.
Learning Quantum Systems
Valentin Gebhart, Raffaele Santagati, Antonio Andrea Gentile, Erik Gauger, David Craig, Natalia Ares, Leonardo Banchi, Florian Marquardt, Luca Pezzè, et al.
Nature Reviews Physics
5
141156
(2023)

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The future development of quantum technologies relies on creating and manipulating quantum systems of increasing complexity, with key applications in computation, simulation and sensing. This poses severe challenges in the efficient control, calibration and validation of quantum states and their dynamics. Although the full simulation of largescale quantum systems may only be possible on a quantum computer, classical characterization and optimization methods still play an important role. Here, we review different approaches that use classical postprocessing techniques, possibly combined with adaptive optimization, to learn quantum systems, their correlation properties, dynamics and interaction with the environment. We discuss theoretical proposals and successful implementations across different multiplequbit architectures such as spin qubits, trapped ions, photonic and atomic systems, and superconducting circuits. This Review provides a brief background of key concepts recurring across many of these approaches with special emphasis on the Bayesian formalism and neural networks.<br><br>
Tunnelinginduced fractal transmission in Valley Hall waveguides
Tirth Shah, Florian Marquardt, Vittorio Peano
Physical Review B
10.1103/PhysRevB.107.054304
(2023)

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The valley Hall effect provides a popular route to engineer robust waveguides for bosonic excitations such as photons and phonons. The almost complete absence of backscattering in many experiments has its theoretical underpinning in a smoothenvelope approximation that neglects large momentum transfer and is accurate only for small bulk band gaps and/or smooth domain walls. For larger bulk band gaps and hard domain walls, backscattering is expected to become significant. Here, we show that in this experimentally relevant regime, the reflection of a wave at a sharp corner becomes highly sensitive to the orientation of the outgoing waveguide relative to the underlying lattice. Enhanced backscattering can be understood as being triggered by resonant tunneling transitions in quasimomentum space. Tracking the resonant tunneling energies as a function of the waveguide orientation reveals a selfrepeating fractal pattern that is also imprinted in the density of states and the backscattering rate at a sharp corner.<br><br>
Investigation of inverse design of multilayer thinfilms with conditional invertible Neural Networks
Alexander Luce, Ali Mahdavi, Heribert Wankerl, Florian Marquardt
Machine Learning: Science and Technology
4(1)
015014
(2023)

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In this work, we apply conditional invertible neural networks (cINN) to inversely design multilayer thinfilms given an optical target in order to overcome limitations of stateoftheart optimization approaches. Usually, stateoftheart algorithms depend on a set of carefully chosen initial thinfilm parameters or employ neural networks which must be retrained for every new application. We aim to overcome those limitations by training the cINN to learn the loss landscape of all thinfilm configurations within a training dataset. We show that cINNs can generate a stochastic ensemble of proposals for thinfilm configurations that are reasonably close to the desired target depending only on random variables. By refining the proposed configurations further by a local optimization, we show that the generated thinfilms reach the target with significantly greater precision than comparable stateoftheart approaches. Furthermore, we tested the generative capabilities on samples which are outside of the training data distribution and found that the cINN was able to predict thinfilms for outofdistribution targets, too. The results suggest that in order to improve the generative design of thinfilms, it is instructive to use established and new machine learning methods in conjunction in order to obtain the most favorable results.<br><br>
Complex decoherencefree interactions between giant atoms
Lei Du, Lingzhen Guo, Yong Li
Physical Review A
107
023705
(2023)

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Giant atoms provide a promising platform for engineering decoherencefree interactions, which is a major task in modern quantum technologies. Here we study systematically how to implement complex decoherencefree interactions among giant atoms resorting to periodic coupling modulations and suitable arrangements of coupling points. We demonstrate that the phase of the modulation, which is tunable in experiments, can be encoded into the decoherencefree interactions and thus enables phasedependent dynamics when the giant atoms constitute an effective closed loop. Moreover, we consider the influence of nonMarkovian retardation effect arising from large separations of the coupling points and study its dependence on the modulation parameters.
From Dyson Models to ManyBody Quantum Chaos
Alexei Andreanov, Matteo Carrega, Jeff Murugan, Jan Olle, Dario Rosa, Ruth Shir
Understanding the mechanisms underlying manybody quantum chaos is one of the big challenges in theoretical physics. We tackle this problem by considering a set of perturbed quadratic SachdevYeKitaev (SYK) Hamiltonians defined on graphs. This allows to disambiguate between operator growth and \emph{delocalization}, showing that the latter is the dominant process in the singleparticle to manybody chaotic transition. Our results are verified numerically with stateoftheart numerical techniques, capable of extracting eigenvalues in a desired energy window of very large Hamiltonians, in this case up to dimension 2<sup>19</sup>×2<sup>19</sup>. Our approach essentially provides a new way of viewing manybody chaos from a singleparticle perspective.
Onchip quantum interference between the origins of a multiphoton state
LanTian Feng, Ming Zhang, Di Liu, YuJie Cheng, GuoPing Guo, DaoXin Dai, GuangCan Guo, M. Krenn, XiFeng Ren
Optica
10(1)
2103.14277
105109
(2023)

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Quantum mechanically, multiple particles can jointly be in a coherent superposition of two or more different states at the same time. This property is called quantum entanglement, and gives rise to characteristic nonlocal interference and stays at the heart of quantum information process. Here, rather than interference of different intrinsic properties of particles, we experimentally demonstrated coherent superposition of two different birthplaces of a fourphoton state. The quantum state is created in four probabilistic photonpair sources, two combinations of which can create photon quadruplets. Coherent elimination and revival of distributed 4photons can be fully controlled by tuning a phase. The stringent coherence requirements are met by using a siliconbased integrated photonic chip that contains four spiral waveguides for producing photon pairs via spontaneous fourwave mixing. The experiment gives rise to peculiar nonlocal phenomena without any obvious involvement of entanglement. Besides several potential applications that exploit the new onchip technology, it opens up the possibility for fundamental studies on nonlocality with spatially separated locations.
Artificial Intelligence and Machine Learning for Quantum Technologies
Mario Krenn, Jonas Landgraf, Thomas Fösel, Florian Marquardt
Physical Review A (107)
010101
(2023)

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In recent years the dramatic progress in machine learning has begun to impact many areas of science and technology significantly. In the present perspective article, we explore how quantum technologies are benefiting from this revolution. We showcase in illustrative examples how scientists in the past few years have started to use machine learning and more broadly methods of artificial intelligence to analyze quantum measurements, estimate the parameters of quantum devices, discover new quantum experimental setups, protocols, and feed back strategies, and generally improve aspects of quantum computing, quantum communication, and quantum simulation. We highlight open challenges and future possibilities and conclude with some speculative visions for the next decade.
2022
Protection of all nondefective twofold degeneracies by antiunitary symmetries in nonHermitian systems
Sharareh Sayyad
Physical Review Research
4(4)
043213
(2022)

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NonHermitian degeneracies are classified as defective exceptional points (EPs) and nondefective de generacies. While in defective EPs, both eigenvalues and eigenvectors coalesce, nondefective degeneracies are characterized merely by the emergence of degenerate eigenvalues. It is also known that all degeneracies are either symmetryprotected or accidental. In this paper, I prove that antiunitary symmetries protect all nondefective twofold degeneracies. By developing a 2D nonHermitian tightbinding model, I have demonstrated that these symmetries comprise various symmetry operations, such as discrete or spatial pointgroup symmetries and Wick’s rotation in the nonHermitian parameter space. Introducing these composite symmetries, I present the protection of nondefective degeneracies in various parameter regimes of my model. This work paves the way to stabilizing nondefective degeneracies and offers a new perspective on understanding nonHermitian band crossings.
Bound states and photon emission in nonHermitian nanophotonics
Zongping Gong, Miguel Bello, Daniel Malz, Flore K. Kunst
Physical Review A
106
053517
(2022)

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We establish a general framework for studying the bound states and the photonemission dynamics of quantum emitters coupled to structured nanophotonic lattices with engineered dissipation (loss). In the singleexcitation sector, the system can be described exactly by a nonHermitian formalism. We have pointed out in the accompanying letter [Gong,et al., arXiv:2205.05479] that a single emitter coupled to a onedimensional nonHermitian lattice may already exhibit anomalous behaviors without Hermitian counterparts. Here we provide further detail on these observations. We also present several additional examples on the cases with multiple quantum emitters or in higher dimensions. Our work unveils the tip of the iceberg of the rich nonHermitian phenomena in dissipative nanophotonic systems.
Deep learning of spatial densities in inhomogeneous correlated quantum systems
Alex Blania, Sandro Herbig, Fabian Dechent, Evert van Nieuwenburg, Florian Marquardt
Machine learning has made important headway in helping to improve the treatment of quantum manybody systems. A domain of particular relevance are correlated inhomogeneous systems. What has been missing so far is a general, scalable deeplearning approach that would enable the rapid prediction of spatial densities for strongly correlated systems in arbitrary potentials. In this work, we present a straightforward scheme, where we learn to predict densities using convolutional neural networks trained on random potentials. While we demonstrate this approach in 1D and 2D lattice models using data from numerical techniques like Quantum Monte Carlo, it is directly applicable as well to training data obtained from experimental quantum simulators. We train networks that can predict the densities of multiple observables simultaneously and that can predict for a whole class of manybody lattice models, for arbitrary system sizes. We show that our approach can handle well the interplay of interference and interactions and the behaviour of models with phase transitions in inhomogeneous situations, and we also illustrate the ability to solve inverse problems, finding a potential for a desired density.
Certification of Genuine Multipartite Entanglement with General and Robust Deviceindependent Witnesses
Chao Zhang, WenHao Zhang, Pavel Sekatski, JeanDaniel Bancal, Michael Zwerger, Peng Yin, GongChu Li, XingXiang Peng, Lei Chen, et al.
Physical Review Letters
129
190503
(2022)

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Genuine multipartite entanglement represents the strongest type of entanglement, which is an essential resource for quantum information processing. Standard methods to detect genuine multipartite entanglement, e.g., entanglement witnesses, state tomography, or quantum state verification, require full knowledge of the Hilbert space dimension and precise calibration of measurement devices, which are usually difficult to acquire in an experiment. The most radical way to overcome these problems is to detect entanglement solely based on the Belllike correlations of measurement outcomes collected in the experiment, namely, device independently. However, it is difficult to certify genuine entanglement of practical multipartite states in this way, and even more difficult to quantify it, due to the difficulty in identifying optimal multipartite Bell inequalities and protocols tolerant to state impurity. In this Letter, we explore a general and robust deviceindependent method that can be applied to various realistic multipartite quantum states in arbitrary finite dimension, while merely relying on bipartite Bell inequalities. Our method allows us both to certify the presence of genuine multipartite entanglement and to quantify it. Several important classes of entangled states are tested with this method, leading to the detection of genuinely entangled states. We also certify genuine multipartite entanglement in weakly entangled GreenbergerHorneZeilinger states, showing that the method applies equally well to less standard states.
Digital Discovery of a Scientific Concept at the Core of Experimental Quantum Optics
Entanglement is a crucial resource for quantum technologies ranging from quantum communication to quantumenhanced measurements and computation. Finding experimental setups for these tasks is a conceptual challenge for human scientists due to the counterintuitive behavior of multiparticle interference and the enormously large combinatorial search space. Recently, new possibilities have been opened by artificial discovery where artificial intelligence proposes experimental setups for the creation and manipulation of highdimensional multiparticle entanglement. While digitally discovered experiments go beyond what has been conceived by human experts, a crucial goal is to understand the underlying concepts which enable these new useful experimental blueprints. Here, we present Halo (Hyperedge Assembly by Linear Optics), a new form of multiphoton quantum interference with surprising properties. Halos were used by our digital discovery framework to solve previously open questions. We  the human part of this collaboration  were then able to conceptualize the idea behind the computer discovery and describe them in terms of effective probabilistic multiphoton emitters. We then demonstrate its usefulness as a core of new experiments for highly entangled states, communication in quantum networks, and photonic quantum gates. Our manuscript has two conclusions. First, we introduce and explain the physics of a new practically useful multiphoton interference phenomenon that can readily be realized in advanced setups such as integrated photonic circuits. Second, our manuscript demonstrates how artificial intelligence can act as a source of inspiration for the scientific discoveries of new actionable concepts in physics.
Theory of LaserAssisted Nuclear Excitation by Electron Capture
The interplay of xray ionization and atomic and nuclear degrees of freedom is investigated theoretically in the process of laserassisted nuclear excitation by electron capture. In the resonant process of nuclear excitation by electron capture, an incident electron recombines into a vacancy in the atomic shell with simultaneous nuclear excitation. Here we investigate the specific scenario in which the free electron and the required atomic shell hole<br>are generated by an xray free electron laser pulse. We develop a theoretical description based on the Feshbach projection operator formalism and consider numerically experimental scenarios at the SACLA xray free electron laser. Our numerical results for excitation of the 29.2 keV nuclear state in <sub>229</sub>Th and the 14.4 keV Mössbauer transition in <sub>57</sub>Fe show low excitation rates but strong enhancement with respect to direct two photon nuclear excitation.
SELFIES and the future of molecular string representations
Mario Krenn, Qianxiang Ai, Senja Barthel, Nessa Carson, Angelo Frei, Nathan C. Frey, Pascal Friederich, Théophile Gaudin, Alberto Alexander Gayle, et al.
Artificial intelligence (AI) and machine learning (ML) are expanding in popularity for broad applications to challenging tasks in chemistry and materials science. Examples include the prediction of properties, the discovery of new reaction pathways, or the design of new molecules. The machine needs to read and write fluently in a chemical language for each of these tasks. Strings are a common tool to represent molecular graphs, and the most popular molecular string representation, SMILES, has powered cheminformatics since the late 1980s. However, in the context of AI and ML in chemistry, SMILES has several shortcomings  most pertinently, most combinations of symbols lead to invalid results with no valid chemical interpretation. To overcome this issue, a new language for molecules was introduced in 2020 that guarantees 100\% robustness: SELFIES (SELFreferencIng Embedded Strings). SELFIES has since simplified and enabled numerous new applications in chemistry. In this manuscript, we look to the future and discuss molecular string representations, along with their respective opportunities and challenges. We propose 16 concrete Future Projects for robust molecular representations. These involve the extension toward new chemical domains, exciting questions at the interface of AI and robust languages and interpretability for both humans and machines. We hope that these proposals will inspire several followup works exploiting the full potential of molecular string representations for the future of AI in chemistry and materials science.
Design of quantum optical experiments with logic artificial intelligence
Alba CerveraLierta, Mario Krenn, Alán AspuruGuzik
Logic Artificial Intelligence (AI) is a subfield of AI where variables can take two defined arguments, True or False, and are arranged in clauses that follow the rules of formal logic. Several problems that span from physical systems to mathematical conjectures can be encoded into these clauses and solved by checking their satisfiability (SAT). In contrast to machine learning approaches where the results can be approximations or local minima, Logic AI delivers formal and mathematically exact solutions to those problems. In this work, we propose the use of logic AI for the design of optical quantum experiments. We show how to map into a SAT problem the experimental preparation of an arbitrary quantum state and propose a logicbased algorithm, called Klaus, to find an interpretable representation of the photonic setup that generates it. We compare the performance of Klaus with the stateoftheart algorithm for this purpose based on continuous optimization. We also combine both logic and numeric strategies to find that the use of logic AI significantly improves the resolution of this problem, paving the path to developing more formalbased approaches in the context of quantum physics experiments.
On scientific understanding with artificial intelligence
Mario Krenn, Robert Pollice, Si Yue Guo, Matteo Aldeghi, Alba CerveraLierta, Pascal Friederich, Gabriel dos Passos Gomes, Florian Häse, Adrian Jinich, et al.
An oracle that correctly predicts the outcome of every particle physics experiment, the products of every possible chemical reaction or the function of every protein would revolutionize science and technology. However, scientists would not be entirely satisfied because they would want to comprehend how the oracle made these predictions. This is scientific understanding, one of the main aims of science. With the increase in the available computational power and advances in artificial intelligence, a natural question arises: how can advanced computational systems, and specifically artificial intelligence, contribute to new scientific understanding or gain it autonomously? Trying to answer this question, we adopted a definition of ‘scientific understanding’ from the philosophy of science that enabled us to overview the scattered literature on the topic and, combined with dozens of anecdotes from scientists, map out three dimensions of computerassisted scientific understanding. For each dimension, we review the existing state of the art and discuss future developments. We hope that this Perspective will inspire and focus research directions in this multidisciplinary emerging field.
Nonreciprocal and chiral singlephoton scattering for giant atoms
YaoTong Chen, Lei Du, Lingzhen Guo, Zhihai Wang, Yan Zhang, Yong Li, JinHui Wu
Communications Physics
5
215
(2022)

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Quantum optics with giant atoms has provided a new paradigm to study photon scatterings. In this work, we investigate the nontrivial singlephoton scattering properties of giant atoms being an effective platform to realize nonreciprocal and chiral quantum optics. For twolevel giant atoms, we identify the condition for nonreciprocal transmission: the external atomic dissipation is further required other than the breaking of timereversal symmetry by local coupling phases. Especially, in the nonMarkovian regime, unconventional revival peaks periodically appear in the reflection spectrum. To explore more interesting scattering behaviors, we extend the twolevel giantatom system to Δtype and ∇ type threelevel giant atoms coupled to double waveguides with different physical mechanisms to realize nonreciprocal and chiral scatterings. Our proposed giantatom structures have potential applications of highefficiency targeted routers that can transport single photons to any desired port deterministically and circulators that can transport single photons between four ports in a cyclic way.
Curiosity in exploring chemical spaces: Intrinsic rewards for deep molecular reinforcement learning
Luca A. Thiede, Mario Krenn, AkshatKumar Nigam, Alán AspuruGuzik
Machine Learning: Science and Technology (3)
035008
(2022)

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Computeraided design of molecules has the potential to disrupt the field of drug and material discovery. Machine learning, and deep learning, in particular, have been topics where the field has been developing at a rapid pace. Reinforcement learning is a particularly promising approach since it allows for molecular design without prior knowledge. However, the search space is vast and efficient exploration is desirable when using reinforcement learning agents. In this study, we propose an algorithm to aid efficient exploration. The algorithm is inspired by a concept known in the literature as curiosity. We show on three benchmarks that a curious agent finds better performing molecules. This indicates an exciting new research direction for reinforcement learning agents that can explore the chemical space out of their own motivation. This has the potential to eventually lead to unexpected new molecules that no human has thought about so far.
Deep Reinforcement Learning for Quantum State Preparation with Weak Nonlinear Measurements
Riccardo Porotti, Antoine Essig, Benjamin Huard, Florian Marquardt
Quantum control has been of increasing interest in recent years, e.g. for tasks like state initialization and stabilization. Feedbackbased strategies are particularly powerful, but also hard to find, due to the exponentially increased search space. Deep reinforcement learning holds great promise in this regard. It may provide new answers to difficult questions, such as whether nonlinear measurements can compensate for linear, constrained control. Here we show that reinforcement learning can successfully discover such feedback strategies, without prior knowledge. We illustrate this for state reparation in a cavity subject to quantumnondemolition detection of photon number, with a simple linear drive as control. Fock states can be produced and stabilized at very high fidelity. It is even possible to reach superposition states, provided the measurement rates for different Fock states can be controlled as well.
Quantum indistinguishability by path identity and with undetected photons
Armin Hochrainer, Mayukh Lahiri, Manuel Erhard, Mario Krenn, Anton Zeilinger
Reviews of Modern Physics
94(2)
025007
(2022)

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Two processes of photonpair creation can be arranged such that the paths of the emitted photons are identical. The path information is thereby not erased but rather never born in the first place due to this path identity. In addition to its implications for fundamental physics, this concept has recently led to a series of impactful discoveries in the fields of imaging, spectroscopy, and quantum information science. Here the idea of path identity is presented and a comprehensive review of recent developments is provided. Specifically, the concept of path identity is introduced based on three defining experimental ideas from the early 1990s. The three experiments have in common that they contain two photonpair sources. The paths of one or both photons from the different sources overlap such that no measurement can recognize from which source they originate. A wide range of noteworthy quantum interference effects (at the single or twophoton level), such as induced coherence, destructive interference of photon pairs, and entanglement generation, are subsequently described. Progress in the exploration of these ideas has stagnated and has gained momentum again only in the last few years. The focus of the review is the new development in the last few years that modified and generalized the ideas from the early 1990s. These developments are overviewed and explained under the same conceptual umbrella, which will help the community develop new applications and realize the foundational implications of this sleeping beauty.
Topological phonon transport in an optomechanical system
Hengjiang Ren, Tirth Shah, Hannes Pfeifer, Christian Brendel, Vittorio Peano, Florian Marquardt, Oskar Painter
Nature Communications
13
3476
(2022)

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Recent advances in cavityoptomechanics have now made it possible to use light not just as a passive measuring device of mechanical motion, but also to manipulate the motion of mechanical objects down to the level of individual quanta of vibrations (phonons). At the same time, microfabrication techniques have enabled smallscale optomechanical circuits capable of onchip manipulation of mechanical and optical signals. Building on these developments, theoretical proposals have shown that larger scale optomechanical arrays can be used to modify the propagation of phonons, realizing a form of topologically protected phonon transport. Here, we report the observation of topological phonon transport within a multiscale optomechanical crystal structure consisting of an array of over 800 cavityoptomechanical elements. Using sensitive, spatially resolved optical readout we detect thermal phonons in a 0.325−0.34GHz band traveling along a topological edge channel, with substantial reduction in backscattering. This represents an important step from the pioneering macroscopic mechanical systems work towards topological phononic systems at the nanoscale, where hypersonic frequency (≳GHz) acoustic wave circuits consisting of robust delay lines and nonreciprocal elements may be implemented. Owing to the broadband character of the topological channels, the control of the flow of heatcarrying phonons, albeit at cryogenic temperatures, may also be envisioned.
Learning Interpretable Representations of Entanglement in Quantum Optics Experiments using Deep Generative Models
Daniel FlamShepherd, Tony Wu, Xuemei Gu, Alba CerveraLierta, Mario Krenn, Alan AspuruGuzik
Nature Machine Intelligence
s42256022004935
(2022)

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Quantum physics experiments produce interesting phenomena such as interference or entanglement, which is a core property of numerous future quantum technologies. The complex relationship between a quantum experiment's structure and its entanglement properties is essential to fundamental research in quantum optics but is difficult to intuitively understand. We present the first deep generative model of quantum optics experiments where a variational autoencoder (QOVAE) is trained on a dataset of experimental setups. In a series of computational experiments, we investigate the learned representation of the QOVAE and its internal understanding of the quantum optics world. We demonstrate that the QOVAE learns an intrepretable representation of quantum optics experiments and the relationship between experiment structure and entanglement. We show the QOVAE is able to generate novel experiments for highly entangled quantum states with specific distributions that match its training data. Importantly, we are able to fully interpret how the QOVAE structures its latent space, finding curious patterns that we can entirely explain in terms of quantum physics. The results demonstrate how we can successfully use and understand the internal representations of deep generative models in a complex scientific domain. The QOVAE and the insights from our investigations can be immediately applied to other physical systems throughout fundamental scientific research.
Realizing exceptional points of any order in the presence of symmetry
Sharareh Sayyad, Flore K. Kunst
Physical Review Research
4(2)
023130
(2022)

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Exceptional points~(EPs) appear as degeneracies in the spectrum of nonHermitian matrices at which the eigenvectors coalesce. In general, an EP of order n may find room to emerge if 2(n−1) real constraints are imposed. Our results show that these constraints can be expressed in terms of the determinant and traces of the nonHermitian matrix. Our findings further reveal that the total number of constraints may reduce in the presence of unitary and antiunitary symmetries. Additionally, we draw generic conclusions for the lowenergy dispersion of the EPs. Based on our calculations, we show that in odd dimensions the presence of sublattice or pseudochiral symmetry enforces nth order EPs to disperse with the (n−1)th root. For two, three and fourband systems, we explicitly present the constraints needed for the occurrence of EPs in terms of system parameters and classify EPs based on their lowenergy dispersion relations.
Deep Learning of Quantum ManyBody Dynamics via Random Driving
Naeimeh Mohseni, Thomas Fösel, Lingzhen Guo, Carlos NavarreteBenlloch, Florian Marquardt
Neural networks have emerged as a powerful way to approach many practical problems in quantumphysics. In this work, we illustrate the power of deep learning to predict the dynamics of a quantummanybody system, where the training is based purely on monitoring expectation values of observables under random driving. The trained recurrent network is able to produce accurate predictions for driving trajectories entirely different than those observed during training. As a proof of principle, here we train the network on numerical data generated from spin models, showing that it can learn the dynamics of observables of interest without needing information about the full quantum state.This allows our approach to be applied eventually to actual experimental data generated from aquantum manybody system that might be open, noisy, or disordered, without any need for a detailedunderstanding of the system. This scheme provides considerable speedup for rapid explorations andpulse optimization. Remarkably, we show the network is able to extrapolate the dynamics to times longer than those it has been trained on, as well as to the infinitesystemsize limit.
TMMFast: A Transfer Matrix Computation Package for Multilayer ThinFilm Optimization: tutorial
Alexander Luce, Ali Mahdavi, Florian Marquardt, Heribert Wankerl
Journal of the Optical Society of America AOptics Image Science and Vision
39(6)
10071013
(2022)

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Achieving the desired optical response from a multilayer thinfilm structure over a broad range of wavelengths and angles of incidence can be challenging. An advanced thinfilm structure can consist of multiple materials with different thicknesses and numerous layers. Design and optimization of complex thinfilm structures with multiple variables is a computationally heavy problem that is still under active research. To enable fast and easy experimentation with new optimization techniques, we propose the Python package Transfer Matrix Method  Fast (TMMFast), which enables parallelized computation of reflection and transmission of light at different angles of incidence and wavelengths through the multilayer thin film. By decreasing computational time, generating datasets for machine learning becomes feasible, and evolutionary optimization can be used effectively. Additionally, the subpackage TMMTorch allows us to directly compute analytical gradients for local optimization by using PyTorch Autograd functionality. Finally, an OpenAI Gym environment is presented, which allows the user to train new reinforcement learning agents on the problem of finding multilayer thinfilm configurations.
Anomalous Behaviors of Quantum Emitters in NonHermitian Baths
Zongping Gong, Miguel Bello, Daniel Malz, Flore K. Kunst
Physical Review Letters
129
223601
(2022)

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Both nonHermitian systems and the behaviour of emitters coupled to structured baths have been studied intensely in recent years. Here we study the interplay of these paradigmatic settings. In a series of examples, we show that a single quantum emitter coupled to a nonHermitian bath displays a number of unconventional behaviours, many without Hermitian counterpart. We first consider a unidirectional hopping lattice whose complex dispersion forms a loop. We identify peculiar bound states inside the loop as a manifestation of the nonHermitian skin effect. In the same setting, emitted photons may display spatial amplification markedly distinct from free propagation, which can be understood with the help of the generalized Brillouin zone. We then consider a nearestneighbor lattice with alternating loss. We find that the longtime emitter decay always follows a power law, which is usually invisible for Hermitian baths. Our work points toward a rich landscape of anomalous quantum emitter dynamics induced by nonHermitian baths.
Upon combining dissipative and nonlinear effects in a bipartite lattice of cavity polaritons, dissipatively stabilized bulk gap solitons emerge, which create a topological interface.
News & Views
Observing polarization patterns in the collective motion of nanomechanical arrays
Juliane Doster, Tirth Shah, Thomas Fösel, Philipp Paulitschke, Florian Marquardt, Eva Weig
Nature Communications
13
2478
(2022)

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In recent years, nanomechanics has evolved into a mature field, with wideranging impact from sensing applications to fundamental physics, and it has now reached a stage which enables the fabrication and study of ever more elaborate devices. This has led to the emergence of arrays of coupled nanomechanical resonators as a promising field of research, serving as model systems to study collective dynamical phenomena such as synchronization or topological transport. From a general point of view, the arrays investigated so far represent scalar fields on a lattice. Moving to a scenario where these could be extended to vector fields would unlock a whole host of conceptually interesting additional phenomena, including the physics of polarization patterns in wave fields and their associated topology. Here we introduce a new platform, a twodimensional array of coupled nanomechanical pillar resonators, whose orthogonal vibration directions encode a mechanical polarization degree of freedom. We demonstrate direct optical imaging of the collective dynamics, enabling us to analyze the emerging polarization patterns and follow their evolution with drive frequency.
Ising machines: Hardware solvers for combinatorial optimization problems
Naeimeh Mohseni, Peter McMahon, Tim Byrnes
Nature Reviews Physics
4
363379
(2022)

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Ising machines are hardware solvers that aim to find the absolute or approximate ground states of the Ising model. The Ising model is of fundamental computational interest because any problem in the complexity class NP can be formulated as an Ising problem with only polynomial overhead, and thus a scalable Ising machine that outperforms existing standard digital computers could have a huge impact for practical applications. We survey the status of various approaches to constructing Ising machines and explain their underlying operational principles. The types of Ising machines considered here include classical thermal annealers based on technologies such as spintronics, optics, memristors and digital hardware accelerators; dynamical systems solvers implemented with optics and electronics; and superconductingcircuit quantum annealers. We compare and contrast their performance using standard metrics such as the groundstate success probability and timetosolution, give their scaling relations with problem size, and discuss their strengths and weaknesses.
Modern applications of machine learning in quantum sciences
Anna Dawid, Julian Arnold, Borja Requena, Alexander Gresch, Marcin Płodzień, Kaelan Donatella, Kim Nicoli, Paolo Stornati, Rouven Koch, et al.
In this book, we provide a comprehensive introduction to the most recent advances in the application of machine learning methods in quantum sciences. We cover the use of deep learning and kernel methods in supervised, unsupervised, and reinforcement learning algorithms for phase classification, representation of manybody quantum states, quantum feedback control, and quantum circuits optimization. Moreover, we introduce and discuss more specialized topics such as differentiable programming, generative models, statistical approach to machine learning, and quantum machine learning.
Directional emission of white light via selective amplification of photon recycling and Bayesian optimization of multilayer thin films
Heribert Wankerl, Christopher Wiesmann, Laura Kreiner, Rainer Butendeich, Alexander Luce, Sandra Sobczyk, Maike Lorena Stern, Elmar Wolfgang Lang
Over the last decades, lightemitting diodes (LED) have replaced common light bulbs in almost every application, from flashlights in smartphones to automotive headlights. Illuminating nightly streets requires LEDs to emit a light spectrum that is perceived as pure white by the human eye. The power associated with such a white light spectrum is not only distributed over the contributing wavelengths but also over the angles of vision. For many applications, the usable light rays are required to exit the LED in forward direction, namely under small angles to the perpendicular. In this work, we demonstrate that a specifically designed multilayer thin film on top of a white LED increases the power of pure white light emitted in forward direction. Therefore, the deduced multiobjective optimization problem is reformulated via a realvalued physicsguided objective function that represents the hierarchical structure of our engineering problem. Variants of Bayesian optimization are employed to maximize this nondeterministic objective function based on ray tracing simulations. Eventually, the investigation of optical properties of suitable multilayer thin films allowed to identify the mechanism behind the increased directionality of white light: angle and wavelength selective filtering causes the multilayer thin film to play ping pong with rays of light.
These are the lecture notes for a course that I am teaching at Zhiyuan College of Shanghai Jiao Tong University, though the first draft was created for a previous course I taught at the University of ErlangenNuremberg in Germany. It has been designed for students who have only had basic training on quantum mechanics, and hence, the course is suited for people at all levels. The notes are a work in progress, meaning that some proofs and many figures are still missing. However, I've tried my best to write everything in such a way that a reader can follow naturally all arguments and derivations even with these missing bits. Quantum optics treats the interaction between light and matter. We may think of light as the optical part of the electromagnetic spectrum, and matter as atoms. However, modern quantum optics covers a wild variety of systems, including superconducting circuits, confined electrons, excitons in semiconductors, defects in solid state, or the centerofmass motion of micro, meso, and macroscopic systems. Moreover, quantum optics is at the heart of the field of quantum information. The ideas and experiments developed in quantum optics have also allowed us to take a fresh look at manybody problems and even highenergy physics. In addition, quantum optics holds the promise of testing foundational problems in quantum mechanics as well as physics beyond the standard model in tablesized experiments. Quantum optics is therefore a topic that no future researcher in quantum physics should miss.
Phase Space Crystal Vibrations: Chiral Edge States with Preserved Timereversal Symmetry
Lingzhen Guo, Vittorio Peano, Florian Marquardt
Physical Review B
105(9)
094301
(2022)

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Chiral transport along edge channels in Chern insulators represents the most robust version of topological transport, but it usually requires breaking of the physical timereversal symmetry. In this work, we introduce a different mechanism that foregoes this requirement, based on the combination of the symplectic geometry of phase space and interactions. Starting from a honeycomb phasespace crystal of atoms, which can be generated by periodic driving of a onedimensional interacting quantum gas, we show that the resulting vibrational lattice waves have topological properties. Our work provides a new platform to study topological manybody physics in dynamical systems.
suggested by editors
Experimental highdimensional GreenbergerHorneZeilinger entanglement with superconducting transmon qutrits
Alba CerveraLierta, Mario Krenn, Alan AspuruGuzik, Alexey Galda
Physical Review Applied
17(2)
024062
(2022)

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Multipartite entanglement is one of the core concepts in quantum information science with broad applications that span from condensed matter physics to quantum physics foundation tests. Although its most studied and tested forms encompass twodimensional systems, current quantum platforms technically allow the manipulation of additional quantum levels. We report the experimental demonstration and certification of a highdimensional multipartite entangled state in a superconducting quantum processor. We generate the threequtrit GreenbergerHorneZeilinger state by designing the necessary pulses to perform highdimensional quantum operations. We obtain the fidelity of 76%±1%, proving the generation of a genuine threepartite and threedimensional entangled state. To this date, only photonic devices have been able to create and certify the entanglement of these highdimensional states. Our work demonstrates that another platform, superconducting systems, is ready to exploit genuine highdimensional entanglement and that a programmable quantum device accessed on the cloud can be used to design and execute experiments beyond binary quantum computation.
2021
Tunneling in the Brillouin Zone: Theory of Backscattering in Valley Hall Edge Channels
Tirth Shah, Florian Marquardt, Vittorio Peano
Physical Review B
104(23)
235431
(2021)

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A large set of recent experiments has been exploring topological transport in bosonic systems,e.g. of photons or phonons. In the vast majority, timereversal symmetry is preserved, and bandstructures are engineered by a suitable choice of geometry, to produce topologically nontrivialbandgaps in the vicinity of highsymmetry points. However, this leaves open the possibility oflargequasimomentum backscattering, destroying the topological protection. Up to now, it has beenunclear what precisely are the conditions where this effect can be sufficiently suppressed. In thepresent work, we introduce a comprehensive semiclassical theory of tunneling transitions in momentum space, describing backscattering for one of the most important system classes, based on thevalley Hall effect. We predict that even for a smooth domain wall effective scattering centres developat locations determined by both the local slope of the wall and the energy. Moreover, our theoryprovides a quantitative analysis of the exponential suppression of the overall reflection amplitudewith increasing domain wall smoothness.
suggested by editors
Phase Space Crystals: Condensed matter in dynamical systems
This book aims to develop a general framework of condensed matter theory in phase space, instead of configuration space, of a dynamical system. Different from Euclidean real space, phase space is embedded with symplectic geometry in classical mechanics or noncommutative geometry in quantum mechanics. Arbitrary lattice Hamiltonians and crystalline manybody states in phase space can be created with the Floquet approach. The book covers topics ranging from dynamical systems, Floquet theory, topological physics to quantum manybody physics and time crystals. The book fills in the blanks in the study of dynamical systems by considering manybody physics in the phase space.
Accelerated NonReciprocal Transfer of Energy Around an Exceptional Point
We develop perturbative methods to study and control dynamical phenomena related to exceptional points in NonHermitian systems. In particular, we show how to find perturbative solutions based on the Magnus expansion that accurately describe the evolution of nonHermitian systems when encircling an exceptional point. This allows us to use the recently proposed Magnusbased strategy for control to design fast nonreciprocal, topological operations whose fidelity error is orders of magnitude smaller than their much slower adiabatic counterparts.
Arbitrary optical wave evolution with Fourier transforms and phase masks
Victor LopézPastor, Jeff S. Lundeen, Florian Marquardt
A large number of applications in classical and quantum photonics require the capability of implementing arbitrary linear unitary transformations on a set of optical modes. In a seminal work by Reck et al. it was shown how to build such multiport universal interferometers with a mesh of beam splitters and phase shifters, and this design became the basis for most experimental implementations in the last decades. However, the design of Reck et al. is difficult to scale up to a large number of modes, which would be required for many applications. Here we present a constructive proof that it is possible to realize a multiport universal interferometer on N modes with a succession of 6N Fourier transforms and 6N+1 phase masks, for any even integer N. Furthermore, we provide an algorithm to find the correct succesion of Fourier transforms and phase masks to realize a given arbitrary unitary transformation. Since Fourier transforms and phase masks are routinely implemented in several optical setups and they do not suffer from the scalability issues associated with building extensive meshes of beam splitters, we believe that our design can be useful for many applications in photonics.
Dynamical phase transitions in quantum spin models with antiferromagnetic longrange interactions
Jad C. Halimeh, Maarten Van Damme, Lingzhen Guo, Johannes Lang, Philipp Hauke
Physical Review B
104(11)
115133
(2021)

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In recent years, dynamical phase transitions and outofequilibrium criticality have been at the forefront of ultracold gases and condensed matter research. Whereas universality and scaling are established topics in equilibrium quantum manybody physics, outofequilibrium extensions of such concepts still leave much to be desired. Using exact diagonalization and the timedependent variational principle in uniform matrix product states, we calculate the time evolution of the local order parameter and Loschmidt return rate in transversefield Ising chains with antiferromagnetic power lawdecaying interactions, and map out the corresponding rich dynamical phase diagram. Anomalous cusps in the return rate, which are ubiquitous at small quenches within the ordered phase in the case of ferromagnetic longrange interactions, are absent within the accessible timescales of our simulations in the antiferromagnetic case, showing that longrange interactions are not a sufficient condition for their appearance. We attribute this to much weaker domainwall binding in the antiferromagnetic case. For quenches across the quantum critical point, regular cusps appear in the return rate and connect to the local order parameter changing sign, indicating the concurrence of two major concepts of dynamical phase transitions. Our results consolidate conclusions of previous works that a necessary condition for the appearance of anomalous cusps in the return rate after quenches within the ordered phase is for topologically trivial local spin flips to be the energetically dominant excitations in the spectrum of the quench Hamiltonian. Our findings are readily accessible in modern trappedion setups and we outline the associated experimental considerations.
Channel discord and distortion
WeiWei Zhang, Yuval R. Sanders, Barry C. Sanders
New Journal of Physics (23)
083025
(2021)

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Discord, originally notable as a signature of bipartite quantum correlation, in fact can be nonzero<br>classically, i.e. arising from noisy measurements by one of the two parties. Here we redefine<br>classical discord to quantify channel distortion, in contrast to the previous restriction of classical<br>discord to a state, and we then show a monotonic relationship between classical (channel) discord<br>and channel distortion. We show that classical discord is equivalent to (doubly stochastic) channel<br>distortion by numerically discovering a monotonic relation between discord and totalvariation<br>distance for a bipartite protocol with one party having a noiseless channel and the other party<br>having a noisy channel. Our numerical method includes randomly generating doubly stochastic<br>matrices for noisy channels and averaging over a uniform measure of input messages. Connecting<br>discord with distortion establishes discord as a signature of classical, not quantum, channel<br>distortion.
Perturbation theory of nearly spherical dielectric optical resonators
Julius Gohsrich, Tirth Shah, Andrea Aiello
Physical Review A
104(2)
023516
(2021)

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Dielectric spheres of various sizes may sustain electromagnetic whisperinggallery modes resonating at optical frequencies with very narrow linewidths. Arbitrary small deviations from the spherical shape typically shift and broaden such resonances. Our goal is to determine these shifted and broadened resonances. A boundarycondition perturbation theory for the acoustic vibrations of nearly circular membranes was developed by Rayleigh more than a century ago. We extend this theory to describe the electromagnetic excitations of nearly spherical dielectric cavities. This approach permits us to avoid dealing with decaying quasinormal modes. We explicitly find the frequencies and the linewidths of the optical resonances for arbitrarily deformed nearly spherical dielectric cavities, as power series expansions by a small parameter, up to and including secondorder terms. We thoroughly discuss the physical conditions for the applicability of perturbation theory.
Optical signatures of the coupled spinmechanics of a levitated magnetic microparticle
Vanessa Wachter, Victor A. S. V. Bittencourt, Shangran Xie, Sanchar Sharma, Nicolas Joly, Philip Russell, Florian Marquardt, Silvia ViolaKusminskiy
Journal of the Optical Society of America BOptical Physics
38(12)
(2021)

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We propose a platform that combines the fields of cavity optomagnonics and levitated optome<br>chanics in order to control and probe the coupled spinmechanics of magnetic dielectric particles. We theoretically study the dynamics of a levitated Faradayactive dielectric microsphere serving as an optomagnonic cavity, placed in an external magnetic field and driven by an external laser. We find that the optically driven magnetization dynamics induces angular oscillations of the particle with low associated damping. Further, we show that the magnetization and angular motion dynamics<br>can be probed via the power spectrum of the outgoing light. Namely, the characteristic frequencies attributed to the angular oscillations and the spin dynamics are imprinted in the light spectrum by two main resonance peaks. Additionally, we demonstrate that a ferromagnetic resonance setup with an oscillatory perpendicular magnetic field can enhance the resonance peak corresponding to<br>the spin oscillations and induce fast rotations of the particle around its anisotropy axis.
Analytic Design of Accelerated Adiabatic Gates in Realistic Qubits: General Theory and Applications to Superconducting Circuits
F Setiawan, Peter Groszkowski, Hugo Ribeiro, Aashish A Clerk
Shortcuts to adiabaticity (STA) is a general methodology for speeding up adiabatic quantumprotocols, and has many potential applications in quantum information processing. Unfortunately,analytically constructing STAs for systems having complex interactions and more than a few levelsis a challenging task. This is usually overcome by assuming an idealized Hamiltonian (e.g., only alimited subset of energy levels are retained, and the rotatingwave approximation (RWA) is made).Here, we develop ananalyticapproach that allows one to go beyond these limitations. Our methodis general and results in analyticallyderived pulse shapes that correct both nonadiabatic errorsas well as nonRWA errors. We also show that our approach can yield pulses requiring a smallerdriving power than conventional nonadiabatic protocols. We show in detail how our ideas can beused to analytically design highfidelity singlequbit “tripod” gates in a realistic superconductingfluxonium qubit.
suggested by editors
Rapid Exploration of Topological Band Structures using Deep Learning
Vittorio Peano, Florian Sapper, Florian Marquardt
Physical Review X
11(2)
021052
(2021)

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The design of periodic nanostructures allows to tailor the transport of photons, phonons, and matter waves for specific applications. Recent years have seen a further expansion of this field by engineering topological properties. However, what is missing currently are efficient ways to rapidly explore and optimize band structures and to classify their topological characteristics for arbitrary unitcell geometries. In this work, we show how deep learning can address this challenge. We introduce an approach where a neural network first maps the geometry to a tightbinding model. The tightbinding model encodes not only the band structure but also the symmetry properties of the Bloch waves. This allows us to rapidly categorize a large set of geometries in terms of their band representations, identifying designs for fragile topologies. We demonstrate that our method is also suitable to calculate strong topological invariants, even when (like the Chern number) they are not symmetry indicated. Engineering of domain walls and optimization are accelerated by orders of magnitude. Our method directly applies to any passive linear material, irrespective of the symmetry class and space group. It is general enough to be extended to active and nonlinear metamaterials.
Machine Learning and Quantum Devices
Florian Marquardt
SciPost Physics (21)
10.21468
(2021)

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These brief lecture notes cover the basics of neural networks and deep learning as well as their applications in the quantum domain, for physicists without prior knowledge. In the first part, we describe training using backpropagation, image classification, convolutional networks and autoencoders.The second part is about advanced techniques like reinforcement learning (for discovering control strategies), recurrent neural networks (for analyzing timetraces), and Boltzmann machines (for learning probability distributions). In the third lecture, we discuss first recent applications to quantum physics, with an emphasis on quantum information processing machines. Finally, the fourth lecture is devoted to the promise of using quantum effects to accelerate machine learning.
Renormalized Mutual Information for Artificial Scientific Discovery
Leopoldo Sarra, Andrea Aiello, Florian Marquardt
Physical Review Letters
126
200601
(2021)

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We derive a welldefined renormalized version of mutual information that allows to estimate the dependence between continuous random variables in the important case when one is deterministically dependent on the other. This is the situation relevant for feature extraction, where the goal is to produce a lowdimensional effective description of a highdimensional system. Our approach enables the discovery of collective variables in physical systems, thus adding to the toolbox of artificial scientific discovery, while also aiding the analysis of information flow in artificial neural networks.
Error suppression in adiabatic quantum computing with qubit ensembles
Naeimeh Mohseni, Marek Narozniak, Alexey N Pyrkov, Valentin Ivannikov, Jonathan P Dowling
npj Quantum Information
7(71)
(2021)

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Incorporating protection against quantum errors into adiabatic quantum computing (AQC) is an important task due to the inevitable presence of decoherence. Here, we investigate an errorprotected encoding of the AQC Hamiltonian, where qubit ensembles are used in place of qubits. Our Hamiltonian only involves total spin operators of the ensembles, offering a simpler route towards errorcorrected quantum computing. Our scheme is particularly suited to neutral atomic gases where it is possible to realize large ensemble sizes and produce ensembleensemble entanglement. We identify a critical ensemble size Nc where the nature of the first excited state becomes a single particle perturbation of the ground state, and the gap energy is predictable by meanfield theory. For ensemble sizes larger than Nc, the ground state becomes protected due to the presence of logically equivalent states and the AQC performance improves with N, as long as the decoherence rate is sufficiently low.
Quantum circuit optimization with deep reinforcement learning
Thomas Fösel, Murphy Yuezhen Niu, Florian Marquardt, Li Li (李力)
A central aspect for operating future quantum computers is quantum circuit optimization, i.e., the search for efficient realizations of quantum algorithms given the device capabilities. In recent years, powerful approaches have been developed which focus on optimizing the highlevel circuit structure. However, these approaches do not consider and thus cannot optimize for the hardware details of the quantum architecture, which is especially important for nearterm devices. To address this point, we present an approach to quantum circuit optimization based on reinforcement learning. We demonstrate how an agent, realized by a deep convolutional neural network, can autonomously learn generic strategies to optimize arbitrary circuits on a specific architecture, where the optimization target can be chosen freely by the user. We demonstrate the feasibility of this approach by training agents on 12qubit random circuits, where we find on average a depth reduction by 27% and a gate count reduction by 15%. We examine the extrapolation to larger circuits than used for training, and envision how this approach can be utilized for nearterm quantum devices.
Floquet theory for temporal correlations and spectra in timeperiodic open quantum systems: Application to squeezed parametric oscillation beyond the rotatingwave approximation
Carlos NavarreteBenlloch, Rafael Garcés, Naeimeh Mohseni, German J. de Valcarcel
Physical Review A
103(2)
023713
(2021)

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Open quantum systems can display periodic dynamics at the classical level either due to external periodic modulations or to selfpulsing phenomena typically following a Hopf bifurcation. In both cases, the quantum fluctuations around classical solutions do not reach a quantumstatistical stationary state, which prevents adopting the simple and reliable methods used for stationary quantum systems. Here we put forward a general and efficient method to compute twotime correlations and corresponding spectral densities of timeperiodic open quantum systems within the usual linearized (Gaussian) approximation for their dynamics. Using Floquet theory, we show how the quantum Langevin equations for the fluctuations can be efficiently integrated by partitioning the time domain into oneperiod duration intervals, and relating the properties of each period to the first one. Spectral densities, like squeezing spectra, are computed similarly, now in a twodimensional temporal domain that is treated as a chessboard with oneperiod × oneperiod cells. This technique avoids cumulative numerical errors as well as efficiently saving computational time. As an illustration of the method, we analyze the quantum fluctuations of a damped parametrically driven oscillator (degenerate parametric oscillator) below threshold and far away from rotatingwave approximation conditions, which is a relevant scenario for modern lowfrequency quantum oscillators. Our method reveals that the squeezing properties of such devices are quite robust against the amplitude of the modulation or the low quality of the oscillator, although optimal squeezing can appear for parameters that are far from the ones predicted within the rotatingwave approximation.
suggested by editors
Engineering Fast HighFidelity Quantum Operations With Constrained Interactions
Thales Figueiredo Roque, Aashish A Clerk, Hugo Ribeiro
npj Quantum Information
7
28
(2021)

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Understanding how to tailor quantum dynamics to achieve a desired evolution is a crucial problemin almost all quantum technologies. We present a very general method for designing highefficiencycontrol sequences that are always fully compatible with experimental constraints on available interactions and their tunability. Our approach reduces in the end to finding control fields by solvinga set of timeindependent linear equations. We illustrate our method by applying it to a numberof physicallyrelevant problems: the strongdriving limit of a twolevel system, fast squeezing in aparametrically driven cavity, the leakage problem in transmon qubit gates, and the acceleration ofSNAP gates in a qubitcavity system.
Squeezed comb states
Namrata Shukla, Stefan Nimmrichter, Barry C. Sanders
Physical Review A
103
012408
(2021)

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Continuousvariable codes are an expedient solution for quantum information processing and quantum communication involving optical networks. Here we characterize the squeezed comb, a finite superposition of equidistant squeezed coherent states on a line, and its properties as a continuousvariable encoding choice for a logical qubit. The squeezed comb is a realistic approximation to the ideal code proposed by Gottesman et al. [D. Gottesman, A. Kitaev, and J. Preskill, Phys. Rev. A 64, 012310 (2001)], which is fully protected against errors caused by the paradigmatic types of quantum noise in continuousvariable systems: damping and diffusion. This is no longer the case for the code space of finite squeezed combs, and noise robustness depends crucially on the encoding parameters. We analyze finite squeezed comb states in phase space, highlighting their complicated interference features and characterizing their dynamics when exposed to amplitude damping and Gaussian diffusion noise processes. We find that squeezed comb states are more suitable and less error prone when exposed to damping, which speaks against standard errorcorrection strategies that employ linear amplification to convert damping into easiertodescribe isotropic diffusion noise.
2020
Oscillating bound states for a giant atom
Lingzhen Guo, Anton Frisk Kockum, Florian Marquardt, Göran Johannson
Physical Review Research
2(4)
043014
(2020)

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We investigate the relaxation dynamics of a single artificial atom interacting, via multiple coupling points, with a continuum of bosonic modes (photons or phonons) in a onedimensional waveguide. In the nonMarkovian regime, where the traveling time of a photon or phonon between the coupling points is sufficiently large compared to the inverse of the bare relaxation rate of the atom, we find that a boson can be trapped and form a stable bound state. As a key discovery, we further find that a persistently oscillating bound state can appear inside the continuous spectrum of the waveguide if the number of coupling points is more than two since such a setup enables multiple bound modes to coexist. This opens up prospects for storing and manipulating quantum information in larger Hilbert spaces than available in previously known bound states.
Spatial localization and pattern formation in discrete optomechanical cavities and arrays
Joaquín RuizRivas, Giuseppe Patera, Carlos NavarreteBenlloch, Eugenio Roldán, German de Valcarcel
New Journal of Physics
22
093076
(2020)

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We investigate theoretically the generation of nonlinear dissipative structures in optomechanical(OM) systems containing discrete arrays of mechanical resonators. We consider both hybridmodels in which the optical system is a continuous multimode field, as it would happen in an OMcavity containing an array of micromirrors, and also fully discrete models in which eachmechanical resonator interacts with a single optical mode, making contact with Ludwig andMarquardt (2013Phys.Rev.Lett.101, 073603). Also, we study the connections between both typesof models and continuous OM models. While all three types of models merge naturally in the limitof a large number of densely distributed mechanical resonators, we show that the spatiallocalization and the pattern formation found in continuous OM models can still be observed for asmall number of mechanical elements, even in the presence of finitesize effects, which we discuss.This opens new venues for experimental approaches to the subject.
Manybody dephasing in a trappedion quantum simulator
Harvey B. Kaplan, Lingzhen Guo, Wen Lin Tan, Arinjoy De, Florian Marquardt, Guido Pagano, Christopher Monroe
Physical Review Letters
125(1218)
120605
(2020)

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How a closed interacting quantum manybody system relaxes and dephases as a function of time is a fundamental question in thermodynamic and statistical physics. In this Letter, we analyze and observe the persistent temporal fluctuations after a quantum quench of a tunable longrange interacting transversefield Ising Hamiltonian realized with a trappedion quantum simulator. We measure the temporal fluctuations in the average magnetization of a finitesize system of spin1/2 particles. We experiment in a regime where the properties of the system are closely related to the integrable Hamiltonian with global spinspin coupling, which enables analytical predictions for the longtime nonintegrable dynamics. The analytical expression for the temporal fluctuations predicts the exponential suppression of temporal fluctuations with increasing system size. Our measurement data is consistent with our theory predicting the regime of manybody dephasing.
Observation of concentrating paraxial beams
Andrea Aiello, Martin Paúr, Bohumil Stoklasa, Zdeněk Hradil, Jaroslav Řeháček, Luis L SánchezSoto
OSA Continuum
3(9)
10.1364/OSAC.400410
23872394
(2020)

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We report the first, to the best of our knowledge, observation of concentrating paraxialbeams of light in a linear nondispersive medium. We have generated this intriguing class of lightbeams, recently predicted by one of us, in both one and twodimensional configurations. As wedemonstrate in our experiments, these concentrating beams display unconventional features, suchas the ability to strongly focus in the focal spot of a thin lens like a plane wave, while keepingtheir total energy finite.
Probing the TavisCummings level splitting with intermediatescale superconducting circuits
Ping Yang, Jan David Brehm, Juha Leppäkangas, Lingzhen Guo, Michael Marthaler, Isabella Boventer, Alexander Stehli, Tim Wolz, Alexey V. Ustinov, et al.
Physical Review Applied (14)
024025
(2020)

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We demonstrate the local control of up to eight twolevel systems interacting strongly with a microwave cavity. Following calibration, the frequency of each individual twolevel system (qubit) is tunable without influencing the others. Bringing the qubits one by one on resonance with the cavity, we observe the collective coupling strength of the qubit ensemble. The splitting scales up with the square root of the number of the qubits, being the hallmark of the TavisCummings model. The local control circuitry causes a bypass shunting the resonator, and a Fano interference in the microwave readout, whose contribution can be calibrated away to recover the pure cavity spectrum. The simulator's attainable size of dressed states is limited by reduced signal visibility, and if uncalibrated by offresonance shifts of subcomponents. Our work demonstrates control and readout of quantum coherent mesoscopic multiqubit system of intermediate scale under conditions of noise.
Kinetics of ManyBody Reservoir Engineering
Hugo Ribeiro, Florian Marquardt
Physical Review Research
2(3)
033231
(2020)

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Recent advances illustrate the power of reservoir engineering in applications to manybody systems, such as quantum simulators based on superconducting circuits. We present a frameworkbased on kinetic equations and noise spectra that can be used to understand both the transientand longtime behavior of many particles coupled to an engineered reservoir in a numberconservingway. For the example of a bosonic array, we show that the nonequilibrium steady state can beexpressed, in a wide parameter regime, in terms of a modified BoseEinstein distribution with anenergydependent temperature.
Condensed matter physics in time crystals
Lingzhen Guo, Pengfei Liang
New Journal of Physics (22)
075003
(2020)

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Time crystals are physical systems whose time translation symmetry is spontaneously broken. Although the spontaneous breaking of continuous timetranslation symmetry in static systems is proved impossible for the equilibrium state, the discrete timetranslation symmetry in periodically driven (Floquet) systems is allowed to be spontaneously broken, resulting in the socalled Floquet or discrete time crystals. While most works so far searching for time crystals focus on the symmetry breaking process and the possible stabilising mechanisms, the manybody physics from the interplay of symmetrybroken states, which we call the condensed matter physics in time crystals, is not fully explored yet. This review aims to summarise the very preliminary results in this new research field with an analogous structure of condensed matter theory in solids. The whole theory is built on a hidden symmetry in time crystals, i.e., the phase space lattice symmetry, which allows us to develop the band theory, topology and strongly correlated models in phase space lattice. In the end, we outline the possible topics and directions for the future research.
Efficient cavity control with SNAP gates
Thomas Fösel, Stefan Krastanov, Florian Marquardt, Liang Jiang
Microwave cavities coupled to superconducting qubits have been demonstrated to be a promising platform for quantum information processing. A major challenge in this setup is to realize universal control over the cavity. A promising approach are selective numberdependent arbitrary phase (SNAP) gates combined with cavity displacements. It has been proven that this is a universal gate set, but a central question remained open so far: how can a given target operation be realized efficiently with a sequence of these operations. In this work, we present a practical scheme to address this problem. It involves a hierarchical strategy to insert new gates into a sequence, followed by a cooptimization of the control parameters, which generates short highfidelity sequences. For a broad range of experimentally relevant applications, we find that they can be implemented with 3 to 4 SNAP gates, compared to up to 50 with previously known techniques.
Chimera states in small optomechanical arrays
Karl Pelka, Vittorio Peano, Andre Xuereb
Physical Review Research (2)
013201
(2020)

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Synchronization of weaklycoupled nonlinear oscillators is a ubiquitous phenomenon that has been observedacross the natural sciences. We study the dynamics of optomechanical arrays—networks of mechanically compliant structures that interact with the radiation pressure force—which are driven to selfoscillation. Thesesystems offer a convenient platform to study synchronization phenomena and have potential technological applications. We demonstrate that this system supports the existence of longlived chimera states, where parts ofthe array synchronize whilst others do not. Through a combined numerical and analytical analysis we show thatthese chimera states can only emerge in the presence of disorder.
Maxwell's lesser demon: A Quantum Engine Driven by Pointer Measurements
Stella Seah, Stefan Nimmrichter, Valerio Scarani
Physical Review Letters
124
100603
(2020)

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We discuss a selfcontained spinboson model for a measurementdriven engine, in which a demongenerates work from random thermal excitations of a quantum spin via measurement and feedbackcontrol. Instead of granting it full direct access to the spin state and to Landauer’s erasure strokes foroptimal performance, we restrict this lesser demon’s action to pointer measurements, i.e. random orcontinuous interrogations of a damped mechanical oscillator that assumes macroscopically distinctpositions depending on the spin state. The engine could reach simultaneously high output powersand efficiencies and can operate in temperature regimes where quantum Otto engines would fail.
Quench dynamics in onedimensional optomechanical arrays
Sadegh Raeisi, Florian Marquardt
Physical Review A
101(2)
023814
(2020)

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Nonequilibrium dynamics induced by rapid changes of external parameters is relevant for a widerange of scenarios across many domains of physics. For waves in spatially periodic systems, quencheswill alter the bandstructure and generate new excitations. In the case of topological bandstructures,defect modes at boundaries can be generated or destroyed when quenching through a topologicalphase transition. Here, we demonstrate that optomechanical arrays are a promising platform forstudying such dynamics, as their bandstructure can be tuned temporally by a control laser. Westudy the creation of nonequilibrium optical and mechanical excitations in 1D arrays, including abosonic version of the SuSchriefferHeeger model. These ideas can be transferred to other systemssuch as driven nonlinear cavity arrays.
Nonreciprocal topological phononics in optomechanical arrays
Claudio Sanavio, Vittorio Peano, André Xuereb
Physical Review B
101(8)
085108
(2020)

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We propose a platform for robust and tunable nonreciprocal phonon transport based on arrays of optomechanical microtoroids. In our approach, timereversal symmetry is broken by the interplay of photonic spinorbit coupling, engineered using a stateoftheart geometrical approach, and the optomechanical interaction. We demonstrate the topologically protected nature of this system by investigating its robustness to imperfections. This type of system could find application in phononbased information storage and signalprocessing devices.
Nonlinear dynamics of weakly dissipative optomechanical systems
Thales Figueiredo Roque, Florian Marquardt, Oleg M. Yevtushenko
New Journal of Physics (22)
013049
(2020)

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Optomechanical systems attract a lot of attention because they provide a novel platform for quantum measurements, transduction, hybrid systems, and fundamental studies of quantum physics. Their classical nonlinear dynamics is surprisingly rich and so far remains underexplored. Works devoted to this subject have typically focussed on dissipation constants which are substantially larger than those encountered in current experiments, such that the nonlinear dynamics of weakly dissipative optomechanical systems is almost uncharted waters. In this work, we fill this gap and investigate the regular and chaotic dynamics in this important regime. To analyze the dynamical attractors, we have extended the "Generalized Alignment Index" method to dissipative systems. We show that, even when chaotic motion is absent, the dynamics in the weakly dissipative regime is extremely sensitive to initial conditions. We argue that reducing dissipation allows chaotic dynamics to appear at a substantially smaller driving strength and enables various routes to chaos. We identify three generic features in weakly dissipative classical optomechanical nonlinear dynamics: the NeimarkSacker bifurcation between limit cycles and limit tori (leading to a comb of sidebands in the spectrum), the quasiperiodic route to chaos, and the existence of transient chaos.
Deterministic generation of hybrid highN N00N states with Rydberg ions trapped in microwave cavities
Naeimeh Mohseni, Carlos NavarreteBenlloch, Shahpoor Saeidian, Jonathan P Dowling
Physical Review A
101(1)
013804
(2020)

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Trapped ions are among the most promising platforms for quantum technologies. They are atthe heart of the most precise clocks and sensors developed to date, which exploit the quantumcoherence of a single electronic or motional degree of freedom of an ion. However, future highprecision quantum metrology will require the use of entangled states of several degrees of freedom.Here we propose a protocol capable of generating highN00N states where the entanglement is sharedbetween the motion of a trapped ion and an electromagnetic cavity mode, a socalled ‘hybrid’configuration. We prove the feasibility of the proposal in a platform consisting of a trapped ionexcited to its circularRydbergstate manifold, coupled to the modes of a highQ microwave cavity.This compact hybrid architecture has the advantage that it can couple to signals of very differentnature, which modify either the ion’s motion or the cavity modes. Moreover, the exact same setupcan be used right after the statepreparation phase to implement the interferometer required forquantum metrology.
2019
Field theory of monochromatic optical beams II. Classical and quantum paraxial fields
Andrea Aiello
Journal of Optics
22(1)
014002
(2019)

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This work is the second part of an investigation aiming at the study of optical wave equations from a fieldtheoretic point of view. Here, we study classical and quantum aspects of scalar fields satisfying the paraxial wave equation. First, we determine conservation laws for energy, linear and angular momentum of paraxial fields in a classical context. Then, we proceed with the quantization of the field. Finally, we compare our result with the traditional ones.
Field theory of monochromatic optical beams I. classical fields
Andrea Aiello
Journal of Optics
22(1)
014001
(2019)

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We study monochromatic, scalar solutions of the Helmholtz and paraxial wave equations from a fieldtheoretic point of view. We introduce appropriate timeindependent Lagrangian densities for which the EulerLagrange equations reproduces either Helmholtz and paraxial wave equations with the $z$coordinate, associated with the main direction of propagation of the fields, playing the same role of time in standard Lagrangian theory. For both Helmholtz and paraxial scalar fields, we calculate the canonical energymomentum tensor and determine the continuity equations relating ``energy'' and ``momentum'' of the fields. Eventually, the reduction of the Helmholtz wave equation to a useful firstorder Dirac form, is presented. This work sheds some light on the intriguing and not so acknowledged connections between angular spectrum representation of optical wavefields, cosmological models and physics of black holes.
Quantum state transfer via acoustic edge states in a 2D optomechanical array
MarcAntoine Lemonde, Vittorio Peano, Peter Rabl, Dimitris G Angelakis
New Journal of Physics
21
113030
(2019)

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We propose a novel hybrid platform where solidstate spin qubits are coupled to the acoustic modes ofa twodimensional array of optomechanical(OM)nano cavities. Previous studies of coupled OMcavities have shown that in the presence of strong optical drivingfields, the interplay between thephotonphonon interaction and their respective intercavity hopping allows the generation oftopological phases of sound and light. In particular, the mechanical modes can enter a Chern insulatorphase where the timereversal symmetry is broken. In this context, we exploit the robust acoustic edgestates as a chiral phononic waveguide and describe a state transfer protocol between spin qubitslocated in distant cavities. We analyze the performance of this protocol as a function of the relevantsystem parameters and show that a highfidelity and purely unidirectional quantum state transfer canbe implemented under experimentally realistic conditions. As a specific example, we discuss theimplementation of such topological quantum networks in diamond based OM crystals where pointdefects such as siliconvacancy centers couple to the chiral acoustic channel via strain.
Collisional quantum thermometry
Stella Seah, Stefan Nimmrichter, Daniel Grimmer, Jader P. Santos, Valerio Scarini, Gabriel T. Landi
Physical Review Letters
123(18)
180602
(2019)

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We introduce a general framework for thermometry based on collisional models, where ancillas probe thetemperature of the environment through an intermediary system. This allows for the generation of correlatedancillas even if they are initially independent. Using tools from parameter estimation theory, we show through aminimal qubit model that individual ancillas can already outperform the thermal CramerRao bound. In addition,when probed collectively, these ancillas may exhibit superlinear scalings of the Fisher information, especiallyfor weak systemancilla interactions. Our approach sets forth the notion of metrology in a sequential interactionssetting, and may inspire further advances in quantum thermometry.
Almost thermal operations: inhomogeneous reservoirs
Angeline Shu, Yu Cai, Stella Seah, Stefan Nimmrichter, Valerio Scarini
Physical Review A
100(4)
042107
(2019)

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The resource theory of thermal operations explains the state transformations that are possible ina very specific thermodynamic setting: there is only one thermal bath, auxiliary systems can onlybe in the corresponding thermal state (free states), and the interaction must commute with the freeHamiltonian (free operation). In this paper we study the mildest deviation: the reservoir particlesare subject to inhomogeneities, either in the local temperature (introducing resource states) or inthe local Hamiltonian (generating a resource operation). For small inhomogeneities, the two modelsgenerate the same channel and thus the same state transformations. However, their thermodynamicsis significantly different when it comes to work generation or to the interpretation of the “secondlaws of thermal operations”.
Accelerated adiabatic quantum gates: optimizing speed versus robustness
Hugo Ribeiro, Aashish A. Clerk
Physical Review A
100(3)
032323
(2019)

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We develop new protocols for highfidelity single qubit gates that exploit and extend theoretical ideas for accelerated adiabatic evolution. Our protocols are compatible with qubit architectures with highly isolated logical states, where traditional approaches are problematic; a prime example are superconducting fluxonium qubits. By using an accelerated adiabatic protocol we can enforce the desired adiabatic evolution while having gate times that are comparable to the inverse adiabatic energy gap (a scale that is ultimately set by the amount of power used in the control pulses). By modelling the effects of decoherence, we explore the tradeoff between speed and robustness that is inherent to shortcutstoadiabaticity approaches.
Macroscopicity of quantum mechanical superposition tests via hypothesis falsification
Björn Schrinski, Stefan Nimmrichter, Benjamin A. Stickler, Klaus Hornberger
Physical Review A
100(3)
032111
(2019)

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We establish an objective scheme to determine the macroscopicity of quantum mechanical superposition tests, which is based on the Bayesian hypothesis falsification of macrorealistic modificationsof quantum theory. The measure uses the raw data gathered in an experiment, taking into accountall measurement uncertainties, and can be used to directly assess any conceivable quantum test.We determine the resulting macroscopicity for three recent tests of quantum physics: doublewellinterference of BoseEinstein condensates, LeggettGarg tests with atomic random walks, and entanglement generation and readout of nanomechanical oscillators.
Kommt der künstliche Physiker?
Thomas Fösel, Florian Marquardt, Talitha Weiß
Physik in unserer Zeit
50(5)
220227
(2019)

Journal
2016 besiegte das Computerprogramm AlphaGo einen der weltbesten Go‐Spieler. Damit rückte eine technische Revolution ins Bewusstsein der breiten Öffentlichkeit: Selbstlernende künstliche neuronale Netze sind zunehmend in der Lage, Menschen bei bestimmten Aufgaben zu schlagen. Zahlreiche Anwendungen, von der Bilderkennung bis zur automatischen Übersetzung, revolutionieren momentan die Technik – und auch Physik und Astronomie bieten viele potenzielle Einsatzmöglichkeiten. In der Astronomie können neuronale Netze das automatische Klassifizieren von Galaxien übernehmen. In der Statistischen Physik sind Magnetisierungsmuster von ferro‐ oder paramagnetischen Zuständen ein Beispiel. Ein anderes Beispiel ist die Suche nach Quantenfehler‐Korrekturstrategien in zukünftigen Quantencomputern. Unsere Forschung konnte zeigen, dass künstliche neuronale Netze mittels Reinforcement Learning hier bereits eigenständig neue Korrekturstrategien entwickeln können.
Perturbation theory of optical resonances of deformed dielectric spheres
Andrea Aiello, Jack G. E. Harris, Florian Marquardt
Physical Review A
100(2)
023837
(2019)

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We analyze the optical resonances of a dielectric sphere whose surface has been slightly deformed in an arbitrary way. Setting up a perturbation series up to second order, we derive both the frequency shifts and modified linewidths. Our theory is applicable, for example, to freely levitated liquid drops or solid spheres, which are deformed by thermal surface vibrations, centrifugal forces or arbitrary surface waves. A dielectric sphere is effectively an open system whose description requires the introduction of nonHermitian operators characterized by complex eigenvalues and not normalizable eigenfunctions. We avoid these difficulties using the KapurPeierls formalism which enables us to extend the popular RayleighSchrödinger perturbation theory to the case of electromagnetic Debye's potentials describing the light fields inside and outside the nearspherical dielectric object. We find analytical formulas, valid within certain limits, for the deformationinduced first and secondorder corrections to the central frequency and bandwidth of a resonance. As an application of our method, we compare our results with preexisting ones finding full agreement.
Nonexponential decay of a giant artificial atom
Gustav Andersson, Baladitya Suri, Lingzhen Guo, Thomas Aref, Per Delsing
In quantum optics, light–matter interaction has conventionally been studied using small atoms interacting with electromagnetic fields with wavelength several orders of magnitude larger than the atomic dimensions1,2. In contrast, here we experimentally demonstrate the vastly different ‘giant atom’ regime, where an artificial atom interacts with acoustic fields with wavelength several orders of magnitude smaller than the atomic dimensions. This is achieved by coupling a superconducting qubit3 to surface acoustic waves at two points with separation on the order of 100 wavelengths. This approach is comparable to controlling the radiation of an atom by attaching it to an antenna. The slow velocity of sound leads to a significant internal timedelay for the field to propagate across the giant atom, giving rise to nonMarkovian dynamics4. We demonstrate the nonMarkovian character of the giant atom in the frequency spectrum as well as nonexponential relaxation in the time domain.
Dynamically Generated Synthetic Electric Fields for Photons
Petr Zapletal, Stefan Walter, Florian Marquardt
Physical Review A
100(2)
023804
(2019)

Journal

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Static synthetic magnetic fields give rise to phenomena including the Lorentz force and the quantum Hall effect even for neutral particles, and they have by now been implemented in a variety of physical systems. Moving towards fully dynamical synthetic gauge fields allows, in addition, for backaction of the particles' motion onto the field. If this results in a timedependent vector potential, conventional electromagnetism predicts the generation of an electric field. Here, we show how synthetic electric fields for photons arise selfconsistently due to the nonlinear dynamics in a driven system. Our analysis is based on optomechanical arrays, where dynamical gauge fields arise naturally from phononassisted photon tunneling. We study open, onedimensional arrays, where synthetic magnetic fields are absent. However, we show that synthetic electric fields can be generated dynamically, which, importantly, suppress photon transport in the array. The generation of these fields depends on the direction of photon propagation, leading to a novel mechanism for a photon diode, inducing nonlinear nonreciprocal transport via dynamical synthetic gauge fields.
Resonance inversion in a superconducting cavity coupled to artificial atoms and a microwave background
Juha Leppäkangas, Jan David Brehm, Ping Yang, Lingzhen Guo, Michael Marthaler, Alexey V. Ustinov, Martin Weides
Physical Review A
99(6)
063804
(2019)

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We demonstrate how heating of an environment can invert the line shape of a driven cavity. We consider a superconducting coplanar cavity coupled to multiple artificial atoms. The measured cavity transmission is characterized by Fanotype resonances with a shape that is continuously tunable by bias current through nearby (magnetic flux) control lines. In particular, the same dispersive shift of the microwave cavity can be observed as a peak or a dip. We find that this Fanopeak inversion is possible due to a tunable interference between a microwave transmission through a background, with reactive and dissipative properties, and through the cavity, affected by biascurrent induced heating. The background transmission occurs due to crosstalk with the multiple control lines. We show how such background can be accounted for by a Jaynes or TavisCummings model with modified boundary conditions between the cavity and transmissionline microwave fields. A dip emerges when cavity transmission is comparable with background transmission and dissipation. We find generally that resonance positions determine system energy levels, whereas resonance shapes give information on system fluctuations and dissipation.
Initialisation of single spin dressed states using shortcuts to adiabaticity
Johannes Kölbl, Arne Barfuss, Mark Kasperczyk, Lucas Thiel, Aashish Clerk, Hugo Ribeiro, Patrick Maletinsky
Physical Review Letters
122(9)
090502
(2019)

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We demonstrate the use of shortcuts to adiabaticity protocols for initialisation, readout, and coherent control of dressed states generated by closedcontour, coherent driving of a single spin. Such dressed states have recently been shown to exhibit efficient coherence protection, beyond what their twolevel counterparts can offer. Our state transfer protocols yield a transfer fidelity of ~ 99.4(2) % while accelerating the transfer speed by a factor of 2.6 compared to the adiabatic approach. We show bidirectionality of the accelerated state transfer, which we employ for direct dressed state population readout after coherent manipulation in the dressed state manifold. Our results enable direct and efficient access to coherenceprotected dressed states of individual spins and thereby offer attractive avenues for applications in quantum information processing or quantum sensing.
Classically Entangled Light
Andrew Forbes, Andrea Aiello, Bienvenu Ndagano
Progress in Optics
64
99153
(2019)

Book Chapter
The concept of entanglement is so synonymous with quantum mechanics that the prefix “quantum” is often deemed unnecessary; there is after all only quantum entanglement. But the hallmark of entangled quantum states is nonseparability, a property that is not unique to the quantum world. On the contrary, nonseparability appears in many physical systems, and pertinently, in classical vector states of light: classical entanglement? Here we outline the concept of classical entanglement, highlight where it may be found, how to control and exploit it, and discuss the similarities and differences between quantum and classical entangled systems. Intriguingly, we show that quantum tools may be applied to classical systems, and likewise that classical light may be used in quantum processes. While we mostly use vectorial structured light throughout the text as our example of choice, we make it clear that the concepts outlined here may be extended beyond this with little effort, which we showcase with a few selected case studies.
2018
Cavity optomagnonics with magnetic textures: coupling a magnetic vortex to light
Jasmin Graf, Hannes Pfeifer, Florian Marquardt, Silvia ViolaKusminskiy
Physical Review B
98(24)
241406
(2018)

Journal

PDF
Optomagnonic systems, where light couples coherently to collective excitations in magnetically ordered solids, are currently of high interest due to their potential for quantum information processing platforms at the nanoscale. Efforts so far, both at the experimental and theoretical level, have focused on systems with a homogeneous magnetic background. A unique feature in optomagnonics is however the possibility of coupling light to spin excitations on top of magnetic textures. We propose a cavityoptomagnonic system with a non homogeneous magnetic ground state, namely a vortex in a magnetic microdisk. In particular we study the coupling between optical whispering gallery modes to magnon modes localized at the vortex. We show that the optomagnonic coupling has a rich spatial structure and that it can be tuned by an externally applied magnetic field. Our results predict cooperativities at maximum photon density of the order of C≈10−2 by proper engineering of these structures.
suggested by editors
Reinforcement Learning with Neural Networks for Quantum Feedback
Thomas Fösel, Petru Tighineanu, Talitha Weiss, Florian Marquardt
Physical Review X
8(3)
031084
(2018)

Journal

PDF
Artificial neural networks are revolutionizing science. While the most prevalent technique involves supervised training on queries with a known correct answer, more advanced challenges often require discovering answers autonomously. In reinforcement learning, control strategies are improved according to a reward function. The power of this approach has been highlighted by spectactular recent successes, such as playing Go. So far, it has remained an open question whether neuralnetworkbased reinforcement learning can be successfully applied in physics. Here, we show how to use this method for finding quantum feedback schemes, where a networkbased "agent" interacts with and occasionally decides to measure a quantum system. We illustrate the utility by finding gate sequences that preserve the quantum information stored in a small collection of qubits against noise. This specific application will help to find hardwareadapted feedback schemes for small quantum modules while demonstrating more generally the promise of neuralnetwork based reinforcement learning in physics.
Quantum nondemolition measurement of mechanical motion quanta
Luca Dellantonio, Oleksandr Kyriienko, Florian Marquardt, Anders S. Sørensen
Nature Communications
9
3621
(2018)

Journal

PDF
The fields of optomechanics and electromechanics have facilitated numerous advances in the areas of precision measurement and sensing, ultimately driving the studies of mechanical systems into the quantum regime. To date, however, the quantization of the mechanical motion and the associated quantum jumps between phonon states remains elusive. For optomechanical systems, the coupling to the environment was shown to make the detection of the mechanical mode occupation difficult, typically requiring the singlephoton strongcoupling regime. Here, we propose and analyse an electromechanical setup, which allows us to overcome this limitation and resolve the energy levels of a mechanical oscillator. We found that the heating of the membrane, caused by the interaction with the environment and unwanted couplings, can be suppressed for carefully designed electromechanical systems. The results suggest that phonon number measurement is within reach for modern electromechanical setups.
Phonon Decoherence of Quantum Dots in Photonic Structures: Broadening of the ZeroPhonon Line and the Role of Dimensionality
Petru Tighineanu, C. L. Dreeßen, C. Flindt, P. Lodahl, A. S. Sorensen
Physical Review Letters
120(25)
257401
(2018)

Journal

PDF
We develop a general microscopic theory describing the phonon decoherence of quantum dots and indistinguishability of the emitted photons in photonic structures. The coherence is found to depend fundamentally on the dimensionality of the structure resulting in vastly different performance for quantum dots embedded in a nanocavity (0D), waveguide (1D), slab (2D), or bulk medium (3D). In bulk, we find a striking temperature dependence of the dephasing rate scaling as T11 implying that phonons are effectively “frozen out” for T≲4 K. The phonon density of states is strongly modified in 1D and 2D structures leading to a linear temperature scaling for the dephasing strength. The resulting impact on the photon indistinguishability can be important even at subKelvin temperatures. Our findings provide a comprehensive understanding of the fundamental limits to photon indistinguishability in photonic structures.
Light polarization measurements in tests of macrorealism
Eugenio Roldan, Johannes Kofler, Carlos NavarreteBenlloch
Physical Review A
97
062117
(2018)

Journal

PDF
According to the world view of macrorealism, the properties of a given system exist prior to and independent of measurement, which is incompatible with quantum mechanics. Leggett and Garg put forward a practical criterion capable of identifying violations of macrorealism, and so far experiments performed on microscopic and mesoscopic systems have always agreed with quantum mechanics. However, a macrorealist can always assign the cause of such violations to the perturbation that measurements effect on such small systems, and hence a definitive test would require using noninvasive measurements, preferably on macroscopic objects, where such measurements seem more plausible. However, the generation of truly macroscopic quantum superposition states capable of violating macrorealism remains a big challenge. In this work we propose a setup that makes use of measurements on the polarization of light, a property that has been extensively manipulated both in classical and quantum contexts, hence establishing the perfect link between the microscopic and macroscopic worlds. In particular, we use LeggettGarg inequalities and the criterion of no signaling in time to study the macrorealistic character of light polarization for different kinds of measurements, in particular with different degrees of coarse graining. Our proposal is noninvasive for coherent input states by construction. We show for states with welldefined photon number in two orthogonal polarization modes, that there always exists a way of making the measurement sufficiently coarse grained so that a violation of macrorealism becomes arbitrarily small, while sufficiently sharp measurements can always lead to a significant violation.
Quantum theory of continuum optomechanics
Peter Rakich, Florian Marquardt
New Journal of Physics
20
045005
(2018)

Journal

PDF
We present the basic ingredients of continuum optomechanics, i.e. the suitable extension of cavityoptomechanical concepts to the interaction of photons and phonons in an extended waveguide. We introduce a realspace picture and argue which coupling terms may arise in leading order in the spatial derivatives. This picture allows us to discuss quantum noise, dissipation, and the correct boundary conditions at the waveguide entrance. The connections both to optomechanical arrays as well as to the theory of Brillouin scattering in waveguides are highlighted. Among other examples, we analyze the 'strong coupling regime' of continuum optomechanics that may be accessible in future experiments.
Active locking and entanglement in type II optical parametric oscillators
Joaquín RuizRivas, Germán J. de Valcarcel, Carlos NavarreteBenlloch
New Journal of Physics
20
023004
(2018)

Journal

PDF
Type II optical parametric oscillators are amongst the highestquality sources of quantumcorrelated light. In particular, when pumped above threshold, such devices generate a pair of bright orthogonallypolarized beams with strong continuousvariable entanglement. However, these sources are of limited practical use, because the entangled beams emerge with different frequencies and a diffusing phase difference. It has been proven that the use of an internal waveplate coupling the modes with orthogonal polarization is capable of locking the frequencies of the emerging beams to half the pump frequency, as well as reducing the phasedifference diffusion, at the expense of reducing the entanglement levels. In this work we characterize theoretically an alternative locking mechanism: the injection of a laser at half the pump frequency. Apart from being less invasive, this method should allow for an easier realtime experimental control. We show that such an injection is capable of generating the desired phase locking between the emerging beams, while still allowing for large levels of entanglement. Moreover, we find an additional region of the parameter space (at relatively large injections) where a mode with well defined polarization is in a highly amplitudesqueezed state.
Snowflake phononic topological insulator at the nanoscale
Christian Brendel, Vittorio Peano, Oskar Painter, Florian Marquardt
Physical Review B (Rapid Communications)
97(2)
020102
(2018)

Journal

PDF
We show how the snowflake phononic crystal structure, which recently has been realized experimentally, can be turned into a topological insulator for mechanical waves. This idea, based purely on simple geometrical modifications, could be readily implemented on the nanoscale.
suggested by editors
Scalable Ion Trap Architecture for Universal Quantum Computation by Collisions
We propose a scalable ion trap architecture for universal quantum computation, which is composed of an array of ion traps with one ion confined in each trap. The neighboring traps are designed capable of merging into one single trap. The universal twoqubit SWAP−−−−−−√ gate is realized by direct collision of two neighboring ions in the merged trap, which induces an effective spinspin interaction between two ions. We find that the collisioninduced spinspin interaction decreases with the third power of two ions' trapping distance. Even with a 200 μm trapping distance between atomic ions in Paul traps, it is still possible to realize a twoqubit gate operation with speed in 0.1 kHz regime. The speed can be further increased up into 0.1 MHz regime using electrons with 10 mm trapping distance in Penning traps.
2017
Cavity optomechanics in a levitated helium drop
L. Childress, M. P. Schmidt, A. D. Kashkanova, C. D. Brown, G. I. Harris, Andrea Aiello, Florian Marquardt, J. G. E. Harris
Physical Review A
96(6)
063842
(2017)

Journal

PDF
We describe a proposal for a type of optomechanical system based on a drop of liquid helium that ismagnetically levitated in vacuum. In the proposed device, the drop would serve three roles: its optical whisperinggallery modes would provide the optical cavity, its surface vibrations would constitute the mechanical element, and evaporation of He atoms from its surface would provide continuous refrigeration. We analyze the feasibility of such a system in light of previous experimental demonstrations of its essential components: magnetic levitation of mmscale and cmscale drops of liquid He, evaporative cooling of He droplets in vacuum, and coupling to highquality optical whisperinggallery modes in a wide range of liquids. We find that the combination of these features could result in a device that approaches the singlephoton strongcoupling regime, due to the high optical quality factors attainable at low temperatures. Moreover, the system offers a unique opportunity to use optical techniques to study the motion of a superfluid that is freely levitating in vacuum (in the case of He4). Alternatively, for a normal fluid drop of He3, we propose to exploit the coupling between the drop's rotations and vibrations to perform quantum nondemolition measurements of angular momentum.
L lines, C points and Chern numbers: understanding band structure topology using polarization fields
Thomas Fösel, Vittorio Peano, Florian Marquardt
New Journal of Physics
19
115013
(2017)

Journal

PDF
Topology has appeared in different physical contexts. The most prominent application is topologically protected edge transport in condensed matter physics. The Chern number, the topological invariant of gapped Bloch Hamiltonians, is an important quantity in this field. Another example of topology, in polarization physics, are polarization singularities, called L lines and C points. By establishing a connection between these two theories, we develop a novel technique to visualize and potentially measure the Chern number: it can be expressed either as the winding of the polarization azimuth along L lines in reciprocal space, or in terms of the handedness and the index of the C points. For mechanical systems, this is directly connected to the visible motion patterns.
General Linearized Theory of Quantum Fluctuations around Arbitrary Limit Cycles
Carlos NavarreteBenlloch, Talitha Weiss, Stefan Walter, Germán J. de Valcarcel
Physical Review Letters
119(13)
133601
(2017)

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The theory of Gaussian quantum fluctuations around classical steady states in nonlinear quantumoptical systems (also known as standard linearization) is a cornerstone for the analysis of such systems. Its simplicity, together with its accuracy far from critical points or situations where the nonlinearity reaches the strong coupling regime, has turned it into a widespread technique, being the first method of choice in most works on the subject. However, such a technique finds strong practical and conceptual complications when one tries to apply it to situations in which the classical longtime solution is time dependent, a most prominent example being spontaneous limitcycle formation. Here, we introduce a linearization scheme adapted to such situations, using the driven Van der Pol oscillator as a test bed for the method, which allows us to compare it with full numerical simulations. On a conceptual level, the scheme relies on the connection between the emergence of limit cycles and the spontaneous breaking of the symmetry under temporal translations. On the practical side, the method keeps the simplicity and linear scaling with the size of the problem (number of modes) characteristic of standard linearization, making it applicable to large (manybody) systems.
Unraveling beam selfhealing
Andrea Aiello, Girish S. Agarwal, Martin Paur, Bohumil Stoklasa, Zdenek Hradil, Jaroslav Rehacek, Pablo de la Hoz, Gerd Leuchs, Luis L. SanchezSoto
Optics Express
25(16)
1914719157
(2017)

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We show that, contrary to popular belief, diffractionfree beams not only may reconstruct themselves after hitting an opaque obstacle but also, for example, Gaussian beams. We unravel the mathematics and the physics underlying the selfreconstruction mechanism and we provide for a novel definition for the minimum reconstruction distance beyond geometric optics, which is in principle applicable to any optical beam that admits an angular spectrum representation. Moreover, we propose to quantify the selfreconstruction ability of a beam via a newly established degree of selfhealing. This is defined via a comparison between the amplitudes, as opposite to intensities, of the original beam and the obstructed one. Such comparison is experimentally accomplished by tailoring an innovative experimental technique based upon ShackHartmann wave front reconstruction. We believe that these results can open new avenues in this field. (C) 2017 Optical Society of America
From KardarParisiZhang scaling to explosive desynchronization in arrays of limitcycle oscillators
Roland Lauter, Aditi Mitra, Florian Marquardt
Physical Review E
96(1)
012220
(2017)

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Phase oscillator lattices subject to noise are one of the most fundamental systems in nonequilibrium physics. We have discovered a dynamical transition which has a significant impact on the synchronization dynamics in such lattices, as it leads to an explosive increase of the phase diffusion rate by orders of magnitude. Our analysis is based on the widely applicable KuramotoSakaguchi model, with local couplings between oscillators. For onedimensional lattices, we observe the universal evolution of the phase spread that is suggested by a connection to the theory of surface growth, as described by the KardarParisiZhang (KPZ) model. Moreover, we are able to explain the dynamical transition both in one and two dimensions by connecting it to an apparent finitetime singularity in a related KPZ lattice model. Our findings have direct consequences for the frequency stability of coupled oscillator lattices.Phase oscillator lattices subject to noise are one of the most fundamental systems in nonequilibrium physics. We have discovered a dynamical transition which has a significant impact on the synchronization dynamics in such lattices, as it leads to an explosive increase of the phase diffusion rate by orders of magnitude. Our analysis is based on the widely applicable KuramotoSakaguchi model, with local couplings between oscillators. For onedimensional lattices, we observe the universal evolution of the phase spread that is suggested by a connection to the theory of surface growth, as described by the KardarParisiZhang (KPZ) model. Moreover, we are able to explain the dynamical transition both in one and two dimensions by connecting it to an apparent finitetime singularity in a related KPZ lattice model. Our findings have direct consequences for the frequency stability of coupled oscillator lattices.
Synchronization of an optomechanical system to an external drive
Ehud Amitai, Niels Loerch, Andreas Nunnenkamp, Stefan Walter, Christoph Bruder
Physical Review A
95(5)
053858
(2017)

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Optomechanical systems driven by an effective bluedetuned laser can exhibit selfsustained oscillations of the mechanical oscillator. These selfoscillations are a prerequisite for the observation of synchronization. Here, we study the synchronization of the mechanical oscillations to an external reference drive. We study two cases of reference drives: (1) an additional laser applied to the optical cavity; (2) a mechanical drive applied directly to the mechanical oscillator. Starting from a master equation description, we derive a microscopic Adler equation for both cases, valid in the classical regime in which the quantum shot noise of the mechanical selfoscillator does not play a role. Furthermore, we numerically show that, in both cases, synchronization arises also in the quantum regime. The optomechanical system is therefore a good candidate for the study of quantum synchronization.
Quantumcoherent phase oscillations in synchronization
Talitha Weiss, Stefan Walter, Florian Marquardt
Physical Review A
95(4)
041802
(2017)

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Recently, several studies have investigated synchronization in quantummechanical limitcycle oscillators. However, the quantum nature of these systems remained partially hidden, since the dynamics of the oscillator's phase was overdamped and therefore incoherent. We show that there exist regimes of underdamped and even quantumcoherent phase motion, opening up new possibilities to study quantum synchronization dynamics. To this end, we investigate the Van der Pol oscillator (a paradigm for a selfoscillating system) synchronized to an external drive. We derive an effective quantum model which fully describes the regime of underdamped phase motion and additionally allows us to identify the quality of quantum coherence. Finally, we identify quantum limit cycles of the phase itself.
ManyParticle Dephasing after a Quench
Thomas Kiendl, Florian Marquardt
Physical Review Letters
118(13)
130601
(2017)

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After a quench in a quantum manybody system, expectation values tend to relax towards longtime averages. However, temporal fluctuations remain in the longtime limit, and it is crucial to study the suppression of these fluctuations with increasing system size. The particularly important case of nonintegrable models has been addressed so far only by numerics and conjectures based on analytical bounds. In this work, we are able to derive analytical predictions for the temporal fluctuations in a nonintegrable model (the transverse Ising chain with extra terms). Our results are based on identifying a dynamical regime of "manyparticle dephasing,"where quasiparticles do not yet relax but fluctuations are nonetheless suppressed exponentially by weak integrability breaking.
Pseudomagnetic fields for sound at the nanoscale
Christian Brendel, Vittorio Peano, Oskar J. Painter, Florian Marquardt
Proceedings of the National Academy of Sciences of the United States of America
114(17)
E3390E3395

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There is a growing effort in creating chiral transport of sound waves. However, most approaches so far have been confined to the macroscopic scale. Here, we propose an approach suitable to the nanoscale that is based on pseudomagnetic fields. These pseudomagnetic fields for sound waves are the analogue of what electrons experience in strained graphene. In our proposal, they are created by simple geometrical modifications of an existing and experimentally proven phononic crystal design, the snowflake crystal. This platform is robust, scalable, and wellsuited for a variety of excitation and readout mechanisms, among them optomechanical approaches.
Generalized nonreciprocity in an optomechanical circuit via synthetic magnetism and reservoir engineering
Kejie Fang, Jie Luo, Anja Metelmann, Matthew H. Matheny, Florian Marquardt, Aashish A. Clerk, Oskar Painter
Synthetic magnetism has been used to control charge neutral excitations for applications ranging from classical beam steering to quantum simulation. In optomechanics, radiationpressureinduced parametric coupling between optical (photon) and mechanical (phonon) excitations may be used to break timereversal symmetry, providing the prerequisite for synthetic magnetism. Here we design and fabricate a silicon optomechanical circuit with both optical and mechanical connectivity between two optomechanical cavities. Driving the two cavities with phasecorrelated laser light results in a synthetic magnetic flux, which, in combination with dissipative coupling to the mechanical bath, leads to nonreciprocal transport of photons with 35 dB of isolation. Additionally, optical pumping with bluedetuned light manifests as a particle nonconserving interaction between photons and phonons, resulting in directional optical amplification of 12 dB in the isolator throughdirection. These results suggest the possibility of using optomechanical circuits to create a more general class of nonreciprocal optical devices, and further, to enable new topological phases for both light and sound on a microchip.
Anderson localization of composite excitations in disordered optomechanical arrays
Thales Figueiredo Roque, Vittorio Peano, Oleg M. Yevtushenko, Florian Marquardt
New Journal of Physics
19
013006
(2017)

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Optomechanical (OMA) arrays are a promising future platform for studies of transport, manybody dynamics, quantum control and topological effects in systems of coupled photon and phonon modes. We introduce disordered OMA arrays, focusing on features of Anderson localization of hybrid photonphonon excitations. It turns out that these represent a unique disordered system, where basic parameters can be easily controlled by varying the frequency and the amplitude of an external laser field. We show that the twospecies setting leads to a nontrivial frequency dependence of the localization length for intermediate laser intensities. This could serve as a convincing evidence of localization in a nonequilibrium dissipative situation.
Noncritical generation of nonclassical frequency combs via spontaneous rotational symmetry breaking
Carlos NavarreteBenlloch, Giuseppe Patera, Germán J. de Valcarcel
Physical Review A
96(4)
043801
(2017)

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Synchronously pumped optical parametric oscillators (SPOPOs) are optical cavities driven by modelocked lasers, and containing a nonlinear crystal capable of downconverting a frequency comb to lower frequencies. SPOPOs have received a lot of attention lately because their intrinsic multimode nature makes them compact sources of quantum correlated light with promising applications in modern quantum information technologies. In this work we show that SPOPOs are also capable of accessing the challenging and interesting regime where spontaneous symmetry breaking confers strong nonclassical properties to the emitted light, which has eluded experimental observation so far. Apart from opening the possibility of studying experimentally this elusive regime of dissipative phase transitions, our predictions will have a practical impact, since we show that spontaneous symmetry breaking provides a specific spatiotemporal mode with large quadrature squeezing for any value of the system parameters, turning SPOPOs into robust sources of highly nonclassical light above threshold.
Linear and angular momenta in tightly focused vortex segmented beams of light
Martin Neugebauer, Andrea Aiello, Peter Banzer
Chinese Optics Letters
15(3)
030003
(2017)

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We investigate the linear momentum density of light, which can be decomposed into spin and orbital parts, in the complex threedimensional field distributions of tightly focused vortex segmented beams. The chosen angular spectrum exhibits two spatially separated vortices of opposite charge and orthogonal circular polarization to generate phase vortices in a meridional plane of observation. In the vicinity of those vortices, regions of negative orbital linear momentum occur. Besides these phase vortices, the occurrence of transverse orbital angular momentum manifests in a vortex chargedependent relative shift of the energy density and linear momentum density.
2016
Topological Quantum Fluctuations and Traveling Wave Amplifiers
Vittorio Peano, Martin Houde, Florian Marquardt, Aashish A. Clerk
Physical Review X
6(4)
041026
(2016)

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It is now well established that photonic systems can exhibit topological energy bands. Similar to their electronic counterparts, this leads to the formation of chiral edge modes which can be used to transmit light in a manner that is protected against backscattering. While it is understood how classical signals can propagate under these conditions, it is an outstanding important question how the quantum vacuum fluctuations of the electromagnetic field get modified in the presence of a topological band structure. We address this challenge by exploring a setting where a nonzero topological invariant guarantees the presence of a parametrically unstable chiral edge mode in a system with boundaries, even though there are no bulkmode instabilities. We show that one can exploit this to realize a topologically protected, quantumlimited traveling wave parametric amplifier. The device is naturally protected against both internal losses and backscattering; the latter feature is in stark contrast to standard traveling wave amplifiers. This adds a new example to the list of potential quantum devices that profit from topological transport.
Classical dynamical gauge fields in optomechanics
Stefan Walter, Florian Marquardt
New Journal of Physics
18
113029
(2016)

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Artificial gauge fields for neutral particles such as photons, recently attracted a lot of attention in various fields ranging from photonic crystals to ultracold atoms in optical lattices to optomechanical arrays. Here we point out that, among all implementations of gauge fields, the optomechanical setting allows for the most natural extension where the gauge field becomes dynamical. The mechanical oscillation phases determine the effective artificial magnetic field for the photons, and once these phases are allowed to evolve, they respond to the flow of photons in the structure. We discuss a simple threesite model where we identify four different regimes of the gaugefield dynamics. Furthermore, we extend the discussion to a twodimensional lattice. Our proposed scheme could for instance be implemented using optomechanical crystals.
Noiseinduced transitions in optomechanical synchronization
Talitha Weiss, Andreas Kronwald, Florian Marquardt
New Journal of Physics
18
013043
(2016)

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We study how quantum and thermal noise affects synchronization of two optomechanical limitcycle oscillators. Classically, in the absence of noise, optomechanical systems tend to synchronize either inphase or antiphase. Taking into account the fundamental quantum noise, we find a regime where fluctuations drive transitions between these classical synchronization states. We investigate how this 'mixed' synchronization regime emerges from the noiseless system by studying the classicaltoquantum crossover and we show how the time scales of the transitions vary with the effective noise strength. In addition, we compare the effects of thermal noise to the effects of quantum noise.
Entanglement rate for Gaussian continuous variable beams
Zhi Jiao Deng, Steven J. M. Habraken, Florian Marquardt
New Journal of Physics
18
063022
(2016)

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We derive a general expression that quantifies the total entanglement production rate in continuous variable systems, where a source emits two entangled Gaussian beams with arbitrary correlators. This expression is especially useful for situations where the source emits an arbitrary frequency spectrum, e.g. when cavities are involved. To exemplify its meaning and potential, we apply it to a fourmode optomechanical setup that enables the simultaneous up and downconversion of photons from a drive laser into entangled photon pairs. This setup is efficient in that both the drive and the optomechanical up and downconversion can be fully resonant.
Topological phase transitions and chiral inelastic transport induced by
the squeezing of light
Vittorio Peano, Martin Houde, Christian Brendel, Florian Marquardt, Aashish A. Clerk
Nature Communications
7
10779
(2016)

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There is enormous interest in engineering topological photonic systems. Despite intense activity, most works on topological photonic states (and more generally bosonic states) amount in the end to replicating a wellknown fermionic singleparticle Hamiltonian. Here we show how the squeezing of light can lead to the formation of qualitatively new kinds of topological states. Such states are characterized by nontrivial Chern numbers, and exhibit protected edge modes, which give rise to chiral elastic and inelastic photon transport. These topological bosonic states are not equivalent to their fermionic (topological superconductor) counterparts and, in addition, cannot be mapped by a local transformation onto topological states found in particleconserving models. They thus represent a new type of topological system. We study this physics in detail in the case of a kagome lattice model, and discuss possible realizations using nonlinear photonic crystals or superconducting circuits.
Coupled spinlight dynamics in cavity optomagnonics
Silvia ViolaKusminskiy, Hong X. Tang, Florian Marquardt
Physical Review A
94(3)
033821
(2016)

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Experiments during the past 2 years have shown strong resonant photonmagnon coupling in microwave cavities, while coupling in the optical regime was demonstrated very recently for the first time. Unlike with microwaves, the coupling in optical cavities is parametric, akin to optomechanical systems. This line of research promises to evolve into a new field of optomagnonics, aimed at the coherent manipulation of elementary magnetic excitations in solidstate systems by optical means. In this work we derive the microscopic optomagnonic Hamiltonian. In the linear regime the system reduces to the wellknown optomechanical case, with remarkably large coupling. Going beyond that, we study the optically induced nonlinear classical dynamics of a macrospin. In the fastcavity regime we obtain an effective equation of motion for the spin and show that the light field induces a dissipative term reminiscent of Gilbert damping. The induced dissipation coefficient, however, can change sign on the Bloch sphere, giving rise to selfsustained oscillations. When the full dynamics of the system is considered, the system can enter a chaotic regime by successive period doubling of the oscillations.
Quantum Nondemolition Measurement of a Quantum Squeezed State Beyond the
3 dB Limit
C. U. Lei, A. J. Weinstein, J. Suh, E. E. Wollman, A. Kronwald, F. Marquardt, A. A. Clerk, K. C. Schwab
Physical Review Letters
117(10)
100801
(2016)

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We use a reservoir engineering technique based on twotone driving to generate and stabilize a quantum squeezed state of a micronscale mechanical oscillator in a microwave optomechanical system. Using an independent backactionevading measurement to directly quantify the squeezing, we observe 4.7±0.9 dB of squeezing below the zeropoint level surpassing the 3 dB limit of standard parametric squeezing techniques. Our measurements also reveal evidence for an additional mechanical parametric effect. The interplay between this effect and the optomechanical interaction enhances the amount of squeezing obtained in the experiment.
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