Aspects Of Multimode Quantum Optomechanics

Transcription

1 Aspects Of Multimode Quantum Optomechanics Item Type text; Electronic Dissertation Authors Seok, HyoJun Publisher The University of Arizona. Rights Copyright is held by the author. Digital access to this material is made possible by the University Libraries, University of Arizona. Further transmission, reproduction or presentation (such as public display or performance) of protected items is prohibited except with permission of the author. Download date 30/04/ :47:17 Link to Item

2 ASPECTS OF MULTIMODE QUANTUM OPTOMECHANICS by HyoJun Seok A Dissertation Submitted to the Faculty of the COLLEGE OF OPTICAL SCIENCES In Partial Fulfillment of the Requirements For the Degree of DOCTOR OF PHILOSOPHY In the Graduate College THE UNIVERSITY OF ARIZONA 2014

3 2 THE UNIVERSITY OF ARIZONA GRADUATE COLLEGE As members of the Dissertation Committee, we certify that we have read the dissertation prepared by HyoJun Seok, titled Aspects of Multimode Quantum Optomechanics and recommend that it be accepted as fulfilling the dissertation requirement for the Degree of Doctor of Philosophy. Dr. Pierre Meystre Date: 29 July 2014 Dr. Ewan M. Wright Date: 29 July 2014 Dr. Brian P. Anderson Date: 29 July 2014 Final approval and acceptance of this dissertation is contingent upon the candidate s submission of the final copies of the dissertation to the Graduate College. I hereby certify that I have read this dissertation prepared under my direction and recommend that it be accepted as fulfilling the dissertation requirement. Dissertation Director: Dr. Pierre Meystre Date: 29 July 2014

4 3 STATEMENT BY AUTHOR This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at the University of Arizona and is deposited in the University Library to be made available to borrowers under rules of the Library. Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgment of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the head of the major department or the Dean of the Graduate College when in his or her judgment the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author. SIGNED: HyoJun Seok

5 4 ACKNOWLEDGEMENTS I have received enduring support and encouragement from a great number of individuals over the past five years. First, it is my privilege to be the last student of Prof. Pierre Meystre and to do physics under his guidance. He has influenced me in many ways including research and life in general and he showed me that physics is a lot of fun. During my Ph.D, I was always inspired by his enthusiasm, vast knowledge and thoroughness in his research. I appreciate my dissertation reviewers Prof. Ewan M. Wright and Prof. Brian P. Anderson for their support, brilliant insights and patience. Prof. Wright has guided me through many discussions about my work with great patience. Prof. Anderson has kindly answered tons of my questions and gave me brilliant ideas and insights regarding physical experiments in general. Special thanks should go to Dr. Francesco Bariani and Dr. Lukas F. Buchmann. As role models for my near future career, they have given me useful advices. Working with Lukas was a lot of fun and he helped me to do research and Francesco went through this dissertation and gave me fruitful comments. I would like to thank my former office mates Dr. Swati Singh, Dr. Steven K. Steinke and the past and present members in our group: Dr. Aravind Chiruvelli, Dr. Ying Dong, Dr. Dan Goldbaum, Prof. Huatang Tan, Prof. Keye Zhang, Prof. Lin Zhang for many stimulating discussions. Finally, I would like to thank my wife, Eunjeong, for sharing my life and being patient with me.

6 5 DEDICATION For those who have guided and taught me.

7 6 TABLE OF CONTENTS LIST OF FIGURES ABSTRACT CHAPTER 1 INTRODUCTION Single-mode optomechanics Linearized regime Weak-coupling regime Strong-coupling regime Single-photon strong-coupling regime Multimode optomechanics Quadratic optomechanics Dissertation format CHAPTER 2 PRESENT STUDY Generation of mechanical squeezing via magnetic dipoles on cantilevers Dynamic stabilization of an optomechanical oscillator Multimode weak-coupling quantum optomechanics Multimode strong-coupling quantum optomechanics APPENDIX A GENERATION OF MECHANICAL SQUEEZING VIA MAG- NETIC DIPOLES ON CANTILEVERS A.1 Introduction A.2 Model system A.2.1 Symmetric case A.2.2 Asymmetric case A.2.3 Experimental considerations A.3 Fluctuations A.3.1 Amplitude fluctuations A.3.2 Phase fluctuations A.3.3 Clamping noise A.4 Detection A.4.1 Continuous optical coupling A.4.2 Delayed detection A.5 Conclusion A.6 Appendix

8 7 TABLE OF CONTENTS Continued A.6.1 Effect of phase fluctuations A.6.2 Equations of motion for optically coupled system APPENDIX B DYNAMIC STABILIZATION OF AN OPTOMECHANICAL OSCILLATOR B.1 Introduction B.2 Classical dynamics B.3 Dynamic stabilization B.4 Classical simulations B.4.1 Undamped case B.4.2 Damped case B.5 Quantum dynamics B.6 Quantum simulations B.6.1 Undamped case B.6.2 Damped case B.6.3 Phase-space distributions B.7 Summary and outlook APPENDIX C OPTICALLY MEDIATED NONLINEAR QUANTUM OP- TOMECHANICS C.1 Introduction C.2 Model C.3 Linear coupling C.3.1 Linearization C.3.2 Elimination of the cavity field C.4 Quadratic coupling C.4.1 Local maxima C.4.2 Local minima C.5 Example two-mode system C.5.1 Resolved sideband regime C.5.2 Doppler regime C.5.3 State transfer C.6 Conclusion APPENDIX D MULTIMODE STRONG-COUPLING QUANTUM OPTOME- CHANICS D.1 Introduction D.2 Basic Model D.3 Classical theory D.3.1 Linear Interactions

9 8 TABLE OF CONTENTS Continued D.3.2 Quadratic Interactions D.4 Quantum effects D.4.1 Heisenberg-Langevin equations D.4.2 Master equation D.5 Results D.5.1 Linear interactions D.5.2 Quadratic interactions D.6 Conclusion

10 9 LIST OF FIGURES 1.1 Fabry-Pérot optical resonator with a moving end mirror. The centerof-mass mode of the moving mirror is coupled to a single cavity field mode via a linear optomechanical interaction Membrane in the middle geometry. Two cavity fields tunnel through a membrane located at the center of a cavity and push the membrane in opposite directions, resulting in a quadratic optomechanical interaction. 35 A.1 Tuning fork magnetically coupled to a quantum-mechanical cantilever. The equilibrium position of the cantilever is equidistant from the two extremities of the nanoscale tuning fork A.2 Coupling frequencies χ 2 (ζ) (red, solid), and χ 1 (ζ)(blue, dashed) in MHz as a function of displacement ζ in nm for parameters of section II.C A.3 ( X 1 ) 2 as a function of the dimensionless time χt for various amount of phase fluctuations: no fluctuations(blue, solid); D = 10 5 χ (green, dashed); D = 10 4 χ (orange, dot-dashed); D = 10 3 χ (red, dotted).. 61 A.4 ( X 1 ) 2 as a function of scaled time with thermal fluctuations for different damping constants and occupation numbers: no fluctuations (blue, solid); n th = 5, γ = 10 2 χ (green, dashed); n th = 10, γ = χ (orange, dot-dashed); and n th = 10, γ = χ (red, dotted) A.5 Schematics of the intracavity optical field optomechanically coupled to the magnetically driven cantilever A.6 Steady-state squeezing of the cavity field as a function of the dimensionless coupling parameters r = g E 0 /κ and s = χ/κ A.7 Steady state squeezing of the cantilever as a function of the dimensionless coupling parameters r = g E 0 /κ and s = χ/κ A.8 Steady state squeezing of the cavity field (red, solid) and cantilever (blue, dashed) as a function of κ/χ in the resolved-side band regime for fixed χ and g E 0 /χ = A.9 Squared quadratures of position for the cavity field (red, solid) and the mechanical oscillator (blue, dashed) as a function of scaled time. The optomechanical coupling is turned on at the dimensionless time χt = 1 and coherently builds up towards g E 0 /χ = 9, κ/χ =

11 10 LIST OF FIGURES Continued A.10 Squared quadratures of position in the cavity field (red, solid) and the mechanical oscillator (blue dashed) with strong coherent optomechanical coupling. Here g E 0 /χ = 9, κ/χ = A.11 Minimum values of squared quadrature variance in the cavity field plotted as a function of g E 0 /κ B.1 Membrane-in-the-middle geometry. Two cavity fields tunnel through a membrane located at the center of a fixed cavity and interact with the membrane in opposite directions, resulting in a quadratic optomechanical interaction B.2 Static mechanical potential for an input power greater than the critical power P c. The parameters are κ/ω m = 10, c /ω m = 0, g (2) 0 x 2 0/ω m = 0.01, P 0 /(E 0 ω L ) = For our parameters, the critical pumping power is P c /(E 0 ω L ) = 62.5 and the effective mechanical frequency is ω 0 /ω m = Here and in all following figures we measure the position, momentum and energy of the mechanical mode in units of the natural length x 0 = /mω m, momentum p 0 = m ω m and energy E 0 = ω m, respectively B.3 Time-dependent potential for the mechanics at t = 0, π/ω (blue solid line), t = π/(2ω) (green dashed line), and t = 3π/(2Ω) (red dotted line). Here, P 0 /(E 0 ω L ) = 66.0, A/P 0 = 1, Ω/ω m = 1.8 and the other parameters are the same as those in Fig. B B.4 Time-averaged potential for several values of the modulation amplitude A with a fixed modulation frequency Ω/ω m = 1.8, A/P 0 = 0.48 (green dashed line), A/P 0 = 0.62 (orange dot-dashed line), A/P 0 = 1 (red dotted line) along with the static potential (blue solid line). Other parameters as in Fig B B.5 Stability domain of an optomechanical oscillator located at the center, for an input power modulated according to Eq. (B.17), for the parameters of Fig. B.3. The oscillator is dynamically stable in the region above the dashed blue line (blue-colored region). The black dots denote the points used in the simulations of Figs. B.7-B.10, and label these points in the figures. In all cases Ω/ω m = 1.8, and (A/P 0 ) = (a) 0, (b) 0.22, (c) 0.56, (d) 0.62, and (e) B.6 Initial energy distribution of the mechanical oscillator (blue solid line) with the mean energy of E = E 0 /2 (gray dashed line)

12 11 LIST OF FIGURES Continued B.7 Trajectories of the classical mechanical oscillator with initial conditions generated at random from the Gaussian distribution function (B.30) with σ x = x 0 / 2, σ p = p 0 / 2 and Ω/ω m = 1.8. The curves follow the labeling of Fig. B.5 with values of (A/P 0 ) given by (a) 0, (b) 0.22, (c) 0.56, (d) 0.62, and (e) 1. Here, γ/ω m = 10 6, T = 0, κ/ω m = 10, c /ω m = 0, g (2) 0 x 2 0/ω m = 0.01, P 0 /(E 0 ω L ) = B.8 Trajectories of the classical mechanical oscillator in the presence of a viscous damping force for a reservoir at zero temperature. Here, γ/ω m = , T = 0. Other parameters as in Fig. B B.9 Time evolution of the spatial probability distribution of the quantum mechanical oscillator initially prepared in the ground state of the bare harmonic trapping potential of frequency ω m. Here, the parameters employed and plot labels are identical to those used in Fig. B B.10 Time evolution of the spatial probability distribution of the damped quantum mechanical oscillator initially prepared in the ground state of the harmonic potential of frequency ω m. Here, the parameters employed and plot labels are identical to those used in Fig. B B.11 Phase-space distributions of the classical oscillator (upper row) and corresponding Wigner quasi-probability distributions of the quantum oscillator (bottom tow) at time ω m t = 100 in the absence of dissipation. Here γ/ω m = 10 6, Ω/ω m = 1.8 and (a, d) A/P 0 = 0, (b, e) A/P 0 = 0.56, (c, f) A/P 0 = 1, κ/ω m = 10, c /ω m = 0, g (2) 0 x 2 0/ω m = 0.01, P 0 /(E 0 ω L ) = B.12 Phase-space distributions of the classical oscillator (upper row) and corresponding Wigner quasi-probability distributions of the quantum oscillator (lower row) at time ω m t = 300. Both classical and quantum oscillators are damped via a reservoir at zero temperature with γ/ω m = Other parameters as in Fig. B C.1 Non-absorptive dielectric membranes inside a Fabry-Pérot resonator which interact with a single mode cavity field C.2 Effective frequencies of the first (red, solid) and second (blue, dashed) mechanical modes in the resolved sideband regime ω 1, ω 2 κ for g 1 > g 2. The dotted lines denote the bare frequencies of the mechanical modes. The shifted frequencies are matched at the intersections. The inset gives a detailed view on the intersections occurring in the red-detuned side. The parameters used are ω 1 = 2π 20MHz, ω 2 = 2π 19.95MHz, κ = 2π 1MHz, g 1 = 2π 0.3MHz, g 2 = 2π 0.12MHz

13 12 LIST OF FIGURES Continued C.3 Upper plot: short time (linear time scale) and lower plot: long time (log time scale) evolution of the motional quadratures of the mechanical modes in the single and two-mode scenarios. The mechanical modes of frequencies ω 1 (red, solid) and ω 2 (blue, dashed) are both cooled down while interacting with each other. For comparison the uncoupled cold damping of the modes ω 1 (orange, dotted) and ω 2 (green, dotdashed) are also shown. Two-mode coupling results in slowing down of the individual cold damping rates. Here ω 1 = 2π 20MHz, ω 2 = 2π 19.99MHz, κ = 2π 0.95MHz, g 1 = 2π 50kHz, g 2 = 2π 10kHz C.4 Qualitative behavior of the effective mechanical frequencies for the first (red, solid) and second (blue, dashed) oscillators in the Doppler regime. The shifted frequencies are matched at the intersections. Here we chose g 1 > g 2. A set of parameters is ω 1 = 2π 100kHz, ω 2 = 2π 93kHz, κ = 2π 1MHz, g 1 = 2π 90kHz, g 2 = 2π 50kHz.133 C.5 Motional quadrature of the moving mirror (red, solid) and the BEC (blue, dashed) in the presence of vacuum noise for ω 1 = 2π 101kHz, ω 2 = 2π 100kHz, κ = 2π 1MHz, g 1 = 2π 100kHz, g 2 = 2π 10kHz.135 C.6 Motional quadrature of the moving mirror (red, solid) and the BEC (blue, dashed) for squeezed vacuum noise with N = 1 and squeezing phase θ s θ c = π/2. Same parameters as in Fig C C.7 State transfer fidelity for squeezed vacuum noise with N = 1 (red, solid), N = 10 (blue, dashed) and vacuum noise (gray, dotted) as a function of the phase difference between the squeezing input and the cavity field. The initial states of the BEC and the moving mirror are the ground state and a thermal state with mean phonon number n = 1, respectively C.8 Same as Fig. C.7, but for the BEC initially in a squeezed state with position quadrature variance of and momentum quadrature variance of 10 and and the moving mirror in a thermal state with mean phonon number n = 1, respectively

14 13 LIST OF FIGURES Continued D.1 Effective potential U eff (x), in units of κ, versus the dimensional position x for a single mechanical oscillator linearly coupled to a cavity mode, and for the several values of the normalized cavity pumping rate: η /κ = 0.14 (red dotted line), η /κ = 0.18 (orange dot-dash line), η /κ = 0.24 (green dashed line), η /κ = 0.34 (blue solid line). The potential with η /κ = 0.18 exhibits two local minima corresponding to stable solutions and one local maximum corresponding to an unstable solution. Here ω m /κ = 0.01, g 0,1 /κ = 0.3, c /κ = D.2 Effective potential U eff (x 1, x 2 ), in units of κ, as a function of the dimensionless positions x 1 and x 2 for two mechanical oscillators linearly coupled to the cavity mode. Here ω m /κ = 0.01, g 0,1 /κ = g 0,2 /κ = 0.3, c /κ = 1.5, η /κ = D.3 Effective potential U eff (x), in units of κ, versus dimensionless position x for a single mechanical oscillator quadratically coupled to the cavity mode for η /κ = 0.05 (red dotted line), η /κ = 0.11 (orange dot-dash line), η /κ = 0.17 (green dashed line), η /κ = 0.20 (blue solid line). Here ω m /κ = 0.01, g 0,1/κ (2) = 0.2, c /κ = D.4 Steady-state dimensionless positions x s for a single mechanical oscillator with quadratic coupling as a function of the normalized cavity pumping rate η /κ. The stable and unstable solutions are denoted by the solid blue and dashed red line, respectively. Inset: zoomed in region around the bifurcation point. Here ω m /κ = 0.01, g 0,1/κ (2) = 0.2, c /κ = D.5 Effective potential U eff (x 1, x 2 ), in units of κ, versus the dimensionless positions x 1 and x 2 for equal and negative quadratic optomechanical coupling coefficients. Here ω m /κ = 0.01, g (2) 0 /κ = 0.2, c /κ = 0.02, η /κ = D.6 Effective potential U eff (x 1, x 2 ), in units of κ, versus the dimensionless positions x 1 and x 2 for quadratic interactions of opposite signs, g 0,1/κ (2) = g 0,2/κ (2) = 0.2, ω m /κ = 0.01, c /κ = 0.02, η /κ = D.7 Expectation value ˆx versus normalized cavity pumping rate η /κ (solid blue line) for a single mechanical oscillator and linear optomechanical interaction. Here ω m /κ = 0.01, g 0,1 /κ = 0.3, c /κ = 1.5, γ/κ = The red dashed curve shows the corresponding classical bistable solution

15 14 LIST OF FIGURES Continued D.8 Steady-state position probability distribution P (x) versus dimensionless position x (solid blue lines) of the mechanical oscillator along with the effective potential U eff (x) (red dash lines), in units of κ, for the normalized cavity pumping rates η /κ (a) 0.14, (b) 0.18, (c) 0.24, (d) 0.34 for a single mechanical oscillator and linear optomechanical interaction. Here ω m /κ = 0.01, g 0,1 /κ = 0.3, c /κ = 1.5, and γ/κ = D.9 Ground state wave function ψ 0 (x 1, x 2 ) as a function of the dimensionless positions x 1 and x 2. Same parameters as in Fig. D D.10 Position representation of the Schmidt basis states x 1 m1 (red solid line) and x 2 m1 (blue dashed line) for the two mechanical oscillators in the ground state ψ 0 (x 1, x 2 ). Same parameters as in Fig. D D.11 Von Neumann entropies and quantum mutual information of the mechanical system as a function of normalized time κt. Green dotted line: entropies of the individual mechanical oscillators; red dashed line: their quantum joint entropy; blue solid line: quantum mutual information of the composite system. Here γ 1 = γ 2 = κ, other parameters as in Fig. D D.12 Normalized time (κt) evolution of the spatial probability P (x, t) for a mechanical mode initially in its ground state, and for the optical pumping rates η /κ (a) 0.05, (b) 0.08, (c) 0.11, (d) 0.14, (e) 0.17, and (f) In each panel the vertical axis is the dimensionless position x, and P (x, t) is color coded. See the potential U eff of Fig. D.3 and Fig. D.4, which are for the same set of parameters, for reference. Here γ = 10 3 κ D.13 Normalized time (κt) evolution of the position probability distribution P (x, t) for an oscillator initially in the cat state (D.48) with β 0 = 1.5 and the relative phases (a) φ 0 = 0, (b) φ 0 = 0, (c) φ 0 = π/2, (d) φ 0 = π, and (e) φ 0 = 3π/2. In each panel the vertical axis is the dimensionless position x, and P (x, t) is color coded. Same parameters as in Fig. D.3, with cavity pumping rate η /κ = The mechanical decay rate is γ = 10 3 κ in panel (a) and γ = 10 6 κ in panels (b)-(e) D.14 Semi-log plot of the position probability distribution P (x = 0, t) P (x = 0, t ) as a function of normalized time (κt) (blue, solid curve). The red straight line is a fit to the long time maxima of the distribution. Same parameters as in Fig. D.13(b)

16 15 LIST OF FIGURES Continued D.15 Ground state of the effective potential describing the motion of the two mechanical modes for the case of quadratic interactions and coupling coefficients of equal magnitude but opposite sign. Same parameters as in Fig. D D.16 Von Neumann entropies of the mechanical oscillators 1 (orange dotdash line) and 2 (green dotted line), their joint quantum entropy (red dash line), and mutual quantum information (blue solid line). Same parameters as in Fig. D.6 with γ 1 /κ = γ 2 /κ = D.17 (a) Normalized time (κt) evolution of the marginal probability distribution P (x 2, t) where the vertical axis is the dimensionless position x, and P (x, t) is color coded, and (b) the effective potential (red line), in units of κ, and probability distribution P (x 2, t) (blue line) for the times indicated in panel (a), with A corresponding to the initial time t = 0. Same parameters as in Fig. D.16, γ 1 /κ = γ 2 /κ =

17 16 ABSTRACT This dissertation aims to investigate systems in which several optical and mechanical degrees of freedom are coupled through optomechanical interactions. Multimode optomechanics creates the prospect of integrated functional devices and it allows us to explore new types of optomechanical interactions which account for collective dynamics and optically mediated mechanical interactions. Owing to the development of fabrication techniques for micro- and nano-sized mechanical elements, macroscopic mechanical oscillators can be cooled to the deep quantum regime via optomechanical interaction. Based on the possibility to control the motion of mechanical oscillators at the quantum level, we design several schemes involving mechanical systems of macroscopic length and mass scales and we explore the nonlinear dynamics of mechanical oscillators. The first scheme includes a quantum cantilever coupled to a classical tuning fork via magnetic dipole-dipole interaction and also coupled to a single optical field mode via optomechanical interaction. We investigate the generation of nonclassical squeezed states in the quantum cantilever and their detection by transferring them to the optical field. The second scheme involves a quantum membrane coupled to two optical modes via optomechanical interaction. We explore dynamic stabilization of an unstable position of a quantum mechanical oscillator via modulation of the optical fields. We then develop a general formalism to fully describe cavity mediated mechanical interactions. We explore a rather general configuration in which multiple mechanical oscillators interact with a single cavity field mode. We specifically consider the situation in which the cavity dissipation is the dominant source of damping so that the cavity field follows the dynamics of the mechanical modes. In particular, we study two limiting regimes with specific applications: the weak-coupling regime

18 17 and single-photon strong-coupling regime. In the weak-coupling regime, we build a protocol for quantum state transfer between mechanical modes. In the single-photon coupling regime, we investigate the nonlinear nature of the mechanical system which generates bistability and bifurcation in the classical analysis and we also explore how these features manifest themselves in interference, entanglement, and correlation in the quantum theory.

19 18 CHAPTER 1 INTRODUCTION The study of the mechanical effects of light dates back to Kepler s postulate of radiative force on objects from the observation of the direction of the tail of a comet [1]. Maxwell s theory of the electromagnetic field validated the fact that light carries momentum and hence exerts forces on matter [2]. Later, Lebedev demonstrated the feeble force from the reflection of light with a light mill arrangement in a high vacuum chamber [3]. The radiative force from conventional light sources was so weak that its practical applications remained obscure. However, the invention of the laser changed this situation dramatically, allowing the use of light as an efficient tool to manipulate and control small objects from micrometer-scale particles to molecules and atoms [4]. When light illuminates neutral atoms or ions, their center-of-mass positions evolve under the action of radiative forces such as the optical dipole force due to the light field profile and the radiation pressure force due to the exchange of linear momentum between atoms and light [5]. The optical dipole force allows the trapping of small objects in a finite region of space and the realization of optical tweezers, which have many applications in physical and biological studies [6, 7]. The radiation pressure force, involving absorption followed by spontaneous emission, is a key ingredient for the laser cooling of neutral atoms or ions [8, 9]. The radiation pressure force combined with other cooling and trapping techniques led to plenty of successful milestones including the realization of the motional ground state of a single ion [10] and Bose-Einstein condensation in a vapor of alkali atoms [11, 12, 13]. Ultracold atomic systems have provided key contributions to improved frequency standards [14, 15] and precision measurements of physical constants [16]. In ad-

20 19 dition, the high degree of controllability and novel detection schemes for ultracold atomic systems pave the way to explore quantum many-body physics [17] and quantum dynamics of the motion of atoms [18], resulting in quantum simulators for condensed matter systems [19, 20] and a wealth of applications in quantum information processing [21, 22]. Conventional laser cooling schemes based on the radiation pressure force rely on a closed atomic transition and a dissipation mechanism provided by spontaneous emission of light. Thus, laser cooling schemes are not applicable to macroscopic objects that not only lack closed internal transitions but also have nonradiative energy relaxation channels. Moreover, the radiation pressure force caused by a single photon is too small to affect the motion of macroscopic objects. Even so, macroscopic objects can possibly be cooled if externally coupled to a considerably large number of photons of a certain frequency and a radiative decay mechanism is provided at the same time. An optical resonator with a harmonically bound end mirror, which is the simplest realization of such a system, is at the foundation of cavity optomechanics, a research area exploring the interaction between the motional degrees of freedom of a mechanical element and the electromagnetic field inside optical or microwave cavities [23, 24, 25, 26, 27, 28, 29]. The mechanical effects of the radiation pressure force on the end mirror can manifest themselves due to the fact that light inside the cavity can be much more intense than the input fields. In addition to the enhanced radiation pressure force, the decay of the cavity field can represent the needed dissipation mechanism to achieve cooling. Cavity optomechanics was first proposed for optical measurement of mechanical displacement with high sensitivity [30, 31] and it became first relevant in the context of gravitational wave detection [32, 33]. Early works in cavity optomechanics addressed the three main effects of the cavity-enhanced radiation pressure force acting on a moving mirror: radiation pressure noise, radiation-induced damping or anti-damping and optical spring effect [30, 31, 32].

21 20 The radiation pressure noise results from random scattering events between the photons inside the cavity and the moving mirror and it is proportional to the square root of the photon number inside the cavity to the extent that the cavity field is driven by a strong external field [33]. In optical measurements of the displacement of the moving mirror, decreasing input powers reduces the radiation pressure force noise but results in significant effects from laser shot noise due to the discrete nature of light which leads to the random arrival of individual photons to detectors and the moving mirror [32, 34]. Balancing the effects of laser shot noise and radiation pressure noise gives rise to the so-called standard quantum limit, the limit of the accuracy of the conventional interferometric measurements of the displacement of the moving mirror [32, 34]. Next, the radiation-induced damping or anti-damping of mechanical motion is due to the fact that the finite cavity field lifetime causes a retardation of the intracavity field with respect to the mechanical motion [30]. The nonconservative nature of the radiation pressure force was first demonstrated in an interferometer where microwave radiation pressure force was exerted on a suspended milligram-scale mirror [31]. Finally, the optical spring effect accounts for a shift in the mechanical vibration frequency and hence induces a significant alteration of the stiffness of the mechanical oscillator. This effect results from the part of the cavity field in-phase with respect to the mechanical motion. For a certain range of input powers, the radiation pressure force becomes highly nonlinear in the mirror position, resulting in optical bistability in the transmission of an optical resonator. The nonlinearity of the radiation pressure force was first demonstrated using a Fabry-Pérot cavity with a suspended milligram-scale mirror in the optical regime [35]. Moving mirrors in the above systems normally experience not only the radiation pressure force but also significant thermal noise from the environment which results in a technical limit for a variety of precision measurements of feeble forces and fields [36]. In order to reduce thermal noise effects, researchers devoted strong efforts

22 21 to develop new fabrication techniques for high finesse optical resonators [37, 38] and mechanical oscillators with high quality factors [39, 40, 41]. In addition to radiation pressure cooling [42], several cooling schemes were proposed and demonstrated including feedback control [43, 44] and cavity-induced photothermal pressure [45]. Due to improvements in fabrication and cooling techniques, cavity optomechanics has prospered and expanded to a variety of mechanical systems, including microtoroid optical resonators [46], capacitors in superconducting LC circuits [47], mirror-coated cantilevers [37], ultracold atoms in an optical cavity [48, 49], suspended mirrors in gravitational wave detectors [50], and others. The mechanical oscillators in these systems operate over a large range of parameters, with mechanical frequencies from a few Hertz [50] to GigaHertz [51], and masses from several attograms [48] to kilograms [50]. The recent demonstrations of cooling of a macroscopic mechanical oscillator to its motional ground state have stimulated interest in bringing macroscopic objects in the deep quantum regime for coherent control and quantum measurement [51, 52, 53, 54]. For the coherent manipulation of mechanical motion, optomechanical systems have made progress in enhancing the optomechanical coupling strength [47, 52, 53, 55]. Advances in quantum optomechanics have enabled the observation of normal-mode splitting [47], optomechanically induced transparency [46, 56], coherent state transfer between photonic and phononic modes [55, 57], asymmetry in the sideband amplitudes indicating the quantum nature of emission and absorption of phonons [58, 59], radiation pressure shot noise [60, 61], conversion between microwave and optical fields [62, 63] and entanglement between microwave and mechanical modes [64]. Quantum manipulation of macroscopic oscillators in a hitherto unachieved parameter regime paves the way for fundamental tests of quantum physics questions, for example quantum superposition state of macroscopic objects [65], decoherence at the boundary of quantum and classical regimes [66], tunnelling of macroscopic objects [67], and possible deviations of

23 22 the Heisenberg uncertainty relations for massive objects [68]. In addition, quantum optomechanics promises a wealth of possible applications in precision measurement of masses, forces and fields by means of mechanical squeezed states [69, 70, 71] and solid-state implementation for quantum information science and networking [72, 73]. Many optomechanical experiments so far can be described by the simple model in which the intensity of a single cavity field mode is coupled to the center-of-mass mode of a mechanical oscillator via the optomechanical interaction and an ensemble of mechanical modes in the substrate is assumed to act as thermal reservoir and accounts for mechanical dissipation. The interaction of optical fields with multiple mechanical modes has recently attracted great attention. The optomechanical interactions between multiple optical and mechanical modes allows for the exploration of reservoir engineering for mechanical modes and hence possible unconventional decoherence processes [74] and the investigation of new types of optomechanical interactions including collective dynamics in optomechanical arrays [75, 76]. In addition, the study of multiple optical and mechanical modes creates the prospect of integrated functional optomechanical devices that have possible applications for quantum information processing and quantum networks in which individual elements implement different functionalities [77, 78]. The goal of this dissertation is to explore a variety of aspects of multimode optomechanical systems. We first consider two specific examples and then develop a general formalism. The first example investigates mechanical squeezing in a nanofabricated system where a quantum cantilever is magnetically coupled to a classical mechanical oscillator and optomechanically coupled to a cavity field mode. The second example explores dynamic stabilization of a mechanical oscillator coupled to two optical fields. Finally, we conduct a formal analysis for optically mediated mechanical interactions and identify a wealth of nonlinear effective interactions between the mechanical modes. This situation is analogous to nonlinear optics situations in

24 23 which multiple optical modes are effectively coupled though a nonlinear medium, with the distinction that photons are replaced by phonons [79, 80]. Throughout this dissertation, we assume in general that mechanical oscillators are pre-cooled to the deep quantum regime, it is thus instructive to review the theory for singlemode optomechanical systems with a particular emphasis on cooling of mechanical motion. 1.1 Single-mode optomechanics Consider a stable optical resonator composed of two highly reflecting fixed mirrors of radii of curvature R 1 and R 2, separated by a distance L in vacuum. The optical modes sustained in the resonator have the following angular mode frequencies [81] ω l,m,n = πc [l + 1π L (m + n + 1) ] cos 1 g 1 g 2, (1.1) where c is the speed of light, the longitudinal mode index l is a positive integer, the transverse mode indices (n, m) are nonnegative integers, and g i is the so-called g-parameter of the resonator, g i = 1 L/R i. Each optical mode acquires a finite spectral linewidth κ due to the imperfect reflectivity of the mirrors, where κ is the full width at half maximum of the spectrum of the mode [81]. To the extent that the optical resonator is driven by an external laser of frequency ω L which is close to one of the resonant frequencies of the optical modes such that ω L ω l,m,n κ, and the field profile of the external field is matched to that of a certain transverse mode, a single cavity mode of frequency ω l,m,n ω L is excited and the rest of the modes remains empty [82]. For simplicity, we assume that the excited mode is the lowestorder transverse mode. This situation allows one to describe the driven optical resonator within a single mode model. After quantization of the single mode [79], one can describe the cavity field as H cav = ω c â â, (1.2)

25 24 Figure 1.1: Fabry-Pérot optical resonator with a moving end mirror. The centerof-mass mode of the moving mirror is coupled to a single cavity field mode via a linear optomechanical interaction. where â is the bosonic annihilation operator of the cavity field of frequency ω c with [â, â ] = 1. We now consider the situation, as depicted in Fig. 1.1, in which the end mirror of the optical resonator is allowed to move and is bound in a harmonic potential [83]. When the cavity is driven by an external field, the radiation pressure force associated with the cavity field changes the position of the moving mirror, inducing a variation in the cavity length. The change of the cavity length modifies the mode frequency and intensity of the cavity field. This feedback mechanism couples the center-ofmass position of the mechanical oscillator to the intensity of the cavity field [83]. Including the dependence on the position of the moving mirror of the cavity field frequency gives the following Hamiltonian for the cavity field, H cav = ω c (x)â â, (1.3) where x is the displacement of the moving mirror from its equilibrium position. Provided that the amplitude of the oscillation of the moving mirror is much smaller than the wavelength of the cavity field, the frequency of the cavity field can be

26 25 expanded as ω c (x) ω c + G (1) x + G (2) x 2 +, (1.4) where ω c is the cavity field frequency in the absence of the mechanical motion, G (1) = ωc(x) x and G (2) = 2 ω c(x) x 2 are expansion coefficients. Combining Eq. (1.3) with the Hamiltonian describing the quantized moving mirror of mass m and frequency ω m, gives the system Hamiltonian H sys = ω c â â + ω m 2 (ˆp2 + ˆx 2 ) g 0 â âˆx, (1.5) where we have kept only the linear term in the mechanical displacement in expansion (1.4). The position and momentum of the moving mirror are measured in units of x 0 = /(mω m ) and p 0 = m ω m, respectively, so that the commutation relation between the dimensionless position (ˆx) and momentum (ˆp) operators is given by [ˆx, ˆp] = i. The single-photon optomechanical coupling coefficient is g 0 = G (1) x 0. In this unit system, the radiation pressure force on the moving mirror reads F RP = g 0 â â, (1.6) where g 0 amounts to the radiation pressure force caused by a single photon and â â is the photon number operator of the cavity field. Though Eq. (1.5) has been derived for the case of a Fabry-Pérot cavity with a moving end mirror, it can be the starting point of the study of a variety of optomechanical systems, including mirror-coated cantilevers [37], LC circuits with an oscillating capacitor [84], microtoroidal dielectric cavities [55], suspended mirrors in gravitational wave detectors [50], and ultracold atoms in optical resonators [48, 49], as long as the variation of the field intensity changes the effective length of the cavity for appropriate conditions [23, 28]. For example, in the case of ultracold atoms in an optical resonator, the collective density excitation of a BEC serves as a mechanical element: the radiation pressure force associated with the cavity field instead of changing the physical length of the optical resonator, varies its optical length by modulating the density of the ultracold atomic cloud [49].

27 26 Eq. (1.5) is valid if the center-of-mass mode of the moving mirror is only coupled to a single cavity field mode. One may ask if the effect of the mechanical motion could break down the single-mode description of the cavity field. Let us consider a simple situation where the photons transmitted through the input coupler at frequency ω L exchange their momenta with the moving mirror. The photons experience three possible scattering processes: Rayleigh, Stokes, and anti-stokes [80]. For Rayleigh scattering, the photons scattered off the moving mirror maintain the same frequency. However, the Stokes (anti-stokes) photons are scattered at frequency ω L ω m (ω L +ω m ). Thus, provided that only one cavity mode is driven by a phase-matched external field and the mechanical frequency is much smaller than the frequency separation of the transverse modes of the optical resonator, scattering of photons from the driven cavity mode into other transverse modes is negligible [85]. In this situation, the single-mode description of the cavity field is valid. In addition to Eq. (1.5) describing the cavity field, the mechanical mode and their mutual optomechanical interaction, one needs to take into account an external field driving the cavity field and the dissipation of the cavity field and mechanical modes so that the optomechanical resonator is a driven and damped system. In fact, the cavity field amplitude dissipates due to the imperfect reflectivity of the mirrors and the mechanical mode amplitude is damped in a variety of ways, including clamping losses and viscous damping [86]. We assume for simplicity that the amplitudes of the cavity field and mechanical modes decay exponentially at rates κ/2 and γ/2, respectively. We assume that the cavity field is driven by an external monochromatic field detuned from the cavity resonance by c = ω L ω c with a pumping rate η. We limit ourselves to the case where the mechanical oscillator has such a high quality factor, Q = ω m /γ, that mechanical oscillation as well as coherent effects from the optomechanical interaction are not significantly degraded by mechanical dissipation and thermal decoherence. In the following sections, we discuss a variety of physical regimes in which the optomechanical coupling exhibits diverse effects.

28 Linearized regime The linearized regime is the regime in which the single-photon optomechanical coupling coefficient is much smaller than the mechanical frequency, g 0 ω m, and the cavity field is driven by a strong external field so that the radiation pressure force associated with the cavity field at the single-photon level is highly enhanced [28]. In this situation, it is suitable to switch to a displacement picture for the cavity field and decompose its operator as the sum of the classical expectation value and the deviations from the expectation value: â = α + ĉ, (1.7) where α is the expectation value of the cavity field amplitude, α = â, and ĉ accounts for its quantum fluctuations with [ĉ, ĉ ] = 1. The cavity photon number operator then becomes â â = α 2 + (α ĉ + αĉ ) + ĉ ĉ. The last term quadratic in ĉ makes contributions smaller by a factor of α compared to that of (α ĉ + αĉ ) and can thus be neglected provided that the magnitude of the mean photon number is much larger than that of the photon number fluctuations, α 2 ĉ ĉ. The radiation pressure force on the mechanical oscillator becomes ˆF RP g 0 α 2 + g 0 (α ĉ + αĉ ). (1.8) Here, the radiation pressure force is linear in ĉ and its quantum fluctuations are amplified by the mean cavity field amplitude. Since the radiation pressure force consists of a nonzero expectation value and a fluctuating component, the mechanics is displaced from equilibrium and experiences fluctuations in the position and momentum. In order to take this into account, we proceed by decomposing the position and momentum operators of the mechanics as a sum of their expectation values and small quantum fluctuations, ˆx = x + δˆx, (1.9) ˆp = p + δˆp, (1.10)

29 28 where x and p are the expectation values of the position and momentum operators of the mechanics, respectively. δˆx and δ ˆp measure quantum fluctuations and their commutation relation is [δˆx, δˆp] = i. Depending on the magnitude of the amplified coupling strength g 0 α compared to that of the cavity linewidth κ, the effects of the radiation pressure force on the mechanics are distinct so that this regime can be separated into two different cases: weak-coupling and strong-coupling [29] Weak-coupling regime This regime corresponds to the situation where the single-photon coupling coefficient is amplified by the classical field amplitude but it is still smaller than the cavity linewidth, g 0 α κ [28]. The fast dissipation of the cavity field destroys the quantum coherence established between the optical and mechanical modes and leads the cavity field to reach its quasi-steady state in a time scale 1/κ [28]. With these considerations in mind, we explore the effects of the mean and fluctuating radiation pressure force on the mechanics Mean radiation pressure force Let us first concentrate on the effects of the mean radiation pressure force on the mechanical oscillator. Depending on the magnitude of the ratio κ/ω m, the mean radiation pressure force exhibits different effects on the mirror oscillation: optical spring effect and radiation pressure damping or anti-damping [89]. In the so-called Doppler regime where the cavity decay rate exceeds the mechanical frequency [89], κ ω m, the cavity field follows the mechanical motion instantaneously and can thus be adiabatically eliminated to yield the mean radiation pressure force [29], F RP g 0 η 2 ( c + g 0 x) 2 + κ 2 /4. (1.11)

30 29 Notice that the radiation pressure force is nonlinear in the mechanical position. The conservative nature of the radiation pressure force allows one to derive the following effective potential governing the dynamics of the mechanical oscillator, U eff (x) = ω [ ] m 2 x2 2 η 2 c + g 0 x arctan. (1.12) κ κ/2 The effective potential accounts for a new equilibrium position and the static optical spring effect or significant modification of the stiffness of the mechanics [29]. It is interesting to note that the effective potential becomes a double-well potential for a finite range of input powers provided that c > 3κ/2 [79]. This situation exhibits classical bistability in the mechanical motion and hence in the intracavity field intensity [35]. In the so-called resolved sideband regime where the cavity decay rate is much smaller than the mechanical frequency [89], κ ω m, the cavity field does not follow the mechanical motion instantaneously but is rather delayed due to the finite cavity lifetime [37]. Noting that the mirror motion is simply sinusoidal in a time scale of 1/κ, one can find the mean radiation pressure force in terms of the position and momentum of the mechanical oscillator, where Ω = Γ = F RP g 0 η 2 Ωx Γp, (1.13) 2 c + κ 2 /4 [ ] g0 η 2 2 c + ω m 2 c + κ 2 /4 ( c + ω m ) 2 + κ 2 /4 + c ω m, (1.14) ( c ω m ) 2 + κ 2 /4 [ ] g0 η 2 2 κ/2 2 c + κ 2 /4 ( c + ω m ) 2 + κ 2 /4 κ/2. (1.15) ( c ω m ) 2 + κ 2 /4 The mean radiation pressure force consists then of three terms with different contributions on the mechanics. The first term in Eq. (1.13) dependent on neither the position nor momentum of the mechanical oscillator describes simply the displacement of the equilibrium position of the mechanics. The second term in Eq. (1.13), linear in the position, accounts for the dynamic optical spring effect or significant

31 30 shift in the mechanical frequency due to dynamical effects of the radiation pressure force [89]. The third term in Eq. (1.13), linear in the momentum, gives rise to damping or anti-damping force, indicating the nonconservative nature of the radiation pressure force due to the retardation of the cavity field [89]. Note that the signs of the optically induced frequency shift Ω and radiation pressure damping rate Γ depend on the pump-cavity detuning. The optically induced frequency shift Ω is the sum of two dispersion curves centered at c = ±ω m. From the dispersive nature, Ω becomes zero if c = ±ω m, while it becomes positive (negative) on the right (left) side of the center, leading to the stiffening (softening) of the mechanics. The radiation pressure damping rate Γ consists of two Lorentzian curves with opposite signs centered at c = ±ω m. Due to the opposite signs of the two curves, their effects on the mirror motion are opposite. For example, if c = ω m, the first term in Eq. (1.15) corresponding to the anti-stokes sideband becomes resonant with the cavity field while the second term is highly suppressed, resulting in a positive damping rate and thus a strongly damped mirror motion. However, if c = +ω m, the second term in Eq. (1.15) corresponding to the Stokes sideband becomes dominant over the anti-stokes sideband, resulting in the radiation pressure force amplifying the mirror motion [23]. It should be noted that the mechanical oscillator is under the action of the mean radiation pressure force as well as its intrinsic damping force. To the extent that the intrinsic damping rate is larger than the magnitude of the radiation pressure damping rate, γ > Γ, the cavity field and mechanics have their steady-state values in the long time limit [89], x ss = g 0 α ss 2, ω m (1.16) p ss = 0, (1.17) α ss = η i( c + g 0 x ss ) + κ/2, (1.18) where x ss, p ss, α ss are the steady-state values of the mean position and momentum

32 31 of the mechanics and the mean cavity field amplitude, respectively Radiation pressure force fluctuations We now explore the effects of the fluctuating radiation pressure force on the mechanics. Assuming that the cavity field and mechanics reach their steady states, the fluctuating radiation pressure force is given by δ ˆF RP = g 0 α ss (ĉ + ĉ ), (1.19) where we have assumed that α ss is real without loss of generality. Following the same approach as before, one can find the fluctuating radiation pressure force in terms of the mechanical fluctuations [29], δ ˆF RP Ωδˆx Γδˆp (1.20) where [ ] Ω = g0α 2 ss 2 c + ω m ( c + ω m ) 2 + κ 2 /4 + c ω m (, (1.21) c ω m ) 2 + κ 2 /4 [ ] Γ = g0α 2 ss 2 κ/2 ( c + ω m ) 2 + κ 2 /4 κ/2 (, (1.22) c ω m ) 2 + κ 2 /4 where c = c + g 0 x ss. In the resolved sideband regime, ω m κ, the same arguments made in the mean field case can be applied. For example, if c = ω m, the resonant anti- Stokes sideband leads to damping of the mechanics or cooling of the motional state of the mechanics. In this situation, the cavity field acts as a viscous fluid so that the mechanical oscillator experiences enough additional friction to be cooled to its motional ground state [87, 88, 89]. If c = +ω m, the resonant Stokes sideband gives rise to the growth of the mechanical energy [29]. In the Doppler regime, ω m κ, the two sidebands resulting from the Raman scatterings are not resolved due to the broad cavity response. The optically induced

33 32 frequency shift and radiation pressure damping rate can be approximated as [29] Ω Doppler = g 2 0α 2 ss 2 c 2 c + κ 2 /4, (1.23) Γ Doppler = g0α 2 ss 2 4ω m c κ ( 2 c + κ 2 /4). (1.24) 2 Note that the dynamic optical spring effect can be significant for the mechanics with a low resonant frequency. However, the radiation pressure damping or anti-damping is negligible since Γ Doppler is proportional to ω m /κ Strong-coupling regime The strong-coupling regime is the regime where the single-photon optomechanical coupling coefficient g 0 is much smaller than the mechanical frequency, g 0 ω m, however, the amplified optomechanical coupling strength is larger than the cavity linewidth [84]. In contrast to the fluctuating radiation pressure force in the weakcoupling regime, the fluctuating radiation pressure force depends on the history of the evolution of the mechanics since the evolutions of the cavity field and mechanics are of the same order. The optomechanical coupling generates coherent mixing between the optical and mechanical modes, resulting in normal mode splitting or polariton states [47]. We assume that the optomechanical system is in steady-state with the mechanical position x ss and cavity field amplitude α ss. In a frame rotating at the driving frequency ω L, the Hamiltonian describing the interaction between the cavity field and mechanical fluctuations is [26] H sys = c ĉ ĉ + ω mˆb ˆb + g(ĉ ˆb + ˆb ĉ + h.c.), (1.25) where g is the amplified optomechanical coupling strength, g = g 0 α ss / 2 and ˆb is the bosonic annihilation operator for the mechanics, ˆb = (δˆx iδˆp)/ 2 with [ˆb, ˆb ] = 1. In the interaction picture with respect to H 0 = c ĉ ĉ + ω mˆb ˆb, the interaction

34 33 Hamiltonian becomes H int = g(ĉ ˆbe i( c ω m)t + ˆb ĉ e i( c+ω m)t + h.c.). (1.26) If c = ω m, the rotating wave approximation, valid for ω m g in the resolved sideband regime, simplifies the interaction Hamiltonian to [90] H int = g(ĉ ˆb + ˆb ĉ). (1.27) This form of Hamiltonian is known as the beam-splitter Hamiltonian in quantum optics [79]. It gives rise to quantum state transfer between the optical and mechanical modes [90]. In particular, Eq. (1.27) can be utilized to realize quantum state swapping between optical and microwave fields [91, 92, 93]. Coherent energy transfer based on this Hamiltonian has been realized in micro-mechanical systems with optical fields [55] and microwaves [57]. If the pump is blue-detuned from the cavity resonance by one mechanical frequency, c = ω m, the rotating wave approximation leads to a different optomechanical interaction [90], H int = g(ĉ ˆb + ˆbĉ). (1.28) This form of Hamiltonian is known as the two-mode squeezing Hamiltonian in quantum optics [79]. It leads to two-mode squeezing as well as quantum entanglement generation between the optical and mechanical modes [90]. The quantum entanglement generation based on this Hamiltonian has recently been demonstrated in micro-mechanical systems with microwaves [64]. 1.3 Single-photon strong-coupling regime The single-photon strong-coupling regime is the regime where the single-photon optomechanical coupling coefficient g 0 is comparable to or larger than the mechanical frequency, g 0 ω m. This regime has not yet been realized in nano-fabricated

35 34 optomechanical systems but it is currently accessible in optomechanical systems involving ultracold atoms in a high finesse optical cavity [48, 49]. In this case, the radiation pressure force of a single photon is able to displace the moving mirror by more than the zero-point fluctuations and thus gives rise to a large shift in the cavity field frequency. Such a strong coupling leads to multiple mechanical sidebands [94] and the modification of the optomechanical effects explored in the linearized regime, including the radiation pressure cooling [95] and optomechanically induced transparency [96]. The optomechanical systems working in this regime allow to investigate the fundamental nonlinearity of the optomechanical coupling and the role of quantum fluctuations in that both optical and mechanical modes operate in the deep quantum regime. The intrinsic nonlinearity of the optomechanical coupling promises in particular the demonstration of photon bunching [97, 98], multiple cooling resonances [95], the generation of optical Schrödinger cat states [99, 100], non-gaussian [94] and nonclassical mechanical states [101]. 1.4 Multimode optomechanics We have so far concentrated on the optomechanical coupling in a variety of different physical regimes assuming that the one cavity field mode is coupled to the center-of-mass mode of the mechanical oscillator. However, all optical resonators possess multiple optical modes and mechanical oscillators have several different normal modes depending on geometry. Under these considerations, one can extend the basic Hamiltonian, Eq. (1.5), to the Hamiltonian describing multiple optical and mechanical modes and their mutual optomechanical interactions as [28] H sys = k ω c,k â kâk + j ω m,jˆb jˆb j g 0,kjl â kâl(ˆb j + ˆb j ), (1.29) j,k,l where â k and ˆb j are the bosonic annihilation operators for the k-th optical and j-th mechanical mode of frequency ω c,k and ω m,j, respectively. The optomechanical coupling coefficients g 0,kjl describing the coupling of the optical modes to the

36 35 Figure 1.2: Membrane in the middle geometry. Two cavity fields tunnel through a membrane located at the center of a cavity and push the membrane in opposite directions, resulting in a quadratic optomechanical interaction. mechanical modes depend on the geometry of the system under consideration. Multimode optomechanical systems open the way for implementation of higher-order optomechanical interactions and the realization of optomechanical devices in which individual elements implement different functionalities Quadratic optomechanics In a simple Fabry-Pérot cavity, the higher-order optomechanical interactions are negligible compared to the linear coupling. However, quadratic optomechanical interactions can be realized in the so-called membrane-in-the-middle setup in which a transmissive membrane situated at the center of a Fabry-Pérot cavity with fixed mirrors is coupled to two optical fields via linear optomechanical interaction in opposite directions, as shown in Fig. 1.2 [102, 103]. The membrane in the middle setup can be viewed as a simple example of a multimode optomechanical system in that two optical fields are coupled to a single mechanical mode. Provided that the membrane is located at an extremum in ω c (x), the linear optomechanical coupling of the membrane to the optical fields can be suppressed, resulting in a dominant quadratic optomechanical interaction.

37 36 In the remainder of this section, we describe a possible realization of such an optomechanical interaction. In Fig. 1.2, a membrane of effective mass m and frequency ω m is located at the center of a cavity and interacts with two cavity fields of the same nominal frequency ω c via linear optomechanical interaction in opposite directions. The Hamiltonian describing the left and right optical fields and their transmission through the membrane is given by H cav = ω c â LâL + ω c â RâR J(â LâR + h.c.), (1.30) where â L(R) denotes the bosonic annihilation operator for the left (right) field with commutation relations, [â j, â k ] = δ jk where k {L, R} and J is the tunnelling rate between the optical modes. The optomechanical interaction is H om = g 0 (â LâL â RâR)(ˆb + ˆb ), (1.31) where g 0 is the single-photon optomechanical coupling. Introducing symmetric and antisymmetric optical normal modes, â + = 1 2 (â L + â R ), (1.32) â = 1 2 (â L â R ), (1.33) the Hamiltonians describing the optical fields and the optomechanical interaction can be written as H cav = ω + â +â + + ω â â, (1.34) H om = g 0 (â +â + â â + )(ˆb + ˆb ). (1.35) Here, ω ± = ω c ± J are the frequencies of the optical normal modes. Note that the optical modes experience a 2J frequency splitting due to the coherent tunnelling interaction. In the regime where the photon tunnelling is much faster than the optomechanical interaction, the latter can be treated as a perturbation [102, 103,

38 37 104, 105]. The system Hamiltonian can be diagonalized formally up to second order in the position of the membrane, H = [ω + + g + (ˆb + ˆb ] ) 2 â +â + + [ω + g (ˆb + ˆb ] ) 2 â â + ω mˆb ˆb, (1.36) where g ± = ± g2 0 2J is the effective optomechanical coupling coefficient and can be identified as the quadratic optomechanical coupling constant since the frequency of the optical fields now depends on the square of the position of the mechanics. It should be noted that the mechanical mode is coupled to the optical normal mode with a higher resonance frequency through the quadratic optomechanical interaction with a positive coupling coefficient and is also coupled to the optical normal mode with a lower resonance frequency with a negative coupling coefficient. In the regime where the frequency splitting of the two normal modes is much larger than their linewidths and the mechanical frequency, it is possible to drive each individual normal mode independently. With an external drive whose frequency is close to the frequency of one of the optical normal modes, one can realize the quadratic optomechanical coupling, H = ω k â kâk + ω mˆb ˆb + gk â kâk(ˆb + ˆb ) 2, (1.37) where k {±}. The quadratic optomechanical interaction can give rise to radiationinduced cooling and squeezing of the membrane [106] and can be utilized for quantum nondemolition detection of the phonon number in the membrane [102, 103]. 1.5 Dissertation format This dissertation addresses some aspects of multimode optomechanical systems. The introductory sections have framed the basic ideas necessary for the remainder of this dissertation. We first concentrate on two specific examples and then develop a general formalism. The first example investigates mechanical squeezing in a nano-fabricated system in which a quantum cantilever is magnetically coupled

39 38 to a classical mechanical oscillator and optomechanically coupled to a cavity field. The research was carried out by myself, Dr. Lukas F. Buchmann, Dr. Swati Singh, Dr. Steven K. Steinke, and my advisor Prof. Pierre Meystre. In the second example, we explore the dynamic stabilization of an unstable quantum mechanical oscillator coupled to two optical fields. The research in this part was performed by myself, Prof. Ewan M. Wright, and Prof. Pierre Meystre. Finally, we conduct a formal analysis for the cavity-mediated mechanical interactions and identify a number of nonlinear effective interactions. We explore specific examples in two limiting regimes, involving quantum state transfer and quantum correlations between mechanical oscillators. The research was done by myself, Dr. Lukas F. Buchmann, Dr. Swati Singh, Prof. Ewan M. Wright, and Prof. Pierre Meystre.

40 39 CHAPTER 2 PRESENT STUDY The four appendices of this dissertation contain the theory, results, and conclusions of this study. The following is a brief summary of the significant results of these papers. 2.1 Generation of mechanical squeezing via magnetic dipoles on cantilevers Appendix A contains the manuscript Generation of mechanical squeezing via magnetic dipoles on cantilevers authored by myself, Dr. Lukas F. Buchmann, Dr. Swati Singh, Dr. Steven K. Steinke, and Prof. Pierre Meystre. This paper appeared in the March 2012 issue of Physical Review A [71]. The main objective of this work is to propose a method to squeeze one of the center-of-mass motional quadratures of a quantum mechanical oscillator below its standard quantum limit. In this work, we consider a system in which a quantum cantilever is coupled to a mesoscopic tuning fork via magnetic dipole-dipole interaction and is also coupled to a cavity field via linear optomechanical interaction. We find that the magnetic dipole-dipole interaction allows the generation of a squeezed state of the quantum cantilever. We also investigate the influence of several sources of noise on the achievable squeezing including the classical noise in the driven tuning fork and the clamping noise in the oscillator. Two detection schemes of the squeezed state of the quantum cantilever based on quantum state transfer to the optical field are considered.

41 Dynamic stabilization of an optomechanical oscillator Appendix B describes the dynamic stabilization of an optomechanical oscillator. The research in this part was carried out by myself, Prof. Ewan M. Wright, and Prof. Pierre Meystre. We consider a mechanical oscillator coupled to a cavity field via quadratic optomechanical interaction with a negative coupling coefficient. The key idea comes from Kapitza s pendulum [107], a prototypical example of dynamic stabilization of a mechanical oscillator in the classical regime. By modulating the radiation pressure force of the cavity field, we show that the mechanical oscillator initially prepared in an unstable configuration can be stabilized for appropriate conditions. We investigate the full dynamics of the mechanical oscillator under the effects of the modulated radiation pressure force and dissipation in the classical and quantum regimes. 2.3 Multimode weak-coupling quantum optomechanics Appendix C contains the manuscript Optically mediated nonlinear quantum optomechanics authored by myself, Dr. Lukas F. Buchmann, Dr. Swati Singh, and Prof. Pierre Meystre. This paper was published in the December 2012 issue of Physical Review A [108]. We develop a general formalism that fully accounts for the cavity-mediated mechanical interactions for both the linear and quadratic coupling cases. The specific situation that we consider amounts to multiple mechanical modes coupled to a single quantized cavity field mode via linear or quadratic optomechanical interactions with identical coupling coefficients. We concentrate on the case in which the optical dissipation is the dominant source of damping, meaning that the optical field can be adiabatically eliminated, resulting in effective multimode interactions between the mechanical resonators. This study concentrates on the situation in which the cavity is strongly driven by an external field and the optomechanical interactions are weak compared to the mechanical harmonic potential so that they

42 41 can be linearized and treated as perturbations. The cavity-mediated mechanical interaction of two modes is investigated in two limiting cases, the resolved sideband and the Doppler regimes. We find that in the case of linear coupling, the coherent contribution to the interaction can be exploited in quantum state swapping protocols, while the incoherent part leads to significant modifications of cold damping or amplification from the single-mode situation. Quadratic coupling can result in a wealth of possible effective nonlinear interactions including the analogs of secondharmonic generation and four-wave mixing in nonlinear optics. As an illustrative application of the formal analysis, we discuss in some detail a two-mode system in which a Bose-Einstein condensate is optomechanically coupled to the moving end mirror of a Fabry-Pérot cavity and we explore the possibility of quantum state transfer between the mechanical modes. 2.4 Multimode strong-coupling quantum optomechanics Appendix D contains the manuscript Multimode strong-coupling quantum optomechanics authored by myself, Dr. Lukas F. Buchmann, Prof. Ewan M. Wright, and Prof. Pierre Meystre. This paper appeared in the December 2013 issue of Physical Review A [109]. We consider multiple mechanical modes coupled to a single quantized cavity field mode via linear or quadratic optomechanical interactions with identical coupling coefficients. We focus specifically on the situation where the cavity field is weakly pumped by an external field and the optomechanical interaction is comparable to or larger than the mechanical harmonic potential so that both the optical and mechanical systems operate in the deep quantum regime. In this regime, we exploit numerical calculations to investigate the nonlinear dynamics of the mechanical modes. Using as examples, systems of one and two mechanical oscillators, we find that the mechanics exhibits nonlinear phenomena including bistability and bifurcations in the classical regime. We also show quantum interference, entanglement, and correlation between the mechanical oscillators in the quantum regime.

43 42 REFERENCES [1] J. Kepler, De Cometis (1619). [2] J. C. Maxwell, Treatise on Electricity and Magnetism (1873). [3] P. Lebedev, Ann. Phys. 6, 433 (1901). [4] A. Ashkin, Science 210, 1081 (1980). [5] P. Meystre, Atom Optics (Springer-Verlag, New York, 2001). [6] A. Ashkin, Phys. Rev. Lett. 25, 1321 (1970). [7] A. Ashkin and J. M. Dziedzic, Science 235, 1517 (1987). [8] T. W. Hansch, and A. L. Schawlow, Opt. Comm. 13, 68 (1975). [9] H. J. Metcalf and P. van der Straten, Laser Cooling and Trapping (Springer- Verlag, New York, 1999). [10] F. Diedrich, J. C. Bergquist, W. M. Itano, and D. J. Wineland, Phys. Rev. Lett. 62, 403 (1989). [11] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Wieman, and E. A. Cornell, Science 269, 198 (1995). [12] K. B. Davis, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, Phys. Rev. Lett. 75, 3969 (1995). [13] C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, Phys. Rev. Lett. 75, 1687 (1995).

44 43 [14] J. Dunningham, K. Burnett, and W. D. Phillips, Phil. Trans. R. Soc. A 363, 2165 (2005). [15] S. A. Diddams et al., Science 293, 825 (2001). [16] S. Gupta, K. Dieckmann, Z. Hadzibabic, and D. E. Pritchard, Phys. Rev. Lett. 89, (2002). [17] I. Bloch and W. Zwerger, Rev. Mod. Phys. 80, 855 (2008). [18] D. Leibfried, R. Blatt, C. Monroe, and D. Wineland, Rev. Mod. Phys. 75, 281 (2003) [19] I. Bloch, J. Dalibard, and S. Nascimbène, Nat. Phys. 8, 267 (2012). [20] R. Blatt and C. F. Roos, Nat. Phys. 8, 277 (2012). [21] J. I. Cirac and P. Zoller, Phys. Rev. Lett. 74, 4091 (1995). [22] D. Wineland, C. Monroe, W. M. Itano, D. Leibfried, B. E. King, and D. M. Meekhof, J. Res. Natl. Inst. Stand. Technol. 103, 259 (1998). [23] T. J. Kippenberg and K. J. Vahala, Optics Express 15, (2007). [24] T. J. Kippenberg and K. J. Vahala, Science 321, 1172 (2008). [25] F. Marquardt and S. M. Girvin, Physics 2, 40 (2009). [26] M. Aspelmeyer et al., J. Opt. Soc. Am. B 27, A189 (2010). [27] M. Aspelmeyer, P. Meystre and K. Schwab, Physics Today 65, 29 (2010). [28] M. Aspelmeyer, T. J. Kippenberg and F. Marquardt, arxiv: (2013). [29] P. Meystre, Annalen der Physik 525, 215 (2013). [30] V. B. Braginsky and A. B. Manukin, Sov. Phys. JETP. 25, 653 (1967).

45 44 [31] V. B. Braginsky, A. B. Manukin, and M. Y. Tikhonov, Sov. Phys. JETP. 31, 829 (1970). [32] C. M. Caves, K. S. Thorne, R. W. P. Drever, V. D. Sandberg, and M. Zimmermann, Rev. Mod. Phys. 52, 341 (1980). [33] V. B. Braginsky, Y. I. Vorontsov, and F. Y. Khalili, Sov. Phys. JETP. 46, 705 (1977). [34] V. B. Braginsky and F. Y. Khalili, Quantum Measurement, (Cambridge, Cambridge, 1992). [35] A. Dorsel, J. D. McCullen, P. Meystre, E. Vignes, and H. Walther, Phys. Rev. Lett. 51, 1550 (1983). [36] M. F. Bocko and R. Onofrio, Rev. Mod. Phys. 68, 755 (1996). [37] S. Gigan et al., Nature 444, 67 (2006). [38] G. Rempe, R. J. Thomson, H. J. Kimble, and R. Lalezari, Opt. Lett. 17, 363 (1992). [39] D. Rugar, R. Budakian, H. J. Mamin, and B. W. Chui, Nature 430, 329 (2004). [40] M. D. LaHaye, O. Buu, B. Camarota, and K. C. Schwab, Science 304, 74 (2004). [41] M. Cai, O. Painter, and K. J. Vahala, Phys. Rev. Lett. 85, 74 (2000). [42] O. Arcizet, P. F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, Nature 444, 71 (2006). [43] S. Mancini, D. Vitali, and P. Tombesi, Phys. Rev. Lett. 80, 688 (1998). [44] P. F. Cohadon, A. Heidmann, and M. Pinard, Phys. Rev. Lett. 83, 3174 (1999).

46 45 [45] C. H. Metzger and K. Karrai, Nature 432, 1002 (2004). [46] S. Weis et al., Science 330, 1520 (2010). [47] J. D. Teufel et al., Nature 471, 204 (2011). [48] S. Gupta et al, Phys. Rev. Lett. 99, (2007). [49] F. Brennecke et al., Science 322, 235 (2008). [50] B. Abbott et al., New. J. Phys. 11, (2009). [51] A. D. O Connell et al., Nature 464, 697 (2010). [52] J. D. Teufel et al., Nature 475, 359 (2011). [53] J. Chan et al., Nature 478, 89 (2011). [54] A. H. Safavi-Naeini et al., Phys. Rev. Lett. 108, (2012). [55] E. Verhagen, S. Deléglise, S. Weis, A. Schliesser, and T. J. Kippenberg, Nature 482, 63 (2012). [56] G. S. Agarwal and S. Huang, Phys. Rev. A 81, (2010). [57] T. A. Palomaki, J. W. Harlow, J. D. Teufel, R. W. Simmonds, and K. W. Lehnert, Nature 495, 210 (2013). [58] A. H. Safavi-Naeini et al., Phys. Rev. Lett. 108, (2012). [59] N. Brahms, T. Botter, S. Schreppler, D. W. C. Brooks, and D. M. Stamper- Kurn, Phys. Rev. Lett. 108, (2012). [60] T. P. Purdy, R. W. Peterson, and C. A. Regal, Science 339, 801 (2013). [61] S. Schreppler et al., Science 344, 1486 (2014).

47 46 [62] R. W. Andrews et al., Nat. Phys. 10, 321 (2014). [63] T. Bagci et al., Nature 507, 81 (2014). [64] T. A. Palomaki, J. D. Teufel, R. W. Simmonds, K. W. Lehnert, Science 342, 710 (2013). [65] W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, Phys. Rev. Lett. 91, (2003). [66] W. H. Zurek, Phys. Today 44, 36 (1991). [67] L. F. Buchmann, L. Zhang, A. Chiruvelli, and P. Meystre, Phys. Rev. Lett. 108, (2012). [68] I. Pikovski, M. R. Vanner, M. Aspelmeyer, M. S. Kim, and Phys. 8, 393 (2012). Č. Brukner, Nat. [69] A. Mari and J. Eisert, Phys. Rev. Lett. 103, (2009). [70] K. Jähne et al., Phys. Rev. A 79, (2009). [71] H. Seok, L. F. Buchmann, S. Singh, S. K. Steinke, and P. Meystre, Phys. Rev. A 85, (2012). [72] P. Rabl et al., Nat. Phys. 6, 602 (2010). [73] K. Stannigel et al., Phys. Rev. Lett. 109, (2012). [74] A. Tomadin, S. Diehl, M. D. Lukin, P. Rabl, and P. Zoller, Phys. Rev. A 86, (2012). [75] G. Heinrich, M. Ludwig, J. Qian, B. Kubala, and F. Marquardt, Phys. Rev. Lett. 107, (2011). [76] M. Zhang et al., Phys. Rev. Lett. 109, (2012).

48 47 [77] D. E. Chang, A. H. Safavi-Naeini, M. Hafezi, and O. Painter, New. J. Phys. 13, (2011). [78] M. Schmidt, M. Ludwig, and F. Marquardt, New. J. Phys. 14, (2005). [79] P. Meystre and M. Sargent III, Elements of Quantum Optics, 4th ed. (Springer- Verlag, Berlin, 2007). [80] R. W. Boyd, Nonlinear Optics, 3rd ed. (Elsevier, Burlington, 2008). [81] M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge, New York, 1999). [82] P. W. Milonni and J. H. Eberly, Lasers (Wiley, Hoboken, 1988). [83] P. Meystre, E. M. Wright, J. D. McCullen, and E. Vignes, J. Opt. Soc. Am. B 2, 1830 (1985). [84] J. D. Teufel, J. W. Harlow, C. A. Regal, and K. W. Lehnert, Phys. Rev. Lett. 101, (2008). [85] C. K. Law, Phys. Rev. A 51, 2537 (1995). [86] A. N. N. Cleland, Foundations of Nanomechanics (Springer-Verlag, Berlin, 2003). [87] I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, Phys. Rev. Lett. 99, (2007). [88] F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, Phys. Rev. Lett. 99, (2007). [89] C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. Aspelmeyer, Phys. Rev. A 77, (2008).

49 48 [90] J. Zhang, K. Peng, and L. Braunstein, Phys. Rev. A 68, (2003). [91] L. Tian and H. Wang, Phys. Rev. A 82, (2010). [92] Y. Wang and A. A. Clerk, Phys. Rev. Lett. 108, (2012). [93] L. Tian, Phys. Rev. Lett. 108, (2012). [94] A. Nunnenkamp, K. Børkje, and S. M. Girvin, Phys. Rev. Lett. 107, (2011). [95] A. Nunnenkamp, K. Børkje, and S. M. Girvin, Phys. Rev. A 85, (R) (2012). [96] A. Kronwald and F. Marquardt, Phys. Rev. Lett. 111, (2013). [97] P. Rabl, Phys. Rev. Lett. 107, (2011). [98] A. Kronwald, M. Ludwig, and F. Marquardt, Phys. Rev. A 87, (2013). [99] S. Mancini, V. I. Man ko, and P. Tombesi, Phys. Rev. A 55, 3042 (1997). [100] S. Bose, K. Jacobs, and P. L. Knight, Phys. Rev. A 56, 4175 (1997). [101] J. Qian, A. A. Clerk, K. Hammerer, and F. Marquardt, Phys. Rev. Lett. 109, (2012). [102] J. D. Thompson et al., Nature 452, 72 (2008). [103] J. C. Sankey, C. Yang, B. M. Zwickl, A. M. Jayich, and J. G. E. Harris, Nat. Phys. 6, 707 (2010). [104] H. Miao, S. Danilishin, T. Corbitt, and Y. Chen, Phys. Rev. Lett. 103, (2009).

50 49 [105] A. A. Gangat, T. M. Stace, and G. J. Milburn, New J. Phys. 13, (2011). [106] A. Nunnenkamp, K. Børkje, J. G. E. Harris, and S. M. Girvin, Phys. Rev. A 82, (R) (2010). [107] P. L. Kapitza, Soviet Phys. JETP 21, 588 (1951). [108] H. Seok, L. F. Buchmann, S. Singh, and P. Meystre, Phys. Rev. A 86, (2012). [109] H. Seok, L. F. Buchmann, E. M. Wright, and P. Meystre, Phys. Rev. A 88, (2013).

51 50 APPENDIX A GENERATION OF MECHANICAL SQUEEZING VIA MAGNETIC DIPOLES ON CANTILEVERS H. Seok, L. F. Buchmann, S. Singh, S. K. Steinke and P. Meystre Published in Physical Review A Copyright 2012 by American Physical Society ABSTRACT A scheme to squeeze the center-of-mass motional quadratures of a quantum mechanical oscillator below its standard quantum limit is proposed and analyzed theoretically. It relies on the dipole-dipole coupling between a magnetic dipole mounted on the tip of a cantilever to equally oriented dipoles located on a mesoscopic tuning fork. We also investigate the influence of several sources of noise on the achievable squeezing, including classical noise in the driving fork and the clamping noise in the oscillator. A detection of the state of the cantilever based on state transfer to a light field is considered. We investigate possible limitations of that scheme. A.1 Introduction Mechanical cantilevers have a long and rich history as force and field meters. In recent decades, microfabrication and nanotechnology have led to advances with the development of tip microscopy [1] that have resulted in significant advances in the measurement of feeble forces [2], the imaging of single atoms, nanoscale magnetic resonance imaging [3], and single spin detection [4], to mention just a few applica-

52 51 tions. And rapid advances in cavity optomechanics [5, 6, 7], recently culminating in the cooling of at least three micromechanical systems [8, 9, 10] to the deep quantum regime with just a fraction of a phonon of center-of-mass excitation left, indicate that mechanical sensing appears to be at the threshold of an important new breakthrough. One potential advantage of operating micromechanical sensors in the quantum regime is that this opens the way to measurement techniques that can circumvent the standard quantum limit. These techniques rely on the capability to locate the unavoidable quantum noise in a quadrature of the field to be measured that does not interact with the measuring apparatus, and to implement back-action evading techniques that prevent that noise from feeding back into the outcome of successive measurements. Such measurement techniques were pioneered by Braginskii and coworkers [11, 12] and became popular in the context of gravitational wave detection [13]. For a comprehensive summary of these early results, see [14, 15]. In quantum optics, the most famous states that permit one to have quadrature variances below the standard quantum limit are single-mode squeezed states, where the variance of one of the quadratures ˆX 1 = (â+â )/2 or ˆX 2 = (â â )/2i of the light field is below 1/4, with ˆX 1 2 ˆX 2 2 1/16. Here â and â are normalized bosonic annihilation and creation operators of the field mode, and ˆX i 2 = ˆX i 2 ˆX i 2. While back-action evading measurement below the amplifier limit [16] and classical noise squeezing of micromechanical oscillators have already been achieved [17, 18], squeezing below the standard quantum limit still remains to be demonstrated. A number of techniques have been proposed, including conditional squeezing using a parametrically coupled electromagnetic cavity driven by one [19] or two sidebands [20, 21] detuned from the cavity resonance by the mechanical oscillation frequency, and are expected to be demonstrated experimentally in the near future. This paper proposes an alternative scheme where a nanoscale cantilever can be prepared in a squeezed state by purely mechanical means via the nonlinear-

53 52 Figure A.1: Tuning fork magnetically coupled to a quantum-mechanical cantilever. The equilibrium position of the cantilever is equidistant from the two extremities of the nanoscale tuning fork. ity provided by the magnetic dipole-dipole interaction between a classically driven fork and the cantilever. We also show that the squeezing can be detected via state transfer to an optical field coupled to the cantilever in an optical resonator configuration [22, 23, 24]. We discuss the impact of the various sources of noise, including both the noise of the classical driving force and the clamping noise of the cantilever, and also comment on limitations to the state transfer scheme resulting from the dynamics of the light field and from the opening of an associated dissipation channel. The remainder of this paper is organized as follows: Section II introduces our model system and demonstrates how the anharmonic potential that describes the magnetic coupling between the classical fork and the quantum mechanical cantilever results in quadrature squeezing of the cantilever motion under appropriate conditions. Section III analyzes the robustness of the system against various sources of noise. Section IV discusses the beam-splitter state transfer mechanism, and section V is a conclusion and outlook. Calculational details are relegated to two appendices.

54 53 A.2 Model system We consider a nanomechanical system consisting of a cantilever magnetically coupled to a nanoscale tuning fork, as shown in Fig. A.1, the coupling being realized via point-like magnetic dipoles located at both extremities of the fork as well as on the cantilever. Apart from this magnetic coupling, the two subsystems are isolated from each other. While the oscillation direction of the fork is taken to be the z-axis, the cantilever s motion of interest is perpendicular to it, along the x-axis. The magnetic dipoles are assumed to point in the positive z-direction. We consider specifically the center-of-mass mode of vibration of the fork, in which the distance between its two extremities remains constant. This arrangement provides a stable mode of operation for experimentally reasonable parameters. Denoting the displacement of the cantilever from its equilibrium position by x c, the interaction energy between the three magnetic dipoles can be written as [ ] V = µ 0 4π d x 2 c 2l 2 + x2c 2l 2 fd c +, (A.1) (x 2 c + l 2 5/2 +) (x 2 c + l 2 ) 5/2 where d f and d c are the dipoles moments of the magnets attached to the tips of the fork and to the cantilever, respectively, and l ± are the distances between the two tips of the fork and the cantilever along the z-direction. A.2.1 Symmetric case We assume first that the equilibrium position of the nanomagnet on the cantilever is equidistant from the two tips of the fork. For that setup the attractive forces from the two dipoles acting on the cantilever cancel each other, but for any departure from that situation this is no longer the case, so that the stability of the system must be enforced by the stiffness of the cantilever.

55 54 For the mode of vibration under consideration we have in the symmetric case where z f l + l + z f, (A.2a) l l z f, (A.2b) denotes the displacement of the tuning fork tips from their equilibrium position and l = (l + + l )/2. For small displacements of the mechanical elements, z f, x c l, the interaction Hamiltonian can be expanded to second order in x c, yielding V µ [ ] 0 d f d c 24z 2 4π l 5 f + 12x 2 c z2 f x2 c, (A.3) l 2 where we ignored a constant term. The first two terms in Eq. (A.3) describe frequency shifts due to the magnetic interaction. For high amplitude driving, the motion of the fork of effective mass m f frequency ω f can be treated classically. The cantilever motion, on the other hand, is much smaller and is therefore treated quantum mechanically, with its displacement x c given in terms of the bosonic annihilation and creation operators ˆb and ˆb by and ˆx c = x 0 (ˆb + ˆb ), (A.4) with x 0 =, 2m c ω c (A.5) where ω c and m c denote the cantilever s frequency and effective mass. The Hamiltonian governing the dynamics of the cantilever is then where H = ω cˆb ˆb + ˆV, ω c = ω c + ω c, (A.6) (A.7) with ω c the frequency shift from the magnetic interaction, ω c 6µ 0d f d c π x 2 0 l 5, (A.8)

56 55 see Eq. (A.3). In terms of ˆb and ˆb, the dipole interaction ˆV becomes where and V = 45µ 0d f d c π z 2 0x 2 0 l 7 (β + β ) 2 (ˆb + ˆb ) 2, (A.9) β = z f 2z 0 + i z 0p f, z 0 =. 2m f ω f (A.10) (A.11) A final simplification is obtained by taking the driving frequency of the fork to be ω f = ω c, (A.12) then invoking the rotating wave approximation and switching to a frame rotating with ω f, where V reduces to the familiar single-mode squeezing Hamiltonian (see e.g. Ref.[25, 26, 27]) V = 1 2 χ s (ˆb 2 e i2φ + ˆb 2 e i2φ ) (A.13) with χ s = 90µ 0d f d c z0x π l 7 β 2 (A.14) and φ the relative phase between the classical fork and the cantilever. It defines which quadrature gets squeezed, and in the following we always consider the quadrature which experiences maximum squeezing, omitting the explicit value of φ. A.2.2 Asymmetric case We now turn to the case where the equilibrium position of the cantilever nanomagnet is displaced by a distance ζ from the center of the fork, the stability of that configuration being guaranteed as before by the mechanical stiffness of the cantilever. We now have l + l + ζ + z f, (A.15a) l l ζ z f, (A.15b)

57 56 and within the same limit as before V µ 0 d f d c 4π l 5 [ 24(ζ + z f ) x 2 c (ζ + z f) 2 x 2 c l 2 ]. (A.16) When compared to the symmetric case, Eq. (A.3), the interaction (A.16) comprises an additional squeezing contribution given by the term proportional to ζz f x 2 c. By driving at twice the cantilever frequency, and invoking the rotating wave approximation, one can access either of the two squeezing interactions separately, in a fashion reminiscent of the situation in parametrically coupled optomechanical resonators [19]. The resulting Hamiltonian is then the sum of two terms of the same form as in Eq. (A.13) with the coupling coefficient of the term oscillating at 2ω c given by χ 2 ( ) = 15µ [ ] 0d f d c 1 π (l ζ) 1 x 2 6 (l + ζ) 0z 6 0 β, (A.17) and that for the term oscillating at ω c by χ 1 (ζ) = 45µ [ 0d f d c π 1 (l ζ) (l + ζ) 7 ] x 2 0z 2 0 β 2. (A.18) The coupling constant χ 1 (ζ) scales as the square of the amplitude of oscillations β 2 of the classical fork, rather than β as is the case for χ 2 (ζ), and hence can be dominant for strong fork driving. However, its dependence on l 7 rather than l 6 indicates that for appropriate values of l the term proportional to χ 2 can be dominant instead, see Fig. 2. A.2.3 Experimental considerations Consider for concreteness a nanomechanical cantilever with natural frequency ω c = 2π 168 khz, effective mass m c = kg, and magnetic dipole moment d c = A m 2. The fork is assumed to have effective mass m f = kg, tips separation 2l = 2.0 µm, and magnetic dipole moments d f = A m 2. These values result in a frequency shift of the cantilever of ω c 2π 72 MHz ω c,

58 57 Χ 1,Χ Ζ Figure A.2: Coupling frequencies χ 2 (ζ) (red, solid), and χ 1 (ζ)(blue, dashed) in MHz as a function of displacement ζ in nm for parameters of section II.C. indicating that the driving frequency of the fork is dominated by the frequency shift ω f ω c. We further assume a fork oscillation amplitude of A f = 2z 0 β = 10 nm, i.e. A f = 0.01l, with associated mean phonon occupation β 2 = , which justifies a classical treatment. For the asymmetric setup, we need to estimate the frequency of the cantilever in the z-direction, which we obtain from ω c (z) /ω c w/h, where h and w are the cantilever s thickness and width. For h = 12 nm and w = 600nm this gives ω (z) c 2π 8.4MHz. With this value and Eqs. (A.15), we can find the critical points of the full interaction potential, Eq. (A.1). For these parameters the maximum value of ζ that yields a stable configuration is 162nm. The resulting strengths of the squeezing interaction are shown in Fig. A.2 as a function of ζ. A.3 Fluctuations It is known that in the absence of fluctuations and starting from the oscillator ground state the squeezing Hamiltonian (A.13) produces a perfect squeezed vacuum with

59 58 average phonon number ˆb ˆb = sinh 2 (χt). (A.19) (In the following we use generically the symbol χ to describe all setups of the previous section.) Clearly, this is unrealistic in the long-time limit. What is missing from the discussion so far is a proper accounting of fluctuations. This section discusses the effects of three sources of technical noise: the amplitude and phase fluctuations of the fork motion and the clamping noise resulting from the attachment of the cantilever to a thermal reservoir. A.3.1 Amplitude fluctuations We assume that the cantilever is slightly displaced from the center of the fork, driven at frequency ω f = 2ω c, so that the squeezing strength is given by Eq. (A.17). The other two cases can be handled similarly and we give the results pertaining to them at the end of this subsection. Assuming as usual that the amplitude of oscillations of the tuning fork can be decomposed into the sum of a constant amplitude and random amplitude fluctuations β (t) = β 0 + δβ(t), (A.20) we find the equation of motion ˆb = [χ 0 + δχ(t)] ˆb, (A.21) where χ 0 is given by Eq. (A.17) and δχ(t) = χ 0 δβ(t). (A.22) β 0 Following Ref. [28], we assume that the mean and two-time correlation functions of the amplitude fluctuations are described by an Ornstein-Uehlenbeck process, so that δχ(t) = 0, (A.23) δχ(t)δχ(t ) = χ2 0 β 0 2 σγe Γ t t, (A.24)

60 59 where σ is proportional to the variance of amplitude fluctuations and Γ is the inverse of the correlation time of the fluctuations. From Ref. [28], we find readily that the variances of the quadratures ˆX 1 and ˆX 2 of the phonon mode are then given by ( ˆX 1 (t)) 2 = e 2χ 0t e 2 t 0 dt δχ(t ) ( ˆX 1 (0)) 2, (A.25) ( ˆX 2 (t)) 2 = e 2χ 0t e 2 t 0 dt δχ(t ) ( ˆX 2 (0)) 2, (A.26) which, upon carrying out the appropriate ensemble averages and taking into account that the initial state of the cantilever is uncorrelated with the fluctuations, yields where ( ˆX 1 (t)) 2 = 1 4 e 2χ 0t+4f(t), (A.27) ( ˆX 2 (t)) 2 = 1 4 e2χ 0t+4f(t), (A.28) f(t) = χ2 0 σ [ e Γt + Γt 1 ]. (A.29) β 0 2 Γ Amplitude fluctuations reduce the rate at which the variances get squeezed. They do not however limit the maximum squeezing that can be generated. Moreover, Eq. (A.29) shows that the lengthening in timescale resulting from amplitude fluctuations scales as χ 2 0/ β 0 2, which is typically a small factor for a high amplitude drive, indicating that in contrast to phase fluctuations to which we turn next, amplitude fluctuations in the classical drive do not significantly affect the dynamics of the system. Finally, we note that if the squeezing factor is given by Eq. (A.18), we have to modify Eq. (A.22) with an additional factor of 2, and the function f(t) of Eq. (A.29) is thus multiplied by a factor of 4. A.3.2 Phase fluctuations Again following Ref. [28], we consider phase fluctuations δφ(t) about the relative phase φ, approximated as a phase diffusion process characterized by the correlation

61 60 functions δφ(t) = δ φ(t) = 0, δφ(t)δφ(t ) = D(t + t t t ), δ φ(t)δ φ(t ) = 2Dδ(t t ), (A.30) where D is the phase diffusion coefficient. From Ref. [29], (see details in Appendix A) we find the variances of the quadratures to be ( ˆX 1 (t)) 2 = 1 [cosh (2χt) exp( Dt/2) sinh (2χt) exp( 5Dt/3)], (A.31) 4 ( ˆX 2 (t)) 2 = 1 [cosh (2χt) exp( Dt/2) + sinh (2χt) exp( 5Dt/3)], (A.32) 4 which reduce to ( ˆX 1 (t)) 2 = 7 48 Dt exp(2χt) exp( 2χt), ( ˆX 2 (t)) 2 = 1 4 exp(2χt) + 7 Dt exp( 2χt), (A.33) 48 for Dt 1. Fig. A.3 depicts the squared quadrature ( ˆX 1 ) 2 as a function of the dimensionless time χt for several values of the diffusion coefficient D, illustrating the disappearance of squeezing for long enough times, an effect familiar from quantum optics. A.3.3 Clamping noise We now turn to a discussion of the effects of clamping noise on cantilever squeezing. We describe the thermal fluctuations coupled into the cantilever via a standard input-output formalism [27], resulting in the Heisenberg-Langevin equations ˆb = χˆb γ 2ˆb + γˆb in. (A.34)

62 61 2 X Χt Figure A.3: ( X 1 ) 2 as a function of the dimensionless time χt for various amount of phase fluctuations: no fluctuations(blue, solid); D = 10 5 χ (green, dashed); D = 10 4 χ (orange, dot-dashed); D = 10 3 χ (red, dotted). Here γ is the damping rate of the cantilever, and ˆb in a noise operator that accounts for thermal fluctuations, with ˆb in (t) = ˆb in (t) = 0, ˆb in (t)ˆb in (t ) = n th δ(t t ), ˆb in (t)ˆb in (t ) = ( n th + 1) δ(t t ), ˆb in (t)ˆb in (t ) = ˆb in (t)ˆb in (t ) = 0, (A.35) and n th = k B T/ ω c. This yields the quadrature evolution equations d dt ( ˆX 1 ) 2 = (2χ + γ)( ˆX 1 ) 2 + γ 4 (2 n th + 1), (A.36) d dt ( ˆX 2 ) 2 = (2χ γ)( ˆX 2 ) 2 + γ 4 (2 n th + 1), (A.37) and for a cantilever initially prepared in its ground state of center-of-mass motion, ( ˆX 1 (t)) 2 = (χ/γ) n th 2[1 + 2(χ/γ)] exp[ (2χ + γ)t] n th 4[1 + 2(χ/γ)], (A.38) ( ˆX 2 (t)) 2 = (χ/γ) n th 2[1 2(χ/γ)] exp[(2χ γ)t] n th 4[1 2(χ/γ)]. (A.39)

63 62 2 X Χt Figure A.4: ( X 1 ) 2 as a function of scaled time with thermal fluctuations for different damping constants and occupation numbers: no fluctuations (blue, solid); n th = 5, γ = 10 2 χ (green, dashed); n th = 10, γ = χ (orange, dot-dashed); and n th = 10, γ = χ (red, dotted). For t, ( ˆX 1 ) 2 approaches a steady state with reduced squeezing, as expected. Figure A.4 shows ( ˆX 1 ) 2 for several thermal occupations of the clamp and two values of the mechanical damping rate γ. Consider for example a cantilever with a quality factor Q m = ω c/2γ = 10 4, γ 2π 3.6 khz in a cryogenic environment at T = 20 mk. The phonon occupation number at ω c is n th 5.7. For a strong squeezing coupling constant of χ = 2π 2.8 MHz, this corresponds to a maximal squeezing of (27 db). A.4 Detection The detection of motional squeezing could be performed along the same lines as in the experiments of Ref. [17], which generated classical squeezing in a parametrically driven mechanical cantilever and characterized it by measuring the two quadratures of oscillations using a fiber-optical interferometer. A more ambitious approach that offers many potential advantages involves a full determination of the cantilever state [30], rather than its covariances only. One promising way to achieve this goal involves

64 63 first transferring that state to an optical field, where detection techniques developed in quantum optics can then be applied. Quantum state transfer has already been the subject of a number of studies [22, 23, 24, 31]. In particular, it is known that the two-mode beam-splitter interaction yields exact state transfer between two harmonic oscillators for appropriate interaction times and in the absence of dissipation, and that this type of interaction can be realized in principle in the optomechanical interaction between the harmonically bound end-mirror of a Fabry-Pérot and a near resonant intracavity light field. To implement that detection scheme, we expand the scheme of Fig. 1 to couple the cantilever to the intracavity light field of a resonator whose moving end-mirror is attached to the cantilever. This could be achieved e.g. with a cantilever of the type used in nanoscale magnetic resonance imaging [3], see Fig. 5. Combined with a fixed large mirror, this can form a high finesse optomechanical resonator. Alternatively, one could also consider an arrangement where the cantilever forms a moving plate of a capacitor in a driven microelectronic LC-circuit [9, 19]. In which case the coupling could actually be stronger, but one would be confronted with the lack of single-photon detectors in the microwave regime. In this section we consider two possible scenarios: In the first one the coupling to the optical field is present at all times, while in the second one a squeezed state of the cantilever is first prepared via magnetic dipole coupling to the classical tuning fork, and an optical field is subsequently turned on. We show that in both approaches the additional dissipation channel associated to the finite transmission of the Fabry- Pérot leads to a significant reduction in squeezing and imperfect state transfer.

65 64 Figure A.5: Schematics of the intracavity optical field optomechanically coupled to the magnetically driven cantilever. A.4.1 Continuous optical coupling With the additional optomechanical coupling of the cantilever to the optical field, the system Hamiltonian becomes H sys = H c + H p + H om + H m + H γ + H κ. (A.40) Here H c = ω 0 â â (A.41) describes the optical cavity mode â of frequency ω 0, H p = i η(â e iω Lt âe iω Lt ) (A.42) accounts for the driving of the cavity by an external field of frequency ω L at rate η = 2P κ/ ω L, with P the input power and κ the cavity linewidth, and H om = gâ â(ˆb + ˆb ) (A.43)

66 65 is the optomechanical coupling between the intracavity field and the cantilever mirror, with single-photon coupling frequency g. Finally, H γ and H κ describe the interaction of the mirror and cavity field to thermal reservoirs and account for dissipation at rates γ and κ, respectively, and H m is the Hamiltonian of the magnetically driven cantilever of the previous sections. Considering a coherent pump of constant amplitude red-detuned from the cavity resonance by ω c, we expand the amplitudes of both the cavity field and cantilever oscillations as the sum of their expectation value and quantum fluctuations, â = â + δâ E 0 + δâ, ˆb = ˆb + δˆb, (A.44) where δâ = δˆb = 0 and we neglect contributions from δâ δâ compared to those from â δâ + â δâ as usual, so that E 0 2 = â â â â. (Note that E 0 is dimensionless.) As is well known, this decomposition allows one to separate the optomechanical interaction Hamiltonian into a classical Kerr-type contribution proportional to E 0 2 and a beam-splitter interaction, so that the cantilever dynamics is approximately described by the Hamiltonian ( ) H = g E0δâδˆb + E 0 δâ δˆb + χ 2 ) (δˆb 2 e iφ + δˆb 2 e iφ + H κ + H γ, (A.45) where E 0 accounts consistently for the Kerr nonlinearity. In steady state, the main consequence of the Kerr effect is a slight shift in the cavity resonance, an effect that can lead under appropriate conditions to optical bistability [32]. Away from this multistable regime, the intracavity amplitude â E 0 of the radiation field is given approximately by (see Appendix B) E 0 = η κ/2 i [ + 2g 2 E 0 2 /ω c]. (A.46) In the interaction picture, after making the rotating wave approximation, and for

67 66 ω c =, the equations of motion for δâ and δˆb are, (see Eqs. (A.73) and (A.74)) δâ = κ 2 δâ + ige 0δˆb + κâ in, (A.47) δˆb = γ 2 δˆb + ige 0δâ χδˆb + γˆb in, (A.48) where we have assumed that the optical field is subject only to shot noise, â in (t)â in(t ) = 0, â in (t)â in (t ) = δ(t t ), â in (t)â in (t ) = 0. (A.49) We consider for concreteness the case E 0 = E 0 e iπ/2, which holds when ω c κ. The equations of motion for the variances of the position quadrature of the optical and phonon modes are then given by V = AV + B, (A.50) with A = B = V = κ 2g E 0 0 g E 0 (κ/2 + γ/2 + χ ) g E 0, 0 2g E 0 (2 χ + γ) κ/4 0, (γ/8)(3n th + 1) ( ˆX o,1 ) 2 ( ˆX om,1 ) 2 ( ˆX m,1 ) 2, (A.51) (A.52) (A.53) where ( ˆX i,1 ) 2, i {o, om, m} are the variance of the position quadrature of the optical field, the covariance of position quadratures of the optical field and cantilever, and the variance of the position quadrature of the cantilever, respectively.

68 67 The steady-state squeezing of the intracavity field follows from some elementary algebra, but its general form is cumbersome and we omit it here. In the physically relevant regime κ γ it reduces to the simple form ( ˆX o,1 ) 2 ( ) = 2r2 + s + 2s 2 4(2r 2 + s)(1 + 2s), (A.54) where r = g E 0 /κ is the classically amplified optomechanical coupling and s = χ/κ. Figure A.6 shows the steady-state squeezing of the optical field as a function of these parameters. It illustrates the monotonic increase in steady-state squeezing as s and r are increased, which is intuitively expected. However, this conclusion needs to be qualified by considering the steady-state limit of the cantilever squeezing. In the same limit, it is likewise easily obtained as ( ˆX m,1 ) 2 ( ) = r 2 2(2r 2 + s)(1 + 2s), (A.55) and is illustrated in Fig. A.7 as a function of r and s. In contrast to the situation for the optical field, we observe now that while increasing s increases the degree of squeezing, as would be expected, increasing the optomechanical coupling results in a decrease in squeezing. Indeed, while in the absence of optomechanical coupling the state of the cantilever mode would be almost perfectly squeezed, this ceases to be the case once the optical coupling is present. However, realistic experimental parameters yield enough squeezing transfer to the cavity field to be successfully detected. The physical origin of this behavior is that in addition to the magnetic squeezing interaction the cantilever is now also subjected to the beam-splitter interaction. It results in a transfer of squeezing to the optical field, where it is now exposed to a dominating decoherence channel associated with the cavity loss rate κ, normally much faster than the mechanical decay rate γ. This is more readily apparent in Fig. A.8, which clearly illustrates how cavity damping decreases the steady-state squeezing of both the cavity field and cantilever for fixed beam-splitter and squeezing coupling constants g E 0 and χ.

69 DX12 H L s 1 2 r Figure A.6: Steady-state squeezing of the cavity field as a function of the dimensionless coupling parameters r = g E0 /κ and s = χ/κ DX12 H L s 1 2 r Figure A.7: Steady state squeezing of the cantilever as a function of the dimensionless coupling parameters r = g E0 /κ and s = χ/κ.

70 69 X Κ Χ Figure A.8: Steady state squeezing of the cavity field (red, solid) and cantilever (blue, dashed) as a function of κ/χ in the resolved-side band regime for fixed χ and g E 0 /χ = 9. A.4.2 Delayed detection The take-home message of the previous section is that while the coupling of the cantilever to the optical cavity allows detection of the squeezing, it does it at the cost of opening up a fast decoherence channel. For reasonable experimental parameters the resulting loss in squeezing is much larger than the limit imposed by thermal losses in the mechanics. This suggests that a better scenario might involve first preparing the cantilever in a strongly squeezed state, and only subsequently coupling it to the optical field. On the other hand, a possible drawback of this approach is that it takes a time of the order of κ 1 to switch on the intracavity optical field, a time during which the optical decoherence channel is already open. As before, we decompose the cantilever phonon field and intracavity optical field as the sum of their expectation value and quantum fluctuations, see Eqs. (A.44), except that a(t) E 0 (t) is now an explicit function of time. The linearization process is questionable for very short times when the intracavity field is still extremely small. However, the optomechanical coupling is normally weak in that case, so that it should not qualitatively change the main features of the system dynamics.

71 70 2 X Χt Figure A.9: Squared quadratures of position for the cavity field (red, solid) and the mechanical oscillator (blue, dashed) as a function of scaled time. The optomechanical coupling is turned on at the dimensionless time χt = 1 and coherently builds up towards g E 0 /χ = 9, κ/χ = 10. As shown in Appendix B, Eqs. (A.73) and (A.74), the Heisenberg equations of motion for δâ(t) and δˆb(t) are approximately given by δâ = [i κ/2] δâ + ige 0 (t)(δˆb + δˆb ) + κâ in, (A.56) δˆb = [ iω c γ/2] δˆb + ig [ E0(t)δâ + E 0 (t)δâ ] 4iχ cos(ω f t + φ)(δˆb + δˆb ) + γˆb in, (A.57) and the evolution of E 0 (t) is determined by Eqs. (A.80)-(A.84). From these equations, it is possible to derive a closed set of equations for the first and second moments of the operators δâ and δˆb, see Appendix B. These equations could not be solved analytically, so this subsection presents selected numerical results that illustrate the main features of the system dynamics. Fig. A.9, which is for a relatively high-loss optical cavity that allows for a fast switching of the optical field, shows the coupled dynamics of the cantilever and optical fields in a situation where the cantilever was first prepared in a squeezed state, before the optical field is switched on at t 0 = χ 1. It illustrates a situation where squeezing transfer suffers from the broad decoherence channel of the optical

72 71 cavity. Thus the squeezing is not efficiently detectable in the cavity field. Note also that since the beam-splitter interaction frequency g E 0 κ in that example, the oscillatory coherent state transfer between the cantilever and the optical field is strongly suppressed, with the energy of the cantilever-field system being rapidly lost through the optical decay channel. A much more significant coherent exchange between the two subsystems requires either a stronger field amplitude E 0, or a slower decay of the light field, so that g E 0 κ. Such an example is illustrated in Fig. A.10, which shows the characteristic coherent state transfer between the phonon and photon field, as expected. One problem here is of course that by decreasing the cavity damping rate, one requires a longer time to turn on the light field to its final value E 0, thereby increasing the role of dissipation. Still, in this situation it is possible to achieve a reasonably good transfer of squeezing from the cantilever to the optical field. During the coherent state transfer, the maximum squeezing in the intracavity field occurs after half an exchange period. In Fig. A.11, the minimum values of the quadrature variance in the cavity field are plotted as a function of g E 0 /κ. As is expected, smaller cavity damping rate or lager coupling strength gives rise to stronger maximum squeezing in the cavity field. A.5 Conclusion In summary, we have presented a theoretical analysis of the motional squeezing of a cantilever magnetically coupled to a classical tuning fork via microscopic magnetic dipoles. We showed that this coupling can result in significant squeezing of a quadrature of motion of the cantilever if appropriately driven by a classical force, and found that the system is robust against various sources of noise, with phase noise in the driving of the classical driving tuning fork the dominant source of decoherence. We proposed a scheme for the detection of the effect based on state transfer to the intracavity field of an optical resonator with one end-mirror formed by the

73 72 2 X Χt Figure A.10: Squared quadratures of position in the cavity field (red, solid) and the mechanical oscillator (blue dashed) with strong coherent optomechanical coupling. Here g E 0 /χ = 9, κ/χ = 1. 2 X 1,m in g E 0 Κ Figure A.11: Minimum values of squared quadrature variance in the cavity field plotted as a function of g E 0 /κ.

74 73 oscillating cantilever. Challenges to the measurement process associated with the additional decoherence channel opened by the coupling to the optical resonator were discussed. It has recently been proposed that pulsed optomechanical configurations permit mapping the quantum state of optomechanical oscillators by using a sequence of appropriately shaped optical pulses separated in time by half a vibration period of the mechanical system [30]. This approach presents several advantages, the first one being that it allows for the use of low finesse optical resonators that permit fast switching and the second being that the oscillator is coupled to the optical dissipation channel for very short times only. Unfortunately, this scheme relies on the mechanical oscillator being subject only to free evolution between the light pulses, which is not the case here since the squeezing interaction is acting at all times. It is not clear how it could rapidly be switched off to make the pulsed detection scheme applicable. Future studies will consider whether adapting pulsed detection scenarios to the present situation may be possible. We will also consider a quantum treatment of the dynamics of the tuning fork, the interaction between these two subsystems on a quantum level, as well as further optomechanical coupling schemes to control and probe the system, including the use of multimode light fields. Acknowledgments This work was supported by the US National Science Foundation, the DARPA ORCHID and QuASAR programs, and the US Army Research Office.

75 74 A.6 Appendix A.6.1 Effect of phase fluctuations This appendix presents details of the evaluation of the effect of phase fluctuations for the case of an asymmetric setup. The symmetric case is analogous. In the presence of phase fluctuations, the Heisenberg evolution of the phonon annihilation operator ˆb(t) becomes d = χe dtˆb(t) iδφ(t)ˆb (t). (A.58) and the full dynamics of the system can be expressed in the general form dy ( dt = A + iδ φ(t)b ) Y, (A.59) where Y is the vector of bilinear operators ˆb 2, ˆb 2, and ˆb ˆb + ˆbˆb. Due to the form of the Heisenberg equations of motion for the problem at hand, the phase fluctuations δφ(t) can only be eliminated in one bilinear entry in Y, and it is therefore impossible to readily perform the statistical average over phase noise. As shown by Wodkievicz [29] this difficulty can be circumvented by solving two systems of matrix equations separately for two particular choices of Y, ˆb 2 and ˆb ˆb + ˆbˆb. For ˆb 2 we have Y = ˆb2 e iδφ(t) (ˆbˆb + ˆb ˆb) e 2iδφ(t)ˆb 2, 0 χ 0 A = 2χ 0 2χ, 0 χ B = 0 1 0, (A.60) (A.61) (A.62)

76 75 while for ˆb ˆb + ˆbˆb Y = ˆbˆb + ˆb ˆb e iδφ(t)ˆb2 e iδφ(t)ˆb 2, 0 2χ 2χ A = χ 0 0, χ B = (A.63) (A.64) (A.65) If the initial state is uncorrelated with the fluctuations, as is physically the case, this form of equations can be solved exactly to give so that Y (t) = e (A DB2 )t Y (0), (A.66) ˆb 2 1 sinh (2χt) exp[ 5Dt/3], 2 (A.67) ˆb ˆb + ˆbˆb cosh (2χt) exp[ Dt/2], (A.68) where we have assumed that D χ for simplicity. A.6.2 Equations of motion for optically coupled system In the rotating frame at frequency ω L the system Hamiltonian (A.40) is H = â â gâ â(ˆb + ˆb ) + i η(â â) + ω cˆb ˆb + χ cos(ω f t + φ)(ˆb + ˆb ) 2 + H γ + H κ, (A.69) where = ω L ω 0, (A.70)

77 76 and the equations of motion for â and ˆb are â = i â + ig(ˆb + ˆb )â κ 2 â + η + κâ in, (A.71) ˆb = iω cˆb + igâ â 2iχ cos(ω f t + φ)(ˆb + ˆb ) γ 2ˆb + γˆb in, (A.72) where φ is the phase difference between the tuning fork and cantilever. Introducing δâ = â â and δˆb = ˆb ˆb yields then for δâ and δˆb the linearized equations of motion δâ [i κ/2] δâ + ig ˆb + ˆb δâ + ig â (δˆb + δˆb ) + κâ in [i κ/2] δâ + ig â (δˆb + δˆb ) + κâ in, (A.73) δˆb [ iω c γ/2] δˆb + ig( â δâ + â δâ ) 2iχ cos(ω f t + φ)(δˆb + δˆb ) + γˆb in. (A.74) The second, approximate form of Eq. (A.73) results from the fact that the mean phonon number is of order unity, which is much smaller than the mean number of intractivity photons ( ). Under these conditions one can neglect the term ig ˆb + ˆb δâ in that equation. From these approximate linearized equations we easily obtain the equations of motion for expectation values of the quadrature operators of â and ˆb, ˆXa = Ŷa κ 2 ˆX a 2g ˆX b Ŷ a + η, (A.75) Ŷ a = ˆX a κ 2 Ŷa + 2g ˆX b ˆXa, (A.76) ˆXb = ω c Ŷb γ 2 ˆX b, (A.77) Ŷ b = [ω c + 4χ cos(ω f t + φ)] ˆX b + g ˆN a γ 2 Ŷb, (A.78) ˆNa = 2η ˆX a κ ˆN a, (A.79) where ˆX a = (â + â )/2, Ŷa = (â â )/2i, and ˆN a is the intracavity photon number operator. In the regime where ω c χ γ and ˆN a E 0 2 1, the change in the

78 77 classical component of the intracavity field due the cantilever oscillations remains very small. It can be ignored in determining the dynamics of the beam-splitter coupling constant g E 0 (t), which is then governed by the approximate equations of motion ˆXa = Ŷa κ 2 ˆX a 2g ˆX b Ŷ a + η, (A.80) Ŷ a = ˆX a κ 2 Ŷa + 2g ˆX b ˆXa, (A.81) ˆXb = ω c Ŷb, (A.82) Ŷ b = ω c ˆX b + g E 0 2, (A.83) E 0 2 = 2η ˆX a κ E 0 2. (A.84) These equations yield the time evolution of â(t), as well as its classical steady-state value. We find, upon factorizing ˆX b Ŷ a, E 0 ss = η κ/2 i [ eff + 2g 2 E 0 2 ss /ω c]. (A.85) In the rotating frame at frequency ω L, the position quadratures of both cantilever and cavity field are defined as ˆX 1,o = 1 2 (δâe i t + δâ e i t ), ˆX 1,m = 1 2 (δˆbe iω c t + δˆb e iω c t ), (A.86) (A.87) In order to calculate their variances we need to have the expectation value of second moments of the fluctuations. Taking quantum averages of the equations of motion

79 78 for these quantities results in the closed set of equations d dt δâ2 = 2i δâ 2 κ δâ 2 + 2ig â δâδˆb + δâδˆb, (A.88) d [ ] dt δâ δâ = ig â δâδˆb + δâδˆb â δâ δˆb + δâ δˆb κ δâ δâ, d dt δˆb 2 = 2iω c δˆb 2 γ δˆb 2 + 2ig d dt δˆb δˆb = ig [ ] â δâδˆb + â δâ δˆb 2iχ cos(ω f t + φ) 2δˆb 2 + δˆb δˆb + δˆbδˆb, [ ] â δâδˆb δâδˆb + â δâ δˆb δâ δˆb γ δˆb δˆb + 2iχ cos(ω f t + φ) δˆb 2 δˆb 2 + γn th, d dt δâδˆb = i( ω c) δâδˆb [(κ + γ)/2] δâδˆb [ ] +ig â δˆb 2 + δˆb δˆb + â δâ 2 + â δâδâ (A.89) (A.90) (A.91) 2iχ cos(ω f t + φ) δâδˆb + δâδˆb, d dt δâδˆb = i( + ω c) δâδˆb [(κ + γ)/2] δâδˆb [ ] +ig â δˆb 2 + δˆbδˆb â δâ 2 â δâδâ +2iχ cos(ω f t + φ) δâδˆb + δâδˆb. (A.92) (A.93) These are the equations that are solved numerically to obtain the figures of section IV.B.

80 79 REFERENCES [1] G. Binnig, C. F. Quate and Ch. Gerber, Phys. Rev. Lett. 56, 930 (1986). [2] M. F. Bocko and R. Onofrio, Rev. Mod. Phys. 68, 755 (1996). [3] C. L. Degen et al., Proc. Natl. Acad. Sci. 106, 1313 (2009). [4] D. Rugar et al., Nature 430, 329 (2004). [5] T. J. Kippenberg and K. J. Vahala, Science 321, 1172 (2008); [6] M. Aspelmeyer et al., JOSA B 27, A189 (2010) [7] F. Marquart and S. M. Girvin, Physics 2, 40 (2009). [8] A. D. O Connell et al., Nature 464, 697 (2010). [9] J.D.Teufel et al., Nature 475, 359 (2011). [10] J. Chan et al., Nature 478, 89 (2011); A. H. Safavi-Naeini et al., arxiv: v1 (2011). [11] V. B. Braginskii and Y. I. Vorontsov, Sov. Phys. Usp. 17, 644 (1975). [12] V. B. Braginskii, Y. I. Vorontsov and F. Y. Khalili, JETP Letters, 27, 276 (1978). [13] V. B. Braginskii, Y. I. Vorontsov and F. Y. Khalili, Sov. Phys. Usp. 46, 705 (1977). [14] V. B. Braginskii and F. Y. Khalili, Quantum Measurement, Cambridge Univ. Press, Cambridge (1992).

81 80 [15] C. Caves et al., Rev. Mod. Phys. 52, 341 (1980). [16] M. F. Bocko and W. W. Johnson, Phys. Rev. Lett, 48, 1371 (1982). [17] D. Rugar et al., Phys. Rev. Lett. 67, 699 (1991). [18] J. Suh et al., Nano Lett. 10, 3990 (2010). [19] M. J. Woolley et al., Phys. Rev. A 78, (2008). [20] V. B. Braginsky, Y. I. Vorontsov and K. S. Thorne, Science 209, 547 (1980). [21] A. A. Clerk, F. Marquardt and K. Jacobs, New J. Phys. 10, (2008). [22] J. Zhang, K. Peng and S. Braunstein, Phys. Rev. A 68, (2003). [23] L. Tian and H. Wang, Phys. Rev. A. 82, (2010). [24] F. Khalili et al., Phys. Rev. Lett. 105, (2010). [25] P. Meystre and M. Sargent III, Elements of Quantum Optics, 4th Edition, Springer Verlag, Berlin, (2007) [26] M. O. Scully and M. Zubairy, Quantum Optics, Cambridge University Press, Cambridge (2008). [27] D. F. Walls and G. J. Milburn, Quantum Optics 2nd Edition, Springer Verlag, Berlin (2007). [28] K. Wodkiewicz and M.S. Zubairy, Phys. Rev. A, 27,2003 (1983). [29] K. Wodkiewicz, J. Math. Phys.20, 1 (1979). [30] M. R. Vanner et al., Proc. Natl. Acad. Sci. 108, (2011). [31] K. Jähne et al., Phys. Rev. A 79, (2009). [32] A. Dorsel et al., Phys. Rev. Lett. 51, 1550 (1983).

82 81 APPENDIX B DYNAMIC STABILIZATION OF AN OPTOMECHANICAL OSCILLATOR ABSTRACT Quantum optomechanics offers the potential to investigate quantum effects in macroscopic quantum systems in extremely well controlled experiments, and in particular to investigate in more detail than previously possible the elusive quantumclassical interface. In this chapter we discuss one such situation, the dynamic stabilization of a mechanical system such as an inverted pendulum. The specific example that we study is the dynamic stabilization of a membrane in the middle mechanical oscillator coupled to a cavity field via a quadratic optomechanical interaction and cavity damping is the dominant source of dissipation. We show that the mechanical oscillator can be dynamically stabilized by a temporal modulation of the radiation pressure force. We investigate this system both in the classical and quantum regimes and identify key differences between these two situations. B.1 Introduction Dynamic stabilization is the process in which an object, unstable with its static potential, is trapped harmonically due to the influence of a high-frequency oscillating force [1]. Dynamic stabilization was first introduced by Kapitza for an inverted classical pendulum stabilized by rapidly oscillating external perturbations [2]. Specifically, Kapitza s inverted pendulum was stabilized by an oscillating pivot in a vertical

83 82 direction. It can be described by the Newton s equation of motion, ] θ = sin θ [ω 20 AΩ2 cos(ωt), (B.1) l where θ is the polar angle from the vertical line and A and Ω are the amplitude and frequency of the vibration of the pivot, respectively. The proper frequency of the pendulum is ω 0 = g/l, where g is the gravitational acceleration and l is the length of the pendulum. The physics underlying Kapitza s pendulum is that large and rapid oscillations of the pivot compared to the proper frequency of the pendulum allow the force acting on the pendulum to alternate between an attractive force and a repulsive force in time, resulting in a net stabilizing force for appropriate conditions [3]. Dynamic stabilization of a quantum system driven by a rapidly oscillating perturbation has been extensively studied in several papers [4, 5, 6]. It has been proposed for the control of a quantum system in the context of atom optics, for example in novel optical trapping [7] and in the stabilization of a Bose-Einstein condensate (BEC) [8, 9, 10, 11]. Such stabilizing mechanisms have also found applications in trapping ions in electromagnetic fields [12], focusing of charged particles in a synchrotron [3, 13], stabilizing spin-1 BEC [14], and the control of the superfluid-mott insulator phase transition [15]. Although Kapitza s pendulum and ultracold atoms have been successful in the realization of dynamic stabilization in the classical and quantum regimes respectively, they are limited in their own regimes and fail to demonstrate dynamic stabilization at the boundary of the classical and quantum regimes. Cavity optomechanics, a research area exploring mechanical degrees of freedom coupled to electromagnetic fields inside optical or microwave cavities, involves a variety of experimental setups in which the mass of a mechanical oscillator ranges from several attograms to kilograms [16, 17, 18, 19, 20, 21, 22]. Recent experimental progress has demonstrated cooling a macroscopic mechanical oscillator to its

84 83 Figure B.1: Membrane-in-the-middle geometry. Two cavity fields tunnel through a membrane located at the center of a fixed cavity and interact with the membrane in opposite directions, resulting in a quadratic optomechanical interaction. motional ground state [23, 24, 25, 26], allowing to explore the quantum nature of massive objects [27, 28]. Cavity optomechanics thus paves the way to investigate dynamic stabilization of a mechanical object in either classical or quantum regime as well as at the boundary of the classical and quantum regimes. In this chapter, we consider an optomechanical system in which a mechanical oscillator is coupled to a cavity field via a quadratic optomechanical interaction. This situation can be realized e.g. in an ensemble of ultracold atoms trapped at the extrema of an optical lattice in a high-finesse cavity [29], a BEC trapped in such a cavity [30], and the so-called membrane-in-the-middle geometry [31, 32], see Fig. B.1. We have shown in a previous paper [33] that a mechanical oscillator can be unstable if it is coupled to a cavity field via a quadratic optomechanical interaction with a negative coupling coefficient. Starting from this unstable configuration, we propose a stabilizing scheme in which the radiation pressure force associated with the cavity field is modulated and explore features of dynamic stabilization of the mechanical motion in both classical and quantum regimes. In particular, we derive a time-averaged potential and demonstrate that the mechanical oscillator can be stabilized within a certain parameter regime. We also show the classical and quantum

85 84 dynamics of the mechanical oscillator under the effects of the oscillating radiation pressure force as well as dissipation. Section B.2 introduces our model system and the unstable configuration. A scheme for dynamic stabilization of the classical system based on a time-averaged potential is proposed in Sec. B.3, and Sec. B.4 provides simulations of the scheme using the full time-dependent potential for the mechanics. A master equation governing the quantum dynamics of the mechanical oscillator is obtained in Sec. B.5, and Sec. B.6 describes numerical simulations to elucidate the features of quantum dynamic stabilization in comparison to the classical case. Summary and outlook are presented in Section B.7. B.2 Classical dynamics We consider an optomechanical system in which a mechanical mode of effective mass m and frequency ω m is coupled to a single cavity field mode of frequency ω c via a quadratic optomechanical interaction. The Hamiltonian describing that system is H = H o + H m + H om + H loss, (B.2) where H o = ω c â â + i (ηe iω Ltâ h.c.) (B.3) describes the cavity field driven by an external field of frequency ω L with a rate η and â denotes the bosonic annihilation operator for the cavity field, with [â, â ] = 1. The mechanical Hamiltonian is H m = ˆp2 2m + U m(ˆx), (B.4) where ˆx and ˆp are the position and momentum operators for the mechanics with the commutation relation [ˆx, ˆp] = i, and U m (ˆx) = mω2 m ˆx 2 2 (B.5)

86 85 is the potential for the bare mechanical oscillator. The quadratic optomechanical interaction Hamiltonian is H om = g (2) 0 â âˆx 2, (B.6) where g (2) 0 is the quadratic optomechanical coupling constant. It is assumed throughout this chapter that g (2) 0 is negative-valued and its realization is studied in detail in our previous paper [33]. Finally, H loss represents the interaction of the optomechanical system with its reservoir and accounts for cavity and mechanical dissipation with rates κ and γ, respectively. In a frame rotating at the laser frequency ω L, the classical equations of motion for the mechanical position x = ˆx and momentum p = ˆp and the dimensionless intracavity field amplitude a = â are readily found from the operator Heisenberg- Langevin equations of motion as where c ẋ = p m, (B.7) ṗ = mωmx 2 2 g (2) 0 a 2 x γp, (B.8) [ ȧ = i c ig (2) 0 x 2 κ ] a + η, (B.9) 2 = ω L ω c is the pump detuning from the cavity resonance, κ is the phenomenological decay rate of the cavity field and γ the mechanical damping rate. In the physically relevant regime κ ω m the cavity field adiabatically follows the mechanical mode, and a 2 2P 0 κ/( ω L ) [ c g (2) 0 x 2 ] 2 + κ 2 /4, (B.10) where P 0 = ω L η 2 /(2κ) is the input power of the cavity. Substituting this expression into Eq. (B.8) then gives ṗ = mω 2 mx 4g (2) 0 P 0 κ/ω L x γp. [ c g (2) 0 x 2 ] 2 + κ2 4 (B.11) In the absence of mechanical dissipation, γ = 0, the above equations of motion for the mechanical system can be put in the canonical form ẋ = p/m, ṗ = Hs x, with

87 86 Hamiltonian H s = p2 2m + U s(x) (B.12) and static mechanical potential [ ] U s (x) = U m (x) 4P 0 c g (2) 0 x 2 arctan. (B.13) ω L κ/2 This highlights the key lesson that adiabatic elimination of the cavity field involves concomitant replacement of the bare mechanical potential U m (x) by the static mechanical potential U s (x) in the dynamics of the mechanical mode. Furthermore transient cases may also be treated, for example, for an input power P 0 (t), giving a time-dependent potential U(x, t) generalizing the static mechanical potential. In general [H m + H om ] should be replaced by the reduced mechanical Hamiltonian [ ] p 2 H r = + U(x, t), (B.14) 2m or its operator generalization, following adiabatic elimination of the cavity field. We shall use this in the next section and also in the quantum theory. The static potential U s (x) exhibits a single minimum at x = 0 if P 0 is less than the critical power ] mωm 2 P c = [ 2c + κ2, (B.15) 4 g (2) 0 κ/ω L 4 whereas a symmetric double-well potential centered on x = 0 results if the input power is greater than P c, see Fig. B.2. In that case the repulsive radiation pressure acting on the oscillator centered at x = 0 is greater than the mechanical restoring force. For small displacement from the origin x = 0, U s (x) can be approximated as an inverted oscillator of frequency [33] ω 0 = ω2 m + 4P 0/ω L g (2) 0 κ m 2 c + κ 2 /4, (B.16) rendering the origin unstable. This is the situation that we consider in the following.

88 87 U s E x x Figure B.2: Static mechanical potential for an input power greater than the critical power P c. The parameters are κ/ω m = 10, c /ω m = 0, g (2) 0 x 2 0/ω m = 0.01, P 0 /(E 0 ω L ) = For our parameters, the critical pumping power is P c /(E 0 ω L ) = 62.5 and the effective mechanical frequency is ω 0 /ω m = Here and in all following figures we measure the position, momentum and energy of the mechanical mode in units of the natural length x 0 = /mω m, momentum p 0 = m ω m and energy E 0 = ω m, respectively. B.3 Dynamic stabilization In this section we investigate a scheme to stabilize a mechanical oscillator at the unstable center illustrated in Fig. B.2. Recalling that Kapitza s pendulum can be stabilized by a rapidly oscillating force, here we propose rapidly modulating the input power P in (t) below and above the critical power: The potential at the center then concomitantly oscillates between a maximum and minimum and the force acting around the center oscillates between repulsive and attractive. We consider specifically a modulation of the form P in (t) = P 0 A sin (Ωt), A < P 0, (B.17) where Ω and A are the frequency and amplitude of the modulation, respectively. Throughout this chapter we choose P 0 > P c so that the mean input power generates a symmetric double-well potential with an unstable center for the mechanics in the

89 88 absence of the modulation. The amplitude of the modulation is chosen positive with the constraint A P 0 so that the input power remains non-negative for all times. The input power (B.17) yields a time-dependent potential U(x, t) = U s (x) + u(x, t) = mω2 m 2 x2 4P 0 + 4A ω L arctan arctan ω L [ c g (2) 0 x 2 κ/2 [ c g (2) 0 x 2 κ/2 ] sin(ωt), ] (B.18) where the second line corresponds to the static portion U s (x) of the potential, which alone produces an unstable centered mechanical mode, and the third line gives its oscillating portion u(x, t). Fig. B.3 shows the static potential U s (x) (solid blue line) along with the time-dependent potential at the times at which it reaches the maximum and minimum powers P in = P 0 + A (red dotted line) P in = P 0 A (green dashed line). Note that the net force at the center x = 0 is attractive for the mechanical potential corresponding to the minimum power and repulsive for the maximum power. The alternating sign of the net force acting at the center is what raises the possibility of dynamic stabilization of the mechanical motion. To develop a physical understanding of how the mechanical mode can be stabilized we derive a time-averaged mechanical potential and identify the parameter regime for the modulation to realize dynamic stabilization of the mechanical motion. The potential (B.18) yields for the mechanical mode the Newton s equation of motion mẍ = F s (x) + f(x, t) = du s(x) dx Here the static force F s (x) which has only spatial dependence, is F s (x) = mω 2 mx u(x, t). (B.19) x 4g (2) 0 P 0 κ/ω L x, (B.20) [ c g (2) 0 x 2 ] 2 + κ2 4 and the oscillating radiation pressure force f(x, t) due to the modulation of the

90 89 U E x x Figure B.3: Time-dependent potential for the mechanics at t = 0, π/ω (blue solid line), t = π/(2ω) (green dashed line), and t = 3π/(2Ω) (red dotted line). Here, P 0 /(E 0 ω L ) = 66.0, A/P 0 = 1, Ω/ω m = 1.8 and the other parameters are the same as those in Fig. B.2. input power is f(x, t) = (4g(2) 0 Aκ/ω L )x [ c g (2) 0 x 2 ] 2 + κ2 4 and is separable into spatial and temporal parts. sin(ωt) = A(x) sin(ωt) (B.21) Following the treatment in Refs. [1, 2] we write the mechanical coordinate as a sum of slow and fast varying variables x = x + ζ, (B.22) where the bar denotes a time average over one oscillation cycle with a period of T = 2π/Ω. Here x is a slowly varying variable with respect to T and describes the macromotion of the mechanics, and ζ is the rapidly oscillating variable with zero mean that describes the micromotion of the mechanics. Substituting Eq. (B.22) into Eq. (B.19) and expanding the right-hand-side of Eq. (B.19) for the rapidly oscillating component ζ to first-order, we obtain m x + m ζ = F s ( x) + ζ df s dx + f( x, t) + ζ f x= x x. x= x (B.23)

91 90 We separate the fast varying portion of this equation as m ζ f( x, t), (B.24) where we also invoked the assumption x ζ, see Refs. [1, 2]. Substituting the formal solution for the rapidly oscillating variable ζ into Eq. (B.23) and taking the time-average over the period T yields then the equation of motion for the slowly varying variable m x = F s ( x) 1 mω 2 T t t T dt f( x, t ) f( x, t ) x = dū(x) d x. (B.25) For simplicity in notation we hereafter replace x with x with the clear understanding that in the time-averaged theory x refers to the slow portion of the mechanical motion. The time-averaged potential governing the macromotion of the mechanics is then given by [ ] Ū(x) = mω2 m 2 x2 4P 0 c g (2) 0 x 2 arctan ω L κ/2 ( ) [ A 2 2g (2) 2 0 κ/ω L + x], (B.26) mω 2 ( c g (2) 0 x 2 ) 2 + κ2 4 where the first two terms on the right-hand-side coincide with the static potential U s (x) and the last term arises from the second term on the right-hand-side of Eq. (B.25) and accounts for the effects of the modulation on the macromotion. The factor A 2 /(mω 2 ) multiplying the last term in Eq. (B.26) implies that a large amplitude of the modulation A can lead to an enhanced effect on the macromotion, while a high frequency Ω tends to diminish the effect of the modulation on the macromotion. For this reason one might be tempted to decrease the modulation frequency to enhance the effect of the modulation. However, in the derivation of the time-averaged potential we assume that the macromotion is much slower than the

92 91 micromotion. The essence of this assumption is basically the same as the adiabatic elimination of a fast variable in quantum optics [34]. This assumption allows one to adiabatically eliminate the rapidly varying variable ζ on a time scale of T = 2π/Ω, resulting in the time-averaged potential Ū(x) governing only the macromotion of the mechanics. Therefore, the modulation frequency Ω must exceed ω 0, the effective frequency of the mechanical motion at the center. If this is not the case the micromotion cannot be separated from the macromotion, and hence the description of the mechanical motion in terms of the time-averaged potential breaks down. Thus A 2 /(mω 2 ) must be large enough that the modulation makes significant contributions on the macromotion of the mechanics even in the high frequency regime. Fig. B.4 shows the static potential (solid blue line) as well as the time-averaged potentials for three different values of the modulation amplitude A with fixed modulation frequency Ω. It illustrates that for a small modulation amplitude the timeaveraged potential remains a double-well potential of reduced depth which retains a local maximum at the center (green dashed line). For large enough modulation amplitude, however, the time-averaged potential develops a local minimum at the center (orange dot-dashed and red dotted lines), indicative dynamic stabilization of the mechanical oscillator, the frequency of the time-averaged stabilized potential increasing with A. The dynamic stability at the equilibrium position x = 0 can be evaluated using the curvature of the potential D = d2 Ū(x) dx 2, x=0 (B.27) where Ū(x) is the time-averaged potential. Positive D ensures that small mechanical oscillations around x = 0 are confined to the trap leading to stability. In contrast negative D indicates that the time-averaged potential acquires a local maximum at x = 0, rendering the mechanical mode unstable. Based on this criterion Fig. B.5 shows the boundary between the unstable and stable regimes (blue dashed line)

93 92 U E x x 0 2 Figure B.4: Time-averaged potential for several values of the modulation amplitude A with a fixed modulation frequency Ω/ω m = 1.8, A/P 0 = 0.48 (green dashed line), A/P 0 = 0.62 (orange dot-dashed line), A/P 0 = 1 (red dotted line) along with the static potential (blue solid line). Other parameters as in Fig B.3. in the (Ω/ω m, A/P 0 ) plane. The mechanical mode at the center is unstable in the regime below the boundary (unshaded region) and becomes stable in the regime above the boundary (blue-colored region). B.4 Classical simulations This section presents selected simulations of the classical dynamics of the system for the driving frequency Ω/ω m = 1.8 (Ω/ω 0 = 7.5), for which the adiabaticity condition Ω ω 0 is fulfilled and (A/P 0 ) = (a) 0, (b) 0.22, (c) 0.56, (d) 0.62, and (e) 1, see Fig. B.5. For the parameters of that figure time-averaged potential develops a minimum at x = 0 for (A/P 0 ) > 0.56, so that cases (a)-(c) are expected to be classically unstable, whereas cases (d)-(e) are stable according to the time-averaged potential. However, by construction that potential captures only the low-frequency mechanical macromotion resulting from the modulation, but not the high-frequency micromotion. To explore the full classical dynamics we now include the time-dependent potential U(x, t) with and without mechanical dissipation and

94 93 Figure B.5: Stability domain of an optomechanical oscillator located at the center, for an input power modulated according to Eq. (B.17), for the parameters of Fig. B.3. The oscillator is dynamically stable in the region above the dashed blue line (blue-colored region). The black dots denote the points used in the simulations of Figs. B.7-B.10, and label these points in the figures. In all cases Ω/ω m = 1.8, and (A/P 0 ) = (a) 0, (b) 0.22, (c) 0.56, (d) 0.62, and (e) 1. compare with expectations for dynamic stabilization based on the time-averaged potential. We start from Newton s equation of motion for the mechanics including the time-dependent force, dissipation, and associated thermal noise mẍ = mωmx 2 4g(2) 0 P in (t)κ/ω L x γp + ξ, [ c g (2) 0 x 2 ] 2 + κ2 4 (B.28) where P in (t) is given by Eq. (B.17), the classical thermal fluctuations possess a two-time correlation function [35] ξ(t)ξ(t ) = 2mγk B T δ(t t ), (B.29) where k B is the Boltzmann constant, and T is the temperature of the heat bath of the mechanical oscillator. In order to explore the full range of dynamics of the mechanical oscillator we consider an ensemble of initial conditions for the position and momentum chosen from Gaussian probability distributions with standard devi-

95 94 P E E E E 0 Figure B.6: Initial energy distribution of the mechanical oscillator (blue solid line) with the mean energy of E = E 0 /2 (gray dashed line). ations σ x = x 0 / 2 and σ p = p 0 / 2, respectively. The joint probability density for the initial positions and momenta is given by and the associated energy distribution reads P (x, p, t = 0) = 1 πx 0 p 0 e x2 /x2 0 e p2 /p2 0, (B.30) with mean energy E = E 0 /2, as shown in Fig. B.6. P (E/E 0 ) = 2e 2E/E 0, (B.31) B.4.1 Undamped case We first consider the situation where the mechanical oscillator has a sufficiently high quality factor Q m = ω m /γ that dissipation can be neglected over time scales of interest. This situation allows the effects of the modulation of the radiation pressure force on the classical mechanical oscillator to be highlighted. For a given set of parameters we generated 1000 trajectories from a random sample of initial positions and momenta generated from the Gaussian probability density (B.30). We display them in same color-coded two-dimensional plot in such a way that the more blue (the

96 95 darker) a region, the more trajectories cross that region: The resulting plots may then be viewed as spatial probability densities (with appropriate normalization.) Fig. B.7 summarize results of such simulations for the parameters marked by black dots in Fig. B.5 and labeled (a-e) in the corresponding figure caption. Fig. B.7(a) shows the dynamics of the mechanics in the static double-well potential, indicating that the mechanics is neither localized at the center nor bound in one of the local potential wells. This arises since the initial mean energy E 0 /2 of the mechanics is higher than the depth of the static double-well potential. The relatively high probability density at the center arises due to the fact that there is a local potential maximum there and the trajectories therefore tend to slow down and linger in the vicinity of the center. As indicated in Fig. B.5 dynamic stabilization should arise for modulation amplitudes (A/P 0 ) > This is borne out, to an extent limited by the impact of micromotion as discussed below, by a comparison of Figs. B.7(b, c) and Figs. B.7(d, e). In the first two cases the modulation amplitude is not large enough to trap the mechanics close to the center. Instead the trajectories explore the spatial extent of the double-well potential spanning the energy range from the potential minimum up to an additional energy of E 0 /2. In contrast, in Figs. B.7(d, e), which are for (A/P 0 ) > 0.56, one can discern the onset of the predicted dynamic stabilization of the probability around the center. Importantly, however, the micromotion makes the sharp unstable to stable transition predicted by the time-averaged potential much less evident. The transition to stability is now much more progressive, with the trapping becoming gradually more pronounced as the modulation amplitude is increased past (A/P 0 ) = Still with this important caveat these results validate the concept of dynamic stabilization of an optomechanical oscillator in the absence of mechanical dissipation.

97 96 x/x 0 (a) (b) (c) (d) (e) Figure B.7: Trajectories of the classical mechanical oscillator with initial conditions generated at random from the Gaussian distribution function (B.30) with σ x = x 0 / 2, σ p = p 0 / 2 and Ω/ω m = 1.8. The curves follow the labeling of Fig. B.5 with values of (A/P 0 ) given by (a) 0, (b) 0.22, (c) 0.56, (d) 0.62, and (e) 1. Here, γ/ω m = 10 6, T = 0, κ/ω m = 10, c /ω m = 0, g (2) 0 x 2 0/ω m = 0.01, P 0 /(E 0 ω L ) = 66. ω m t B.4.2 Damped case To explore the effects of mechanical dissipation, Fig. B.8 repeats the same simulations as in Fig. B.7 but now including damping of the mechanical oscillator via coupling to a reservoir at zero temperature. For the simulations in Figs. B.8(b, c) the modulation amplitude (A/P 0 ) 0.56 and each trajectory ultimately gets trapped in one or the other of the wells of the time-averaged double-well potential, plus some high-frequency micromotion. The amplitude of the micromotion is quite large due to fact that the amplitude of the time-dependent radiation pressure force

98 97 x/x 0 (a) (b) (c) (d) (e) Figure B.8: Trajectories of the classical mechanical oscillator in the presence of a viscous damping force for a reservoir at zero temperature. Here, γ/ω m = , T = 0. Other parameters as in Fig. B.7. ω m t appearing in Eq. (B.21) A(x) = depends on the mechanical displacement x. 4g (2) 0 Aκ/ω L x, (B.32) [ c g (2) 0 x 2 ] 2 + κ 2 /4 It is small near the center but can become significant around the minima of the double-well potential. Turning next to the cases with (A/P 0 ) > 0.56 shown in Figs. B.8(d, e), the trajectories damp into the center consistent with the idea of dynamic stabilization. These results show that the concept of dynamic stabilization survives the inclusion of mechanical dissipation, the stable-unstable transition following closely the predictions based on the time-averaged potential.

99 98 B.5 Quantum dynamics To properly account for fluctuations and noise in the quantum regime, we find it convenient to work in the Schrödinger picture, where the combined field-mechanics system is described by the master equation [34] ρ(t) = i [H, ρ(t)] + (L m + L o )ρ(t), (B.33) where H = H o + H m + H om, see Eqs. (B.3)-(B.6), L m and L o are standard Lindblad forms that describe the dissipation of the mechanics and the cavity field due to the coupling to their respective reservoirs, which are assumed for simplicity to be at zero temperature T = 0. For fast dissipation of the optical field, κ ω m we assume that decoherence prohibits the build up of quantum correlation between the two subsystems, so that the total density operator can be factorized as ρ(t) ρ m (t) ρ o (t). (B.34) By taking partial traces over the mechanics and the optical field, it is then possible to get reduced master equations for the two subsystems, ρ m = i [H m + g (2) 0 â â ˆx 2, ρ m ] + L m ρ m, (B.35) ρ o = i [H o + g (2) 0 â â ˆx 2, ρ o ] + L o ρ o. (B.36) Note that because of the approximate absence of correlations between the two subsystems it is only the mean photon number that appears in the master equation for the reduced density operator of the mechanics, and likewise only the expectation value ˆx 2 that appears in the master equation for the field mode. This indicates that the frequency of the field mode is shifted by the optomechanical interaction from ω c to ω c + g (2) 0 ˆx 2, in keeping with expectations from the classical analysis. The next step is to adiabatically eliminate the optical field. Building upon the lessons learned from the classical case this entails replacing [H m + H om ] =

100 99 [H m + g (2) 0 â â ˆx 2 ] in Eq. (B.35) with the reduced Hamiltonian [ ] ˆp 2 H r = + U(ˆx, t), (B.37) 2m see the discussion surrounding Eq. (B.14). To gain some insight into how this replacement manifests itself in the quantum theory it is instructive to look at the quantum averaged reduced potential U(ˆx, t), with U(ˆx, t) given by Eq. (B.18) with x ˆx. Then, consistent with the fact that only ˆx 2 appears in the master equation (B.36) for the field mode, we factorize products ˆx 2n = ˆx 2 n, n = 0, 1, 2,..., yielding the result U(ˆx, t) = U( ˆx 2, t). Given that we consider a potential that is symmetric around the origin, and taking ˆx = 0 for a symmetric initial condition, then x = ˆx 2, and the quantum averaged reduced potential U( x, t) is the same as the classical one with the classical mechanical displacement replaced by the root-mean-square displacement. In this way the properties of the classical reduced potential also manifest themselves in the quantum theory, meaning that quantum dynamic stabilization is also a possibility. Bringing the above results together yields the effective master equation for the mechanics ρ m = i [ ] ˆp 2 2m + U(ˆx, t), ρ m + L m ρ m. (B.38) Then expanding the density matrix for the mechanics in the position representation as ρ m (t) = dx dx ρ m (x, x, t) x x, and substituting into the master equation (B.38) yields the equation of motion [ ( ) i t ρ m(x, x 2, t) = 2m x 2 2 x 2 ( x x + x i (U(x, t) U(x, t)) + γ 2 + γ ( 4mω m x + ) 2 γmω m x 4 (x x ) 2 ) x + 1 ] ρ m (x, x, t). (B.39) (B.40)

101 100 x/x 0 (a) (b) (c) (d) (e) Figure B.9: Time evolution of the spatial probability distribution of the quantum mechanical oscillator initially prepared in the ground state of the bare harmonic trapping potential of frequency ω m. Here, the parameters employed and plot labels are identical to those used in Fig. B.7. ω m t B.6 Quantum simulations We used a finite difference method to solve the second-order partial differential equation Eq. (40) on a finite spatial grid, making sure that the density matrix is negligible at the edges of the grid and allowing the norm of the density matrix to be conserved to a high degree of accuracy. For all the following simulations it is assumed that the mechanical oscillator is initially prepared in the quantum mechanical ground state of the bare harmonic trapping potential with frequency ω m and average energy E 0 /2, thus allowing comparison with the classical simulations.

102 101 B.6.1 Undamped case As in the classical case we first consider the case where the mechanical damping rate is small enough compared to the mechanical frequency that it may be neglected over the time scale of our simulations, and the evolution of the system is Hamiltonian. Fig. B.9 shows plots of the spatial probability density P (x, t) = ρ m (x, x, t), (B.41) that are in on-to-one correspondence with the classical results shown in Fig. B.7. Recalling that for the chosen parameters classical dynamic stabilization arises for modulation amplitudes (A/P 0 ) > 0.56, classical stabilization is expected in plots (d, e). Fig. B.9(a) shows the quantum dynamics for the case of the static double-well potential. The main distinctions with respect to the classical result in Fig. B.7(a) are the pronounced quantum interferences resulting from the oscillations of the mechanics wave packet between the two potential wells. For modulation amplitudes (A/P 0 ) 0.56, Figs. B.9(b, c), the probability density is spatially extended and, similarly to the classical case, bounded by the potential barriers given by the timeaveraged double-well potential evaluated at the average energy E 0 /2 of the initial condition. The quantum carpet patterns resulting from quantum interferences of mechanical wave packets in bound potentials are a distinct, and expected, feature in all of the above cases in comparison to the classical case [36]. Furthermore, similarly to the classical case the micromotion is largest at the boundary of the spatial probability distribution of the mechanics since the amplitude of the timedependent radiation pressure force A(x) in Eq. (B.32) is the largest at that point. For modulation amplitudes above the threshold for dynamic stabilization, Figs. B.9(d, e), the spatial width of P (x, t) decreases. As in the classical case the micromotion softens the transition from unstable to stable regime. Except for the quantum interferences characteristic of wave packet dynamics in a potential well,

103 102 x/x 0 (a) (b) (c) (d) (e) Figure B.10: Time evolution of the spatial probability distribution of the damped quantum mechanical oscillator initially prepared in the ground state of the harmonic potential of frequency ω m. Here, the parameters employed and plot labels are identical to those used in Fig. B.8 ω m t the quantum and classical cases yield quite similar probability densities. Our results therefore validate the concept of dynamic stabilization of a quantum optomechanical oscillator in the absence of damping. B.6.2 Damped case The contrast between the classical and quantum cases is more pronounced in the presence of damping, as shown in Fig. B.10 which is in one-to-one correspondence with the classical results in Fig. B.8. For the case of the static mechanical potential with no applied modulation, see Fig. B.10(a), P (x, t) is asymptotically split with dual peaks at the local minima of the underlying double-well potential, in agreement with expectations from the classical theory. This splitting persists for lower

104 103 values of the modulation amplitude, see Fig. B.10(b). In section B.4, Figs. B.8(c, d) illustrated that in the case of the classical oscillator damped by a reservoir at zero temperature, a sharp transition occurs from the unstable to stable regime at (A/P 0 ) = In contrast, the transition to dynamic stabilization is much more gradual in the quantum theory as illustrated by Figs. B.10(c, d) which straddle the threshold with little change in features. This is a purely quantum effect: while a damped classical oscillator at zero temperature does not experience any noise, the corresponding quantum oscillator experiences quantum noise which blurs the classical stability transition. Dynamic stabilization still occurs for sufficiently large modulation amplitudes, see Fig. B.10(e) for (A/P 0 ) = 1. Once again we see that for sufficiently large modulation amplitudes the probability densities showing dynamic stabilization from the quantum and classical theories are quite similar, modulo the expected quantum interferences. B.6.3 Phase-space distributions Further information and insight regarding dynamic stabilization in the classical and quantum domains can be obtained from the corresponding phase-space distributions. For the classical case this is constructed by plotting the ensemble of trajectories in the (p, x) plane and interpreting the density of trajectories as the probability density. For example, Fig. B.11(a) shows the classical phase-space distribution corresponding to the results in Fig. B.7(a) for a time ω m t = 100. For the quantum case the phasespace distribution is obtained from the Wigner quasi-probability distribution W (x, p, t) = 1 π x + y ˆρ m (t) x y e 2ipy dy, (B.42) and Fig. B.11(d) shows the quantum phase-space distribution corresponding to the results in Fig. B.9(a) for a time ω m t = 100. Note that the quantum Wigner distribution displays negative regions, seen as blue, these being signatures of non-classicality. In Fig. B.11 the upper row shows the classical phase-space distributions for (a)

105 104 A/P 0 = 0, the case of the static mechanical potential, (b) A/P 0 = 0.56, and (c) A/P 0 = 1, for a time ω m t = 100, all other parameters being the same as before. The lower row of plots labeled (d)-(f) are the corresponding quantum phase-space distributions. Comparing the upper and lower rows, we see that the classical and quantum plots share broad structural features while displaying marked differences in detail. For example, Fig. B.11(a) for the static mechanical potential shows the classic figure-eight phase-space portrait characteristic of a double-well potential, while Fig. B.11(d) reflects similar structure plus oscillatory structures and negative regions that are uniquely quantum. The same comments apply to Figs. B.11(b, e) for A/P 0 = 0.56, which is below the threshold for dynamic stabilization. For the results shown in Figs. B.11(c, f) for A/P 0 = 1 we see that fluctuations in the displacement x around the origin are reduced with respect to the other examples, which is consistent with the fact that dynamic stabilization is expected in this case. Note that although the fluctuations in the displacement are reduced there remain large positive and negative variations in the momentum p. This may be traced to the micromotion which, although it has small spatial extent, reflected by the reduced displacement fluctuations in Figs. B.11(c, f), can nonetheless be associated with a largetime-oscillating momentum due to its high frequency. As for the orientation of the phase-space distributions in Figs. B.11(b, e) or (c, f), this depends on the specific choice of dimensionless interaction time ω m t since the micromotion is synced with the applied modulation. The impact of quantum noise on the behavior of the quantum oscillator is explored in Fig. B.12, which is for the same parameters as used in Fig. B.11 but including damping at T = 0 and for a time ω m t = 300. In all cases the spatial extent of the classical phase-space distributions are much narrower than their quantum counterparts which reflects the absence of noise in the classical case alluded to earlier. In Fig. B.12(a, d) for the static mechanical potential the phase-space distributions show equal peaks around the minima of the double-well potential, whereas

106 105 Figure B.11: Phase-space distributions of the classical oscillator (upper row) and corresponding Wigner quasi-probability distributions of the quantum oscillator (bottom tow) at time ω m t = 100 in the absence of dissipation. Here γ/ω m = 10 6, Ω/ω m = 1.8 and (a, d) A/P 0 = 0, (b, e) A/P 0 = 0.56, (c, f) A/P 0 = 1, κ/ω m = 10, c /ω m = 0, g (2) 0 x 2 0/ω m = 0.01, P 0 /(E 0 ω L ) = 66. dynamic stabilization is clearly evident in Fig. B.12(c, f). B.7 Summary and outlook We have investigated a scheme for the dynamic stabilization of a mechanical oscillator based on an optomechanical variation of the Kapitza pendulum problem that involves the modulation of the radiation pressure force. We derived the timeaveraged potential and explored the parameter regime for dynamic stabilization. We investigated the full dynamics of the mechanical mode both in the classical and quantum regimes, identifying the important role of micromotion in both situations. The presence of dissipation due to the coupling of the mechanics to a reservoir at T = 0 illustrated the fundamental difference in the onset of stability in both cases,

107 106 Figure B.12: Phase-space distributions of the classical oscillator (upper row) and corresponding Wigner quasi-probability distributions of the quantum oscillator (lower row) at time ω m t = 300. Both classical and quantum oscillators are damped via a reservoir at zero temperature with γ/ω m = Other parameters as in Fig. B.11. a direct consequence of the impact of quantum fluctuations. In the adiabatic approximation for the optical field, we neglected the impact of the radiation pressure noise on the mechanics, resulting in the reduced mechanical potential. Though this approximation is valid in the case of a strongly driven cavity field with a fast decay, which prevents the build up of quantum correlation between the cavity field and the mechanics, the role of the radiation pressure noise on the dynamics of the mechanics is also worth investigating. In addition, since the stabilizing scheme relies on the modulation of the input power, it leads to large micromotion of the stabilized mechanical motion. Future studies will consider the contributions of the fluctuations of the radiation pressure force on dynamic stabilization of the mechanics as well as further stabilizing schemes based on phase and frequency modulations of the input.

108 107 REFERENCES [1] L. D. Landau and E. M. Lifshitz, Mechanics (Pergamon, Oxford, 1960). [2] P. L. Kapitza, Soviet Phys. JETP 21, 588 (1951). [3] R. P. Feynman, R. B. Leighton, and M. L. Sands, The Feynman Lectures on Physics, Vol.II (Addison-Wesley, Reading, 1964). [4] T. P. Grozdanov and M. J. Raković, Phys. Rev. A 38, 1739 (1988). [5] S. Rahav, I. Gilary, and S. Fishman, Phys. Rev. Lett. 91, (2003). [6] S. Rahav, I. Gilary, and S. Fishman, Phys. Rev. A 68, (2003). [7] I. Gilary, N. Moiseyev, S. Rahav, and S. Fishman, J. Phys. A: Math. Gen. 36, L409 (2003). [8] H. Saito and M. Ueda, Phys. Rev. Lett. 90, (2003). [9] F. Kh. Abdullaev, J. G. Caputo, R. A. Kraenkel, and B. A. Malomed, Phys. Rev. A 67, (2003). [10] H. Saito, R. G. Hulet, and M. Ueda, Phys. Rev. A 76, (2007). [11] W. Zhang, B. Sun, M. S. Chapman, and L. You, Phys. Rev. A 81, (2010). [12] W. Paul, Rev. Mod. Phys. 62, 531 (1990). [13] E. D. Courant, M. S. Livingston, and H. S. Snyder, Phys. Rev. 88, 1190 (1952). [14] T. M. Hoang, C. S. Gerving, B. J. Land, M. Anquez, C. D. Hamley, and M. S. Chapman, Phys. Rev. Lett. 111, (2013).

109 108 [15] A. Zenesini, H. Lignier, D. Ciampini, O. Morsch, and E. Arimondo, Phys. Rev. Lett. 102, (2009). [16] T. J. Kippenberg and K. J. Vahala, Optics Express 15, (2007). [17] T. J. Kippenberg and K. J. Vahala, Science 321, 1172 (2008). [18] F. Marquardt and S. M. Girvin, Physics 2, 40 (2009). [19] M. Aspelmeyer et al., J. Opt. Soc. Am. B 27, A189 (2010). [20] M. Aspelmeyer, P. Meystre and K. Schwab, Physics Today 65, 29 (2010). [21] P. Meystre, Annalen der Physik 525, 215 (2013). [22] M. Aspelmeyer, T. J. Kippenberg and F. Marquardt, arxiv: (2013). [23] A. D. O Connell et al., Nature 464, 697 (2010). [24] J. D. Teufel et al., Nature 475, 359 (2011). [25] J. Chan et al., Nature 478, 89 (2011). [26] A. H. Safavi-Naeini et al., Phys. Rev. Lett. 108, (2012). [27] I. Pikovski, M. R. Vanner, M. Aspelmeyer, M. S. Kim, and Č. Brukner, Nature Physics 8, 393 (2012). [28] L. F. Buchmann, L. Zhang, A. Chiruvelli, and P. Meystre, Phys. Rev. Lett. 108, (2012). [29] T. P. Purdy et al., Phys. Rev. Lett. 105, (2010). [30] F. Brennecke et al., Science 322, 235 (2008). [31] J. D. Thompson et al., Nature 452, 72 (2008).

110 109 [32] J. C. Sankey et al., Nature Physics 6, 707 (2010). [33] H. Seok, L. F. Buchmann, E. M. Wright, and P. Meystre, Phys. Rev. A 88, (2013). [34] P. Meystre and M. Sargent III, Elements of Quantum Optics, 4th ed. (Springer- Verlag, Berlin, 2007). [35] C. W. Gardiner and P. Zoller, Quantum Noise, 3rd ed. (Springer-Verlag, Berlin, 2010). [36] A. E. Kaplan, I. Mazolli, W. E. Lamb, and W. P. Schleich, Phys. Rev. A 61, (2000).

111 110 APPENDIX C OPTICALLY MEDIATED NONLINEAR QUANTUM OPTOMECHANICS H. Seok, L. F. Buchmann, S. Singh, and P. Meystre Published in Physical Review A Copyright 2012 by American Physical Society ABSTRACT We consider theoretically the optomechanical interaction of several mechanical modes with a single quantized cavity field mode for linear and quadratic coupling. We focus specifically on situations where the optical dissipation is the dominant source of damping, in which case the optical field can be adiabatically eliminated, resulting in effective multimode interactions between the mechanical modes. In the case of linear coupling, the coherent contribution to the interaction can be exploited e.g. in quantum state swapping protocols, while the incoherent part leads to significant modifications of cold damping or amplification from the single-mode situation. Quadratic coupling can result in a wealth of possible effective interactions including the analogs of second-harmonic generation and four-wave mixing in nonlinear optics, with specific forms depending sensitively on the sign of the coupling. The cavity-mediated mechanical interaction of two modes is investigated in two limiting cases, the resolved sideband and the Doppler regime. As an illustrative application of the formal analysis we discuss in some detail a two-mode system where a Bose- Einstein condensate is optomechanically linearly coupled to the moving end mirror of a Fabry-Pérot cavity.

112 111 C.1 Introduction Quantum optomechanics has recently witnessed a rapid succession of key advances, with important milestones including the cooling of several systems to the quantum ground state of vibration [1, 2, 3, 4], the demonstration of strong optomechanical coupling [3, 5, 6] and of quantum coherent coupling between optical and phonon fields [7], the realization of optomechanical systems from ultracold atomic ensembles [8, 9], and impressive advances toward quantum state transfer [10, 11, 12, 13, 14, 15], to mention just a few examples. These developments hint at the promise for optomechanics applications in quantum metrology [16, 17], quantum information science [18, 19, 20, 21], and also in possible tests of fundamental physics questions [22, 23]. So far, though, the focus of most advances has been on single mode dynamics of the oscillator [24, 25, 26]. In complete analogy with the situation in quantum optics, however, single-mode quantum acoustics fails to capture many important aspects, including most obviously perhaps propagation phenomena, nonlinear acoustical effects, focusing and defocusing of acoustic waves, etc. The interaction of optical fields with mechanical arrays has recently gained increased attention [27, 28]. Mechanical elements dressed by optical fields couple indirectly to each other through these fields, and particular nonlinear features have been studied [29, 30]. Specific phenomena such as multimode cooling [31, 32] and synchronization [33, 34] have been discussed and hold promise for the entanglement of mechanical oscillators [35, 36] and of hybrid optomechanical systems [37, 38]. However a general theory of optically mediated nonlinear phenomena in mulitimode quantum optomechanical systems that also fully accounts for the effects of quantum noise has not yet been developed. Within this general context, our paper represents a step toward the development of nonlinear quantum acoustics. Our main objective is to present a rather general

113 112 formalism that fully accounts for the quantum nature of the light field and to outline a number of key consequences. The specific situation that we consider is a multimode mechanical field perhaps provided by several mechanical oscillators or atomic samples trapped inside an optical lattice, the mode coupling being provided by a common optical field. For concreteness we concentrate on linear and quadratic optomechanical coupling and single-mode optical fields, but the generalization to more complex couplings and multimode fields is straightforward. We identify the onset of a number of nonlinear quantum optomechanical effects, including many situations familiar from nonlinear optics. These include second-harmonic generation and four-wave mixing processes, corresponding to χ (2) and χ (3) nonlinear susceptibilities. In particular, optical four-wave mixing phenomena lead to a rich spectrum of effects that have found numerous applications in applied and fundamental studies, including for example sum frequency generation, high harmonic generation, optical parametric amplification, self-focusing and defocusing, the generation and propagation of optical solitons, phase conjugation, the generation of quantum mechanical squeezing and entangled photon pairs, and more [39, 40]. Our analysis paves the way to the investigation of a similar range of effects and applications in quantum optomechanics and quantum acoustics, and as such, opens up a rich direction of investigations that will be carried out in several follow-up studies. As a specific example that illustrates some of the potential of our general approach, section V discusses the specific example of a two-mode system that can lead to quantum state transfer in a two-mode hybrid quantum optomechanical system, a Bose-Einstein condensate coupled to a mechanical oscillator. The physics underlying optically mediated nonlinear quantum optomechanics is analogous to nonlinear atom optics, a result of coherent cold collisions that lead to coherent nonlinear effects [41, 42] in quantum degenerate atomic gases. These effects include e.g. matter-wave four-wave mixing, soliton generation, the creation of entangled atom pairs, and more. The two- or few-body interactions describing atomic

114 113 Figure C.1: Non-absorptive dielectric membranes inside a Fabry-Pérot resonator which interact with a single mode cavity field. collisions are effective interactions that result from the elimination of the underlying exchange of photons. While collisions are usually thought of as largely incoherent processes, resulting e.g. in decoherence and dephasing in atomic ensembles, this needs not be the case: In the nonlinear atom optics situations demonstrated in quantum degenerate atomic systems they result in coherent matter wave mixing. In analogy to that situation, one can expect similar effects to occur in quantum acoustics. Our main result is that this is indeed the case, and that incoherent effects due to quantum fluctuations associated primarily with the elimination of the electromagnetic field need not overwhelm coherent effects. This paves the way to investigate nonlinear effects in phonon physics reminiscent of those encountered in nonlinear optics and nonlinear atom optics, with the promise of similarly exciting applications. The remaining of the paper is organized as follows: Our starting point is a relatively general discussion of optically mediated effective interactions between many mechanical modes, discussing in some detail the quantum noise and damping associated with that interaction, both for linear and quadratic optomechanical coupling.

115 114 We identify a number of specific forms of the effective interaction that lead to qualitatively different dynamics, in particular a regime where the coupling of the quantum fluctuations of the mechanical modes is intrinsically nonlinear. We then illustrate the general analysis with an example involving just two mechanical modes, with emphasis on two limiting cases, the resolved sideband and the Doppler regimes. A specific example is given by the state transfer between two mechanical modes, a Bose-Einstein condensate coupled to the center-of-mass mode of a moving end mirror of a Fabry-Pérot resonator through a cavity field. We comment on the optimization of the transfer fidelity using squeezed input light as the optically mediating light field and conclude with some general remarks and outlook. C.2 Model We consider a multimode optomechanical system described in terms of a set of N modes of effective masses m j, bare frequencies ω j and damping rates γ j. The interaction between these modes is assumed to be mediated by a single mode of the optical resonator, of nominal frequency ω c and driven by a monochromatic external field of frequency ω L. A realization of this system could be a high-finesse Fabry- Pérot resonator with a series of internal non-absorptive dielectric membranes such as depicted in Fig. C.1, a collection of ultracold atomic samples trapped at or near the minima of an optical lattice inside such a resonator, one or several Bose-Einstein condensates trapped inside a Fabry-Pérot cavity, perhaps with a moving end-mirror, or a number of similar setups operating either in the optical or in the microwave regime. The Hamiltonian governing the system is where H = H opt + H m + H om + H loss, H opt = ω c à à + i (ηe iω Lt à η e iω Lt Ã), (C.1) (C.2)

116 115 describes the cavity field mode, pumped externally by a field of frequency ω L at a rate η, H m = N j=1 ω j ˆB j ˆB j (C.3) describes the mechanical modes, with [ ˆB j, ˆB k ] = δ jk. The optomechanical interaction is in the linear case and H om = à à H om = à à in the quadratic case, with g 0,j and g (2) 0,j N g 0,j ( ˆB j + ˆB j ) j=1 N j=1 (C.4) g (2) 0,j ( ˆB j + ˆB j )2 (C.5) linear and quadratic single-photon optomechanical coupling coefficients, respectively. Finally, H loss describes the interaction of the cavity field and the mechanics with their respective reservoirs and accounts for dissipation. For simplicity, we will assume that all mechanical modes either couple purely linearly or quadratically. C.3 Linear coupling This section discusses the case of linear coupling including modifications to cold damping and optical spring effects. With à = Âe iω Lt (C.6) and for the optomechanical coupling (C.4) the Heisenberg-Langevin equations of motion in the standard input-output formalism [43] are  = [ i c κ 2 ]  + η + i  N g 0,j ( ˆB j + ˆB j ) + κâ in, j=1 ˆB j = iω j ˆBj γ j 2 ( ˆB j ˆB j ) + ig 0,j  + i ξ j, (C.7)

117 116 where c = ω L ω c, (C.8) κ and γ j are the decay rates of the cavity field and mechanical modes, respectively, and â in and ξ j the associated Markovian quantum noise operators. Their two-time correlation functions will be given explicitly later on. C.3.1 Linearization If the system is driven by a classical field, it is useful to decompose the various field operators as the sum of their expectation values and small quantum fluctuations, whose dynamics are treated to lowest order only. Specifically we expand the operators  and ˆB j as  =  + δâ α + â, (C.9a) ˆB j = ˆB j + δ ˆB j β j + b j, (C.9b) where [â, â ] = 1, [ b j, b k ] = δ jk and we assume that their expectation values α and β j are much larger than the amplitudes of the fluctuations, for example, α 2 â â. (Note that rigorously speaking this assumes all quantum contributions to have zero expectation value. However, the treatment remains valid for fluctuations with nonzero, but small expectation values [44].) This yields the mean field equations of motion with steady-state values α = i α κ α + η, (C.10) 2 β j = iω j β j γ j 2 (β j βj ) + ig 0,j α 2, (C.11) α s = η i + κ/2 = n c, (C.12) (β j + β j ) s = 2g 0,j ω j α s 2, (C.13)

118 117 where N = c + g 0,j (β j + βj ). (C.14) j=1 Here we have chosen the phase of η such that α s is real without loss of generality for the case of cw pumping. As is well known, dynamical back-action gives rise to frequency shifts the optical spring effect and radiative damping or amplification for each mechanical oscillator [45, 46, 47]. In order to capture these effects explicitly, we switch to an interaction picture with the transformations bj = ˆb j e iν jt, (C.15) ξ j = ˆξ j e iν jt (C.16) where ν j = ω j + Ω j, (C.17) and the expressions for the dynamical shifts Ω j will be determined self-consistently later on [48]. The equations of motion for the fluctuation operators then read â = [ ˆb j = [ i κ ] â + i 2 iω j γ j 2 N g j (ˆb j e iνjt + ˆb j eiν jt ) + κâ in, j=1 (C.18) ] ˆbj + γ j 2 ˆb j e2iν jt + ig j e iν jt (â + â) + iˆξ j, (C.19) where we have introduced the amplified optomechanical coupling strengths, g j = g 0,j nc. (C.20) C.3.2 Elimination of the cavity field In the physically relevant regime where the optical loss is much larger than the mechanical decay rates, κ γ j, the cavity field follows the dynamics of the mechanical

119 118 oscillators adiabatically. For time scales long compared to κ 1, the formal solution of Eq. (C.18) reads â(t) [ ] N e iν jtˆbj g j ( + ν j ) + iκ/2 + e iνjtˆb j ( + ν j ) + iκ/2 ˆf in (t), (C.21) j=1 where the noise operator ˆf in (t) is given by ˆf in (t) = κ t 0 dτ e (i κ/2)(t τ)â in (τ). (C.22) Substituting the formal solution into the equations of motion for the mechanical oscillators, Eq. (C.19), gives ˆb j = [ iω j γ j 2 N k=1 ] ˆbj + γ j 2 ˆb j e2iν jt N k=1 [ g jk iω k + Γ ] k e i(ν j ν k )tˆbk 2 [ g jk iω k Γ ] k e i(ν j+ν k )tˆb k 2 + ig je iνjt ( ˆf in + ˆf in ) + iˆξ j, (C.23) where g jk = g j /g k. (C.24) and the radiation-induced frequency shifts Ω k and damping coefficients Γ k are given by Ω k = g 2 k [ Γ k 2 = g2 k [ ] ν k ( ν k ) 2 + κ 2 /4 + + νk (, (C.25a) + ν k ) 2 + κ 2 /4 κ/2 ( + ν k ) 2 + κ 2 /4 κ/2 ( ν k ) 2 + κ 2 /4 ]. (C.25b) Due to the appearance of the shifted frequencies ν k ( ) in the denominators on its right hand side, Eq. (C.25a) is a fifth-order polynomial equation for Ω k ( ), so that it is not Lorentzian in general. In the strong coupling regime g k > κ/2, it becomes multivalued for a range of effective detunings, a feature that can be interpreted in terms of normal mode splitting [5]. In this situation the mechanical and

120 119 optical modes are strongly hybridized and it is not meaningful to treat them separately. Also, the counter-rotating terms in Eqs. (C.23), combined in particular with a driving field blue detuned from the resonator resonance, can result in instabilities in the strong coupling regime and a concomitant would lead to the break-down of the linearization process. The strong coupling and blue detuned regimes will be the object of future work. Here we focus on the simpler case of weak coupling, g k κ. In this limit the frequency shifts Ω k remain small compared to the bare frequencies of the oscillators, so that we can approximate ν k by ω k on the right hand side of Eqs. (C.25), see Eq. (C.17). The frequency shifts Ω k are then the sum of two dispersion curves while the decay rates Γ k consist of two Lorentzians with opposite signs, all of which are centered around = ±ω k. The weak coupling regime also allows us to drop the counter-rotating terms in Eqs. (C.23). The resulting equations of motion are ˆb j [ iω j γ j 2 ] ˆbj N k=1 +ig j e iν jt ( ˆf in + ˆf in ) + iˆξ j, [ g jk iω k + Γ ] k e i(ν j ν k )tˆbk 2 (C.26) These equations describe a system of N pairwise, linearly coupled oscillators, with coherent contributions proportional to the optically induced frequency shifts Ω k, cold damping coefficients Γ k familiar from the single-mode situation, and associated noise operators ( ˆf in + ˆf in ) and ˆξ j. The fact that each mechanical element experiences a different frequency shift and radiative decay, depending on its optomechanical coupling strength and the pump-cavity detuning, is the key ingredient that allows one to couple oscillators with different bare frequencies on-demand, by choosing appropriate pump-powers and detunings. We discuss selected consequences of this coupled dynamics in section V for the case of a two-mode system. But first, the next section extends our discussion to the case of quadratic optomechanical coupling.

121 120 C.4 Quadratic coupling We now consider the quadratic optomechanical coupling see Eq. (C.5), where g (2) 0,j H om = à à N j=1 g (2) 0,j ( ˆB j + ˆB j )2, (C.27) are single-photon coupling coefficients. In that case the Heisenberg-Langevin equations of motion are, in the frame of the laser frequency ω L,  = [ i c κ 2 ]  + η i  N j=1 g (2) 0,j ( ˆB j + ˆB j )2 + κâ in, ˆB j = iω j ˆBj γ j 2 ( ˆB j ˆB j ) 2ig(2) 0,j  Â( ˆB j + ˆB j ) + i ξ j with à =  exp( iω Lt) as before. With the expansion (C.9) we have α i (2) α κ 2 α + η, [ β j iϖ j + γ ] [ j β j 2ig (2) 0,j 2 α 2 γ ] j βj, 2 where the effective detuning and shifted mechanical frequencies are now (2) c N j=1 (C.28) g (2) 0,j (β j + β j ) 2, (C.29) ϖ j = ω j + 2g (2) 0,j α 2, (C.30) and consistently with the linearization procedure we have further neglected the vacuum expectation value contribution. The steady-state expectation value of the cavity field amplitude is η α s = i (2) + κ/2, (C.31) and the steady-state displacements of the mechanical oscillators are found from the N coupled fifth-order equations ( 4g (2) 0,j α s 2 + ω j ) (β j + β j ) s = 0. (C.32)

122 121 Zero displacement, (β j + β j ) s = 0, is always a solution to these equations, and for g (2) 0,j > 0, which is the case for membranes located at minima of the intracavity field, it is the only stable solution. However that solution may become unstable for negative couplings, g (2) 0,j < 0, occurring when oscillators are located at cavity-field maxima. In this case, each oscillator settles in one of the stable positions, x j = ± 1 2 (β j + β j ) s, (C.33) located symmetrically around the intracavity field maximum. As we will see in the following sections, these two geometries result in qualitative consequences for the behavior of the system. C.4.1 Local maxima To simplify the discussion we assume that all coupling constants g (2) 0,j have the same sign, and consider first the case where they are negative and the pumping is strong enough to render the x j = 0 solution unstable. The stable steady state is degenerate, as each oscillator can have a negative or positive mean displacement. A particular choice of steady state will break the symmetry of the sign of the displacements and influence the physics of the system. Experimentally, it could be controlled by addressing the individual oscillators through an additional field or preparing them slightly displaced towards the desired steady-state position. The Heisenberg- Langevin equations of motion for the fluctuations can be approximated to lowest non-vanishing order, as â [ ˆb j [ i (2) κ ] â 4i 2 iϖ j + γ j 2 N j=1 ] ˆbj [ 2ig (2) 0,j n c γ j 2 g (2) j x j (ˆb j + ˆb j ) + κâ in, (C.34) ] ˆb j 4ig(2) j x j (â + â ) + i ξ j, (C.35) where we have introduced the amplified quadratic optomechanical strengths g (2) j = g (2) 0,j nc. (C.36)

123 122 Assuming that the resulting frequencies of the mechanical oscillators are ν j = ϖ j + Ω j, and following the same approach as before the equations of motion for the mechanical modes become ˆb j = [ iω j γ j 2 N k=1 g (2) jk ] [ ˆbj 2ig (2) 0,j n c γ j 2 ] ˆb j e2iν jt N k=1 [ g (2) jk iω k + Γ ] k e i(ν j ν k )tˆbk 2 [ iω k Γ ] k e i(ν j+ν k )tˆb k 2 4ig(2) j x j e iνjt ( ˆf in + ˆf in ) + iˆξ j, (C.37) where the noise operator associated with the adiabatically eliminated optical field is and ˆf in (t) = κ t 0 g (2) jk dτ e (i (2) κ/2)(t τ)â in (τ) = g(2) j x j x k g (2) k (C.38). (C.39) We then find for the radiation-induced frequency shifts Ω k and damping coefficients Γ k ( Ω k = Γ k 2 = ( 4g (2) k 4g (2) k ) [ ] 2 x (2) ν k k ( (2) ν k ) 2 + κ 2 /4 + (2) + ν k (, (C.40) (2) + ν k ) 2 + κ 2 /4 ) [ ] 2 x κ/2 k ( (2) + ν k ) 2 + κ 2 /4 κ/2 (. (C.41) (2) ν k ) 2 + κ 2 /4 In the weak coupling regime, g (2) j x j κ, we can neglect the fast oscillating terms on the right-hand side of Eq. (C.37) except the term proportional to g (2) 0,j n c. The equations of motion for the mechanics further simplify to ˆb j = [ iω j γ j 2 ] ˆbj 2ig (2) 0,j n cˆb j e2iν jt 4ig (2) j x j e iν jt ( ˆf in + ˆf in ) + iˆξ j. N k=1 [ g (2) jk iω k + Γ ] k e i(ν j ν k )tˆbk 2 (C.42)

124 123 These equations have the same form as in the case of linear optomechanical coupling of the mechanical oscillators except for the second term which is a secondharmonic generation process, a direct acoustic analog of optical second harmonic generation and a direct consequence of the nonlinear nature of the quadratic optomechanical coupling. Another notable new feature in this regime is the flexibility to choose the sign and strength of the coupling coefficients g (2) jk through the equilibrium positions x j and x k, see Eq. (C.39). C.4.2 Local minima The situation is changed qualitatively in the case where the mechanical oscillators are located at minima of the cavity field. The single-photon, single mode quadratic optomechanical coupling coefficients are then positive and the cavity field results in a tighter confinement through the static optical spring effect. The only stable steadystate displacement is given by x j = 0 so that the effects described in the previous section are highly suppressed. On the other hand, a number of novel features appear as a result of the fact that the first non-vanishing couplings are of higher order in the quantum fluctuations. In particular, instead of the second-harmonic generation discussed in the previous section, the dominant nonlinear effects are akin to fourwave mixing in optics. To lowest order in the fluctuations, the Heisenberg-Langevin equations of motion become â [ ˆb j [ i (2) κ ] â i 2 iϖ j + γ j 2 N j=1 g (2) j (ˆb 2 2 j + ˆb j + 2ˆb jˆb j ) + κâ in, ] ˆbj [ 2ig (2) 0,j n c γ j 2 where the effective detuning (2) is now (2) = c ] ˆb j 2ig(2) j (â + â )(ˆb j + ˆb j ) + i ξ j, N j=1 g (2) 0,j. (C.43)

125 124 Applying the same arguments and approximations as before to adiabatically eliminate the optical field dynamics, we find the coupled equations of motion for the mechanical modes ˆb j = [ iω j γ j 2 1 N i 2 k=1 N k=1 N k=1 g (2) jk ] [ ˆbj 2ig (2) 0,j n c γ j 2 [ iω k + Γ k 2 ] ˆb j e2iν jt ] (e 2iν ktˆb2 kˆbj + e 2i(ν j ν k )tˆb2 kˆb j ) [ g (2) jk iω k Γ ] k 2iν (e ktˆb 2 k 2 ˆb j + e 2i(ν j+ν k )tˆb 2 ˆb k j ) g (2) jk Λ k(ˆb kˆb kˆbj + ˆb kˆb kˆb j e2iν jt ) 2ig (2) j ( ˆf in + ˆf in )(ˆb j + ˆb j e2iν jt ) + iˆξ j, where the mode-coupling is now given by (C.44) g (2) jk = g(2) j /g (2) k, (C.45) the frequency shifts and radiative damping coefficients are [ ] Ω k = (2g (2) (2) 2ν k k )2 ( (2) 2ν k ) 2 + κ 2 /4 + (2) + 2ν k (, (2) + 2ν k ) 2 + κ 2 /4 [ ] Γ k = (2g (2) κ/2 k 2 )2 ( (2) + 2ν k ) 2 + κ 2 /4 κ/2 (, (2) 2ν k ) 2 + κ 2 /4 and Λ k = (2g (2) (2) 2 k )2 ( (2) ) 2 + κ 2 /4, (C.46) ˆf in (t) = κ t 0 dτe (i (2) κ/2)(t τ)â in (τ). (C.47) Note that in Eq. (C.44) the quantum noise stemming from the elimination of the optical field is now multiplicative. Since we can neglect fast rotating terms in the weak coupling regime, Eqs. (C.44)

126 125 further simplify to ˆb j = [ iω j γ j 2 ] ˆbj i 2 N k=1 g (2) 0,jk Λ kˆb kˆb kˆbj 1 2 2ig (2) j ( ˆf in + ˆf in )(ˆb j + ˆb j e2iν jt ) + iˆξ j. N k=1 [ g (2) 0,jk iω k + Γ ] k e 2i(ν j ν k )tˆb2 2 kˆb j (C.48) These coupled nonlinear equations are indicative of four-wave mixing processes, and are formally similar to situations driven by χ (3) nonlinear susceptibilities in nonlinear classical and quantum optics [39, 40]. In that context, four-wave mixing phenomena are known to lead to a rich spectrum of effects that have found numerous applications in applied and fundamental studies. They include for example sum frequency generation, high harmonic generation, optical parametric amplification, self-focusing and defocusing, the generation and propagation of optical solitons, phase conjugation, the generation of quantum mechanical squeezing and entangled photon pairs, and more [39, 40]. The analysis of this section paves the way to investigate a similar range of studies in quantum optomechanics and quantum acoustics, and as such, opens the way to an intriguing and rich direction of investigation that will be carried out in follow-up studies. One particularly attractive feature of optomechanics in this context is the ease with which the sign and strength of the nonlinear interactions can be adjusted in combination with the functionalization of these devices. The following sections discuss a few illustrative examples, concentrating for now on the simple situation of a two-mode system with linear optomechanical coupling only, and demonstrating a example of functionalization that involves the optically mediated coupling of a nanomechanical system and a quantum degenerate atomic system. C.5 Example two-mode system Elementary multimode effects can be studied in the case where two mechanical modes interact linearly with a common cavity field, i.e. N = 2. In the weak

127 126 coupling regime the Heisenberg-Langevin Eqs. (C.26) become ˆb 1 = 1 [ 2 Γ e,1ˆb 1 iω c,1 + 1 ] 2 Γ c,1 ˆb 2 = 1 [ 2 Γ e,2ˆb 2 iω c,2 + 1 ] 2 Γ c,2 where we have introduced the effective decay rate ˆb2 e i(ν 1 ν 2 )t + ig 2 e iν 1t ( ˆf in + ˆf in ) + iˆξ 1, ˆb1 e i(ν 1 ν 2 )t + ig 2 e iν 2t ( ˆf in + ˆf in ) + iˆξ 2,(C.49) Γ e,j = Γ j + γ j, (C.50) the cross-damping rate and the cross-coupling frequency Γ c,j = g jk Γ k, (C.51) Ω c,j = g jk Ω k. (C.52) The cross-damping rate is associated with effects such as sympathetic cooling or heating of coupled oscillators, while the cross-coupling frequency is associated with the coherent aspects of multimode coupling, such as e.g. quantum state transfer. Note that when the two effective mechanical frequencies are matched, ν 1 = ν 2, the cross-coupling frequencies and damping coefficients become equal Ω c,1 = Ω c,2, Γ c,1 = Γ c,2. In terms of the motional quadratures ˆX j = 1 2 (ˆb j e iνjt + ˆb j eiν jt ), (C.53) ˆP j = 1 2i (ˆb j e iνjt ˆb j eiν jt ), (C.54) Eqs. (C.49) become û = Mû + ˆΞ, (C.55)

128 127 ( where û = ˆX1, ˆP 1, ˆX 2, ˆP ) T 2, the drift matrix M is given by Γ e,1 2ν 1 Γ c,1 2Ω c,1 M = 1 2ν 1 Γ e,1 2Ω c,1 Γ c,1, (C.56) 2 Γ c,2 2Ω c,2 Γ e,2 2ν 2 2Ω c,2 Γ c,2 2ν 2 Γ e,2 and the noise operator matrix ˆΞ is 0 ˆξ 1 e ˆΞ + g 1 ˆFin =, (C.57) 0 ˆξ 2 e iν2t + g 2 ˆFin with the effective optical noise operator ˆF in = κ t 0 dτ e (i κ/2)(t τ)â in (τ) + adj.. (C.58) In the following we limit ourselves to the case of Gaussian states and assume that the mechanical elements are coupled to statistically independent Markovian heat baths at low temperature T characterized by the two-time correlations functions [49] ξ j (t) ξ j (t ) + ξ j (t ) ξ j (t) 2 γ j (2 n th,j + 1)δ(t t ), (C.59) where n th,j = [exp( ω j /k B T ) 1] 1. From Eqs. (C.9) the first moments of all position and momentum quadratures are always zero, and one readily derives a closed set of differential equations for the second moments of the oscillator quadratures. The equation of motion for the covariance matrix V ij = 1 2 û iû j +û j û i can be written as [50] V = MV + V M T + D, (C.60) where M is the drift matrix, and the inhomogeneous term D is given by D ij = 1 ˆΞ 2 i û j + û j ˆΞi + ˆΞ j û i + û iˆξj. This system of equations can be solved exactly in

129 128 Ν 1,Ν 2 a.u Ω 1 Ω Κ Figure C.2: Effective frequencies of the first (red, solid) and second (blue, dashed) mechanical modes in the resolved sideband regime ω 1, ω 2 κ for g 1 > g 2. The dotted lines denote the bare frequencies of the mechanical modes. The shifted frequencies are matched at the intersections. The inset gives a detailed view on the intersections occurring in the red-detuned side. The parameters used are ω 1 = 2π 20MHz, ω 2 = 2π 19.95MHz, κ = 2π 1MHz, g 1 = 2π 0.3MHz, g 2 = 2π 0.12MHz. principle but the resulting solutions lack transparency. More physical insight into the coupled dynamics of the oscillator can be gained in the resolved sideband limit and in the Doppler limit, two cases to which we now turn. C.5.1 Resolved sideband regime The resolved sideband regime is characterized by slow optical resonator damping compared to the mechanical frequencies ω 1, ω 2 κ. (We assume without loss of generality that ω 1 > ω 2 in the following.) It is known from the standard theory of single-mode cavity optomechanics that in the absence of coupling each oscillator experiences a frequency shift Ω k ( ) comprised of the sum of two dispersion curves centered around = ±ω k with characteristic widths κ, see Fig. C.2, while their damping coefficients Γ k ( ) are the sum of a positive and negative Lorentzian, also of width κ, and centered at these same detunings [51].

130 129 The red and blue detunings for which the two oscillators are brought into resonance with each other are given by the intersections of the two curves in Fig. C.2. There are two such intersections on the red-detuned side of the optical resonance, one near = ω k and one on the wing of the dispersion curve. Two more intersections are located similarly on the blue-detuned side. At these points we have ν 1 = ν 2 ν and therefore Ω c,1 = Ω c,2 Ω c and Γ c,1 = Γ c,2 Γ c from Eq. (C.52). It is for those detunings that the coupling between the two mechanical oscillators is most effective. The detailed dynamics of resonant mode coupling depends on whether the resonance condition ν 1 = ν 2 occurs near the center or on the wing of the dispersion curves that comprise Ω k ( ). Near the resonance = ±ω k optical damping (or amplification) dominates over the optical spring effect, see Eqs. (C.25a) and (C.25b), and to first approximation one can safely neglect that effect compared to those of Γ c,k by setting Ω c,k = 0, see Eqs. (C.51) and (C.52). The drift matrix M reduces to Γ e,1 2ν Γ c 0 M = 1 2ν Γ e,1 0 Γ c (C.61) 2 Γ c 0 Γ e,2 2ν 0 Γ c 2ν Γ e,2 and the dominant consequence of the effective optomechanical coupling between the oscillators is the modification of their rate of cold damping or amplification compared to their uncoupled values. One such example is shown in Fig. C.3, which plots the evolution of the variances of the motional quadratures of the mechanical modes when their effective frequencies are matched around the center of the Lorentzian curve on the red side of the cavity field. The mechanical modes are both assumed to be initially in thermal equilibrium. In the absence of cross interaction, both oscillators are subject to cold damping (dotted and dot-dashed lines) with cooling rates that differ since g 1 g 2. The coupling changes the situation significantly, with both cooling rates now markedly slower (solid and dashed lines). This is a result of

131 130 the exchange of thermal excitations between the two oscillators that inhibits their direct optical cooling. (Note that our numerical calculations suggest that the longtime limit and thus the cooling limit does not change.) Performing a normal-mode analysis on the two modes reveals that the two out-of-phase oscillations decouple from the cavity field, while the two in-phase modes are cooled with an increased damping rate. We now turn to the resonances ν 1 = ν 2 that occur on the wings of the dispersion curves. In these cases the optical spring effect dominates over cold damping (or amplification), a consequence of the faster decay of the Lorentzian Γ k ( ) (with 1/ 2 ) compared to the dispersive nature of Ω k ( ), which decreases as 1/. That situation is therefore characterized by the dominance of coherent exchange between the two oscillators, and we approximate the corresponding drift matrix by γ 1 /2 ν 0 Ω c ν γ 1 /2 Ω c 0 M =. (C.62) 0 Ω c γ 2 /2 ν Ω c 0 ν γ 2 /2 In this regime, and provided that the noise sources associated with the optical field can be managed, one can realize coherent effects such as quantum state transfer between the two mechanical modes. We return to this effect in the Doppler regime, which we consider in the following. C.5.2 Doppler regime In the Doppler regime, ω 1, ω 2 κ, the width of the optical cavity is so large that it washes out the two distinct features of Ω k ( ) and Γ k ( ), characteristic of the resolved sideband regime, and one can approximate Ω j g 2 j Γ j g 2 j κ 2 /4, (C.63) 2 κν j ( 2 + κ 2 /4) 2. (C.64)

132 131 X c t X c t Figure C.3: Upper plot: short time (linear time scale) and lower plot: long time (log time scale) evolution of the motional quadratures of the mechanical modes in the single and two-mode scenarios. The mechanical modes of frequencies ω 1 (red, solid) and ω 2 (blue, dashed) are both cooled down while interacting with each other. For comparison the uncoupled cold damping of the modes ω 1 (orange, dotted) and ω 2 (green, dotdashed) are also shown. Two-mode coupling results in slowing down of the individual cold damping rates. Here ω 1 = 2π 20MHz, ω 2 = 2π 19.99MHz, κ = 2π 0.95MHz, g 1 = 2π 50kHz, g 2 = 2π 10kHz.

133 132 The qualitative behavior of the effective frequencies ν j (j = 1, 2) is plotted in Fig. C.4 for g 1 > g 2 and shows the intersections of the two curves on the red-detuned side of the cavity resonance. (They would lie on the blue-detuned side for g 2 > g 1 ). Since Γ j ( ) decays with the third power of the cavity linewidth while Ω j ( ) scales as 1/κ 2, it follows that for the large cavity damping rates, characteristic of the Doppler regime, we can neglect in first approximation the effects of cold damping (or amplification) and at resonance we find again γ 1 /2 ν 0 Ω c ν γ 1 /2 Ω c 0 M =. (C.65) 0 Ω c γ 2 /2 ν Ω c 0 ν γ 2 /2 Note that the argument leading to this result is different in the resolved sideband limit, where the radiation induced damping can be neglected if the effective detuning is outside of the mechanical sideband, and in the Doppler regime, in which case it is negligible due to the large cavity linewidth. C.5.3 State transfer The previous discussion showed that both, the resolved sideband and the Doppler regimes, provide situations where cold damping can be neglected and the two oscillators are resonantly coupled by the effective coherent interaction H e = Ω c (ˆb 1ˆb 2 + ˆb 1ˆb 2 ). (C.66) This is the familiar beam splitter Hamiltonian, which is known in particular to lead to quantum state swapping between the two oscillators. However, an important issue is that the optical field mediating the interaction also results in the appearance of quantum noise, further limiting the fidelity of state transfer in addition to the clamping noise associated with the mechanical coupling to thermal reservoirs.

134 133 Ν 1,Ν 2 a.u. Ω 1 Ω Κ Figure C.4: Qualitative behavior of the effective mechanical frequencies for the first (red, solid) and second (blue, dashed) oscillators in the Doppler regime. The shifted frequencies are matched at the intersections. Here we chose g 1 > g 2. A set of parameters is ω 1 = 2π 100kHz, ω 2 = 2π 93kHz, κ = 2π 1MHz, g 1 = 2π 90kHz, g 2 = 2π 50kHz. Clamping noise can be reduced in principle in cryogenic environments or perhaps in levitated structures [52]. What we show in this brief section is that the fundamental quantum noise associated with the optical field can also be reduced in principle by the use of squeezed input fields with correlation functions [53] â in (t)â in (t ) = e iω L(t+t ) Mδ(t t ), (C.67) â in (t)â in (t ) = e iω L(t+t ) M δ(t t ), (C.68) â in (t)â in(t ) = Nδ(t t ), (C.69) â in (t)â in (t ) = (N + 1)δ(t t ), (C.70) where ω L is the central frequency of a squeezing device, and positive-valued N and complex-valued M are the squeezing parameters which define the strength of the squeezing as well as its direction in phase space, given by the complex phase of M.

135 134 In the following, we assume an ideal squeezed state which satisfies M = N(N + 1)e 2iθs. (C.71) The equations of motion for vacuum input noise are recovered by substituting N 0, M 0. For concreteness we consider the explicit situation of a hybrid two-mode mechanical system where one of the modes is a recoil-induced side-mode of a Bose-Einstein condensate [8] trapped inside an optical resonator with an oscillating end-mirror, and the other is the center-of-mass mode of oscillation of that mirror. As discussed in Ref. [14], in the case where a system is prepared in such a way that the shifted optomechanical frequencies of the two oscillators ν 1 and ν 2 are equal, Ω c,1 = Ω c,2 = Ω c, the equations of motion for the quadratures of the coupled mechanical oscillators reduce to d dt ˆX 1 0 ν 0 Ω c ˆX 1 ˆP 1 ν 0 Ω c 0 ˆP 1 = + ˆX 2 0 Ω c 0 ν ˆX 2 ˆP 2 Ω c 0 ν 0 ˆP 2 0 g 1 ˆFin 0 g 2 ˆFin where the subscripts 1 and 2 refer to the oscillating mirror and the condensate side-mode respectively, and the cross-coupling frequency is Ω c = g 1 g κ 2 /4., (C.72) For a squeezed input field the two-time correlation function of the effective noise can be approximated as ˆF in (t) ˆF in (t ) + ˆF in (t ) ˆF in (t) 2 ( M cos[2(θ s θ c )] + N κ 2 + κ 2 /4 ) δ(t t ), (C.73) with θ c = arg(i + κ/2). Here we have assumed that intrinsic mechanical decoherence is negligible for the time scales of interest.

136 135 X c t Figure C.5: Motional quadrature of the moving mirror (red, solid) and the BEC (blue, dashed) in the presence of vacuum noise for ω 1 = 2π 101kHz, ω 2 = 2π 100kHz, κ = 2π 1MHz, g 1 = 2π 100kHz, g 2 = 2π 10kHz. Figure C.5 shows the co-evolution of the variance of one of the motional quadratures of the two oscillators. In this example the resonator is driven by a coherent field and the initial states of the BEC and the moving mirror are the ground state and a thermal state with mean phonon number n = 1, respectively. As expected, the fidelity of the coherent quantum state transfer, which would be unity at times (π/2 + pπ)ω 1 c, with p integer, in the absence of noise, is increasingly reduced by the quantum fluctuations of the light field, even without intrinsic mechanical decoherence mechanisms. The use of a squeezed optical field to reduce the detrimental effect of quantum noise is illustrated in Fig. C.6, where the improved fidelity is readily apparent in particular at time t = π/2ω c. As illustrated in Figs. C.7 and C.8, the state transfer fidelity depends strongly on the phase difference between the cavity field and a squeezing device. For θ s = θ c ± π/2, the fluctuations in the position quadrature of the cavity field are minimized so that very high fidelity state transfer is achieved, F = for N = 10 in our example. As would be intuitively expected, we also

137 136 X c t Figure C.6: Motional quadrature of the moving mirror (red, solid) and the BEC (blue, dashed) for squeezed vacuum noise with N = 1 and squeezing phase θ s θ c = π/2. Same parameters as in Fig C.5. find that strong squeezing leads to an increased sensitivity of the state transfer fidelity on that relative phase. As a comparison to Fig. C.7, which is for a BEC side-mode initially in the ground state and the oscillating mirror in a thermal state with n = 1, Fig. C.8 displays the fidelity of state transfer where the BEC is initially in a squeezed state with ˆX 2 2 = and ˆP 2 2 = 10. The transfer fidelity in that case exhibits an increased dependence of the relative phase between the cavity field and a squeezing device, as would be expected. In the optimal case θ s = θ c ±π/2 and N = 10, the fidelity of squeezed state transfer is F = 0.978, which is somewhat lower than for a vacuum state, a consequence of the higher sensitivity of squeezed states to decoherence. C.6 Conclusion In summary, we have analyzed the optically mediated interaction between multimode phonon fields in cavity optomechanical systems where it is possible to adi-

138 137 F Θ s Θ c Figure C.7: State transfer fidelity for squeezed vacuum noise with N = 1 (red, solid), N = 10 (blue, dashed) and vacuum noise (gray, dotted) as a function of the phase difference between the squeezing input and the cavity field. The initial states of the BEC and the moving mirror are the ground state and a thermal state with mean phonon number n = 1, respectively. F Θ s Θ c Figure C.8: Same as Fig. C.7, but for the BEC initially in a squeezed state with position quadrature variance of and momentum quadrature variance of 10 and and the moving mirror in a thermal state with mean phonon number n = 1, respectively.

139 138 abatically eliminate the electromagnetic field. A number of possible forms of this interaction were identified, including multimode beam-splitter interactions and, in the case of quadratic optomechanical coupling, a plethora of possible nonlinear effective interactions, such as analogs of second-harmonic generation and four-wave mixing in nonlinear optics. The specific forms of the effective interaction depend sensitively on the sign of the couplings. Importantly, we found that incoherent effects due to quantum fluctuations associated primarily with the elimination of the electromagnetic field need not overwhelm coherent mode-coupling effects. As a result one can expect that analogs of nonlinear effects such as four-wave mixing, soliton generation, the creation of entangled phonon pairs, and others should be achievable in quantum acoustics. Our treatment was kept sufficiently generic that it can be applied to a wide variety of systems and parameter regimes, and as a concrete example we have discussed quantum state transfer between a momentum side-mode of a Bose-Einstein condensate and the center-of-mass mode of oscillation of the end mirror of a Fabry-Pérot interferometer. We showed how decoherence effects can be reduced significantly in that case by the use of squeezed optical fields. Future work will expand these results to more general situations and apply it to the study of practical situations where the effective mode coupling is nonlinear and investigate particular effects such as phase conjugation, entanglement, and the creation of entangled phonon pairs, for example. We will also exploit pulsed optomechanics ideas to further expand the toolbox of quantum optomechanics. Acknowledgments We thank E. M. Wright for stimulating discussions. This work is supported in part by the National Science Foundation, ARO, and the DARPA ORCHID and QuASAR programs through grants from AFOSR and ARO.

140 139 REFERENCES [1] J. D. Teufel et al., Nature 475, 359 (2011). [2] J. Chan et al., Nature 478, 89 (2011). [3] A. D. O Connell et al., Nature 464, 697 (2010). [4] A. H. Safavi-Naeini et al., Phys. Rev. Lett. 108, (2012). [5] S. Gröblacher et al., Nature 460, 724 (2009). [6] J. D. Teufel et al., Nature 471, 204 (2011). [7] E. Verhagen et al., Nature 482, 63 (2012). [8] F. Brennecke et al., Science 322, 235 (2008). [9] S. Gupta et al., Phys. Rev. Lett. 99, (2007). [10] J. Zhang, K. Peng and S. L. Braunstein, Phys. Rev. A 68, (2003). [11] L. Tian, and H. Wang, Phys. Rev. A 82, (2010). [12] Y. Wang and A. A. Clerk, Phys. Rev. Lett. 108, (2012). [13] L. Tian, Phys. Rev. Lett. 108, (2012). [14] S. Singh et al., Phys. Rev. A 86, (2012). [15] H. Seok et al., Phys. Rev. A 85, (2012). [16] J. Suh et al., Nano Lett. 10, 3990 (2010).

141 140 [17] A. A. Clerk, F. Marquardt, and J. G. E. Harris, Phys. Rev. Lett. 104, (2010). [18] P. Rabl et al., Nature Physics 6, 602 (2010). [19] V. Fiore et al., Phys. Rev. Lett. 107, (2011). [20] K. Stannigel et al., Phys. Rev. Lett. 109, (2012). [21] K. Stannigel et al., Phys. Rev. A 84, (2011). [22] I. Pikovski et al., Nature Physics 8, 393 (2012). [23] L. F. Buchmann et al., Phys. Rev. Lett. 108, (2012). [24] T. J. Kippenberg and K. J. Vahala, Science 321, 1172 (2008). [25] F. Marquardt and S. M. Girvin, Physics 2, 40 (2009). [26] M. Aspelmeyer et al., J. Opt. Soc. Am. B 27, A189 (2010). [27] G. Heinrich and F. Marquardt, EPL (2011). [28] G. Heinrich et al., C. R. Physique 12, 837 (2011). [29] P. Kómár et al., arxiv: v1 (2012). [30] A. Tomadin et al., Phys. Rev. A 86, (2012). [31] M. Bhattacharya and P. Meystre, Phys. Rev. A 78, (R) (2008). [32] F. Massel et al., Nature Commun. 3, 987 (2012). [33] G. Heinrich et al., Phys. Rev. Lett. 107, (2011). [34] C. A. Holmes, C. P. Meaney, and G. J. Milburn, Phys. Rev. E 85, (2012).

142 141 [35] M. J. Hartmann and M. B. Plenio, Phys. Rev.Lett. 101, (2008). [36] M. Ludwig, K. Hammerer, and F. Marquardt, Phys. Rev. A 82, (2010). [37] G. De Chiara, M. Paternostro, and G. M. Palma, Phys. Rev. A 83, (2011). [38] B. Rogers, M. Paternostro, G. M. Palma, G. De Chiara, Phys. Rev. A 86, (2012). [39] R. W. Boyd, Nonlinear Optics, 3rd ed. (Elsevier, 2008). [40] P. Meystre and M. Sargent III, Elements of Quantum Optics, 4th ed. (Springer- Verlag, 2007). [41] G. Lenz, P. Meystre and E. M. Wright, Phys. Rev. Lett. 71, 3271 (1993). [42] L. Deng et al., Nature 398, 218 (1999). [43] M. J. Collett and C. W. Gardiner, Phys. Rev. A 30, 1386 (1984). [44] U. Akram, N. Kiesel, M. Aspelmeyer and G. J. Milburn, New J. Phys. 12, (2010). [45] I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippenberg, Phys. Rev. Lett 99, (2007). [46] F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin, Phys. Rev. Lett 99, (2007). [47] C. Genes et al., Phys. Rev. A 77, (2008). [48] K. Jähne et al., Phys. Rev. A 79, (2009). [49] R. Benguria and M. Kac, Phys. Rev. Lett. 46, 1 (1981).

143 142 [50] A. Mari and J. Eisert, Phys. Rev. Lett. 103, (2009). [51] T. J. Kippenberg and K. J. Vahala, Optics Express 15, (2007). [52] S. Singh et al., Phys. Rev. Lett. 105, (2010). [53] C. W. Gardiner, Phys. Rev. Lett. 56, 1917 (1986).

144 143 APPENDIX D MULTIMODE STRONG-COUPLING QUANTUM OPTOMECHANICS H. Seok, L. F. Buchmann, E. M. Wright, and P. Meystre Published in Physical Review A Copyright 2013 by American Physical Society ABSTRACT We study theoretically the dynamics of multiple mechanical oscillators coupled to a single cavity field mode via linear or quadratic optomechanical interactions. We focus specifically on the strong coupling regime where the cavity decays much faster than the mechanical modes, and the optomechanical coupling is comparable to or larger than the mechanical frequency, so that both the optical and mechanical systems operate in the deep quantum regime. Using the examples of one and two mechanical oscillators we show that the system can classically exhibit bistability and bifurcations, and we explore how these manifest themselves in interference, entanglement, and correlation in the quantum theory, while revealing the impact of decoherence of the mechanical system due to cavity fluctuations and coherent driving. D.1 Introduction Cavity optomechanics has undergone tremendous growth owing to recent advances of nano- and micro-fabrication, culminating in the cooling of macroscopic mechanical objects to near the ground state [1, 2, 3, 4]. Further developments promise a variety of applications, for example precision measurements of feeble forces, masses and

145 144 displacements [5, 6, 7], coherent control of quantum states for quantum information science [8, 9, 10] and fundamental tests of quantum mechanics with macroscopic objects [11, 12, 13], to mention just a few. The linear optomechanical coupling of a cavity field to the position of a mechanical mode is typically modeled using a Fabry-Pérot cavity with a moving end mirror, and allows one to elucidate such effects as optical bistability, optical spring effects, and radiation-induced cooling of the mechanical oscillators [14, 15, 16, 17, 18, 19, 20]. Alternative experimental setups such as the membrane-in-the-middle geometry [21, 22, 23], cold atoms in a cavity [24, 25, 26, 27], and photonic crystal structures [3, 4], facilitate the investigation of quadratic or higher-order optomechanical interactions [28, 29], and opens up the exploration of multimode optomechanics [30, 31, 32, 33, 34, 35, 36, 37] including the possibility of entanglement between the mechanical modes [38, 39, 40, 41]. Most experiments to date are carried out in a regime where the optomechanical coupling is weak compared to the mechanical frequency and the cavity linewidth, and can be treated as a perturbation. When the cavity is driven by a classical field, the optomechanical coupling in this regime can be linearized, resulting in effects such as beam splitter swapping and parametric amplification. This linearized optomechanical interaction also accounts e.g. for radiation-induced cooling or amplification [15, 16, 17, 18, 19, 20], normal-mode splitting [42], and optomechanically induced transparency [43, 44, 45]. However, the linearized treatment needs to be revisited if the optomechanical interaction frequency becomes comparable to the frequency of harmonic trapping potential of the mechanics, in which case its intrinsic nonlinearity becomes of crucial importance even for cavity fields that only contain very few photons [46]. Here the role of quantum fluctuations can become of central importance. The so realized nonlinear optomechanics promises the generation of non-gaussian states [47], nonclassical mechanical steady-state [48], and quantum state control and optical

146 145 switching at a single photon level [49]. Such a strong-coupling regime has not been realized in nano-fabricated optomechanical systems, but optomechanical systems involving ultracold atomic clouds [24, 25, 26, 27] can currently operate in this regime. In this paper, we study theoretically optomechanical interactions in the singlephoton strong-coupling regime in which the optomechanical coupling is comparable to or stronger than the harmonic trapping potential of the mechanics, and cavity dissipation is the dominant source of damping. We first investigate the classical effective potential of the mechanical oscillators coupled to a single mode cavity field via a strong linear or quadratic coupling. The classical theory is used to elucidate selected examples that are then explored in the quantum theory. A quantum master equation is solved numerically to explore the dynamics of the mechanical oscillators and the effects of cavity fluctuations on the mechanics for the cases of one or two mechanical oscillators. We demonstrate that for linear coupling a single mechanical oscillator in the deep quantum regime does not exhibit the bistability that exists in the classical regime, a result that parallels the situation familiar from cavity QED [50, 51]. In the quadratic coupling case the cavity fields leads to splitting and recombination dynamics of the spatial wave-functions of the oscillators, realizing a quantum interferometer. We also demonstrate the contribution of cavity fluctuations on the coherence of the mechanical system and propose a way to increase the lifetime of that coherence. The remainder of this paper is organized as follows. Section II presents our basic model, introducing both linear and quadratic coupling of multimode mechanical systems to a single-mode optical field. Section III reviews how the adiabatic elimination of the optical field results in effective nonlinear dynamics for the mechanics and discusses its most important properties in the classical limit, concentrating of the form of the effective potential governing this dynamics for both linear and quadratic optomechanical couplings, and in the single-mode and two-mode cases. Section IV addresses the extension of the treatment to the quantum regime and discusses the

147 146 additional multiplicative quantum noise resulting from the elimination of the optical field. Section V then presents a number of examples for one and two mechanical oscillators and explores how quantum fluctuations manifest themselves in interference, entanglement, and correlations in the quantum theory, while revealing the impact of decoherence of the mechanical system due to cavity fluctuations. Finally Section VI is a summary and outlook. D.2 Basic Model In this section we introduce the quantized Hamiltonian for our system and evaluate the corresponding Heisenberg-Langevin equations of motion. We consider an optomechanical system composed of N identical mechanical oscillators, with effective mass m and frequency ω m, that are coupled via optomechanical interactions to a single cavity mode. The Hamiltonian describing the system is H = H opt + H m + H om + H loss, (D.1) where the cavity field Hamiltonian H opt = ω c â â + i (ηe iω Ltâ η e iω Ltâ), (D.2) describes the cavity mode of frequency ω c driven by an external field of frequency ω L with pumping parameter η, and H m = ω m 2 N (ˆp 2 k + ˆx 2 k) k=1 (D.3) is the mechanical Hamiltonian for the N identical modes, ˆx k and ˆp k being the dimensionless position and momentum operators for the k-th mirror, respectively. Here the dimensionless position and momentum operators can be obtained as their dimensional counterparts in units of x 0 = /(mω m ), p 0 = m ω m, (D.4)

148 147 so that [ˆx j, ˆp k ] = iδ jk. (D.5) Linear optomechanical interactions are described by the Hamiltonian H om = â â N g 0,kˆx k, k (D.6) whereas quadratic optomechanical interactions are accounted for using H om = â â N k g (2) 0,kˆx2 k, (D.7) where g 0,k and g (2) 0,k are the linear and quadratic single-photon coupling coefficients, respectively. Finally, H loss describes the interaction of the cavity field and the mechanical modes with their respective reservoirs and accounts for dissipation. For both linear and quadratic optomechanical interactions the Heisenberg- Langevin equations of motion for the cavity and mechanical modes may be evaluated using the standard input-output formalism [52], and we adopt a frame rotating with the laser frequency ω L. For the case of linear interactions described by the Hamiltonian (D.6) the operator Heisenberg-Langevin equations are where c ˆx j = ω mˆp j, (D.8) ˆp j = ω mˆx j + g 0,j â â γ 2 ˆp j + ˆξ, (D.9) [ ] N â = i c + g 0,kˆx k â κ 2 â + η + κâ in, (D.10) k = ω L ω c is the detuning between the pump and cavity frequencies, and κ(γ) and â in (ˆξ) are the cavity (mechanics) decay rate and corresponding noise operator. In a similar manner for the case of quadratic interactions described by

149 148 the Hamiltonian (D.7) the Heisenberg-Langevin equations are given by ˆx j = ω mˆp j, (D.11) ˆp j = (ω m + 2g (2) 0,j â â)ˆx j γ 2 ˆp j + ˆξ, (D.12) [ ] N â = i c â κ 2 â + η + κâ in. (D.13) k g (2) 0,kˆx2 k This completes the description of our basic model. D.3 Classical theory We first outline aspects of the classical theory that will be useful to frame the results of the quantum theory discussed in the next sections. We specifically consider the regime in which the cavity decay rate is much larger than all other system rates, including the mechanical frequency and decay rate, and the single-photon optomechanical coupling coefficients. In this regime we derive effective potentials for the mechanics for the cases of both linear and quadratic interactions. These allow us to identify interesting operating conditions for each case. D.3.1 Linear Interactions The classical theory applies when the cavity field and mechanical modes are sufficiently highly excited that fluctuations around their mean-field values may be neglected. In this limit the quantum operators may be replaced by their c-number expectation values, â α, ˆx j x j, ˆp j p j. For the case of linear interactions this leads to the mean-field equations ẋ j = ω m p j, (D.14) ṗ j = ω m x j + g 0,j α 2 γ 2 p j, (D.15) [ ] N α = i c + g 0,k x k α κ α + η. (D.16) 2 k

150 149 If the cavity decay rate κ is much larger than the decay rate of the mechanical modes and the coupling strengths, κ {γ, g 0,j }, the cavity field may be adiabatically eliminated on time scales greater than 1/κ to yield α(t) Substituting Eq. (D.17) into Eq. (D.15) then gives ṗ j = ω m x j + η i[ c + N k g 0,kx k (t)] + κ/2. (D.17) [ c + N k g 0,j η 2 g 0,kx k ] 2 + κ2 4 γ 2 p j, (D.18) where the first term of the right-hand-side is the harmonic restoring force for the mechanics, the second term describes the radiation pressure force due to the linear optomechanical coupling, and the last term is the mechanical damping force. Note that the mechanical modes are coupled through the radiation pressure force from the shared optical field. Solving Eq. (D.18) with ṗ j = 0 yields a set of N coupled equations for the mechanical mode positions x j,s in steady-state ( ) 2 N c + g 0,k x k,s + κ2 x j,s = g 0,j η 2. (D.19) 4 ω m k For the case that the coupling coefficients are identical, g 0,k g 0, the steady-state positions of the mechanical modes must also be identical x j,s = x s, since the quantity in the bracket in Eq. (D.19) has the same value for all mechanical modes, as does the right hand side of the equation. Here x s satisfies the cubic polynomial equation ] g0n 2 2 x 3 s + 2 c g 0 N x 2 s + [ 2c + κ2 x s g 0 η 2 = 0. (D.20) 4 ω m The number of physical solutions for the steady-state position x s depends on the discriminant of Eq. (D.20). For a positive discriminant there can be three solutions, with possible multistability, whereas a single solution results for a negative discriminant. Since the formula for the discriminant is rather long and complicated we

151 150 make use of an alternative procedure to assess the number of the physical solutions: A necessary condition for the discriminant in Eq. (D.20) to be positive is that the first derivative of the equation with respect to x s should have two distinct roots, yielding the condition c 3 κ > 2. (D.21) When this condition is satisfied the mechanics can exhibit three steady-state solutions for appropriate values of the cavity pumping rate η. The stability of the steady-state position of the mechanics can be investigated by linearizing the equations of motion (D.14) and (D.18) for small mechanical fluctuations. Here we employ the alternative approach of examining the effective potential U eff for the mechanics in terms of which Eq. (D.18) can be written as ṗ j = 1 U eff x j in the absence of dissipation, where U eff = ω m 2 N k x 2 k 2 η 2 κ arctan [ c + N k κ/2 g 0 k x k ]. (D.22) We next present two examples of the effective potential to highlight interesting features of the classical model with linear interactions, with a view to exploring these further in the quantum theory. As a first example Fig. D.1 shows the effective potential U eff (x) as a function of the dimensionless mechanical position x for a single mode, N = 1, and for a variety of values of the normalized cavity pumping rate η /κ, with κ the field decay rate. The ratio of the cavity detuning to the decay rate is chosen as c /κ = 1.5, meaning that the condition in Eq. (D.21) is satisfied and multiple solutions are possible, other fixed parameters being given in the figure caption. Here and in all the following figures, we use dimensionless positions. What Fig. D.1 illustrates is that for the lower values of the cavity pumping rate the effective potential has a single stable minimum, the dimensionless position x of the minimum increasing monotonically from zero with increasing pumping rate. For large enough pumping rates, however, a double-well effective potential arises

152 151 U eff ħκ x 0.2 Figure D.1: Effective potential U eff (x), in units of κ, versus the dimensional position x for a single mechanical oscillator linearly coupled to a cavity mode, and for the several values of the normalized cavity pumping rate: η /κ = 0.14 (red dotted line), η /κ = 0.18 (orange dot-dash line), η /κ = 0.24 (green dashed line), η /κ = 0.34 (blue solid line). The potential with η /κ = 0.18 exhibits two local minima corresponding to stable solutions and one local maximum corresponding to an unstable solution. Here ω m /κ = 0.01, g 0,1 /κ = 0.3, c /κ = 1.5. with two stable minima and one unstable maximum (see the orange dot-dash curve for η /κ = 0.18), indicating bistability in the mechanical response. For still larger pump rates the effective potential again exhibits a single stable minimum position displaced from the origin. Thus, as is well known, even for a single mechanical mode and linear interactions a bistable mechanical response can arise [21, 53]. We return to this example in Sec. D.5.1 to discuss the impact of quantum fluctuations on that behavior. As a second example we consider the case of two mechanical modes with linear interactions and with equal values of the coupling strengths, g 0,1 = g 0,2. Figure D.2 shows the effective potential U eff (x 1, x 2 ) versus the dimensionless positions x 1 and x 2 of the two modes for the parameters given in the figure caption. What is interesting about this case is that there are two stable minima of equal depth, situated on

153 x 2 x Figure D.2: Effective potential U eff (x 1, x 2 ), in units of κ, as a function of the dimensionless positions x 1 and x 2 for two mechanical oscillators linearly coupled to the cavity mode. Here ω m /κ = 0.01, g 0,1 /κ = g 0,2 /κ = 0.3, c /κ = 1.5, η /κ = the line x 1 = x 2 by virtue of the equal coupling constants. We shall explore the possibility that in the quantum regime this effective potential can lead to entangled quantum states in Sec. D.5.1. D.3.2 Quadratic Interactions Following the same procedure of replacing the quantum operators with their c- number expectation values to realize the classical theory, we find the following mean-

154 153 field equations of motion for the case of quadratic interactions ẋ j = ω m p j, (D.23) ṗ j = (ω m + 2g (2) 0,j α 2 )x j γ 2 p j, (D.24) [ ] N α = i c α κ 2 α + η. (D.25) k g (2) 0,k x2 k Then upon adiabatically eliminating the cavity mode field as before we obtain the equations of motion for the mechanical modes ṗ j = ω m x j 2g (2) 0,j η 2 [ c N k=1 g(2) 0,k x2 k ]2 + κ2 4 x j γ 2 p j. (D.26) In contrast to the case of linear interactions the radiation pressure force due to the quadratic interactions does not displace the mechanical oscillators, but rather shifts their mechanical frequencies, as is well known. Solving Eq. (D.26) with ṗ j = 0 yields for the steady-state displacements x j,s of the mechanical modes ω m + 2g (2) 0,j η 2 ( c N k g(2) 0,k x2 k,s ) 2 + κ 2 4 x j,s = 0. (D.27) In case all mechanical oscillators are located at local minima of the intracavity intensity, we have g (2) 0,j > 0 and the term in the square bracket in Eq. (D.27) is always positive. In that case, the only solution has zero displacement for all the mechanical modes. However, if the mechanical oscillators are located at local maxima of the intracavity intensity, we have g (2) 0,j < 0 and for certain parameter regimes Eq. (D.27) allows non-zero displacements. For the case of identical coupling coefficients g (2) 0,k = g (2) 0 the non-zero displacements obey the equation N k x 2 k,s = 1 g (2) 0 c ± 2 g (2) 0 η 2 ω m κ2 4. (D.28)

155 154 U eff ħκ x Figure D.3: Effective potential U eff (x), in units of κ, versus dimensionless position x for a single mechanical oscillator quadratically coupled to the cavity mode for η /κ = 0.05 (red dotted line), η /κ = 0.11 (orange dot-dash line), η /κ = 0.17 (green dashed line), η /κ = 0.20 (blue solid line). Here ω m /κ = 0.01, g 0,1/κ (2) = 0.2, c /κ = The stability of the steady-state positions can also be investigated via a qualitative analysis on the effective potential of the mechanics. For quadratic interactions the effective potential obtained from Eq. (D.26) in the absence of dissipation is [ U eff = ω N m x 2 k 2 η 2 c ] N k arctan g(2) 0,k x2 k. (D.29) 2 κ κ/2 k Next we present three examples of the effective potential that highlight interesting features of the classical model with quadratic interactions, with a view to exploring the quantum version of the examples. For a first example we consider a single mechanical mode with quadratic coupling to the cavity mode. The steady-state solutions x k,s = x s for this case are determined by Eq. (D.28) with N = 1, which yields a fifth-order polynomial equation with up to five solutions (x s = 0 is always a solution). In contrast to the case of linear coupling, the necessary condition for the mechanical oscillator to have multiple equilibrium

156 155 x s 2 x s η 1 κ η 2 κ η κ η κ 1 2 Figure D.4: Steady-state dimensionless positions x s for a single mechanical oscillator with quadratic coupling as a function of the normalized cavity pumping rate η /κ. The stable and unstable solutions are denoted by the solid blue and dashed red line, respectively. Inset: zoomed in region around the bifurcation point. Here ω m /κ = 0.01, g 0,1/κ (2) = 0.2, c /κ = position is simply that the single-photon coupling coefficient be negative, that is, that the mirror be trapped at a maximum of the intracavity intensity. With a negative coupling coefficient the cavity pumping rate η determines the number of physical solutions for x s. More specifically, below the first critical pumping rate η 1 = ωm κ 2 8 g (2) 0,1 (D.30) the mechanical oscillator experiences a flattened harmonic-like effective potential and has only one stable equilibrium position at x = 0. If the cavity pumping rate is increased such that where the second critical pumping rate η 2 is given by η 2 = η 1 < η < η 2, (D.31) ωm 2 g (2) 0,1 [ ] 2 c + κ2, (D.32) 4

157 156 the oscillator experiences a potential with five extrema, three stable equilibrium positions, including x = 0, and two unstable equilibrium positions. For still stronger pumping rates larger than η 2, the zero displacement becomes unstable and the mechanical mode experiences a symmetric double well potential with only two stable equilibrium positions. Figure D.3 illustrates these features and shows the effective potential U eff (x) versus dimensionless position x for a single mechanical oscillator quadratically coupled to the cavity mode, at a maximum of the intracavity intensity. For our parameters the critical cavity pumping rates are η 1 /κ = and η 2 /κ = An alternative view of these results is shown in Fig. D.4 where we plot the allowed steady-state dimensionless positions x s as a function of the normalized cavity pumping rate for the parameters of Fig. D.3. The inset shows an expanded view of the plot around the critical cavity pumping rates. What these results show is that around the critical pumping rates the system undergoes a subcritical bifurcation. For larger cavity pumping rates the system displays two stable and energetically degenerate solutions. In any given realization of the system we expect that one or other of the two solutions will arise with equal probability if the system is initialized from noise. We explore the quantum dynamics in that regime in Sec. D.5.2. For our second and third examples we consider two mechanical modes quadratically coupled to the cavity mode, either with both mechanical oscillators located at maxima of the intracavity intensity in which case the coupling constants are both negative g (2) 0,1 = g (2) 0,2 g (2) 0, (D.33) or with one oscillator located at a maximum and the other at a minimum, so that the coupling coefficients have opposite signs g (2) 0,1 = g (2) 0,2 g (2) 0. (D.34) The effective potential U eff (x 1, x 2 ) versus the dimensionless positions x 1 and x 2 of

158 157 U eff κ x 2 x 1 Figure D.5: Effective potential U eff (x 1, x 2 ), in units of κ, versus the dimensionless positions x 1 and x 2 for equal and negative quadratic optomechanical coupling coefficients. Here ω m /κ = 0.01, g (2) 0 /κ = 0.2, c /κ = 0.02, η /κ = 0.3. the mechanical oscillators can be obtained from Eq. (D.29). An example of the first case, g (2) 0,1 = g (2) 0,2, is shown in Fig. D.5. The key feature is that for large enough cavity pumping rates, the effective potential for the mechanical oscillators changes from a harmonic shape into a sombrero or Higgs potential, where the potential minimum is realized on a circle with radius R = 1 c ± g (2) 0 2 g (2) 0 η 2 ω m κ2 4. (D.35) Finally, Fig. D.6 illustrates the effective potential U eff (x 1, x 2 ) for the case in which one of the mechanical oscillators is located at a local minimum and the other oscillator is located at a local maximum of the intracavity intensity, g (2) 0,1 = g (2) 0,2, with the cavity pumping rate chosen large enough to change the harmonic trapping potential to a double well potential for x 2 and to stiffen the harmonic trapping potential for x 1. We shall explore quantum features of the two mode double well

159 158 U eff κ x 2 x 1 Figure D.6: Effective potential U eff (x 1, x 2 ), in units of κ, versus the dimensionless positions x 1 and x 2 for quadratic interactions of opposite signs, g 0,1/κ (2) = g 0,2/κ (2) = 0.2, ω m /κ = 0.01, c /κ = 0.02, η /κ = potential in Sec. D.5.2. D.4 Quantum effects We have so far investigated mean-field solutions of the mechanical modes, neglecting the effects of cavity field fluctuations on the mechanics. This analysis is valid when the cavity is sufficiently strongly pumped, and the cavity photon number is large enough, that the displacement of the mechanical oscillator is large compared to the natural harmonic oscillator length. However, this needs not be the case in the single photon strong coupling regime, where small photon numbers can still lead to optomechanical couplings comparable to the mechanical frequency, and both cavity and mechanical systems can be in the deep quantum regime. In this case quantum fluctuations become significant and can enter the quantum dynamics of the mechanics in interesting ways.

160 159 The remainder of this article presents numerical results that illustrate the dynamics of one or two mechanical modes coupled to the single cavity mode via either linear or quadratic optomechanical interactions in the single-photon strong-coupling regime g 0,j ω m > γ. We concentrate on the prevalent case where the cavity decay rate κ is much larger than the single-photon coupling coefficient, so that κ g 0,j ω m γ. In this regime, the cavity field follows the dynamics of the mechanical mode and nonlinear quantum effects can be observed at a single-photon level. This regime can be realized in optomechanical systems involving ultracold atoms [24, 25, 26, 27] but has not yet been reached in current state-of-the-art micromechanical systems. For concreteness in our quantum simulations we adopt representative parameters from the experiment in Ref. [25], namely, the cavity decay rate κ = 2π 2.6 MHz, mechanical frequency ω m = 2π 15.2 khz, and effective single-photon coupling coefficient g 0,1 = 2π 0.5 MHz. In units such that κ = 1, these are equivalent to ω m = , g 0,1 = We concentrate on the case where the laser is red-detuned with respect to the cavity resonance. Before presenting results based on a direct numerical integration of the master equation for the oscillator-light system that is, without adiabatic elimination of the optical field we discuss briefly the Heisenberg-Langevin equations of motion for the mechanical oscillators in order to capture more intuitively perhaps the effects of cavity fluctuations on the dynamics of the mechanical mode. D.4.1 Heisenberg-Langevin equations Starting from the quantum mechanical Heisenberg-Langevin equations for the cavity and mechanical modes of Sec. D.2, in the regime where the cavity decay is the dominant rate we may adiabatically eliminate the cavity mode while retaining the quantum noise terms. For the case of linear coupling this yields the effective

161 160 Heisenberg-Langevin equations of motion for the mechanical oscillator ˆx j = ω mˆp j, (D.36) g 0,j η ˆp 2 j = ω mˆx j + ( c + ) 2 N k g 0,kˆx k + κ2 /4 + g 0,j η ˆζ ( i c + ) + H.c. γ + g 0,j ˆζ ˆζ N k g 0,kˆx k + κ/2 2 ˆp j + ˆξ, (D.37) where the cavity noise operator ˆζ involving the cavity input noise â in is ˆζ(t) κ t 0 dτe (i c κ/2)(t τ) â in (τ). (D.38) Following the same procedure for the case of quadratic coupling yields the effective Heisenberg-Langevin equations ˆx j = ω mˆp j, (D.39) ˆp j = ω mˆx j 2g (2) 0,j ( η 2 c 2 ˆx j N k k) g(2) 0,kˆx2 + κ2 /4 2g (2) 0,j η ˆζ ( i c N k g(2) 0,kˆx2 k ) + H.c. ˆx j + κ/2 2g (2) 0,j ˆζ ˆζ ˆxj γ 2 ˆp j + ˆξ. (D.40) For both linear and quadratic interactions the second term on the right-hand-side of the equations of motion (D.37) and (D.40) for the mechanical momenta is independent of the cavity noise, but rather derives from the effective potential. In contrast, the third and fourth terms explicitly involve the additional random forces due to the quantum fluctuations of the optical field. These are in addition to the intrinsic random forces associated with their direct coupling to a heat bath. Importantly, in both the case of linear and quadratic coupling the additional noise experienced by the mechanical modes is multiplicative, with consequences that will be discussed in the next section.

162 161 D.4.2 Master equation The master equation describing the evolution of the total density operator prior to adiabatic elimination of the cavity field is [54] d dt ˆρ = i [Ĥ, ˆρ] + κ 2 D[â]ˆρ + γ 2 N D[ˆb k ]ˆρ, k=1 (D.41) where ˆb k is the annihilation operator for the k-th mechanical mode, and we have assumed that the cavity and mechanical modes are both coupled to reservoirs at zero temperature for simplicity, so that D[ô]ρ = (2ôρô ô ôρ ρô ô). (D.42) For small enough numbers of photons and phonons and small number of mechanical modes the size of the relevant Hilbert space remains manageably small and it is possible to solve that master equation directly by brute force, without resort to the adiabatic elimination of the cavity field. We proceed by expanding the density matrix in the Fock states basis {n a, n b1,.., n bn } as ˆρ = ρ na,ma,n b1,m b1,.. n a, n b1,.., n bn m a, m b1,.., m bn, and verify that the Hilbert space is large enough to avoid boundary issues and that the norm of the density operator is preserved at all times. The master equation of Eqs. (D.41) and (D.42) is of the standard form familiar from quantum optics. However, given the strong-coupling regime considered here one may inquire whether it would be more appropriate to use a master equation formulated in the dressed state basis that incorporates the coherent mixing between the photon and phonon modes [55]. In particular, in the strong coupling regime the mixing between photons and phonons that characterizes the dressed states obtained in the limit of zero dissipation is significant. If the cavity and mechanical decay rates were small compared to the mechanical frequency and single-photon

163 162 coupling coefficient the dressed state approach would be the way to proceed as they would display a reasonable robustness against dissipation. However here it is assumed from the outset that the cavity decay rate κ is much larger than both the mechanical frequency and the single-photon coupling coefficient. In that case the dressed states are not robust against cavity dissipation and quickly decay, and the more appropriate master equation has the standard form of Eqs. (D.41) and (D.42), which does not reflect the coherent mixing. We remark that in obtaining this master equation a Markovian approximation was employed, implying that the cavity noise and oscillator noise are broadband. In the event that the noise was narrow band the master equation would also need to be revisited. D.5 Results D.5.1 D Linear interactions Single-mode mechanics This subsection considers the case of a single mechanical mode linearly coupled to the optical field, using the same parameters as in Fig. D.1, and addresses how quantum fluctuations, in particular the multiplicative noise of Eq. (D.37), impact the mean-field bistable behavior. It is known that deep in the quantum regime quantum fluctuations eradicate the possibility that the system dwells in one or other of the two classically allowed states [50]. This was demonstrated experimentally at the single atom and single photon level in cavity QED experiments with ultracold atomic beams [51, 56]. Not surprisingly, a similar situation occurs here for the center-ofmass of the mechanics. This is illustrated in Fig. D.7 which shows the quantum expectation value x of the dimensionless position operator for the mechanical oscillator in steady state versus the normalized cavity pumping rate η /κ (solid blue line), along with the classically allowed positions (red dash line). Recall that the negative slope region of the classical solution is unstable.

164 163 x Η Κ Figure D.7: Expectation value ˆx versus normalized cavity pumping rate η /κ (solid blue line) for a single mechanical oscillator and linear optomechanical interaction. Here ω m /κ = 0.01, g 0,1 /κ = 0.3, c /κ = 1.5, γ/κ = The red dashed curve shows the corresponding classical bistable solution. (a) P (x), U eff (x)/ κ 0.4 (b) (c) x (d) Figure D.8: Steady-state position probability distribution P (x) versus dimensionless position x (solid blue lines) of the mechanical oscillator along with the effective potential U eff (x) (red dash lines), in units of κ, for the normalized cavity pumping rates η /κ (a) 0.14, (b) 0.18, (c) 0.24, (d) 0.34 for a single mechanical oscillator and linear optomechanical interaction. Here ω m /κ = 0.01, g 0,1 /κ = 0.3, c /κ = 1.5, and γ/κ =

165 164 To further clarify the washing out of the mechanical bistability the solid blue lines in Fig. D.8 show the steady-state position probability distribution P (x) P (x, t ) = x ˆρ m (t ) x, (D.43) where ˆρ m (t ) is the reduced density matrix for the mechanical subsystem in the steady state and x is the eigenstate of the dimensionless position operator ˆx, for several values of the normalized cavity pumping rate η /κ, the dashed red lines being the corresponding effective classical potential U eff (x) in Eq. (D.22). As expected, Fig. D.8(b) displays a bimodal probability density for a cavity pumping rate for which the classical theory predicts bistability. We note, however, that the absolute peak of the P (x) distribution does not correspond to the absolute minimum of the classical potential, as would be expected on the basis of additive noise. This can be intuited by realizing that, for a single mechanical mode, the cavity noise operator ˆζ in the third term of Eq. (D.37) appears in conjunction with a cavity resonant denominator involving the mode position operator, meaning that this noise source is multiplicative. The (classical) lower branch of the bistability curve therefore corresponds to lower intracavity fields than the upper branch, and therefore less quantum noise. For this reason, the shallower minimum of the potential is rendered more stable than the deeper minimum against quantum noise. As such, this behavior is a direct consequence of the multiplicative nature of the noise. D Two-mode mechanics We now turn to the case of two mechanical modes of equal frequency ω m and equal linear optomechanical coupling to the optical field mode. To set the stage we first ignore cavity field fluctuations and determine the quantum mechanical ground state wave function ψ 0 (x 1, x 2 ) of the effective potential U eff (x 1, x 2 ), given by ω [ ] m d 2 + d2 ψ 2 dx 2 1 dx U eff (x 1, x 2 )ψ 0 = E 0 ψ 0, 2 (D.44)

166 165 ψ 0 x 2 x 1 Figure D.9: Ground state wave function ψ 0 (x 1, x 2 ) as a function of the dimensionless positions x 1 and x 2. Same parameters as in Fig. D.2. where E 0 is the energy eigenvalue, using the imaginary time propagation method. The ground state wave function, assumed real and positive, is plotted in Fig. D.9. As expected from the effective potential, it has two peaks localized at the local minima of U eff (x 1, x 2 ). We next make use of the Schmidt decomposition of ψ 0 (x 1, x 2 ) in order to determine whether the two mechanical oscillators in the ground state can be separated or not. It is known that subsystems are entangled if their Schmidt number, or the number of nonzero Schmidt coefficients, is greater than unity [57]. Decomposing ψ 0 (x 1, x 2 ) as ψ 0 = λ i i m1 i m2, (D.45) i where λ i are the Schmidt coefficients with respect to the basis i m1 i m2, m 1 and m 2 labeling the two mechanical oscillators, we find for the case at hand that the non-zero Schmidt coefficients λ 1 = 0.96, λ 2 = 0.29, λ 3 = 0.02, and λ 4 = 0.02 in descending order. Since the Schmidt number is greater than unity the ground state of the two

167 166 x 1 m1, x 2 m x Figure D.10: Position representation of the Schmidt basis states x 1 m1 (red solid line) and x 2 m1 (blue dashed line) for the two mechanical oscillators in the ground state ψ 0 (x 1, x 2 ). Same parameters as in Fig. D.2. oscillators is entangled, with the state dominated by the first two Schmidt states 1 m1 1 m2 and 2 m1 2 m2. Their position representation is illustrated in Fig. D.10. So far we have neglected the effects of cavity mode fluctuations. In order to ascertain the contribution of cavity fluctuations and decoherence on the dynamics of the mechanical system, we finally determine the evolution of the initial state ψ 0 (x 1, x 2 ) including cavity and mechanical damping, and quantify the correlations between the mechanical oscillators through their quantum mutual information I(ˆρ m ) = S(ˆρ m1 ) + S(ˆρ m2 ) S(ˆρ m ). (D.46) Here S(ˆρ m ) is the quantum joint entropy or simply entropy of the composite mechanical system, and S(ˆρ m1 ) and S(ˆρ m2 ) are the von Neumann entropies of the individual mechanical oscillators, with S(ˆρ) = Tr[ˆρ ln ˆρ]. (D.47) Figure D.11 shows the time evolution of the entropy of the individual mechanical oscillators (green dotted line), which is the same for both oscillators, their joint

168 167 S Ρ, I Ρ Κ t Figure D.11: Von Neumann entropies and quantum mutual information of the mechanical system as a function of normalized time κt. Green dotted line: entropies of the individual mechanical oscillators; red dashed line: their quantum joint entropy; blue solid line: quantum mutual information of the composite system. Here γ 1 = γ 2 = κ, other parameters as in Fig. D.2. quantum entropy (red dashed lines), and their quantum mutual information (blue solid line). The initial ground-state entanglement between the two mechanical oscillators is apparent from the fact that the quantum entropies of the mechanical subsystems are nonzero while their joint entropy vanishes [57]. Both the joint entropy and the entropy of the individual oscillators tend to increase with time due to both the random radiation pressure variations arising from cavity intensity fluctuations and mechanical damping. We also note that the quantum mutual information between the two oscillators (blue solid line) is maintained in the long-time limit, indicative of the correlations between the two mechanical subsystems resulting from their optically mediated interaction via a common cavity mode.

169 168 D.5.2 D Quadratic interactions Single-mode mechanics As in Sec. D.3 we first consider a single mechanical mode with negative single-photon optomechanical coupling coefficient. We determined (see Fig. D.3) that classically the system undergoes a subcritical bifurcation for sufficiently large cavity pumping rate. Here we investigate the impact of quantum fluctuations on the associated dynamics. Figure D.12 shows the normalized time (κt) evolution of the oscillator spatial probability distribution P (x, t) for a variety of cavity pumping rates, obtained by direct numerical solution of the master equation for the oscillator-field system. In this example the mechanical mode is initially in its ground state and the optical field in the vacuum, and the cavity pumping rate is switched on suddenly at t = 0 to a constant value η. The successive panels show the effect of increased η /κ. For pumping below the bifurcation point which occurs at η /κ 0.08 for the parameters of the figure see panels (a) and (b), the variance in position of the mechanics increases as a consequence of the flattening of the effective potential as the bifurcation point is approached from below, see Fig. D.3. For cavity pumping rates past the bifurcation point, see panels (c)-(f), U eff is a double-well potential with the zero displacement point x = 0 unstable and two stable and degenerate minima. From a classical perspective, above that point we expect fluctuations to drive the system into one or other of these two minima. Panels (d)-(f) of Fig. D.12 show that quantum mechanically the mechanical mode, initially localized around x = 0, undergoes oscillations involving both potential minima. Taking panel (f) as an example, we see that the initial quantum wave packet splits symmetrically between both wells, reverses at the turning point of the double-well potential at κt 200, and the split wave-packet components recombine at κt 400. Bifurcation-induced wave-packet splitting and subsequent recombination requires

170 169 x (a) (b) (c) (d) (e) (f) Figure D.12: Normalized time (κt) evolution of the spatial probability P (x, t) for a mechanical mode initially in its ground state, and for the optical pumping rates η /κ (a) 0.05, (b) 0.08, (c) 0.11, (d) 0.14, (e) 0.17, and (f) In each panel the vertical axis is the dimensionless position x, and P (x, t) is color coded. See the potential U eff of Fig. D.3 and Fig. D.4, which are for the same set of parameters, for reference. Here γ = 10 3 κ. κt

171 170 the cavity pumping rate to be sufficiently above the bifurcation point so that the split wave-packet components become well separated spatially: This conforms to the usual notion that a quantum phase transition will be smoothed out close to the bifurcation point, in comparison to a classical bifurcation that occurs discretely. This basic process can repeat several times but with diminishing contrast due to the combined action of optical and mechanical decoherence, see panels (e)-(f). The wave-packet splitting and subsequent recombination shown between κt = [0, 400] in panel (f) of Fig. D.12 is reminiscent of a Mach-Zehnder interferometer. This suggests exploiting this system for observing quantum interferences, provided that decoherence remains manageable. To explore the relative contributions of the optical and mechanical damping to decoherence we now calculate the evolution of the mechanical mode initially prepared in a coherent superposition of wave packets localized at non-zero displacements ±β 0 with different relative phase φ 0 ψ m (0) = 1 2 ( β 0 + e iφ 0 β 0 ), (D.48) and for two different values of the mechanical damping. The idea of this simulation is that the two components of the coherent superposition are representative of the wave-packet components resulting from the bifurcation-induced splitting at κt 200 in panel (f) of Fig. D.12. The subsequent time evolution of the probability density P (x, t) for these cat states is plotted in Fig. D.13 for a cavity pumping rate beyond the bifurcation point. Panel (a) is for γ = 10 3 κ and φ 0 = 0, while the subsequent panels are for γ = 10 6 κ and various values of φ 0. The bottom four panels show that for the case of negligible mechanical damping (on the time scale of the plots) the probability near zero displacement depends on the initial relative phase φ 0, a clear signature of a quantum interference effect. Interferences reappear periodically for longer times, but with a slowly decreasing amplitude due to the decoherence resulting from the quantum fluctuations of the optical field. What is perhaps surprising is however that the interferences subsist

172 171 x (a) (b) (c) (d) (e) Figure D.13: Normalized time (κt) evolution of the position probability distribution P (x, t) for an oscillator initially in the cat state (D.48) with β 0 = 1.5 and the relative phases (a) φ 0 = 0, (b) φ 0 = 0, (c) φ 0 = π/2, (d) φ 0 = π, and (e) φ 0 = 3π/2. In each panel the vertical axis is the dimensionless position x, and P (x, t) is color coded. Same parameters as in Fig. D.3, with cavity pumping rate η /κ = The mechanical decay rate is γ = 10 3 κ in panel (a) and γ = 10 6 κ in panels (b)-(e). κt

173 172 for remarkably long times, thousands of cavity decay times κ 1. That quantum coherence can persist on such long time scales is attributed to the coherent pumping of the cavity mode. It is known, see e.g. Refs. [58, 59], that the coherent pumping of Schrödinger cats can result in maintaining their coherence for arbitrarily long times. In the specific case of coherently driven micromasers for example, it was shown that the onset of these superpositions resembles a second-order phase transition, with the control parameter being the ratio of the atomic injection rate to the cavity damping rate. In contrast, panels (a) and (b) illustrate the effect intuitively expected from mechanical dissipation. All parameters are identical in these panels, except that γ = 10 3 κ in (a) and γ = 10 6 κ in (b). As expected, the first interference peak visible in panel (b) is already significantly reduced in case (a) after a time of about 0.2γ 1, and all but extinguished after a time γ 1. Returning to the remarkably slow optically induced decoherence, Fig. D.14 shows on a semi-log scale P (x = 0, t) P (x = 0, t ) versus normalized time κt for a case of negligible mechanical damping, γ = 10 6 κ, the red dashed straight line being a fit through the peak maxima that illustrates an effective exponential decay rate about 3 order of magnitude slower than κ 1 for κt > 1000, but that starts off faster and non-exponentially [60]. We attribute the initial decay to the multiplicative nature of the noise due to cavity mode fluctuations appearing in the third term on the right-hand-side of Eq. (D.40). First, we observe that this term gives rise to fluctuations in the frequency experienced by the mechanical mode, and it is known that such frequency fluctuations can translate into an effective decay [54]. Second, it has a resonant denominator that assumes its smallest value, and hence gives the largest loss, when the mechanical mode has a position in the vicinity of the minima of the double-well effective potential, whereas the loss will be relatively small when the oscillator is in the vicinity of x = 0. With reference to Fig. D.13(b) we see that between the first and second peaks, that occur at κt 190 and κt 540, the probability density

174 173 P (0, t) P (0, t ) κt Figure D.14: Semi-log plot of the position probability distribution P (x = 0, t) P (x = 0, t ) as a function of normalized time (κt) (blue, solid curve). The red straight line is a fit to the long time maxima of the distribution. Same parameters as in Fig. D.13(b). P (x, t) undergoes a transient and recurs close to the initial form P (x, 0), which is centered around the minima of the effective potential. Thus there is sizable loss between the first two peaks. However, between subsequent pairs of neighboring peaks there is less of a recurrence, and the decay rate between peaks decreases with increasing time. Eventually P (x, t) approaches a near steady-state and the loss rate becomes exponential. We remark that optical decoherence can also be reduced by using a pulsed rather than a continuous wave laser to excite the cavity, pulsed optomechanics having been previously studied in the context of squeezing of the position uncertainty of a mechanical oscillator [61]. Thus, using the bifurcation-induced wave-packet splitting followed by evolution of the subsequent wave packet in the harmonic potential of the mechanical mode offers a route to observing quantum interference effects alluded to here over times much longer than the optical decoherence time but of course shorter than the inverse mechanical decay rate. This suggests that a single mode with

175 174 quadratic optomechanical coupling, coherent driving and extremely slow mechanical damping is a viable candidate for producing quantum interference effects resulting from the bifurcation induced splitting. D Two-mode mechanics D Equal Couplings We finally turn to the case of two mechanical modes with quadratic interactions, considering as in section D.3 both the cases of equal negative coupling coefficients and of coupling coefficients of equal magnitude but opposite sign. In the first case the classical dynamics of the mechanics is captured by the effective potential U eff (x 1, x 2 ) of Fig. D.5, which has the form of a sombrero (or Higgs) potential for sufficiently large cavity pumping rate. To exploit its rotational symmetry we introduce the angular momentum in the (x 1, x 2 ) plane ˆL φ = ˆx 1ˆp 2 ˆx 2ˆp 1, (D.49) with Heisenberg-Langevin equation of motion ˆL φ = γ 2 ˆL φ + ˆξ φ, (D.50) and we have introduced the noise operator ˆξ φ = ˆx 1 ˆξ2 ˆx 2 ˆξ1. (D.51) As expected from the symmetry of the potential, ˆL φ is a constant of motion in the absence of mechanical dissipation. Importantly, the angular momentum is insensitive to cavity fluctuations and the associated decoherence, assuming as we have done that both mechanical modes are subject to the same optomechanical coupling. From Eq. (D.50) we have ˆL φ (t) = e γt/2 ˆL φ (0). (D.52) showing that the mean angular momentum decays to zero due to mechanical phase diffusion with the characteristic time scale of γ 1.

176 175 ψ 0 x 2 x 1 Figure D.15: Ground state of the effective potential describing the motion of the two mechanical modes for the case of quadratic interactions and coupling coefficients of equal magnitude but opposite sign. Same parameters as in Fig. D.6. D Opposite Couplings Perhaps more interesting is the situation where the two modes have optomechanical coupling constants of equal magnitude but opposite signs and are governed classically by the effective potential of Fig. D.6. The corresponding quantum mechanical ground state is plotted in Fig. D.15. Not surprisingly, it is symmetric about both the lines x 1 = 0 and x 2 = 0, and exhibits two peaks localized at the local minima of U eff. Physically this state corresponds to the oscillator 2 being in a cat state, whereas the oscillator 1 is a Gaussian centered at the origin x 1 = 0. The Schmidt number for this ground state is found numerically to be equal to one, which implies that it is separable. Figure D.16 shows the time evolution of the von Neumann entropies of oscillators 1 (orange dot-dashed line) and 2 (green dotted line), their joint entropy (red dashed line), and mutual quantum information (blue solid line). All entropies being initially equal to zero confirms that the ground state is separable, with both subsystems in pure states [57]. Under the influence of quantum noise from both the

177 176 S Ρ, I Ρ Κ t Figure D.16: Von Neumann entropies of the mechanical oscillators 1 (orange dot-dash line) and 2 (green dotted line), their joint quantum entropy (red dash line), and mutual quantum information (blue solid line). Same parameters as in Fig. D.6 with γ 1 /κ = γ 2 /κ = optical field and the mechanics the entropy of the oscillators then increases, with oscillator 1 experiencing an entropy increase that is much slower than oscillator 2., This is not surprising, since due to its cat-like nature the second oscillator is expected to be much more sensitive to decoherence. Eventually the increase in entropy of oscillator 2 reverses, and asymptotically the mechanics reaches a situation where entropy is distributed almost equally between the two oscillators. As was the case for linear optomechanical coupling, see Fig. D.11, the growth of their quantum mutual information (blue solid line) with time shows that the cavity fluctuations in fact correlate the two initially uncorrelated mechanical oscillators, that is, dissipation builds correlation via interaction of the two oscillators with a common light field or bath. The absence of mutual coherence between the two oscillators in the ground state begs the question of the extent to which the spatial coherence of oscillator 2 which is initially in a cat-like state associated with the double-well effective potential along

178 177 (a) x 2 B C D κt (b) A P (x 2 ), U eff (x 2 )/ κ B x 2 C D Figure D.17: (a) Normalized time (κt) evolution of the marginal probability distribution P (x 2, t) where the vertical axis is the dimensionless position x, and P (x, t) is color coded, and (b) the effective potential (red line), in units of κ, and probability distribution P (x 2, t) (blue line) for the times indicated in panel (a), with A corresponding to the initial time t = 0. Same parameters as in Fig. D.16, γ 1 /κ = γ 2 /κ = 10 3.