Dynamics of coupled multimode optomechanical systems - OPUS4

Nanomechanics interacting with light: Dynamics of coupled multimode optomechanical systems Georg Heinrich

July 2011

Nanomechanics interacting with light: Dynamics of coupled multimode optomechanical systems Georg Heinrich

Der Naturwissenschaftlichen Fakultät der Friedrich-Alexander-Universität Erlangen-N¨ urnberg zur Erlangung des Doktorgrades Dr. rer. nat. vorgelegt von Georg Heinrich aus Heidelberg

Als Dissertation genehmigt von der Naturwissenschaftlichen Fakultät der Friedrich-Alexander-Universität Erlangen-N¨ urnberg Tag der m¨ undlichen Pr¨ ufung: 27. Oktober 2011 Vorsitzender der Promotionskommission: Prof. Dr. Rainer Fink Erstgutachter: Prof. Dr. Florian Marquardt Zweitgutachter: Prof. Dr. Tania Monteiro

Contents Abstract 1 Introduction 1.1 Standard optomechanical systems . . . . . . . . . . . . . . . . . . . . . 1.1.1 The basic optomechanical setup . . . . . . . . . . . . . . . . . . 1.1.2 A universal model applicable to a wide range of experiments . 1.1.3 Displacement measurements: a universal mechanical transducer 1.2 Classical dynamics of the standard optomechanical system . . . . . . . 1.2.1 Modification of mechanical properties . . . . . . . . . . . . . . 1.2.2 Optical retardation: dynamical back-action . . . . . . . . . . . 1.2.3 The nonlinear regime: Dynamical multistability . . . . . . . . . 1.3 Quantum optomechanics . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Coupled multimode optomechanical systems . . . . . . . . . . . . . . . 1.4.1 Trends and developments . . . . . . . . . . . . . . . . . . . . . 1.4.2 This thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xi

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2 Strongly driven two- and multilevel dynamics in optomechanics 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 System Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 Mechanically driven non-equilibrium photon dynamics . . . . . . 2.2.3 Equation of motion and transmission to the right . . . . . . . . . 2.3 Direct simulations: transmission spectra for increasing driving strength 2.4 Photon dynamics resembling atomic multilevel physics . . . . . . . . . . 2.4.1 Excitation process and internal dynamics . . . . . . . . . . . . . 2.4.2 Two-level dynamics with time-dependent coupling . . . . . . . . 2.5 Analysis for weak and modest mechanical driving . . . . . . . . . . . . . 2.5.1 Transmission without mechanical driving . . . . . . . . . . . . . 2.5.2 Mechanically assisted photon transfer . . . . . . . . . . . . . . . 2.5.3 Mechanically driven Rabi dynamics . . . . . . . . . . . . . . . . 2.5.3.1 Interpretation of the full expression for x ¯0 g. . . . . . 2.5.4 Autler-Townes splitting . . . . . . . . . . . . . . . . . . . . . . . 2.5.5 The emergence of multiphonon processes . . . . . . . . . . . . . . 2.5.5.1 Internal dynamics: multiphonon couplings . . . . . . . 2.5.5.2 Cavity excitation: Tuning multiphonon processes . . . . 2.6 The strong mechanical driving regime . . . . . . . . . . . . . . . . . . .

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17 18 19 19 20 20 21 22 22 22 24 25 26 27 27 28 30 30 32 33

vi

CONTENTS . . . . . .

33 34 36 36 39 40

3 Self-oscillations: Dynamical back-action in terms of multilevel dynamics 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Coupled multimode optomechanical setup . . . . . . . . . . . . . . . . . . . . 3.3 Landau-Zener-Stueckelberg dynamics acting back on its driving mechanism . 3.4 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Effective optomechanical damping rate Γopt . . . . . . . . . . . . . . . . . . . 3.5.1 Damping rate Γopt from the net mechanical power input . . . . . . . . 3.5.2 Results on Γopt for given mechanical oscillations . . . . . . . . . . . . 3.6 Back-action driven mechanical self-oscillations . . . . . . . . . . . . . . . . . . 3.6.1 Steady state conditions for the dynamics’ attractors . . . . . . . . . . 3.6.2 Two-level dynamics determining dynamical back-action . . . . . . . . 3.6.3 Interpretation using Floquet Theory . . . . . . . . . . . . . . . . . . . 3.7 Characterizing the parameter space . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

41 42 42 44 44 44 44 45 46 46 46 46 48 48

4 Coupled multimode optomechanics in the microwave regime 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Two-resonators setup with linear mechanical coupling . . . . . . . . . . 4.2.1 General structure of the Hamiltonian . . . . . . . . . . . . . . . . 4.2.2 Deriving the Hamiltonian from a microscopic picture . . . . . . . 4.3 Coupling Frequency comparable to the mechanical frequency . . . . . . 4.3.1 Example: Mechanically driven photon dynamics . . . . . . . . . 4.4 Coupling to the square of displacement . . . . . . . . . . . . . . . . . . . 4.4.1 Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Quantum non-demolition Fock state detection: general principle 4.5 Fock state detection in the microwave regime . . . . . . . . . . . . . . . 4.5.1 Signal-to-noise ratio to detect a mechanical quantum jump . . . 4.5.2 Appropriate setup design . . . . . . . . . . . . . . . . . . . . . . 4.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.7

2.6.1 Landau-Zener physics . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Landau-Zener-Stueckelberg oscillations in an optomechanical system 2.6.3 Physical description: Multiphonon transition picture . . . . . . . . . 2.6.4 Analytical description: Resonance approximation . . . . . . . . . . . 2.6.5 Tuning the laser frequency . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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51 52 53 53 53 56 56 58 58 60 60 61 62 62

5 Collective Phenomena: Synchronization in optomechanical arrays 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Optomechanical crystals and optomechanical arrays . . . . . . . . 5.1.1.1 Collective phenomena of the classical, nonlinear dynamics 5.1.1.2 Self-induced oscillations of a single optomechanical cell . 5.1.2 Synchronization: a universal feature in nonlinear science . . . . . . 5.1.2.1 The Kuramoto model . . . . . . . . . . . . . . . . . . . . 5.1.2.2 Synchronization in optomechanical arrays . . . . . . . . . 5.2 The Hopf model: a reduced description for single optomechanical cells . . 5.2.1 Equations of motion of a single optomechanical Hopf oscillator . .

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63 64 64 65 66 66 66 67 67 68

CONTENTS 5.2.2

5.3

5.4 5.5

5.6

5.7 5.8

Dependence on microscopic parameters . . . . . . . . . . . . . . . . . 5.2.2.1 Bifurcation threshold and steady state amplitude . . . . . . . 5.2.2.2 Amplitude decay rate . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Numerical simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . External periodic forcing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Synchronization to an external force . . . . . . . . . . . . . . . . . . . 5.3.2 Amplitude dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . An experimentally feasible optomechanical array setup . . . . . . . . . . . . . Synchronization of two coupled optomechanical cells . . . . . . . . . . . . . . 5.5.1 Two coupled optomechanical cells: lowest-order phase coupling . . . . 5.5.2 Amplitude dynamics coupling to the phase . . . . . . . . . . . . . . . 5.5.2.1 Dispersion of the mechanical frequency . . . . . . . . . . . . 5.5.2.2 Higher-order phase coupling . . . . . . . . . . . . . . . . . . 5.5.3 Effective Kuramoto-type model for coupled optomechanical cells . . . 5.5.3.1 Amplitude modulations in terms of phase dynamics . . . . . 5.5.3.2 Amplitude-mediated phase coupling . . . . . . . . . . . . . . 5.5.3.3 Effective slow phase description . . . . . . . . . . . . . . . . 5.5.4 Synchronization Phenomena: Comparison with numerical results . . . 5.5.4.1 Transition to phase locking; in-phase and anti-phase synchronization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5.4.2 Mechanical frequency spectra . . . . . . . . . . . . . . . . . . Synchronization in optomechanical arrays . . . . . . . . . . . . . . . . . . . . 5.6.1 Extended Kuramoto-type model . . . . . . . . . . . . . . . . . . . . . 5.6.1.1 Amplitude modulations in terms of phase dynamics . . . . . 5.6.1.2 Amplitude-mediated phase coupling . . . . . . . . . . . . . . 5.6.1.3 Generalized effective slow phase description . . . . . . . . . . 5.6.2 Nearest-neighbor mechanical coupling . . . . . . . . . . . . . . . . . . 5.6.3 Effective global coupling via an extended optical mode . . . . . . . . . 5.6.4 An array setup involving only a single extended optical mode . . . . . A remaining challenge: γ an adjustable parameter . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

6 Conclusion

vii 68 68 69 70 72 72 74 75 76 77 78 78 79 79 79 80 81 82 83 85 86 87 87 88 90 90 93 94 95 96 97

A Appendix A.1 General structure of the mechanically driven light field dynamics . . . . . . . A.2 Effective phase model for four nearest-neighbor coupled optomechanical cells A.3 Amplitude decay rate - an adjustable parameter . . . . . . . . . . . . . . . .

99 99 102 103

Bibliography

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Acknowledgments

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Titel und Zusammenfassung in deutscher Sprache

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List of Publications

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viii

LIST OF FIGURES

List of Figures 1.1 1.2 1.3 1.4

The standard optomechanical system . . . . . . . . . . . . . . . . . . . Dynamical back-action and static multistability . . . . . . . . . . . . . Attractor diagram for self-induced oscillations in the nonlinear regime Coupled multimode optomechanical systems realized in experiments .

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3 7 10 13

2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 2.10 2.11

Setup realizing mechanically driven coherent photon dynamics . . . Transmission spectrum resembling the optical resonances . . . . . . Transmission spectrum for increasing mechanical driving strength . . Autler-Townes splitting in an optomechanical system . . . . . . . . . Multiphonon interactions and higher order splittings . . . . . . . . . Tunable excitation processes . . . . . . . . . . . . . . . . . . . . . . . Landau-Zener physics of a single photon in the strong driving regime Landau-Zener-Stueckelberg oscillations in an optomechanical system Multiphonon transition picture and formation of LZS oscillations . . Resonance approximation for strong mechanical driving . . . . . . . Dependence on laser detuning . . . . . . . . . . . . . . . . . . . . . .

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3.1 3.2 3.3 3.4

Setup: driven multi-level photon dynamics back-acting on the mechanics Effective optomechanical damping for given mechanical oscillations . . . Attractor diagram for phonon lasing oscillations . . . . . . . . . . . . . . Complete parameter space: effective damping and attractor diagram . .

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4.1 4.2 4.3 4.4 4.5

Coupled multimode microwave setup with linear mechanical coupling . . . Discretized circuit diagram for two coupled microwave resonators . . . . . Transmission spectrum: mechanically driven dynamics . . . . . . . . . . . Coupled multimode microwave setup with quadratic mechanical coupling Signal-to-noise ratio to measure a quantum jump . . . . . . . . . . . . . .

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5.1 5.2 5.3 5.4 5.5 5.6 5.7 5.8

Optomechanical crystal structures and optomechanical arrays . . . . Hopf bifurcation for a single optomechanical cell . . . . . . . . . . . Slow mechanical amplitude dynamics of a single optomechanical cell Synchronization of an optomechanical cell to an external force . . . . Amplitude dynamics of an optomechanical cell driven by an external Potential setup of an optomechanical array and FEM simulations . . Phase-locking of two coupled optomechanical cells . . . . . . . . . . Mechanical frequency spectra for two coupled optomechanical cells .

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ABSTRACT 5.9

Mechanical frequency spectra for an optomechanical array with nearest-neighbor mechanical interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.10 Optomechanical array with nearest-neighbor mechanical interaction - comparison to the effective Kuramoto-type model . . . . . . . . . . . . . . . . . . . . 5.11 Optomechanical array with global coupling . . . . . . . . . . . . . . . . . . . 5.12 Array setup involving only a single extended optical mode . . . . . . . . . . .

90 92 93 94

Abstract In this thesis we theoretically investigate the dynamics of coupled multimode optomechanical systems. Progress in nanofabrication recently enabled to design micro- and nanomechanical resonators whose motion can be coupled to electromagnetic fields. Given the enormous optical control that has been achieved for atomic systems, this led to significant interest in using light to manipulate and control macroscopic mechanical objects. This stimulated the field of optomechanics that by now has evolved into a fast developing area of research at the intersection between nanophysics and quantum optics. At first, in Chapter 1, we introduce the reader to the field of optomechanics. The standard optomechanical setup consists of a laser-driven optical cavity whose resonance frequency is changed due to the motion of a mechanical object. We review the essential features of this intriguingly simple system whose dynamics is described in terms of a single optical mode coupled a single mechanical one. An exciting new development recently introduced optomechanical setups that involve several coupled optical and vibrational modes, so-called coupled multimode optomechanical systems. We highlight these recent trends pointing the way towards integrated optomechanical circuits. This leads us to the subject matter of this thesis, the dynamics of coupled multimode optomechanical systems. Up to now research mainly focused on the steady-state dynamics of optomechanical systems. Going beyond this standard approach, in Chapter 2, we consider a time-dependent drive of the mechanics and investigate non-equilibrium photon dynamics driven by mechanical motion. It turns out that the most striking effects can be observed for coupled multimode optomechanical setups consisting of several coupled optical (and vibrational) modes. In this case, mechanical driving allows one to deliberately transfer photons between different optical modes in a coherent fashion. More generally, the mechanically driven coherent photon dynamics, that we introduce, in principle enables to realize all kinds of strongly driven twoand multilevel dynamics known from atomic physics in the light fields of optomechanical systems. For instance, for a recently developed setup and experimentally feasible parameters, we predict the possibility of observing an Autler-Townes splitting indicative of Rabi dynamics as well as Landau-Zener-Stueckelberg oscillations. The application of external mechanical driving can thus open up the whole domain of strongly driven two- and multilevel phenomena to the field of optomechanics. In Chapter 3, we investigate the dynamical back-action of coupled multimode optomechanical systems, i.e. light acting back on the mechanics after having been influenced by the mechanical motion. Generally, back-action effects have been of particular interest for the field of optomechanics. This dynamics, for instance, enables optomechanical cooling and can induce mechanical self-oscillations akin to lasing. Going beyond the regime of the linearized light-field dynamics, we analyze how mechanically driven multi-mode photon dynamics (cf. Chapter 2) acts back on the mechanics via radiation forces. We show that even for

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ABSTRACT

two optical modes Landau-Zener-Stueckelberg oscillations of the light field drastically change the nonlinear attractor diagram of the resulting phonon lasing oscillations. More generally, our findings illustrate the generic effects of Landau-Zener physics on back-action induced self-oscillations. Besides the optical domain, the standard optomechanical system has recently been implemented by replacing the optical cavity by a superconducting microwave resonator. Motivated by this development, in Chapter 4, we investigate coupled multimode optomechanics in the microwave regime. We point out that 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. This has several implications and so these systems have advantages that go beyond bulk refrigerator cooling and on-chip integration. 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. Finally, building on the fascinating development to diminish and on-chip integrate optomechanical systems (based on photonic crystal structures for instance), in Chapter 5, we start to investigate the collective nonlinear dynamics in arrays of coupled optomechanical cells. Here, each unit consists of a laser-driven optical and a mechanical mode. Beyond a certain threshold of the laser input power, each cell, implementing the standard optomechanical system, shows a Hopf bifurcation towards a regime of self-induced mechanical oscillations. We show that the phases of many such coupled optomechanical oscillators, even in case of different bare initial frequencies, can lock to each other, synchronizing the dynamics to a collective oscillation frequency. We present different regimes for the dynamics and derive an effective Kuramoto-type model that allows one to explain and predict most of the features that will be observable in future experiments.

Chapter 1

Introduction The interaction of light with matter has been at the heart of the development of modern physics. The inception of this new era, marked by the invention of quantum mechanics at the beginning of the 20th century, was stimulated by addressing various effects involving both phenomena such as the black body radiation problem or the photoelectric effect. Since then studying the interplay between light and matter has enormously advanced the fundamental understanding of physics, led to profound applications and benefited various other scientific fields. At the theory’s early stage, quantum mechanics, initially developed to describe the physics of an atom, proved to be highly successful at explaining the optical spectra of light emitted from atomic systems. Later, the description of a single atom interacting with light was further completed by developing the full quantum theory of light. This initiated the field of quantum optics, leading to groundbreaking inventions such as laser technology. In general, experimentalists achieved a remarkable experimental control of atomic systems using radiation fields. For instance, laser-cooling of optically trapped ions and neural atoms allowed observing the ground state of mechanical motion, improving the measurement of atomic spectra or enhancing atomic clocks with all their implications on metrology. Furthermore, this approach introduced the crucial technology that enabled to achieve Bose-Einstein condensation seventy years after its predication. Given these achievements to optically control atomic systems, there has been increasing interest in manipulating macroscopic objects using light. After pioneering work of Braginsky and co-workers in the late 1960s, recent progress in nanofabrication has enabled to design micro- and nanoscale devices whose mechanical motion can interact with radiation pressure forces. These optomechanical systems have been the focus of considerable experimental and theoretical research during the past few years. By now optomechanical systems are used to perform ultra-sensitive force and displacement measurements that are able to test fundamental concepts in various fields of research. Prospects are to use mechanical motion as a universal coupling scheme to connect miscellaneous hybrid components (such as spins, superconducting quantum bits, cold atoms, etc.) or to fabricate integrated optomechanical circuits that might be used for classical and quantum information processing, amplification and storage. With respect to fundamental questions, laser cooling of nanomechanical oscillators to the quantum ground state promises novel tests of quantum mechanics in a new regime of large-scale structures. Goals range from observing quantum dynamics in the mechanics, creating entanglement between light and mechanical

2

1. Introduction

degrees of freedom, realizing non-classical states, to finally controlling the quantum state of mechanical motion. This might eventually implement the physicist’s dream of a mechanical Schrödinger cat state, i.e. a macroscopic mechanical object, such as a nanomechanical cantilever, being in a superposition of two positions. In this first chapter we introduce the reader to the field of optomechanics and lead him towards the subject matter of this thesis, coupled multimode optomechanical systems. We note that some excellent recent reviews on optomechanics can be found in [1–3].

1.1 1.1.1

Standard optomechanical systems The basic optomechanical setup

According to Maxwell, an electromagnetic field can exert a radiation pressure force. The first two experiments to observe radiation pressure forces in a lab were independently conducted in 1901 by Nichols together with Hull [4] and by Lebedev [5]. Remarkably, early speculations on this effect actually go back to Johannes Kepler who tried to address the question why the tail of a comet generally points away from the sun. Today we know that the dust part of a comet is indeed pushed due to radiation pressure forces. To get an idea of this tiny effect, imagine a photon bouncing off a mirror. In this process, it reverses its direction and transfers a momentum 2~k, where k is the wave vector. For a ray of light with power P , the rate of photons impinging on the device is N˙ = P/hν. Accordingly, the radiation pressure force exerted by the light field is Frad = 2P/c. As this expression involves the speed of light c, the force strength is generally very small. As an example one might again consider sunlight. Its power on earth after passing through the atmosphere is roughly 1 kW/m2 . Thus, the sun’s radiation pressure force per square meter is approximately 10−5 N. Note that this is ten orders of magnitude smaller than the atmospheric pressure. Not surprisingly, optomechanical experiments usually involve ultra-low-weight mechanical objects where light can have a significant impact on the mechanics. After pioneering work of Braginsky and co-workers [6, 7] in the late 1960s, the upsurge of the field and its recent fast development, both in experiments and theory, is mainly triggered by the advancement of nanotechnology that allows one to fabricate such ultra-sensitive devices. To control and manipulate macroscopic objects using light, free-space experiments turn out to be inadequate. Thus, the mechanical device, such as a micromirror that is mounted on top of a doubly clamped microbeam [8,9] or attached to an atomic force microscope cantilever [10], is generally integrated into an optical cavity, see Fig 1.1. The standard optomechanical system can be depicted as a one-sided, laser-driven Fabry-Perot cavity with a movable end mirror. The optical cavity implements two important functions. First, it resonantly enhances the light-field intensity. This significantly increases the radiation pressure effects on the mechanics as each photon repeatedly bounces off the mirrors before it finally leaves the cavity. Second, using an optical resonator, the light-field intensity, and thus the radiation pressure force acting on the mechanics, sensitively depends on the mechanics’ position, i.e. the length of the Fabry-Perot interferometer. Quantitatively, the standard optomechanical system is described in terms of an optical mode whose resonance frequency is changed due to the displacement of a mechanical degree of freedom x ˆ. In its quantized form [13, 14], the system’s Hamiltonian reads x ˆ ˆ ˆL + H ˆκ + H ˆ Γ. H = ~ω0 1 − a ˆ† a ˆ + ~Ωˆb†ˆb + H (1.1) l

1.1 Standard optomechanical systems

radiation laser drive

force optical cavity

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movable mirror

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cantilever

fixed mirror (a)

(c)

Figure 1.1: The standard optomechanical system. (a) Schematic setup consisting of a laserdriven optical cavity with a movable mirror. Usually, the end-mirror is attached to a mechanically oscillating object, such as a cantilever. The light intensity stored inside an optical mode exerts a radiation pressure force on the mechanics whose motion conversely changes the cavity’s resonance frequency via the mechanical displacement x. (b, c) Scanning electron micrographs of two experimental realizations. (b) Micromirror attached to a doubly-clamped beam fabricated for the experiment in [11]. (Courtesy of Vienna Center for Quantum Science and Technology (VCQ)) (c) Micromirror mounted on an atomic force microscope cantilever reported in [12]. (Courtesy of Dustin Kleckner et al.)

The optical as well as the mechanical mode are described in terms of two harmonic oscillators characterized by the annihilation operators a ˆ and ˆb, respectively. The displacement x ˆ = † ˆ ˆ xzp (b + b) of the mechanical degree of freedom with frequency Ω, where xzp denotes the zeropoint displacement, modulates the optical resonance frequency ω0 . Note that the gradient ˆ represents a force and Fˆrad = (~ω0 /l)ˆ −∂x H a† a ˆ is the radiation pressure force exerted by ˆ =a the number of photons circulating inside the cavity, N ˆ† a ˆ. The last terms added to the ˆ L, H ˆ κ and H ˆ Γ , describe the laser drive, the photon decay out of the cavity Hamiltonian, H and the mechanical damping of the system, respectively. We note that the system depicted in Fig. 1.1a generally possesses several optical as well as mechanical modes. For an experiment, while the optical mode is determined by the choice of the laser drive frequency, a specific mechanical degree of freedom can be addressed by selecting an adequate optical detuning between the external laser and the optical resonance, see details below. For the setup depicted in Fig. 1.1a, if the end-mirror moves to the right, the length of the Fabry-Perot cavity l is increased, i.e. the optical resonance frequency decreases. Likewise, if the mechanics moves in the opposite direction, the optical resonance frequency increases. Note that the system is continuously driven by a laser at ωL and photons decay out of the cavity. The modulation of the optical resonance in terms of the mechanics’ motion directly translates to a variation of the cavity’s light-field intensity. This leads to a concomitant change of the radiation pressure force acting on the mechanics. As we will see, this apparently simple mechanism gives rise to a wealth of dynamical back-action effects between the optical field and the mechanics that allows one to control and manipulate mechanical motion using light.

4

1.1.2

1. Introduction

A universal model applicable to a wide range of experiments

Although at first sight the system depicted in Fig. 1.1a might look special, it actually describes a very general situation. In fact, the standard optomechanical model applies to any situation where a resonance (here the optical mode) is driven (by a laser), and whose resonance frequency is changed due to the displacement of a mechanical degree of freedom. Thus, this basic setup characterizes a very universal nonlinear and non-equilibrium scenario. Given this generality, a wide range of different experimental realizations of the standard optomechanical system has been implemented by now. Setups based on micro- and nanomechanical cantilevers integrated into Fabry-Perot cavities [8–10,15,16] most closely resemble the scheme depicted in Fig. 1.1a. Another approach uses microtoroids [17, 18], i.e. micrometerscale, on-chip disk structures fabricated from silica. In this case, light circulating inside a whispering gallery mode couples to mechanical breathing modes of the structure. Generally, toroidal microcavities can achieve very high optical finesse [19]. A different realization uses a dielectric membrane [20] with a typical thickness of 50 nm that is placed in the middle between two fixed mirrors. Experimentally, this approach is beneficial as it allows separating optical and mechanical properties. Generally, optomechanical systems need to achieve both high optical finesse and good mechanical quality factors, i.e. low-loss and high-frequency mechanics, for instance to observe coherent interaction. Thus, typical designs, such as micromirrors attached to mechanical beams or cantilevers, need to integrate both challenging concepts into a single device. This technical hassle is circumvented by this new approach using fixed mirrors and a membrane that is independent of the optical elements. Other conceptual advantages of this setup will be discussed further below. Besides micro- and nanomechanical objects, we note that also the LIGO project, attempting to measure gravitational waves, implements optomechanical schemes on remarkably different scales [21, 22]. Based on the standard model’s generality, optomechanical concepts have also been realized for systems operated in completely different regimes, not including any optical element at all. This includes mechanical objects coupled to LC circuits [23] or superconducting microwave resonators [24, 25]. In the latter case, the optical cavity is replaced by a superconducting microwave resonator whose central conductor is capacitively coupled to a nanomechanical beam. One advantage of these systems is that they naturally operate in a cryogenic environment. This is important to eventually observe quantum phenomena of macroscopic mechanical objects. Furthermore, on-chip integration enables to combine these systems with other on-chip elements such as superconducting qubits. Optomechanical effects have also been observed for setups consisting of mechanical objects coupled to single electron transistors [26–32]. An even more exotic scheme to realize an optomechanical system is to use clouds of cold atoms [33–37]. For these designs, the moving end-mirror of the standard setup (Fig. 1.1) is replaced by an ultra-cold atomic gas that is confined within a Fabry-Perot cavity. In this case, the optical field couples to the center of mass motion of the atomic ensemble realizing the standard optomechanical interaction (Eq. 1.1). The typical number of atoms in such an ensemble is on the order of 105 . Note that the mass of typical mechanical objects used for optomechanical experiments range from 10−12 to 10−10 kg and thus these devices usually consist of approximately 1013 to 1015 atoms. The size of these atomic cloud objects therefore lies between the size of conventional micro- and nanomechanical beams and the limiting case of a single atom (10−25 kg) where ground state cooling has already been achieved in the 1980s [38]. Due to the strong coupling to the cavity field, the high mechanical quality factors and the small masses, atomic cloud systems are ranked prime candidates to observe quantum

1.2 Classical dynamics of the standard optomechanical system

5

phenomena of mechanical objects containing a large number of atoms. Last but not least, we note the exciting new development to design optomechanical systems based on photonic crystals [39,40]. These so-called “optomechanical crystals” are periodically patterned silicon structures that yield both localized optical and mechanical modes. Thus, they combine the concept of photonic and phononic crystals into an on-chip design. This new approach adds to other on-chip realizations pointing the way towards integrated optomechanical circuits [41,42]. The prospect to couple various custom-designed optomechanical elements on a chip is a major motivation for this thesis. We will further highlight these designs when introducing coupled multimode optomechanical systems further below and also in Chapter 5. Despite their diversity, all these schemes share the dynamical properties of the standard optomechanical system (Fig. 1.1a). They all implement and try to improve the nonlinear light-mechanics interaction described by Eq. (1.1). Eventually, this interaction might be used to measure and control quantum effects of macroscopic mechanical objects.

1.1.3

Displacement measurements: a universal mechanical transducer

Before actually turning to discuss dynamical features of the light-mechanics interaction, we want to highlight the general importance of optomechanical systems for “classical” sensing applications. In principle, measuring the mechanics’ position is straightforward. The mechanical displacement modifies the cavity’s resonance frequency, resulting in an optical phase shift [43]. In experiments, this can easily be detected using a Pound-Drever-Hall scheme [44]. The combination of high-optical finesse and ultra-light-weight micro- and nanomechanical devices yields an enormous sensitivity for various measurements. For instance, following the pioneering work of Braginsky and co-workers in the late 1960s, who sensitively measured lightinduced mechanical damping via the decay of a mechanical oscillator [6,7], the newly available nanofabricated mechanical objects now allow one to routinely measure the random thermal motion of the mechanics and even cool these Brownian fluctuations, see Section 1.2.2. By now these systems have been able to resolve forces in the zeptonewton (10−21 N) regime [45], masses in the 10−24 kg domain, actually going towards single-molecule nanomechanical mass spectrometry [46, 47], and displacements in the attometer (10−18 m) regime [48]. Such highly sensitive systems allowed measuring fundamental physical effects such as single electron spins [49], persistent currents [50] or the Casimir force [51]. Furthermore, micro- and nanomechanical resonators can generally be coupled to various physical systems via functionalizing the mechanical device. Therefore, optomechanical systems are applicable as universal mechanical transducers for a wide range of ultra-sensitive sensing applications. Currently this versatility to couple to various physical systems is explored with respect to potential applications in quantum information processing [52]. For instance, one might think of building a mechanical quantum bus interconnecting various quantum systems [53].

1.2

Classical dynamics of the standard optomechanical system

We now turn to discuss some of the essential features of the standard optomechanical dynamics that has intensively been studied over recent years. From the Hamiltonian (Eq. (1.1)) we generally find the Heisenberg equations of motion for the cavity operator a ˆ and the mechanical displacement operator x ˆ. To introduce the essential dynamical features of the standard optomechanical system (Fig. 1.1a), we first consider its purely classical dynamics. To do so,

6

1. Introduction

we replace the operator a ˆ by the complex light-field amplitude α(t) and x ˆ by the classical displacement x [54, 55]. The classical equations of motion then read m¨ x = −mΩ2 x − mΓx˙ + ~G|α|2 , h κi κ α˙ = i(∆ + Gx) − α + αmax , 2 2

(1.2) (1.3)

where κ denotes the cavity decay rate, Γ is the mechanical damping of the mechanics, ∆ = ωL − ω0 denotes the laser’s detuning from the optical resonance, and αmax is the maximum light-field amplitude achieved at resonance that is set by the laser input power. In view of the wide range of various implementations of the standard optomechanical dynamics, here we considered a general optomechanical coupling G defined in terms of the cavity’s frequency shift per displacement x. For the standard Fabry-Perot realization we have G = ω0 /l, see Eq. (1.1).

1.2.1

Modification of mechanical properties

Intrinsically, the mechanical part of the system is an harmonic oscillator. However, as the radiation pressure force Frad = ~G|α|2 depends on the displacement x, it modifies the mechanical properties of the system. Much of the basic physics can be understood from the system’s linearized dynamics (see for instance [15] where a remarkable experiment is nicely described in terms of a well-illustrated linearized model). The linearized optomechanical dynamics only considers small fluctuations of the cavityfield around its classical steady-state value, α(t) = α ¯ + δα(t). The constant light-field contribution α ¯ yields an average radiation pressure force that simply leads to a constant deflection of the mechanics. Considering the mechanical dynamics around such a static displacement x ¯, we write x(t) = x ¯ + δx(t). Changes of the mechanical properties manifest themselves in terms of the linear mechanical response of the system to an applied external force Fext (t). In the frequency domain, the linear response f δx(ω) = χef xx (ω)Fext (ω),

(1.4)

is given in terms of an effective susceptibility f χef xx (ω) =

m(Ω2

−

ω2)

1 , − imωΓ + Σ(ω)

(1.5)

f where Σ(ω) denotes the “optomechanical self-energy”. Note that for Σ = 0, χef xx simply is the susceptibility of an harmonic oscillator. In the generic form of Eq. (1.5), Σ(ω) describes the modification of mechanical properties due to the light-mechanics interaction. Generally, Σ(ω) comprises real and imaginary contributions. While the real part affects the mechanical frequency, the imaginary part changes the effective mechanical damping, see Eq. (1.5). Here we want to focus on the physical origin of these effects explaining these phenomena physically without discussing quantitative details of Σ(ω) that can be found in the literature. We start with effects modifying the mechanical frequency, i.e. the real part of Σ(ω). Effects due to ImΣ(ω) will be discussed in Section 1.2.2. As pointed out, the radiation pressure force Frad = ~G|α|2 depends on the displacement x, see Fig. 1.2a,b. Looking at Eq. (1.2), we see that the force gradient generally changes the mechanics’ spring constant. This modification of the mechanical frequency is known as

2.0


7

1.5

1 2.0 1.0

1 0 1

0.5

0 0 −1

1.5 0.0

(a)

(b) −1.0

−0.5

0.0

0.5

1.0

−1 −1

−2 1.0

−2

−2 −3

cooling

0.5

heating

−3 −4 −4

−5

−4 −5

0.0

(c)

−1.0

−0.5

0.0

0

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5

6

−5 0

1

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3

4

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6

0

1

2

3

4

5

6

(d)

Figure 1.2: Dynamical back-action and static multistability. (a, b) The radiation pressure force Frad is proportional to the light-field intensity I that depends on the mechanics’ position x. (c) For infinitesimally slow mechanical motion x(t) the radiation pressure force follows the Lorentzian curve (adiabatic limit). For a finite velocity however, due to the finite cavity ring-down time κ−1 , the force is delayed, i.e. it is smaller (larger) than the adiabatic limits H for motion towards [black arrow] (away [blue arrow]) from the resonance. As Frad dx 6= 0, this results in heating and cooling of the mechanics. (d) The effective static potential Veff to the right hand side of Eq. (1.2) gives rise to several multistable positions (blue arrows). the “optical spring effect”. If the cavity is fast compared to the mechanics (κ Ω), the mechanical object experiences the quasi-static light-field intensity of a laser-driven optical mode, see Fig. 1.2b. For a laser blue-detuned from resonance (∆ > 0), the gradient of the Lorentzian-shaped radiation pressure force is negative. Thus, the light-mechanics interaction increases the mechanical spring constant, making the device stiffer. Accordingly, for reddetuning (∆ < 0) the mechanical stiffness is softened. Deviations from this simple description occur if the mechanics moves fast compared to the light-field dynamics (κ < Ω). In this case the change of mechanical frequency depending on detuning ∆ gets more involved. We note that the optical spring effect allows one to significantly tune mechanical frequencies [56]. Experiments already achieved an increase of the intrinsic frequency by a factor of more than 25 for a gram-scale mirror [22]. For very strong optomechanical coupling (G¯ α > κ), the mechanical resonance splits into two peaks [57, 58]. In this so-called “strong coupling regime”, the normal modes of the system establish a new set of dynamical variables involving both the light-field and the mechanics. Thus, the mechanical resonance hybridizes with the driven optical resonance. This hybridization is observable in terms of a normal-mode splitting. The strong coupling regime has recently been reached in experiments [59].

1.2.2

Optical retardation: dynamical back-action

As pointed out, the optical force gradient, that is introduced via the displacement-dependent light-field intensity, modifies the mechanical frequency. In addition to this, there is another crucial dynamical feature of optomechanical systems: the light-field generally responds with a time lag that leads to a finite delay of the radiation force. For a generic optomechanical

8

1. Introduction

systems, the optical delay is determined by the cavity decay rate κ. Essentially, due to the optical ring-down time κ−1 , photons need a while to leak out of the cavity. As a result, the light intensity inside the cavity, and thus the concomitant radiation pressure force, is delayed in its response to a new mechanical position x. The same dynamics also results for other systems based on bolometric (i.e. photothermal) forces [15]. For instance, a bimorph cantilever might absorb some photons depending on the light-field intensity. In this case the finite delay time is determined by the thermal conductance. To understand the physical implications of this effect, note that the radiation force depending on displacement has a Lorentzian shape, see Fig. 1.2b. Assume the cavity to be driven at the bare optical mode’s frequency, ωL = ω0 , and imagine the mechanical oscillator to be placed to the left of the bare optical resonance, see Fig. 1.2c, i.e. x < 0 in Eq. (1.1). In this case, the length of the Fabry-Perot cavity is reduced and the optical mode’s frequency is higher than the laser’s driving frequency. Thus, the laser is red-detuned with respect to the cavity. Now consider small oscillations of the mechanics that might be due to its thermal motion. When moving towards the resonance, the radiation force performs work and supplies energy to the cantilever. In contrast, energy is extracted from the mechanics if it moves in the opposite direction. For the limiting case of infinitesimally slow motion (adiabatic limit), the system dynamics exactly follows the Lorentzian curve shown in Fig. 1.2c. However, due to the time lag of the optical field to adjust to a new mechanical position x, the light-field intensity is lower than for the adiabatic limit when moving towards resonance. Conversely, it is higher for the opposite direction of motion. Altogether, there is a net amount of energy H extracted from the mechanics, Frad dx < 0 [57, 60], see Fig. 1.2c. This adds an additional optomechanical damping Γopt that enters Eq. (1.5) in terms of the imaginary part of the optomechanical self-energy Σ(ω). Positioning the mechanical oscillator on the other side ofHthe resonance (Fig. 1.2c) leads to negative optomechanical damping Γopt . Here we have Frad dx > 0, i.e. heating of the mechanics, that yields parametric amplification. Once this negative damping overcomes the intrinsic mechanical friction, the system can enter a regime of self-induced mechanical oscillations [54], that will be discussed further below. Note that for a given mechanical displacement the regimes of heating (∆ > 0) and cooling (∆ < 0) can be tuned in terms of the laser detuning ∆. These effects are commonly referred to as “dynamical backaction” as they involve the light-field acting back on the mechanics after being influenced by the mechanical motion itself. Cavity-assisted optomechanical cooling, based on the intrinsic effect discussed above, has been implemented in several experiments [8,9,15,18,20,22]. Previously, optomechanical cooling using an active feedback has also been achieved [10,61–63]. See also [64] for a comparison of the two approaches and [65] on an alternative optomechanical cooling scheme. For operations in the heating regime (Fig. 1.2c), self-induced oscillations have been observed in [17,55,66,67]. Starting from a bulk temperature T , the classical picture of cavity-assisted cooling in principle allows one to reach arbitrarily low temperatures. Following this naive description, the effective temperature of the mechanical mode considered for cooling is given by Tef f =

Γ T, Γ + Γopt

where the optomechanical damping Γopt linearly scales with laser intensity. At sufficiently low temperatures, however, photon shot noise due to the discreteness of light, counteracts cooling. Thus, such unavoidable, genuine quantum effects have to be considered in a full


9

quantum theory of optomechanical cooling [57,68]. We will come back to this when discussing quantum optomechanics further below.

1.2.3

The nonlinear regime: Dynamical multistability

Beyond the linearized optomechanical dynamics that was discussed so far, the standard optomechanical system (Eq. (1.1)) displays an enormously rich nonlinear dynamics. In this regime, the system can develop various multistabilities, both static as well as unique dynamical ones. In the following, we will present some of these remarkable features. To understand the static multistability that can arise for the standard optomechanical system, we note that for sufficiently large displacements x, the light-field intensity of the Fabry-Perot cavity (Fig. 1.1a) comprises several resonances, see Fig. 1.2d. While the distance between the Lorentzians is λ/2, where λ is the wavelength of the driving laser, the peaks’ width is determined by the cavity’s finesse F, i.e. λ/(2F). Integrating the radiation force to the right-hand side of Eq. (1.2), we find several steps in the resulting potential Vrad (x), see Fig. 1.2d, whose size scales with the light-field intensity and the optomechanical coupling G. Together with the harmonic potential of the mechanical oscillator, this yields several barriers in the effective static potential Veff (x), giving rise to various stable positions for the mechanics (Fig. 1.2d). In one of the field’s pioneering experiments, the group of H. Walther at the MPQ in Garching measured this radiation-pressure-induced static bistability in the 1980s [69]. In addition to the static multistability, the optomechanical dynamics also features a dynamical multistability that was predicted recently [54]. As pointed out, for a laser bluedetuned from the optical resonance (∆ > 0), the dynamical back-action of the system yields a negative optomechanical damping that corresponds to light-induced mechanical heating. Once this antidamping exceeds the intrinsic mechanical friction, the system features selfinduced mechanical oscillations. In this regime, the amplitude of any small oscillation will increase exponentially until it finally saturates due to nonlinear effects. The mechanics then settles into self-induced, sinusoidal oscillations at its intrinsic frequency Ω. The oscillation amplitude A and mean position x ¯ depend on the system’s parameters such as mechanical damping, cavity decay rate, laser detuning or laser drive power [54]. Note that this behavior is similar to a laser above its lasing threshold. In contrast to the optical analog, here mechanical oscillations are pumped via the light field. It turns out, that the dynamical attractors, i.e. the stable steady-state solutions for the oscillation amplitude A and the mean position x ¯, can be found from two basic conditions [54, 70]. First, in steady-state, the total force on the mechanics has to vanish on average (force balance). Second, the net mechanical power input, supplied by the radiation pressure force during one oscillation period, has to match the total power loss due to friction (power balance). Given this description, for any set of parameters we can plot a map of all possible solutions that satisfy these two conditions, see Fig. 1.3. In particular, several stable solutions for the system’s oscillatory dynamics can coexist simultaneously, introducing a new kind of dynamical multistability in this highly nonlinear system. By now, experimentalists have started to investigate this multistability [67]. Also the influence of quantum effects on the optomechancial instability has been explored [70]. We note that the analytical description presented here applies to the regime where radiation pressure effects during one oscillation period are still small. For very large laser input powers, the system can enter a regime of chaotic motion that was observed experimentally in [71]. Nevertheless, this chaotic domain is still essentially unexplored.

10

1. Introduction 13

the cantilever

2 1 −1

0

1

3

2

pl

Am

ing etun detuning D

itu

de

cantilever energy

0

(a) (a)

100

3

2

-1

0

1

15

Mechanical oscillation amplitude

Power feed into power fed into mechanics

Mechanical oscillation amplitude

2.1. CLASSICAL SOLUTION

(b) (b)

15

10 10

5 5

+ -

0 0

ï

0

Detuning from resonance ï 0

Detuning from resonance

Figure 2.1: Classical self-induced oscillations of the coupled cavity-cantilever system. The radiation Figure Attractor diagram oscillations in the nonlinear regime. (a) Averpressure 1.3: acting on the cantilever provides an average mechanicalfor power self-induced input of P . The ratio P /P of this power P vs. the loss due to mechanical friction, P , is shown as a function age ofmechanical power input Prad , supplied the detuning ∆ and the cantilever’s oscillation energy E , at fixed laser input power P.by The the radiation pressure force, depending on laser oscillation energy E = mω A /2 is shown in units of E , where E /E = (A/x ) . Self-induced oscillations requireoscillation P = P . This condition is fulfilled along the horizontal cutStable at detuning ∆ and amplitude [70]. self-induced oscillations with amplitude A P /P = 1 (see black line and the inset depicting the same plot, viewed from above). These solutions are stable if the ratio P /P decreases with increasing oscillation amplitude A. The occur if P the floorequals the power loss due to mechanical friction Pfric (blue contour line). (Courblue regions atrad of the plot indicate that P is negative, resulting in cooling. The cavity rate is κ = 0.5ω , the mechanical damping is chosen as Γ /ω = 1.47 · 10 , and the tesydecay ofpower Max Ludwig et al.) (b) Density plot of Prad for slightly different parameters that input as P = 6.05 · 10 ; these parameters are also used in figures 2.2, 2.3, 2.4, and 2.6, and will be referred to as Γ and P . displays several multistable solutions for the oscillation amplitude A (black contour lines). This new kind of dynamical multistability was first published in [54]. (Courtesy of Florian Marquardt et al.) rad

rad

fric

rad

fric

2 M

M

M

2

0

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rad

M

FWHM

2

fric

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fric

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−3

∗ M

1.3

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fric

M

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−3

∗

Quantum optomechanics

When entering the quantum regime, fundamental new effects alter the classical dynamics described until now. Observing and controlling these phenomena constitutes another major motivation for the field of optomechanics. In this thesis, we mainly focus on classical lightmechanics interaction that displays a variety of dynamical features and still is the most easily accessible regime for experiments. Here, we give a short overview on some of the major quantum effects that start to play a role when entering the quantum domain. As pointed out, the standard optomechanical system represents a highly sensitive scheme to measure mechanical displacement. For such weak displacement measurements [43], the Heisenberg uncertainty principle sets a fundamental constraint. In an experiment that records mechanical motion in terms of an optical phase shift (see Section 1.1.3), we can distinguish two fundamental noise sources, i.e. imprecision noise and measurement back-action noise. Due to photon shot noise, imprecision noise limits the precision to detect the phase of the optical output field (which is proportional to the oscillator’s position). Nevertheless, this accuracy can be improved by increasing the light-field intensity, i.e. the coupling between the mechanics and the optical resonator that constitutes the motion’s detector. If there was only imprecision noise, this would in principle allow one to measure mechanical motion over time with arbitrary precision. However, for increasing mechanics-detector coupling, i.e. high light-field intensity, eventually measurement back-action noise arises, in accordance with the constraints imposed by the Heisenberg uncertainty principle. In fact, due to their discrete nature, the photons being reflected from the mirror (Fig. 1.1a) imprint an unavoidable

1.3 Quantum optomechanics

11

jitter on the signal that actually limits the precision to measure the intrinsic mechanical motion [43]. The best overall accuracy is achieved at the so-called standard quantum limit (SQL) where both contributions, due to imprecision and measurement back-action noise, are equal. Physically, at the SQL the mechanical position can be resolved within the mechanical ground state uncertainty after averaging the signal over a damping period. Improving the measurement imprecision in experiments to reach the SQL has been a major goal [72] that has recently been achieved [73,74]. Detecting measurement back-action effects, that might be achieved by measuring correlations [75], however still constitutes an outstanding experimental challenge. We note that back-action-evading measurements can be performed by recording only one of the two quadratures of mechanical motion [76, 77]. Genuine quantum effects also alter the classical picture of optomechanical cooling. In particular, photon shot noise eventually counteracts the classical cooling process described in Section 1.2.2, and needs to be considered in a full quantum description [57,68]. For the quantum picture, it is convenient to switch from the classical time-domain that nicely illustrates the origin of dynamical back-action (see Fig. 1.2) to a frequency representation. Generally, for the standard setup (Fig. 1.1), the mechanical motion causes phase and amplitude modulations of the optical field. Similar to Raman scattering in a solid, this yields Stokes and anti-Stokes sidebands displaced from the optical carrier (at the laser drive frequency ωL ) by the mechanical frequency ±Ω. In the quantum picture, the lower sideband, i.e. the Stokes line, is due to a process where a photon from the optical drive loses energy, producing a phonon ~Ω added to the mechanics. Likewise, the anti-Stokes process increases the photon energy in terms of a phonon ~Ω, extracting energy from the mechanics. Thus, these processes correspond to heating and cooling, respectively. To obtain an effective optomechanical cooling or heating rate, there needs to be an asymmetry in the strength of both processes. Generally, the phase and amplitude modulations of the optical field lead to an interference of the Stokes and anti-Stokes sideband that can cause one process to be more pronounced than the other. The strength of the individual process is determined by the cavity’s resonance that can be tuned in terms of the laser’s frequency mismatch with respect to the optical resonance ∆. For red-detuning (∆ < 0), the anti-Stokes line is closer to the optical resonance that yields cooling. In contrast, for blue-detuning (∆ > 0), there is a larger density of states seen by the Stokes process leading to an effective optomechanical heating (see for instance the review in [1]). The corresponding rates and their tunability via the cavity’s resonance can nicely be analyzed in terms of the “quantum noise approach” [43, 57]. From the full quantum description, it turns out that optomechanical cooling to the quantum ground state, i.e. an occupation of the mechanical mode of less than one, can only be achieved for a setup in the resolved-sideband regime where the cavity decay rate is smaller than the mechanical frequency [57,68]. In this case, the cavity line width is smaller than the spacing between Stokes and anti-Stokes so that the heating process is significantly suppressed. In addition, to reach the ground state, the light-field intensity has to be high enough such that the cavity-assisted cooling compensates any reheating that is due to the thermal environment. By now several groups have reached the resolved-sideband regime and cooled the motion of a macroscopic mechanical object close to the quantum mechanical ground state [11, 25, 72, 78–80]. Ground state cooling has recently been achieved for a mechanical resonator with an enormously high frequency of several GHz using conventional cryogenic refrigeration, demonstrating a remarkable control of single phonon excitations [81]. Being in the quantum regime, an essential goal is to prepare and measure nonclassical states of the light field and/or the mechanics. It has been pointed out that the standard optomechanical setup (Fig. 1.1) corresponds to an optical system involving a nonlinear medium

12

1. Introduction

whose index of refraction is modulated by the light-field intensity. This might be used to produce squeezed light that, for instance, can allow one to reduce intensity fluctuations [82, 83]. Conversely, using the light field to periodically modulate the mechanical spring constant (Section 1.2.1), one could generate squeezed states of the mechanics. Another goal in this respect is to detect entanglement. In principle, entanglement between the mechanics and the light field arises naturally. For the optical field being in a superposition of number states (e.g. a coherent state), each of these Fock states corresponds to a different radiation pressure force displacing the mechanics by a different amount. This would correspond to a macroscopic mechanical object being in a superposition state of different positions, i.e. a “Schrödinger cat state”, see for instance [84–87]. Entanglement between various mechanical objects can be achieved via the light field. To observe such effects in an experiment, thermal fluctuations generally need to be very low and the experimental proof involves correlation measurements using different probes. Another very convincing demonstration of the quantum nature of a macroscopic mechanical resonator would be to directly measure is energy quantization [88,89]. This can be achieved using a new optomechanical setup that was recently realized in [20, 90]. This scheme also allows one to measure phonon shot noise, i.e. quantum energy fluctuations around an average phonon number of a mechanically driven, ground-state–cooled mechanical oscillator [91]. Another perspective is to couple mechanical oscillators to atomic systems an even single atoms. This then introduces an interface to control and manipulate the mechanics’ quantum state using well established concepts from atomic physics [92–96]. In summary, quantum optomechanics tries to address the fundamental question whether quantum mechanics, that provides an excellent description of the microscopic domain, can equally well be applied to macroscopic mechanical systems.

1.4

Coupled multimode optomechanical systems

Although any mechanical device features several mechanical modes and every cavity has a discrete spectrum of optical resonances, the dynamics of the standard optomechanical system only involves a single optical mode coupled to a single mechanical degree of freedom. As we have pointed out, this intriguingly simple model displays a wide range of dynamical features that can be observed in a variety of experimental implementations. Here, for any specific experimental system, the optical mode is selected in terms of the frequency of the driving laser. The mechanical mode is then determined by the frequency one considers in the mechanical displacement spectrum that, for instance, is measurable via an optical phase shift of the cavity. To optomechanically cool a specific mechanical degree of freedom, the mechanical mode can be selected via the detuning between the laser drive frequency and the optical resonance. Until very recently, the physics of multiple coupled mechanical as well as optical modes did not play an important role in optomechanical experiments. One exception was the measurement of the impact of several mechanical modes on the nonlinear dynamics of self-induced oscillations [67]. Experimental progress is now changing this situation.

1.4.1

Trends and developments

Generally, we can identify two main experimental trends with respect to the design of optomechanical setups. The first one is diminution of the system’s size. The second one is on-chip integration. We note that both trends are not totally independent as on-chip integration often

1.4 Coupled multimode optomechanical systems

ARTICLES

13

a

(a)

(b)

optical feed

microtoroid

op microtoroids

with the na consideratio

b

(c)

5 µm

tic a

l fe

ed

nanobeams

c

50 µm

Figure 1.4: Coupled multimode optomechanical systems already realized in experiments. (a) Figure 1 | Evanescent coupling of nanomechanical oscillators to an 50 nm thick dielectric membrane used in the “membrane-in-the-middle” setup of[20]. Placed optical microresonator. a, Schematic the experiment, showing a tapered-fibre-interfaced optical microresonator dispersively between two high-finesse mirrors, the membrane couples various optical modes residing in coupled to an array of nanomechanical oscillators. b, Scanning electron micrograph (false the two cavity halves to its left and to its right, respectively. (Courtesy of Thompson et with colour) of an array of doubly clamped SiN nanostring oscillators dimensions nm × (300–500) × (15–40) µm. c, Scanning al.) (b) Setup consisting of two coupled microtoroids realizing a 110 phonon lasernmthat involves a electron micrograph (false colour) of a toroid silica microcavity acting as an tunable optical two-level system [97]. (Courtesy of Ivan Grudinin etnear-field al.) (c) Setup consisting optomechanical sensor. of nanobeams coupled to a microtoroid [74]. (Courtesy of Georg Anetsberger et al.) 14,28,29

resolved-sideband regime . By detuned excitation, dynamical backaction mediated by the optical dipole force is demonstrated, which leads to radiation-pressure-induced coherent oscillations the nanomechanical oscillator, whereas thermal effects are (but not always; like in case of on-chip stripline microwaveof resonators [24,25]) goes along with negligible. Equally important, the combination of picogram and high-quality-factor nanostrings with an ultrahigh-optical-finesse a reduction of the system’s overall scale including both the resonator and the 30mechanics. microresonator provides a route to the remarkable regime where Scaling down the mechanics has been an issue from the early daysquantum of thebackaction field asis small radiation-pressure the dominant force noise on the mechanical oscillator even at room temperature mechanical objects are generally more sensitive to radiation pressure forces. Furthermore, and might thus allow quantum optomechanical experiments such 31 the quantum ground state of a low-mass mechanical device possesses a larger zero-point as ponderomotive squeezing , quantum non-demolition (QND) 32,33 measurements of photons or optomechanical entanglement34 at displacement xzp . This is important to finally observe quantum effects of a macroscopic ambient temperature. mechanical object, note x ˆ = xzp (ˆb† + ˆb) in Eq. (1.1). Experimentalist carried on this Figure 1a shows have a schematic of the experimental setup. We use an array of nanomechanical oscillators in the idea and have recently started to scale down the optical resonator as well. This significantly form of high-Q, tensile stressed and doubly clamped SiN 15,30 decreased the overall size of the standard optomechanicalstrings system. microand have typical suchStarting as shown from in Fig. 1b. The strings dimensions of 110 nm × (300–800) nm × (15–40) µm, effective nanomechanical devices integrated into cm-scaled Fabry-Perot cavities (e.g. [8–10, 15]), much masses of meff = 0.9–5 pg and fundamental resonance frequencies Ω = 6.5–16 MHz cavities with mechanical quality factors of smaller optomechanical systems have been designed, based on microtoroid (several m /2π Q = 104 –105 (see Supplementary Information). Following a special ten µm in diameter) [17,18], photonic crystal structures (several µm in size) [39,40] and other fabrication process (see Supplementary Information) indeed allows using tightly confined optical modes of toroid silica on-chip optical designs (also on a µm-scale) [41,42]. In addition, thethestandard optomechanical microcavities as near-field probes (see Fig. 1c) that interact with the system has also been implemented in the microwave regime using superconducting nanomechanical oscillator through√ their stripline evanescent field decaying −1 on a length scale of α ≈ (λ/2π)/ n2 − 1 (that is, approximately resonators (cm-scale) [24, 25]. 238 nm for the refractive index of silica n = 1.44 and a vacuum One of the main motivations to scale-down and integrate the optomechanical wavelength of λstandard ≈ 1,550 nm used throughout this work).

system on a chip has been to increase the light-mechanics interaction via tightly confined Optomechanical coupling rate optical and mechanical modes (see also Chapter 5). ThisFirst, is crucial for sensing applica- coupling of we study both the strength of the optomechanical the nanomechanical oscillators to the optical mode of a 58-µmtions as well as to eventually observe genuine quantum effects of the mechanics. Furthermore, diameter microcavity (showing an unloaded optical linewidth of on-chip integration allows one to combine optomechanics4.9with other elements. Forpresence of a MHz, that is, aon-chip finesse of F = 230,000). The dielectric oscillator in the evanescent cavity field, at a distance a microwave setup, for instance, such a combination recently afforded ultra-sensitive disx0 to the microresonator surface, can in principle give rise to placement measurements with measurement imprecision below that at standard quantum both a reactive andthe dispersive contribution to the optical-cavity 35 . The former would be characterized by increased cavity limit [73]. For applications, one future goal is to use thisresponse design flexibility to realize optical losses owing to scattering or absorption, given by a positiondependent cavity linewidth κ(x0 )/2π. The latter can be described by on-chip systems that integrate several functions, such as signal detection, processing, storing an optical-frequency shift �ω0 (x0 )/2π = (ω0 (x0 ) − ω0 )/2π caused and amplification. by the increased effective refractive index sampled by the evanescent fraction of the mode (ω0 denotes the unperturbed cavity frequency Beyond improvements of the standard setup and applications, diminution and on-chip integration also open up a whole new route for the field of optomechanics, i.e. towards 910 coupled multimode optomechanical systems. These systems comprise several mechanical as well as optical modes that interact with each other. Using nanofabrication, they might be deliberately designed. Going beyond the standard setup, these systems possess genuinely

where Vnan (refractive volume Vcav interface di is denoted detailed an the nanome gallery mo inset) to th experiment the nanostr unless othe with a nano the expecte the distance measured d with the va we do not Fig. 2a) even of changes upper boun nanomecha coupling is described b aˆ † aˆ denotes fluctuations of the strin �int = H �0 + H g (x0 ) = dω0 Experim pling rates o obtained by shifts, that coupling ra For compar nanomecha sheet of Si coupling (s able coupli microcaviti g ∝ (Vnano / which in o that is, l = analytical e rates can b microcavity photonic-cr allowed rem be obtained sufficiently the resolved parable to could not b in contrast, As the nano losses to the allows com high mecha regime, wh evading me

NATURE

14

1. Introduction

new dynamical features. For instance, it has been pointed out that for applications, coupled multimode setups allow one to increase measurement sensitivity [98]. Some coupled multimode optomechanical systems have already been implemented in experiments, see Fig. 1.4. One experimental realization in this respect is the “membrane-in-themiddle” setup [20], where a dielectric membrane (Fig. 1.4a) is placed in the middle between two high-finesse mirrors separating the cavity into two halves. Photon tunneling through the mechanical device couples optical modes residing in the left and the right half of the cavity, respectively. The frequency of the resulting hybridized optical modes can possess a fundamentally different dependence on mechanical displacement than the standard system [20, 99]. In particular, this setup allows one to achieve an optomechanical coupling to the square of mechanical displacement. Potentially this will soon enable to observe quantum jumps between mechanical Fock states [20,88,90,99,100] or to measure “phonon shot noise” [91]. Both measurements fundamentally depend on the existence of two coupled optical modes. Another experimental realization of a coupled multimode setup (Fig. 1.4b) recently interconnected two microtoroids implementing a phonon laser that involves a tunable optical two-level system [97]. Another setup has coupled nanomechanical beams to a microtoroid [74] (Fig. 1.4c).

1.4.2

This thesis

In this thesis we investigate dynamical features and implications of coupled multimode optomechanical systems. We generally focus on specific setups where system parameters have been worked out. Nevertheless, the general concepts apply to a variety of optomechanical implementations. Going beyond the steady-state dynamics, that is usually considered for optomechanical system up to now, in Chapter 2, we ask the question of what happens if we subject the mechanics to an external period drive. It turns out that for coupled multimode setups this enables to deliberately transfer photons between individual optical modes. As we will see, this new domain of mechanically driven coherent photon dynamics in principle allows one to transfer the whole realm of strongly driven multi-level dynamics, that is known from atomic physics, to optomechanical systems. Based on our work on mechanically driven coherent photon dynamics between optical modes, in Chapter 3, we ask how the dynamical backaction (Section 1.2.2) changes for multimode optomechanical systems. In particular, we develop the fully nonlinear theory of phonon lasing (self-induced mechanical oscillations) in such a setup and interpret the corresponding attractor diagram (Fig. 1.3). In Chapter 4, we turn towards the microwave regime, where recently the analog of the standard optomechanical system (Eq. (1.1)) has been realized [24]. We investigate implications of coupled multimode optomechanical systems in the new domain. As we will see, these systems have advantages that go beyond bulk refrigerator cooling and on-chip integration that are usually considered to be main reasons for doing optomechanical experiments using microwave resonators. We introduce a microwave scheme that allows one to perform QND Fock state detection and calculate the signal-to-noise ratio to detect a single quantum jump. Finally, based on the new concept to design small, on-chip optomechanical systems using photonic crystal structures for instance, in Chapter 5, we consider assemblies of many coupled “optomechanical cells”. We term an optomechanical cell to be a single optical mode coupled to a single mechanical one, i.e. the standard optomechanical system. With the prospect of fabricating such coupled designs in the near future, we start to investigate the collective

1.4 Coupled multimode optomechanical systems dynamics of such “optomechanical arrays”.

15

16

1. Introduction

Chapter 2

Strongly driven two- and multilevel dynamics in optomechanics 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. Up to now it has been customary to focus on the steady-state dynamics of these systems. For instance, using an optical drive and recording the transmission or reflection, the mechanics is generally influenced and detected in steady state. Going beyond the usual approach, we ask the question of what happens if one subjects such a system to strong external mechanical driving. It turns out that the most impressive effects of periodic, time-dependent driving on the light field can be discerned in optomechanical systems that consist of several coupled optical (and vibrational) modes. In general, we propose an approach to investigate non-equilibrium photon dynamics driven by mechanical motion in coupled multimode optomechanical systems. As we will show, mechanical driving allows one to deliberately transfer photons between different optical modes, in a coherent fashion. Despite the concept’s generality, we concentrate on a specific setup that has recently been realized in the group of Jack Harris at Yale University. It consists of a movable membrane placed in the middle between two fixed, high-finesse mirrors. For this scheme, experimental parameters have been worked out such that all parameter regimes reported in this chapter can be realized using current-state technology. Basically, in such a system one can identify the presence of a photon in the left or the right half of the cavity with the two states of a two-level system. As we will see, external driving then opens up the whole domain of strongly driven quantum systems to the field of optomechanics. Essentially, the mechanical drive is working to shuttle photons from left to right. We will show that this leads to effects such as mechanically driven photonic Rabi oscillations and Autler-Townes splittings in the spectroscopy of the light passing through the setup. At larger mechanical driving strength, intricate interference effects, known from atomic systems as Landau-Zener-Stueckelberg oscillations, show up. The present setup would for the first time allow one to observe this fundamental dynamical interference phenomenon in an optomechanical system (or, indeed, in any optical system). More generally, beyond the specific setup considered here, the concept of mechanically driven coherent photon dynamics also applies to larger optomechanical systems involving several optical and mechanical modes. In principle, one may think of all the multi-level

18

2. Strongly driven two- and multilevel dynamics in optomechanics

dynamics known from atomic physics (like Λ- and V -type systems), replacing lasers and atomic levels by mechanical drives and optical modes, respectively. This is all the more relevant due to the recently introduced idea of optomechanical circuits, where multiple optical (and vibrational) modes may be combined and coupled on a chip. Thus, the whole realm of strongly driven multi-level atom dynamics may be transferred and adapted to optomechanical systems. The main results of this chapter, in particular the ones on the strongly driven regime showing Landau-Zener-Stueckelberg dynamics, have been published in • Georg Heinrich, J. G. E. Harris and Florian Marquardt, Photon shuttle: Landau-Zener-St¨ uckelberg dynamics in an optomechanical system, Physical Review A 81, 011801(R) (2010) Copyright (2010) by the American Physical Society. Some more details on this strongly driven dynamics can be found in the more general review articles that have been published in • Georg Heinrich, Max Ludwig, Huaizhi Wu, K. Hammerer and Florian Marquardt, Dynamics of coupled multimode and hybrid optomechanical systems, Comptes Rendus Physique (2011), doi:10.1016/j.crhy.2011.02.004 (in press), • Max Ludwig, Georg Heinrich and Florian Marquardt, Examples of quantum dynamics in optomechanical systems, in Quantum Communication and Quantum Networking (Springer 2010) - proceedings of QuantumCom 2009. Further results complementing the regimes of weak and modest mechanical driving, that have been added to this thesis, might be published elsewhere.

2.1

Introduction

Optomechanical systems couple mechanical degrees of freedom to radiation fields and constitute a rapidly evolving field of current research (reviewed in [2]). These systems provide new means to manipulate both the light field and the mechanical motion. Apart from the hope of eventually exploring the quantum regime of mechanical motion, there have been several studies of the complex nonlinear dynamics of these systems [17, 54, 55, 67, 70]. An exciting recent development is the introduction of setups with multiple coupled optical (and vibrational) modes, pointing the way towards integrated optomechanical circuits [20, 39–41]. In this chapter, we show how the application of an external mechanical drive to these structures can open up the whole domain of strongly driven two- and multilevel systems to the field of optomechanics. As a concrete example of such a mechanically driven coherent photon dynamics, we consider the system recently realized in [20, 90], where we show that a vibrating membrane inside an optical cavity can shuttle photons between two optical modes. We predict effects such as mechanically driven photonic Rabi oscillations, Autler-Townes splittings and Landau-Zener-Stueckelberg (LZS) oscillations visible in the transmission spectrum. Landau-Zener (LZ) transitions [101,102] and LZS oscillations [103] have originally been studied in atomic systems [104–106], but lately they have also been applied to quantum dots and superconducting qubits [107–110]. Some purely optical setups [111, 112] have also mimicked

2.2 Setup

19 3

aL

aR

Opt. frequency

ω

ωL

aL 2g

0

−1

aR

−2

−3

(b)

x0

1

−3

x(t) (a)

ω+

2

ω−

−2

−1

−A +A 0

1

Displacement

2

3

x

Figure 2.1: (a) Setup: a dielectric membrane couples two modes aL , aR inside a cavity. The left hand side is excited by a laser ωL while the transmission to the right is recorded. (b) Optical resonance frequency as function of displacement: the membrane’s displacement linearly changes the bare mode frequencies (dashed). Due to the coupling g, there is an avoided crossing of the eigenfrequencies ω± (black). The membrane is driven, with x(t) = A cos(Ωt) + x0 (blue). (This figure has previously been published in [113].) two-level and standard LZ dynamics, but not LZS oscillations. More generally, the mechanically driven coherent photon dynamics in multimode optomechanical systems introduced in this chapter will allow one to realize analogs to driven atomic multi-level systems, such as V -type and Λ-type level schemes and effects such as coherent trapping or electromagnetically induced transparency.

2.2 2.2.1

Setup System Hamiltonian

We consider the optomechanical system depicted in Fig. 2.1a. A highly-reflective, dielectric membrane is placed in the middle between two high-finesse mirrors. Due to its dielectric properties, the membrane couples modes of left and right cavity half, respectively. This involves photon tunneling through the device. Such an optomechanical system has recently been designed in [20] and is now used for many optomechanical experiments. We take into account two optical modes aL , aR in the left and the right cavity half (each of length l), respectively. The system Hamiltonian then reads x(t) x(t) † ˆ Hsys = ~ω0 1 − a ˆL a ˆL + ~ω0 1 + a ˆ†R a ˆR l l ˆ drive + H ˆ decay . + ~g a ˆ†L a ˆR + a ˆ†R a ˆL + H

(2.1)

Here, a ˆ†L a ˆL and a ˆ†R a ˆR are the photon number operators for the two optical modes whose resonance frequency ω0 is changed due to the displacement x of the membrane. The coupling g describes photon tunneling through the membrane. Due to the coupling, there is an avoided

20


p crossing in the optical resonance frequency ω± (x) = ± g 2 + (ω0 x/l)2 , see Fig. 2.1b. To investigate photon dynamics, the cavity needs to be excited in terms of an external laser ˆ drive . Furthermore, H ˆ decay describes photon decay out of drive added to the Hamiltonian, H the optical modes.

2.2.2

Mechanically driven non-equilibrium photon dynamics

We propose to drive the membrane with mechanical frequency Ω and resulting amplitude A around a mean position x0 , x(t) = A cos(Ωt) + x0 , (2.2) and investigate the photon dynamics. We point out that here Ω need not coincide with the membrane’s eigenfrequency but depends only on the driving. Note that this mechanically driven system differs from the steady-state situation that is usually considered for optomechanics so far. We are particularly interested in the regime where the timescale of photon exchange is comparable to the timescale of the mechanical motion (g ' Ω). Recently the coupling frequency g/2π has been significantly reduced by exploiting properties of transverse modes [114], and it is tunable down to 200 kHz at present. The mechanical eigenfrequencies of typical 1 mm×1 mm×50 nm membranes range between 100 kHz and 1 MHz. Thus, this new regime, that we will start to investigate here, is accessible in experiments using current-state technology. For g ' Ω, due to the fast mechanics, the system’s dynamics can no longer be treated as quasi-static in terms of the parabola branches depicted in Fig. 2.1b. Instead, non-equilibrium dynamics of photons driven by the membrane must be taken into account. For an experiment, we propose to optically drive the left hand side of the cavity with a laser of tunable frequency ωL and investigate the photon dynamics in terms of the cavity transmission to the right. This observable is easily measured using a photo detector placed on the other side of the cavity, see Fig. 2.1a. The mechanical driving Eq. (2.2) might be realized by mounting the membrane on a piezo actuator.

2.2.3

Equation of motion and transmission to the right

To take into account photon decay and laser driving of the optical system, the optical modes need to be coupled to the environment modeled in terms of a bath of external, bosonic modes. Doing so, input/output theory conveniently describes the effective dynamics. The equation of motion for the optical mode a î reads i √ d 1 h ˆ sys (t) − κ a a î (t) = î (t) − κ ˆbin a î (t), H i (t), dt i~ 2 with the cavity decay rate κ for each of the modes and the input field ˆbin i . Most generally ˆbin = bin e−iωL t + ξî (t) comprises both, classical coherent laser driving at frequency ωL with i i amplitude bin and noise contributions added in terms of the noise operator ξî . In the following, we are interested in the purely classical (large-amplitude) light field dynamics. Given the system Hamiltonian and considering an exclusive coherent drive of the left mode, the equation of motions for the average fields aL = hˆ aL i and aR = hˆ aR i read √ d 1 κ aL = [−¯ x(t) aL + g aR ] − aL − κ bin L (t) dt i 2 d 1 κ aR = [+¯ x(t) aR + g aL ] − aR , (2.3) dt i 2

2.3 Direct simulations: transmission spectra for increasing driving strength

21

−i∆L t bin is the laser drive with amplitude bin . To ease the notation, we used where bin L (t) = e a rotating frame with laser detuning from resonance

∆L = ωL − ω0 . Note that the displacement x(t) always enters the Hamiltonian together with the constant factor, ωopt /l, translating the displacement x into a frequency shift x ¯ = (ωopt /l)x. In the following, we will express all lengths in terms of frequency and define A¯ = (ωopt /l)A, x ¯0 = (ωopt /l)x0 . We note that for an optical cavity with ωopt /2π = 3 · 1014 Hz (λopt = 1000 nm) and l = 1 cm, we find ωopt /l = 30 MHz/nm. To study the photon dynamics, we calculate the transmission that is an effortlessly accessible observable in experiments. As we will show below, the transmission to the right, T (t) = κhˆ a†R (t)ˆ aR (t)i/(bin )2 , can be expressed as Z T (t) = κ 2

t

−∞

0

−i∆L t0 −(κ/2)(t−t0 )

GR (t, t ) e

2 0

dt ,

(2.4)

where the phase factor includes the laser drive ∆L and cavity decay κ, while the Green’s function GR (t, t0 ) describes the amplitude for a photon to enter the cavity from the left at time t0 and to be found in the right cavity mode later at time t. Technically, GR (t, t0 ) is found by setting κ = 0 in Eq. (2.3) and solving for aR (t) with the initial conditions aL (t0 ) = 1, aR (t0 ) = 0 (see Section 2.4 and Appendix A.1).

2.3

Direct simulations: transmission spectra for increasing driving strength

Before starting a rigorous analysis of the dynamics, we first want to get an idea of some of the features that can be observed for the mechanically driven optomechanical system, Fig. 2.1a. For this reason, we first conduct some numerical simulations for the transmission spectrum according to Eq. (2.4). Fig. 2.2a displays the time-averaged transmission depending on mean position x ¯0 and laser ¯ detuning ∆L without mechanical driving (A = 0). The spectrum displays the two hyperbola branches ω± (Fig. 2.1b). Transmission is largest at the avoided crossing where photons can most easily tunnel from the left into the right mode. For mechanical driving, the spectrum features mechanical sidebands, see Fig. 2.2b. These excitations are due to energy exchange between the light field and the oscillating membrane in terms of multiples of the mechanical oscillation frequency Ω. The transmission spectrum gets more involved if sideband excitations can cross the underlying photon branches (that are separated by 2g at the avoided crossing, see Fig. 2.1b). Fig. 2.3 displays results for Ω = 4g. In this case, the first mechanical sideband can intersect the upper photon branch. For increasing mechanical driving strength A¯ (Fig. 2.3a-d), the sideband becomes more intense and avoided level crossings emerge between the mechanical excitation and the hyperbola branches. The size of all splittings, including the original anticrossing ω± , depends on the driving strength. In Fig. 2.3d the latter is tuned to zero. In the following, we will work out a complete description of these phenomena. This will explain the parameter dependence indicated in Fig. 2.3.


Detuning ∆L /g

22

44 22

44 %

22

100

00

2g

50

00

0

−2 -2

−2 -2

−4 -4

(a)

Ω

−4 -4

−4 -4

−2 -2

00

22

44

(b)

−4 -4

−2 -2

00

22

44

¯0 /g Mean position x Figure 2.2: Transmission spectrum resembling the optical resonances. Density plot for the time-averaged transmission as function of mean position x0 and laser detuning ∆L . Cavity decay rate κ = 0.1g. (a) Without mechanical drive (A¯ = 0), the spectrum corresponds to the hyperbola branches ω± depicted in Fig. 2.1b. (b) Considering a mechanical drive with A¯ = 0.5 Ω, Ω = 0.5 g, the spectrum comprises mechanical sidebands due to excitations in terms of multiphonon transitions mΩ.

2.4 2.4.1

Photon dynamics resembling atomic multilevel physics Excitation process and internal dynamics

To understand the dynamics that initiate the phenomena displayed in Fig. 2.3, we note that the observation of transmission through the cavity is generally determined by two subsequent processes. First, the laser must excite the left mode aL to insert photons into the cavity. Second, the internal dynamics must be able to transfer photons from the left into the right mode such that transmission can actually be observed. In general, both processes are inelastic and require energy to be transferred between the light fields and the oscillating membrane. Thus, for sufficiently strong mechanical driving, both the cavity excitation as well as the internal dynamics will involve multiphonon transitions in terms of multiples of the mechanical driving frequency. For a quantitative analysis, we now show that the internal dynamics describing photon transfer between the two halves of the cavity can be mapped to the dynamics of a two-state system, represented by the two optical modes of the system.

2.4.2

Two-level dynamics with time-dependent coupling

For an analytical description, we first find the general structure of the light field dynamics. For an external mechanical drive x(t) = A cos(Ωt) + x0 , the formal solution to Eq. (2.3) can be expressed as Z t 2 2 in 2 0 −i∆L t0 −(κ/2)(t−t0 ) 0 |ai (t)| = κ(b ) Gi (t, t ) e dt , (2.5) −∞

see Appendix A.1. Here, the Green’s function Gi (t, t0 ) describes the amplitude for a photon to enter the left mode at time t0 and to be found in the left or right mode (i = L, R) later at

2.4 Photon dynamics resembling atomic multilevel physics

3

3

22

22 %

11 00

Detuning ∆L /g

23

100

11

50

00

0

−1 -1

2gJ1

2gJ0

−1 -1

−2 -2

−2 -2

(a)

(b)

−3

−3

−3

−2

-2

−1

-1

0

0

1

1

2

2

3

3

−3

−2 -2

−1 -1

00

11

22

3

3

2gJ1

22

22

11

2gJ1

11

2gJ0

00 −1 -1

−1 -1

−2 -2

−2 -2

(c)

−3

−3 −3

-2

−2

-1 −1

0 0

1 1

2 2

2gJ0

00

3

(d) −3

-2−2

-1−1

00

11

22

3

¯0 /g Mean position x Figure 2.3: Transmission spectrum for increasing mechanical driving strength. Density plot of the time-averaged transmission for cavity decay rate κ = 0.1 g and a mechanical drive at frequency Ω = 4 g. (a) Driving strength A¯ = 0.05 Ω, (b) A¯ = 0.2 Ω, (c) A¯ = 0.8 Ω, (d) A¯ = 1.2 Ω. The first mechanical sideband can intersect the photon branches, cf. Fig 2.1b. At these intersections, avoided crossings arise, whose gap is determined by the mechanical drive ¯ according to 2gJ1 (A/Ω), see analysis below. The original spacing between the two photon ¯ branches is renormalized according to 2gJ0 (A/Ω). J0 and J1 denote Bessel functions.

24


time t. From Eq. (2.3), Gi (t, t0 ) is found to be 0

Gi (t, t0 ) = a ˜i (t, t0 )e−iφ(t ) ,

(2.6)

¯ where φ(t0 ) = (A/Ω) sin(Ωt0 ), and a ˜R (t, t0 ) is a solution to d i dt

a ˜R a ˜L

=

x ¯0 ge−2iφ(t)

ge+2iφ(t) −¯ x0

a ˜R a ˜L

,

(2.7)

with t ≥ t0 and initial condition a ˜R (t0 , t0 ) = 0, a ˜L (t0 , t0 ) = 1 (see Appendix A.1). For the transmission this yields Eq. (2.4). Each factor in Eq. (2.6) describes one of the processes discussed in Section 2.4.1. For the excitation of the left mode, ¯ X A −imΩt0 0 e−iφ(t ) = Jm e (2.8) Ω m describes possible multiphonon transitions mΩ in addition to excitations due to the laser at ∆L , see Eq. (2.5). The strength of the individual process mΩ is determined by a Bessel function Jm . On the other hand, a ˜i (t, t0 ) describes the internal dynamics that shuttles photons, originally inserted into the left mode at time t0 , between the two halves of the cavity. Corresponding to Eq. (2.7) this process can be expressed in terms of a two-level system with time-dependent coupling X 2A¯ 2iφ(t) ge = Jn einΩt . (2.9) Ω n

Given this general structure of the light field dynamics, we now discuss mechanically driven ¯ coherent photon dynamics for various regimes of the driving strength A.

2.5

Analysis for weak and modest mechanical driving

To obtain an analytical description and physical understanding of mechanically driven coher¯ 1. To appreciate ent photon dynamics, we start considering weak mechanical driving, A/Ω the general structure of the dynamics presented in Section 2.5.2, that is in fact independent T ˜R a ˜L of any specific driving strength, it is useful to define |ψi = a and H0 =

x ¯0 g0 , g0 −¯ x0

(2.10)

¯ where g0 = gJ0 (2A/Ω). Thus, we can rewrite Eq. (2.7) as Schrödinger Equation i

d |ψ; t, t0 i = [H0 + V (t)] |ψ; t, t0 i, dt

(2.11)

separating time-independent and time-dependent contributions of the two-level system. Given T 0 , t0 i = 0 1 this notation, the initial condition reads |ψ; t . √ We note that for 0 < y n + 1 and a positive integer k, the asymptotic form of the 1 x k Bessel function Jk (y) reads Jk (y) ' k! 2 . Thus, for weak mechanical driving, i.e. small

2.5 Analysis for weak and modest mechanical driving

25

¯ arguments 2A/Ω, we can neglect contributions with |m|, |n| > 1 in Eq. (2.8) and Eq. (2.9). Using J−n (y) = (−1)n Jn (y), we approximate V (t) ' g1

0 eiΩt − e−iΩt −iΩt iΩt e −e 0

(2.12)

where g1 = gJ1 (2A/Ω). With this we can find an analytical description of the weak mechanical driving regime.

2.5.1

Transmission without mechanical driving

To be able to compare our results and highlight features that are due to the mechanical drive, we first consider the static case in the absence of mechanical driving (A¯ = 0). Then we have g0 = g, g1 = 0 and |ψi is simply determined by the eigenstates and eigenvalues of Eq. (2.10), i.e. H0 |±i = E± |±i, where

q E± = ± x ¯20 + g02 .

(2.13)

T T . They characterize a single and |2i = 0 1 We denote the states |1i = 1 0 photon being in the right and left cavity half, respectively. The initial condition then reads |ψ; t0 , t0 i = |2i and from the time evolution of Eq. (2.11) we find 0

a ˜R (t, t0 ) = h1|+ie−iE+ (t−t ) h+|2i 0

+ h1|−ie−iE− (t−t ) h−|2i.

(2.14)

0 Considering this for the Green’s function, Eq. (2.6), where e−iφ(t ) = 1 as A¯ = 0, we calculate the transmission according to Eq. (2.4). This yields for the transmission in the absence of mechanical driving,

h1|+ih+|2i T =κ κ + 2 + i (E+ − ∆L ) 2

κ 2

h1|−ih−|2i 2 . + i (E− − ∆L )

(2.15)

According to Eq. (2.15), there can be transmission if the laser at frequency ∆L is in resonance with the cavity’s eigenvalues E± . The resonance width is set by the cavity decay rate κ. Note that for A¯ = 0 we have g0 = g and E± equal the hyperbola branches ω± depicted in Fig. 2.1b. The weight of the two resonances is set by A± = h1|±ih±|2i and depends on the mean position x ¯0 . Physically, A± is the amplitude to excite the cavity state |±i by a photon inserted into the left mode h±|2i, and to leave the cavity out of the right mode h1|±i. The transmission is unity right at the anti-crossing, see Fig. 2.2a. Here the eigenstates |±i are symmetric and√antisymmetric superpositions of a photon inside the left and right mode, |±i = (|1i ± |2i)/ 2. Thus, photons, inserted into the left mode, can most easily tunnel to the right. Away from the anti-crossing, the internal process, transferring photons between the two cavity halves, is suppressed and the transmission reduces. For x ¯0 g, we have |+i = |1i, |−i = |2i such that A± = 0, and there is no transmission at all.

26

2.5.2


Mechanically assisted photon transfer

In case of mechanical driving (A¯ 6= 0), energy can be exchanged between the mechanics’ motion and the light field. As pointed out, in addition to the laser at frequency ∆L , the phase 0 factor e−iφ(t ) of the Green’s function Eq. (2.6) describes excitations of the cavity in terms of multiphonon transitions mΩ, see Eq. (2.4) and (2.5). This yields mechanical sidebands in the transmission spectrum. The strength of the individual transition mΩ is set by Bessel ¯ , see Eq. (2.8). function Jm A/Ω Besides the excitation process, the mechanical drive also modifies the internal dynamics due to V (t), Eq. (2.11). To find the general structure of the solution |ψ; t, t0 i, we use the interaction picture representation 0

|ψiI = eiH0 (t−t ) |ψi and expand in terms of the eigenbasis |±i, |ψiI = |ψi =

X

P

n=± cn (t)|ni.

We then have

0

e−iEn (t−t ) cn (t)|ni,

(2.16)

n=±

where the coefficients are determined according to i

X d 0 cn (t) = Vnm ei(En −Em )(t−t ) cm (t), dt m

(2.17)

with Vnm = hn|V (t)|mi. To meet the initial condition |ψ, t0 , t0 i = |2i, we demand c+ (t = t0 ) = h+|2i,

c− (t = t0 ) = h−|2i.

(2.18)

According to Eq. (2.16) we then have 0

a ˜R (t, t0 ) = c+ (t, t0 )e−iE+ (t−t ) h1|+i 0

+ c− (t, t0 )e−iE− (t−t ) h1|−i.

(2.19)

Much of the qualitative photon dynamics becomes clear from this general structure of the dynamics, Eq. (2.19) and (2.17). In the absence of mechanical driving (A¯ = 0), we have V (t) = 0 and c± (t, t0 ) are constant. In this case Eq. (2.19) recovers Eq. (2.14). For A¯ 6= 0, Eq. (2.17) initiates a dynamics for the coefficients c± (t, t0 ). If the mechanical drive is resonant with the frequency difference of the two optical modes, this introduces an additional exchange of photons between left and right. In contrast to A¯ = 0 (Fig. 2.2a), this mechanically assisted photon transfer allows one to have high transmission even far away from the original anticrossing, see Fig. 2.3. If the drive frequency Ω is non-resonant, c± (t) remain constant and we find transmission for ∆L = E± . Note however that unlike for the case A¯ = 0, we generally have g0 6= g, i.e. the original anti-crossing between the two photon branches (Fig. 2.1b) gets ¯ renormalized according to g0 = 2gJ0 (2A/Ω). As shown in Fig. 2.3d, it can even be tuned to zero. Thus, the effective optical coupling for photons between the two cavity halves can be tuned in terms of the mechanical drive.


2.5.3

27

Mechanically driven Rabi dynamics

Quantitatively, for weak mechanical driving, V (t) can be approximated as in Eq. (2.12). Using rotating wave approximation, Eq. (2.17) reads 0

c˙+ = ig1 e−i∆E t c− ei(∆E −Ω)t 0

c˙− = ig1 e+i∆E t c+ e−i(∆E −Ω)t ,

(2.20)

where we have defined ∆E = E+ − E− . Eq. (2.20) corresponds to the Rabi dynamics of a driven two-level system (see for instance [115]). Note however the additional phase factors 0 e±i∆E t in Eq. (2.20) and the initial condition (2.18). This is important to find the correct t0 -dependence of a ˜i (t, t0 ). The general solution to Eq. (2.20) that satisfies Eq. (2.18) is found to be g1 −iΩt0 1 ωR − (∆E − Ω)/2 0 0 h+|2i + e h−|2i eiωR (t−t ) c+ (t, t ) = (2.21) 2 ωR ωR g1 −iΩt0 ωR + (∆E − Ω)/2 0 0 h+|2i − e h−|2i e−iωR (t−t ) ei(∆E −Ω)(t−t )/2 + ωR ωR 1 ωR + (∆E − Ω)/2 g1 +iΩt0 0 0 c− (t, t ) = h−|2i + e h+|2i eiωR (t−t ) (2.22) 2 ωR ωR g1 +iΩt0 ωR − (∆E − Ω)/2 0 −iωR (t−t0 ) h−|2i − e h+|2i e e−i(∆E −Ω)(t−t )/2 , + ωR ωR where we defined the Rabi frequency ωR =

p (g1 )2 + (∆E − Ω)2 /4.

(2.23)

¯ Note that ωR depends on the mechanical driving strength A¯ via g1 = gJ1 (2A/Ω), and the detuning of the mechanical frequency Ω from the frequency difference of the two optical modes ∆E . The latter, ∆E = E+ − E− , is determined by the mechanics’ position x ¯0 . The results on c± (t, t0 ) determine the Green’s functions GL (t, t0 ) and GR (t, t0 ), see Eq. (2.6) as well as Eq. (2.17) and (2.19). Thus, we have found a general, analytical expression for the light field dynamics aL (t), aR (t) for the case of modest mechanical driving, Eq. (2.5). We note that these results also allow one to calculate the back-action of the photon dynamics on the mechanics in terms of radiation pressure forces. Accordingly, these expressions will also enter the description of self-induced mechanical oscillations in multimode optomechanical systems that we will discuss in Chapter 3. 2.5.3.1

Interpretation of the full expression for x ¯0 g.

For an interpretation of the analytical result, we discuss the case x ¯0 g. This will highlight some of the essential features of mechanically driven coherent photon dynamics while keeping the discussion as simple as possible. For x ¯0 g, we have |+i = |1i, |−i = |2i and the laser can only excite the |−i state of the cavity, (h+|2i = 0, h−|2i = 1, see Eq. (2.21) and (2.22)), while transmission to the right is governed exclusively out of the |+i state (h1|+i = 1, h1|−i = 0, see Eq. (2.19)). This significantly simplifies the expression on GR (t, t0 ) in Eq. (2.6) and we find, g1 0 0 GR (t, t0 ) = −i sin ωR (t − t0 ) e−iΩ(t+t )/2 e−iφ(t ) . (2.24) ωR

28


Note that in case of no mechanical driving (A¯ = 0), for x ¯0 g, there is no transmission at all, cf. the discussion in Section 2.5.1. 0 As pointed out in Section 2.4.1, e−iφ(t ) describes cavity excitations in terms of multi¯ phonon transitions mΩ in addition to the laser, see Eq. (2.8). For weak driving A/Ω 1, 0) −iφ(t ¯ ¯ we have J0 (A/Ω) ' 1 Jm (A/Ω), m 6= 0 and we can consider e ' 1 (note the asymptotic form of Jk (y) discussed in connection with Eq. (2.12)). With this, we determine the transmission (Eq. (2.4)) for x ¯0 g, 2 κ 2 g 2 1 1 1 Ω − κ Ω . (2.25) T = κ 2 ωR 2 + i − 2 − ωR − ∆L + i − + ω − ∆ R L 2 2 According to the overall factor, 2 g12 g1 , = ωR (g1 )2 + (∆E − Ω)2 /4

¯ where g1 = gJ1 (2A/Ω), there can only be transmission if the mechanical drive frequency Ω is in resonance with the frequency difference between the two optical modes ∆E = E+ − E− . In this case, mechanically driven Rabi dynamics can shuttle photons between the left and the right mode, cf. the qualitative discussion in Section 2.5.2. This corresponds to the two spots of high transmission in Fig. 2.3a found from numerical simulation. The width of the ¯ Note that for mechanical resonance is set by g1 and depends on the driving strength A. x ¯0 g, the frequency of the left cavity mode is −¯ x0 and for ∆E = Ω we have −¯ x0 = −Ω/2. Using this, we can rewrite the expression to the right hand side of Eq. (2.25) as κ 2

+i

− Ω2

1 − − ωR − ∆L

κ 2

+i

− Ω2

1 = + ωR − ∆L

2iωR

κ 2

, 2 2 + i [−¯ x 0 − ∆ L ] + ωR

For very weak driving g1 κ/2, the transmission to the right in case of mechanical (∆E = Ω) and optical resonance (laser set to the left mode’s frequency, i.e. ∆L = −¯ x0 ), reads 2 ¯ T = (4g A/κΩ) .

Thus, for very weak driving the transmission quadratically increases with the mechanical ¯ driving strength A.

2.5.4

Autler-Townes splitting

¯ the Rabi frequency ωR , Eq. (2.23), increases. According For increasing drive amplitude A, to Eq. (2.25), for ωR > κ/2, the transmission spectrum splits into two peaks. In case of mechanical resonance (∆E = Ω), we find high transmission at ∆L = −¯ x0 ± g1 . Thus, the size ¯ of the splitting is set by 2gJ1 (2A/Ω), see Fig. 2.3a-d. We note that each peak can yield unity transmission. In general, for resonant driving, the transmission spectrum shows a splitting of the two hyperbola branches ω± , see Fig. 2.4a. As shown above, the mechanical drive induces Rabi oscillations between the two photon branches, at a Rabi frequency ωR leading to a corresponding splitting in the spectroscopic picture. For modest driving A¯ < Ω, we can linearize the coupling (Eq. (2.12)) and find A¯ g1 ' g . (2.26) Ω


29

3 2

Detuning ∆L /g

∆L /g

2g1

2

1

%

Ω(¯ x0 )

%

50

0

25

-1

0

-2

100

0

1

50

-2 -1

0

0

1

2

x ¯0 /g

(c)

-1

a -2

b -3

probe -3

(a)

-2

-1

0

1

¯0 /g Mean position x

2

c

3

(b)

Figure 2.4: Autler-Townes splitting. (a) Density plot for the time-averaged transmission depending on mean position x ¯0 and laser detuning ∆L for cavity decay rate κ = 0.1 g. The transmission spectrum shows an Autler-Townes splitting of the cavity frequency ω± due to mechanical motion, A¯ = 0.2 Ω. For every position x ¯0 , the mechanical drive frequency is set p 2 2 to be Ω = 2 g + x ¯0 such that it is always resonant with the frequency difference between ¯ the two optical modes at ω± . The splitting is set by the Rabi frequency g1 ' g A/Ω, see Eq. (2.26). (b) Schematic of the standard Autler-Townes phenomena known from quantum optics: an atomic two-level system a ↔ b, driven by an intense laser at ΩL , is probed via a second, low intensity beam whose frequency is close to a second transistion b ↔ c sharing the common level b; ∆ab 6= ∆bc . Due to the strong drive, atom-photon interaction splits both levels a, b (right) leading to two distinct transitions for the probe laser (blue arrows). The Autler-Townes doublet can be observed in the absorption spectrum of the probe. (c) Plot as in (a) but for stronger drive A¯ = 1.6 Ω. Mechanical sidebands, displaced by ±Ω, become visible and interact. (Panel (a) and (c) of this figure have previously been published in [113].)

30


This corresponds to an Autler-Townes splitting [111, 116] known from quantum optics, see illustration in Fig. 2.4b. In the standard scenario, an intensively laser-driven atomic transition gets dressed due to the interaction with the photon field (see for instance [117]). Similarly, in our case, the resonance frequencies of the two optical modes, that constitute the two-level system, split due to the interaction with the mechanical drive. Note that the excitation of the left cavity mode in terms of the laser at ∆L precisely corresponds to probing the mechanically driven two-level system via an additional transition like illustrated in Fig. 2.4b. To record transmission, the left cavity mode first needs to be excited, i.e. a photon is “absorbed” into the cavity. Subsequently, the mechanically assisted process transfers photons from left to the right, enabling high transmission even farther away from the anti-crossing (cf. Fig. 2.2a for A¯ = 0). Note that transmission is largest if the laser is in resonance with the left mode, i.e. ∆L = −¯ x0 for |¯ x0 | g, see Fig. 2.4a. This facilitates the excitation of the cavity. For strongly driven atomic systems, a phenomena related to the Autler-Townes splitting is the so-called Mollow triplet [118, 119]. Compared to Fig. 2.4b, in such an experiment there is no probe beam sampling transitions to a third level c. In contrast, the spectrum of the resonance fluorescence of the strongly driven atomic system a ↔ b is measured [119]. For sufficiently high laser intensities, transitions between the dressed states (Fig. 2.4b) result in three peaks for the fluorescence spectrum, the so-called Mollow triplet. For our optomechanical system, the probe in Fig. 2.4b corresponds to the excitation of the cavity in terms of the laser ∆L . Photons residing in distinct modes constitute the two-level system. Thus, this part is crucial and there is no direct, easily accessible analogue to the Mollow triplet in our case. Increasing the drive amplitude beyond the regime of Autler-Townes splittings, see Fig. 2.4c, the dynamics becomes more involved as mechanical sidebands arise and interact with each other. In the following, we turn towards this regime and even stronger mechanical driving.

2.5.5

The emergence of multiphonon processes

If the mechanical drive is intensified beyond the regime where we can approximate the coupling as in Eq. (2.12), we expect two qualitative trends for the dynamics. First, in addition to the laser, the strongly driven mechanics will allow one to excite the cavity in terms of multiphonon transitions. For the transmission spectrum, this will cause additional mechanical sidebands. Second, also the internal dynamics, governed by the time-dependent coupling (Eq. (2.9)), will be modified by multiphonon interactions. In particular, the mechanically assisted photon transfer (Section 2.5.2) might involve several phonons at once. 2.5.5.1

Internal dynamics: multiphonon couplings

Fig. 2.5 shows numerical results for the transmission spectrum, that involves several mechanical sidebands, for increasing mechanical driving strength. In addition to level repulsions governed by single-phonon interactions, see Eq. (2.12) and Fig. 2.3, we observe higher order splittings where the photon branches ω± can be connected by two phonon transitions 2Ω. We ¯ several terms arise for the time-dependent note that for sufficiently large driving amplitude A, coupling, Eq. (2.9). From the general structure of the internal dynamics (Eq. (2.17)) we see ¯ that the strength of the n-phonon interaction is governed by gJn (2A/Ω). The two-phonon ¯ process thus leads to a splitting whose size is determined by 2gJ2 (2A/Ω), see Fig. 2.5a-c. Given our analysis, that accounts for multiple phonon interactions and the renormalization


3

3

2gJ2

22

22

11

11

2Ω

Detuning ∆L /g

00

00

−1 -1

−1 -1

−2 -2

−2 -2

−3

(a) −3

−3

−2 -2

−1 -1

00

11

22

3

(b)−3

3

3

22

22

11

11

00

00

−1 -1

−1 -1

−2 -2

−2 -2

−3

(c) −3

31

−2 -2

−1 -1

00

11

22

3

Ω(¯ x0 )

−3

−2 -2

−1 -1

00

11

22

3

(d) −3 -2−2

Mean position

−1 -1

00

11

22

3

x ¯0 /g

Figure 2.5: Multiphonon interactions and higher order splittings. (a)-(c) Transmission spec¯ (a) trum for fixed mechanical drive frequency Ω = 2g and increasing driving strength A: A¯ = 0.5Ω, (b) A¯ = 0.8Ω, (c) A¯ = 1.2Ω. Red lines display the optical resonance frequency ω± in case of no mechanical driving (see Fig. 2.1b) - lower branch (solid), upper branch (dotted). White lines show the corresponding mechanical sidebands ω± ± mΩ. In addition to level repulsions including the first sideband, splittings due to interactions involving a two-phonon process (yellow arrow in panel (a)) become visible. The size of the corresponding anti-crossing ¯ varies with A¯ and is determined by 2gJ2 (A/Ω). Due to the drive, the original splitting ω± gets renormalized as indicated by red arrows, cf. Fig. 2.3. Accordingly, this displaces the sidebands (white arrows). (d) Transmission spectrum with adjusted mechanical drive frequency p Ω(¯ x0 ). For every position x ¯0 , we set Ω = 2 g 2 + x ¯20 such that it is always resonant with ¯ the difference ω+ − ω− (yellow), cf. Fig. 2.4c. The driving strength is A/Ω = 1.2. Thus, for x ¯0 = 0, (c) and (d) are equal. (The cavity decay rate κ = 0.1 g is the same for all panels.)

32


of the resonance frequency ω± , we generally have a good understanding of the transmission ¯ see the illustrations spectrum and its intricate evolution for increasing driving strength A, ¯ a complete analytical description is in Fig. 2.5a-c. We note that for large amplitudes A, complicated due to various, time-dependent terms that become relevant for the coupling in Eq. (2.9). Below we will find an exact, quantitative description for the strong driving regime, see Section 2.6.4. p In Fig. 2.4 we considered an adjusted driving frequency Ω = 2 g 2 + x ¯20 , i.e. Ω is tuned in resonance with ω+ − ω− for every position x ¯0 . This nicely illustrated the Autler-Townes splitting (Fig. 2.4a) but results became less intuitive for stronger driving (Fig. 2.4c). It is thus instructive to compare Fig. 2.5c, for fixed Ω, to a similar scenario with resonantly adjusted mechanical drive frequency Ω(¯ x0 ) shown in Fig. 2.5d. Parameters are chosen such that panel c and d are identical for x ¯0 = 0. Given our understanding of Fig. 2.5a-c we also comprehend Fig. 2.5d. Moving away from x ¯0 = 0, the driving frequency Ω(¯ x0 ) increases. In Fig. 2.5c ¯ ¯ this would displace the sidebands. Noting the level repulsions set by gJn (2A/Ω), where A/Ω is kept constant throughout the plot, this allows us to qualitatively understand the general structure of Fig. 2.5d for an adjusted drive frequency Ω(¯ x0 ), cf. also Fig. 2.4c.

2.5.5.2

Cavity excitation: Tuning multiphonon processes

For increasing mechanical drive, excitations in terms of multiphonon transitions become important. For the spectrum this yields additional sidebands that have already been observed in Fig. 2.5. Quantitatively, this is described by Eq. (2.8). In particular we note that the ¯ strength of a specific process mΩ is determined by the Bessel function Jm (A/Ω) and can be tuned via the driving strength. 0 The expression Eq. (2.25) (¯ x0 g) was derived for weak driving where eiφ(t ) ' 1. If in addition we consider single phonon excitations (|m| ≤ 1) the result for the transmission modifies and we have

T

=

κ 2 g 2 1

2

+ J1 e−iΩt −

J1 e+iΩt

ωR

κ 2

κ 2

× J0

+i

+i

− Ω2 − Ω2

κ 2

+i

− Ω2

1 − − ωR − ∆L

1 − − ωR − (∆L + Ω)

1 − − ωR − (∆L − Ω)

κ 2

κ 2

+i

+i

κ 2

+i − Ω2 − Ω2

− Ω2

1 + ωR − ∆ L

!

1 + ωR − (∆L + Ω)

!

1 + ωR − (∆L − Ω)

! 2 ,

¯ where Jm = Jm (A/Ω). ¯ The first term proportional to J0 (A/Ω) describes direct laser excitations excluding any ¯ phonon transitions. The other terms, proportional to J1 (A/Ω), lead to additional resonances at E− ± Ω, indicated in Fig. 2.6a. Note that for x ¯0 g we have E− = ω− = −¯ x0 and on mechanical resonance (∆E = Ω), where T 6= 0, we have −Ω/2 = −¯ x0 . Each resonance is split by ωR as discussed above. Remarkably, if we increase the drive to the first minimum of ¯ J0 (A/Ω) we can completely turn off direct laser excitations, see Fig. 2.6b. In this case the cavity can only be excited if the process involves phonon transitions.

Detuning ∆L /g

2.6 The strong mechanical driving regime

3ω+

44 22

−4 -4

(a)

ω+

22

+Ω

00 −2 -2

ω− −4 -4

−3¯ x0

44

00 -2 −2

33

3ω− −2 -2

−Ω 00

22

+¯ x0

−4 -4

44

(b)

Mean position

−4 -4

−2 -2

00

22

44

x ¯0 /g

Figure 2.6: Tunable excitation processes. Transmission spectrum for cavity decay rate κ = 0.1 g and adjusted drive frequency Ω = ω+ − ω− , as in Fig. 2.4c and Fig. 2.5d. (a) Driving strength A¯ = 1.6Ω. The spectrum comprises direct laser excitations as well as excitations including multiphonon transitions mΩ like the single-phonon process ω− ± Ω that is indicated by yellow arrows. Due to the adjustment of the drive frequency we have ω− − Ω(¯ x0 ) = 3ω− , ω− + Ω(¯ x0 ) = ω+ . (b) A¯ = 2.4Ω. At this driving strength, direct laser excitations without any phonon contribution are tuned to zero. Usually this direct process occurs if the laser is in resonance with the left mode’s frequency, i.e. ∆L = −¯ x0 for |¯ x0 | g, cf. (a).

2.6

The strong mechanical driving regime

Having discussed weak (A¯ Ω) and modest (A¯ ' Ω) mechanical driving of the coupled multimode optomechanical systems in Fig. 2.1, we now turn towards the strongly driven regime (A¯ Ω). Physically, for mechanical oscillations with sufficiently large amplitudes, the system repeatedly traverses the anti-crossing of the optical resonance frequency in Fig. 2.1b. Generally, if a parameter in a two-level system is non-adiabatically swept through an avoided crossing, the system may undergo an LZ transition into the other eigenstate [101, 102]. We will now see how such Landau-Zener physics affects the mechanically driven optomechanical system.

2.6.1

Landau-Zener physics

Landau-Zener (LZ) transitions [101, 102] are essential to the dynamics of many physical systems. In the usual model, a parameter in a two-state Hamiltonian is swept through an avoided level crossing where the two bare eigenstates |1i and |2i hybridize. Depending on the speed of the transition, the splitting might be passed adiabatically, diabatically or in a coherent mixture. In general, for a system that is prepared far away from the degeneracy point d in state |1i, and whose eigenenergy difference is swept at a constant energy change rate v = dt (E1 − E2 ) , the probability for a transition into the other state is given by the LZ formula π∆2 P1→1 = exp − , (2.27) 2~v where ∆ is the coupling strength. Thus, a sufficiently large sweep velocity, ∆2 . hv/π 2 , is needed to access the non-adiabatic regime.

34


2.5 1.01 0.8 2.0

0.8

0.6 0.6 0.4 0.4 0.2 0.2

1.5 0.0

0 0 0

0.1 0.1

(a)

0.2 0.2

0.3 0.3

0.4 0.4

0.5 0.5

(c)

time

(d)

1.0' 1.0 !&% 0.8 !&$ 0.6

0.5!&# 0.4

!&" 0.2

0.0! 0.0

(b)

! 0

1

"2

3

# 4

5

$6

7

% 8

time

−0.5

Figure 2.7: LZ physics of a single photon for strong mechanical driving. (a,b) Timedependence of the internal dynamics a ˜i (t, 0) according to Eq. (2.7) in units of the mechanical ¯ oscillation−1.0 period T = 2π/Ω. Parameters: A/Ω = 40, g/Ω = 5, x ¯0 = 0 (i.e. symmetric 2 sweep). |˜ aR (t, 0)| describes the amplitude for a single photon, inserted into the left mode 2 the right mode 4 at time t. At 6 t0 = 0, the mechanics 8 at time t0 = 0, to0be found inside is at its right turning point. (a) For fast mechanical sweeping there is a LZ transition when the system transverses the avoided crossing at t = T /4. (b) Due to the periodic mechanical drive we have repeated transitions of the anti-crossing. (c) The first LZ transition splits the photon state. (d) The two contributions gather different phases and interfere the next time the system transverses the avoided crossing. Beyond this standard LZ problem, the dynamics becomes more elaborate if repeated transitions are take into account. For a periodic modulation of the parameter, the first LZ transition splits the state into a coherent superposition α|1i + β|2i. Due to the difference in energy, the system afterwards accumulates a relative phase between states |1i and |2i. Thus, when returned to the avoided crossing, the system undergoes quantum interference with itself during the second LZ transition. This leads to interference patterns for the state population, so called Stueckelberg oscillations [103]. Originally, Landau-Zener-Stueckelberg (LZS) dynamics was studied in atomic systems [104–106]. Recently, the concept has been applied to superconducting qubits [107]. We note that currently, there is growing interest in LZ and LZS dynamics concerning topics such as state preparation and entanglement [108,109], cooling or qubit spectroscopy [110].

2.6.2

Landau-Zener-Stueckelberg oscillations in an optomechanical system

As we have seen, the mechanically driven photon dynamics of the optomechanical system in Fig. 2.1 is characterized by the single-photon Green’s function, Eq. (2.6). The dynamics can be mapped to a two-level system, Eq. (2.7). Thus, for a photon inserted into the left mode and a sufficiently fast sweep of the mechanics through the avoided crossing, we can have a


35


1.5

%

1.0

x ¯0 = Ω

0.5

x ¯0 = Ω/2

1.26 0.63 0.00

0.0

! "2 ¯ J−1 (A/Ω)

-0.5

-1.0

! "2 ¯ J−2 (2A/Ω)

-1.5 54.0

56.0

58.0

60.0

62.0

64.0

66.0

¯ Amplitude A/g Figure 2.8: LZS oscillations in an optomechanical system. Density plot for the time-averaged transmission as function of average displacement x ¯0 and mechanical drive amplitude A¯ Ω, g. Further parameters are laser detuning ∆L = 0, mechanical frequency Ω/2π = 0.2g and cavity decay rate κ = 0.2g. Finite transmission is only observed if x ¯0 is a multiple of Ω, i.e. when the resonance conditions Eq. (2.28) and (2.29) for multiphonon transitions are met. The transmission is modulated according to the product of two Bessel functions. 2 (A/Ω), ¯ For the case x ¯0 = −Ω both are depicted in the plot’s plane. Red: ∼ Jm due to the 2 ¯ excitation process. Yellow: ∼ Jn (2A/Ω), due to LZS dynamics. (This figure has previously been published in [113].)

LZ transition between the two optical modes, see Fig 2.7a. Note that the condition for nonadiabaticity actually only requires the inverse coupling, g −1 , to coincide with the time needed for sweeping through the anti-crossing. This can be shortened via increasing the oscillation amplitude. For periodic mechanical driving, we face iterated LZ transitions, see Fig. 2.7b. In fact, the first LZ transition splits the photon state into a coherent superposition (Fig. 2.7c); the two amplitudes gather different phases and interfere the next time the system transverses the avoided crossing (Fig. 2.7d). For two-state systems, the resulting interference patterns in the state population are known as Stueckelberg oscillations [103]. Note that in Fig. 2.7b the internal dynamics a ˜R (t, t0 ) is shown for the case of a single photon inserted into the cavity at time t0 = 0. At this time the mechanics is at its right turning point. Generally, the timeevolution a ˜i (t, t0 ) depends on t0 . For photons inserted at different times, the initial position of the mechanics is changed. In fact, varying t0 in Fig. 2.7b affects the interference between consecutive LZ transitions and can modify the pattern of the time-evolution. Note that the actual optomechanical system is continuously driven by a laser, while photons decay out of the cavity. The light field dynamics and the transmission are calculated via

36


Eq. (2.4) and Eq. (2.5), respectively. Both expressions involve an integration over t0 . This verifies the intuitive fact that the overall dynamics is described in terms of a coherent sum of various contributions due to photons being inserted into the cavity at different times t0 . Fig. 2.8 shows numerical results on the transmission for A¯ Ω, g, considering experimentally accessible parameters. For g/2π = 1 MHz, l = 1 cm and ω0 /2π = 3 · 1014 Hz; A¯ = 60g corresponds to an oscillation amplitude A = 2 nm. For typical membranes with a width of 50 nm, this driving strength is still within the linear regime of their mechanical motion. In Fig. 2.8 we note that transmission can only be observed if the mean position x ¯0 is a multiple ¯ of Ω. Furthermore, there is a modulation as a function of A. In the following we explain these features and analytically analyze the strong driving regime.

2.6.3

Physical description: Multiphonon transition picture

We first give an intuitive description of why finite transmission T in Fig. 2.8 can only be observed if x ¯0 is a multiple of Ω, and we comment later on the modulation as a function of ¯ A. As pointed out in Section 2.4.1, transmission is determined by two subsequent processes: first, the laser has to excite the left mode; secondly, the internal dynamics must be able to transfer photons into the right one. In general, both processes are inelastic and therefore require energy to be exchanged between the light field and the mechanics. Excitation process. - The left mode’s frequency is oscillating around the time-averaged value −¯ x0 . Hence, the resonance condition to excite the left mode reads ∆L + mΩ = −¯ x0 ,

(2.28)

see Fig. 2.9a. Here, mΩ is an adequate multiphonon transition. The width of the individual resonances is determined by κ. Internal dynamics. - The subsequent process displays the physics of LZS dynamics: for a photon inserted into the left mode, the first LZ transition splits the photon state into a coherent superposition, the two amplitudes gather different phases and interfere the next time the system transverses the avoided crossing. The condition for constructive interference can also be phrased in terms of an additional multiphonon transition that transfers a photon from the left mode with average frequency −¯ x0 to the right one at +¯ x0 (Fig. 2.9a), nΩ = 2¯ x0 .

(2.29)

Transmission can only be observed if both conditions are met. Note that the coupling g between modes does not enter here. We will come back to this point later.

2.6.4

Analytical description: Resonance approximation

¯ in the To derive these resonance conditions as well as to understand the dependence on A, following, we calculate an approximate, analytical expression for the transmission in the strongly driven regime. As we have seen in Section 2.4, the Green’s function GR (t, t0 ), required for the transmission (Eq. (2.4)), can be separated into two factors describing mechanical cavity excitations and the intracavity photon dynamics, respectively. According to Eq. (2.8), an m-phonon transition for the excitation, Eq. (2.28), is determined by the Bessel function ¯ Jm (A/Ω). In contrast, an nΩ-phonon transition of the internal dynamics, Eq. (2.29), is ¯ characterized by Jn (2A/Ω).


nΩ

x ¯0 /g

+¯ x0

mΩ

0 −¯ x0

aR

4

x ¯0 = 3Ω

2 0 -2

%

80

-4 40

∆L

-6

x0 (a)

x ¯0 = 5Ω x ¯0 = 4Ω

6

aL

Mean position

Optical frequency

ω − ω0

37

Displacement

0

0

(b)

5

10

¯ Amplitude A/g

Figure 2.9: Multiphonon transition picture and formation of LZS oscillations. (a) To see transmission, two processes are involved: First (magenta, labeled mΩ), excitation of the left cavity mode by the laser drive at ∆L , supported by m phonons. Second (red, labeled nΩ), a suitable n-quanta multiphonon transition to transfer a photon from the left into the right mode. (b) Density plot for the time-averaged transmission as a function of x ¯0 and A¯ ' g. Further parameters as in Fig. 2.8. (This figure has previously been published in [113].) In the case of LZS dynamics (i.e. strong driving), P for sufficiently large amplitudes only inΩt (see Eq. (2.7) and ¯ one of the harmonics of the time-dependent coupling g n Jn (2A/Ω)e (2.9)) will be in resonance with the system. This corresponds to leading-order perturbation theory within the Floquet approach [120] applied to Eq. (2.7), see also Chapter 3. In this case Eq. (2.7) simplifies to the problem of a two-state system with harmonic drive at nΩ and effective coupling constant ¯ gn = gJn (2A/Ω), (2.30) cf. Eq. (2.12) for the weak driving regime. To estimate when this approximation becomes appropriate, we note that for a driven undamped two-state system the width of the power-broadened resonance is set by the Rabi frequency. Thus, Eq. (2.7) yields a series of resonance peaks at x ¯0 = nΩ/2, and they become separated if 4gn < Ω, see Fig. 2.10a. Using the asymptotic form of the Bessel function, r 2 nπ π Jn (y) ' cos y − − , πy 2 4 ¯ that is valid for large arguments A/Ω 1, we get an upper bound on |gn | and we find the resonance approximation to hold whenever ¯ g 2 < (π/16)AΩ.

(2.31)

This is clearly fulfilled for the parameters in Fig. 2.8. Note the resemblance of this expression to the criterion for non-adiabatic transitions that follows from the standard LZ formula (see ¯ corresponds to the sweep velocity. Eq. 2.27), where v = AΩ

1.5

38


1.5

1.0

1.0

(a)

(b)

!'% 0.8 !'$ 0.6

0.5 !'# 0.4

0.5

!'" 0.2

0.0! 0.0

0.0 −10

−5

0

5

0!

10

2"

4#

6$

% 8

&! 10

time

−0.5 Figure 2.10: Resonance approximation for strong mechanical driving. (a) Schematic: The 2 4 6 at nΩ. 8The power 10 time-dependent coupling Eq. (2.9) is a sum0 of harmonic contributions broadened resonance of a harmonically driven two-level system is set by a Lorentzian whose width is determined by the coupling strength gn . When the FWHM 4gn is smaller than the peaks’ spacing, we can approximate the coupling Eq. (2.9) by a simple harmonic drive gn einΩt . (b) Full numerical simulation of |˜ aR (t, 0)|2 (Eq. (2.7)) (blue) vs. the analytical result in Eq. (2.32), |˜ aR (t, 0)|2 = |GR (t, 0)|2 , that involves the resonance approximation (red). ¯ Parameters: A/Ω = 60, Ω/2π = 0.2g, x ¯0 /g = 0.04. Compare (b) to Fig. 2.7b where the ¯ see Eq. (2.31). resonance approximation does not yet apply as g 2 ≮ (π/16)AΩ,

Given the resonance approximation, we find for the Green’s function G(t, t0 ) = −i

gn 0 0 sin ωn (t − t0 ) e−inΩ(t+t )/2 e−iφ(t ) , ωn

(2.32)

q with ωn = (gn )2 + (¯ x0 − nΩ/2)2 , cf. Eq. (2.24) that was found for weak mechanical driving. Fig. 2.10b illustrates that this approximation nicely mimics the full dynamics of consecutive ¯ LZ transitions. Note that ωn contains gn that, for A/Ω 1, is much smaller than the bare splitting g. This explains why the resonance conditions (Eq. (2.28) and (2.29)) involve the modes’ bare optical frequency ±¯ x0 instead of the adiabatic eigenfrequency ω± . Physically, when the system is swept fast and with large amplitudes through the avoided crossing, the time it spends within the coupling region is very small. Thus, it effectively experiences much less of the coupling and the bare modes’ frequency ±¯ x0 is hardly affected. We insert Eq. (2.32) into Eq. (2.4), taking into account the sum over independent contributions with n quanta. In the resolved sideband regime (Ω > κ), the integration of Eq. (2.4) selects a specific m for the excitation process, see Eq. (2.28). We find an approximate expression (displayed here for the special case ∆L = 0, where n = 2m)

T

 ¯ 2 X A κ  = Jm g Ω m

1 g2

2 ¯ J2m 2 A Ω  h i h i2  . ¯ 2 κ 2 A + (¯ x0 − mΩ) + J2m 2 Ω 2

(2.33)

This analytical expression fully captures the numerical results shown in Fig. 2.8. It illustrates 2 (A/Ω) ¯ that the transmission is determined by the product of two Bessel functions. While Jm 2 ¯ is due to the excitation process, Eq. (2.28), Jn (2A/Ω) is due to the interference between ¯ consecutive LZ transitions, Eq. (2.29). Note that in contrast to Jm (A/Ω), the LZS dynamics,


39

1.5

1.51.5

x ¯0 = Ω

1.2



0.5

0.6

x ¯0 = Ω/2

0.5

1.0

%

1.0

0.0

x ¯0 = 0

0.0 -0.5 -1.0

i

ii

iii

iv

(a)

0.5

1.0

1.5

Laser detuning ∆L /Ω

0.5

0.0

0.0

−0.5

−0.5

−1.0

−1.0

-1.5 −1.5 1.51.5

54

56

58

60

62

64

iii

1.0

0.5

56

58

60

62

64

66

iv

1.0

0.5

0.0

−0.5

−0.5

−1.0

−1.0

54

(b)

−1.5 66 54 1.5

0.00.0

54

2.0

ii

1.0

0.0

-1.5 −1.5

-1.5 0.0

1.5

i

56

58

60

60

62

64

−1.5 66 54

66 54

56

58

¯ Amplitude A/g

60

60

62

64

66

66

Figure 2.11: Dependence on laser detuning. Density plot for the time-averaged transmission. (a) Numerical results as a function of mean position x ¯0 and laser detuning ∆L for strong ¯ mechanical driving with A = 54g. Further parameters as in Fig. 2.8. For m = −1, the dashed yellow line indicates where Eq. (2.28) is met; likewise the dotted lines show x ¯0 = nΩ/2 ¯ where Eq. (2.29) is fulfilled. If resonant, details of the transmission depend on Jm (A/Ω) and ¯ Jn (2A/Ω), cf. Fig. 2.8. The indices (m, n) are set according to Eq. (2.28) and (2.29). (b) Analytical results according to Eq. (2.33) as a function of mean position x ¯0 and amplitude A¯ for various resonant laser detunings as indicated in (a): (i) ∆L = 0. (ii) ∆L = Ω/2. (iii) ∆L = Ω. (iv) ∆L = 3Ω/2. Except for the variation in ∆L , each panel is identical to Fig. 2.8. (Panel (a) and parts of panel (b) have previously been published in [121] and [122], respectively.) ¯ characterized by Jn (2A/Ω), involves 2A¯ as it is determined by the phase difference gathered between LZ transitions. To complete the description, Fig. 2.9b shows numerical results on the transmission and ¯ (while keeping Ω as in Fig. 2.8). As its amplitude modulations for smaller values of A/g before we see resonances of width κ for x ¯0 being a multiple of Ω and regions of excitation ¯ determined by Jm (A/Ω). Within these regions, we note the emergence of the already familiar substructure that is due to LZS dynamics. Note that LZS oscillations can only be observed if the system actually transverses the avoided crossing, i.e. for A¯ > x ¯0 .

2.6.5

Tuning the laser frequency

So far we considered a constant laser detuning of ∆L = 0. To discuss this final tuning parameter of the optomechanical setup, Fig. 2.11a presents numerical results for the timeaveraged transmission (A¯ Ω, g) as a function of mean position x ¯0 and laser detuning ∆L . Parameters are chosen such that the left boundary of Fig. 2.11a, i.e. the plot at ∆L = 0, equals the left boundary of Fig. 2.8, i.e. the plot at A¯ = 54g. For ∆L = 0, the conditions in Eq. (2.28) and (2.29) are met for x ¯0 being a multiple of Ω. Note that, for the parameters ¯ ¯ used, |J±1 (A/Ω)| is near a maximum while |J0 (A/Ω)| is close to a minimum, cf. the Bessel

40


functions plotted in Fig. 2.8. Thus, for ∆L = 0, the transmission in Fig. 2.11a is significantly smaller for x ¯0 = 0 than for x ¯0 = ±Ω. For m = −1, Eq. (2.28) is met along the yellow, dashed line in Fig. 2.11a. If we increase ∆L we thus tune out of resonance. In this case, the transmission in Fig. 2.8 would vanish everywhere. For ∆L = Ω/2, Eq. (2.28) and (2.29) can be met for x ¯0 being an odd multiple of Ω/2. For ∆L = Ω we recover the resonances of Fig. 2.8. Note however that the details of the ¯ ¯ transmission spectrum are set by Jm (2A/Ω) and Jn (2A/Ω). This sensitively depends on the indices of the Bessel functions (m, n) that are determined according to Eq. (2.28) p and (2.29). The periodicity of Jk (y) becomes clear from its asymptotic form, Jk (y) ' 2/πy cos(y − kπ/2 − π/4), that is valid for large arguments. Thus, as ∆L = Ω involves a different index m, the entire plot Fig. 2.8 would be shifted in A¯ by πΩ/2, while all the resonances for x ¯0 are identical. To illustrate this modulation of the transmission pattern in Fig. 2.8, Fig. 2.11b shows analytical results according to Eq. (2.33) for various resonant values of the laser detuning ∆L (see indication in Fig. 2.11a). Panel (i) for ∆L = 0 exactly matches the results of the full numerical simulation in Fig. 2.8. This confirms the excellent agreement of Eq. (2.33). For ∆L 6= 0, the indices (m, n) of the Bessel functions in Eq. (2.33) change according to Eq. (2.28) and (2.29). This modifies the transmission pattern. Note the shift of A¯ by πΩ/2 between (i) ∆L = 0 and (iii) ∆L = Ω that was pointed out above. Also compare the left boundary of each panel, i.e. each plot at A¯ = 54g, to the numerical results in Fig. 2.11a.

2.7

Conclusion

To conclude, we have introduced mechanically driven coherent photon dynamics for multimode optomechanical systems. For a specific setup we predicted effects such as mechanically driven photonic Rabi oscillations, Autler-Townes splittings and features of Landau-ZenerStueckelberg dynamics whose observation is within reach of current experiments. We note that the same photon dynamics will enter when describing self-induced mechanical oscillations in coupled multimode optomechanical systems, see Chapter 3. Future optomechanical circuits [39–41] that integrate multiple optical and vibrational modes on a chip will thus allow one to transfer the whole realm of strongly driven multi-level dynamics from atomic physics into the world of optomechanics. The tunability and custom design of optomechanical systems, as well as their ability to couple to each other, will offer a new regime in which to explore these phenomena.

Chapter 3

Self-oscillations: Dynamical back-action in terms of multilevel dynamics Over the last few years optomechanics has evolved into a vivid, fast-growing area of research. Within this field, dynamical back-action effects, i.e. light acting back on the mechanics after having been influenced by the mechanical motion, have been of special interest. In particular, these effects enable to optomechanically cool or heat mechanical degrees of freedom. Operated in the regime where the light-field interaction yields anitdamping of the mechanics, this can induce mechanical self-oscillations akin to lasing. Until recently, research mainly focussed on systems that are described in terms of a single optical mode coupled to a single mechanical one. As pointed out, this has now changed with the design of new setups whose dynamics involve several coupled optical and vibrational modes. In this chapter we therefore start to investigate dynamical back-action and self-induced mechanical oscillations (phonon lasing) in coupled multimode optomechanical systems, going beyond the regime of the linearized light-field dynamics. As we have seen in Chapter 2, here mechanical motion can initiate all kinds of strongly driven two- and multilevel phenomena in the light-fields, such as Landau-Zener-Stueckelberg oscillations. Via the radiation pressure force this dynamics acts back on the mechanics. As we will see, even for two modes this drastically changes the dynamical back-action that drives mechanical lasing oscillations. We focus on a specific optomechanical setup where all our results can be realized experimentally using current state-of-the-art technology. Nevertheless, our findings are generic and the analysis applies to any situation where self-induced oscillations (of other nanomechanical structures or even microwave modes) are pumped by a parametrically coupled two-level system that undergoes LZS dynamics (e.g. current-driven double quantum dot setups, superconducting single-electron transistors, or Cooper-pair boxes). This chapter has been published, essentially in the form presented here, in • Huaizhi Wu, Georg Heinrich and Florian Marquardt, The effect of Landau-Zener dynamics on phonon lasing, arXiv:1102.1647 preprint (2011) (submitted).

42

3.1

3. Self-oscillations: Dynamical back-action in terms of multilevel dynamics

Introduction

The exploration of nanomechanical objects and their interaction with light constitutes the rapidly evolving field of optomechanics (see [2] for a recent review). The key element of any optomechanical system is a laser-driven optical mode whose resonance frequency shifts in response to the displacement of a mechanical object. The photon dynamics conversely acts back on the mechanics in terms of a radiation pressure force. These dynamical back-action effects, mediated by the light field, can cool or amplify mechanical motion, and even drive the system into a regime of self-induced mechanical oscillations [17, 54, 55, 66, 67, 70, 123] akin to lasing. An exciting new development has introduced optomechanical setups with multiple coupled optical and vibrational modes [20, 40, 124]. These systems allow one to realize sophisticated measurement schemes [90,98,99], to study collective phenomena [125], or to mechanically drive coherent photon dynamics [113]. For applications, they furthermore stimulate prospects of integrated optomechanical circuits [41, 42]. Recently, phonon lasing for an optomechanical setup involving a tunable optical two-level system has been demonstrated [97]. Hence, implementing a nanomechanical analog of a laser has finally been achieved [126]. Here, we develop the fully nonlinear theory of phonon lasing (self-induced mechanical oscillations) in such multimode optomechanical setups. In particular, we will point out that the mechanical oscillations may induce Landau-Zener physics with respect to the optical two-level system, and that this has a strong effect on the dynamical back-action. The resulting phenomena drastically change the nonlinear attractor diagram, i.e. the relation between the mechanical lasing amplitudes and the experimentally tunable parameters. We will refer to an existing optomechanical setup [20] where our predictions can be verified experimentally. However, most of our analysis and discussion are applicable to the quite generic situation where self-induced oscillations are pumped by a parametrically coupled, driven two-level system. Our findings thus are also relevant for nanomechanical structures or microwave modes whose oscillations are amplified by coupling to, e.g., current-driven double quantum dot setups, superconducting single-electron transistors, or Cooper-pair boxes [127].

3.2

Coupled multimode optomechanical setup

We consider the system depicted in Fig. 3.1a. A dielectric membrane is placed in the middle between two high-finesse mirrors [20]. Transmission through the membrane couples the optical modes of the left and right half of the cavity, respectively. Focussing on two nearly degenerate modes, the Hamiltonian of the cavity reads x ˆ † x ˆ † ˆ Hcav = ~ω0 1 − a ˆ a ˆL + ~ω0 1 + a ˆ a ˆR l L l R +~g a ˆ†L a ˆR + a ˆ†R a ˆL . (3.1)

Here, a ˆ†L a ˆL and a ˆ†R a ˆR are the photon number operators of the left and right cavity mode, ω0 is the modes’ frequency for x = 0 (where the two modes are degenerate), and 2l is the length of the full cavity. The membrane’s displacement x ˆ linearly changes the modes’ bare frequencies, while the optical p coupling g leads to an avoided crossing for the system’s two optical resonances, ω± = g 2 + (ω0 x/l)2 (Fig. 3.1b). Thus, mechanical oscillations x ˆ(t) periodically sweep the system along the hyperbola branches ω± .

3.2 Coupled multimode optomechanical setup

(a)

43

opt. freq.

optical frequency

(b)

displacement

0 (c)

1 Time (

2

3 )

(d)

Figure 3.1: (a) Setup. A moveable membrane, placed inside a cavity, couples two optical modes aL , aR via transmission. (b) Optical resonance frequencies vs. displacement. The membrane’s displacement linearly changes the bare modes’ frequencies (dashed). Due to the photon coupling g there is an avoided crossing for the resonance frequencies ω± (black). Mechanical oscillations x(t) = A cos(Ωt) + xa periodically sweep the system along the photon branches (red). (c) Cavity resonance frequency ω± (x(t)) depending on time. For nonadiabatic sweeps through the anti-crossing, repeated LZ transitions (highlighted regions) split the photon state. After each passage, the two contributions gather a phase difference that leads to subsequent interference. The resulting LZS oscillations in the light field act back on the mechanics via the radiation pressure force. (d) For sufficiently large back-action-induced anti-damping, the system enters a regime of mechanical self-oscillations (phonon lasing). (This figure has previously been published in [128].)

44

3.3


Landau-Zener-Stueckelberg dynamics acting back on its driving mechanism

We focus on the experimentally accessible, non-adiabatic regime [99, 113] where fast periodic sweeping through the avoided crossing results in consecutive Landau-Zener (LZ) transitions [101,102]. For a photon inserted into the left mode, the first transition splits the photon state into a coherent superposition, the two contributions gather different phases and interfere the next time the system traverses the avoided crossing (Fig. 3.1c). For a two-state system, the resulting interference patterns are known as Landau-Zener-Stueckelberg (LZS) oscillations [103]. These have been demonstrated in many setups, ranging from atomic systems [104–106] to quantum dots and superconducting qubits [107–110]. In all of these situations, LZS effects are produced by a fixed external periodic driving. In contrast, here we address the case where LZS oscillations act back on the mechanism that drives them (i.e. the mechanical motion), via the radiation pressure force. We will see that LZS interference strongly influences this back-action force and thereby drastically affects the mechanical self-oscillations that occur when this force overcomes the internal friction (Fig. 3.1d). More generally, the following discussion thus illustrates the effect of LZS dynamics on back-action induced instabilities.

3.4

Equations of motion

ˆ cav /∂ x Given the radiation pressure force Fˆrad = −∂ H ˆ, the coupled equations of motion for the displacement x ˆ(t) and a î (t) (i = L, R), read ˆ ¨ ˆR ) − Ω2 (ˆ x − x0 ) − Γx ˆ˙ + ξ(t), ˆL − a ˆ†R a x ˆ = A0 (ˆ a†L a

(3.2)

i √ 1 h ˆ cav − κ a a ˆ˙ i = a î , H î − κˆbin i (t), i~ 2

(3.3)

where we used input-output theory for the light fields and set A0 = ~ω0 /lm. The membrane has a mechanical frequency Ω, an intrinsic damping rate Γ and a rest position x0 . Photons decay at a rate κ out of the cavity. We assume the left mode a ˆL to be driven by a laser at frequency ωL ; the input fields ˆbin (t) contain this contribution. In the following, we will i consider purely classical (large-amplitude) nonlinear dynamics and replace the operators a î (t) in by thepcoherent light amplitudes αi (t). The classical input fields then read βR = 0, βLin = e−iωL t Pin /~ωL , where Pin is the laser input power, and the mechanical Langevin force will be neglected (ξ ≈ 0). For convenience, we define the laser detuning ∆L = ωL − ω0 .

3.5 3.5.1

Effective optomechanical damping rate Γopt Damping rate Γopt from the net mechanical power input

The radiation pressure force gives rise to a time-averaged net mechanical power input hFrad xi. ˙ In analogy to the intrinsic friction Γ, see Eq. (3.2), we can define hFrad xi ˙ = −mΓopt hx˙ 2 i such that we obtain an effective optomechanical damping rate Γopt = −

A0 2 2 h |α (t)| − |α (t)| xi. ˙ L R hx˙ 2 i

(3.4)

3.5 Effective optomechanical damping rate Γopt

(a)

1.28

45

(b)

1 -1.28

(c)

!0

(d) -1 -1.0

(b)

(d)

(c)

-0.5

0.0

0.5

1.0

Figure 3.2: (a) Effective optomechanical damping Γopt for given mechanical oscillations x ¯(t) = ¯ = A¯ cos(Ωt) + x ¯a as a function of mean position x ¯a and laser detuning ∆L . Parameters: A/Ω 0.5, g/Ω = 0.2, κ/Ω = 0.1. Mechanical sidebands (dashed), displaced by multiples of Ω, show cooling (blue; Γopt > 0) and heating (red; Γopt < 0). The value of |Γopt | is largest if the optical modes’ frequency difference is in resonance with the mechanical frequency Ω; position (b) and (c). For finite amplitude, this yields an Autler-Townes splitting (see circled regions). Rate Γopt in units of 2ω0 Pin /mΩ3 l2 . (b) Creation (heating) or (c) destruction (cooling) of a phonon upon transferring a photon from left to right. (d) At the degeneracy point, the bare optical frequencies are swept past each other in an oscillatory fashion (cf. Fig. 3.1b). (This figure has previously been published in [128].)

For Γopt > 0 (Γopt < 0) the light-field interaction damps (anti-damps) the mechanics. For given oscillations x(t) = A cos(Ωt) + xa , Γopt can be calculated via the periodic light field dynamics αL (t), αR (t) that is found by solving Eq. (3.3); see also Eq. (3.6) further below. Note that our Γopt is amplitude-dependent, and the usual linearized case [57, 90] is recovered for A → 0. In the following we will express displacement in terms of frequency, x ¯(t) = (ω0 /l)x(t) ¯ (see Eq. (3.1)); likewise for A, x ¯a .

3.5.2

Results on Γopt for given mechanical oscillations

Fig. 3.2a shows results for Γopt in this setup, at moderate amplitudes A. Optomechanical damping and heating is largest if the optical modes’ frequency difference is in resonance with the mechanical frequency Ω [97, 123]. In this case, photon transfer from the laser-driven left mode into the right mode involves absorption (or emission) of a phonon, that yields strong mechanical heating (or cooling), see Fig. 3.2b-c. For finite amplitudes, we observe an Autler¯ Townes (AT) splitting [116] that scales as 2g A/Ω [113]. Given Γopt , we now turn to discuss back-action driven mechanical self-oscillations (phonon lasing) of the membrane.

46

3.6 3.6.1


Back-action driven mechanical self-oscillations Steady state conditions for the dynamics’ attractors

For suitable laser input powers, the radiation pressure force only weakly affects the mechanics over one oscillation period and the mechanics approximately performs sinusoidal oscillations at its unperturbed eigenfrequency Ω; x(t) = A cos(Ωt) + xa . The possible attractors of the dynamics (A, xa ) have to meet two conditions [54, 70]. First, the time-averaged total force must vanish: h¨ xi = 0. Second, the overall mechanical power input due to radiation pressure must equal the power loss due to friction, h¨ xxi ˙ = 0. From Eq. 3.2, the power balance h¨ xxi ˙ =0 is equivalent to Γopt (A, xa ) = −Γ. (3.5)

The force balance h¨ xi = 0 yields hFrad (t)i = mΩ2 (xa − x0 ), i.e. the radiation pressure force displaces the membrane’s average position xa from its rest position x0 . In general, one solves the force balance to find xa = xa (A, x0 ) and uses this to calculate Γopt (A, xa ) [54, 70]. For high quality mechanics (Ω/Γ 1), the power balance (Eq. (3.5)) is met for weak radiation pressure forces where xa ' x0 . For clarity, we will focus on this case. Otherwise, attractor diagrams get deformed slightly [54].

3.6.2

Two-level dynamics determining dynamical back-action

Fig. 3.3a displays the effective optomechanical damping Γopt depending on laser-detuning ∆L and amplitude A. The structure of this diagram is drastically different from the standard case with one optical mode [54, 70]. There are “ridges” of high Γopt which display an oscillatory shape (clarified in the inset). A physical understanding of Fig. 3.3 can be found from the general structure of the light field dynamics that enters the optomechanical damping, Eq. (3.4). The light-field dynamics that enters our calculation of the dynamical back-action is identical to what was discussed in Chapter 2. As we have seen, for given mechanical oscillations x(t) = A cos(Ωt) + xa , the formal solution to Eq. (3.3) can be expressed as Z 2 κPin t 0 −κ(t−t0 )/2 −i∆L t0 0 |αi (t)| = Gi (t, t )e e dt , ~ωL −∞ 2

(3.6)

where the Green’s function Gi (t, t0 ) describes the amplitude for a photon entering the left mode at time t0 and to be found in the left or right one (i = L, R) at time t. From Eq. (3.3) 0 ¯ Gi (t, t0 ) is found to be Gi (t, t0 ) = a ˜i (t, t0 )e−iφ(t ) where φ(t0 ) = (A/Ω) sin(Ωt0 ) and a ˜i (t, t0 ) is a solution to d a ˜R x ¯a ge+2iφ(t) a ˜R i = (3.7) ˜L a ˜L ge−2iφ(t) −¯ xa dt a

with t ≥ t0 and initial condition a ˜R (t0 , t0 ) = 0, a ˜L (t0 , t0 ) = 1. Thus, the internal photon 0 dynamics between the two modes a ˜i (t, t ) is expressed in terms of a two-level system with a 2iφ(t) time-dependent coupling ge . For the details on Eq. (3.6) and (3.7) see Appendix A.1.

3.6.3

Interpretation using Floquet Theory

With ψ = (˜ aR , a ˜L )T , Eq. (3.7) is the Schrödinger equation including a time-periodic Hamiltonian, H(t + T ) = H(T ). In this case it is appropriate to consider the time-evolution

3.6 Back-action driven mechanical self-oscillations

5 1.07

47

3

4 3

2

-0.1

1

-0.35

-1.07

2 1 0

(a)

0 -2

0

2 -0.2 0.2

-1

0

1

(b)

Figure 3.3: Attractor diagram for phonon lasing oscillations (regime Ω > 2g). (a) Effective ¯ optomechanical damping Γopt as a function of laser detuning ∆L and oscillation amplitude A, for a membrane positioned at the degeneracy point x ¯a = 0; other parameters as in Fig. 3.2. For large amplitudes, the interference between consecutive LZ transitions (Fig. 3.1c) leads to LZS oscillations. They result in ridges of high Γopt , whose oscillatory shape can be understood via the Floquet eigenvalues ± (middle panel) for the periodic light field dynamics. The ridges ¯ where ± (A) ¯ ≈ ±gJ0 (2A/Ω) ¯ are located at ∆L = mΩ + j (A), involves a Bessel function. ¯ (b) Blow-up of framed region in (a). The contour lines Dashed lines indicate |∆L | = A. at Γopt (A, xa ) = −Γ (Eq. 3.5) denote possible attractors (allowed amplitude values: solid – stable / dashed – unstable) for the mechanical self-oscillations generated by back-action; plotted for two different values of −Γ, as indicated. (Γ, Γopt in units of 2ω0 Pin /mΩ3 l2 ) (This figure has previously been published in [128].)

48


operator for one period, ψ(t0 + T ) = U (T )ψ(t0 ), and its two eigenvalues, the so-called Floquet eigenvalues ± : U (T )χ± = exp(−i± T )χ± . U (T ) is obtained by integrating Eq. (3.7). Using Floquet theory [120], we find the general structure of the Green’s function Gi (t, t0 ) = P 0 n,n0 ,j −iΩ(nt−n0 t0 ) −ij (t−t0 ) e e , where Cin,n ,j are time-independent coefficients. Then, j,n,n0 Ci ¯ via Eq. (3.6) we obtain pronounced resonances in Γopt located at ∆L = mΩ + ± (A), corresponding to the ridges in Fig. 3.3. The interference between consecutive LZ transitions the coupling between modes in terms of Bessel functions Jn : ge2iφ(t) = P renormalizes inΩt ¯ g n Jn (2A/Ω)e (Eq. 3.7). This results in an oscillatory modulation of the Floquet eigen¯ At certain amplitudes, these vanish due to total destructive interference, see values ± (A). Fig. 3.3a. The oscillatory shape of the ridges in Γopt then directly determines the attractor diagram for the self-induced oscillations, via the power balance (Eq. (3.5)), see Fig. 3.3b. Regarding the global structure of Fig. 3.3a, Γopt tends to be large near ∆L = ±A¯ (dashed lines). This is because then the left mode gets into resonance with the laser at the motion’s turning point. For larger amplitudes, we recover the predictions for the standard optomechanical setup [54] (checkerboard in Fig. 3.3a).

3.7

Characterizing the parameter space

So far, we discussed dynamical back-action effects for parameters where the mechanical frequency is larger than the optical splitting, Ω > 2g (Fig. (3.2) and (3.3)). In general, the parameter space can be subdivided as shown in Fig. 3.4a. Multimode dynamics that goes beyond the standard scenario [54, 70] can only be observed if the photon lifetime inside the cavity is larger than the timescale for photons to tunnel between modes, 2g > κ (colored region, Fig. 3.4a). Otherwise, photons inserted into the left mode decay before the second mode affects the dynamics and we recover the standard results [54,70]. Within the new region (colored in Fig. 3.4a), the most interesting regime is where mechanical sidebands can in fact be resolved, i.e. κ < Ω. Above, we had focussed on the sector 2g < Ω within this regime. Now Fig. 3.4b displays Γopt in the opposite sector where 2g > Ω. Here, several mechanical sidebands lie within the avoided crossing. With respect to self-induced mechanical oscillations, these sidebands and their interaction yield an intricate web of multistable attractors, see Fig. 3.4c. The global asymptotics of these structures (green lines) can be found from the quasistatic approximation, i.e. from the time-averaged transition frequency: q ∆L = 2hω+ (t)i = 4 g 2 + A¯2 E(π/2, k)/π, where k =

3.8

q A¯2 / g 2 + A¯2 and E( π2 , k) is the complete elliptic integral of the second kind.

Conclusion

To conclude, we have investigated self-induced mechanical oscillations (phonon lasing) in a multimode optomechanical system. The mechanical motion drives Stueckelberg oscillations in the light field of two coupled optical modes, and this drastically modifies the attractor diagram. The additional influence of quantum (and thermal) noise could be analyzed along the lines of [54, 70]. Our example, which can be realized in present optomechanical setups, illustrates the potential of Landau-Zener physics to appreciably alter lasing behavior.

3.8 Conclusion

49

5 1

4

(2)

(1)

1

(a)

3 -0.03

5

0.50

2 0.61

0

1

-0.50

-5 -5 (b)

-0.61

0

0 -5

5

0

5

(c)

Figure 3.4: (a) Overview of the parameter space. Multimode dynamics leads to effects beyond the standard scenario when the optical splitting can be resolved: 2g > κ (colored region). Parameter set (1) corresponds to the one in Figs. 3.2, 3.3; set (2) is considered in (b-c). (b) Effective optomechanical damping Γopt for given mechanical oscillations x ¯(t) = A¯ cos(Ωt)+ x ¯a , as a function of mean position x ¯a and laser detuning ∆L (compare Fig. 3.2). Parameters: ¯ A/Ω = 1.5, g/Ω = 2.3, κ/Ω = 0.2. (c) Attractor diagram. Effective optomechanical damping Γopt as a function of laser detuning ∆L and oscillation amplitude A¯ for a membrane positioned at the degeneracy point, x ¯a = 0. Further parameters as in (b). The solid contour line Γopt (A, xa ) = −Γ indicates the stable attractors for self-induced oscillations. Green (thick) lines show the asymptotic behavior. (Γopt in units of 2ω0 Pin /mΩ3 l2 ) (This figure has previously been published in [128].)

50


Chapter 4

Coupled multimode optomechanics in the microwave regime The combination of quantum optics and nanomechanics has resulted in tremendous progress during the past few years and established the fast-growing field of optomechanics. As pointed out, the standard setup studied in this field is an optical cavity with a movable end-mirror. The light stored inside the laser-driven cavity exerts a radiation pressure force on the mirror, whose motion in turn acts back on the dynamics of the light field. Besides optics, the standard optomechanical scheme has recently been realized in on-chip microwave experiments. Here the optical cavity is replaced by a superconducting microwave resonator whose central conductor capacitively couples to the motion of a nanomechanical beam. So far, only single mode systems have been considered for optomechanics in the microwave regime. Given our interest in coupled multimode optomechanical setups, we thus ask what are the implications for such systems in the microwave domain. As we will see, these designs have advantages that go beyond bulk refrigerator cooling and on-chip integration that are usually considered to be main reasons for doing optomechanical experiments using microwave resonators. In particular, in contrast to similar systems in the optical realm, they possess coupling frequencies, governing photon exchange between microwave modes, that are naturally in the range of typical mechanical frequencies. This has several implications both for the classical and the prospective quantum regime. As an example, focussing on the currently in experiments accessible regime of classical mechanical motion, we demonstrate how this enables new ways to manipulate the microwave field using mechanically driving coherent photon dynamics introduced in Chapter 2. In addition to the more advanced dynamics, coupled multimode systems can furthermore realize fundamentally different optomechanical coupling schemes. In the second part of this chapter, we propose a multimode setup that allows one to couple the microwave photon number to the square of mechanical displacement. This enables quantum non-demolition Fock state detection of the mechanics. For experimentally realistic parameters, we calculate the signal-to-noise ratio for detecting an individual quantum jump from the mechanical ground state to the first excited state. The same scheme also allows one to measure phonon shot noise. Both experiments would constitute a major breakthrough. So far, they have only been considered for optical systems.

52

4. Coupled multimode optomechanics in the microwave regime This chapter has been published, essentially in the form presented here, in • Georg Heinrich and Florian Marquardt, Coupled multimode optomechanics in the microwave regime, Europhysics Letters 93, 18003 (2011).

4.1

Introduction

Significant interest in the interaction and dynamics of systems comprising micro- and nanomechanical resonators coupled to electromagnetic fields, as well as the prospect to eventually measure and control the quantum regime of mechanical motion, has stimulated the rapidly evolving field of optomechanics (see [2] for a recent review). In the standard setup, the light field, stored inside an optical cavity, exerts a radiation pressure force on a movable end-mirror whose motion changes the cavity frequency and thus acts back on the photon dynamics. This way, the photon number inside the optical mode is linearly coupled to the displacement of a mechanical object. Beyond the standard approach, new developments have introduced optical setups with multiple coupled light and vibrational modes pointing the way towards integrated optomechanical circuits [20, 39–41, 74]. These systems allow one to study elaborate interactions between mechanical motion and light such as mechanically driven coherent photon dynamics that introduces the whole realm of driven two- and multi-level systems to the field of optomechanics [113]. Coupled multimode setups furthermore allow one to increase measurement sensitivity [98] and enable fundamentally different coupling schemes. Accordingly, recent experiments achieved coupling the photon number to the square and quadruple of mechanical displacement [20, 99]. Such different coupling schemes are needed, for instance, to afford quantum non-demolition (QND) Fock state detection of a mechanical resonator [20, 88–90]. Besides optics, recent progress has made it possible to realize optomechanical systems in the microwave regime [24, 77]. In this case the optical cavity is replaced by a superconducting microwave resonator whose central conductor capacitively couples to the motion of a nanomechanical beam. This optomechanical approach constitutes a new path to perform onchip experiments measuring and manipulating nanomechanical motion that adds to electrical concepts using single electron transistors [29,129,130], superconducting quantum interference devices [131, 132], driven RF circuits [23] or a Cooper-pair-box [81, 133]. One advantage of on-chip optomechanics is to use standard bulk refrigerator techniques in addition to laser cooling schemes [25, 134]. This recently enabled cooling a single vibrational mode close to the quantum mechanical ground state [80]. Furthermore, nonlinear circuit elements can be integrated. This afforded ultra-sensitive displacement measurements with measurement imprecision below that at the standard quantum limit [73]. Here we go beyond single mode systems, that have been considered for microwave optomechanics so far, and propose setups with coupled microwave resonators. As we show, the microwave regime is especially promising for coupled multimode optomechanics as it offers explicit advantages compared to the optical realm. In particular, it allows one to conveniently access a new regime where the coupling frequency between microwave modes is comparable to mechanical frequencies. This has several implications both for the classical and the prospective quantum regime. In addition, on-chip fabrication and the use of bulk refrigerator techniques are further advantages that directly relate to the prospects to perform quantum measurements of mechanical motion.

4.2 Two-resonators setup with linear mechanical coupling

53

Accordingly, we start with a two-resonators setup with linear mechanical coupling. The derivation of the Hamiltonian allows one to discuss the feasibility of a new regime for optomechanics. Selecting one example that focus on the domain of classical mechanical motion that is accessible in experiments at present, we demonstrate how this new regime, for instance, allows one to manipulate the microwave field in terms of mechanical driving. For simplicity, here we consider two-resonator setups but our findings directly translate to larger systems. In the last part of this letter we present a multimode microwave scheme that allows coupling to the square of displacement. As an example regarding prospective quantum measurements we discuss QND Fock state detection using this device.

4.2

Two-resonators setup with linear mechanical coupling

We consider the coplanar device geometries depicted in Fig. 4.1a with two identical superconducting microwave resonators aL , aR . The central conductors of aL and aR are assumed to adjoin for a length dg that is much smaller than the total wave guides’ length d (Fig. 4.1b). A nanomechanical beam, connected to the ground plane, is placed at the other end of aL (Fig. 4.1c). In the following, we will first derive the Hamiltonian and then turn to a discussion of implications and prospects of multimode optomechanics in the microwave regime.

4.2.1

General structure of the Hamiltonian

Heuristically, the form of the Hamiltonian for the system depicted in Fig. 4.1 can be guessed. Considering two microwave modes with frequency ωL , ωR and photon number operator a†L aL , a†R aR in the left and right resonator, respectively, its general structure reads H = ~ (ωL − x) a†L aL + ~ωR a†R aR + ~g a†L aR + a†R aL .

(4.1)

Due to the resonators’ alignment there is a tunnel coupling of photons between the two modes described in terms of a coupling frequency g. Furthermore, via the left resonator’s capacitance, the motion of the mechanical beam x shifts its resonance frequency where = −∂ω/∂x is the optomechanical frequency pull per displacement. For an explicit discussion of the microwave regime, however, we have to derive Eq. (4.1). For instance, this will yield the parameters’ dependence.

4.2.2

Deriving the Hamiltonian from a microscopic picture

Rigorously, the Hamiltonian of an electric circuit can be derived from its Lagrangian (see for instance [136]). The latter can conveniently be expressed in terms of a flux variable φ(x, t) ≡

Z

t

dτ V (x, τ ),

−∞

where V (x, t) = ∂t φ(x, t) is the voltage at position x and time t. Without the nanomechanical beam, the circuit diagram looks as depicted in Fig. 4.2. As the length dg of the region where aL and aR adjoin is much smaller than the resonators’ length d, the capacitive coupling between both central conductors can be considered in terms of a constant, total capacitance

4. Coupled multimode optomechanics in the microwave regime

feed line

dg ! d

aL

aR

aL

x

aL

aR

(b)

(c)

ground

frequency

feed line

(a)

transmission line

54

(d)

aL 2g

aR displacement

aL (e)

x

aR

x

Figure 4.1: Schematic device geometry for two superconducting microwave resonators aL , aR with a nanomechanical beam coupled to aL . (a) The two resonators (each of length d) are coupled to external feed and transmission lines (green). (b) The central conductors of aL and aR (red) capacitively couple due to a small region of length dg where the resonators adjoin. (c) At the other end of aL a small mechanical beam, connected to ground (blue), is placed. Its displacement x affects the line capacitance of resonator aL changing its resonance frequency. (d) System’s resonance frequency as function of displacement: the beam’s displacement x linearly changes the bare mode frequency of aL while the one of aR is unaffected (dashed). Due to the coupling g between modes there is an avoided crossing 2g in the eigenfrequencies (blue). (e) Analogous optical setup: a static, dielectic membrane, placed inside a cavity with a movable mirror, couples two separate optical modes aL , aR . (This figure has previously been published in [135].)

4.2 Two-resonators setup with linear mechanical coupling

55

G. Given the circuit diagram, each node defines an equation of motion for the flux at that position. This allows one to derive the corresponding Lagrangian [136]. For Fig. 4.2 we find !2 X G X ˙ L = LL + LR + φR,k − φ˙ L,k , d k

k

where φL,k , φR,k refer to the uncoupled (G = 0) normal modes of the flux in the left and right resonator, respectively; and # " 2 X c kπ 1 2 2 φj,k φ˙ + Lj = 2 j,k 2l d k

(j = L, R) describes the corresponding Lagrangian for separate resonators, each with line inductance l and line capacitance c [137]. To transform to the Hamiltonian, we consider the canonically conjugated momentum πL[R],k = ∂L/∂ φ˙ L[R],k . For G/d c (see discussion below), the expression for πL[R],k simplifies to πL[R],k = cφ˙ L[R],k . In the following we will restrict to a single mode in each resonator (k = 1), and drop the label referring to the mode index. Then, the Legendre transformation transforms LL , LR into two harmonic oscillators of frequency ωL and ωR . For the coupling we consider (φ˙ R − φ˙ L )2 with q φ˙ L[R] = πL[R] /c = i ~ωL[R] /2c(a†L[R] − aL[R] ),

see [137]. As the microwave frequency is by far the fastest time-scale involved in the system, we can use a rotating wave approximation and find for the Hamiltonian, G G † H = ~ωL 1 − aL aL + ~ωR 1 − a†R aR dc dc √ G † + ~ ωL ωR a aR + a†R aL , dc L where we neglected the vacuum energy. We note that the frequencies ωL , ωR , originally defined for uncoupled modes, are lowered by a constant value. This shift can be neglected by simply redefining the resonators’ frequencies. More important is the coupling between modes. From our derivation we can read off its frequency g =

√

ωL ωR

G . dc

(4.2)

Finally we take into account the nanomechanical beam. For the resonator aL in Fig. 4.1, the beam’s motion changes the capacitance between the central conductor and the ground plane. We find the optomechanical frequency pull per displacement k = −∂ωk /∂x from the √ resonance frequency of a microwave resonator, ωk = kπ/d lc. This yields k = (∂Cb /∂x) · Z · ωk2 / (2πk) , where Cb denotes the total capacitance between the central conductor and the beam. Z = p l/c is the line impedance. Altogether this yields the Hamiltonian whose form was already stated in Eq. (4.1). We note that according to Eq. (4.2) with ωL (x), the coupling frequency g in principle depends on displacement x. However, for typical parameters the dependence is negligible and g can be considered to be constant. The resonance frequency of Eq. (4.1) is depicted in Fig. 4.1d.

56


φ−N

φ−2

φ−1

φ0

G

φ1

φ2

φ3

φN

a Figure 4.2: Discretized circuit diagram for the system depicted in Fig. 4.1 without nanomechanical beam. The flux variable φ(x, t) is discretized as φn (t) at node x = na. The two central conductors are coupled via a total capacitance G. (This figure has previously been published in [135].)

4.3

Coupling Frequency comparable to the mechanical frequency

Given Eq. (4.2), the coupling frequency between the two resonator modes reads g/ω0 = (cg /c) · (dg /d), where we defined the coupling line capacitance cg = G/dg along the length of the coupling region dg . In general, cg will be much smaller than the line capacitance between each central conductor and the ground plane c: first of all, the distance between the two central conductors is significantly larger than the distance between a single conductor and the adjacent ground plane. Second, the capacitance between the central strip lines is shielded by the grounded region in between. Here we crudely assume cg /c = 10−2 . For d in the cm range and dg ' 0.1mm (dg is chosen such that a several 10 µm long nanomechanical beam can be fabricated in between the region where the resonators align), we have dg /d = 10−2 and the coupling between modes is g/ω0 = 10−4 where ω0 will be in the GHz range. Common eigenfrequencies of nanomechanical beams are Ω = 100 kHz - 10 MHz. Hence, due to their much smaller photon frequency, coupled multimode optomechanical systems in the microwave regime naturally possess coupling frequencies in the range of typical mechanical frequencies (g ' Ω). The relevance of this regime for instance to realize all kinds of driven two- and multi-level photon dynamics in optomechanical systems has been pointed out in Chapter 2.

4.3.1

Example: Mechanically driven photon dynamics

As an example to emphasize the characteristics of coupled optomechanical systems in the microwave regime and to demonstrate implications of g ' Ω even in the presently accessible regime of classic mechanical motion, we will discuss how the microwave field in the setup of Fig. 4.1 can be manipulated in terms of mechanical driving (see [138] for a universal mechanical actuation scheme). Experimentally, the impact can be most easily observed in terms of the transmission spectrum. We assume the left resonator aL to be driven at frequency ωL via the feed line, while the transmission down the transmission line is recorded. We consider the coupling of the left (right) resonator to the feed (transmission) line in terms of the the resonators’ decay rate κ. Given the Hamiltonian (Eq. (4.1)), using input/output theory, the

4.3 Coupling Frequency comparable to the mechanical frequency 4

2 0 %

-2

100 50 0

-4 -4

(a)

Detuning ∆L /g

Detuning ∆L /g

4

57

2g1

2 0 %

-2

-2

0

2

0

-4

4

(b)

Ω

43

-4


86

-2

0

2

4


Figure 4.3: Transmission spectrum for the setup depicted in Fig. 4.1 for resonators’ decay rate κ = 0.1g: density plot for the time-averaged transmission depending on mean mechanical displacement x ¯0 = x0 , and frequency detuning ∆L = ωL − ω0 of the feed line’s microwave drive at ωL (ω0 denotes the left mode’s bare frequency for x = 0). (a) Without mechanical drive (x(t) = x0 ); the spectrum is given by the resonance frequency depicted in Fig. 4.1d. (b) For mechanical driving (x(t) = A cos(Ωt) + x0 ) with amplitude A¯ = A = 1.5Ω and frequency Ω = 3g; mechanical sidebands displaced by Ω appear and intersect the original photon branches where, due to mechanically driven Rabi dynamics, high transmission and additional anticrossings arise. The gap 2g1 is determined by the Bessel function J1 according ¯ to 2g1 = 2gJ1 (A/Ω). (This figure has previously been published in [135].) equation of motion for the averaged fields αL = haL i, αR = haR i read d αL = dt d αR = dt

√ 1 κ (−x(t)αL + gαR ) − αL − κbin L (t) i 2 1 κ gαL − αR , i 2

(4.3)

−i∆L t bin describes the electromagnetic drive along the feed line with amplitude where bin L (t) = e in b and frequency ωL . Here we used a rotating frame with laser detuning from resonance 2 ∆L = ωL − ω0 . The transmission T (t) = κha†R (t)aR (t)i/ bin can be expressed as

Z T (t) = κ 2

t

−∞

0

G(t, t )e

−i∆L t0 −(κ/2)(t−t0 )

2 0

dt ,

(4.4)

where the phase comprises the feed line’s drive and resonators’ decay, while the Green’s function G(t, t0 ) describes the amplitude for a photon to enter the left resonator’s mode aL at time t0 and to be found in the right one aR later at time t. We take into account two scenarios: first, the beam is at rest given a constant displacement x(t) = x0 ; second, the beam is mechanically driven to oscillate with amplitude A and frequency Ω around the mean position x0 , x(t) = x0 + A cos(Ωt). Fig. 4.3a shows numerical results of the transmission spectrum without mechanical driving. The spectrum corresponds to the system’s resonance frequency depicted in Fig. 4.1d where the resonance width is set by the resonators’ decay rate κ. In contrast, Fig. 4.3b shows the transmission including mechanical driving with Ω = 3g, i.e. g ' Ω being characteristic for coupled microwave optomechanics.

58


In this case, the mechanics moves on a time-scale that is comparable to the one governing photon exchange between modes. This results in new, nonequilibrium photon dynamics that goes beyond the usual adiabatic case g Ω. To understand the main features of Fig. 4.3b we note that, in general, two processes are involved to observe transmission, see (4.4): first, the left resonator aL must be excited by −i∆L t bin ; second, the internal dynamics must be able to the electromagnetic drive bin L (t) = e transfer photons from aL to aR . From Eq. (4.3) the solution G(t, t0 ) can be found to be 0

G(t, t0 ) = α ˜ R (t, t0 )e−iφ(t )

(4.5)

¯ where φ(t0 ) = (A/Ω) sin(Ωt0 ) and α ˜ R (t, t0 ) is a solution to the driven two state problem d 1 α ˜L −¯ x0 ge−iφ(t) α ˜L = , (4.6) ˜R α ˜R 0 dt α i ge+iφ(t) with t ≥ t0 and initial condition α ˜ L (t0 , t0 ) = 1, α ˜ R (t0 , t0 ) = 0. Note that we expressed ¯ displacement in terms of frequency; A = A, x ¯0 = x0 . For A¯ 6= 0, in addition to the 0 −i∆t electromagnetic drive (see e in Eq. (4.4)), the mechanical driving can excite aL in terms of multiples of the mechanical frequency mΩ. This mechanical excitation is described by the P 0 −imΩt0 in Eq. (4.5) and leads to mechanical sidebands ¯ phase factor e−iφ(t ) = m Jm (A/Ω)e in the spectrum [cf. Fig. 4.3b]. Note that the individual process mΩ is described by a ¯ Bessel function Jm (A/Ω) and can be tuned by the driving strength. Beyond the modified excitation, the driving significantly changes the internal dynamics of the microwave fields, see Eq. (4.6). In particular the mechanical motion can initiate mechanically driven Rabi dynamics exchanging photons between aL and aR that leads to high transmission if the mechanical drive at Ω is in resonance with the modes’ frequency difference. For sufficiently strong driving, the mechanically assisted process leads to additional anticrossings in the spectrum resembling Autler-Townes splittings known from quantum optics (see marker in Fig. 4.3b). From Eq. ¯ (4.6) we find that the spacing of this first additional splitting scales according to 2gJ1 (A/Ω) and can likewise be tuned by the mechanical driving strength. All this illustrates how, due to g ' Ω, the microwave field can extensively be manipulated by mechanical motion in terms of mechanically driven coherent photon dynamics.

4.4

Coupling to the square of displacement

We present a modified scheme comprising coupled microwave resonators that allows one to couple the photon number to the square of mechanical displacement (Fig. 4.4a-b). In contrast to the setup in Fig. 4.1, here the nanomechanical beam is placed in the region between the two resonators, such that its motion affects both simultaneously (Fig. 4.4b).

4.4.1

Hamiltonian

According to our previous results, the Hamiltonian reads H = ~ (ω0 − x) a†L aL + ~ (ω0 + x) a†R aR + ~g a†L aR + a†R aL ,

(4.7)

where g = ω0 cg dg /cd, see Eq. (4.2) p and Eq. (4.1). Fig. 4.4c illustrates the system’s resonance frequency ω± (x) = ω0 ± g 2 + (x)2 as function of displacement. Naturally all the

4.4 Coupling to the square of displacement

ground

(b)

aL

x φ0 L (e)

dg � d frequency

aR

C

aR

x

aL transmission line

feed line

(a)

59

ω+

aL

2g

aR ω−

C

φ1

(c)

L

displacement

aR

aL (d)

x

x

Figure 4.4: Schematic device geometry for two microwave resonators aL , aR with a nanomechanical device coupled to both of them. (a) Two stripline resonators (each of length d) are coupled to external feed and transmission lines (green). The central conductors of aL , aR (red) are capacitively coupled in a small region of length dg where the wave guides adjoin. (b) Between the two resonators a small mechanical beam, connected to ground (blue), is placed. Its displacement x affects the line capacitance of both, aL and aR . (c) System’s resonance frequency as function of displacement: the beam’s displacement linearly changes the bare modes’ frequency of aL and aR (dashed). Due to the coupling g between the resonators, there is an avoided crossing 2g in the eigenfrequencies ω± (blue). (d) Analogous optical setup with a movable dielectric membrane placed in the middle of a cavity [20]. (e) Schematic realization with two microwave LC circuits where a central plate is grounded and resonates against two others that build the LC circuits. √ In the notation of Fig. 4.2 we get for the coupling frequency g = ωLC G/C with ωLC = 1/ LC. (This figure has previously been published in [135].)

Georg Heinrich and Florian Marquardt

60


Table 1: Results on the signal-to-noise ratio Σ, eq. (9), for two sets of experimental parameters that would allow to observe an individual quantum jump from the mechanical ground state to the first excited state (Σ ! 1). Further parameter ωc /2π = 5 GHz.

! (MHz/nm) 65 70

m (pg) Ω/2π (MHz) Q/105 10 11 3.5 10 11 5.0

κ/Ω 1/70 1/100

g/2π (MHz) 0.5 0.5

Pin (pW) x0 (pm) 200 0.5 50 0.5

nadd 1 1

T (mK) 20 20

Σ 1.0 1.2

microwave system however, of an LC circuit by Σ, the Eq. DFG(4.8), (NIM, for SFB two 631, sets Emmy-Noether program), Figure 4.5: Results onconsisting the signal-to-noise ratio of experimental where the plates of a parallel-plate condensator mechaniGIF and DIP. parameters that would allow one to observe an individual quantum jump from the mechanical cally resonate, achieves ! = 65 MHz/nm [32]. Our proposal ground tostate the by first excited state ≥ 1). Further parameter ωc /2π = 5 GHz. (This transfers this to scheme stacking three such(Σ plates, table has previously been realistic published in [135].) see fig. 4(e). For experimentally parameters [32], REFERENCES a calculation of Σ yields that a setup with this optomechanical coupling would allow to detect an individual [1] Marquardt F. and Girvin S. M., Physics, 2 (2009) 40. [2] Thompson J. D. et al., Nature, 452 (2008) 72. quantum jump from the mechanical ground state to the characteristics of coupled multimode optomechanics in the microwave regime, that have been [3] Li M. et al., Nature, 456 (2008) 480. first excited state, see table 1. Note that, in contrast discussed, the hyperbola-shaped avoided level crossing allows [4] Eichenfield M. et al., Nature, 459 (2009) 550.one to to the setup apply. discussed In in particular, [2], the parameters here are [5] Eichenfield M. etsee al., Nature, 462 2. (2009) 78. already the small-splitting regime g 2), see Section 5.6. For the dynamics of the phase difference δϕ = ϕ2 − ϕ1 we find δ ϕ˙ = −δΩ − C cos(δϕ) − K sin(2δϕ). (5.36) The coupling constants are given by C = (ξ12 − ξ21 )/2 and K = (ξ12 + ξ21 )2 /8γ. The first two terms to the right hand side of Eq. (5.35) and (5.36) are the ones already found in Eq. (5.21) and (5.22), respectively. The last term is due to the amplitude-mediated phase coupling (Eq. (5.26)). In contrast to the standard Kuramoto model, 2δϕ appears. The implications of this modification will be discussed in Section 5.5.4. In the following, we focus on the case of nearly identical optomechanical cells where the coupling C can be neglected (C/δΩ = k/2mΩ2 1, see discussion in Section 5.5.1) and K = k 2 /2m2 Ω2 γ. Synchronization occurs if δ ϕ˙ = 0 possesses a solution, i.e. |δΩ| ≤ K. According to our analytical results, the critical mechanical coupling to synchronize reads p kc = 2m2 Ω2 γδΩ, (5.37)

where γ is the amplitude decay rate that depends on the microscopic cell parameters, see Section 5.2.2. The Hopf model was the starting point of our analytical analysis. We point out that for δΩ < γ, we generally find a very good agreement between the Hopf dynamics and the effective Kuramoto-type model. This can be checked numerically. In Section 5.5.4 we will explicitly compare the dynamics of the full coupled optomechanical system, the Hopf model and the Kuramoto-type phase description. For δΩ > γ, deviations occur via terms of higher order in δΩ/γ. Corresponding to the results in Eq. (5.33) and (5.34), this starts with −(δΩ/γ)K (cos(2δϕ) + 1) in Eq. (5.36). We point out that the results on the effective amplitude-mediated coupling (Eq. (5.33) and (5.34)) were derived from our solution on δAi (ϕ1 , ϕ2 ) that was found assuming δAi /A¯i to be small, see Eq. (5.29). According to Eq. (5.30), the size of the amplitude modulations around the limit cycle scales as |δA1 | ' k A¯2 /2mΩγ. On thep other hand, for a given frequency difference δΩ, the critical coupling scales as kc = 2mΩγ δΩ/2γ (see Eq. (5.37)). Thus, at the transition to synchronization (k = kc ), the amplitude modulations roughly scale as s |δA| 1 δΩ ' . (5.38) 2 γ A¯ k=kc− This simple estimate, based on our analysis considering an expansion in δA/A¯ (Eq. (5.25)), yields δA ' A¯ for δΩ > γ. Thus, at the transition to synchronization, there will be further contributions to the amplitude dynamics δAi (t) that are not considered by our analytical analysis. According to Eq. (5.26), we therefore expect further phase couplings that are not included in the effective Kuramoto-type model so far.

5.5.4

Synchronization Phenomena: Comparison with numerical results

In Section 5.5.3 we derived an effective Kuramoto-type model for the slow phase dynamics of coupled optomechanical cells. To verify this description, we now turn towards simulations

synchronization

0.8

0.4

sin(δϕ)

(a)

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time 0.0

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(b)

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5.5 Synchronization of two coupled optomechanical cells

0.025

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0.06

0.05

0.04

n sy

c

on hr

0.03

d ize

→

→

0

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83

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hro

d nize

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Figure 5.7: Phase-locking of two mechanically coupled optomechanical cells. (a) Phase particle in the effective Kuramoto potential U (δϕ); de-synchronized (left), phase-locked (right). (b) Time-average hsin δϕi as function of mechanical coupling k. Insets illustrate the explicit time-dependence of sin(δϕ) at the points indicated. When k exceeds a threshold kc (colored region), the phase difference δϕ(t) between the oscillations locks to a constant value despite different bare mechanical frequencies, here δΩ = 0.003 Ω1 . Both in-phase (δϕ → 0) and antiphase (δϕ → π) synchronization regimes are observed. (c) hsin(δϕ)i in the plane mechanical coupling k vs. frequency difference δΩ, including a comparison of the critical coupling kc (white, solid) with the one from a Hopf model with one fit parameter (green, dash) and the effective Kuramoto-type model (blue, dash-dot). Cell parameters as in Fig. 5.4. (This figure has previously been published in [125].) of the full optomechanical dynamics and compare them to predictions from our analytical treatment. As we will see, the Kuramoto-type model allows one to explain and predict most of the features that will be observable in future experiments. 5.5.4.1

Transition to phase locking; in-phase and anti-phase synchronization

In contrast to the standard Kuramoto model, 2δϕ appears in our effective slow-phase description for coupled optomechanical Hopf oscillators, Eq. (5.35) (and Eq. (5.36)). The additional factor of 2 was due to the amplitude-mediated phase coupling (Eq. (5.26)). Thus, depending on parameters, δ ϕ˙ = 0 can have two distinct, stable solutions that lead to both in-phase and anti-phase synchronization, respectively. This corresponds to two distinct minima in the effective potential that can be used to rewrite Eq. (5.36), δ ϕ˙ = −U 0 (δϕ). The description in terms of a phase particle sliding down a washboard potential (Fig. 5.7a) is similar to that of an overdamped Josephson junction that is driven by a current bias set by δΩ. As mentioned above, we focus on the case of nearly identical cells where the coupling C can be neglected (C/δΩ = k/2mΩ2 1), and K = k 2 /2m2 Ω2 γ. To test whether the features predicted by Eq. (5.36) are observed in the full optomechanical system, we directly simulate the motion of two coupled optomechanical cells (Eq. (5.2) and (5.3)) and increase the coupling k for a fixed frequency difference δΩ = Ω2 − Ω1 . The results are displayed in Fig. 5.7b. For small mechanical couplings, each cell runs at its intrinsic frequency Ωi . Thus, sin(δϕ(t)) (see insets of Fig. 5.7b) sinusoidally oscillates at δΩ and the

84

5. Collective Phenomena: Synchronization in optomechanical arrays

phase lag hsin(δϕ)i averages to zero. For increasing k, the phases start to interact due to the effective Kuramoto coupling K. The phase difference δϕ(t) commence to get pulled towards a constant value. Nevertheless, the phases still slip away from each other. Altogether, this leads to a distortion of the sinusoidal oscillations sin(δϕ(t)), causing a finite value for the timeaverage hsin(δϕ(t))i. Not yet synchronized, the period of phase slips increases as the coupling k is further increased and the cells’ average frequencies get pulled towards each other. Beyond a threshold kc (Eq. (5.37)), the frequencies and phases lock, indicated by a kink in hsin δϕi. According to our analytical results, in the phase-locked regime (where δ ϕ˙ = 0), we have sin(2δϕ) = δΩ/K. Correspondingly, as the coupling increases further, |sin(δϕ)| decreases as the phases are pulled towards each other (in-phase locking, δϕ → 0) or away from each other (anti-phase locking, δϕ → π). As predicted by our analytical treatment, we indeed observe both in-phase and anti-phase synchronization for coupled optomechanical cells. Whether the system synchronizes towards δϕ → 0 or δϕ → π also depends on the initial conditions. Our simulations initially start with a system at rest and consider an instantaneous switch-on of the laser input power. Note that this procedure is used for every data point in Fig. 5.7. Remarkably, numerical simulations with randomly chosen initial conditions always confirm the same qualitative behavior: right above the synchronization threshold, we always find anti-phase locking while its in-phase counterpart is exclusively observed for larger couplings (cf. Fig. 5.7b). Different initial conditions however change the domains of exclusive in-phase and anti-phase synchronization. Thus, whether the system settles into the stable solution δϕ → 0 or δϕ → π is not completely random for random initial conditions, as we would have expected from the effective Kuramoto-type model and also numerical simulations of the full coupled Hopf dynamics. We note that in principle the C cos(δϕ)-term in Eq. (5.22) can yield an exclusive antiphase locking just above the synchronization threshold. Qualitatively, this is similar to what is observed for the optomechanical simulations. Rewriting the first terms of Eq. (5.36) as in Eq. (5.23), we see that δ ϕ˙ = 0 can be met for slighly smaller coupling strengths k if δϕ ' π, compared to δϕ ' 0. However, analyzing the parameters, we have to notice that this effect is completely negligible. In fact, as pointed out, we have C/δΩ 1. Thus, the reduced description is actually not capable to explain the domains of exclusive in-phase and antiphase synchronization that nonetheless depend on initial conditions. There might be other dynamical effects, that have been lost when switching from optomechanics to the reduced Hopf description, that favor one specific final solution, ϕ → 0, π. Besides this, the effective phase description well explains the essential features of Fig. 5.7b. Fig. 5.7c shows hsin(δϕ)i as a function of mechanical coupling k and frequency difference δΩ. The density plot displays the different regimes of the system’s dynamics, i.e. unsynchronized, in-phase and anti-phase phase-locked. The synchronization threshold kc of the optomechanical simulation is plotted versus δΩ (white, solid line). Numerically a very accurate way to determine kc is to simulate δϕ(t) for a given frequency difference and tuning k until maxt (| sin(δϕ(t))|) sharply drops below one. As pointed out, the transition from anti-phase to in-phase synchronization indicated by the dotted line depends on the initial conditions. The dependence of the synchronization threshold kc on the frequency difference δΩ can be compared to the predictions of our analytical treatment. In Fig. 5.7c the threshold of the full optomechanical simulation is contrasted to both analytical results from the effective Kuramoto-type model and full numerical simulations of the Hopf dynamics (Eq. (5.4) and √ (5.5)). At small δΩ, the observed behavior kc ∝ δΩ is correctly reproduced by the effective Kuramoto-type model, Eq. (5.37). For δΩ > γ deviations occur via terms of higher order

5.5 Synchronization of two coupled optomechanical cells

85

in δΩ/γ not included in the analytical description, see discussion in Section 5.5.3.3. These produce a linear slope kc ∝ δΩ, see Fig. 5.7c. Indeed, this behavior of the optomechanical system is reproduced by a full numerical simulation of the Hopf dynamics. Comparing the effective Kuramoto-type model and Hopf, we note that for δΩ < γ the results of the analytical description well match numerical simulations of the full Hopf dynamics. Note that for δΩ < γ, the approximations leading to Eq. (5.35) generally apply. We point out that the simulation in Fig. 5.7 shows results for experimentally realistic parameters (see Section 5.4) using a laser input power well above the bifurcation threshold. This allows one to observe the essential features of synchronization in an appropriate range of frequency detuning δΩ. To achieve quantitative agreement of the Hopf model with microscopic results in Fig. 5.7, its parameter γ has to be treated as an adjustable parameter (here γ = 0.02 Ω). We note that if the “real” value of γ is used, i.e. the one that was determined for a single optomechanical cell to describe its slow amplitude dynamics (see Fig. 5.3), the coupled optomechanical system generally synchronizes earlier, i.e. for smaller couplings, than predicted by the Hopf description. Although this effect is not understood microscopically, we point out that the renormalization of γ is due to the mutual interaction of the cells. It is absent in case of driving the system in terms of an external force. We will further discuss this in Section 5.7 and give more details on the analysis in Appendix A.3. Treating γ as a fit parameter for our analytical description, we generally find a very good match between the Hopf dynamics, the derived effective-Kuramoto type model and optomechanics, cf. Fig. 5.7 and also the excellent agreements of the mechanical frequency spectrum in Fig. 5.8 that we will discuss in the following. 5.5.4.2

Mechanical frequency spectra

In experiments, a convenient observable would be the RF frequency spectrum of the light intensity emanating from the cells, |α|2 (ω). Fig. 5.8a shows the spectrum as a function of frequency difference δΩ and fixed mechanical coupling between cells (additional peaks are produced by nonlinear mixing). We note that experimentally, mechanical frequencies can be tuned via the optical spring effect. The comparison to results from a simulation of Eq. (5.35) illustrates an excellent agreement with the effective Kuramoto-type model. Around δΩ = 0, we recover in-phase and anti-phase synchronization that differ in the synchronization frequency. According to our analytical description (Eq. (5.35)), if we add ϕ˙ 1 + ϕ˙ 2 to determine the common frequency of the synchronized system, we find ξ12 + ξ21 ¯ ¯ Ω(π) − Ω(0) = ' k/mΩ. 2 In-phase and anti-phase synchronization can thus easily be distinguished in experiments by simply measuring the mechanical spectrum. Although mechanical frequencies can be tuned in terms of the optical spring effect, the 2 most easily tunable parameter however is the laser drive power (∝ αmax ), see Fig. 5.8b. Synchronization sets in right at the Hopf bifurcation while at higher drive, we find a transition towards de-synchronization. Again, this can be explained from our analytical results. We know that γ increases away from the Hopf bifurcation (i.e. for higher drive), leading to a concomitant decrease in the effective Kuramoto coupling K ∝ 1/γ (inset Fig. 5.8b), and finally a loss of synchronization (K < δΩ).

86


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3 0. 0 5 2 0. 0 2 0. 0 5 1 0. 0 1 0. 0 5 0 0. 0 0

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(a)

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Figure 5.8: Mechanical frequency spectra I(ω) of intensity fluctuations, I(t) = |α1 (t)|2 + |α2 (t)|2 , for two mechanically coupled cells. (a) Frequency locking upon changing the detuning δΩ = Ω2 − Ω1 between the mechanical frequencies P(magenta, dashed). Top: spectrum I(ω) from optomechanics. Bottom: spectral peaks of i cos(ϕi (t)) from a simulation of the effective Kuramoto-type model, Eq. (5.35). Depending on initial conditions, we find in-phase (blue) and anti-phase (red) synchronization. (k/mΩ21 = 0.015). (b) Spectrum I(ω) and effective Kuramoto coupling K (Eq. (5.36)) vs. laser input power. (k/mΩ21 = 0.01, δΩ = 0.005Ω1 ). (c) Example of trajectories Gxi (t)/κ displaying strong amplitude modulation, not described by the Kuramoto model (at power indicated by the triangle in (b)). (Color scale indicates |I(ω)| in units of the peak height at ω = 0 for a system with G = 0; δ-peaks are broadened for clarity; parameters as in Fig. 5.4 and 5.7) (This figure has previously been published in [125].)

When the system is synchronized, the amplitude of each cell is generally fixed. In some regimes, however, we observe strong amplitude modulation, see Fig. 5.8c. From our investigations we note that this effect only occurs in small regions at the transition between regimes of in-phase and anti-phase synchronization (cf. Fig. 5.8c) and if the coupling between cells in terms of k/mΩ is larger than γ. Note that γ quantifies the force that confines the Hopf oscillator to its limit cycle. Thus, this phenomenon, that is not described by the Kuramoto model, only arises if the coupling between cells is large compared to the limit cycle’s restoring force.

5.6

Synchronization in optomechanical arrays

Having studied the dynamics of two coupled optomechanical cells, i.e. the smallest unit for optomechanical arrays, we now turn towards the collective dynamics of larger assemblies. Again we first find the slow phase description where we extend the effective Kuramoto-type model to take into account arbitrary couplings between many optomechanical cells. We then compare these results to full optomechanical simulations. In particular, we consider a design that allows one to achieve a global coupling between individual cells. Note that global couplings are often discussed with respect to the standard Kuramoto model, nevertheless, they are mostly difficult to implement in experiments.

5.6 Synchronization in optomechanical arrays

5.6.1

87

Extended Kuramoto-type model

In Section 5.5 we derived and verified an effective Kuramoto-type model for the case of two coupled optomechanical cells. We want to generalize this description to account for larger assemblies of arbitrarily coupled units. Thus, we consider N optomechanical cells where each unit is described by Eq. (5.2) and (5.3). To these equations we add mechanical couplings in terms of a general coupling matrix set by individual spring constants kij , X m¨ xi = · · · + kij (xj − xi ). j6=i

An optical coupling scheme that yields effective P long-range interactions kij will be presented in Section 5.6.3. Considering the force Fi = j6=i kij Aj cos(ϕj ) that acts on cell i, the phase and amplitude dynamics of the Hopf model (Eq. (5.4) and (5.5)) reads ∂t ϕi = −Ωi +

X kij Aj cos(ϕj ) cos(ϕi ), mi Ωi Ai j6=i

∂t Ai = −γ(Ai − A¯i ) + 5.6.1.1

(5.39)

X kij Aj j6=i

mi Ωi

cos(ϕj ) sin(ϕi ).

(5.40)

Amplitude modulations in terms of phase dynamics

In Section 5.5 we discovered the importance of the amplitude dynamics and studied its influence for the case of two coupled optomechanical cells. Similarly, here we try to find an approximation δAi (ϕ1 , . . . , ϕN ) that expresses the amplitude modulations Ai (t) = A¯i +δAi (t), around each cell’s limit cycle A¯i , in terms of the phase dynamics ϕi (t) exclusively. As above, we approximate the formal solution of Eq. (5.40) that reads   Z t X kij A¯j + δAj 0 δAi = e−γ(t−t )  cos(ϕj ) sin(ϕi ) dt0 . m Ω i i −∞ j6=i

For δAi A¯i , we can consider

δAi =

X

∆Aj,i ,

j6=i

where ∆Aj,i

Z kij A¯j t −γ(t−t0 ) e = cos(ϕj (t0 )) sin(ϕi (t0 )) dt0 . mi Ωi −∞

(5.41)

describes the contribution to the amplitude dynamics of cell i that is due to its coupling to cell j. For j = 2 and i = 1, Eq. (5.41) coincides with Eq. (5.29). Similarly as above, we conduct the integration of Eq. (5.41), ! kij A¯j 1 ei(ϕj (t)+ϕi (t)) ei(ϕj (t)−ϕi (t)) − − c.c. . (5.42) ∆Aj,i ' mi Ωi 4i γ − i(Ωj + Ωi ) γ − i(Ωj − Ωi ) This expression is nothing but Eq. (5.30) with 2 → j, 1 → i. Note however that in contrast to the case N = 2, the overall amplitude modulation δAi of each cell i is not only driven by a single contribution ∆Aj,i , originating from one other cell j, but by all oscillators it is coupled P to, δAi = j6=i ∆Aj,i .

88


5.6.1.2

Amplitude-mediated phase coupling

Given the solution δAi (ϕ1 , . . . , ϕN ), we eliminate the amplitude dynamics from Eq. (5.39) by expanding Aj /Ai up to linear order, Aj Ai

=

A¯j A¯j 1 + δA − δAi j A¯i A¯i A¯2i

and find   X kij X ¯j ¯j X A A 1  ∂t ϕi = −Ωi + + ¯ ∆Ak,j − ¯2 ∆Ak,i  cos(ϕj ) cos(ϕi ). mi Ωi A¯i Ai Ai j6=i

k6=j

(5.43)

k6=i

To arrive at the slow phase description, we have to time-average Eq. (5.43) keeping only the slow contributions of the dynamics. Like for the case N = 2, this yields direct phase interactions between ϕj and ϕi , i.e. (A¯j /A¯i )hcos(ϕj ) cos(ϕi )i, that do not involve any amplitude modulation. In addition, there are amplitude-mediated contributions (1/A¯i )hδAj cos(ϕj ) cos(ϕi )i, (A¯j /A¯2i )hδAi cos(ϕj ) cos(ϕi )i, P P that comprise the amplitude dynamics δAj = k6=i ∆Ak,j and δAi = j6=i ∆Ak,i corresponding to the phases ϕj and ϕi , respectively. Both terms share the same structure. Thus, to describe the amplitude-mediated coupling, we generally have to find h∆Ak,j cos(ϕj ) cos(ϕi )i. To keep track of the individual time-scales and to distinguish fast from slow phase contri¯ + ϕ˜i . Using the result of Eq. (5.42) we have butions, we again write ϕi (t) = −Ωt ! ¯ h∆Ak,j cos(ϕj ) cos(ϕi )i ei(ϕ˜k +ϕ˜j −2Ωt) ei(ϕ˜k −ϕ˜j ) 1 = h − − c.c. 4i γ − i(Ωk + Ωj ) γ − i(Ωk − Ωj ) (kjk A¯k /mj Ωj ) 1 i(ϕ˜j +ϕ˜i −2Ωt) ¯ × e + ei(ϕ˜j −ϕ˜i ) + c.c. i. (5.44) 4

Note that, in contrast to Eq. (5.32) for N = 2, thisP expression involves three indices k, j, i. For N > 2, the amplitude dynamics of cell j, δAj = k6=j ∆Ak,j , involves several contributions from every cell k it is coupled to. Thus, the time-averaged hϕ˙ i i for arbitrary N (Eq. (5.43)) generally comprises terms like Eq. (5.44) with k 6= i, j. ¯ average to zero. Evaluating Eq. (5.44) and keeping Fast oscillations involving exp(iΩt) only the slow phase dynamics ϕ˜i , we find h∆Ak,j cos(ϕj ) cos(ϕi )i (kjk A¯k /8mj Ωj )

= − −

γ sin(ϕk − ϕi ) + (Ωk + Ωj ) cos(ϕk − ϕi ) γ 2 + (Ωk + Ωj )2 γ sin(ϕk − 2ϕj + ϕi ) + (Ωk − Ωj ) cos(ϕk − 2ϕj + ϕi ) γ 2 + (Ωk − Ωj )2 γ sin(ϕk − ϕi ) + (Ωk − Ωj ) cos(ϕk − ϕi ) , (5.45) γ 2 + (Ωk − Ωj )2


89

where finally we reinserted ϕi . Note that Eq. (5.45) only involves phase differences. For k = i, we recover the amplitude-mediated coupling of Eq. (5.33) that was previously found for N = 2. This is the case where the amplitude modulation ∆Ak,j , to the left hand side of Eq. (5.45), is driven by the same cell i that enters the expression via the phase dynamics cos(ϕj ) cos(ϕi ), see Eq. (5.43). It is instructive to compare this result to the case of two coupled cells. At first, we note 2 + 2Ω ¯ ¯ 2 ) in Eq. (5.33). Like in the first term in Eq. (5.45) that corresponds to the term 2Ω/(γ case of N = 2, due to the fact that γ, |Ωi −Ωj | (Ωi +Ωk ) for any combination of indices, this contribution is significantly smaller than the other terms and thus it is perfectly negligible. We will therefore drop the first term of Eq. (5.45) in the following. More important are the other terms. Compared to Eq. (5.33), the more general expression Eq. (5.45) is due to the diverse combinations of exponential phase terms in Eq. (5.44) that now involves three indices k, j, i. Note that Eq. (5.44) always yields terms of the form Im

e−iα a + ib

= −

a sin α + b cos α . a2 + b2

While the slow phase contributions to right hand side of Eq. (5.32) always comprise terms with α = 0, the analogous ones found from the more general Eq. (5.44), have α = ϕk − ϕi . This yield deviations if k 6= i, i.e. if the amplitude modulation ∆Ak,j to the right hand side of Eq. (5.45) is driven by a cell k other then i that enters due to the phase dynamics cos(ϕj ) cos(ϕi ), see discussion above. For a quantitative description we focus on the regime where the frequency difference between cells is small compared to the amplitude decay rate, |Ωk −Ωj | γ, see also Eq. (5.38) (δΩ γ). In this case, Eq. (5.45) simplifies and we have h∆Ak,j cos(ϕj ) cos(ϕi )i = −

1 kjk A¯k [sin(ϕk − 2ϕj + ϕi ) + sin(ϕk − ϕi )] . 8 mj Ωj γ

Using this result for the time average of Eq. (5.43), the slow phase dynamics reads ∂ t ϕi

X kij A¯j 1 = −Ωi + cos(ϕj − ϕi ) mi Ωi A¯i 2 j6=i X kij 1 X 1 kjk A¯k − [sin(ϕk − 2ϕj + ϕi ) + sin(ϕk − ϕi )] + mi Ωi A¯i 8 mj Ωj γ k6=j j6=i X kij A¯j X 1 kik A¯k − − [sin(ϕk − 2ϕi + ϕj ) + sin(ϕk − ϕj )] . mi Ωi A¯2i 8 mi Ωi γ j6=i

k6=i

We point out that the terms that do not depend on ϕi cancel, i.e. X X kij kik A¯j A¯k 1 sin(ϕk − ϕj ) = 0. mi Ωi mi Ωi 8γ A¯2i j6=i k6=i

Note that sin(ϕk − ϕj ) vanishes if j = k. Otherwise, if j 6= k, for every combination of indices (j, k) there is always the inverted contribution (k, j) that enters the sum with a reversed sign canceling (j, k).


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frequency

Figure 5.9: Mechanical frequency spectra for an 1D optomechanical array consisting of four optomechanical cells with nearest-neighbor mechanical as sketched in Fig. 5.1d. P interaction 2 (a) Spectrum I(ω) of intensity fluctuations I(t) = i |αi (t)| from an optomechanical simulation starting with random initial conditions at k/mΩ2 = 0.04 and adiabatically decreasing coupling strength. Yellow arrows indicate the cells’ intrinsic mechanical frequencies Ωi , i.e. [0.98, 0.99, 1.01, 1.02]Ω. The spectrum displays full synchronization, partial synchronization and a fully unsynchronized state. (b) Spectra of the individual oscillators’ trajectories xi (t) corresponding to the simulation in (a). (Density plot as in Fig. 5.8; δ-peaks are broadened for clarity; parameters as in Fig. 5.4 and 5.7). 5.6.1.3

Generalized effective slow phase description

Thus, we finally arrive at an effective Kuramoto-type model for N coupled optomechanical Hopf oscillators, given an arbitrary coupling kij between the individual cells, X ξij X X ξij ξik ∂t ϕi = −Ωi + cos(ϕj − ϕi ) + sin(ϕk + ϕj − 2ϕi ) 2 8γ j6=i

+

j6=i k6=i

X X ξij ξjk j6=i k6=j

8γ

(sin(2ϕj − ϕk − ϕi ) − sin(ϕk − ϕi )) ,

(5.46)

where ξij = kij A¯j /mi Ωi A¯i . Note that for two coupled cells (N = 2, kij = k) we have ∂t ϕ1 = −Ω1 + +

ξ12 ξ12 ξ12 cos(ϕ2 − ϕ1 ) + sin(ϕ2 + ϕ2 − 2ϕ1 ) 2 8γ

ξ12 ξ21 (sin(2ϕ2 − ϕ1 − ϕ1 ) − sin(ϕ1 − ϕ1 )) 8γ

and we recover the result of Eq. (5.35).

5.6.2

Nearest-neighbor mechanical coupling

In general, the extended effective Kuramoto-type model (Eq. (5.46)) allows one to take into account various coupling schemes for optomechanical cells in arbitrary array structures. Note


91

that the general coupling between cell i and j, i.e. the coupling constant kij , determines ξij in Eq. (5.46). In Section 5.4, we quantified the mechanical coupling strength between two cells localized on a one-dimensional beam, see Fig. 5.6. These results immediately generalize to larger 1D assemblies of cells with nearest-neighbor mechanical interaction. To take into account such an array, we first consider the system, schematically depicted in Fig. 5.1d. The setup corresponds to four identical cells with different intrinsic frequencies Ωi arranged on a 1D beam, cf. Fig. 5.6. This yields a nearest-neighbor mechanical interaction between cells (kij = kδi,j−1 ). Using full dynamical simulations of the optomechanical system (Eq. (5.2) and(5.3)), we calculate the mechanical frequency spectrum experimentally observable via the light-field emanating from the individual cells, cf. Fig. 5.8. The results are shown in Fig. 5.9a. For this simulation we used random initial conditions starting at large interactions (k/mΩ2 = 0.04) and subsequently decreased the mechanical coupling adiabatically. Numerically, for every new simulation that considers a slightly smaller value of k, the steady-state solution of the pervious calculation is used as initial condition. Despite the difference of the cells’ intrinsic frequencies (Fig. 5.9a), at large couplings, the mechanical spectrum only displays a single peak corresponding to a fully synchronized state. Here, all cells phase-lock to a common frequency. Decreasing the mechanical coupling, synchronization is gradually lost until finally, at sufficiently low couplings, each cell rotates at its intrinsic frequency Ωi . Note that, apart from being fully phase-locked and fully unsynchronized, that were the only two scenarios observable for a system consisting of two cells, the array with nearest-neighbor mechanical interaction additionally displays partial synchronization. The dynamics of partial synchronization can be further highlighted if we look at the frequency spectrum of each oscillator’s trajectory xi (t), see Fig. 5.9b. At low couplings, each spectrum xi (ω) only displays the individual cell’s intrinsic frequency, i.e. all oscillators rotate independently. Increasing the coupling beyond k/mΩ2 = 0.01, we see that at first the third (i = 3) and the fourth (i = 4) cell synchronize to a common frequency while the others still oscillate separately. Increasing the coupling further, also the second cell’s frequency (i = 2) gets pulled towards the frequency of i = 3 and i = 4. Only for large couplings, all oscillators share the same frequency and the array is fully synchronized. For the simulations shown in Fig. 5.9, we decreased the mechanical coupling starting with random initial conditions at large k. In contrast, Fig. 5.10a displays identical optomechanical simulations however reversing the sweep direction. Here we start at k = 0 and adiabatically increase the interaction. We note that for low and modest coupling strength, the spectra of Fig. 5.9a and Fig. 5.10a essentially coincide. However, using this procedure, we do not find the fully synchronized state that was observed in Fig. 5.9a for large values of k. Thus, the optomechanical array with nearest-neighbor mechanical interaction possesses several steady-state solutions for the system’s dynamics, such as full synchronization and partial synchronization, that can co-exist simultaneously. Due to this, the final result of the dynamics also depends on initial conditions. We now compare this dynamics of the full optomechanical simulation to results obtained from the effective Kuramoto-type model. Given the general expression in Eq. (5.46), we find the corresponding phase model for four coupled optomechanical cells. Although it is straight forward to obtain the corresponding phase equations, they are displayed in Appendix A.2 for clarity. Note that although we only have nearest-neighbor mechanical interaction between the optomechanical cells, there is an effective phase coupling going beyond nearest-neighbor interaction in the Kuramoto-type model. This is due to the amplitude-mediated phase interaction


mechanical coupling

92 0.04

0.04

0.03

0.03

0.02

0.02

0.01

0.01

0.00

optomechanics 0.90

(a)

0.95

0.00 1.00

frequency

1.05

1.10

effective Kuramoto 0.90

(b)

0.95

1.00

1.05

1.10

frequency

Figure 5.10: Mechanical frequency spectrum of an optomechanical array with nearestneighbor mechanical interaction compared to P the effective Kuramoto-type model. (a) Spectrum of I(ω) of intensity fluctuations I(t) = i |αi (t)|2 from an optomechanical simulation identical to Fig. 5.9 but with reversed sweep direction Pstarting at k = 0. Here, the fully synchronized solution does not occur. (b) Spectrum of i cos(ϕi (t)) from a simulation of the effective Kuramoto-type model, Eq. (5.46), corresponding to the optomechanical calculation displayed in (a). Parameters as in Fig. 5.4 and 5.7; cf. Fig. 5.9.

(see Section 5.6.1.2). Physically, one cell’s phase dynamics couples to another oscillator’s amplitude dynamics that in turn can act back on the phase dynamics of a third cell leading to an effective next-nearest-neighbor coupling. Fig. 5.10b shows the results of numerical calculations of the effective Kuramoto-type model that corresponds to the optomechanical simulation displayed in Fig. 5.10a. Like in case of the full optomechanical dynamics, we find transitions to partial synchronization. Furthermore, the dynamics of the reduced description also depends on initial conditions and the details of the sweep. Using modified initial values, also the effective Kuramoto-type model can yield a full synchronized state at large couplings corresponding to what has been observed for the optomechanical dynamics in Fig. 5.9a. Generally, comparing the effective Kuramoto-type model and the full optomechanical system, we find the essential features of the dynamics. The comparison in Fig. 5.9 yields qualitatively similar results. An exact quantitative match, however, is also complicated due to the system’s sensitivity to initial conditions. Note that due to the translation to an exclusive phase model, there is no simple description of how given optomechanical initial conditions (such as laser input power, position and momentum) translate to initial phases for the effective Kuramoto-type simulation. In the following section we discuss a globally coupled optomechanical array where this complication does not occur and where we find a nice quantitative match between the dynamics of an even larger optomechanical array and the effective Kuramoto-type model.

5.6 Synchronization in optomechanical arrays 1.01

optical mode mechanical mode

(a)

order para.

extended optical mode

...

93

0.1 0.1

0.00

0.002 0.002

0.004 0.004

0.006 0.006

0.008 0.008

0.01 0.01

global coupling

(b)

Figure 5.11: Optomechanical array with global coupling. (a) Schematic of a global coupling scheme realized via one fast, extended optical mode that couples to the mechanics of each cell i. (b) Order parameter χ as a function of global coupling strength kg for an array of N = 10 cells, comparing result of a full optomechanical simulation (blue) to results from the effective Kuramoto-type model in Eq. (5.46) (red, dashed) using one fit parameter, γ = 0.036Ω. The cells’ mechanical frequencies are chosen equidistant from 0.98Ω to 1.02Ω. (This figure has previously been published in [125].)

5.6.3

Effective global coupling via an extended optical mode

Although mechanical couplings are important and constitute one possibility to design interactions between optomechanical cells, this approach has two drawbacks. On the one hand, mechanical couplings are difficult to tune in experiments. As pointed out above, it is much more convenient to tune optical parameters, such as the cell’s laser drive power or the mechanical frequency using the optical spring effect, see Fig. 5.8. On the other hand, mechanical interactions mediated by the geometry are usually short range, see Fig. 5.6. Particularly for large arrays, one might be interested in a long-range, global coupling between cells (kij = kg ). Such a coupling scheme is often considered for the standard Kuramoto model, but generally it is difficult to implement in experiments. Given the enormous design flexibility of optomechanical crystal structures, a long-range, global coupling between cells can be designed in terms of a fast (κext Ωi ), extended optical mode αext that couplesPto the mechanics of each cell, see Fig. 5.11a. The light-field intensity is then modulated by j xj , and the light force thus generates a global, effective mechanical coupling 8∆ kg = −~G2 |¯ αext |2 2 , 4∆ + κ2ext where |¯ aext |2 is the average number of circulating photons inside the extended optical mode. Thus, this long-range global coupling kg is tunable via the laser input power. Considering such a coupling, Fig. 5.11b shows results on the order parameter 2 1 X χ = h eiϕk i N k

for an array of ten globally coupled optomechanical cells (Fig. 5.11a). For small couplings, each cell oscillates independently, the phases ϕk are random, and thus χ = 1/N . For larger coupling, we find a regime where the phases are not fully phase-locked but the phase factors become anticorrelated, decreasing χ. At large coupling kg , there is a transition to phaselocking, where finally all cells are synchronized with χ ' 1. All these features are reproduced nicely by the effective Kuramoto-type model, Eq. (5.46), (Fig. 5.11b).


(a)

(b) optical mode mechanical mode

(c) 2.0

-2.0

laser input power

94

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0.8 0.6

0.2

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00.0

0.8

time

1.2

1.0

frequency 1.00

Figure 5.12: Array setup involving only a single extended optical mode. (a) Schematic setup: several mechanical modes couple to a single extended optical one. (b) Spectrum of intensity fluctuations I(ω) vs. laser input power for an array of five mechanical modes coupled to a common optical mode. At large drive, a regime of chaotic motion is entered. Due to the presence of multiple attractors, the regimes observed in such a diagram may depend on how the parameters are swept. (c) Example of a trajectory in the chaotic regime (at power indicated by blue triangle in (b)). (Density plot as in Fig. 5.8; δ-peaks are broadened for clarity; parameters are ∆ = κ = 100Γ = Ω1 , as in Fig. 5.4 and 5.7; yellow triangles indicate the intrinsic mechanical frequencies Ωi ) (This figure has previously been published in the preprint version [153] of [125]) 0.0

5.6.4

0.1

0.2

An array setup involving only a single extended optical mode

Another way of designing a global coupling might be to consider identical optical modes that combine into extended ’molecular’ modes, one of which is then driven by a single laser via an evanescently coupled tapered fibre (Fig. 5.6). Thus, this scheme, displayed in Fig. 5.12a, only considers a single optical mode that fulfills two functions: it supplies the optical drive to induce mechanical self-oscillations and it initiates an optical coupling between oscillators. Note that this approach significantly differs from the previous optomechanical array setup that used a fast, extended optical mode to design an effective global coupling between optomechanical cells (each consisting of an optical mode coupled to a mechanical one). For large arrays this scheme might be particularly practical as the design only involves a single optical mode. For efficient excitation of self-induced oscillations, one has to ensure that the detuning ∆ is equal to all the mechanical frequencies Ωj in the array, to within |∆ − Ωj | < κ. For arrays of reasonable size, the splittings between adjacent optical molecular modes will be more than 102 times larger than κ, such that we can ignore all but one optical mode. This setup then leads to a global coupling of many nanoresonators to a single OM, such that   X κ κ α˙ = i(∆ + Gj xj ) −  α + αmax , 2 2 j

and the force on each resonator is given by −~Gj |α|2 .

5.7 A remaining challenge: γ an adjustable parameter

95

For illustration, we chose N = 5 cells (Fig. 5.12b)1 . As before, we find synchronization regimes. In addition, at higher drive, a transition into chaos takes place. Analyzing the timeevolution in more detail, we observe transient fluctuations in amplitude and phase, with a strong sensitivity on changes in initial conditions (Fig. 5.12c). Note that in a single optomechanical cell one may also find chaotic behavior [71], but for far larger driving strengths.

5.7

A remaining challenge: γ an adjustable parameter

Towards the end of this chapter, we want to point out a remaining challenge with respect to the Hopf model. Recall that the Hopf description (Eq. (5.4) and (5.5)) was essential for our analytical analysis of synchronization behavior in coupled optomechanical systems. It describes the effective slow phase and amplitude dynamics that follow via eliminating the light field. Thus, this model significantly reduces the system’s complexity and focusses on the essential dynamical properties of a single optomechanical cell driven towards selfinduced mechanical oscillations. Starting from this description, it allowed us to derive an effective Kuramoto-type model that enabled to explain and predict the main features of synchronization. The exclusive phase model then provided an analytical understanding of the coupled optomechanical dynamics. Furthermore the phase description can significantly reduce the computation time needed to simulate systems consisting of many coupled cells. Despite these achievements, we needed to keep one adjustable parameter to obtain quantitative agreement between the full dynamics of coupled optomechanical cells and the corresponding Hopf description. In fact, when considering a renormalized amplitude decay rate γ, that is fitted to match the optomechanical result, the analytical model yields the correct critical coupling for transitions to synchronization (Fig. 5.7), the excellent agreement with respect to the mechanical frequency spectrum (Fig. 5.8) or all the features of the order parameter of an optomechanical array (Fig. 5.11). Note that the modified value of γ enters the Kuramoto-type description that is derived from Hopf model. Conceptually, it is however unsatisfying to treat γ as an adjustable parameter. In fact, for a single optomechanical cell, γ characterizes its slow amplitude dynamics that is generally set in terms of the microscopic cell parameters, see Section 5.2. Accordingly, the amplitude decay rate should either be determined analytically via Eq. (5.10) (valid close to the bifurcation threshold), or numerically by simulating the amplitude’s exponential decay towards steadystate self-oscillations, see Fig. 5.3. For a single optomechanical cell driven by an external periodic force (Section 5.3), the ¯ actually perfectly Hopf dynamics, considering the microscopically derived parameters (γ, A), matches the full optomechanical dynamics. This is true for the cell’s phase as well as for its amplitude dynamics, see Fig. 5.4 and Fig. 5.5. Note that γ enters the magnitude of the amplitude modulations around the limit cycle A¯ illustrating an excellent agreement in Fig. 5.5a. Generally, the perfect match between Hopf and optomechanics for the case of external periodic forcing (Section 5.3) is always found for the “real”, microscopically derived value of γ - as it should be. In contrast, for coupled optomechanical cells close to the synchronization regime, deviations occur between optomechanics and the Hopf description. In Appendix A.3 we point out further aspects that have been investigated in this respect. This suggests that deviations occur due to the mutual interaction between optomechanical cells that might not be fully captured by the Hopf model up to now. Note that in Section 5.2 we only considered 1

The simulations shown in Fig. 5.12 and the investigation of the chaotic regime were done by Max Ludwig.

96


a single cell adding a force to it. A future task will be to microscopically derive a Hopf model for mutually interacting optomechanical cells. This potentially modified slow phase and amplitude description might then account for the deviations between the current Hopf model and the full coupled optomechanical dynamics that so far have only been treated in terms of an effective, adjustable value of γ.

5.8

Conclusion

To conclude, we have introduced optomechanical arrays as a new system to study collective oscillator dynamics, with room-temperature operation in integrated nano-fabricated circuits and with novel possibilities for readout and control, complementing existing research on Josephson arrays [146], laser arrays [147] and other nanomechanical structures [148, 149]. Recent experiments on 2D optomechanical crystals [154] could form the basis for investigating collective dynamics in 2D settings with various coupling schemes. Applications in metrology and time-keeping may benefit from phase noise suppression via synchronization [155]. Variations of the optomechanical arrays investigated here may also be realized in other designs based on existing setups, like multiple membranes in an optical cavity [20] or arrays of toroidal microcavities [72,74]. Inspired by our work, several groups now started working on synchronization in optomechanical systems, see for instance [156].

Chapter 6

Conclusion In this thesis, we theoretically investigated the dynamics of coupled multimode optomechanical systems. Going beyond the standard approach, that up to now mainly focused on the steady-state dynamics of optomechanical systems, we analyzed non-equilibrium photon dynamics driven by mechanical motion. For coupled multimode optomechanical systems, consisting of several interacting optical (and vibrational) modes, this scheme allows one to deliberately transfer photons in a coherent fashion. More generally, we showed that this mechanically driven coherent photon dynamics in principle enables to realize all kinds of strongly driven two- and multilevel dynamics, know from atomic physics, in the light-fields of optomechanical systems. This opened up the whole domain of strongly driven two- and multilevel phenomena to the field of optomechanics. Furthermore, we investigated the dynamical back-action of coupled multimode optomechanical systems, i.e. light acting back on the mechanics after having been influenced by the mechanical motion. This dynamics is significantly influenced by mechanically driven photon dynamics between individual modes. We explored light-induced mechanical self-oscillations (phonon lasing) in these systems and developed the fully nonlinear theory of phonon lasing for such coupled multimode optomechanical setups. We showed 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. More generally, our findings illustrated the generic effects of Landau-Zener physics on back-action induced self-oscillations. Based on recent experiments that implemented the standard optomechanical setup using superconducting microwave resonators, we considered coupled multimode optomechanics in the microwave regime. In contrast to similar systems in the optical realm, the coupling frequency governing photon exchange between microwave modes turned out to be naturally comparable to typical mechanical frequencies. In particular we proposed 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 predicted the possibility to measure an individual quantum jump from the mechanical ground state to the first excited state. Finally, building on the fascinating development to diminish and on-chip integrate optomechanical systems, based on photonic crystal structures for instance, we started to explore the collective nonlinear dynamics in arrays of coupled optomechanical cells. We showed that the phase of many such coupled optomechanical oscillators, even in case of different bare initial

98

6. Conclusion

frequencies, can lock to each other, synchronizing the dynamics to a collective oscillation frequency. In particular, we derived an effective Kuramoto-type model that allows one to explain and predict most of the features that will be observable in future experiments.

Appendix A

Appendix A.1

General structure of the mechanically driven light field dynamics

Given the system Hamiltonian for the coupled multimode optomechanical system in Eq. (2.1), we find the equation of motion using input/output theory, see Eq. (2.3). According to this, the purely classical light field dynamics for the average fields read √ d 1 x(t) κ aL (t) = ω0 ~ 1 − aL + g~ aR − aL (t) − κ bin L (t) dt i~ l 2 √ d 1 x(t) κ aR (t) = ω0 ~ 1 + aR + g~ aL − aR (t) − κ bin R (t), dt i~ l 2 where bin i (t) denote the classical input fields driving the optical modes (i = L, R). Transforming to a rotating frame with d d ai eiω0 t = ai eiω0 t − iω0 eiω0 t ai , dt dt the equations of motion read d aL (t) 1 −¯ x(t) − i κ2 = g dt aR (t) i with

g x ¯(t) − i κ2

aL (t) + Bin (t) aR (t)

(A.1)

in √ b (t) Bin (t) = − κeiω0 t L . bin R (t)

Given an external mechanical drive x ¯(t) = A¯ cos(Ωt) + x ¯0 , the formal solution to Eq. (A.1) is Z t aL (t) = U (t, t0 )Bin (t0 ), dt0 (A.2) aR (t) −∞

where the time evolution operator U (t, t0 ) is a solution to the homogenous differential equation d U11 (t, t0 ) U12 (t, t0 ) U11 (t, t0 ) U12 (t, t0 ) = M (t) (A.3) U21 (t, t0 ) U22 (t, t0 ) dt U21 (t, t0 ) U22 (t, t0 )

100 with

A. Appendix

1 M (t) = i

and initial condition

−¯ x(t) − i κ2 g

g x ¯(t) − i κ2

(A.4)

U (t0 , t0 ) = 1 at time t0 (t ≥ t0 ). Ui1 (t, t0 ) describes the amplitude for a photon to enter the left mode at time t0 and to be found in the left (i = 1) or right (i = 2) mode later at time t. In contrast, Ui2 (t, t0 ) describes the same amplitudes but for a photon initially inserted into the right mode. Due to the fact that the equation of motion (Eq. (A.1)) is linear in ai (t), the entire dynamics is characterized by the single-photon Green’s function Uij (t, t0 ). As a result, the light field dynamics is mapped to a two-level system (Eq. (A.3) and (A.4)) where each level corresponds to one of the optical modes involved. This mapping, based on the linearity of the equations of motions, is quite universal. Thus, going beyond the case of two modes, this also allows one to similarly describe larger coupled systems in terms of multi-level dynamics. For the system depicted in Fig. 2.1a, where the laser exclusively drives the left hand side −iωL t bin , bin (t) = 0), U (t, t0 ) is sufficient to describe the light field of the cavity (bin i1 R L (t) = e dynamics. If we define 0

GL (t, t0 ) = e(κ/2)(t−t ) U11 (t, t0 ) (κ/2)(t−t0 )

0

GR (t, t ) = e

0

U21 (t, t )

the general solution ai (t) according to Eq. (A.2) reads Z √ in t 0 0 ai (t) = − κb Gi (t, t0 )e−(κ/2)(t−t ) e−i∆L t dt0

(A.5) (A.6)

(A.7)

−∞

with laser detuning ∆L = ωL − ω0 and Gi (t, t0 ) is a solution to d 1 −¯ GL (t, t0 ) GL (t, t0 ) x(t) g = GR (t, t0 ) g x ¯(t) dt GR (t, t0 ) i

(A.8)

with initial condition GL (t0 , t0 ) = 1, GR (t0 , t0 ) = 0 (t ≥ t0 ). Note that the initial condition is unchanged when transforming from Ui1 (t, t0 ) to Gi (t, t0 ). According to input/output theory, √ the output mode to the right of the cavity reads bout κaR . The transmitted power R = out 2 2 in 2 is proportional to |bR | . Thus, the transmission is given by T = |bout R | /(b ) and from Eq. (A.7) we find the result stated in Eq. (2.4). With respect to a physical interpretation and further analytic treatment of the dynamics, it is appropriate to rewrite this result. Transforming to a non-uniformly rotating frame Z t 0 0 0 ¯ a ˜L (t, t ) = exp −i A cos(Ωt ) dt GL (t, t0 ) 0 Z t a ˜R (t, t0 ) = exp +i A¯ cos(Ωt0 ) dt0 GR (t, t0 ) 0

and considering the mechanical drive x ¯(t) = A¯ cos(Ωt) + x ¯0 in Eq. (A.8), we have Z t 1 −(A¯ cos(Ωt) + x ¯0 )aL + gaR a ˜˙ L = −iA¯ cos(Ωt)˜ aL + exp −i A¯ cos(Ωt0 ) dt0 i 0 Z t 1 = −x0 a ˜L + g exp −2i A¯ cos(Ωt0 ) dt0 a ˜R i 0

A.1 General structure of the mechanically driven light field dynamics and similarly a ˜˙ L , such that d 1 a ˜L −¯ x0 ge−2iφ(t) a ˜L = ˜R a ˜R +¯ x0 dt a i ge+2iφ(t)

101

(A.9)

¯ with φ(t) = (A/Ω) sin(Ωt). The initial conditions GL (t0 , t0 ) = 1, GR (t0 , t0 ) = 0 translate to t0 -dependent conditions in the non-uniformly rotating frame, 0) 1 a ˜L (t0 , t0 ) sin(Ωt −i A . =e Ω 0 a ˜R (t0 , t0 ) Note that as Eq. (A.9) is linear we can still solve it using a ˜L (t0 , t0 ) = 1, a ˜R (t0 , t0 ) = 0 and 0) −iφ(t multiply the result by e . Thus, we finally find Z i √ in ±iφ(t) t h 0 0 0 ai (t) = − κb e a ˜i (t, t0 )e−iφ(t ) e−(κ/2)(t−t ) e−i∆L t dt0 , −∞

where a ˜i (t, t0 ) is a solution to Eq. (A.9) with initial condition a ˜L (t0 , t0 ) = 1, a ˜R (t0 , t0 ) = 0 (t ≥ t0 ). This is the form of the solution stated in Eqs. (2.5-2.7). Note that the phase factor e±iφ(t) drops when considering |ai (t)|2 . Using this representation in terms of the nonuniformly rotating frame a ˜i (t, t0 ) allows one to clearly distinguish mechanical contribution to the excitation process and to the internal dynamics. Furthermore, the internal dynamics is mapped to a two-level system with time-dependent coupling.

102

A.2

A. Appendix

Effective phase model for four nearest-neighbor coupled optomechanical cells

For clarity we display the equations of motion of the effective Kuramoto-type phase model for the case of four identical optomechanical cells arranged in a 1D array with intrinsic frequency Ωi and nearest-neighbor mechanical interaction (kij = kδi,j−1 ). Given the general expression in Eq. (5.46) we find ˜ sin [2(ϕ2 − ϕ1 )] ∂t ϕ1 = −Ω1 + C˜ cos (ϕ2 − ϕ1 ) + K ˜ K − [sin (ϕ3 − 2ϕ2 + ϕ1 ) + sin (ϕ3 − ϕ1 )] , 2 ˜ sin [2(ϕ3 − ϕ2 )] + K ˜ sin [2(ϕ1 − ϕ2 )] ∂t ϕ2 = −Ω2 + C˜ cos (ϕ3 − ϕ2 ) + C˜ cos (ϕ1 − ϕ2 ) + K ˜ K ˜ [sin (ϕ3 − 2ϕ2 + ϕ1 )] , − [sin (ϕ4 − 2ϕ3 + ϕ2 ) + sin (ϕ4 − ϕ2 )] + K 2 ˜ sin [2(ϕ4 − ϕ3 )] + K ˜ sin [2(ϕ2 − ϕ3 )] ∂t ϕ3 = −Ω3 + C˜ cos (ϕ4 − ϕ3 ) + C˜ cos (ϕ2 − ϕ3 ) + K ˜ K ˜ [sin (ϕ4 − 2ϕ3 + ϕ2 )] , − [sin (ϕ1 − 2ϕ2 + ϕ3 ) + sin (ϕ1 − ϕ3 )] + K 2 ˜ sin [2(ϕ3 − ϕ4 )] ∂t ϕ4 = −Ω4 + C˜ cos (ϕ3 − ϕ4 ) + K ˜ K [sin (ϕ2 − 2ϕ3 + ϕ4 ) + sin (ϕ2 − ϕ4 )] , − 2 ˜ = k 2 /4m2 Ω2 γ. where C˜ = k/2mΩ and K Note that for each equation ∂t ϕi , the first line displays terms involving only nearestneighbor phase interactions. One might have guessed these terms given the results found previously for the case of two coupled optomechanical cells, cf. Eq. (5.35) and (5.36). However, despite the fact that there is only a nearest-neighbor mechanical interaction between cells in this scenario, due to the amplitude-mediated coupling (see Section 5.6.1.2), there are phase couplings going beyond nearest-neighbor in the effective Kuramoto-type model, cf. second line of each equation ∂t ϕi .

A.3 Amplitude decay rate - an adjustable parameter

A.3

103

Amplitude decay rate - an adjustable parameter

We want to present some further aspects that have been investigated with respect to the treatment of γ. To find a quantitative match between optomechanics and the reduced description as shown in Fig. 5.7 (Phase-locking of two coupled optomechanical cells), Fig. 5.8 (mechanical frequency spectra tuning various parameters), Fig. 5.10 (array with nearest-neighbor mechanical interaction) or Fig. 5.11 (order parameter of a globally coupled optomechanical array), the amplitude decay rate γ needed to be treated as an adjustable parameter. As explained in Section 5.7, although the reduced description is very successful explaining the essential features of synchronization in these systems, this constitutes a remaining challenge for the model. In particular, it remains an open question why for coupled optomechanical cells an adjusted parameter is needed to cope with the deviation between optomechanics and Hopf, while for the case of a single periodically forced cell, we generally find a perfect correspondence of both descriptions. Note that Fig. 5.4 and Fig. 5.5 display an excellent quantitative match for the oscillator’s phase and the amplitude dynamics, respectively. Thus, if we look at a single cell, the Hopf model perfectly describes both the effective slow-phase and amplitude dynamics that follow after eliminating the light field. A deviation only occurs for coupled optomechanical cells. We would like to emphasize that this discrepancy displays between simulations of the full optomechanical dynamics and results found for the reduced Hopf model. Thus, it is not due to any approximation that was made when deriving the effective Kuramoto-type model. For a single optomechanical cell that is periodically driven by an external force, we compare the full time-dependent dynamics of both optomechanics and the Hopf model in Fig. 5.4a and Fig. 5.5a. Here, the reduced description considering the “real”, microscopically derived value of γ shows an excellent agreement with the full optomechanical results. For this set of experimentally feasible parameters, γ was determined numerically by fitting the cell’s slow amplitude dynamics in Fig. 5.3. Remarkably, the Hopf simulation fully captures the distortion of the periodic oscillations of δϕ(t) and A(t) that occur close to the transition to the phase-locked regime; see Fig. 5.4a and Fig. 5.5a, respectively. Also note that the size of the amplitude modulations scales as 1/γ (see Eq. (5.16) and (5.17)). Thus, using a different value of γ for the Hopf simulation (i.e. one that deviates from the “real”, microscopic one found in Fig. 5.3), we would not find the excellent match that is observed in Fig. 5.5. Similarly, also for the case of two coupled cells, we considered full optomechanical simulations and corresponding calculations of the Hopf model. Analogous to the plots in Fig. 5.4a and Fig. 5.5a, we compared both the slow phase ϕi (t) and amplitude dynamics Ai (t), where i = 1, 2 denotes the two cells. Considering the “real”, microscopically derived value of the amplitude decay rate γ (Fig. 5.3), we found deviations between optomechanics and the Hopf model for couplings close to the transition to synchronization. For weak couplings far away from the synchronized regime, sin(δϕ(t)) (with phase difference δϕ = ϕ2 − ϕ1 ) as well as ¯ display an δAi (t) = Ai (t) − A¯ (i.e. the amplitude modulation around the limit cycle A) harmonically oscillating behavior with period 2π/δΩ, where δΩ = Ω2 − Ω1 is the frequency difference. This corresponds to the behavior that is observed for a single optomechanical cell exposed to an external force, see Fig. 5.4a and Fig. 5.5a. In this regime of weak coupling that is far away from synchronization, we can find an exact match between the Hopf model and the full optomechanical dynamics considering the “real”, microscopic value of γ. Increasing the coupling between the cells, similar to the case of periodic forcing (Fig. 5.5a), the harmonic os-

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cillations get distorted as the oscillators’ frequencies are pulled towards each other. However, in contrast to Fig. 5.4a and Fig. 5.5a, for two coupled optomechanical cells, the optomechanical dynamics at some point hurries ahead. Thus, the optomechanical system synchronizes earlier than predicted by the reduced Hopf description. Considering an adjusted value of γ that is smaller than the “real”, microscopically derived parameter, leads to a smaller threshold to synchronize in the Hopf model. This effectively considers the deviation between both descriptions, that is observed at the transition to synchronization for coupled optomechanical cells. A perfect match as found in the case of an externally forced single cell is however not achieved in this way. We further investigated this deviation. In particular, we tried to find parameters where the Hopf simulations, using the microscopically derived value of γ, fully match the optomechanical dynamics of two coupled optomechanical cells. In addition to the parameters of a single cell (i.e. laser detuning ∆0 = ∆/Ω, cavity decay rate κ0 = κ/Ω, mechanical damping rate Γ0 = Γ/Ω and laser input power P = ~G2 αmax /Ω3 ), the dynamics is characterized by the mechanical coupling k and the frequency difference δΩ between cells. Note that we assumed weak forces acting on each cell in oder to be able to characterize the cells’ dynamics in terms of Hopf oscillators. To check the influence of the interaction strength k on the observed deviation between Hopf and optomechanics, we considered simulations with weaker mechanical coupling, that significantly reduced the force strength acting on each cell. To ensure that the transition to synchronization was actually observed for smaller forces, the frequency difference between the cells needed to be reduced accordingly, see Eq. (5.37). Nevertheless, the deviation between Hopf and optomechanics for coupled optomechanical cells still occurred for these weaker forces. In particular, we compared the force strength Fi (t) = kAi (t) of simulations for coupled cells to analogous computations for a single optomechanical cell with external forcing (same cell parameters). It turned out that even for coupling forces much weaker than the ones shown in Fig. 5.4 and Fig. 5.5, the calculations still displayed the deviations occurring for the coupled system. This illustrates that the considered coupling is generally not too high for the Hopf model to apply as it matched the phase-locking behavior a single cells involving even higher force strengths. Besides the coupling, we also investigated the cells’ oscillation frequency. As pointed out in Section 5.5.2.1, we explored whether the amplitude dynamics has any effect on the mechanical oscillation frequency. Furthermore, we took into account the optical spring effect that generally modifies the cells’ intrinsic mechanical frequency. However, given the cell parameters (in particular a fixed laser drive power) it only shifts the oscillation frequency by a constant that is not affected by external periodic forcing (as demonstrated by the perfect match of the single cell’s dynamics including a constant shift due to the optical spring effect). For two identical cells, the frequency difference δΩ is then still set by the intrinsic frequencies. In that sense, the optical spring effect is actually more an hassle for phase-locking to an external force, where one has to consider the precise shifted mechanical frequency to find the excellent match of the Hopf dynamics shown in Fig. 5.4 and Fig. 5.5. As pointed out in Section 5.3.2, phase-locking to an external force is therefore an excellent method to determine the precise oscillation frequency of an optomechanical cell. Another attempt to find a better match between Hopf and optomechanics for two cells, using the microscopically derived value of γ, was to tune the cell parameters. Decreasing the laser input power, for instance, we moved the system closer to the Hopf bifurcation, see Section 5.2. Note that being close to the onset of self-induced oscillations complicates the numerical simulation of synchronization as pointed out in Section 5.2.3. However, we

A.3 Amplitude decay rate - an adjustable parameter

105

did not get any significant improvement. In fact, the factor that is needed to adjust γ, such that the Hopf model quantitatively describes the transition to synchronization of the optomechanical system, remains more or less the same. This way, we determined the panel in Fig. 5.8b displaying the effective Kuramoto constant K, that involves γ, as a function of laser input power. Being closer to the bifurcation threshold also decreases the oscillation ¯ reduces the mechanical modulation amplitude A¯ (see Eq. (5.11)). A smaller value of GA/κ of the light-field intensity back-acting on the mechanics. This decreases the (already small) size of the higher harmonics (Fig. 5.3c) that occur for self-induced oscillations. As pointed out, these contributions are not included in the Hopf model. Using our insights on how the ¯ Hopf parameters can be tuned via the microscopic cell parameters, we tried to decrease GA/κ while keeping γ fixed as explained in Section 5.2.2.2. We also considered fast cavities κ 1. Nevertheless, we did not find any significant improvement on the match between Hopf and optomechanics using the microscopic value of γ for coupled cells. To explain the deviation between Hopf and optomechanics, the higher harmonics that show for the amplitude dynamics of self-induced mechanical oscillations (see Fig. 5.3c) might be of particular interest. Due to this dynamics, the force kAi (t), acting on an optomechanical cell in a coupled system, actually possesses weak modulations in terms of higher harmonics that are not included in the Hopf description. To see whether these small higher order effects can account for the deviation between Hopf and optomechanics (that is indeed exclusively observed for coupled cells), we Fourier decomposed the amplitude dynamics shown in Fig. 5.3c and added the resulting higher contributions to an external force acting on a single cell. Nevertheless, comparing this simulation to phase-locking involving only a single, well-defined frequency (as in Fig. 5.4 and Fig. 5.5), we did not observe any relevant difference in the synchronization behavior. If the higher harmonics shown in Fig. 5.3c were responsible for the deviation between Hopf and optomechanics, we should have found earlier synchronization as observed for the coupled optomechanical system. Altogether this suggests that the discrepancy arises due to the mutual interaction between cells. As pointed out in Section 5.7, a revised Hopf model might eventually explain this deviation microscopically. So far it is considered by keeping γ to be an adjustable parameter in our model that then well describes the essential features of synchronization in coupled optomechanical array systems.

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Acknowledgments

Acknowledgments I am deeply grateful to my advisor Florian Marquardt, who patiently guided me through this thesis. His insights and extraordinary physical understanding, that he shared with me, laid the foundation for this work, while his support and goodwill continuously motivated me. Florian, I very much enjoyed working with you and being a part of your remarkable group. I would like to thank the cluster of excellence ”Nanosystems Initiative Munich”, the Center for NanoScience, the Arnold-Sommerfeld-Center, the Ludwig-Maximilians-University Munich and the Friedrich-Alexander-University Erlangen-Nuremberg for their financial and infrastructural support. I acknowledge various fruitful cooperations in connection with this thesis. In particular, I would like to thank Jack Harris for the collaboration on mechanically driven coherent photon dynamics, Huaizhi Wu for the work on dynamical back-action in coupled multimode optomechanical system as well as Max Ludwig, Jiang Qian and Björn Kubala for the joint exploration and numerous discussions on the collective dynamics in optomechanical arrays. I thank Jack Harris and Hong Tang as well as both their groups at Yale University for having me as their guest in New Haven and sharing their insights. Furthermore, I acknowledge enriching conversations with Max Ludwig, Clemens Neuenhahn and Jiang Qian not only during our shared journeys from Munich to Erlangen. I also would like to thank all colleagues and former office mates in particular Michael Möckel, Hamed Saberi, Ferdinand Helmer, Stefan Keßler and Markus Heyl for their company and all their assistance. I especially thank my parents B¨ arbel and Norbert for all their aid and dedication in all the years. Finally and most importantly, I want to thank my wife Katharina for her true support and continuous encouragement.

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Deutsche Zusammenfassung Nanomechanik im Wechselspiel mit Licht: Dynamik gekoppelter optomechanischer Multimodensysteme In dieser Arbeit wird die Dynamik von gekoppelten optomechanischen Multimodensystemen untersucht. Die Fabrikation von Nanostrukturen hat es ermöglicht mikro- und nanomechanische Resonatoren herzustellen, deren mechanische Bewegung an elektromagnetische Strahlung koppelt. Angesichts der enormen optischen Kontrolle, die bereits f¨ ur atomare Systeme existiert, hat dies zu großem Interesse an der Manipulation von makroskopischen mechanischen Objekten mit Hilfe von Licht gef¨ uhrt. Die Optomechanik hat sich dadurch in den letzten Jahren zu einem eigenst¨ andigen, schnell wachsenden Forschungsgebiet an der Schnittstelle zwischen Nanophysik und Quantenoptik entwickelt. Zunächst wird der Leser in Kapitel 1 in das Feld der Optomechanik eingef¨ uhrt. Das optomechanische Standardsystem besteht aus einer lasergetriebenen optischen Kavität, deren Resonanzfrequenz durch die Bewegung eines mechanischen Elements verändert wird. Wir ge¨ ben einen Uberblick u altigen Eigenschaften dieses verbl¨ uffend einfachen Systems, ¨ber die vielf¨ das allgemein durch eine einzige optische und eine einzige mechanische Mode beschrieben ¨ wird. Uber dieses Standardsystem hinaus wurden k¨ urzlich einige optomechanische Systeme entwickelt, deren Dynamik durch mehrere gekoppelte optische und mechanische Moden bestimmt ist. Wir beleuchten diese neue Entwicklung, die einen Weg in Richtung integrierter optomechanischer Schaltkreise weisen. Diese f¨ uhrt uns zur Thematik dieser Arbeit; die Dynamik gekoppelter optomechanischer Multimodensysteme. Die bisherige Forschung hat sich hauptsächlich auf die Gleichgewichtsdynamik von op¨ tomechanischen Systemen konzentiert. Uber diese gängige Vorgehensweise hinaus betrachten wir in Kapitel 2 einen periodischen, zeitabhängigen Antrieb der Mechanik und analysieren die durch mechanische Bewegung getriebene Photonendynamik ausserhalb ihres Gleichgewichts. F¨ ur gekoppelte optomechanische Multimodensysteme, deren Dynamik durch mehrere gekoppelte optische (und mechanische) Moden bestimmt ist, ergeben sich dabei wesentlich neue Effekte. Insbesondere erlaubt dieser Ansatz Photonen zwischen verschiedenen optischen Moden kontrolliert und auf koh¨ arente Weise zu transferieren. In Prinzip ermöglicht die mechanisch getriebene Photonendynamik diverse stark getriebene Zwei- und Mehr-Niveau-Phänomene der Atomphysik in den Lichtfeldern optomechanischer Systeme zu realisieren. Mechanisches Treiben er¨ offnet der Optomechanik damit das gesamte Feld der stark getriebenen Zwei- und Mehr-Nieveau-Dynamik. In Kapitel 3 untersuchen wir die dynamische R¨ uckwirkung (back-action) von gekoppelten optomechanischen Multimodensystemen, d.h. Licht wirkt zur¨ uck auf die Mechanik nachdem es durch die mechanische Bewegung beeinflusst wurde. Diese Effekte sind f¨ ur die Optomecha-

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nik von besonderem Interesse. So ermöglicht diese Dynamik beispielsweise optomechanisches K¨ uhlen und kann mechanische Selbstoszillationen induzieren. Letztere Oszillationen ähneln ¨ dem Verhalten eines Lasers. Uber die linearisierte Lichtfelddynamik hinausgehend analysieren wir wie die mechanisch getriebene Photonendynamik zwischen verschiedenen Moden (vgl. Kapitel 2) mittels Strahlungskr¨ aften auf die Mechanik zur¨ uckwirkt. Wir zeigen, dass bereits f¨ ur zwei optische Moden Landau-Zener-St¨ uckelberg Oszillationen des Lichtfeldes das nichtlineare Attraktordiagramm der resultierenden Phonon-Laser-Oszillationen drastisch ändern. Grundsätzlich zeigen unserer Ergebnisse die allgemeinen Effekte von Landau-Zener Physik auf durch R¨ uckwirkung induzierte Selbstoszillationen. Das optomechanische Standardsystem wurde k¨ urzlich realisiert indem die optische Kavität durch einen supraleitenden Mikrowellenresonator ersetzt wurde. Motiviert durch diese Entwicklung untersuchen wir in Kapital 4 gekoppelte optomechanische Multimodensysteme im Mikrowellenregime. Im Unterschied zu ähnlichen optischen Systemen ist die Kopplungsfrequenz, die den Photonenaustausch zwischen Mikrowellenmoden charakterisiert, naturgemäß vergleichbar mit typischen mechanischen Frequenzen. Dies hat verschiedene Konsequenzen und diese Systeme besitzen dadurch Vorteile, die u ¨ber die Verwendung standarisierter K¨ uhlsysteme und Integration auf einem Chip hinausgehen. Insbesondere untersuchen wir zwei Systeme bei denen das elektromagnetische Feld linear beziehungsweise quadratisch an die Auslenkung eines nanomechanischen Biegebalkens gekoppelt ist. Das letztere Schema ermöglicht die QND-Detektion von Fockzuständen. F¨ ur experimentell realisierbare Parameter berechnen wir die M¨ oglichkeit der Detektion eines einzelnen Quantensprungs vom mechanischen Grundzustand zum ersten angeregten Zustand. Beispielsweise durch die Nutzung photonischer Kristalle konnten optomechanische Systeme zum einen deutlich verkleinert und zum anderen auf Chips integriert werden. Ausgehend von dieser neuen Entwicklung beginnen wir in Kapitel 5 damit die kollektive nichtlineare Dynamik in Anordnungen von gekoppelten optomechanischen Zellen zu untersuchen. Hierbei besteht jede Einheit aus einer lasergetriebenen optischen und einer mechanischen Mode. Jenseits eines bestimmten Grenzwertes f¨ ur die Lasereingangsleistung weist jede Zelle, die f¨ ur sich genommen das optomechanische Standardsystem darstellt, eine Hopf-Bifurkation zu selbstinduzierten mechanischen Oszillationen auf. Wir zeigen, dass die Phasen vieler solcher optomechanischen Oszillatoren, auch f¨ ur den Fall verschiedener intrinsischer Zellfrequenzen, auf verschiedenen Phasenabst¨ ande einrasten können. Dadurch ist das System auf eine gemeinsame, kollektive Oszillationsfrequenz synchronisiert. Wir beschreiben verschiedene Regime der Dynamik und leiten ein effektives Kuramoto-Modell her, das es erlaubt die wesentlichen Merkmale der gekoppelten optomechanischen Dynamik zu beschreiben.

List of Publications • Georg Heinrich, Max Ludwig, Jiang Qian, Björn Kubala and Florian Marquardt, Collective Dynamics in optomechanical arrays, Physical Review Letters 107, 043603 (2011) • Huaizhi Wu, Georg Heinrich and Florian Marquardt, The effect of Landau-Zener dynamics on phonon lasing, arXiv:1102.1647 preprint (2011) (submitted) • Georg Heinrich and Florian Marquardt, Coupled multimode optomechanics in the microwave regime, Europhysics Letters 93, 18003 (2011) • Georg Heinrich, Max Ludwig, Huaizhi Wu, K. Hammerer and Florian Marquardt, Dynamics of coupled multimode and hybrid optomechanical systems, Comptes Rendus Physique (2011), doi:10.1016/j.crhy.2011.02.004 (in press) • Max Ludwig, Georg Heinrich and Florian Marquardt, Examples of quantum dynamics in optomechanical systems, in Quantum Communication and Quantum Networking (Springer 2010) - proceedings of QuantumCom 2009 • Georg Heinrich, J. G. E. Harris and Florian Marquardt, Photon shuttle: Landau-Zener-St¨ uckelberg dynamics in an optomechanical system, Physical Review A 81, 011801(R) (2010) • Georg Heinrich and F. K. Wilhelm, Current fluctuations in rough superconducting tunnel junctions, Phys. Rev. B 80, 214536 (2009) - previous work not in this thesis.

Dynamics of coupled multimode optomechanical systems - OPUS4

Dynamics of coupled multimode optomechanical systems - OPUS4

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