Machine Learning and Quantum Devices

Florian Marquardt

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 back-propagation, 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.

Microsphere kinematics from the polarization of tightly focused nonseparable light

Stefan Berg-Johansen, Martin Neugebauer, Andrea Aiello, Gerd Leuchs, Peter Banzer, Christoph Marquardt

Recently, it was shown that vector beams can be utilized for fast kinematic sensing via measurements of their global polarization state [Optica 2(10), 864 (2015)]. The method relies on correlations between the spatial and polarization degrees of freedom of the illuminating field which result from its nonseparable mode structure. Here, we extend the method to the nonparaxial regime. We study experimentally and theoretically the far-field polarization state generated by the scattering of a dielectric microsphere in a tightly focused vector beam as a function of the particle position. Using polarization measurements only, we demonstrate position sensing of a Mie particle in three dimensions. Our work extends the concept of back focal plane interferometry and highlights the potential of polarization analysis in optical tweezers employing structured light.

Oscillating bound states for a giant atom

Lingzhen Guo, Anton Frisk Kockum, Florian Marquardt, Göran Johannson

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 one-dimensional waveguide. In the non-Markovian 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.

Many-body dephasing in a trapped-ion quantum simulator

Harvey B. Kaplan, Lingzhen Guo, Wen Lin Tan, Arinjoy De, Florian Marquardt, Guido Pagano, Christopher Monroe

How a closed interacting quantum many-body 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 long-range interacting transverse-field Ising Hamiltonian realized with a trapped-ion quantum simulator. We measure the temporal fluctuations in the average magnetization of a finite-size system of spin-1/2 particles. We experiment in a regime where the properties of the system are closely related to the integrable Hamiltonian with global spin-spin coupling, which enables analytical predictions for the long-time 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 many-body dephasing.

Observation of concentrating paraxial beams

Andrea Aiello, Martin Paúr, Bohumil Stoklasa, Zdeněk Hradil, Jaroslav Řeháček, Luis L Sánchez-Soto

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 two-dimensional 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.

Topological phonon transport in an optomechanical system

Hengjiang Ren, Tirth Shah, Hannes Pfeifer, Christian Brendel, Vittorio Peano, Florian Marquardt, Oskar Painter

Recent advances in cavity-optomechanics 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 small-scale optomechanical circuits capable of on-chip 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 cavity-optomechanical elements. Using sensitive, spatially resolved optical read-out 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 non-reciprocal elements may be implemented. Owing to the broadband character of the topological channels, the control of the flow of heat-carrying phonons, albeit at cryogenic temperatures, may also be envisioned.

Probing the Tavis-Cummings level splitting with intermediate-scale superconducting circuits

Ping Yang, Jan David Brehm, Juha Leppäkangas, Lingzhen Guo, Michael Marthaler, Isabella Boventer, Alexander Stehli, Tim Wolz, Alexey V. Ustinov, et al.

We demonstrate the local control of up to eight two-level systems interacting strongly with a microwave cavity. Following calibration, the frequency of each individual two-level 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 Tavis-Cummings 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 off-resonance shifts of sub-components. Our work demonstrates control and readout of quantum coherent mesoscopic multi-qubit system of intermediate scale under conditions of noise.

Kinetics of Many-Body Reservoir Engineering

Hugo Ribeiro, Florian Marquardt

Recent advances illustrate the power of reservoir engineering in applications to many-body sys-tems, 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 long-time behavior of many particles coupled to an engineered reservoir in a number-conservingway. For the example of a bosonic array, we show that the non-equilibrium steady state can beexpressed, in a wide parameter regime, in terms of a modified Bose-Einstein distribution with anenergy-dependent temperature.

Condensed matter physics in time crystals

Lingzhen Guo, Pengfei Liang

Time crystals are physical systems whose time translation symmetry is spontaneously broken. Although the spontaneous breaking of continuous time-translation symmetry in static systems is proved impossible for the equilibrium state, the discrete time-translation symmetry in periodically driven (Floquet) systems is allowed to be spontaneously broken, resulting in the so-called 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 many-body physics from the interplay of symmetry-broken 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.

Renormalized Mutual Information for Artificial Scientific Discovery

Leopoldo Sarra, Andrea Aiello, Florian Marquardt

We derive a well-defined 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 low-dimensional effective description of a high-dimensional 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.

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 number-dependent 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 co-optimization of the control parameters, which generates short high-fidelity 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.

Engineering Fast High-Fidelity Quantum Operations With Constrained Interactions

Thales Figueiredo Roque, Aashish A Clerk, Hugo Ribeiro

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 high-efficiencycontrol sequences that are always fully compatible with experimental constraints on available inter-actions and their tunability. Our approach reduces in the end to finding control fields by solvinga set of time-independent linear equations. We illustrate our method by applying it to a numberof physically-relevant problems: the strong-driving limit of a two-level system, fast squeezing in aparametrically driven cavity, the leakage problem in transmon qubit gates, and the acceleration ofSNAP gates in a qubit-cavity system.

Maxwell's lesser demon: A Quantum Engine Driven by Pointer Measurements

Stella Seah, Stefan Nimmrichter, Valerio Scarani

We discuss a self-contained spin-boson model for a measurement-driven engine, in which a demongenerates work from random thermal excitations of a quantum spin via measurement and feedbackcontrol. Instead of granting it full direct access to the spin state and to Landauer’s erasure strokes foroptimal performance, we restrict this lesser demon’s action to pointer measurements, i.e. random orcontinuous interrogations of a damped mechanical oscillator that assumes macroscopically distinctpositions depending on the spin state. The engine could reach simultaneously high output powersand efficiencies and can operate in temperature regimes where quantum Otto engines would fail.

Quench dynamics in one-dimensional optomechanical arrays

Sadegh Raeisi, Florian Marquardt

Non-equilibrium 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 Su-Schrieffer-Heeger model. These ideas can be transferred to other systemssuch as driven nonlinear cavity arrays.

Nonlinear dynamics of weakly dissipative optomechanical systems

Thales Figueiredo Roque, Florian Marquardt, Oleg M. Yevtushenko

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 Neimark-Sacker 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.

Field theory of monochromatic optical beams I. classical fields

Andrea Aiello

We study monochromatic, scalar solutions of the Helmholtz and paraxial wave equations from a field-theoretic point of view. We introduce appropriate time-independent Lagrangian densities for which the Euler-Lagrange 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 energy-momentum tensor and determine the continuity equations relating ``energy'' and ``momentum'' of the fields. Eventually, the reduction of the Helmholtz wave equation to a useful first-order 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

This work is the second part of an investigation aiming at the study of optical wave equations from a field-theoretic 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.

Arbitrary optical wave evolution with Fourier transforms and phase masks

Victor Lopéz-Pastor, 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.

Rapid Exploration of Topological Band Structures using Deep Learning

Vittorio Peano, Florian Sapper, Florian Marquardt

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 unit cell 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 tight-binding model. This allows us to exploit any underlying space group and predict the symmetries of Bloch waves. We demonstrate how that helps to rapidly categorize a large set of geometries in terms of their band representations, identifying designs for fragile topologies. Engineering of domain walls and optimization are also accelerated by orders of magnitude. The approach is general enough to permit future applications to the geometry discovery in other classes of materials (e.g. active and nonlinear metamaterials).

Quantum state transfer via acoustic edge states in a 2D optomechanical array

Marc-Antoine Lemonde, Vittorio Peano, Peter Rabl, Dimitris G Angelakis

We propose a novel hybrid platform where solid-state spin qubits are coupled to the acoustic modes ofa two-dimensional array of optomechanical(OM)nano cavities. Previous studies of coupled OMcavities have shown that in the presence of strong optical drivingfields, the interplay between thephoton-phonon interaction and their respective inter-cavity hopping allows the generation oftopological phases of sound and light. In particular, the mechanical modes can enter a Chern insulatorphase where the time-reversal 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 high-fidelity 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 silicon-vacancy centers couple to the chiral acoustic channel via strain.

Collisional quantum thermometry

Stella Seah, Stefan Nimmrichter, Daniel Grimmer, Jader P. Santos, Valerio Scarini, Gabriel T. Landi

We introduce a general framework for thermometry based on collisional models, where ancillas probe thetemperature of the environment through an intermediary system. This allows for the generation of correlatedancillas even if they are initially independent. Using tools from parameter estimation theory, we show through aminimal qubit model that individual ancillas can already outperform the thermal Cramer-Rao bound. In addition,when probed collectively, these ancillas may exhibit superlinear scalings of the Fisher information, especiallyfor weak system-ancilla interactions. Our approach sets forth the notion of metrology in a sequential interactionssetting, and may inspire further advances in quantum thermometry.

Almost thermal operations: inhomogeneous reservoirs

Angeline Shu, Yu Cai, Stella Seah, Stefan Nimmrichter, Valerio Scarini

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

Error suppression in adiabatic quantum computing with qubit ensembles

Naeimeh Mohseni, Marek Narozniak, Alexey N Pyrkov, Valentin Ivannikov, Jonathan P Dowling

In the standard approach to adiabatic quantum computing (AQC), quantum information storedon qubits are adiabatically evolved to find the ground state of a problem Hamiltonian. Here weinvestigate a variation of AQC where qubit ensembles are used in place of qubits. We identify twodistinct regimes for a given problem Hamiltonian under thismapping as a function of the ensemblesizeN. At a critical ensemble sizeNc, the nature of the first excited state changes from beingmacroscopically distinct spin configuration to a single particle perturbation of the ground state.AboveNcthe minimum gap for large ensembles is well predicted by mean-field theory and theAQC performance improves withN, realizing error-suppression due to duplication of the quantuminformation. While belowNcthe performance is mixed, and can increase withN. For randomlychosen problem instances Nc tends to be smaller than realistic ensemble sizes, hence we expect theensemble version of AQC to work well in a great majority of cases. Our approach shows it is possibleto perform AQC without the necessity of controlling individual qubits, allowing an alternative routetowards implementing AQC.

Accelerated adiabatic quantum gates: optimizing speed versus robustness

Hugo Ribeiro, Aashish A. Clerk

We develop new protocols for high-fidelity 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 shortcuts-to-adiabaticity approaches.

Macroscopicity of quantum mechanical superposition tests via hypothesis falsification

Björn Schrinski, Stefan Nimmrichter, Benjamin A. Stickler, Klaus Hornberger

We establish an objective scheme to determine the macroscopicity of quantum mechanical super-position tests, which is based on the Bayesian hypothesis falsification of macrorealistic modificationsof quantum theory. The measure uses the raw data gathered in an experiment, taking into accountall measurement uncertainties, and can be used to directly assess any conceivable quantum test.We determine the resulting macroscopicity for three recent tests of quantum physics: double-wellinterference of Bose-Einstein condensates, Leggett-Garg tests with atomic random walks, and en-tanglement generation and read-out of nanomechanical oscillators.

Kommt der künstliche Physiker?

Thomas Fösel, Florian Marquardt, Talitha Weiß

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

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 non-Hermitian operators characterized by complex eigenvalues and not normalizable eigenfunctions. We avoid these difficulties using the Kapur-Peierls formalism which enables us to extend the popular Rayleigh-Schrödinger perturbation theory to the case of electromagnetic Debye's potentials describing the light fields inside and outside the near-spherical dielectric object. We find analytical formulas, valid within certain limits, for the deformation-induced first- and second-order 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.

Non-exponential 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 time-delay for the field to propagate across the giant atom, giving rise to non-Markovian dynamics4. We demonstrate the non-Markovian character of the giant atom in the frequency spectrum as well as non-exponential relaxation in the time domain.

Dynamically Generated Synthetic Electric Fields for Photons

Petr Zapletal, Stefan Walter, Florian Marquardt

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 time-dependent vector potential, conventional electromagnetism predicts the generation of an electric field. Here, we show how synthetic electric fields for photons arise self-consistently due to the nonlinear dynamics in a driven system. Our analysis is based on optomechanical arrays, where dynamical gauge fields arise naturally from phonon-assisted photon tunneling. We study open, one-dimensional 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

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 Fano-type 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 Fano-peak 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 bias-current 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 Tavis-Cummings model with modified boundary conditions between the cavity and transmission-line 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.

Chimera states in small disordered optomechanical arrays

Karl Pelka, Vittorio Peano, Andre Xuereb

Synchronization of weakly-coupled non-linear oscillators is a ubiquitous phenomenon that has been observedacross the natural sciences. We study the dynamics of optomechanical arrays—networks of mechanically com-pliant structures that interact with the radiation pressure force—which are driven to self-oscillation. Thesesystems offer a convenient platform to study synchronization phenomena and have potential technological ap-plications. We demonstrate that this system supports the existence of long-lived 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.

Initialisation of single spin dressed states using shortcuts to adiabaticity

Johannes Kölbl, Arne Barfuss, Mark Kasperczyk, Lucas Thiel, Aashish Clerk, Hugo Ribeiro, Patrick Maletinsky

We demonstrate the use of shortcuts to adiabaticity protocols for initialisation, readout, and coherent control of dressed states generated by closed-contour, coherent driving of a single spin. Such dressed states have recently been shown to exhibit efficient coherence protection, beyond what their two-level 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 bi-directionality 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 coherence-protected 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

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.

Cavity optomagnonics with magnetic textures: coupling a magnetic vortex to light

Jasmin Graf, Hannes Pfeifer, Florian Marquardt, Silvia Viola-Kusminskiy

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 cavity-optomagnonic 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.

Reinforcement Learning with Neural Networks for Quantum Feedback

Thomas Fösel, Petru Tighineanu, Talitha Weiss, Florian Marquardt

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 neural-network-based reinforcement learning can be successfully applied in physics. Here, we show how to use this method for finding quantum feedback schemes, where a network-based "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 hardware-adapted feedback schemes for small quantum modules while demonstrating more generally the promise of neural-network based reinforcement learning in physics.

Deterministic generation of hybrid high-N00N states with Rydberg ions trapped in microwave cavities

Naeimeh Mohseni, Shahpoor Saeidian, Jonathan P Dowling, Carlos Navarrete-Benlloch

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 high-precision quantum metrology will require the use of entangled states of several degrees of freedom.Here we propose a protocol capable of generating high-N00N states where the entanglement is sharedbetween the motion of a trapped ion and an electromagnetic cavity mode, a so-called ‘hybrid’configuration. We prove the feasibility of the proposal in a platform consisting of a trapped ionexcited to its circular-Rydberg-state manifold, coupled to the modes of a high-Q 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 state-preparation phase to implement the interferometer required forquantum metrology.

Quantum nondemolition measurement of mechanical motion quanta

Luca Dellantonio, Oleksandr Kyriienko, Florian Marquardt, Anders S. Sørensen

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 single-photon strong-coupling 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 Zero-Phonon Line and the Role of Dimensionality

Petru Tighineanu, C. L. Dreeßen, C. Flindt, P. Lodahl, A. S. Sorensen

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 sub-Kelvin 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 Navarrete-Benlloch

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 Leggett-Garg 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 well-defined 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

We present the basic ingredients of continuum optomechanics, i.e. the suitable extension of cavity-optomechanical concepts to the interaction of photons and phonons in an extended waveguide. We introduce a real-space 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 Ruiz-Rivas, Germán J. de Valcarcel, Carlos Navarrete-Benlloch

Type II optical parametric oscillators are amongst the highest-quality sources of quantum-correlated light. In particular, when pumped above threshold, such devices generate a pair of bright orthogonally-polarized beams with strong continuous-variable 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 wave-plate 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 phase-difference 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 real-time 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 amplitude-squeezed state.

Snowflake phononic topological insulator at the nanoscale

Christian Brendel, Vittorio Peano, Oskar Painter, Florian Marquardt

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.

Scalable Ion Trap Architecture for Universal Quantum Computation by Collisions

Pengfei Liang, Lingzhen Guo

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 two-qubit SWAP−−−−−−√ gate is realized by direct collision of two neighboring ions in the merged trap, which induces an effective spin-spin interaction between two ions. We find that the collision-induced spin-spin 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 two-qubit 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.

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

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 whispering-gallery 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 mm-scale and cm-scale drops of liquid He, evaporative cooling of He droplets in vacuum, and coupling to high-quality optical whispering-gallery modes in a wide range of liquids. We find that the combination of these features could result in a device that approaches the single-photon strong-coupling 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 He-4). Alternatively, for a normal fluid drop of He-3, 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

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 Navarrete-Benlloch, Talitha Weiss, Stefan Walter, Germán J. de Valcarcel

The theory of Gaussian quantum fluctuations around classical steady states in nonlinear quantum-optical 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 long-time solution is time dependent, a most prominent example being spontaneous limit-cycle 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 (many-body) systems.

Unraveling beam self-healing

Andrea Aiello, Girish S. Agarwal, Martin Paur, Bohumil Stoklasa, Zdenek Hradil, Jaroslav Rehacek, Pablo de la Hoz, Gerd Leuchs, Luis L. Sanchez-Soto

We show that, contrary to popular belief, diffraction-free 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 self-reconstruction 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 self-reconstruction ability of a beam via a newly established degree of self-healing. 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 Shack-Hartmann wave front reconstruction. We believe that these results can open new avenues in this field. (C) 2017 Optical Society of America

From Kardar-Parisi-Zhang scaling to explosive desynchronization in arrays of limit-cycle oscillators

Roland Lauter, Aditi Mitra, Florian Marquardt

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 Kuramoto-Sakaguchi model, with local couplings between oscillators. For one-dimensional 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 Kardar-Parisi-Zhang (KPZ) model. Moreover, we are able to explain the dynamical transition both in one and two dimensions by connecting it to an apparent finite-time 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 Kuramoto-Sakaguchi model, with local couplings between oscillators. For one-dimensional 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 Kardar-Parisi-Zhang (KPZ) model. Moreover, we are able to explain the dynamical transition both in one and two dimensions by connecting it to an apparent finite-time 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

Optomechanical systems driven by an effective blue-detuned laser can exhibit self-sustained oscillations of the mechanical oscillator. These self-oscillations 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 self-oscillator 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.

Quantum-coherent phase oscillations in synchronization

Talitha Weiss, Stefan Walter, Florian Marquardt

Recently, several studies have investigated synchronization in quantum-mechanical limit-cycle 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 quantum-coherent 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 self-oscillating 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.

Many-Particle Dephasing after a Quench

Thomas Kiendl, Florian Marquardt

After a quench in a quantum many-body system, expectation values tend to relax towards long-time averages. However, temporal fluctuations remain in the long-time 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 "many-particle 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

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 well-suited for a variety of excitation and readout mechanisms, among them optomechanical approaches.

Generalized non-reciprocity 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, radiation-pressure-induced parametric coupling between optical (photon) and mechanical (phonon) excitations may be used to break time-reversal 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 phase-correlated laser light results in a synthetic magnetic flux, which, in combination with dissipative coupling to the mechanical bath, leads to non-reciprocal transport of photons with 35 dB of isolation. Additionally, optical pumping with blue-detuned light manifests as a particle non-conserving interaction between photons and phonons, resulting in directional optical amplification of 12 dB in the isolator through-direction. These results suggest the possibility of using optomechanical circuits to create a more general class of non-reciprocal 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

Optomechanical (OMA) arrays are a promising future platform for studies of transport, many-body 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 photon-phonon 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 two-species setting leads to a non-trivial frequency dependence of the localization length for intermediate laser intensities. This could serve as a convincing evidence of localization in a non-equilibrium dissipative situation.

Linear and angular momenta in tightly focused vortex segmented beams of light

Martin Neugebauer, Andrea Aiello, Peter Banzer

We investigate the linear momentum density of light, which can be decomposed into spin and orbital parts, in the complex three-dimensional 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 charge-dependent relative shift of the energy density and linear momentum density.

Noncritical generation of nonclassical frequency combs via spontaneous rotational symmetry breaking

Carlos Navarrete-Benlloch, Giuseppe Patera, Germán J. de Valcarcel

Synchronously pumped optical parametric oscillators (SPOPOs) are optical cavities driven by mode-locked lasers, and containing a nonlinear crystal capable of down-converting 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.

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

We use a reservoir engineering technique based on two-tone driving to generate and stabilize a quantum squeezed state of a micron-scale mechanical oscillator in a microwave optomechanical system. Using an independent backaction-evading measurement to directly quantify the squeezing, we observe 4.7±0.9  dB of squeezing below the zero-point 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.

Coupled spin-light dynamics in cavity optomagnonics

Silvia Viola-Kusminskiy, Hong X. Tang, Florian Marquardt

Experiments during the past 2 years have shown strong resonant photon-magnon 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 solid-state systems by optical means. In this work we derive the microscopic optomagnonic Hamiltonian. In the linear regime the system reduces to the well-known optomechanical case, with remarkably large coupling. Going beyond that, we study the optically induced nonlinear classical dynamics of a macrospin. In the fast-cavity 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 self-sustained oscillations. When the full dynamics of the system is considered, the system can enter a chaotic regime by successive period doubling of the oscillations.

Topological phase transitions and chiral inelastic transport induced by the squeezing of light

Vittorio Peano, Martin Houde, Christian Brendel, Florian Marquardt, Aashish A. Clerk

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 well-known fermionic single-particle 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 non-trivial 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 particle-conserving 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.

Topological Quantum Fluctuations and Traveling Wave Amplifiers

Vittorio Peano, Martin Houde, Florian Marquardt, Aashish A. Clerk

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 bulk-mode instabilities. We show that one can exploit this to realize a topologically protected, quantum-limited 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.

Entanglement rate for Gaussian continuous variable beams

Zhi Jiao Deng, Steven J. M. Habraken, Florian Marquardt

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 four-mode optomechanical setup that enables the simultaneous up- and down-conversion of photons from a drive laser into entangled photon pairs. This setup is efficient in that both the drive and the optomechanical up- and down-conversion can be fully resonant.

Noise-induced transitions in optomechanical synchronization

Talitha Weiss, Andreas Kronwald, Florian Marquardt

We study how quantum and thermal noise affects synchronization of two optomechanical limit-cycle oscillators. Classically, in the absence of noise, optomechanical systems tend to synchronize either in-phase or anti-phase. 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 classical-to-quantum 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.

Classical dynamical gauge fields in optomechanics

Stefan Walter, Florian Marquardt

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 three-site model where we identify four different regimes of the gauge-field dynamics. Furthermore, we extend the discussion to a two-dimensional lattice. Our proposed scheme could for instance be implemented using optomechanical crystals.

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 zero-point motion. Although the zero-point fluctuations are unavoidable, they can be manipulated. Using microwave frequency radiation pressure, we have manipulated the thermal fluctuations of a micrometer-scale mechanical resonator to produce a stationary quadrature-squeezed state with a minimum variance of 0.80 times that of the ground state. We also performed phase-sensitive, back-action evading measurements of a thermal state squeezed to 1.09 times the zero-point 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.

Optomechanical Dirac physics

M. Schmidt, V. Peano, F. Marquardt

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 Dirac-type 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 photon-phonon polaritons.

Magnon dark modes and gradient memory

Xufeng Zhang, Chang-Ling Zou, Na Zhu, Florian Marquardt, Liang Jiang, Hong X. Tang

Extensive efforts have been expended in developing hybrid quantum systems to overcome the short coherence time of superconducting circuits by introducing the naturally long-lived spin degree of freedom. Among all the possible materials, single-crystal 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 well-defined magnon modes, which allows flexible and precise tuning. Here we demonstrate that by dissipation engineering, a non-Markovian 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 long-lifetime, multimode quantum memories.

Pattern phase diagram for two-dimensional arrays of coupled limit-cycle oscillators

Roland Lauter, Christian Brendel, Steven J. M. Habraken, Florian Marquardt

Arrays of coupled limit-cycle oscillators represent a paradigmatic example for studying synchronization and pattern formation. We find that the full dynamical equations for the phase dynamics of a limit-cycle oscillator array go beyond previously studied Kuramoto-type equations. We analyze the evolution of the phase field in a two-dimensional 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 nano-and optomechanical oscillator arrays, as well as for any array of coupled limit-cycle oscillators that have undergone a Hopf bifurcation. The possible observation in optomechanical arrays is discussed briefly.

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 free-standing 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 wavelength-conversion 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 time-dependent in situ tunability without the need for individual electrical addressing of localized optical modes. (C) 2015 Optical Society of America

Nonlinear Radiation Pressure Dynamics in an Optomechanical Crystal

Alex G. Krause, Jeff T. Hill, Max Ludwig, Amir H. Safavi-Naeini, Jasper Chan, Florian Marquardt, Oskar Painter

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 T-b approximate to 10 K, highly nonlinear driving of mechanical motion is observed via continuous wave optical pumping. Introduction of a time-dependent (modulated) optical pump is used to steer the system towards an otherwise inaccessible dynamically stable attractor in which mechanical self-oscillation occurs for an optical pump red detuned from the cavity resonance. An analytical model incorporating thermo-optic effects due to optical absorption heating is developed and found to accurately predict the measured device behavior.

Topological Phases of Sound and Light

V. Peano, C. Brendel, M. Schmidt, F. Marquardt

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.

Position-Squared Coupling in a Tunable Photonic Crystal Optomechanical Cavity

Taofiq K. Paraiso, Mahmoud Kalaee, Leyun Zang, Hannes Pfeifer, Florian Marquardt, Oskar Painter

We present the design, fabrication, and characterization of a planar silicon photonic crystal cavity in which large position-squared optomechanical coupling is realized. The device consists of a double-slotted photonic crystal structure in which motion of a central beam mode couples to two high-Q 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 position-squared vacuum coupling rate as large as (g) over tilde'/2 pi = 245 Hz is inferred between the optical supermodes and the fundamental in-plane 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.

Intracavity Squeezing Can Enhance Quantum-Limited Optomechanical Position Detection through Deamplification

V. Peano, H. G. L. Schwefel, Ch. Marquardt, F. Marquardt

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 signal-to-noise 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 quantum-limited detectors based on optomechanical devices. Our approach can be straightforwardly extended to quantum nondemolition qubit detection.

Quantum simulation of expanding space-time with tunnel-coupled condensates

Clemens Neuenhahn, Florian Marquardt

We consider two weakly interacting quasi-1D condensates of cold bosonic atoms. It turns out that a time-dependent variation of the tunnel-coupling between those condensates is equivalent to the spatial expansion of a one-dimensional toy-Universe, with regard to the dynamics of the relative phase field. The dynamics of this field is governed by the quantum sine-Gordon 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 sine-Gordon breathers in the relative phase field, seeded by quantum

Cavity optomechanics

Markus Aspelmeyer, Tobias J. Kippenberg, Florian Marquardt

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 radiation-pressure 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 cavity-quantum-optomechanics 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

Niels Loerch, Jiang Qian, Aashish Clerk, Florian Marquardt, Klemens Hammerer

Optomechanical systems can exhibit self-sustained 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 Fokker-Planck 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 single-photon strong-coupling regime, for strong cavity drive and rather large limit-cycle amplitudes. The approach taken here provides a natural starting point for further studies of quantum effects in optomechanics.

Single-site-resolved measurement of the current statistics in optical lattices

Stefan Kessler, Florian Marquardt

At present, great effort is spent on the experimental realization of gauge fields for quantum many-body systems in optical lattices. At the same time, the single-site-resolved detection of individual atoms has become a new powerful experimental tool. We discuss a protocol for the single-site-resolved measurement of the current statistics of quantum many-body systems, which makes use of a bichromatic optical superlattice and single-site 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 on-site interactions for the current pattern and the ground-state symmetry for small two-dimensional lattices with artificial magnetic fields.

Focus on optomechanics

Ivan Favero, Florian Marquardt

We provide a brief overview of the various topics addressed in this 'focus on' collection on optomechanics.

Decoherence in a double-dot Aharonov-Bohm interferometer: Numerical renormalization group study

Bjoern Kubala, David Roosen, Michael Sindel, Walter Hofstetter, Florian Marquardt

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 finite-frequency transport and confirm the restoration of coherence in the dc limit.

Dissipative optomechanical squeezing of light

Andreas Kronwald, Florian Marquardt, Aashish A. Clerk

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.

Cavity Optomechanics Nano- and Micromechanical Resonators Interacting with Light Introduction

Markus Aspelmeyer, Tobias J. Kippenberg, Florian Marquardt

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

Basic Theory of Cavity Optomechanics

Aashish A. Clerk, Florian Marquardt

This chapter provides a brief basic introduction to the theory used to describe cavity-optomechanical 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 self-oscillations and attractor diagram). Finally, wediscuss the quantum theory of optomechanical cooling, using the powerful and versatile quantum noise approach.

Optomechanically Induced Transparency in the Nonlinear Quantum Regime

Andreas Kronwald, Florian Marquardt

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 single-photon coupling strengths significantly smaller than the cavity linewidth.

Full photon statistics of a light beam transmitted through an optomechanical system

Andreas Kronwald, Max Ludwig, Florian Marquardt

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 single-photon strong coupling, this allows us to predict a transition from sub-Poissonian to super-Poissonian 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

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

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 so-called self-trapping 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 single-photon optomechanical coupling strength parameters g are a function of the optical field intensities, in contrast to usual set-ups 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 = 20-500 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 light-matter 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 single-optical mode optomechanical systems, bistabilities are independent of intracavity intensity and can occur for very weak laser driving amplitudes.

Photonic Cavity Synchronization of Nanomechanical Oscillators

Mahmood Bagheri, Menno Poot, Linran Fan, Florian Marquardt, Hong X. Tang

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 steady-state bosonic squeezing via dissipation

Andreas Kronwald, Florian Marquardt, Aashish A. Clerk

We discuss how large amounts of steady-state 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.

Gain-tunable optomechanical cooling in a laser cavity

Li Ge, Sanli Faez, Florian Marquardt, Hakan E. Tuereci

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 Many-Body Dynamics in Optomechanical Arrays

Max Ludwig, Florian Marquardt

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 laser-driven 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 many-body 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 phase-coherent 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.

Creation and dynamics of remote spin-entangled pairs in the expansion of strongly correlated fermions in an optical lattice

Stefan Kessler, Ian P. McCulloch, Florian Marquardt

We consider the nonequilibrium dynamics of an interacting spin-1/2 fermion gas in a one-dimensional optical lattice after switching off the confining potential. In particular, we study the creation and the time evolution of spatially separated, spin-entangled fermionic pairs. The time-dependent density-matrix renormalization group is used to simulate the time evolution and evaluate the two-site 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 time-dependent modulation of the tunnelling amplitude can enhance the production of spin entanglement. Moreover, we discuss the prospects of experimentally observing these phenomena using spin-dependent single-site detection.

Observing the Nonequilibrium Dynamics of the Quantum Transverse-Field Ising Chain in Circuit QED

Oliver Viehmann, Jan von Delft, Florian Marquardt

We show how a quantum Ising spin chain in a time-dependent 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 many-body 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

The effect of Landau-Zener dynamics on phonon lasing

Huaizhi Wu, Georg Heinrich, Florian Marquardt

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 multi-mode photon dynamics that acts back on the mechanics via radiation forces. We show that even for two optical modes Landau-Zener-Stueckelberg oscillations of the light field drastically change the nonlinear attractor diagram of the resulting phonon lasing oscillations. Our findings illustrate the generic effects of Landau-Zener physics on back-action induced self-oscillations.

Optomechanical circuits for nanomechanical continuous variable quantum state processing

Michael Schmidt, Max Ludwig, Florian Marquardt

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.

Thermalization of interacting fermions and delocalization in Fock space

Clemens Neuenhahn, Florian Marquardt

We investigate the onset of "eigenstate thermalization" and the crossover to ergodicity in a system of one-dimensional fermions with increasing interaction. We analyze the fluctuations in the expectation values of most relevant few-body 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.

Stroboscopic observation of quantum many-body dynamics

Stefan Kessler, Andreas Holzner, Ian P. McCulloch, Jan von Delft, Florian Marquardt

Recent experiments have demonstrated single-site resolved observation of cold atoms in optical lattices. Thus, in the future it may be possible to take repeated snapshots of an interacting quantum many-body system during the course of its evolution. Here we address the impact of the resulting quantum (anti-)Zeno physics on the many-body dynamics. We use the time-dependent density-matrix 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 one-dimensional lattice of spinless fermions with nearest-neighbor interactions, we find regimes for which many-particle configurations are stabilized or destabilized, depending on the interaction strength and the time between observations.

Quantum Signatures of the Optomechanical Instability

Jiang Qian, A. A. Clerk, K. Hammerer, Florian Marquardt

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 self-induced oscillations. We show that there are also distinct signatures in the photon-photon 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.

Enhanced Quantum Nonlinearities in a Two-Mode Optomechanical System

Max Ludwig, Amir H. Safavi-Naeini, Oskar Painter, Florian Marquardt

In cavity optomechanics, nanomechanical motion couples to a localized optical mode. The regime of single-photon 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 photon-phonon and the photon-photon 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.

Localized Phase Structures Growing Out of Quantum Fluctuations in a Quench of Tunnel-coupled 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 sine-Gordon 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 sine-Gordon equation. Both depend on the initial relative phase between the condensates, which is considered as a tunable parameter.

Observation of spontaneous Brillouin cooling

Gaurav Bahl, Matthew Tomes, Florian Marquardt, Tal Carmon

Although bolometric- and ponderomotive-induced deflection of device boundaries are widely used for laser cooling, the electrostrictive Brillouin scattering of light from sound was considered an acousto-optical amplification-only process(1-7). It was suggested that cooling could be possible in multi-resonance Brillouin systems(5-8) when phonons experience lower damping than light(8). However, this regime was not accessible in electrostrictive Brillouin systems(1-3,5,6) as backscattering enforces high acoustical frequencies associated with high mechanical damping(1). Recently, forward Brillouin scattering(3) in microcavities(7) has allowed access to low-frequency 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 anti-Stokes 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

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 two-mode cooling of levitated nanospheres in a self-trapping regime. We identify a structure of overlapping, multiple cooling resonances and strong cooling even when one mode is blue-detuned. 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 single-resonance cooling rates.

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 end-mirrors 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 Landau-Zener-Stueckelberg 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 cavity-mediated mechanical coupling. Such a setup would allow the well-developed 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.

Quantum-mechanical theory of optomechanical Brillouin cooling

Matthew Tomes, Florian Marquardt, Gaurav Bahl, Tal Carmon

We analyze how to exploit Brillouin scattering of light from sound for the purpose of cooling optomechanical devices and present a quantum-mechanical 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 anti-Stokes resonance.

Coupled multimode optomechanics in the microwave regime

Georg Heinrich, Florian Marquardt

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

Superradiant Phase Transitions and the Standard Description of Circuit QED

Oliver Viehmann, Jan von Delft, Florian Marquardt

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 no-go theorem for such phase transitions known from cavity QED applies to circuit QED systems as well. We generalize the no-go theorem to the case of (artificial) atoms with many energy levels and thus make it more applicable for realistic cavity or circuit QED systems.

Collective Dynamics in Optomechanical Arrays

Georg Heinrich, Max Ludwig, Jiang Qian, Bjoern Kubala, Florian Marquardt

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 Kuramoto-type model.

QUANTUM MECHANICS The gentle cooling touch of light

Florian Marquardt

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.

Optimal control of circuit quantum electrodynamics in one and two dimensions

R. Fisher, F. Helmer, S. J. Glaser, F. Marquardt, T. Schulte-Herbrueggen

Optimal control can be used to significantly improve multi-qubit 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 two-dimensional (2D) arrays of cavities and qubits. For high-fidelity 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

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.

Introduction to quantum noise, measurement, and amplification

A. A. Clerk, M. H. Devoret, S. M. Girvin, Florian Marquardt, R. J. Schoelkopf

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, optical-quantum 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 linear-response framework. This approach is shown how it relates to the standard Haus-Caves 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.

Single-atom 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 micron-sized 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 well-developed 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 strong-coupling 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

We propose the implementation of fast resonant gates in circuit quantum electrodynamics for quantum information processing. We show how a suitable utilization of three-level superconducting qubits inside a resonator constitutes a key tool to perform diverse two-qubit resonant gates, improving the operation speed when compared to slower dispersive techniques. To illustrate the benefit of resonant two-qubit gates in circuit quantum electrodynamics, we consider the implementation of a two-dimensional cluster state in an array of N x N superconducting qubits by using resonant controlled-phase and one-qubit 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.

ac conductance through an interacting quantum dot

Bjoern Kubala, Florian Marquardt

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 well-known Meir-Wingreen 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 resonant-level model and deriving the result for the lowest order of electron-phonon 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

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 full-counting statistics.

Electron-plasmon scattering in chiral one-dimensional systems with nonlinear dispersion

M. Heyl, S. Kehrein, F. Marquardt, C. Neuenhahn

We investigate systems of spinless one-dimensional chiral fermions realized, e. g., in the arms of electronic Mach-Zehnder 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 high-energy electrons scatter off plasmons (density excitations). This leads to an exponential decay of the single-particle Green's function even at zero temperature with an energy-dependent rate. As a consequence of this electron-plasmon scattering channel, we observe the coherent excitation of a plasmon wave in the wake of a high-energy electron resulting in the buildup of a monochromatic sinusoidal density pattern.

Photon shuttle: Landau-Zener-Stuckelberg dynamics in an optomechanical system

Georg Heinrich, J. G. E. Harris, Florian Marquardt

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 Autler-Townes splitting indicative of Rabi dynamics. For large drive, we show that this system displays Landau-Zener-Stueckelberg dynamics originally known from atomic two-state systems.

Universal Dephasing in a Chiral 1D Interacting Fermion System

Clemens Neuenhahn, Florian Marquardt

We consider dephasing by interactions in a one-dimensional chiral fermion system (e.g., a quantum Hall edge state). For finite-range interactions, we calculate the spatial decay of the Green's function at fixed energy, which sets the contrast in a Mach-Zehnder interferometer. Using a physically transparent semiclassical ansatz, we find a power-law 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.

Dimensional crossover of the dephasing time in disordered mesoscopic rings

M. Treiber, O. M. Yevtushenko, F. Marquardt, J. von Delft, I. V. Lerner

We study dephasing by electron interactions in a small disordered quasi-one-dimensional (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 T-2/3,T-1) to zero-dimensional (0D) behavior (tau(-1)(phi)alpha T-2) as T drops below the Thouless energy. The crossover to 0D, predicted earlier for two-dimensional and three-dimensional systems, has so far eluded experimental observation. The ring geometry holds promise of meeting this long-standing challenge, since the crossover manifests itself not only in the smooth part of the magnetoconductivity but also in the amplitude of Altshuler-Aronov-Spivak oscillations. This allows signatures of dephasing in the ring to be cleanly extracted by filtering out those of the leads.

Quantum nondemolition photon detection in circuit QED and the quantum Zeno effect

Ferdinand Helmer, Matteo Mariantoni, Enrique Solano, Florian Marquardt

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.

Measurement-based synthesis of multiqubit entangled states in superconducting cavity QED

Ferdinand Helmer, Florian Marquardt

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 three-qubit W and Greenberger-Horne-Zeilinger 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.

OPTOMECHANICS

B. Kubala, M. Ludwig, F. Marquardt

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.

Recent progress in open quantum systems: Non-Gaussian noise and decoherence in fermionic systems

Clemens Neuenhahn, B. Kubala, B. Abel, Florian Marquardt

We review our recent contributions to two topics that have become of interest in the field of open, dissipative quantum systems: non-Gaussian noise and decoherence in fermionic systems. Decoherence by non-Gaussian 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, many-body 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 Mach-Zehnder interferometer. This model can be solved employing the technique of bosortization as well as a physically transparent semiclassical method. (C) 2009 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim

Dephasing rate formula in the many-body context

Doron Cohen, Jan von Delft, Florian Marquardt, Yoseph Imry

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 many-fermion 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 strong-coupling case when interactions appreciably modify the spectra. This is relevant for the situation in disordered metals, where screening is important.

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 on-chip planar microwave resonators are arranged in a two-dimensional 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 fault-tolerant quantum-computing architecture. Copyright (C) EPLA, 2009

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

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 high-finesse 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.

Strong dispersive coupling of a high-finesse cavity to a micromechanical membrane (vol 452, pg 72, 2008)

J. D. Thompson, B. M. Zwickl, A. M. Jayich, Florian Marquardt, S. M. Girvin, J. G. E. Harris

Correction

Decoherence by quantum telegraph noise: A numerical evaluation

Benjamin Abel, Florian Marquardt

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 harmonic-oscillator bath (Gaussian noise). When the coupling strength crosses a certain temperature-dependent threshold, we observe coherence oscillations in the strong-coupling regime. Moreover, we present the time evolution of the echo signal in a spin-echo experiment. Our analysis relies on a numerical evaluation of the exact solution for the density matrix of the qubit.

Back-action evasion and squeezing of a mechanical resonator using a cavity detector

A. A. Clerk, F. Marquardt, K. Jacobs

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 back-action 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.

Optomechanics: Push towards the quantum limit

Florian Marquardt

Optomechanical set-ups use radiation pressure to manipulate macroscopic mechanical objects. Two experiments transfer this concept to the fields of superconducting microwave circuits and cold-atom physics.

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

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 empty-cavity 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 less-intuitive 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 non-demolition measurements of the membrane's phonon number in the low signal-to-noise regime where the phonon lifetime is comparable to the QND readout time.

Strong dispersive coupling of a high-finesse 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(1-4), and would provide a new arena in which to explore the boundary between quantum and classical physics(5-7). Experiments so far have achieved sufficient optomechanical coupling to laser- cool mechanical devices(8-12), 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(14-16) 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 50-nm- 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.

Measuring the size of a quantum superposition of many-body states

Florian Marquardt, Benjamin Abel, Jan von Delft

We propose a measure for the "size" of a quantum superposition of two many-body states with (supposedly) macroscopically distinct properties by counting how many single-particle 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 Greenberger-Horne-Zeilinger-like states. We apply our measure to the experimentally relevant, nontrivial example of a superconducting three-junction flux qubit put into a superposition of left- and right-circulating supercurrent states, and we find the size of this superposition to be surprisingly small.

Self-induced oscillations in an optomechanical system driven by bolometric backaction

Constanze Metzger, Max Ludwig, Clemens Neuenhahn, Alexander Ortlieb, Ivan Favero, Khaled Karrai, Florian Marquardt

We have explored the nonlinear dynamics of an optomechanical system consisting of an illuminated Fabry-Perot 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.

Dephasing by electron-electron interactions in a ballistic Mach-Zehnder interferometer

Clemens Neuenhahn, Florian Marquardt

We consider a ballistic Mach-Zehnder 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 power-law 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.

The optomechanical instability in the quantum regime

Max Ludwig, Bjoern Kubala, Florian Marquardt

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 self-sustained 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 quantum-to-classical transition in some detail.

Decoherence in weak localization. I. Pauli principle in influence functional

Florian Marquardt, Jan von Delft, R. A. Smith, Vinay Ambegaokar

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 quantum-mechanical 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 path-integral 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 energy-averaged 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 Bethe-Salpeter equation for the Cooperon.

Self-consistent calculation of the electron distribution near a quantum point contact in the integer quantum Hall effect

A. Siddiki, F. Marquardt

In this work we implement the self-consistent Thomas-Fermi-Poisson approach to a homogeneous two-dimensional 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 self-consistent (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.

Quantum theory of cavity-assisted sideband cooling of mechanical motion

Florian Marquardt, Joe P. Chen, A. A. Clerk, S. M. Girvin

We present a quantum-mechanical 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 good-cavity (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.

Controlled dephasing of electrons by non-gaussian 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 gaussian-phase 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, lobe-type fashion as the detector current is varied, and with a peculiar V-shaped dependence on the detector's partitioning. We ascribe these features to the non-gaussian 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.

Efficient on-chip source of microwave photon pairs in superconducting circuit QED

Florian Marquardt

We describe a scheme for the efficient generation of microwave photon pairs by parametric down-conversion in a superconducting transmission line resonator coupled to a Cooper-pair box serving as an artificial atom. By properly tuning the first three levels with respect to the cavity modes, the down-conversion probability may reach the percentage level at good fidelity. We show this by numerically simulating the dissipative quantum dynamics of the coupled cavity-box 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 on-chip source of entangled microwave photons, e.g., using Franson's idea of energy-time entanglement.

Coherence oscillations in dephasing by non-Gaussian shot noise

Izhar Neder, Florian Marquardt

A non-perturbative treatment is developed for the dephasing produced by the shot noise of a one-dimensional electron channel. It is applied to two systems: a charge qubit and the electronic Mach-Zehnder 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 non-Gaussian 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 cond-mat/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.

Decoherence in weak localization. II. Bethe-Salpeter calculation of the cooperon

Jan von Delft, Florian Marquardt, R. A. Smith, Vinay Ambegaokar

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 Bethe-Salpeter equation for the cooperon, which includes self-energy and vertex terms on an equal footing and is free from both infrared and ultraviolet divergences. We then approximately solve this Bethe-Salpeter 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 divergence-free expression for the decay function F(t), it is sufficient to calculate (C) over bar (1)(t), the cooperon in the position-time representation to first order in the interaction. Paper II is independent of Paper I and can be read without detailed knowledge of the latter.

Equations of motion approach to decoherence and current noise in ballistic interferometers coupled to a quantum bath

Florian Marquardt

We present a technique for treating many particles moving inside a ballistic interferometer, under the influence of a quantum-mechanical environment (phonons, photons, Nyquist noise, etc.). Our approach is based on solving the coupled Heisenberg equations of motion of the many-particle system and the bath, and it is inspired by the quantum Langevin method known for the Caldeira-Leggett model. As a first application, we treat a fermionic Mach-Zehnder 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 single-particle model, both the Pauli principle as well as the contribution of hole-scattering processes become important, and are automatically taken into account in this method.

Dynamical multistability induced by radiation pressure in high-finesse micromechanical optical cavities

F Marquardt, JGE Harris, SM Girvin

We analyze the nonlinear dynamics of a high-finesse 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.

Correlation-induced resonances in transport through coupled quantum dots

V Meden, F Marquardt

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 correlation-induced 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 double-dot setups.

Spin relaxation in a quantum dot due to Nyquist noise

F Marquardt, VA Abalmassov

We calculate electron and nuclear spin relaxation rates in a quantum dot due to the combined action of Nyquist noise and electron-nuclei hyperfine or spin-orbit interactions. The relaxation rate is linear in the resistance of the gate circuit and, in the case of spin-orbit 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.

Fermionic Mach-Zehnder interferometer subject to a quantum bath

F Marquardt

We study fermions in a Mach-Zehnder interferometer, subject to a quantum-mechanical environment leading to inelastic scattering, decoherence, renormalization effects, and time-dependent 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 leading-order perturbation theory. The dependence of the shot noise on the Aharonov-Bohm 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.

Many-fermion generalization of the Caldeira-Leggett model

F Marquardt, DS Golubev

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 well-known Caldeira-Leggett model to the case of many fermions. Using the method of bosonization, we calculate one- and two-particle 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 center-of-mass and relative motion, the Pauli principle, and the bath-induced 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.

Perturbative corrections to the Gutzwiller mean-field solution of the Mott-Hubbard model

C Schroll, F Marquardt, C Bruder

We study the Mott-insulator transition of bosonic atoms in optical lattices. Using perturbation theory, we analyze the deviations from the mean-field 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 short-range correlations in the one-particle density matrix. These corrections are important for experimentally observed expansion patterns, both for bulk lattices and in a confining trap potential.

Effects of dephasing on shot noise in an electronic Mach-Zehnder interferometer

F Marquardt, C Bruder

We present a theoretical study of the influence of dephasing on shot noise in an electronic Mach-Zehnder 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 shot-noise 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(1-T ) 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.

Influence of dephasing on shot noise in an electronic Mach-Zehnder interferometer

F Marquardt, C Bruder

We analyze shot noise under the influence of dephasing in an electronic Mach-Zehnder 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(1-T) 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.

Relaxation and dephasing in a many-fermion generalization of the Caldeira-Leggett model

F Marquardt, DS Golubev

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 well-known Caldeira-Leggett 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.

Electron-nuclei spin relaxation through phonon-assisted hyperfine interaction in a quantum dot

VA Abalmassov, F Marquardt

We investigate the inelastic spin-flip rate for electrons in a quantum dot due to their contact hyperfine interaction with lattice nuclei. In contrast to other works, we obtain a spin-phonon 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 zero-temperature 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 triplet-singlet transition, and we provide numerical estimates for realistic system parameters. The mechanism turns out to be less efficient than electron-nuclei spin relaxation involving piezoelectric electron-phonon coupling in a GaAs quantum dot.

Dephasing in sequential tunneling through a double-dot interferometer

F Marquardt, C Bruder

We analyze dephasing in a model system where electrons tunnel sequentially through a symmetric interference setup consisting of two single-level 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 which-way 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."

Visibility of the Aharonov-Bohm effect in a ring coupled to a fluctuating magnetic flux

F Marquardt, C Bruder

We consider the visibility of the Aharonov-Bohm effect for cotunneling transport through a clean one-channel 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 V-3, 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.

Separation quality of a geometric ratchet

C Keller, F Marquardt, C Bruder

We consider an experimentally relevant model of a geometric ratchet in which particles undergo drift and diffusive motion in a two-dimensional 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 master-equation 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.

Visibility of the Aharonov-Bohm effect in a ring coupled to a fluctuating magnetic flux (vol 126, pg 1325, 2002)

F Marquardt, C Bruder

CORRECTION

Non-Markoffian effects of a simple nonlinear bath

H Gassmann, F Marquardt, C Bruder

We analyze a model of a nonlinear bath consisting of a single two-level 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, non-Markoffian 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 z-component 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 weak-coupling approximation, and the substitution of a linear bath for the original nonlinear bath.

Aharonov-Bohm ring with fluctuating flux

F Marquardt, C Bruder

We consider a noninteracting system of electrons on a clean one-channel Aharonov-Bohm ring that is threaded by a fluctuating magnetic flux. The flux derives from a Caldeira-Leggett bath of harmonic oscillators. We address the influence of the bath on the following properties: one- and two-particle Green's functions, dephasing, persistent current, and visibility of the Aharonov-Bohm 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.

Superposition of two mesoscopically distinct quantum states: Coupling a Cooper-pair box to a large superconducting island

F Marquardt, C Bruder

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 "Cooper-pair box," i.e., it is an effective two-level 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.