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.
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.
Complex decoherencefree interactions between giant atoms
Giant atoms provide a promising platform for engineering decoherencefree interactions which
<br><br><br>is a major task in modern quantum technologies. Here we study systematically how to implement
<br><br><br>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
<br><br><br>enables the AharonovBohm effect of photons when the giant atoms constitute an effective closed loop. In particular, we consider the influence of nonMarkovian retardation effect arising from large separations of the coupling points and study its dependence on the modulation parameters.
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.
To realize the full potential of quantum technologies, finding good strategies to control quantum information processing devices in real time becomes increasingly important. Usually these strategies require a precise understanding of the device itself, which is generally not available. Modelfree reinforcement learning circumvents this need by discovering control strategies from scratch without relying on an accurate description of the quantum system. Furthermore, important tasks like state
<br><br>preparation, gate teleportation and error correction need feedback at time scales much shorter than the coherence time, which for superconducting circuits is in the microsecond range. Developing and training a deep reinforcement learning agent able to operate in this realtime feedback regime has been an open challenge. Here, we have implemented such an agent in the form of a latencyoptimized deep neural network on a fieldprogrammable gate array (FPGA). We demonstrate its use to efficiently initialize a superconducting qubit into a target state. To train the agent, we use
<br><br>modelfree reinforcement learning that is based solely on measurement data. We study the agent’s performance for strong and weak measurements, and for threelevel readout, and compare with simple strategies based on thresholding. This demonstration motivates further research towards adoption of
<br><br>reinforcement learning for realtime feedback control of quantum devices and more generally any physical system requiring learnable lowlatency feedback control.
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<br>$^{229}\text{Th}$ and the 14.4 keV M\"ossbauer transition in $^{57}\text{Fe}$<br>show low excitation rates but strong enhancement with respect to direct two<br>photon nuclear excitation.<br>
Investigation of inverse design of multilayer thinfilms with conditional invertible Neural Networks
Alexander Luce, Ali Mahdavi, Heribert Wankerl, Florian Marquardt
The task of designing optical multilayer thinfilms regarding a given target is currently solved using gradientbased optimization in conjunction with methods that can introduce additional thinfilm layers. Recently, Deep Learning and Reinforcement Learning have been been introduced to the task of designing thinfilms with great success, however a trained network is usually only able to become proficient for a single target and must be retrained if the optical<br>targets are varied. In this work, we apply conditional Invertible Neural Networks (cINN) to inversely designing multilayer thinfilms given an optical target. Since the cINN learns the energy landscape of all thinfilm configurations within the training dataset, we show that cINNs can generate a stochastic ensemble of proposals for thinfilm configurations that 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 stateofthe art approaches. Furthermore, we tested the generative capabilities on samples which are outside the training data distribution and found that the cINN was able to predict thinfilms for<br>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<br>favorable results.
Deeplearning approach for large atomic structure calculations
Highprecision atomic structure calculations require accurate modelling of<br>electronic correlations involving large multiconfiguration wave function<br>expansions. Here we develop a deeplearning approach which allows to preselect<br>the most relevant configurations out of large basis sets until the targeted<br>precision is achieved. Our method replaces a large multiconfiguration<br>DiracHartreeFock computation by a series of smaller ones performed on an<br>iteratively expanding basis subset managed by a convolutional neural network.<br>The results for several examples with manyelectron atoms show that deep<br>learning can significantly reduce the required computational memory and running<br>time and renders possible largescale computations on otherwise unaccessible<br>basis sets.<br>
Tunnelinginduced fractal transmission in Valley Hall waveguides
The Valley Hall effect provides a popular route to engineer robust waveguides<br>for bosonic excitations such a photons and phonons. The almost complete absence<br>of backscattering in many experiments has its theoretical underpinning in a<br>smoothenvelope approximation that neglects large momentum transfer and is<br>accurate only for small bulk band gaps and/or smooth domain walls. For larger<br>bulk band gaps and hard domain walls backscattering is expected to become<br>significant. Here, we show that in this experimentally relevant regime, the<br>reflection of a wave at a sharp corner becomes highly sensitive on the<br>orientation of the outgoing waveguide relative to the underlying lattice.<br>Enhanced backscattering can be understood as being triggered by resonant<br>tunneling transitions in quasimomentum space. Tracking the resonant tunneling<br>energies as a function of the waveguide orientation reveals a selfrepeating<br>fractal pattern that is also imprinted in the density of states and the<br>backscattering rate at a sharp corner.<br>
Topologically Protected Transport in Engineered Mechanical Systems
Tirth Shah, Christian Brendel, Vittorio Peano, Florian Marquardt
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.
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.
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 Quantum Systems
Valentin Gebhart, Raffaele Santagati, Antonio Andrea Gentile, Erik Gauger, David Craig, Natalia Ares, Leonardo Banchi, Florian Marquardt, Luca Pezzè, et al.
Quantum technologies hold the promise to revolutionise our society with<br>groundbreaking applications in secure communication, highperformance<br>computing and ultraprecise sensing. One of the main features in scaling up<br>quantum technologies is that the complexity of quantum systems scales<br>exponentially with their size. This poses severe challenges in the efficient<br>calibration, benchmarking and validation of quantum states and their dynamical<br>control. While the complete simulation of largescale quantum systems may only<br>be possible with a quantum computer, classical characterisation and<br>optimisation methods (supported by cutting edge numerical techniques) can still<br>play an important role.<br> Here, we review classical approaches to learning quantum systems, their<br>correlation properties, their dynamics and their interaction with the<br>environment. We discuss theoretical proposals and successful implementations in<br>different physical platforms such as spin qubits, trapped ions, photonic and<br>atomic systems, and superconducting circuits. This review provides a brief<br>background for key concepts recurring across many of these approaches, such as<br>the Bayesian formalism or Neural Networks, and outlines open questions.<br>
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 TMMFast which enables parallelized computation of reflection and transmission of light at different angles of incidence and wavelengths through the multilayer thinfilm.<br>By decreasing computational time, generating datasets for machine learning becomes feasible and evolutionary optimization can be used effectively. Additionally, the subpackage TMMTorch allows to directly compute analytical<br>gradients for local optimization by using PyTorch Autograd functionality. Finally, an OpenAi Gym environment is presented which allows the user to train reinforcement learning agents on the problem of finding multilayer thinfilm configurations.
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 which aim to find the absolute or approximate ground states of the Ising model. The Ising model is of fundamental computational interest because it is possible to formulate any problem in the complexity class NP as an Ising problem with only polynomial overhead. A scalable Ising machine that outperforms existing standard digital computers could have a huge impact for practical applications for a wide variety of optimization problems. In this review, we survey the current 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<br>spintronics, optics, memristors, and digital hardware accelerators; dynamicalsystems 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<br>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 these Lecture Notes, we provide a comprehensive introduction to the most<br>recent advances in the application of machine learning methods in quantum<br>sciences. We cover the use of deep learning and kernel methods in supervised,<br>unsupervised, and reinforcement learning algorithms for phase classification,<br>representation of manybody quantum states, quantum feedback control, and<br>quantum circuits optimization. Moreover, we introduce and discuss more<br>specialized topics such as differentiable programming, generative models,<br>statistical approach to machine learning, and quantum machine learning.<br>
These are the lecture notes for a course that I am teaching at Zhiyuan College of Shanghai Jiao<br>Tong University (available at www.youtube.com/derekkorg), 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<br>for people at all levels (say, from the end of the bachelor all the way into the PhD). The notes are<br>a work in progress, meaning that some proofs and many figures are still missing. However, I’ve<br>tried my best to write everything in such a way that a reader can follow naturally all arguments<br>and derivations even with these missing bits. Also a few chapters are left to add, including one<br>on mathematical methods to analyze the dynamics of open systems, and another introducing the plethora of current experimental platforms where the tools and ideas developed in these notes are being currently implemented.
Efficient approaches to quantum control and feedback are essential for quantum technologies, from sensing to quantum computation. Pure control tasks have been successfully solved using optimization techniques, including methods like 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. There, modelfree reinforcement learning (RL) has recently proven a powerful new ansatz. What is missing is a way to combine the best of both approaches for scenarios that go beyond weak measurements. In this work, we introduce feedbackGRAPE, which borrows concepts from modelfree RL 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 on a JaynesCummings model with feedback, where it yields interpretable feedback strategies for state preparation and stabilization in the presence of noise. This approach could be employed for discovering strategies in a wide range of feedback tasks, from calibration of multiqubit devices to linearoptics quantum computation strategies, quantumenhanced sensing with adaptive measurements, and quantum error correction.
Nonreciprocal and chiral singlephoton scattering for giant atoms
YaoTong Chen, Lei Du, Lingzhen Guo, Zhihai Wang, Yan Zhang, Yong Li, JinHui Wu
In this work, we investigate the nontrivial singlephoton scattering properties of giant atoms cou<br>pled to waveguides that can be an effective platform for realising nonreciprocal and chiral quantum optics. For the twolevel giantatom setup, 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 of such a twolevel giantatom system. To explore more interesting scattering behaviours, we further extend the twolevel giantatom system to ∆type and<br>∇type threelevel giant atoms coupled to double waveguides without external atomic dissipation.<br>We analyse the different physical mechanisms for the nonreciprocal and chiral scattering properties of the ∆type and ∇type giant atoms. Our proposed giantatom structures have potential applications of highefficient singlephoton targeted router and circulator for quantum information precessing.
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
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.
Playing Ping Pong with Light: Directional Emission of White Light
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<br>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<br>rays of light.
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.
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.
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.
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.
Genuine multipartite entanglement represents the strongest type of entanglement, which is an essential resource for quantum information<br>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<br>entanglement solely based on the Belllike correlations of measurement outcomes collected in the experiment, namely, deviceindependently (DI). However, it is difficult to certify genuine entanglement of practical multipartite states in<br>this way, and even more difficult to quantify it, due to the difficulty to identify optimal multipartite Bell inequalities and protocols tolerant to state impurity. In this work, we explore a general and robust DI method which can be applied to various realistic multipartite quantum state 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 weaklyentangled GHZ states, thus showing that the method applies equally well to less standard states.
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.
Selflearning Machines based on Hamiltonian Echo Backpropagation
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.
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.
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 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.
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.
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.
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.
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)

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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)

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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)

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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)

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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)

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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)

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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)

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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)

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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)

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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)

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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)

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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 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.
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.
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.
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.
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.
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.
2015
Quantum squeezing of motion in a mechanical resonator
E. E. Wollman, C. U. Lei, A. J. Weinstein, J. Suh, A. Kronwald, F. Marquardt, A. A. Clerk, K. C. Schwab
According to quantum mechanics, a harmonic oscillator can never be completely at rest. Even in the ground state, its position will always have fluctuations, called the zeropoint motion. Although the zeropoint fluctuations are unavoidable, they can be manipulated. Using microwave frequency radiation pressure, we have manipulated the thermal fluctuations of a micrometerscale mechanical resonator to produce a stationary quadraturesqueezed state with a minimum variance of 0.80 times that of the ground state. We also performed phasesensitive, backaction evading measurements of a thermal state squeezed to 1.09 times the zeropoint level. Our results are relevant to the quantum engineering of states of matter at large length scales, the study of decoherence of large quantum systems, and for the realization of ultrasensitive sensing of force and motion.
Nonlinear Radiation Pressure Dynamics in an Optomechanical Crystal
Alex G. Krause, Jeff T. Hill, Max Ludwig, Amir H. SafaviNaeini, Jasper Chan, Florian Marquardt, Oskar Painter
Physical Review Letters
115(23)
233601
(2015)

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Utilizing a silicon nanobeam optomechanical crystal, we investigate the attractor diagram arising from the radiation pressure interaction between a localized optical cavity at lambda(c) = 1542 nm and a mechanical resonance at omega(m)/2 pi = 3.72 GHz. At a temperature of Tb approximate to 10 K, highly nonlinear driving of mechanical motion is observed via continuous wave optical pumping. Introduction of a timedependent (modulated) optical pump is used to steer the system towards an otherwise inaccessible dynamically stable attractor in which mechanical selfoscillation occurs for an optical pump red detuned from the cavity resonance. An analytical model incorporating thermooptic effects due to optical absorption heating is developed and found to accurately predict the measured device behavior.
Intracavity Squeezing Can Enhance QuantumLimited Optomechanical
Position Detection through Deamplification
V. Peano, H. G. L. Schwefel, Ch. Marquardt, F. Marquardt
Physical Review Letters
115(24)
243603
(2015)

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It has been predicted and experimentally demonstrated that by injecting squeezed light into an optomechanical device, it is possible to enhance the precision of a position measurement. Here, we present a fundamentally different approach where the squeezing is created directly inside the cavity by a nonlinear medium. Counterintuitively, the enhancement of the signaltonoise ratio works by deamplifying precisely the quadrature that is sensitive to the mechanical motion without losing quantum information. This enhancement works for systems with a weak optomechanical coupling and/or strong mechanical damping. This can allow for larger mechanical bandwidth of quantumlimited detectors based on optomechanical devices. Our approach can be straightforwardly extended to quantum nondemolition qubit detection.
Optomechanical Dirac physics
M. Schmidt, V. Peano, F. Marquardt
New Journal of Physics
17
023025
(2015)

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Recent progress in optomechanical systems may soon allow the realization of optomechanical arrays, i.e. periodic arrangements of interacting optical and vibrational modes. We show that photons and phonons on a honeycomb lattice will produce an optically tunable Diractype band structure. Transport in such a system can exhibit transmission through an optically created barrier, similar to Klein tunneling, but with interconversion between light and sound. In addition, edge states at the sample boundaries are dispersive and enable controlled propagation of photonphonon polaritons.
Topological Phases of Sound and Light
V. Peano, C. Brendel, M. Schmidt, F. Marquardt
Physical Review X
5(3)
031011
(2015)

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Topological states of matter are particularly robust, since they exploit global features of a material's band structure. Topological states have already been observed for electrons, atoms, and photons. It is an outstanding challenge to create a Chern insulator of sound waves in the solid state. In this work, we propose an implementation based on cavity optomechanics in a photonic crystal. The topological properties of the sound waves can be wholly tuned in situ by adjusting the amplitude and frequency of a driving laser that controls the optomechanical interaction between light and sound. The resulting chiral, topologically protected phonon transport can be probed completely optically. Moreover, we identify a regime of strong mixing between photon and phonon excitations, which gives rise to a large set of different topological phases and offers an example of a Chern insulator produced from the interaction between two physically distinct particle species, photons and phonons.
PositionSquared Coupling in a Tunable Photonic Crystal Optomechanical
Cavity
Taofiq K. Paraiso, Mahmoud Kalaee, Leyun Zang, Hannes Pfeifer, Florian Marquardt, Oskar Painter
Physical Review X
5(4)
041024
(2015)

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We present the design, fabrication, and characterization of a planar silicon photonic crystal cavity in which large positionsquared optomechanical coupling is realized. The device consists of a doubleslotted photonic crystal structure in which motion of a central beam mode couples to two highQ optical modes localized around each slot. Electrostatic tuning of the structure is used to controllably hybridize the optical modes into supermodes that couple in a quadratic fashion to the motion of the beam. From independent measurements of the anticrossing of the optical modes and of the dynamic optical spring effect, a positionsquared vacuum coupling rate as large as (g) over tilde'/2 pi = 245 Hz is inferred between the optical supermodes and the fundamental inplane mechanical resonance of the structure at omega(m)/2 pi = 8.7 MHz, which in displacement units corresponds to a coupling coefficient of g'/2 pi = 1 THz/nm(2). For larger supermode splittings, selective excitation of the individual optical supermodes is used to demonstrate optical trapping of the mechanical resonator with measured (g) over tilde'/2 pi = 46 Hz.
Magnon dark modes and gradient memory
Xufeng Zhang, ChangLing Zou, Na Zhu, Florian Marquardt, Liang Jiang, Hong X. Tang
Nature Communications
6
8914
(2015)

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Extensive efforts have been expended in developing hybrid quantum systems to overcome the short coherence time of superconducting circuits by introducing the naturally longlived spin degree of freedom. Among all the possible materials, singlecrystal yttrium iron garnet has shown up recently as a promising candidate for hybrid systems, and various highly coherent interactions, including strong and even ultrastrong coupling, have been demonstrated. One distinct advantage in these systems is that spins form welldefined magnon modes, which allows flexible and precise tuning. Here we demonstrate that by dissipation engineering, a nonMarkovian interaction dynamics between the magnon and the microwave cavity photon can be achieved. Such a process enables us to build a magnon gradient memory to store information in the magnon dark modes, which decouple from the microwave cavity and thus preserve a long lifetime. Our findings provide a promising approach for developing longlifetime, multimode quantum memories.
Optomechanical creation of magnetic fields for photons on a lattice
M. Schmidt, S. Kessler, V. Peano, O. Painter, F. Marquardt
Recently, there has been growing interest in the creation of artificial magnetic fields for uncharged particles, such as cold atoms or photons. These efforts are partly motivated by the resulting desirable features, such as transport along edge states that is robust against backscattering. We analyze how the optomechanical interaction between photons and mechanical vibrations can be used to create artificial magnetic fields for photons on a lattice. The ingredients required are an optomechanical crystal, i. e., a freestanding photonic crystal with localized vibrational and optical modes, and two laser beams with the right pattern of phases. One of the two schemes analyzed here is based on optomechanical modulation of the links between optical modes, while the other is a lattice extension of optomechanical wavelengthconversion setups. We analyze both schemes theoretically and present numerical simulations of the resulting optical spectrum, photon transport in the presence of an artificial Lorentz force, edge states, and the photonic Aharonov Bohm effect. We discuss the requirements for experimental realizations. Finally, we analyze the completely general situation of an optomechanical system subject to an arbitrary optical phase pattern and conclude that it is best described in terms of gauge fields acting in synthetic dimensions. In contrast to existing nonoptomechanical approaches, the schemes analyzed here are very versatile, since they can be controlled fully optically, allowing for timedependent in situ tunability without the need for individual electrical addressing of localized optical modes. (C) 2015 Optical Society of America
Pattern phase diagram for twodimensional arrays of coupled limitcycle
oscillators
Roland Lauter, Christian Brendel, Steven J. M. Habraken, Florian Marquardt
Physical Review E
92(1)
012902
(2015)

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Arrays of coupled limitcycle oscillators represent a paradigmatic example for studying synchronization and pattern formation. We find that the full dynamical equations for the phase dynamics of a limitcycle oscillator array go beyond previously studied Kuramototype equations. We analyze the evolution of the phase field in a twodimensional array and obtain a "phase diagram" for the resulting stationary and nonstationary patterns. Our results are of direct relevance in the context of currently emerging experiments on nanoand optomechanical oscillator arrays, as well as for any array of coupled limitcycle oscillators that have undergone a Hopf bifurcation. The possible observation in optomechanical arrays is discussed briefly.
Quantum simulation of expanding spacetime with tunnelcoupled condensates
Clemens Neuenhahn, Florian Marquardt
New Journal of Physics
17
125007
(2015)

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We consider two weakly interacting quasi1D condensates of cold bosonic atoms. It turns out that a timedependent variation of the tunnelcoupling between those condensates is equivalent to the spatial expansion of a onedimensional toyUniverse, with regard to the dynamics of the relative phase field. The dynamics of this field is governed by the quantum sineGordon equation. Thus, this analogy could be used to 'quantum simulate' the dynamics of a scalar, interacting quantum field on an expanding background. We discuss how to observe the 'freezing' of quantum fluctuations during an accelerating expansion in a possible experiment. We also analyze an experimental protocol to study the formation of sineGordon breathers in the relative phase field, seeded by quantum
2014
Singlesiteresolved measurement of the current statistics in optical
lattices
Stefan Kessler, Florian Marquardt
Physical Review A
89(6)
061601
(2014)

Journal

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At present, great effort is spent on the experimental realization of gauge fields for quantum manybody systems in optical lattices. At the same time, the singlesiteresolved detection of individual atoms has become a new powerful experimental tool. We discuss a protocol for the singlesiteresolved measurement of the current statistics of quantum manybody systems, which makes use of a bichromatic optical superlattice and singlesite detection. We illustrate the protocol by a numerical study of the current statistics for interacting bosons in one and two dimensions and discuss the role of the onsite interactions for the current pattern and the groundstate symmetry for small twodimensional lattices with artificial magnetic fields.
Synchronizing a singleelectron shuttle to an external drive
Michael Möckel, Darren R. Southworth, Eva M. Weig, Florian Marquardt
New Journal of Physics
16
043009
(2014)

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The nanomechanical singleelectron shuttle is a resonant system in which a suspended metallic island oscillates between and impacts at two electrodes. This setup holds promise for onebyone electron transport and the establishment of an absolute current standard. While the charge transported per oscillation by the nanoscale island will be quantized in the Coulomb blockade regime, the frequency of such a shuttle depends sensitively on many parameters, leading to <br>drift and noise. Instead of considering the nonlinearities introduced by the impact events as a nuisance, here we propose to exploit the resulting nonlinear dynamics to realize a highly precise oscillation frequency via synchronization of the shuttle selfoscillations to an external signal. We link the established phenomenological description of synchronization based on the ADLER equation to the microscopic nonlinear dynamics of the electron shuttle by calculating the effective ADLER constant analytically in terms of the microscopic parameters.
Cavity optomechanics
Markus Aspelmeyer, Tobias J. Kippenberg, Florian Marquardt
Reviews of Modern Physics
86(4)
13911452
(2014)

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The field of cavity optomechanics is reviewed. This field explores the interaction between electromagnetic radiation and nanomechanical or micromechanical motion. This review covers the basics of optical cavities and mechanical resonators, their mutual optomechanical interaction mediated by the radiationpressure force, the large variety of experimental systems which exhibit this interaction, optical measurements of mechanical motion, dynamical backaction amplification and cooling, nonlinear dynamics, multimode optomechanics, and proposals for future cavityquantumoptomechanics experiments. In addition, the perspectives for fundamental quantum physics and for possible applications of optomechanical devices are described.
Laser Theory for Optomechanics: Limit Cycles in the Quantum Regime
Physical Review X
4(1)
011015
(2014)

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Optomechanical systems can exhibit selfsustained limit cycles where the quantum state of the mechanical resonator possesses nonclassical characteristics such as a strongly negative Wigner density, as was shown recently in a numerical study by Qian et al. [Phys. Rev. Lett. 109, 253601 (2012)]. Here, we derive a FokkerPlanck equation describing mechanical limit cycles in the quantum regime that correctly reproduces the numerically observed nonclassical features. The derivation starts from the standard optomechanical master equation and is based on techniques borrowed from the laser theory due to Haake and Lewenstein. We compare our analytical model with numerical solutions of the master equation based on Monte Carlo simulations and find very good agreement over a wide and so far unexplored regime of system parameters. As one main conclusion, we predict negative Wigner functions to be observable even for surprisingly classical parameters, i.e., outside the singlephoton strongcoupling regime, for strong cavity drive and rather large limitcycle amplitudes. The approach taken here provides a natural starting point for further studies of quantum effects in optomechanics.
Decoherence in a doubledot AharonovBohm interferometer: Numerical
renormalization group study
Bjoern Kubala, David Roosen, Michael Sindel, Walter Hofstetter, Florian Marquardt
Physical Review B
90(3)
035417
(2014)

Journal

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Coherence in electronic interferometers is typically believed to be restored fully in the limit of small voltages, frequencies, and temperatures. However, it is crucial to check this essentially perturbative argument by nonperturbative methods. Here we use the numerical renormalization group to study ac transport and decoherence in an experimentally realizable model interferometer, a parallel double quantum dot coupled to a phonon mode. The model allows us to clearly distinguish renormalization effects from decoherence. We discuss finitefrequency transport and confirm the restoration of coherence in the dc limit.
Focus on optomechanics
Ivan Favero, Florian Marquardt
New Journal of Physics
16
085006
(2014)

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We provide a brief overview of the various topics addressed in this 'focus on' collection on optomechanics.
Dissipative optomechanical squeezing of light
Andreas Kronwald, Florian Marquardt, Aashish A. Clerk
New Journal of Physics
16
063058
(2014)

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We discuss a simple yet surprisingly effective mechanism which allows the generation of squeezed output light from an optomechanical cavity. In contrast to the well known mechanism of 'ponderomotive squeezing', our scheme generates squeezed output light by explicitly using the dissipative nature of the mechanical resonator. We show that our scheme has many advantages over ponderomotive squeezing; in particular, it is far more effective in the good cavity limit commonly used in experiments. Furthermore, the squeezing generated in our approach can be directly used to enhance the intrinsic measurement sensitivity of the optomechanical cavity; one does not have to feed the squeezed light into a separate measurement device. As our scheme is very general, it could also e. g. be implemented using superconducting circuits.
Basic Theory of Cavity Optomechanics
Aashish A. Clerk, Florian Marquardt
Quantum Science and Technology
(2014)

Book Chapter
This chapter provides a brief basic introduction to the theory used to describe cavityoptomechanical systems. This can serve as background information to understand the other chapters of the book. We first review the Hamiltonian and show how it can be approximately brought into quadratic form. Then we discuss the classical dynamics both in the linear regime (featuring optomechanical damping, optical spring, strong coupling, and optomechanically induced transparency) and in the nonlinear regime (optomechanical selfoscillations and attractor diagram). Finally, wediscuss the quantum theory of optomechanical cooling, using the powerful and versatile quantum noise approach.
Cavity Optomechanics Nano and Micromechanical Resonators Interacting with Light Introduction
Markus Aspelmeyer, Tobias J. Kippenberg, Florian Marquardt
Quantum Science and Technology
(2014)

Book Chapter
We briefly guide the reader through the chapters of the book, highlighting the connections between the various approaches to cavity optomechanics.
Cavity Optomechanics Nano and Micromechanical Resonators Interacting with Light Preface
Markus Aspelmeyer, Tobias J. Kippenberg, Florian Marquardt
Quantum Science and Technology
(2014)
2013
Optomechanically Induced Transparency in the Nonlinear Quantum Regime
Andreas Kronwald, Florian Marquardt
Physical Review Letters
111(13)
133601
(2013)

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Optomechanical systems have been shown both theoretically and experimentally to exhibit an analogon to atomic electromagnetically induced transparency, with sharp transmission features that are controlled by a second laser beam. Here we investigate these effects in the regime where the fundamental nonlinear nature of the optomechanical interaction becomes important. We demonstrate that pulsed transistorlike switching of transmission still works even in this regime. We also show that optomechanically induced transparency at the second mechanical sideband could be a sensitive tool to see first indications of the nonlinear quantum nature of the optomechanical interaction even for singlephoton coupling strengths significantly smaller than the cavity linewidth.
The effect of LandauZener dynamics on phonon lasing
Huaizhi Wu, Georg Heinrich, Florian Marquardt
New Journal of Physics
15
123022
(2013)

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Optomechanical systems couple light to the motion of nanomechanical objects. Intriguing new effects are observed in recent experiments that involve the dynamics of more than one optical mode. There, mechanical motion can stimulate strongly driven multimode photon dynamics that acts back on the mechanics via radiation forces. We show that even for two optical modes LandauZenerStueckelberg oscillations of the light field drastically change the nonlinear attractor diagram of the resulting phonon lasing oscillations. Our findings illustrate the generic effects of LandauZener physics on backaction induced selfoscillations.
Observing the Nonequilibrium Dynamics of the Quantum TransverseField Ising Chain in Circuit QED
Oliver Viehmann, Jan von Delft, Florian Marquardt
Physical Review Letters
110(3)
030601
(2013)

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We show how a quantum Ising spin chain in a timedependent transverse magnetic field can be simulated and experimentally probed in the framework of circuit QED with current technology. The proposed setup provides a new platform for observing the nonequilibrium dynamics of interacting manybody systems. We calculate its spectrum to offer a guideline for its initial experimental characterization. We demonstrate that quench dynamics and the propagation of localized excitations can be observed with the proposed setup and discuss further possible applications and modifications of this circuit QED quantum simulator. DOI: 10.1103/PhysRevLett.110.030601
Dynamics of levitated nanospheres: towards the strong coupling regime
T. S. Monteiro, J. Millen, G. A. T. Pender, Florian Marquardt, D. Chang, P. F. Barker
New Journal of Physics
15
015001
(2013)

Journal

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The use of levitated nanospheres represents a new paradigm for the optomechanical cooling of a small mechanical oscillator, with the prospect of realizing quantum oscillators with unprecedentedly high quality factors. We investigate the dynamics of this system, especially in the socalled selftrapping regime, where one or more optical fields simultaneously trap and cool the mechanical oscillator. The determining characteristic of this regime is that both the mechanical frequency omega(M) and singlephoton optomechanical coupling strength parameters g are a function of the optical field intensities, in contrast to usual setups where omega(M) and g are constant for the given system. We also measure the characteristic transverse and axial trapping frequencies of different sized silica nanospheres in a simple optical standing wave potential, for spheres of radii r = 20500 nm, illustrating a protocol for loading single nanospheres into a standing wave optical trap that would be formed by an optical cavity. We use these data to confirm the dependence of the effective optomechanical coupling strength on sphere radius for levitated nanospheres in an optical cavity and discuss the prospects for reaching regimes of strong lightmatter coupling. Theoretical semiclassical and quantum displacement noise spectra show that for larger nanospheres with r greater than or similar to 100 nm a range of interesting and novel dynamical regimes can be accessed. These include simultaneous hybridization of the two optical modes with the mechanical modes and parameter regimes where the system is bistable. We show that here, in contrast to typical singleoptical mode optomechanical systems, bistabilities are independent of intracavity intensity and can occur for very weak laser driving amplitudes.
Photonic Cavity Synchronization of Nanomechanical Oscillators
Physical Review Letters
111(21)
213902
(2013)

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Synchronization in oscillatory systems is a frequent natural phenomenon and is becoming an important concept in modern physics. Nanomechanical resonators are ideal systems for studying synchronization due to their controllable oscillation properties and engineerable nonlinearities. Here we demonstrate synchronization of two nanomechanical oscillators via a photonic resonator, enabling optomechanical synchronization between mechanically isolated nanomechanical resonators. Optical backaction gives rise to both reactive and dissipative coupling of the mechanical resonators, leading to coherent oscillation and mutual locking of resonators with dynamics beyond the widely accepted phase oscillator (Kuramoto) model. In addition to the phase difference between the oscillators, also their amplitudes are coupled, resulting in the emergence of sidebands around the synchronized carrier signal.
Arbitrarily large steadystate bosonic squeezing via dissipation
Andreas Kronwald, Florian Marquardt, Aashish A. Clerk
Physical Review A
88(6)
063833
(2013)

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We discuss how large amounts of steadystate quantum squeezing (beyond 3 dB) of a mechanical resonator can be obtained by driving an optomechanical cavity with two control lasers with differing amplitudes. The scheme does not rely on any explicit measurement or feedback, nor does it simply involve a modulation of an optical spring constant. Instead, it uses a dissipative mechanism with the driven cavity acting as an engineered reservoir. It can equivalently be viewed as a coherent feedback process, obtained by minimally perturbing the quantum nondemolition measurement of a single mechanical quadrature. This shows that in general the concepts of coherent feedback schemes and reservoir engineering are closely related. We analyze how to optimize the scheme, how the squeezing scales with system parameters, and how it may be directly detected from the cavity output. Our scheme is extremely general, and could also be implemented with, e.g., superconducting circuits.
Full photon statistics of a light beam transmitted through an
optomechanical system
Andreas Kronwald, Max Ludwig, Florian Marquardt
Physical Review A
87(1)
013847
(2013)

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In this paper, we study the full statistics of photons transmitted through an optical cavity coupled to nanomechanical motion. We analyze the entire temporal evolution of the photon correlations, the Fano factor, and the effects of strong laser driving, all of which show pronounced features connected to the mechanical backaction. In the regime of singlephoton strong coupling, this allows us to predict a transition from subPoissonian to superPoissonian statistics for larger observation time intervals. Furthermore, we predict cascades of transmitted photons triggered by multiphoton transitions. In this regime, we observe Fano factors that are drastically enhanced due to the mechanical motion. DOI: 10.1103/PhysRevA.87.013847
Creation and dynamics of remote spinentangled pairs in the expansion of
strongly correlated fermions in an optical lattice
Stefan Kessler, Ian P. McCulloch, Florian Marquardt
New Journal of Physics
15
053043
(2013)

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We consider the nonequilibrium dynamics of an interacting spin1/2 fermion gas in a onedimensional optical lattice after switching off the confining potential. In particular, we study the creation and the time evolution of spatially separated, spinentangled fermionic pairs. The timedependent densitymatrix renormalization group is used to simulate the time evolution and evaluate the twosite spin correlation functions, from which the concurrence is calculated. We find that the typical distance between entangled fermions depends crucially on the onsite interaction strength, and that a timedependent modulation of the tunnelling amplitude can enhance the production of spin entanglement. Moreover, we discuss the prospects of experimentally observing these phenomena using spindependent singlesite detection.
Gaintunable optomechanical cooling in a laser cavity
Li Ge, Sanli Faez, Florian Marquardt, Hakan E. Tuereci
Physical Review A
87(5)
053839
(2013)

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We study the optical cooling of the cavity mirror in an active laser cavity. We find that the optical damping rate is vanishingly small for an incoherently pumped laser above threshold. In the presence of an additional external coherent drive however, the optical damping rate can be enhanced substantially with respect to that of a passive cavity. We show that the strength of the incoherent pump provides the means to tune the optical damping rate and the steady state phonon number. The system is found to undergo a transition from the weak optomechanical coupling regime to the strong optomechanical coupling regime as the strength of the incoherent pump is varied.
Quantum ManyBody Dynamics in Optomechanical Arrays
Max Ludwig, Florian Marquardt
Physical Review Letters
111(7)
073603
(2013)

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We study the nonlinear driven dissipative quantum dynamics of an array of optomechanical systems. At each site of such an array, a localized mechanical mode interacts with a laserdriven cavity mode via radiation pressure, and both photons and phonons can hop between neighboring sites. The competition between coherent interaction and dissipation gives rise to a rich phase diagram characterizing the optical and mechanical manybody states. For weak intercellular coupling, the mechanical motion at different sites is incoherent due to the influence of quantum noise. When increasing the coupling strength, however, we observe a transition towards a regime of phasecoherent mechanical oscillations. We employ a Gutzwiller ansatz as well as semiclassical Langevin equations on finite lattices, and we propose a realistic experimental implementation in optomechanical crystals.
2012
Optomechanical circuits for nanomechanical continuous variable quantum
state processing
Michael Schmidt, Max Ludwig, Florian Marquardt
New Journal of Physics
14
125005
(2012)

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We propose and analyze a nanomechanical architecture where light is used to perform linear quantum operations on a set of many vibrational modes. Suitable amplitude modulation of a single laser beam is shown to generate squeezing, entanglement and state transfer between modes that are selected according to their mechanical oscillation frequency. Current optomechanical devices based on photonic crystals, as well as other systems with sufficient control over multiple mechanical modes, may provide a platform for realizing this scheme.
Quantum Signatures of the Optomechanical Instability
Jiang Qian, A. A. Clerk, K. Hammerer, Florian Marquardt
Physical Review Letters
109(25)
253601
(2012)

Journal

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In the past few years, coupling strengths between light and mechanical motion in optomechanical setups have improved by orders of magnitude. Here we show that, in the standard setup under continuous laser illumination, the steady state of the mechanical oscillator can develop a nonclassical, strongly negative Wigner density if the optomechanical coupling is comparable to or larger than the optical decay rate and the mechanical frequency. Because of its robustness, such a Wigner density can be mapped using optical homodyne tomography. This feature is observed near the onset of the instability towards selfinduced oscillations. We show that there are also distinct signatures in the photonphoton correlation function g((2))(t) in that regime, including oscillations decaying on a time scale not only much longer than the optical cavity decay time but even longer than the mechanical decay time.
Stroboscopic observation of quantum manybody dynamics
Stefan Kessler, Andreas Holzner, Ian P. McCulloch, Jan von Delft, Florian Marquardt
Physical Review A
85(1)
011605
(2012)

Journal

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Recent experiments have demonstrated singlesite resolved observation of cold atoms in optical lattices. Thus, in the future it may be possible to take repeated snapshots of an interacting quantum manybody system during the course of its evolution. Here we address the impact of the resulting quantum (anti)Zeno physics on the manybody dynamics. We use the timedependent densitymatrix renormalization group to obtain the time evolution of the full wave function, which is then periodically projected in order to simulate realizations of stroboscopic measurements. For the example of a onedimensional lattice of spinless fermions with nearestneighbor interactions, we find regimes for which manyparticle configurations are stabilized or destabilized, depending on the interaction strength and the time between observations.
Enhanced Quantum Nonlinearities in a TwoMode Optomechanical System
Max Ludwig, Amir H. SafaviNaeini, Oskar Painter, Florian Marquardt
Physical Review Letters
109(6)
063601
(2012)

Journal

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In cavity optomechanics, nanomechanical motion couples to a localized optical mode. The regime of singlephoton strong coupling is reached when the optical shift induced by a single phonon becomes comparable to the cavity linewidth. We consider a setup in this regime comprising two optical modes and one mechanical mode. For mechanical frequencies nearly resonant to the optical level splitting, we find the photonphonon and the photonphoton interactions to be significantly enhanced. In addition to dispersive phonon detection in a novel regime, this offers the prospect of optomechanical photon measurement. We study these quantum nondemolition detection processes using both analytical and numerical approaches.
Thermalization of interacting fermions and delocalization in Fock space
Clemens Neuenhahn, Florian Marquardt
Physical Review E
85(6)
060101
(2012)

Journal

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We investigate the onset of "eigenstate thermalization" and the crossover to ergodicity in a system of onedimensional fermions with increasing interaction. We analyze the fluctuations in the expectation values of most relevant fewbody operators with respect to eigenstates. It turns out that these are intimately related to the inverse participation ratio of eigenstates displayed in the operator eigenbasis. Based on this observation, we find good evidence that eigenstate thermalization should set in even for vanishingly small perturbations in the thermodynamic limit.
Observation of spontaneous Brillouin cooling
Gaurav Bahl, Matthew Tomes, Florian Marquardt, Tal Carmon
Although bolometric and ponderomotiveinduced deflection of device boundaries are widely used for laser cooling, the electrostrictive Brillouin scattering of light from sound was considered an acoustooptical amplificationonly process(17). It was suggested that cooling could be possible in multiresonance Brillouin systems(58) when phonons experience lower damping than light(8). However, this regime was not accessible in electrostrictive Brillouin systems(13,5,6) as backscattering enforces high acoustical frequencies associated with high mechanical damping(1). Recently, forward Brillouin scattering(3) in microcavities(7) has allowed access to lowfrequency acoustical modes where mechanical dissipation is lower than optical dissipation, in accordance with the requirements for cooling(8). Here we experimentally demonstrate cooling via such a forward Brillouin process in a microresonator. We show two regimes of operation for the electrostrictive Brillouin process: acoustical amplification as is traditional and an electrostrictive Brillouin cooling regime. Cooling is mediated by resonant light in one pumped optical mode, and spontaneously scattered resonant light in one antiStokes optical mode, that beat and electrostrictively attenuate the Brownian motion of the mechanical mode.
Optomechanical cooling of levitated spheres with doubly resonant fields
G. A. T. Pender, P. F. Barker, Florian Marquardt, J. Millen, T. S. Monteiro
Physical Review A
85(2)
021802
(2012)

Journal

PDF
Optomechanical cooling of levitated dielectric particles represents a promising new approach in the quest to cool small mechanical resonators toward their quantum ground state. We investigate twomode cooling of levitated nanospheres in a selftrapping regime. We identify a structure of overlapping, multiple cooling resonances and strong cooling even when one mode is bluedetuned. We show that the best regimes occur when both optical fields cooperatively cool and trap the nanosphere, where cooling rates are over an order of magnitude faster compared to corresponding singleresonance cooling rates.
Localized Phase Structures Growing Out of Quantum Fluctuations in a Quench of Tunnelcoupled Atomic Condensates
Clemens Neuenhahn, Anatoli Polkovnikov, Florian Marquardt
We investigate the relative phase between two weakly interacting 1D condensates of bosonic atoms after suddenly switching on the tunnel coupling. The following phase dynamics is governed by the quantum sineGordon equation. In the semiclassical limit of weak interactions, we observe the parametric amplification of quantum fluctuations leading to the formation of breathers with a finite lifetime. The typical lifetime and density of these "quasibreathers" are derived employing exact solutions of the classical sineGordon equation. Both depend on the initial relative phase between the condensates, which is considered as a tunable parameter.
2011
Collective Dynamics in Optomechanical Arrays
Georg Heinrich, Max Ludwig, Jiang Qian, Bjoern Kubala, Florian Marquardt
Physical Review Letters
107(4)
043603
(2011)

Journal

PDF
Optomechanical systems couple light stored inside an optical cavity to the motion of a mechanical mode. Recent experiments have demonstrated setups, such as photonic crystal structures, that in principle allow one to confine several optical and vibrational modes on a single chip. Here we start to investigate the collective nonlinear dynamics in arrays of coupled optomechanical cells. We show that such "optomechanical arrays" can display synchronization, and that they can be described by an effective Kuramototype model.
Superradiant Phase Transitions and the Standard Description of Circuit QED
Oliver Viehmann, Jan von Delft, Florian Marquardt
Physical Review Letters
107(11)
113602
(2011)

Journal

PDF
We investigate the equilibrium behavior of a superconducting circuit QED system containing a large number of artificial atoms. It is shown that the currently accepted standard description of circuit QED via an effective model fails in an important aspect: it predicts the possibility of a superradiant phase transition, even though a full microscopic treatment reveals that a nogo theorem for such phase transitions known from cavity QED applies to circuit QED systems as well. We generalize the nogo theorem to the case of (artificial) atoms with many energy levels and thus make it more applicable for realistic cavity or circuit QED systems.
Quantum mechanics: The gentle cooling touch of light
Laser light has been used to cool a nanomechanical resonator to its lowest energy state. The result opens the door to testing the principles of quantum mechanics and to applications in quantum information processing.
Dynamics of coupled multimode and hybrid optomechanical systems
Georg Heinrich, Max Ludwig, Huaizhi Wu, K. Hammerer, Florian Marquardt
Recent experimental developments have brought into focus optomechanical systems containing multiple optical and mechanical modes interacting with each other. Examples include a setup with a movable membrane between two endmirrors and "optomechanical crystal" devices that support localized optical and mechanical modes in a photonic crystal type structure. We discuss how mechanical driving of such structures results in coherent photon transfer between optical modes, and how the physics of LandauZenerStueckelberg oscillations arises in this context. Another area where multiple modes are involved are hybrid systems. There, we review the recent proposal of a single atom whose mechanical motion is coupled to a membrane via the light field. This is a special case of the general principle of cavitymediated mechanical coupling. Such a setup would allow the welldeveloped tools of atomic physics to be employed to access the quantum state of the 'macroscopic' mechanical mode of the membrane. (C) 2011 Academie des sciences. Published by Elsevier Masson SAS. All rights reserved.
Quantummechanical theory of optomechanical Brillouin cooling
Matthew Tomes, Florian Marquardt, Gaurav Bahl, Tal Carmon
Physical Review A
84(6)
063806
(2011)

Journal

PDF
We analyze how to exploit Brillouin scattering of light from sound for the purpose of cooling optomechanical devices and present a quantummechanical theory for Brillouin cooling. Our analysis shows that significant cooling ratios can be obtained with standard experimental parameters. A further improvement of cooling efficiency is possible by increasing the dissipation of the optical antiStokes resonance.
Coupled multimode optomechanics in the microwave regime
The motion of micro and nanomechanical resonators can be coupled to electromagnetic fields. This allows one to explore the mutual interaction and introduces new means to manipulate and control both light and mechanical motion. Such optomechanical systems have recently been implemented in nanoelectromechanical systems involving a nanomechanical beam coupled to a superconducting microwave resonator. Here, we propose optomechanical systems that involve multiple, coupled microwave resonators. In contrast to similar systems in the optical realm, the coupling frequency governing photon exchange between microwave modes is naturally comparable to typical mechanical frequencies. For instance this enables new ways to manipulate the microwave field, such as mechanically driving coherent photon dynamics between different modes. In particular we investigate two setups where the electromagnetic field is coupled either linearly or quadratically to the displacement of a nanomechanical beam. The latter scheme allows one to perform QND Fock state detection. For experimentally realistic parameters we predict the possibility to measure an individual quantum jump from the mechanical ground state to the first excited state. Copyright (C) EPLA, 2011
2010
Optimal control of circuit quantum electrodynamics in one and two
dimensions
R. Fisher, F. Helmer, S. J. Glaser, F. Marquardt, T. SchulteHerbrueggen
Optimal control can be used to significantly improve multiqubit gates in quantum information processing hardware architectures based on superconducting circuit quantum electrodynamics. We apply this approach not only to dispersive gates of two qubits inside a cavity, but, more generally, to architectures based on twodimensional (2D) arrays of cavities and qubits. For highfidelity gate operations, simultaneous evolutions of controls and couplings in the two coupling dimensions of cavity grids are shown to be significantly faster than conventional sequential implementations. Even under experimentally realistic conditions speedups by a factor of three can be gained. The methods immediately scale to large grids and indirect gates between arbitrary pairs of qubits on the grid. They are anticipated to be paradigmatic for 2D arrays and lattices of controllable qubits.
Entanglement of mechanical oscillators coupled to a nonequilibrium
environment
Max Ludwig, K. Hammerer, Florian Marquardt
Physical Review A
82(1)
012333
(2010)

Journal

PDF
Recent experiments aim at cooling nanomechanical resonators to the ground state by coupling them to nonequilibrium environments in order to observe quantum effects such as entanglement. This raises the general question of how such environments affect entanglement. Here we show that there is an optimal dissipation strength for which the entanglement between two coupled oscillators is maximized. Our results are established with the help of a general framework of exact quantum Langevin equations valid for arbitrary bath spectra, in and out of equilibrium. We point out why the commonly employed Lindblad approach fails to give even a qualitatively correct picture.
Electronplasmon scattering in chiral onedimensional systems with nonlinear dispersion
We investigate systems of spinless onedimensional chiral fermions realized, e. g., in the arms of electronic MachZehnder interferometers, at high energies. Taking into account the curvature of the fermionic spectrum and a finite interaction range, we find a new scattering mechanism where highenergy electrons scatter off plasmons (density excitations). This leads to an exponential decay of the singleparticle Green's function even at zero temperature with an energydependent rate. As a consequence of this electronplasmon scattering channel, we observe the coherent excitation of a plasmon wave in the wake of a highenergy electron resulting in the buildup of a monochromatic sinusoidal density pattern.
Photon shuttle: LandauZenerStuckelberg dynamics in an optomechanical system
Georg Heinrich, J. G. E. Harris, Florian Marquardt
Physical Review A
81(1)
011801
(2010)

Journal

PDF
The motion of micro and nanomechanical resonators can be coupled to electromagnetic fields. Such optomechanical setups allow one to explore the interaction of light and matter in a new regime at the boundary between quantum and classical physics. We propose an approach to investigate nonequilibrium photon dynamics driven by mechanical motion in a recently developed setup with a membrane between two mirrors, where photons can be shuttled between the two halves of the cavity. For modest driving strength we predict the possibility of observing an AutlerTownes splitting indicative of Rabi dynamics. For large drive, we show that this system displays LandauZenerStueckelberg dynamics originally known from atomic twostate systems.
Introduction to quantum noise, measurement, and amplification
A. A. Clerk, M. H. Devoret, S. M. Girvin, Florian Marquardt, R. J. Schoelkopf
Reviews of Modern Physics
82(2)
11551208
(2010)

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The topic of quantum noise has become extremely timely due to the rise of quantum information physics and the resulting interchange of ideas between the condensed matter and atomic, molecular, opticalquantum optics communities. This review gives a pedagogical introduction to the physics of quantum noise and its connections to quantum measurement and quantum amplification. After introducing quantum noise spectra and methods for their detection, the basics of weak continuous measurements are described. Particular attention is given to the treatment of the standard quantum limit on linear amplifiers and position detectors within a general linearresponse framework. This approach is shown how it relates to the standard HausCaves quantum limit for a bosonic amplifier known in quantum optics and its application to the case of electrical circuits is illustrated, including mesoscopic detectors and resonant cavity detectors.
ac conductance through an interacting quantum dot
Bjoern Kubala, Florian Marquardt
Physical Review B
81(11)
115319
(2010)

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We investigate the linear ac conductance for tunneling through an arbitrary interacting quantum dot in the presence of a finite dc bias. In analogy to the wellknown MeirWingreen formula for the dc case, we are able to derive a general formula for the ac conductance. It can be expressed entirely in terms of local correlations on the quantum dot in the form of a Keldysh block diagram with four external legs. We illustrate the use of this formula as a starting point for diagrammatic calculations by considering the ac conductance of the noninteracting resonantlevel model and deriving the result for the lowest order of electronphonon coupling. We show how known results are recovered in the appropriate limits.
Quantum Measurement of Phonon Shot Noise
A. A. Clerk, Florian Marquardt, J. G. E. Harris
Physical Review Letters
104(21)
213603
(2010)

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We provide a full quantum mechanical analysis of a weak energy measurement of a driven mechanical resonator. We demonstrate that measurements too weak to resolve individual mechanical Fock states can nonetheless be used to detect the nonclassical energy fluctuations of the driven mechanical resonator, i.e., "phonon shot noise". We also show that the third moment of the oscillator's energy fluctuations provides a far more sensitive probe of quantum effects than the second moment, and that measuring the third moment via the phase shift of light in an optomechanical setup directly yields the type of operator ordering postulated in the theory of fullcounting statistics.
Singleatom cavity QED and optomicromechanics
M. Wallquist, K. Hammerer, P. Zoller, C. Genes, M. Ludwig, F. Marquardt, P. Treutlein, J. Ye, H. J. Kimble
In a recent publication [ K. Hammerer, M. Wallquist, C. Genes, M. Ludwig, F. Marquardt, P. Treutlein, P. Zoller, J. Ye, and H. J. Kimble, Phys. Rev. Lett. 103, 063005 ( 2009)] we have shown the possibility to achieve strong coupling of the quantized motion of a micronsized mechanical system to the motion of a single trapped atom. In the proposed setup the coherent coupling between a SiN membrane and a single atom is mediated by the field of a high finesse cavity and can be much larger than the relevant decoherence rates. This makes the welldeveloped tools of cavity quantum electrodynamics with single atoms available in the realm of cavity optomechanics. In this article we elaborate on this scheme and provide detailed derivations and technical comments. Moreover, we give numerical as well as analytical results for a number of possible applications for transfer of squeezed or Fock states from atom to membrane as well as entanglement generation, taking full account of dissipation. In the limit of strongcoupling the preparation and verification of nonclassical states of a mesoscopic mechanical system is within reach.
Resonant quantum gates in circuit quantum electrodynamics
G. Haack, F. Helmer, M. Mariantoni, F. Marquardt, E. Solano
Physical Review B
82(2)
024514
(2010)

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We propose the implementation of fast resonant gates in circuit quantum electrodynamics for quantum information processing. We show how a suitable utilization of threelevel superconducting qubits inside a resonator constitutes a key tool to perform diverse twoqubit resonant gates, improving the operation speed when compared to slower dispersive techniques. To illustrate the benefit of resonant twoqubit gates in circuit quantum electrodynamics, we consider the implementation of a twodimensional cluster state in an array of N x N superconducting qubits by using resonant controlledphase and onequbit gates, where the generation time grows linearly with N. For N = 3, and taking into account decoherence mechanisms, a fidelity over 60% for the generation of this cluster state is obtained.
2009
Universal Dephasing in a Chiral 1D Interacting Fermion System
Clemens Neuenhahn, Florian Marquardt
Physical Review Letters
102(4)
046806
(2009)

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We consider dephasing by interactions in a onedimensional chiral fermion system (e.g., a quantum Hall edge state). For finiterange interactions, we calculate the spatial decay of the Green's function at fixed energy, which sets the contrast in a MachZehnder interferometer. Using a physically transparent semiclassical ansatz, we find a powerlaw decay of the coherence at high energies and zero temperature (T=0), with a universal asymptotic exponent of 1, independent of the interaction strength. We obtain the dephasing rate at T > 0 and the fluctuation spectrum acting on an electron.
Recent progress in open quantum systems: NonGaussian noise and decoherence in fermionic systems
Clemens Neuenhahn, B. Kubala, B. Abel, Florian Marquardt
Physica Status Solidi B
246(5)
10181023
(2009)

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We review our recent contributions to two topics that have become of interest in the field of open, dissipative quantum systems: nonGaussian noise and decoherence in fermionic systems. Decoherence by nonGaussian noise, i.e. by an environment that cannot be approximated as a bath of harmonic oscillators, is important in nanostructures (e.g. qubits) where there might be strong coupling to a small number of fluctuators. We first revisit the pedagogical example of dephasing by classical telegraph noise. Then we address two models where the quantum nature of the noise becomes essential: "quantum telegraph noise" and dephasing by electronic shot noise. In fermionic systems, manybody aspects and the Pauli principle have to be taken care of when describing the loss of phase coherence. This is relevant in electronic quantum transport through metallic and semiconducting structures. Specifically, we recount our recent results regarding dephasing in a chiral interacting electron liquid, as it is realized in the electronic MachZehnder interferometer. This model can be solved employing the technique of bosortization as well as a physically transparent semiclassical method. (C) 2009 WILEYVCH Verlag GmbH & Co. KGaA, Weinheim
Dephasing rate formula in the manybody context
Doron Cohen, Jan von Delft, Florian Marquardt, Yoseph Imry
Physical Review B
80(24)
245410
(2009)

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We suggest a straightforward approach to the calculation of the dephasing rate in a fermionic system, which correctly keeps track of the crucial physics of Pauli blocking. Starting from Fermi's golden rule, the dephasing rate can be written as an integral over the frequency transferred between system and environment, weighted by their respective spectral densities. We show that treating the full manyfermion system instead of a single particle automatically enforces the Pauli principle. Furthermore, we explain the relation to diagrammatics. Finally, we show how to treat the more involved strongcoupling case when interactions appreciably modify the spectra. This is relevant for the situation in disordered metals, where screening is important.
Quantum nondemolition photon detection in circuit QED and the quantum Zeno effect
Ferdinand Helmer, Matteo Mariantoni, Enrique Solano, Florian Marquardt
Physical Review A
79(5)
052115
(2009)

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We analyze the detection of itinerant photons using a quantum nondemolition measurement. An important example is the dispersive detection of microwave photons in circuit quantum electrodynamics, which can be realized via the nonlinear interaction between photons inside a superconducting transmission line resonator. We show that the back action due to the continuous measurement imposes a limit on the detector efficiency in such a scheme. We illustrate this using a setup where signal photons have to enter a cavity in order to be detected dispersively. In this approach, the measurement signal is the phase shift imparted to an intense beam passing through a second cavity mode. The restrictions on the fidelity are a consequence of the quantum Zeno effect, and we discuss both analytical results and quantum trajectory simulations of the measurement process.
Dimensional crossover of the dephasing time in disordered mesoscopic rings
M. Treiber, O. M. Yevtushenko, F. Marquardt, J. von Delft, I. V. Lerner
Physical Review B
80(20)
201305
(2009)

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We study dephasing by electron interactions in a small disordered quasionedimensional (1D) ring weakly coupled to leads. We use an influence functional for quantum Nyquist noise to describe the crossover for the dephasing time tau(phi)(T) from diffusive or ergodic 1D (tau(1)(phi)alpha T2/3,T1) to zerodimensional (0D) behavior (tau(1)(phi)alpha T2) as T drops below the Thouless energy. The crossover to 0D, predicted earlier for twodimensional and threedimensional systems, has so far eluded experimental observation. The ring geometry holds promise of meeting this longstanding challenge, since the crossover manifests itself not only in the smooth part of the magnetoconductivity but also in the amplitude of AltshulerAronovSpivak oscillations. This allows signatures of dephasing in the ring to be cleanly extracted by filtering out those of the leads.
Measurementbased synthesis of multiqubit entangled states in superconducting cavity QED
Ferdinand Helmer, Florian Marquardt
Physical Review A
79(5)
052328
(2009)

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Entangled multiqubit states may be generated through a dispersive collective quantum nondemolition measurement of superconducting qubits coupled to a microwave transmission line resonator. Using the quantum trajectory approach, we analyze the stochastic measurement traces that would be observed in experiments. We illustrate the synthesis of threequbit W and GreenbergerHorneZeilinger states, and we analyze how the fidelity and the entanglement evolve in time during the measurement. We discuss the influence of decoherence and relaxation, as well as of imperfect control over experimental parameters. We show that the desired states can be generated on time scales much faster than the qubit decoherence rates.
Cavity grid for scalable quantum computation with superconducting circuits
F. Helmer, M. Mariantoni, A. G. Fowler, J. von Delft, E. Solano, F. Marquardt
We propose an architecture for quantum computing based on superconducting circuits, where onchip planar microwave resonators are arranged in a twodimensional grid with a qubit at each intersection. This allows any two qubits on the grid to be coupled at a swapping overhead independent of their distance. We demonstrate that this approach encompasses the fundamental elements of a scalable faulttolerant quantumcomputing architecture. Copyright (C) EPLA, 2009
OPTOMECHANICS
B. Kubala, M. Ludwig, F. Marquardt
NATO Science for Peace and Security Series B  Physics and Biophysics
(2009)

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We review recent progress in the field of optomechanics, where one studies the effects of radiation on mechanical motion. The paradigmatic example is in optical cavity with a movable mirror. where the radiation pressure can induce cooling. amplification and nonlinear dynamics of the mirror.
Strong Coupling of a Mechanical Oscillator and a Single Atom
K. Hammerer, M. Wallquist, C. Genes, M. Ludwig, F. Marquardt, P. Treutlein, P. Zoller, J. Ye, H. J. Kimble
Physical Review Letters
103(6)
063005
(2009)

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We propose and analyze a setup to achieve strong coupling between a single trapped atom and a mechanical oscillator. The interaction between the motion of the atom and the mechanical oscillator is mediated by a quantized light field in a laser driven highfinesse cavity. In particular, we show that high fidelity transfer of quantum states between the atom and the mechanical oscillator is in reach for existing or near future experimental parameters. Our setup provides the basic toolbox from atomic physics for coherent manipulation, preparation, and measurement of micromechanical and nanomechanical oscillators.
2008
Selfinduced oscillations in an optomechanical system driven by bolometric backaction
Constanze Metzger, Max Ludwig, Clemens Neuenhahn, Alexander Ortlieb, Ivan Favero, Khaled Karrai, Florian Marquardt
Physical Review Letters
101(13)
133903
(2008)

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We have explored the nonlinear dynamics of an optomechanical system consisting of an illuminated FabryPerot cavity, one of whose end mirrors is attached to a vibrating cantilever. The backaction induced by the bolometric light force produces negative damping such that the system enters a regime of nonlinear oscillations. We study the ensuing attractor diagram describing the nonlinear dynamics. A theory is presented that yields quantitative agreement with experimental results. This includes the observation of a regime where two mechanical modes of the cantilever are excited simultaneously.
The optomechanical instability in the quantum regime
Max Ludwig, Bjoern Kubala, Florian Marquardt
New Journal of Physics
10
095013
(2008)

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We consider a generic optomechanical system, consisting of a driven optical cavity and a movable mirror attached to a cantilever. Systems of this kind (and analogues) have been realized in many recent experiments. It is well known that these systems can exhibit an instability towards a regime where the cantilever settles into selfsustained oscillations. In this paper, we briefly review the classical theory of the optomechanical instability, and then discuss the features arising in the quantum regime. We solve numerically a full quantum master equation for the coupled system, and use it to analyze the photon number, the cantilever's mechanical energy, the phonon probability distribution and the mechanical Wigner density, as a function of experimentally accessible control parameters. When a suitable dimensionless 'quantum parameter' is sent to zero, the results of the quantum mechanical model converge towards the classical predictions. We discuss this quantumtoclassical transition in some detail.
Dispersive optomechanics: a membrane inside a cavity
A. M. Jayich, J. C. Sankey, B. M. Zwickl, C. Yang, J. D. Thompson, S. M. Girvin, A. A. Clerk, F. Marquardt, J. G. E. Harris
New Journal of Physics
10
095008
(2008)

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We present the results of theoretical and experimental studies of dispersively coupled (or 'membrane in the middle') optomechanical systems. We calculate the linear optical properties of a high finesse cavity containing a thin dielectric membrane. We focus on the cavity's transmission, reflection and finesse as a function of the membrane's position along the cavity axis and as a function of its optical loss. We compare these calculations with measurements and find excellent agreement in cavities with emptycavity finesses in the range 10(4)10(5). The imaginary part of the membrane's index of refraction is found to be similar to 10(4). We calculate the laser cooling performance of this system, with a particular focus on the lessintuitive regime in which photons 'tunnel' through the membrane on a timescale comparable to the membrane's period of oscillation. Lastly, we present calculations of quantum nondemolition measurements of the membrane's phonon number in the low signaltonoise regime where the phonon lifetime is comparable to the QND readout time.
Decoherence by quantum telegraph noise: A numerical evaluation
Benjamin Abel, Florian Marquardt
Physical Review B
78(20)
201302
(2008)

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We investigate the time evolution of a charge qubit subject to quantum telegraph noise produced by a single electronic defect level. We obtain results for the time evolution of the coherence that are strikingly different from the usual case of a harmonicoscillator bath (Gaussian noise). When the coupling strength crosses a certain temperaturedependent threshold, we observe coherence oscillations in the strongcoupling regime. Moreover, we present the time evolution of the echo signal in a spinecho experiment. Our analysis relies on a numerical evaluation of the exact solution for the density matrix of the qubit.
Backaction evasion and squeezing of a mechanical resonator using a cavity detector
A. A. Clerk, F. Marquardt, K. Jacobs
New Journal of Physics
10
095010
(2008)

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We study the quantum measurement of a cantilever using a parametrically coupled electromagnetic cavity which is driven at the two sidebands corresponding to the mechanical motion. This scheme, originally due to Braginsky et al (Braginsky V, Vorontsov Y I and Thorne K P 1980 Science 209 547), allows a backaction free measurement of one quadrature of the cantilever's motion, and hence the possibility of generating a squeezed state. We present a complete quantum theory of this system, and derive simple conditions on when the quantum limit on the added noise can be surpassed. We also study the conditional dynamics of the measurement, and discuss how such a scheme (when coupled with feedback) can be used to generate and detect squeezed states of the oscillator. Our results are relevant to experiments in optomechanics, and to experiments in quantum electromechanics employing stripline resonators coupled to mechanical resonators.
Measuring the size of a quantum superposition of manybody states
Florian Marquardt, Benjamin Abel, Jan von Delft
Physical Review A
78(1)
012109
(2008)

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We propose a measure for the "size" of a quantum superposition of two manybody states with (supposedly) macroscopically distinct properties by counting how many singleparticle operations are needed to map one state onto the other. This definition gives sensible results for simple, analytically tractable cases and is consistent with a previous definition restricted to GreenbergerHorneZeilingerlike states. We apply our measure to the experimentally relevant, nontrivial example of a superconducting threejunction flux qubit put into a superposition of left and rightcirculating supercurrent states, and we find the size of this superposition to be surprisingly small.
Optomechanical setups use radiation pressure to manipulate macroscopic mechanical objects. Two experiments transfer this concept to the fields of superconducting microwave circuits and coldatom physics.
Dephasing by electronelectron interactions in a ballistic MachZehnder interferometer
Clemens Neuenhahn, Florian Marquardt
New Journal of Physics
10
115018
(2008)

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We consider a ballistic MachZehnder interferometer for electrons propagating chirally in one dimension (such as in an integer quantum Hall effect edge channel). In such a system, dephasing occurs when the finite range of the interaction potential is taken into account. Using the tools of bosonization, we discuss the decay of coherence as a function of propagation distance and energy. We supplement the exact solution by a semiclassical approach that is physically transparent and is exact at high energies. In particular, we study in more detail the recently predicted universal powerlaw decay of the coherence at high energies, where the exponent does not depend on the interaction strength. In addition, we compare against Keldysh perturbation theory, which works well for small interaction strength at short propagation distances.
Strong dispersive coupling of a highfinesse cavity to a micromechanical membrane
J. D. Thompson, B. M. Zwickl, A. M. Jayich, Florian Marquardt, S. M. Girvin, J. G. E. Harris
Macroscopic mechanical objects and electromagnetic degrees of freedom can couple to each other through radiation pressure. Optomechanical systems in which this coupling is sufficiently strong are predicted to show quantum effects and are a topic of considerable interest. Devices in this regime would offer new types of control over the quantum state of both light and matter(14), and would provide a new arena in which to explore the boundary between quantum and classical physics(57). Experiments so far have achieved sufficient optomechanical coupling to laser cool mechanical devices(812), but have not yet reached the quantum regime. The outstanding technical challenge in this field is integrating sensitive micromechanical elements ( which must be small, light and flexible) into high finesse cavities ( which are typically rigid and massive) without compromising the mechanical or optical properties of either. A second, and more fundamental, challenge is to read out the mechanical element's energy eigenstate. Displacement measurements ( no matter how sensitive) cannot determine an oscillator's energy eigenstate(13), and measurements coupling to quantities other than displacement(1416) have been difficult to realize in practice. Here we present an optomechanical system that has the potential to resolve both of these challenges. We demonstrate a cavity which is detuned by the motion of a 50nm thick dielectric membrane placed between two macroscopic, rigid, high finesse mirrors. This approach segregates optical and mechanical functionality to physically distinct structures and avoids compromising either. It also allows for direct measurement of the square of the membrane's displacement, and thus in principle the membrane's energy eigenstate. We estimate that it should be practical to use this scheme to observe quantum jumps of a mechanical system, an important goal in the field of quantum measurement.
2007
Decoherence in weak localization. II. BetheSalpeter calculation of the cooperon
Jan von Delft, Florian Marquardt, R. A. Smith, Vinay Ambegaokar
Physical Review B
76(19)
195332
(2007)

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This is the second in a series of two papers (Papers I and II) on the problem of decoherence in weak localization. In Paper I, we discussed how the Pauli principle could be incorporated into an influence functional approach for calculating the cooperon propagator and the magnetoconductivity. In the present paper, we check and confirm the results so obtained by diagrammatically setting up a BetheSalpeter equation for the cooperon, which includes selfenergy and vertex terms on an equal footing and is free from both infrared and ultraviolet divergences. We then approximately solve this BetheSalpeter equation by the ansatz (C) over bar (t)=(C) over bar (0)(t)e(F(t)), where the decay function F(t) determines the decoherence rate. We show that in order to obtain a divergencefree expression for the decay function F(t), it is sufficient to calculate (C) over bar (1)(t), the cooperon in the positiontime representation to first order in the interaction. Paper II is independent of Paper I and can be read without detailed knowledge of the latter.
Quantum theory of cavityassisted sideband cooling of mechanical motion
Florian Marquardt, Joe P. Chen, A. A. Clerk, S. M. Girvin
Physical Review Letters
99(9)
093902
(2007)

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We present a quantummechanical theory of the cooling of a cantilever coupled via radiation pressure to an illuminated optical cavity. Applying the quantum noise approach to the fluctuations of the radiation pressure force, we derive the optomechanical cooling rate and the minimum achievable phonon number. We find that reaching the quantum limit of arbitrarily small phonon numbers requires going into the goodcavity (resolved phonon sideband) regime where the cavity linewidth is much smaller than the mechanical frequency and the corresponding cavity detuning. This is in contrast to the common assumption that the mechanical frequency and the cavity detuning should be comparable to the cavity damping.
Efficient onchip source of microwave photon pairs in superconducting circuit QED
Florian Marquardt
Physical Review B
76(20)
205416
(2007)

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We describe a scheme for the efficient generation of microwave photon pairs by parametric downconversion in a superconducting transmission line resonator coupled to a Cooperpair box serving as an artificial atom. By properly tuning the first three levels with respect to the cavity modes, the downconversion probability may reach the percentage level at good fidelity. We show this by numerically simulating the dissipative quantum dynamics of the coupled cavitybox system and discussing the effects of dephasing and relaxation in the solid state environment. The setup analyzed here might form the basis for a future onchip source of entangled microwave photons, e.g., using Franson's idea of energytime entanglement.
Controlled dephasing of electrons by nongaussian shot noise
Izhar Neder, Florian Marquardt, Moty Heiblum, Diana Mahalu, Vladimir Umansky
In a 'controlled dephasing' experiment, an interferometer loses its coherence owing to entanglement of the interfering electron with a controlled quantum system, which effectively is equivalent to path detection. In previous experiments, only partial dephasing was achieved owing to weak interactions between many detector electrons and the interfering electron, leading to a gaussianphase randomizing process. Here, we report the opposite extreme, where interference is completely destroyed by a few (that is, one to three) detector electrons, each of which has a strong randomizing effect on the phase. We observe quenching of the interference pattern in a periodic, lobetype fashion as the detector current is varied, and with a peculiar Vshaped dependence on the detector's partitioning. We ascribe these features to the nongaussian nature of the noise, which is also important for qubit decoherence. In other words, the interference seems to be highly sensitive to the full counting statistics of the detector's shot noise.
Selfconsistent calculation of the electron distribution near a quantum point contact in the integer quantum Hall effect
A. Siddiki, F. Marquardt
Physical Review B
75(4)
045325
(2007)

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In this work we implement the selfconsistent ThomasFermiPoisson approach to a homogeneous twodimensional electron system. We compute the electrostatic potential produced inside a semiconductor structure by a quantum point contact (QPC) placed at the surface of the semiconductor and biased with appropriate voltages. The model is based on a semianalytical solution of the Laplace equation. Starting from the calculated confining potential, the selfconsistent (screened) potential and the electron densities are calculated for finite temperature and magnetic field. We observe that there are mainly three characteristic rearrangements of the incompressible edge states which will determine the current distribution near a QPC.
Decoherence in weak localization. I. Pauli principle in influence functional
Florian Marquardt, Jan von Delft, R. A. Smith, Vinay Ambegaokar
Physical Review B
76(19)
195331
(2007)

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This is the first in a series of two papers, in which we revisit the problem of decoherence in weak localization. The basic challenge addressed in our work is to calculate the decoherence of electrons interacting with a quantummechanical environment while taking proper account of the Pauli principle. First, we review the usual influence functional approach valid for decoherence of electrons due to classical noise, showing along the way how the quantitative accuracy can be improved by properly averaging over closed (rather than unrestricted) random walks. We then use a heuristic approach to show how the Pauli principle may be incorporated into a pathintegral description of decoherence in weak localization. This is accomplished by introducing an effective modification of the quantum noise spectrum, after which the calculation proceeds analogous to the case of classical noise. Using this simple but efficient method, which is consistent with much more laborious diagrammatic calculations, we demonstrate how the Pauli principle serves to suppress the decohering effects of quantum fluctuations of the environment, and essentially confirm the classic result of Altshuler, Aronov, and Khmelnitskii [J. Phys. C 15, 7367 (1982)] for the energyaveraged decoherence rate, which vanishes at zero temperature. Going beyond that, we employ our method to calculate explicitly the leading quantum corrections to the classical decoherence rates and to provide a detailed analysis of the energy dependence of the decoherence rate. The basic idea of our approach is general enough to be applicable to the decoherence of degenerate Fermi systems in contexts other than weak localization as well. Paper II will provide a more rigorous diagrammatic basis for our results by rederiving them from a BetheSalpeter equation for the Cooperon.
Coherence oscillations in dephasing by nonGaussian shot noise
Izhar Neder, Florian Marquardt
New Journal of Physics
9
112
(2007)

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A nonperturbative treatment is developed for the dephasing produced by the shot noise of a onedimensional electron channel. It is applied to two systems: a charge qubit and the electronic MachZehnder interferometer (MZI), both of them interacting with an adjacent partitioned electronic channel acting as a detector. We find that the visibility (interference contrast) can display oscillations as a function of detector voltage and interaction time. This is a unique consequence of the nonGaussian properties of the shot noise, and only occurs in the strong coupling regime, when the phase contributed by a single electron exceeds p. The resulting formula reproduces the recent surprising experimental observations reported in (I Neder et al 2006 Preprint condmat/0610634), and indicates a general explanation for similar visibility oscillations observed earlier in the MZI at large bias voltage. We explore in detail the full pattern of oscillations as a function of coupling strength, voltage and time, which might be observable in future experiments.
2006
Correlationinduced resonances in transport through coupled quantum dots
V Meden, F Marquardt
Physical Review Letters
96(14)
146801
(2006)

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We investigate the effect of local electron correlations on transport through parallel quantum dots. The linear conductance as a function of gate voltage is strongly affected by the interplay of the interaction U and quantum interference. We find a pair of novel correlationinduced resonances separated by an energy scale that depends exponentially on U. The effect is robust against a small detuning of the dot energy levels and occurs for arbitrary generic tunnel couplings. It should be observable in experiments on the basis of presently existing doubledot setups.
Equations of motion approach to decoherence and current noise in ballistic interferometers coupled to a quantum bath
Florian Marquardt
Physical Review B
74(12)
125319
(2006)

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We present a technique for treating many particles moving inside a ballistic interferometer, under the influence of a quantummechanical environment (phonons, photons, Nyquist noise, etc.). Our approach is based on solving the coupled Heisenberg equations of motion of the manyparticle system and the bath, and it is inspired by the quantum Langevin method known for the CaldeiraLeggett model. As a first application, we treat a fermionic MachZehnder interferometer. In particular, we discuss the dephasing rate and present full analytical expressions for the leading corrections to the current noise, brought about by the coupling to the quantum bath. In contrast to a singleparticle model, both the Pauli principle as well as the contribution of holescattering processes become important, and are automatically taken into account in this method.
Dynamical multistability induced by radiation pressure in highfinesse micromechanical optical cavities
F Marquardt, JGE Harris, SM Girvin
Physical Review Letters
96(10)
103901
(2006)

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We analyze the nonlinear dynamics of a highfinesse optical cavity in which one mirror is mounted on a flexible mechanical element. We find that this system is governed by an array of dynamical attractors, which arise from phase locking between the mechanical oscillations of the mirror and the ringing of the light intensity in the cavity. We develop an analytical theory to map out the diagram of attractors in parameter space, derive the slow amplitude dynamics of the system, including thermal fluctuations, and suggest a scheme for exploiting the dynamical multistability in the measurement of small displacements.
2005
Fermionic MachZehnder interferometer subject to a quantum bath
We study fermions in a MachZehnder interferometer, subject to a quantummechanical environment leading to inelastic scattering, decoherence, renormalization effects, and timedependent conductance fluctuations. We present a method to derive both the loss of interference contrast as well as the shot noise, using equations of motion and leadingorder perturbation theory. The dependence of the shot noise on the AharonovBohm phase acquires an unexpected average phase shift, due to correlations between the fluctuating renormalized phase shift and the output current. We discuss the limiting behaviours at low and high voltages, compare with simpler models of dephasing, and present implications for experiments.
Spin relaxation in a quantum dot due to Nyquist noise
F Marquardt, VA Abalmassov
Physical Review B
71(16)
165325
(2005)

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We calculate electron and nuclear spin relaxation rates in a quantum dot due to the combined action of Nyquist noise and electronnuclei hyperfine or spinorbit interactions. The relaxation rate is linear in the resistance of the gate circuit and, in the case of spinorbit interaction, it depends essentially on the orientations of both the static magnetic field and the fluctuating electric field, as well as on the ratio between Rashba and Dresselhaus interaction constants. We provide numerical estimates of the relaxation rate for typical system parameters, compare our results with other, previously discussed mechanisms, and show that the Nyquist mechanism can have an appreciable effect for experimentally relevant systems.
Manyfermion generalization of the CaldeiraLeggett model
F Marquardt, D S Golubev
Physical Review A
72(2)
022113
(2005)

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We analyze a model system of fermions in a harmonic oscillator potential under the influence of a dissipative environment: The fermions are subject to a fluctuating force deriving from a bath of harmonic oscillators. This represents an extension of the wellknown CaldeiraLeggett model to the case of many fermions. Using the method of bosonization, we calculate one and twoparticle Green's functions of the fermions. We discuss the relaxation of a single extra particle added above the Fermi sea, considering also dephasing of a particle added in a coherent superposition of states. The consequences of the separation of centerofmass and relative motion, the Pauli principle, and the bathinduced effective interaction are discussed. Finally, we extend our analysis to a more generic coupling between system and bath, which results in complete thermalization of the system.
2004
Relaxation and dephasing in a manyfermion generalization of the CaldeiraLeggett model
F Marquardt, D S Golubev
Physical Review Letters
93(13)
130404
(2004)

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We analyze a model system of fermions in a harmonic oscillator potential under the influence of a fluctuating force generated by a bath of harmonic oscillators. This represents an extension of the wellknown CaldeiraLeggett model to the case of many fermions. Using the method of bosonization, we calculate Green's functions and discuss relaxation and dephasing of a single extra particle added above the Fermi sea. We also extend our analysis to a more generic coupling between system and bath that results in complete thermalization of the system.
Electronnuclei spin relaxation through phononassisted hyperfine interaction in a quantum dot
V A Abalmassov, F Marquardt
Physical Review B
70(7)
075313
(2004)

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We investigate the inelastic spinflip rate for electrons in a quantum dot due to their contact hyperfine interaction with lattice nuclei. In contrast to other works, we obtain a spinphonon coupling term from this interaction by taking directly into account the motion of nuclei in the vibrating lattice. In the calculation of the transition rate the interference of first and second orders of perturbation theory turns out to be essential. It leads to a suppression of relaxation at long phonon wavelengths, when the confining potential moves together with the nuclei embedded in the lattice. At higher frequencies (or for a fixed confining potential), the zerotemperature rate is proportional to the frequency of the emitted phonon. We address both the transition between Zeeman sublevels of a single electron ground state as well as the tripletsinglet transition, and we provide numerical estimates for realistic system parameters. The mechanism turns out to be less efficient than electronnuclei spin relaxation involving piezoelectric electronphonon coupling in a GaAs quantum dot.
Effects of dephasing on shot noise in an electronic MachZehnder interferometer
F Marquardt, C Bruder
Physical Review B
70(12)
125305
(2004)

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We present a theoretical study of the influence of dephasing on shot noise in an electronic MachZehnder interferometer. In contrast to phenomenological approaches, we employ a microscopic model where dephasing is induced by the fluctuations of a classical potential. This enables us to treat the influence of the environment's fluctuation spectrum on the shot noise. We compare against the results obtained from a simple classical model of incoherent transport, as well as those derived from the phenomenological dephasing terminal approach, arguing that the latter runs into a problem when applied to shotnoise calculations for interferometer geometries. From our model, we find two different limiting regimes: If the fluctuations are slow as compared to the time scales set by voltage and temperature, the usual partition noise expression T(1T ) is averaged over the fluctuating phase difference. For the case of "fast" fluctuations, it is replaced by a more complicated expression involving an average over transmission amplitudes. The full current noise also contains other contributions, and we provide a general formula, as well as explicit expressions and plots for specific examples.
Perturbative corrections to the Gutzwiller meanfield solution of the MottHubbard model
C Schroll, F Marquardt, C Bruder
Physical Review A
70(5)
053609
(2004)

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We study the Mottinsulator transition of bosonic atoms in optical lattices. Using perturbation theory, we analyze the deviations from the meanfield Gutzwiller ansatz, which become appreciable for intermediate values of the ratio between hopping amplitude and interaction energy. We discuss corrections to number fluctuations, order parameter, and compressibility. In particular, we improve the description of the shortrange correlations in the oneparticle density matrix. These corrections are important for experimentally observed expansion patterns, both for bulk lattices and in a confining trap potential.
Influence of dephasing on shot noise in an electronic MachZehnder interferometer
F Marquardt, C Bruder
Physical Review Letters
92(5)
056805
(2004)

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We analyze shot noise under the influence of dephasing in an electronic MachZehnder interferometer, of the type that was realized recently [Yang Ji et al., Nature (London) 422, 415 (2003)]. Using a model of dephasing by a fluctuating classical field, we show how the usual partition noise expression T(1T) is modified. We study the dependence on the power spectrum of the field, which is impossible in simpler approaches such as the dephasing terminal, against which we compare. We remark on shot noise as a tool to distinguish thermal smearing from genuine dephasing.
2003
Dephasing in sequential tunneling through a doubledot interferometer
F Marquardt, C Bruder
Physical Review B
68(19)
195305
(2003)

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We analyze dephasing in a model system where electrons tunnel sequentially through a symmetric interference setup consisting of two singlelevel quantum dots. Depending on the phase difference between the two tunneling paths, this may result in perfect destructive interference. However, if the dots are coupled to a bath, it may act as a whichway detector, leading to partial suppression of the phase coherence and the reappearance of a finite tunneling current. In our approach, the tunneling is treated in leading order whereas coupling to the bath is kept to all orders [using P(E) theory]. We discuss the influence of different bath spectra on the visibility of the interference pattern, including the distinction between "mere renormalization effects" and "true dephasing."
2002
Visibility of the AharonovBohm effect in a ring coupled to a fluctuating magnetic flux (vol 126, pg 1325, 2002)
F Marquardt, C Bruder
Journal of Low Temperature Physics
128(34)
UNSP 00222291/02/08000163/0
163163
(2002)
CORRECTION
NonMarkoffian effects of a simple nonlinear bath
H Gassmann, F Marquardt, C Bruder
Physical Review E
66(4)
041111
(2002)

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We analyze a model of a nonlinear bath consisting of a single twolevel system coupled to a linear bath (a classical noise force in the limit considered here). This allows us to study the effects of a nonlinear, nonMarkoffian bath in a particularly simple situation. We analyze the effects of this bath onto the dynamics of a spin by calculating the decay of the equilibrium correlator of the zcomponent of the spin. The exact results are compared with those obtained using three commonly used approximations: a Markoffian master equation for the spin dynamics, a weakcoupling approximation, and the substitution of a linear bath for the original nonlinear bath.
Separation quality of a geometric ratchet
C Keller, F Marquardt, C Bruder
Physical Review E
65(4)
041927
(2002)

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We consider an experimentally relevant model of a geometric ratchet in which particles undergo drift and diffusive motion in a twodimensional periodic array of obstacles, and which is used for the continuous separation of particles subject to different forces. The macroscopic drift velocity and diffusion tensor are calculated by a Monte Carlo simulation and by a masterequation approach, using the corresponding microscopic quantities and the shape of the obstacles as input. We define a measure of separation quality and investigate its dependence on the applied force and the shape of the obstacles.
AharonovBohm ring with fluctuating flux
F Marquardt, C Bruder
Physical Review B
65(12)
125315
(2002)

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We consider a noninteracting system of electrons on a clean onechannel AharonovBohm ring that is threaded by a fluctuating magnetic flux. The flux derives from a CaldeiraLeggett bath of harmonic oscillators. We address the influence of the bath on the following properties: one and twoparticle Green's functions, dephasing, persistent current, and visibility of the AharonovBohm effect in cotunneling transport through the ring. For the bath spectra considered here (including Nyquist noise of an external coil), we find no dephasing in the linear transport regime at zero temperature.
Visibility of the AharonovBohm effect in a ring coupled to a fluctuating magnetic flux
F Marquardt, C Bruder
Journal of Low Temperature Physics
126(34)
13251337
(2002)

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We consider the visibility of the AharonovBohm effect for cotunneling transport through a clean onechannel ring coupled to a fluctuating magnetic flux. We concentrate on the modification of the destructive interference at Phi(0)/2 by the fluctuating flux, since changes in the magnitude of the current away from this point can also be caused by renormalization effects and do not necessarily indicate dephasing. For fluctuations arising from the Nyquist noise in an external coil at T = 0, the suppression of the destructive interference shows up only in a contribution proportional to V3, and therefore does not affect the linear conductance. In this sense, the Nyquist bath does not lead to dephasing in the linear transport regime at zero temperature in our model.
2001
Superposition of two mesoscopically distinct quantum states: Coupling a Cooperpair box to a large superconducting island
F Marquardt, C Bruder
Physical Review B
63(5)
054514
(2001)

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We consider a system of two superconducting islands, each of which is coupled to a bulk superconductor via Josephson tunneling. One of the islands represents a "Cooperpair box," i.e., it is an effective twolevel system. The other island has a smaller charging energy and approximates a harmonic oscillator. A capacitive interaction between the islands results in a dependence of the oscillator frequency on the quantum state of the box. Placing the latter in a coherent superposition of its eigenstates and exciting coherent oscillations in the large island will lead to a phase shift of these oscillations depending on the box quantum state, thereby producing a coherent superposition of two "mesoscopically distinct" quantum states in the large island.
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