Optoacoustic Entanglement in a Continuous Brillouin-Active Solid State System

Changlong Zhu, Claudiu Genes, Birgit Stiller

Entanglement in hybrid quantum systems comprised of fundamentally different degrees of freedom, such as light and mechanics, is of interest for a wide range of applications in quantum technologies. Here, we propose to engineer bipartite entanglement between traveling acoustic phonons in a Brillouin active solid state system and the accompanying light wave. The effect is achieved by applying optical pump pulses to state-of-the-art waveguides, exciting a Brillouin Stokes process. This pulsed approach, in a system operating in a regime orthogonal to standard optomechanical setups, allows for the generation of entangled photon-phonon pairs, resilient to thermal fluctuations. We propose an experimental platform where readout of the optoacoustics entanglement is done by the simultaneous detection of Stokes and anti-Stokes photons in a two-pump configuration. The proposed mechanism presents an important feature in that it does not require initial preparation of the quantum ground state of the phonon mode.

Purcell-modified Doppler cooling of quantum emitters inside optical cavities

J. Lyne, N.S. Bassler, S. Park, G. Pupillo, C. Genes

Standard cavity cooling of atoms or dielectric particles is based on the action of dispersive optical forces in high-finesse cavities. We investigate here a complementary regime characterized by large cavity losses, resembling the standard Doppler cooling technique. For a single two-level emitter a modification of the cooling rate is obtained from the Purcell enhancement of spontaneous emission in the large cooperativity limit. This mechanism is aimed at cooling quantum emitters without closed transitions, which is the case for molecular systems, where the Purcell effect can mitigate the loss of population from the cooling cycle. We extend our analytical formulation to the many-particle case governed by small individual coupling, but exhibiting large collective coupling.

Hybrid architectures for terahertz molecular polaritonics

Ahmed Jaber, Michael Reitz, Avinash Singh, Ali Maleki, Yongbao Xin, Brian T. Sullivan, Ksenia Dolgaleva, Robert W. Boyd, Claudiu Genes, et al.

Atoms and their different arrangements into molecules are nature’s building blocks. In a regime of strong coupling, matter hybridizes with light to modify physical and chemical properties, hence creating new building blocks that can be used for avant-garde technologies. However, this regime relies on the strong confinement of the optical field, which is technically challenging to achieve, especially at terahertz frequencies in the far-infrared region. Here we demonstrate several schemes of electromagnetic field confinement aimed at facilitating the collective coupling of a localized terahertz photonic mode to molecular vibrations. We observe an enhanced vacuum Rabi splitting of 200 GHz from a hybrid cavity architecture consisting of a plasmonic metasurface, coupled to glucose, and interfaced with a planar mirror. This enhanced light-matter interaction is found to emerge from the modified intracavity field of the cavity, leading to an enhanced zero-point electric field amplitude. Our study provides key insight into the design of polaritonic platforms with organic molecules to harvest the unique properties of hybrid light-matter states.

Generalized energy gap law: An open system dynamics approach to non-adiabatic phenomena in molecules

Nico S. Baßler, Michael Reitz, Raphael Holzinger, A. Vibók, G. J. Halász, Burak Gurlek, Claudiu Genes

Non-adiabatic molecular phenomena, arising from the breakdown of the Born-Oppenheimer approximation, govern the fate of virtually all photo-physical and photochemical processes and limit the quantum efficiency of molecules and other solid-state embedded quantum emitters. A simple and elegant description, the energy gap law, was derived five decades ago, predicting that the non-adiabatic coupling between the excited and ground potential landscapes lead to non-radiative decay with a quasi-exponential dependence on the energy gap. We revisit and extend this theory to account for crucial aspects such as vibrational relaxation, dephasing, and radiative loss. We find a closed analytical solution with general validity which indicates a direct proportionality of the non-radiative rate with the vibrational relaxation rate at low temperatures, and with the dephasing rate of the electronic transition at high temperatures. Our work establishes a connection between nanoscale quantum optics, open quantum system dynamics and non-adiabatic molecular physics.

Cooperative effects in dense cold atomic gases including magnetic dipole interactions

Nico S. Baßler, I. Varma, M. Proske, P. Windpassinger, K. P. Schmidt, Claudiu Genes

We theoretically investigate cooperative effects in cold atomic gases exhibiting both electric and magnetic dipole-dipole interactions, such as occurs, for example, in clouds of dysprosium atoms. After introducing a general framework capturing both the quantum degenerate and nondenegerate cases, we focus on the emergence of tailorable spin models in the quantum nondegenerate regime. In the low-excitation limit, we provide analytical and numerical results detailing the effect of magnetic interactions on the directionality of scattered light and characterize sub and superradiant effects.

Scaling Law for Kasha’s Rule in Photoexcited Molecular Aggregates

Raphael Holzinger, Nico S. Baßler, Helmut Ritsch, Claudiu Genes

We study the photophysics of molecular aggregates from a quantum optics perspective, with emphasis on deriving scaling laws for the fast nonradiative relaxation of collective electronic excitations, referred to as Kasha’s rule. Aggregates exhibit an energetically broad manifold of collective states with delocalized electronic excitations originating from near-field dipole–dipole exchanges between neighboring monomers. Photoexcitation at optical wavelengths, much larger than the monomer–monomer average separation, addresses almost exclusively symmetric collective states, which for an arrangement known as H-aggregate show an upward hypsochromic shift. The extremely fast subsequent nonradiative relaxation via intramolecular vibrational modes populates lower energy, subradiant states, resulting in effective inhibition of fluorescence. Our analytical treatment allows for the derivation of an approximate scaling law of this relaxation process, linear in the number of available low-energy vibrational modes and directly proportional to the dipole–dipole interaction strength between neighboring monomers.

Nonlinear optovibronics in molecular systems

Q. Zhang, M. Asjad, Michael Reitz, Christian Sommer, Burak Gurlek, Claudiu Genes

We analytically tackle optovibronic interactions in molecular systems driven by either classical or quantum light fields. In particular, we examine a simple model of molecules with two relevant electronic levels, characterized by potential landscapes with different positions of minima along the internuclear coordinates and of varying curvatures. Such systems exhibit an electron-vibron interaction, which can be composed of linear and quadratic terms in the vibrational displacement. By employing a combination of conditional displacement and squeezing operators, we present analytical expressions based on a quantum Langevin equations approach, to describe the emission and absorption spectra of such nonlinear molecular systems. Furthermore, we examine the imprint of the quadratic interactions onto the transmission properties of a cavity-molecule system within the collective strong-coupling regime of cavity quantum electrodynamics.

Metasurface-Based Hybrid Optical Cavities for Chiral Sensing

Nico S. Baßler, Andrea Aiello, Kai P. Schmidt, Claudiu Genes, Michael Reitz

Quantum metasurfaces, i.e., two-dimensional subwavelength arrays of quantum emitters, can be employed as mirrors towards the design of hybrid cavities, where the optical response is given by the interplay of a cavity-confined field and the surface modes supported by the arrays. We show that stacked layers of quantum metasurfaces with orthogonal dipole orientation can serve as helicity-preserving cavities. These structures exhibit ultranarrow resonances and can enhance the intensity of the incoming field by orders of magnitude, while simultaneously preserving the handedness of the field circulating inside the resonator, as opposed to conventional cavities. The rapid phase shift in the cavity transmission around the resonance can be exploited for the sensitive detection of chiral scatterers passing through the cavity. We discuss possible applications of these resonators as sensors for the discrimination of chiral molecules. Our approach describes a new way of chiral sensing via the measurement of particle-induced phase shifts.

Classical phase synchronization in dissipative non-Hermitian coupled systems

Jonas Rohn, Kai Phillip Schmidt, Claudiu Genes

We study the interplay between non-Hermitian dynamics and classical phase synchronization in a system of N bosonic modes commonly coupled to an auxiliary, driven mode. For any set of non-Hermitian bipartite interactions between the auxiliary and other modes, the system evolves towards a phase synchronized state. We provide analytical and numerical evidence of such classical phase synchronization for systems ranging from a few modes to the macroscopic limit of large N and analyze the effects of inhomogeneous frequency broadening and robustness under the action of external thermal noise.

Theory of phase-adaptive parametric cooling

Alekhya Ghosh, Pardeep Kumar, Christian Sommer, Fidel G. Jimenez, Vivishek Sudhir, Claudiu Genes

We propose an adaptive phase technique for the parametric cooling of mechanical oscillators. Our scheme calls for a sequence of periodic adjustments of the phase of a parametric modulation of the mechanical oscillator that is conditioned on measurements of its two quadratures. The technique indicates an exponential loss of thermal energy at initial high occupancies, similar in performance to other optomechanical techniques such as cold-damping or cavity self-cooling. As the quantum ground state is approached, the phase adaptive scheme leads to residual occupancies at the level of a few phonons owing to the competition between parametric amplification of quantum fluctuations and the feedback action.

Linear optical elements based on cooperative subwavelength emitter arrays

Nico S. Baßler, Michael Reitz, Kai P. Schmidt, Claudiu Genes

We describe applications of two-dimensional subwavelength quantum emitter arrays as efficient optical elements in the linear regime. For normally incident light, the cooperative optical response, stemming from emitter-emitter dipole exchanges, allows the control of the array’s transmission, its resonance frequency, and bandwidth. Operations on fully polarized incident light, such as generic linear and circular polarizers as well as phase retarders can be engineered and described in terms of Jones matrices. Our analytical approach and accompanying numerical simulations identify optimal regimes for such operations and reveal the importance of adjusting the array geometry and of the careful tuning of the external magnetic fields amplitude and direction.<br><br>

Spectral theorem for dummies: A pedagogical discussion on quantum probability and random variable theory

Andrea Aiello

John von Neumann's spectral theorem for self-adjoint operators is a cornerstone of quantum mechanics. Among other things, it also provides a connection between expectation values of self-adjoint operators and expected values of real-valued random variables. This paper presents a plain-spoken formulation of this theorem in terms of Dirac's bra and ket notation, which is based on physical intuition and provides techniques that are important for performing actual calculations. The goal is to engage students in a constructive discussion about similarities and differences in the use of random variables in classical and quantum mechanics. Special emphasis is given on operators that are simple functions of noncommuting self-adjoint operators. The presentation is self-contained and includes detailed calculations for the most relevant results.

Helicity, chirality, and spin of optical fields without vector potentials

Andrea Aiello

Helicity H, chirality C, and spin angular momentum S are three physical observables that play an important role in the study of optical fields. These quantities are closely related, but their connection is hidden by the use of four different vector fields for their representation, namely, the electric and magnetic fields E and B, and the two transverse potential vectors C⊥ and A⊥. Helmholtz's decomposition theorem restricted to solenoidal vector fields entails the introduction of a bona fide inverse curl operator, which permits one to express the above three quantities in terms of the observable electric and magnetic fields only. This yields clear expressions for H,C, and S, which are automatically gauge invariant and display electric-magnetic democracy.

Cooperative subwavelength molecular quantum emitter arrays

Raphael Holzinger, Sue Ann Oh, Michael Reitz, Helmut Ritsch, Claudiu Genes

Dipole-coupled subwavelength quantum emitter arrays respond cooperatively to external light fields as they may host collective delocalized excitations (a form of excitons) with super- or subradiant character. Deeply subwavelength separations typically occur in molecular ensembles, where in addition to photon-electron interactions, electron-vibron couplings and vibrational relaxation processes play an important role. We provide analytical and numerical results on the modification of super- and subradiance in molecular rings of dipoles including excitations of the vibrational degrees of freedom. While vibrations are typically considered detrimental to coherent dynamics, we show that molecular dimers or rings can be operated as platforms for the preparation of long-lived dark superposition states aided by vibrational relaxation. In closed ring configurations, we extend previous predictions for the generation of coherent light from ideal quantum emitters to molecular emitters, quantifying the role of vibronic coupling onto the output intensity and coherence.

A non-separability measure for spatially disjoint vectorial fields

Andrea Aiello, Xiao-Bo Hu, Valeria Rodríguez-Fajardo, Andrew Forbes, Raul I. Hernandez-Aranda, Benjamin Perez-Garcia, Carmelo Rosales-Guzmán

Vectorial forms of structured light that are non-separable in their spatial and polarisation degrees of freedom have become topical of late, with an extensive toolkit for their creation and control. In contrast, the toolkit for quantifying their non-separability, the inhomogeneity of the polarisation structure, is less developed and in some cases fails altogether. To overcome this, here we introduce a new measure for vectorial light, which we demonstrate both theoretically and experimentally. We consider the general case where the local polarisation homogeneity can vary spatially across the field, from scalar to vector, a condition that can arise naturally if the composite scalar fields are path separable during propagation, leading to spatially disjoint vectorial light. We show how the new measure correctly accounts for the local path-like separability of the individual scalar beams, which can have varying degrees of disjointness, even though the global vectorial field remains intact. Our work attempts to address a pressing issue in the analysis of such complex light fields, and raises important questions on spatial coherence in the context of vectorially polarised light.

One more time on the helicity decomposition of spin and orbital optical currents

Andrea Aiello

The helicity representation of the linear momentum density of a light wave is well understood for monochromatic optical fields in both paraxial and non-paraxial regimes of propagation. In this note we generalize such representation to nonmonochromatic optical fields. We find that, differently from the monochromatic case, the linear momentum density, aka the Poynting vector divided by c2, does not separate into the sum of right-handed and left-handed terms, even when the so-called electric–magnetic democracy in enforced by averaging the electric and magnetic contributions. However, for quasimonochromatic light, such a separation is approximately restored after time-averaging. This paper is dedicated to Sir Michael Berry on the occasion of his 80th birthday.

Cooperative quantum phenomena in light-matter platforms

Michael Reitz, Christian Sommer, Claudiu Genes

Quantum cooperativity is evident in light-matter platforms where quantum-emitter ensembles are interfaced with confined optical modes and are coupled via the ubiquitous electromagnetic quantum vacuum. Cooperative effects can find applications, among other areas, in topological quantum optics, in quantum metrology, or in quantum information. This tutorial provides a set of theoretical tools to tackle the behavior responsible for the onset of cooperativity by extending open quantum system dynamics methods, such as the master equation and quantum Langevin equations, to electron-photon interactions in strongly coupled and correlated quantum-emitter ensembles. The methods are illustrated on a wide range of current research topics such as the design of nanoscale coherent-light sources, highly reflective quantum metasurfaces, or low intracavity power superradiant lasers.

A Simple Field Theoretic Description of Single-Photon Nonlocality

Andrea Aiello

We present a simple yet rigorous field theoretic demonstration of the nonlocality of a single-photon field. The formalism used allows us to calculate the electric field of a single-photon light beam sent through a beam splitter, which directly demonstrates that it is the light field, rather than the photon itself regarded as a particle, that exhibits nonlocality. Our results are obtained without using either inequalities or specific measurement apparatuses, so that they have perfectly general validity.

Perturbation theory of nearly spherical dielectric optical resonators

Julius Gohsrich, Tirth Shah, Andrea Aiello

Dielectric spheres of various sizes may sustain electromagnetic whispering-gallery modes resonating at optical frequencies with very narrow linewidths. Arbitrary small deviations from the spherical shape typically shift and broaden such resonances. Our goal is to determine these shifted and broadened resonances. A boundary-condition perturbation theory for the acoustic vibrations of nearly circular membranes was developed by Rayleigh more than a century ago. We extend this theory to describe the electromagnetic excitations of nearly spherical dielectric cavities. This approach permits us to avoid dealing with decaying quasinormal modes. We explicitly find the frequencies and the linewidths of the optical resonances for arbitrarily deformed nearly spherical dielectric cavities, as power series expansions by a small parameter, up to and including second-order terms. We thoroughly discuss the physical conditions for the applicability of perturbation theory.

Molecular polaritonics in dense mesoscopic disordered ensembles

Christian Sommer, Michael Reitz, Francesca Mineo, Claudiu Genes

We study the dependence of the vacuum Rabi splitting (VRS) on frequency disorder, vibrations, near-field effects, and density in molecular polaritonics. In the mesoscopic limit, static frequency disorder alone can already introduce a loss mechanism from polaritonic states into a dark state reservoir, which we quantitatively describe, providing an analytical scaling of the VRS with the level of disorder. Disorder additionally can split a molecular ensemble into donor-type and acceptor-type molecules and the combination of vibronic coupling, dipole-dipole interactions, and vibrational relaxation induces an incoherent FRET (Förster resonance energy transfer) migration of excitations within the collective molecular state. This is equivalent to a dissipative disorder and has the effect of saturating and even reducing the VRS in the mesoscopic, high-density limit. Overall, this analysis allows to quantify the crucial role played by dark states in cavity quantum electrodynamics with mesoscopic, disordered ensembles.

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.

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, 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.

Excitation transport with collective radiative decay

Francesca Mineo, Claudiu Genes

We investigate a one-dimensional quantum emitter chain where transport of excitations and correlations takes place via nearest neighbor, dipole-dipole interactions. In the presence of collective radiative emission, we show that a phase imprinting wavepacket initialization procedure can lead to subradiant transport and can preserve quantum correlations. In the context of cavity mediated transport, where emitters are coupled to a common delocalized optical mode, we analyze the effect of frequency disorder and nonidentical photon-emitter couplings on excitation transport.

Analytical model of the deformation-induced inertial dynamics of a magnetic vortex

Myoung-Woo Yoo, Francesca Mineo, Joo-Von Kim

We present an analytical model to account for the deformation-induced inertial dynamics of a magnetic vortex. The model is based on a deformation of the vortex core profile based on the Döring kinetic field, whereby the deformation amplitudes are promoted to dynamical variables in a collective-coordinate approach that provides a natural extension to the Thiele model. This extended model describes complex transients due to inertial effects and the variation of the effective mass with velocity. The model also provides a quantitative description of the inertial dynamics leading up to vortex core reversal, which is analogous to the Walker transition in domain wall dynamics. Our work paves the way for a standard prescription for describing the inertial effects of topological magnetic solitons

Floquet engineering of molecular dynamics via infrared coupling

Michael Reitz, Claudiu Genes

We discuss Floquet engineering of dissipative molecular systems through periodic driving of an infrared-active vibrational transition, either directly or via a cavity mode. Following a polaron quantum Langevin equations approach, we derive correlation functions and stationary quantities showing strongly modified optical response<br>of the infrared-dressed molecule. The coherent excitation of molecular vibrational modes, in combination with the modulation of electronic degrees of freedom due to vibronic coupling can lead to both enhanced<br>vibronic coherence as well as control over vibrational sideband amplitudes. The additional coupling to an infrared cavity allows for the controlled suppression of undesired sidebands, an effect stemming from the Purcell enhancement of vibrational relaxation rates.

Multimode cold-damping optomechanics with delayed feedback

Christian Sommer, Alekhya Ghosh, Claudiu Genes

We investigate the role of time delay in cold-damping optomechanics with multiple mechanical resonances.<br>For instantaneous electronic response, it was recently shown by C. Sommer and C. Genes [Phys. Rev. Lett. 123,<br>203605 (2019)] that a single feedback loop is sufficient to simultaneously remove thermal noise from many<br>mechanical modes. While the intrinsic delayed response of the electronics can induce single-mode and mutual<br>heating between adjacent modes, we propose to counteract such detrimental effects by introducing an additional<br>time delay to the feedback loop. For lossy cavities and broadband feedback, we derive analytical results for the<br>final occupancies of the mechanical modes within the formalism of quantum Langevin equations. For modes<br>that are frequency degenerate collective effects dominate, mimicking behavior similar to Dicke super- and<br>subradiance. These analytical results, corroborated with numerical simulations of both transient and steady state<br>dynamics, allow us to find suitable conditions and strategies for efficient single-mode or multimode feedback<br>optomechanics.

Molecule-photon interactions in phononic environments

Michael Reitz, Christian Sommer, Burak Gürlek, Vahid Sandoghdar, Diego-Martin Cano, Claudiu Genes

Molecules constitute compact hybrid quantum optical systems that can interface photons, electronic degrees of freedom, localized mechanical vibrations, and phonons. In particular, the strong vibronic interaction between electrons and nuclear motion in a molecule resembles the optomechanical radiation pressure Hamiltonian. While molecular vibrations are often in the ground state even at elevated temperatures, one still needs to get a handle on decoherence channels associated with phonons before an efficient quantum optical network based on optovibrational interactions in solid-state molecular systems could be realized. As a step towards a better understanding of decoherence in phononic environments, we take here an open quantum system approach to the nonequilibrium dynamics of guest molecules embedded in a crystal, identifying regimes of Markovian versus non-Markovian vibrational relaxation. A stochastic treatment, based on quantum Langevin equations, predicts collective vibron-vibron dynamics that resembles processes of sub- and super-radiance for radiative transitions. This in turn leads to the possibility of decoupling intramolecular vibrations from the phononic bath, allowing for enhanced coherence times of collective vibrations. For molecular polaritonics in strongly confined geometries, we also show that the imprint of optovibrational couplings onto the emerging output field results in effective polariton cross-talk rates for finite bath occupancies.

Ising model in a light-induced quantized transverse field

Jonas Rohn, Max Hörmann, Claudiu Genes, Kai Phillip Schmidt

We investigate the influence of light-matter interactions on correlated quantum matter by studying the<br>paradigmatic Dicke-Ising model. This type of coupling to a confined, spatially delocalized bosonic light mode,<br>such as provided by an optical resonator, resembles a quantized transverse magnetic field of tunable strength. As<br>a consequence, the symmetry-broken magnetic state breaks down for strong enough light-matter interactions to<br>a paramagnetic state. The nonlocal character of the bosonic mode can change the quantum phase transition in<br>a drastic manner, which we analyze quantitatively for the simplest case of the Dicke-Ising chain geometry.<br>The results show a direct transition between a magnetically ordered phase with zero photon density and a<br>magnetically polarized phase with superradiant behavior of the light. Our predictions are equally valid for the<br>dual quantized Ising chain in a conventional transverse magnetic field.

Ensemble-induced strong light-matter coupling of a single quantum emitter

Stefan Schütz, Johannes Schachenmayer, David Hagenmüller, Gavin K. Brennen, Thomas Volz, Vahid Sandoghdar, Thomas W. Ebbesen, Claudiu Genes, Guido Pupillo

We discuss a technique to strongly couple a single target quantum emitter to a cavity mode, which is enabled by virtual excitations of a nearby mesoscopic ensemble of emitters. A collective coupling of the latter to both the cavity and the target emitter induces strong photon nonlinearities in addition to polariton formation, in contrast to common schemes for ensemble strong coupling. We demonstrate that strong coupling at the level of a single emitter can be engineered via coherent and dissipative dipolar interactions with the ensemble, and provide realistic parameters for a possible implementation with <br>SiV− defects in diamond. Our scheme can find applications, amongst others, in quantum information processing or in the field of cavity-assisted quantum chemistry.

Prospects of reinforcement learning for the simultaneous damping of many mechanical modes

Christian Sommer, Muhammad Asjad, Claudiu Genes

We apply adaptive feedback for the partial refrigeration of a mechanical resonator, i.e. with the aim to<br>simultaneously cool the classical thermal motion of more than one vibrational degree of freedom. The<br>feedback is obtained from a neural network parametrized policy trained via a reinforcement learning<br>strategy to choose the correct sequence of actions from a fnite set in order to simultaneously reduce<br>the energy of many modes of vibration. The actions are realized either as optical modulations of the<br>spring constants in the so-called quadratic optomechanical coupling regime or as radiation pressure<br>induced momentum kicks in the linear coupling regime. As a proof of principle we numerically illustrate<br>efcient simultaneous cooling of four independent modes with an overall strong reduction of the total<br>system temperature.

Partial optomechanical refrigeration via multi-mode cold-damping feedback

Christian Sommer, Claudiu Genes

We provide a fully analytical treatment for the partial refrigeration of the thermal motion of a quantum mechanical resonator under the action of feedback. As opposed to standard cavity optomechanics where the aim is to isolate and cool a single mechanical mode, the aim here is to extract the thermal energy from many vibrational modes within a large frequency bandwidth. We consider a standard cold-damping technique, where homodyne readout of the cavity output field is fed into a feedback loop that provides a cooling action directly applied on the mechanical resonator. Analytical and numerical results predict that low final occupancies are achievable independent of the number of modes addressed by the feedback, as long as the cooling rate is smaller than the intermode frequency separation. For resonators exhibiting a few nearly degenerate pairs of modes, cooling is less efficient and a weak dependence on the number of modes is obtained. These scalings hint toward the design of frequency-resolved mechanical resonators, where efficient refrigeration is possible via simultaneous cold-damping feedback.

Partial Optomechanical Refrigeration via Multimode Cold-Damping Feedback

Christian Sommer, Claudiu Genes

We provide a fully analytical treatment for the partial refrigeration of the thermal motion of a quantum mechanical resonator under the action of feedback. As opposed to standard cavity optomechanics where the aim is to isolate and cool a single mechanical mode, the aim here is to extract the thermal energy from many vibrational modes within a large frequency bandwidth. We consider a standard cold-damping technique, where homodyne readout of the cavity output field is fed into a feedback loop that provides a cooling action directly applied on the mechanical resonator. Analytical and numerical results predict that low final occupancies are achievable independent of the number of modes addressed by the feedback, as long as the cooling rate is smaller than the intermode frequency separation. For resonators exhibiting a few nearly degenerate pairs of modes, cooling is less efficient and a weak dependence on the number of modes is obtained. These scalings hint toward the design of frequency-resolved mechanical resonators, where efficient refrigeration is possible via simultaneous cold-damping feedback.

Laser refrigeration of gas filled hollow-core fibres

Christian Sommer, Nicolas Y. Joly, Helmut Ritsch, Claudiu Genes

We evaluate prospects, performance and temperature limits of a new approach to macroscopic scale laser refrigeration. The considered<br>refrigeration device is based on exciplex-mediated frequency up-conversion inside hollow-core fibers pressurized with a dopant - buffer<br>gas mixture. Exciplexes are excited molecular states formed by two atoms (dopant and buffer) which do not form a molecule in the<br>ground state but exhibit bound states for electronically excited states. The cooling cycle consists of absorption of laser photons during<br>atomic collisions inducing light assisted exciplex formation followed by blue-shifted spontaneous emission on the atomic line of the bare<br>dopant atoms after molecular separation. This process, closely related to reversing the gain mechanism in excimer lasers, allows for a large<br>fraction of collision energy to be extracted in each cycle. The hollow-core fiber plays a crucial role as it allows for strong light-matter<br>interactions over a long distance, which maximizes the cooling rate per unit volume and the cooling efficiency per injected photon while<br>limiting re-absorption of spontaneously emitted photons channeled into unguided radiation modes. Using quantum optical rate equations<br>and refined dynamical simulations we derive general conditions for efficient cooling of both the gas and subsequently of the surrounding<br>solid state environment. Our analytical approach is applicable to any specific exciplex system considered and reveals the shape of the<br>exciplex potential landscapes as well as the density of the dopant as crucial tuning knobs. The derived scaling laws allow for the identification<br>of optimal exciplex characteristics that help to choose suitable gas mixtures that maximize the refrigeration efficiency for specific<br>applications.<br>

Cavity Quantum Electrodynamics with Frequency-Dependent Reflectors

Ondrej Černotík, Aurelien Dantan, Claudiu Genes

We present a general framework for cavity quantum electrodynamics with strongly frequency-dependent mirrors. The method is applicable to a variety of reflectors exhibiting sharp internal resonances as can be realized, for example, with photonic-crystal mirrors or with two-dimensional atomic arrays around subradiant points. Our approach is based on a modification of the standard input-output formalism to explicitly include the dynamics of the mirror’s internal resonance. We show how to directly extract the interaction parameters from the comparison with classical transfer matrix theory and how to treat the non-Markovian dynamics of the cavity field mode introduced by the mirror’s internal resonance. As an application within optomechanics, we illustrate how a non-Markovian Fano-resonance cavity with a flexible photonic-crystal mirror can provide both sideband resolution as well as strong heating suppression in optomechanical cooling. This approach, amenable to a wide range of systems, opens up possibilities for using hybrid frequency-dependent reflectors in cavity quantum electrodynamics for engineering novel forms of light-matter interactions.

Langevin Approach to Quantum Optics with Molecules

Michael Reitz, Christian Sommer, Claudiu Genes

We investigate the interaction between light and molecular systems modeled as quantum emitters coupled to a multitude of vibrational modes via a Holstein-type interaction. We follow a quantum Langevin equations approach that allows for analytical derivations of absorption and fluorescence profiles of molecules driven by classical fields or coupled to quantized optical modes. We retrieve analytical expressions for the modification of the radiative emission branching ratio in the Purcell regime and for the asymmetric cavity transmission associated with dissipative cross talk between upper and lower polaritons in the strong coupling regime. We also characterize the Förster resonance energy transfer process between donor-acceptor molecules mediated by the vacuum or by a cavity mode.

Enhanced collective Purcell effect of coupled quantum emitter systems

David Plankensteiner, Christian Sommer, Michael Reitz, Helmut Ritsch, Claudiu Genes

Cavity-embedded quantum emitters show strong modifications of free space radiation properties such as an enhanced decay known as the Purcell effect. The central parameter is the cooperativity C - the ratio of the square of the coherent cavity coupling strength over the product of cavity and emitter decay rates. For a single emitter, C is independent of the transition dipole moment and dictated by geometric cavity properties such as finesse and mode waist. In a recent work Phys. Rev. Lett. 119, 093601 (2017) we have shown that collective excitations in ensembles of dipole-dipole coupled quantum emitters show a disentanglement between the coherent coupling to the cavity mode and spontaneous free space decay. This leads to a strong enhancement of the cavity cooperativity around certain collective subradiant antiresonances. Here, we present a quantum Langevin equations approach aimed at providing results beyond the classical coupled dipoles model. We show that the subradiantly enhanced cooperativity imprints its effects onto the cavity output field quantum correlations while also strongly increasing the cavity-emitter system's collective Kerr nonlinear effect.

Super- and subradiance of clock atoms in multimode optical waveguides

Laurin Ostermann, Clément Meignant, Claudiu Genes, Helmut Ritsch

The transversely confined propagating modes of an optical fiber mediate virtually infinite range energy exchanges among atoms placed within their field, which adds to the inherent free space dipole–dipole coupling. Typically, the single atom free space decay rate largely surpasses the emission rate into the guided fiber modes. However, scaling up the atom number as well as the system size amounts to entering a collective emission regime, where fiber-induced superradiant spontaneous emission dominates over free space decay. We numerically study this super- and subradiant decay of highly excited atomic states for one or several transverse fiber modes as present in hollow core fibers. As particular excitation scenarios we compare the decay of a totally inverted state to the case of π/2 pulses applied transversely or along the fiber axis as in standard Ramsey or Rabi interferometry. While a mean field approach fails to correctly describe the initiation of superradiance, a second-order approximation accounting for pairwise atom–atom quantum correlations generally proves sufficient to reliably describe superradiance of ensembles from two to a few hundred particles. In contrast, a full account of subradiance requires the inclusion of all higher order quantum correlations. Considering multiple guided modes introduces a natural effective cut-off for the interaction range emerging from the dephasing of different fiber contributions.

Interference effects in hybrid cavity optomechanics

Ondrej Černotík, Claudiu Genes, Aurelien Dantan

Radiation pressure forces in cavity optomechanics allow for efficient cooling of vibrational modes of macroscopic mechanical resonators, the manipulation of their quantum states, as well as generation of optomechanical entanglement. The standard mechanism relies on the cavity photons directly modifying the state of the mechanical resonator. Hybrid cavity optomechanics provides an alternative approach by coupling mechanical objects to quantum emitters, either directly or indirectly via the common interaction with a cavity field mode. While many approaches exist, they typically share a simple effective description in terms of a single force acting on the mechanical resonator. More generally, one can study the interplay between various forces acting on the mechanical resonator in such hybrid mechanical devices. This interplay can lead to interference effects that may, for instance, improve cooling of the mechanical motion or lead to generation of entanglement between various parts of the hybrid device. Here, we provide such an example of a hybrid optomechanical system where an ensemble of quantum emitters is embedded into the mechanical resonator formed by a vibrating membrane. The interference between the radiation pressure force and the mechanically modulated Tavis--Cummings interaction leads to enhanced cooling dynamics in regimes in which neither force is efficient by itself. Our results pave the way towards engineering novel optomechanical interactions in hybrid optomechanical systems.

Attosecond Control of Restoration of Electronic Structure Symmetry

Chun Mei Liu, Jörn Manz, Kenji Ohmori, Christian Sommer, Nobuyuki Takei, Jean Christophe Tremblay, Yichi Zhang

Laser pulses can break the electronic structure symmetry of atoms and molecules by preparing a superposition of states with different irreducible representations. Here, we discover the reverse process, symmetry restoration, by means of two circularly polarized laser pulses. The laser pulse for symmetry restoration is designed as a copy of the pulse for symmetry breaking. Symmetry restoration is achieved if the time delay is chosen such that the superposed states have the same phases at the temporal center. This condition must be satisfied with a precision of a few attoseconds. Numerical simulations are presented for the C6H6 molecule and Rb-87 atom. The experimental feasibility of symmetry restoration is demonstrated by means of high-contrast time-dependent Ramsey interferometry of the Rb-87 atom.

Spatially Adiabatic Frequency Conversion in Optoelectromechanical Arrays

Ondrej Černotík, Sahand Mahmoodian, Klemens Hammerer

Faithful conversion of quantum signals between microwave and optical frequency domains is crucial for building quantum networks based on superconducting circuits. Optoelectromechanical systems, in which microwave and optical cavity modes are coupled to a common mechanical oscillator, are a promising route towards this goal. In these systems, efficient, low-noise conversion is possible using a mechanically dark mode of the fields, but the conversion bandwidth is limited to a fraction of the cavity linewidth. Here, we show that an array of optoelectromechanical transducers can overcome this limitation and reach a bandwidth that is larger than the cavity linewidth. The coupling rates are varied in space throughout the array so that the mechanically dark mode of the propagating fields adiabatically changes from microwave to optical or vice versa. This strategy also leads to significantly reduced thermal noise with the collective optomechanical cooperativity being the relevant figure of merit. Finally, we demonstrate that the bandwidth enhancement is, surprisingly, largest for small arrays; this feature makes our scheme particularly attractive for state-of-the-art experimental setups.

Ramsey interferometry of Rydberg ensembles inside microwave cavities

Christian Sommer, Claudiu Genes

We study ensembles of Rydberg atoms in a confined electromagnetic environment such as is provided by a microwave cavity. The competition between standard free space Ising type and cavity-mediated interactions leads to the emergence of different regimes where the particle -particle couplings range from the typical van der Waals r(-6) behavior to r(-3) and to r-independence. We apply a Ramsey spectroscopic technique to map the two-body interactions into a characteristic signal such as intensity and contrast decay curves. As opposed to previous treatments requiring high-densities for considerable contrast and phase decay (Takei et al 2016 Nat. Comms. 7 13449; Sommer et al 2016 Phys. Rev. A 94 053607), the cavity scenario can exhibit similar behavior at much lower densities.

Energy transfer and correlations in cavity-embedded donor-acceptor configurations

Michael Reitz, Francesca Mineo, Claudiu Genes

The rate of energy transfer in donor-acceptor systems can be manipulated via the common interaction with the confined electromagnetic modes of a micro-cavity. We analyze the competition between the near-field short range dipole-dipole energy exchange processes and the cavity mediated long-range interactions in a simplified model consisting of effective two-level quantum emitters that could be relevant for molecules in experiments under cryogenic conditions. We find that free-space collective incoherent interactions, typically associated with sub-and superradiance, can modify the traditional resonant energy transfer scaling with distance. The same holds true for cavity-mediated collective incoherent interactions in a weak-coupling but strong-cooperativity regime. In the strong coupling regime, we elucidate the effect of pumping into cavity polaritons and analytically identify an optimal energy flow regime characterized by equal donor/acceptor Hopfield coefficients in the middle polariton. Finally we quantify the build-up of quantum correlations in the donor-acceptor system via the two-qubit concurrence as a measure of entanglement.

Ultrafast Coherent Control of Condensed Matter with Attosecond Precision

Hiroyuki Katsuki, Nobuyuki Takei, Christian Sommer, Kenji Ohmori

Coherent control is a technique to manipulate wave functions of matter with light. Coherent control of isolated atoms and molecules in the gas phase is well-understood and developed since the 1990s, whereas its application to condensed matter is more difficult because its coherence lifetime is shorter. We have recently applied this technique to condensed matter samples, one of which is solid para-hydrogen (p-H2). Intramolecular vibrational excitation of solid p-H2 gives an excited vibrational wave function called a “vibron”, which is delocalized over many hydrogen molecules in a manner similar to a Frenkel exciton. It has a long coherence lifetime, so we have chosen solid p-H2 as our first target in the condensed phase. We shine a time-delayed pair of femtosecond laser pulses on p-H2 to generate vibrons. Their interference results in modulation of the amplitude of their superposition. Scanning the interpulse delay on the attosecond time scale gives a high interferometric contrast, which demonstrates the possibility of using solid p-H2 as a carrier of information encoded in the vibrons. In the second example, we have controlled the terahertz collective phonon motion, called a “coherent phonon”, of a single crystal of bismuth. We employ an intensity-modulated laser pulse, whose temporal envelope is modulated with terahertz frequency by overlap of two positively chirped laser pulses with their adjustable time delay. This modulated laser pulse is shined on the bismuth crystal to excite its two orthogonal phonon modes. Their relative amplitudes are controlled by tuning the delay between the two chirped pulses on the attosecond time scale. Two-dimensional atomic motion in the crystal is thus controlled arbitrarily. The method is based on the simple, robust, and universal concept that in any physical system, two-dimensional particle motion is decomposed into two orthogonal one-dimensional motions, and thus, it is applicable to a variety of condensed matter systems. In the third example, the double-pulse interferometry used for solid p-H2 has been applied to many-body electronic wave functions of an ensemble of ultracold rubidium Rydberg atoms, hereafter called a “strongly correlated ultracold Rydberg gas”. This has allowed the observation and control of many-body electron dynamics of more than 40 Rydberg atoms interacting with each other. This new combination of ultrafast coherent control and ultracold atoms offers a versatile platform to precisely observe and manipulate nonequilibrium dynamics of quantum many-body systems on the ultrashort time scale. These three examples are digested in this Account.

Cavity-assisted mesoscopic transport of fermions: Coherent and dissipative dynamics

David Haggenmüller, Stefan Schütz, Johannes Schachenmayer, Claudiu Genes, Guido Pupillo

We study the interplay between charge transport and light-matter interactions in a confined geometry by considering an open, mesoscopic chain of two-orbital systems resonantly coupled to a single bosonic mode close to its vacuum state. We introduce and benchmark different methods based on self-consistent solutions of nonequilibrium Green's functions and numerical simulations of the quantum master equation, and derive both analytical and numerical results. It is shown that in the dissipative regime where the cavity photon decay rate is the largest parameter, the light-matter coupling is responsible for a steady-state current enhancement scaling with the cooperativity parameter. We further identify different regimes of interest depending on the ratio between the cavity decay rate and the electronic bandwidth. Considering the situation where the lower band has a vanishing bandwidth, we show that for a high-finesse cavity, the properties of the resonant Bloch state in the upper band are transferred to the lower one, giving rise to a delocalized state along the chain. Conversely, in the dissipative regime with low-cavity quality factors, we find that the current enhancement is due to a collective decay of populations from the upper to the lower band.

Cavity Antiresonance Spectroscopy of Dipole Coupled Subradiant Arrays

David Plankensteiner, Christian Sommer, Helmut Ritsch, Claudiu Genes

An array of N closely spaced dipole coupled quantum emitters exhibits super-and subradiance with characteristic tailorable spatial radiation patterns. Optimizing the emitter geometry and distance with respect to the spatial profile of a near resonant optical cavity mode allows us to increase the ratio between light scattering into the cavity mode and free space emission by several orders of magnitude. This leads to distinct scaling of the collective coherent emitter-field coupling vs the free space decay as a function of the emitter number. In particular, for subradiant states, the effective cooperativity increases much faster than the typical linear proportional to N scaling for independent emitters. This extraordinary collective enhancement is manifested both in the amplitude and the phase profile of narrow collective antiresonances appearing at the cavity output port in transmission spectroscopy.

Light-matter interactions in multi-element resonators

Claudiu Genes, Aurelien Dantan

We investigate light-matter interactions in multi-element optical resonators and provide a roadmap for the identification of structural resonances and the description of the interaction of single extended cavity modes with quantum emitters or mechanical resonators. Using a first principle approach based on the transfer matrix formalism we analyze, both numerically and analytically, the static and dynamical properties of three-and four-mirror cavities. We investigate in particular conditions under which the confinement of the field in specific subcavities allows for enhanced light-matter interactions in the context of cavity quantum electrodynamics and cavity optomechanics.

Cavity-Enhanced Transport of Charge

David Hagenmueller, Johannes Schachenmayer, Stefan Schutz, Claudiu Genes, Guido Pupillo

We theoretically investigate charge transport through electronic bands of a mesoscopic one-dimensional system, where interband transitions are coupled to a confined cavity mode, initially prepared close to its vacuum. This coupling leads to light-matter hybridization where the dressed fermionic bands interact via absorption and emission of dressed cavity photons. Using a self-consistent nonequilibrium Green's function method, we compute electronic transmissions and cavity photon spectra and demonstrate how light-matter coupling can lead to an enhancement of charge conductivity in the steady state. We find that depending on cavity loss rate, electronic bandwidth, and coupling strength, the dynamics involves either an individual or a collective response of Bloch states, and we explain how this affects the current enhancement. We show that the charge conductivity enhancement can reach orders of magnitudes under experimentally relevant conditions.

Laser noise imposed limitations of ensemble quantum metrology

D. Plankensteiner, J. Schachenmayer, H. Ritsch, Claudiu Genes

Laser noise is a decisive limiting factor in high precision spectroscopy of narrow lines using atomic ensembles. In an idealized Doppler and differential-light-shift-free magic wavelength lattice configuration, it remains as one distinct principal limitation beyond collective atomic decay. In this work we study the limitations originating from laser phase and amplitude noise in an idealized Ramsey pulse interrogation scheme with uncorrelated atoms. Phase noise leads to a saturation of the frequency sensitivity with increasing atom number while amplitude noise implies a scaling 1/root T with T being the interrogation time. We employ a technique using decoherence-free subspaces first introduced in Dorner (2012 New J. Phys. 14 043011) which can restore the scaling with the square root of the inverse particle number 1/root N. Similar results and improvements are obtained numerically for a Rabi spectroscopy setup.

Direct observation of ultrafast many-body electron dynamics in an ultracold Rydberg gas

Nobuyuki Takei, Christian Sommer, Claudiu Genes, Guido Pupillo, Haruka Goto, Kuniaki Koyasu, Hisashi Chiba, Matthias Weidemueller, Kenji Ohmori

Many-body correlations govern a variety of important quantum phenomena such as the emergence of superconductivity and magnetism. Understanding quantum many-body systems is thus one of the central goals of modern sciences. Here we demonstrate an experimental approach towards this goal by utilizing an ultracold Rydberg gas generated with a broadband picosecond laser pulse. We follow the ultrafast evolution of its electronic coherence by time-domain Ramsey interferometry with attosecond precision. The observed electronic coherence shows an ultrafast oscillation with a period of 1 femtosecond, whose phase shift on the attosecond timescale is consistent with many-body correlations among Rydberg atoms beyond mean-field approximations. This coherent and ultrafast many-body dynamics is actively controlled by tuning the orbital size and population of the Rydberg state, as well as the mean atomic distance. Our approach will offer a versatile platform to observe and manipulate non-equilibrium dynamics of quantum many-body systems on the ultrafast timescale.

Time-domain Ramsey interferometry with interacting Rydberg atoms

Christian Sommer, Guido Pupillo, Nobuyuki Takei, Shuntaro Takeda, Akira Tanaka, Kenji Ohmori, Claudiu Genes

We theoretically investigate the dynamics of a gas of strongly interacting Rydberg atoms subject to a time-domain Ramsey interferometry protocol. The many-body dynamics is governed by an Ising-type Hamiltonian with long-range interactions of tunable strength. We analyze and model the contrast degradation and phase accumulation of the Ramsey signal and identify scaling laws for varying interrogation times, ensemble densities, and ensemble dimensionalities.

Selective protected state preparation of coupled dissipative quantum emitters

D. Plankensteiner, L. Ostermann, H. Ritsch, Claudiu Genes

Inherent binary or collective interactions in ensembles of quantum emitters induce a spread in the energy and lifetime of their eigenstates. While this typically causes fast decay and dephasing, in many cases certain special entangled collective states with minimal decay can be found, which possess ideal properties for spectroscopy, precision measurements or information storage. We show that for a specific choice of laser frequency, power and geometry or a suitable configuration of control fields one can efficiently prepare these states. We demonstrate this by studying preparation schemes for strongly subradiant entangled states of a chain of dipole-dipole coupled emitters. The prepared state fidelity and its entanglement depth is further improved via spatial excitation phase engineering or tailored magnetic fields.

Conductivity in organic semiconductors hybridized with the vacuum field

E. Orgiu, J. George, J. A. Hutchison, E. Devaux, J. F. Dayen, B. Doudin, F. Stellacci, C. Genet, J. Schachenmayer, et al.

Much effort over the past decades has been focused on improving carrier mobility in organic thin-film transistors by optimizing the organization of the material or the device architecture. Here we take a different path to solving this problem, by injecting carriers into states that are hybridized to the vacuum electromagnetic field. To test this idea, organic semiconductors were strongly coupled to plasmonic modes to form coherent states that can extend over as many as 105 molecules and should thereby favour conductivity. Experiments show that indeed the current does increase by an order of magnitude at resonance in the coupled state, reflecting mostly a change in field-effect mobility. A theoretical quantum model confirms the delocalization of thewavefunctions of the hybridized states and its effect on the conductivity. Our findings illustrate the potential of engineering the vacuum electromagnetic environment to modify and to improve properties of materials.

A Realization of a Quasi-Random Walk for Atoms in Time-Dependent Optical Potentials

Torsten Hinkel, Helmut Ritsch, Claudiu Genes

We consider the time dependent dynamics of an atom in a two-color pumped cavity, longitudinally through a side mirror and transversally via direct driving of the atomic dipole. The beating of the two driving frequencies leads to a time dependent effective optical potential that forces the atom into a non-trivial motion, strongly resembling a discrete random walk behavior between lattice sites. We provide both numerical and analytical analysis of such a quasi-random walk behavior.

Cavity-Enhanced Transport of Excitons

Johannes Schachenmayer, Claudiu Genes, Edoardo Tignone, Guido Pupillo

We show that exciton-type transport in certain materials can be dramatically modified by their inclusion in an optical cavity: the modification of the electromagnetic vacuum mode structure introduced by the cavity leads to transport via delocalized polariton modes rather than through tunneling processes in the material itself. This can help overcome exponential suppression of transmission properties as a function of the system size in the case of disorder and other imperfections. We exemplify massive improvement of transmission for excitonic wave packets through a cavity, as well as enhancement of steady-state exciton currents under incoherent pumping. These results may have implications for experiments of exciton transport in disordered organic materials. We propose that the basic phenomena can be observed in quantum simulators made of Rydberg atoms, cold molecules in optical lattices, as well as in experiments with trapped ions.

Transmissive optomechanical platforms with engineered spatial defects

Edoardo Tignone, Guido Pupillo, Claudiu Genes

Linear optomechanical photon-phonon couplings in the membrane-in-the-middle setup can be enhanced by taking a multielement approach as it was recently shown [A. Xuereb, C. Genes, and A. Dantan, Phys. Rev. Lett. 109, 223601 ( 2012)]. The particular example considered consists of a periodic array of membranes embedded in a high-finesse optical cavity and operating in the transmissive regime, i.e., around resonances of the compound cavity-membrane system. Here we propose further improvements in such a setup by breaking the translational invariance of the array, i.e., by considering quasiperiodic arrays with engineered quadratic spatial defects in the membrane positions. The localization of light modes induced by the defect combined with the access of the aforementioned transmissive regime window can lead to additional enhancement of the strength of both linear and quadratic optomechanical couplings.

Protected subspace Ramsey spectroscopy

L. Ostermann, D. Plankensteiner, H. Ritsch, Claudiu Genes

We study a modified Ramsey spectroscopy technique employing slowly decaying states for quantum metrology applications using dense ensembles. While closely positioned atoms exhibit super-radiant collective decay and dipole-dipole induced frequency shifts, recent results [L. Ostermann, H. Ritsch, and C. Genes, Phys. Rev. Lett. 111, 123601 (2013)] suggest the possibility to suppress such detrimental effects and achieve an even better scaling of the frequency sensitivity with interrogation time than for noninteracting particles. Here we present an in-depth analysis of this "protected subspace Ramsey technique" using improved analytical modeling and numerical simulations including larger three-dimensional (3D) samples. Surprisingly we find that using subradiant states of N particles to encode the atomic coherence yields a scaling of the optimal sensitivity better than 1/root N . Applied to ultracold atoms in 3D optical lattices we predict a precision beyond the single atom linewidth.

Hybrid cavity mechanics with doped systems

Aurelien Dantan, Bhagya Nair, Guido Pupillo, Claudiu Genes

We investigate the dynamics of a mechanical resonator in which is embedded an ensemble of two-level systems interacting with an optical cavity field. We show that this hybrid approach to optomechanics allows for enhanced effective interactions between the mechanics and the cavity field, leading, for instance, to ground-state cooling of the mechanics, even in regimes, like the unresolved sideband regime, in which standard radiation pressure cooling would be inefficient.

Reconfigurable Long-Range Phonon Dynamics in Optomechanical Arrays

Andre Xuereb, Claudiu Genes, Guido Pupillo, Mauro Paternostro, Aurelien Dantan

We investigate periodic optomechanical arrays as reconfigurable platforms for engineering the coupling between multiple mechanical and electromagnetic modes and for exploring many-body phonon dynamics. Exploiting structural resonances in the coupling between light fields and collective motional modes of the array, we show that tunable effective long-range interactions between mechanical modes can be achieved. This paves the way towards the implementation of controlled phononic walks and heat transfer on densely connected graphs as well as the coherent transfer of excitations between distant elements of optomechanical arrays.

Nonclassical States of Light and Mechanics

Klemens Hammerer, Claudiu Genes, David Vitali, Paolo Tombesi, Gerard Milburn, Christoph Simon, Dirk Bouwmeester

This chapter reports on theoretical protocols for generating nonclassical states of light and mechanics. Nonclassical states are understood as squeezed states, entangled states or states with negativeWigner function, and the nonclassicality can refer either to light, to mechanics, or to both, light and mechanics. In all protocols nonclassicality arises from a strong optomechanical coupling. Some protocols rely in addition on homodyne detection or photon counting of light.

Hybrid Mechanical Systems

Philipp Treutlein, Claudiu Genes, Klemens Hammerer, Martino Poggio, Peter Rabl

We discuss hybrid systems in which a mechanical oscillator is coupled to another (microscopic) quantum system, such as trapped atoms or ions, solid-state spin qubits, or superconducting devices. We summarize and compare different coupling schemes and describe first experimental implementations. Hybrid mechanical systems enable new approaches to quantum control of mechanical objects, precision sensing, and quantum information processing.

Collectively enhanced optomechanical coupling in periodic arrays of scatterers

Andre Xuereb, Claudiu Genes, Aurelien Dantan

We investigate the optomechanical properties of a periodic array of identical scatterers placed inside an optical cavity and extend our previous results [Xuereb, Genes, and Dantan, Phys. Rev. Lett. 109, 223601 (2012)]. We show that operating at the points where the array is transmissive results in linear optomechanical coupling strengths between the cavity field and collective motional modes of the array that may be several orders of magnitude larger than is possible with an equivalent reflective ensemble. We describe and interpret these effects in detail and investigate the nature of the scaling laws of the coupling strengths for the different transmissive points in various regimes.

Enhanced optomechanical readout using optical coalescence

Claudiu Genes, Andre Xuereb, Guido Pupillo, Aurelien Dantan

We present a scheme to strongly enhance the readout sensitivity of the squared displacement of a mobile scatterer placed in a Fabry-Perot cavity. We investigate the largely unexplored regime of cavity electrodynamics in which a highly reflective element positioned between the end mirrors of a symmetric Fabry-Perot resonator strongly modifies the cavity response function, such that two longitudinal modes with different spatial parity are brought close to frequency degeneracy and interfere in the cavity output field. In the case of a movable middle reflector we show that the interference in this generic "optical coalescence" phenomenon gives rise to an enhanced frequency shift of the peaks of the cavity transmission that can be exploited in optomechanics.

Protected State Enhanced Quantum Metrology with Interacting Two-Level Ensembles

Laurin Ostermann, Helmut Ritsch, Claudiu Genes

Ramsey interferometry is routinely used in quantum metrology for the most sensitive measurements of optical clock frequencies. Spontaneous decay to the electromagnetic vacuum ultimately limits the interrogation time and thus sets a lower bound to the optimal frequency sensitivity. In dense ensembles of two-level systems, the presence of collective effects such as superradiance and dipole-dipole interaction tends to decrease the sensitivity even further. We show that by a redesign of the Ramsey-pulse sequence to include different rotations of individual spins that effectively fold the collective state onto a state close to the center of the Bloch sphere, partial protection from collective decoherence is possible. This allows a significant improvement in the sensitivity limit of a clock transition detection scheme over the conventional Ramsey method for interacting systems and even for noninteracting decaying atoms.

Strong Coupling and Long-Range Collective Interactions in Optomechanical Arrays

Andre Xuereb, Claudiu Genes, Aurelien Dantan

We investigate the collective optomechanics of an ensemble of scatterers inside a Fabry-Perot resonator and identify an optimized configuration where the ensemble is transmissive, in contrast to the usual reflective optomechanics approach. In this configuration, the optomechanical coupling of a specific collective mechanical mode can be several orders of magnitude larger than the single-element case, and long-range interactions can be generated between the different elements since light permeates throughout the array. This new regime should realistically allow for achieving strong single-photon optomechanical coupling with massive resonators, realizing hybrid quantum interfaces, and exploiting collective long-range interactions in arrays of atoms or mechanical oscillators.

Atom-membrane cooling and entanglement using cavity electromagnetically induced transparency

Claudiu Genes, Helmut Ritsch, Michael Drewsen, Aurelien Dantan

We investigate a hybrid optomechanical system composed of a micromechanical oscillator as a movable membrane and an atomic three-level ensemble within an optical cavity. We show that a suitably tailored cavity field response via electromagnetically induced transparency (EIT) in the atomic medium allows for strong coupling of the membrane's mechanical oscillations to the collective atomic ground-state spin. This facilitates ground-state cooling of the membrane motion, quantum state mapping, and robust atom-membrane entanglement even for cavity widths larger than the mechanical resonance frequency.

Optical lattices with micromechanical mirrors

K. Hammerer, K. Stannigel, C. Genes, P. Zoller, P. Treutlein, S. Camerer, D. Hunger, T. W. Haensch

We investigate a setup where a cloud of atoms is trapped in an optical lattice potential of a standing-wave laser field which is created by retroreflection on a micromembrane. The membrane vibrations itself realize a quantum mechanical degree of freedom. We show that the center-of-mass mode of atoms can be coupled to the vibrational mode of the membrane in free space. Via laser cooling of atoms a significant sympathetic cooling effect on the membrane vibrations can be achieved. Switching off laser cooling brings the system close to a regime of strong coherent coupling. This setup provides a controllable segregation between the cooling and coherent dynamics regimes, and allows one to keep the membrane in a cryogenic environment and atoms at a distance in a vacuum chamber.

Optomechanical approach to cooling of small polarizable particles in a strongly pumped ring cavity

R. J. Schulze, C. Genes, H. Ritsch

Cavity cooling of an atom works best on a cyclic optical transition in the strong coupling regime near resonance, where small-cavity photon numbers suffice for trapping and cooling. A straightforward application to the cooling of the translational motion of other polarizable particles without sharply defined two-level transitions (such as molecules) fails as optical pumping transfers the particle into uncoupled states. An alternative operation in the far-off-resonant regime generates only very slow cooling due to the reduced field-particle coupling. We suggest one can overcome this by using a strongly driven ring cavity operated in the sideband cooling regime. The dynamics can be mapped onto the optomechanics setup with a movable mirror and allows one to take advantage of a collectively enhanced field-particle coupling by large photon numbers. A linearized analytical treatment confirmed by full numerical quantum simulations predicts fast cooling despite the off-resonant small single-particle-single-photon coupling. Even ground-state translational cooling (in the external potential) can be obtained by tuning the cavity field close to the Anti-stokes sideband for sufficiently high trapping frequency. Numerical simulations show quantum jumps of the particle between the lowest two trapping levels, which can be directly and continuously monitored via scattered light intensity detection.

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.

Phase-noise induced limitations on cooling and coherent evolution in optomechanical systems

P. Rabl, C. Genes, K. Hammerer, M. Aspelmeyer

We present a detailed theoretical discussion of the effects of ubiquitous laser noise on cooling and the coherent dynamics in optomechanical systems. Phase fluctuations of the driving laser induce modulations of the linearized optomechanical coupling as well as a fluctuating force on the mirror due to variations of the mean cavity intensity. We first evaluate the influence of both effects on cavity cooling and find that for a small laser linewidth, the dominant heating mechanism arises from intensity fluctuations. The resulting limit on the final occupation number scales linearly with the cavity intensity both under weak- and strong-coupling conditions. For the strong-coupling regime, we also determine the effect of phase noise on the coherent transfer of single excitations between the cavity and the mechanical resonator and obtain a similar conclusion. Our results show that conditions for optical ground-state cooling and coherent operations are experimentally feasible and thus laser phase noise does pose a challenge but not a stringent limitation for optomechanical systems.

Micromechanical oscillator ground-state cooling via resonant intracavity optical gain or absorption

C. Genes, H. Ritsch, D. Vitali

We suggest possibilities for manipulation and ground-state cooling of micromechanical oscillators by resonant coupling of the mirror vibrations to narrow optical transitions of a designed material ensemble within a cavity mode. Particles, modeled as a two-level ensemble, create intracavity narrow bandwidth loss or gain and induce tailored asymmetric structuring of the cavity noise spectrum interacting with the oscillator. This facilitates cooling via inhibition of the Stokes-scattering process or enhancement of anti-Stokes scattering even for short low-finesse optical cavities.

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.

Cavity-assisted squeezing of a mechanical oscillator

K. Jaehne, C. Genes, K. Hammerer, M. Wallquist, E. S. Polzik, P. Zoller

We investigate the creation of squeezed states of a vibrating membrane or a movable mirror in an optomechanical system. An optical cavity is driven by the squeezed light and couples via the radiation pressure to the membrane or mirror, effectively providing a squeezed heat bath for the mechanical oscillator. Under the conditions of laser cooling to the ground state, we find an efficient transfer of squeezing with roughly 60% of light squeezing conveyed to the membrane or mirror (on a dB scale). We determine the requirements on the carrier frequency and the bandwidth of squeezed light. Beyond the conditions of ground-state cooling, we predict mechanical squashing to be observable in current systems.

Sub-Planck-scale structures in a vibrating molecule in the presence of decoherence

Suranjana Ghosh, Utpal Roy, Claudiu Genes, David Vitali

We study the effect of decoherence on the sub-Planck scale structures of the vibrational wave packet of a molecule. The time evolution of these wave packets is investigated under the influence of a photonic or phononic environment. We determine the master equation describing the reduced dynamics of the wave packet and analyze the sensitivity of the sub-Planck structures against decoherence in the case of a hydrogen iodide (HI) molecule.

Quantum Effects in Optomechanical Systems

C. Genes, A. Mari, D. Vitali, P. Tombesi

The search for experimental demonstration of the quantum behavior of macroscopic mechanical resonators is a fast growing field of investigation and recent results suggest that the generation of quantum states of resonators with a mass at the microgram scale is within reach. In this chapter we give an overview of two important topics within this research field: cooling to the motional ground state and the generation of entanglement involving mechanical, optical, and atomic degrees of freedom. We focus on optomechanical systems where the resonator is coupled to one or more driven cavity modes by the radiation-pressure interaction. We show that robust stationary entanglement between the mechanical resonator and the output fields of the cavity can be generated, and that this entanglement can be transferred to atomic ensembles placed within the cavity. These results show that optomechanical devices are interesting candidates for the realization of quantum memories and interfaces for continuous variable quantum-communication networks.

Simultaneous cooling and entanglement of mechanical modes of a micromirror in an optical cavity

Claudiu Genes, David Vitali, Paolo Tombesi

Laser cooling of a mechanical mode of a resonator by the radiation pressure of a detuned optical cavity mode has been recently demonstrated by various groups in different experimental configurations. Here, we consider the effect of a second mechanical mode with a close but different resonance frequency. We show that the nearby mechanical resonance is simultaneously cooled by the cavity field, provided that the difference between the two mechanical frequencies is not too small. When this frequency difference becomes smaller than the effective mechanical damping of the secondary mode, the two cooling processes interfere destructively similarly to what happens in electromagnetically induced transparency, and cavity cooling is suppressed in the limit of identical mechanical frequencies. We show that also the entanglement properties of the steady state of the tripartite system crucially depend upon the difference between the two mechanical frequencies. If the latter is larger than the effective damping of the second mechanical mode, the state shows fully tripartite entanglement and each mechanical mode is entangled with the cavity mode. If instead, the frequency difference is smaller, the steady state is a two-mode biseparable state, inseparable only when one splits the cavity mode from the two mechanical modes. In this latter case, the entanglement of each mechanical mode with the cavity mode is extremely fragile with respect to temperature.

Robust entanglement of a micromechanical resonator with output optical fields

C. Genes, A. Mari, P. Tombesi, D. Vitali

We perform an analysis of the optomechanical entanglement between the experimentally detectable output field of an optical cavity and a vibrating cavity end-mirror. We show that by a proper choice of the readout (mainly by a proper choice of detection bandwidth) one cannot only detect the already predicted intracavity entanglement but also optimize and increase it. This entanglement is explained as being generated by a scattering process owing to which strong quantum correlations between the mirror and the optical Stokes sideband are created. All-optical entanglement between scattered sidebands is also predicted, and it is shown that the mechanical resonator and the two sideband modes form a fully tripartite-entangled system capable of providing practicable and robust solutions for continuous-variable quantum-communication protocols.

Emergence of atom-light-mirror entanglement inside an optical cavity

C. Genes, D. Vitali, P. Tombesi

We propose a scheme for the realization of a hybrid, strongly quantum-correlated system formed of an atomic ensemble surrounded by a high-finesse optical cavity with a vibrating mirror. We show that the steady state of the system shows tripartite and bipartite continuous variable entanglement in experimentally accessible parameter regimes, which is robust against temperature.

Ground-state cooling of a micromechanical oscillator: Comparing cold damping and cavity-assisted cooling schemes

C. Genes, D. Vitali, P. Tombesi, S. Gigan, M. Aspelmeyer

We provide a general framework to describe cooling of a micromechanical oscillator to its quantum ground state by means of radiation-pressure coupling with a driven optical cavity. We apply it to two experimentally realized schemes, back-action cooling via a detuned cavity and cold-damping quantum-feedback cooling, and we determine the ultimate quantum limits of both schemes for the full parameter range of a stable cavity. While both allow one to reach the oscillator's quantum ground state, we find that back-action cooling is more efficient in the good cavity limit, i.e., when the cavity bandwidth is smaller than the mechanical frequency, while cold damping is more suitable for the bad cavity limit. The results of previous treatments are recovered as limiting cases of specific parameter regimes.

Self-cooling of a movable mirror to the ground state using radiation pressure

A. Dantan, C. Genes, D. Vitali, M. Pinard

We show that one can cool a micromechanical oscillator to its quantum ground state using radiation pressure in an appropriately detuned cavity (self-cooling). From a theory based on Heisenberg-Langevin equations we find that optimal self-cooling occurs in the good cavity regime, when the cavity bandwidth is smaller than the mechanical frequency, but still larger than the effective mechanical damping. In this case the intracavity field and the vibrational mechanical mode coherently exchange their fluctuations, thus reducing the mirror temperature by several orders of magnitude. We also present dynamical calculations which show how to access the mirror temperature from a homodyne measurement of the fluctuations of the reflected field.

Atomic entanglement generation with reduced decoherence via four-wave mixing

C. Genes, P. R. Berman

In most proposals for the generation of entanglement in large ensembles of atoms via projective measurements, the interaction with the vacuum is responsible for both the generation of the signal that is detected and the spin depolarization or decoherence. In consequence, one must usually work in a regime where the information acquisition via detection is sufficiently slow (weak measurement regime) such as not to strongly disturb the system. We propose here a four-wave mixing scheme where, owing to the pumping of the atomic system into a dark state, the polarization of the ensemble is not critically affected by spontaneous emission. In the language of spin squeezing, the removal of the limitations imposed by spontaneous emission allows one to work in a strong signal regime where the Heisenberg limit can be reached.

Cooperative spin decoherence and population transfer

C. Genes, P. R. Berman

An ensemble of multilevel atoms is a good candidate for a quantum information storage device. The information is encrypted in the collective ground state atomic coherence, which, in the absence of external excitation, is decoupled from the vacuum and therefore decoherence free. However, in the process of manipulation of atoms with light pulses (writing, reading), one inadvertently introduces a coupling to the environment, i.e., a source of decoherence. The dissipation process is often treated as an independent process for each atom in the ensemble, an approach which fails at large atomic optical depths where cooperative effects must be taken into account. In this paper, the cooperative behavior of spin decoherence and population transfer for a system of two, driven multilevel atoms is studied. Not surprisingly, an enhancement in the decoherence rate is found, when the atoms are separated by a distance that is small compared to an optical wavelength; however, it is found that this rate increases even further for somewhat larger separations for atoms aligned along the direction of the driving field's propagation vector. A treatment of the cooperative modification of optical pumping rates and an effect of polarization swapping between atoms is also discussed, lending additional insight into the origin of the collective decay.

Generating conditional atomic entanglement by measuring photon number in a single output channel

C Genes, PR Berman

The polarization analysis of quantized probe light transmitted through an atomic ensemble has been used to prepare entangled collective atomic states. In a "balanced" detection configuration, where the difference signal from two detection ports is analyzed, the continuous monitoring of a component of the Stokes field vector provides a means for conditional projective measurements on the atomic system. Here, we make use of classical driving fields, in the pulsed regime, and of an "unbalanced" detection setup (single detector) where the effective photon number of scattered photons is the detected observable. Conditional atomic spin squeezed states and superpositions of such squeezed states can be prepared in this manner.

Spin squeezing via atom-cavity field coupling

Claudiu Genes, PR Berman, AG Rojo

Spin squeezing via atom-field interactions is considered within the context of the Tavis-Cummings model. An ensemble of N two-level atoms interacts with a quantized cavity field. For all the atoms initially in their ground states, it is shown that spin squeezing of both the atoms and the field can be achieved provided the initial state of the cavity field has coherence between number states differing by 2. Most of the discussion is restricted to the case of a cavity field initially in a coherent state, but initial squeezed states for the field are also discussed. Optimal conditions for obtaining squeezing are obtained. An analytic solution is found that is valid in the limit that the number of atoms is much greater than unity and is also much larger than the average number of photons, alpha(2), initially in the coherent state of the cavity field. In this limit, the degree of spin squeezing increases with increasing a, even though the field more closely resembles a classical field for which no spin squeezing could be achieved.