Optomechanics - light and motion in the nanoworld

Transcription

1 Optomechanics - light and motion in the nanoworld Florian Marquardt [just moved from: Ludwig-Maximilians-Universität München] University of Erlangen-Nuremberg and Max-Planck Institute for the Physics of Light (Erlangen) DIP project Dynamics of Electrons and Collective Modes in Nanostructures SFB 631

2 Radiation pressure Johannes Kepler De Cometis, 1619 (Comet Hale-Bopp; by Robert Allevo)

3 Radiation pressure Johannes Kepler De Cometis, 1619

4 Radiation pressure Nichols and Hull, 1901 Lebedev, 1901 Nichols and Hull, Physical Review 13, 307 (1901)

5 Radiation forces Trapping and cooling Optical tweezers Optical lattices

6 Optomechanics on different length scales 4 km LIGO Laser Interferometer Gravitational Wave Observatory

7 Optomechanics on different length scales 4 km Mirror on cantilever Bouwmeester lab, Santa Barbara LIGO Laser Interferometer Gravitational Wave Observatory atoms

8 Optomechanical systems input laser optical cavity cantilever

9 Optomechanical systems radiation pressure force input laser optical cavity cantilever

10 Basic physics: Statics F rad (x) =2I(x)/c λ 2F λ/2 Experimental proof of static bistability: A. Dorsel, J. D. McCullen, P. Meystre, E. Vignes and H. Walther: Phys. Rev. Lett. 51, 1550 (1983) V rad (x) V eff = V rad + V HO x hysteresis

11 Basic physics: dynamics F quasistatic finite sweep!rate sweep x finite cavity ring-down rate γ delayed response to cantilever motion finite optical ringdown time delayed response to cantilever motion F Fdx < 0 > 0 Höhberger-Metzger and Karrai, Nature 432, 1002 (2004): 300K to 17K [photothermal force] cooling 0 x heating (amplification) C. Höhberger!Metzger and K. Karrai, Nature 432, 1002 (2004) (with photothermal force instead of radiation pressure)

12 The zoo of optomechanical (and analogous) systems Karrai (Munich) Bouwmeester (Santa Barbara) Vahala (Caltech) Kippenberg (MPQ), Carmon,... Harris (Yale) LKB group (Paris) Mavalvala (MIT) Wineland (NIST) Lehnert (NIST) Schwab (Cornell) Stamper-Kurn (Berkeley) cold atoms Aspelmeyer (Vienna)...

13 Optomechanics: general outlook Bouwmeester (Santa Barbara/Leiden) Fundamental tests of quantum mechanics in a new regime: entanglement with macroscopic objects, unconventional decoherence? [e.g.: gravitationally induced?] Mechanics as a bus for connecting hybrid components: superconducting qubits, spins, photons, cold atoms,... Kapitulnik lab (Stanford) Tang lab (Yale) Precision measurements [e.g. testing deviations from Newtonian gravity due to extra dimensions] Optomechanical circuits & arrays Exploit nonlinearities for classical and quantum information processing, storage, and amplification; study collective dynamics in arrays

14 Towards the quantum regime of mechanical motion Schwab and Roukes, Physics Today 2005 nano-electro-mechanical systems Superconducting qubit coupled to nanoresonator: Cleland & Martinis 2010 optomechanical systems

15 Optomechanics (Outline) Introduction Displacement detection Linear optomechanics Nonlinear dynamics System Bath Quantum theory of cooling Interesting quantum states Towards Fock state detection Coupling to the motion of a single atom

16 Optical displacement detection input laser optical cavity cantilever reflection phase shift

17 Thermal fluctuations of a harmonic oscillator Classical equipartition theorem: Possibilities: extract temperature! Direct time-resolved detection Analyze fluctuation spectrum of x

18 Fluctuation spectrum

19 Fluctuation spectrum

20 Fluctuation spectrum

21 Fluctuation spectrum Wiener-Khinchin theorem

22 Fluctuation spectrum Wiener-Khinchin theorem area yields variance of x:

23 Fluctuation-dissipation theorem General relation between noise spectrum and linear response susceptibility susceptibility (classical limit)

24 Fluctuation-dissipation theorem General relation between noise spectrum and linear response susceptibility susceptibility (classical limit) for the damped oscillator:

25 Displacement spectrum T=300 K Experimental curve: Gigan et al., Nature 2006

26 Measurement noise

27 Measurement noise meas Two contributions to phase noise of 1. measurement imprecisionlaser beam (shot noise limit!) 2. measurement back-action: fluctuating force on system noisy radiation pressure force

28 Standard Quantum Limit (measured) + backaction noise + imprecision noise true spectrum

29 Standard Quantum Limit (measured) + backaction noise + imprecision noise imprecision noise (measured) full noise intrinsic fluctuations backaction noise true spectrum coupling to detector (intensity of measurement beam)

30 Standard Quantum Limit (measured) + backaction noise + imprecision noise imprecision noise (measured) full noise intrinsic fluctuations backaction noise true spectrum coupling to detector (intensity of measurement beam) Best case allowed by quantum mechanics: Standard quantum limit (SQL) of displacement detection...as if adding the zero-point fluctuations a second time: adding half a photon

31 Notes on the SQL weak measurement : integrating the signal over time to suppress the noise trying to detect slowly varying quadratures of motion : Heisenberg is the reason for SQL! no limit for instantaneous measurement of x(t)! SQL means: detect down to on a time scale Impressive:!

32 Enforcing the SQL (Heisenberg) in a weak optical measurement reflection phase shift: (here: free space) fluctuations: N photons arrive in time t Poisson distribution for a coherent laser beam 1. Uncertainty in phase estimation: 2. Fluctuating force: momentum transfer Uncertainty product: Heisenberg

33 Optomechanics (Outline) Introduction Displacement detection Linear optomechanics Nonlinear dynamics System Bath Quantum theory of cooling Interesting quantum states Towards Fock state detection Coupling to the motion of a single atom

34 Equations of motion cantilever input laser optical cavity

35 Equations of motion cantilever input laser optical cavity mechanical frequency mechanical damping equilibrium position radiation pressure detuning from resonance cavity decay rate laser amplitude

36 Linearized optomechanics (solve for arbitrary ) Optomechanical frequency shift ( optical spring ) Effective optomechanical damping rate

37 Linearized dynamics Effective optomechanical damping rate Optomechanical frequency shift ( optical spring ) laser detuning cooling heating/ amplification softer stiffer

38 Linearized dynamics Effective optomechanical damping rate Optomechanical frequency shift ( optical spring ) laser detuning cooling heating/ amplification softer stiffer

39 Cooling by damping T=300 K Thompson, Zwickl, Jayich, Marquardt, Girvin, Harris, Nature 72, 452 (2008)

40 Optomechanics (Outline) Introduction Displacement detection Linear optomechanics Nonlinear dynamics System Bath Quantum theory of cooling Interesting quantum states Towards Fock state detection Coupling to the motion of a single atom

41 Self-induced oscillations F Fdx < 0 > 0 x total

42 Self-induced oscillations F Fdx < 0 > 0 x total Beyond some laser input power threshold: instability Cantilever displacement x Time t Amplitude A

43 Self-induced oscillations x 0 Cantilever motion x(t) A Cavity intensity Γ α Output intensity Time FM, Harris, Girvin, PRL 96, (2006)

44 Self-induced oscillations cavity intensity oscillations: 0 x 0 Cantilever motion x(t) A Γ [analytical solution] α Cavity intensity 1. force balance: 0 0 Output intensity Time 2. power balance: FM, Harris, Girvin, PRL 96, (2006) Two equations for two unknowns:

45 Attractor diagram power fed into the cantilever 1 0 cantilever energy detuning FM, Harris, Girvin, PRL 96, (2006) Ludwig, Kubala, FM, NJP 2008

46 Attractor diagram power fed into the cantilever 1 power balance 0 cantilever energy detuning FM, Harris, Girvin, PRL 96, (2006) Ludwig, Kubala, FM, NJP 2008

47 Attractor diagram power fed into the cantilever 1 power balance 0 cantilever energy detuning FM, Harris, Girvin, PRL 96, (2006) Ludwig, Kubala, FM, NJP 2008

48 Attractor diagram 15 Color scale: power fed into cantilever Cantilever oscillation amplitude detuning FM, Harris, Girvin, PRL 96, (2006)

49 Attractor diagram 15 Color scale: power fed into cantilever Cantilever oscillation amplitude Multistability detuning FM, Harris, Girvin, PRL 96, (2006)

50 Attractor diagram 15 Color scale: power fed into cantilever Cantilever oscillation amplitude detuning FM, Harris, Girvin, PRL 96, (2006)

51 Attractor diagram 15 Color scale: power fed into cantilever Cantilever oscillation amplitude detuning FM, Harris, Girvin, PRL 96, (2006)

52 Attractor diagram 15 Color scale: power fed into cantilever Cantilever oscillation amplitude detuning FM, Harris, Girvin, PRL 96, (2006)

53 First experimental observation of attractor diagram Time-delayed bolometric force: Low optical finesse light intensity follows instantaneously: Ludwig, Neuenhahn, Metzger, Ortlieb, Favero, Karrai, FM, Phys. Rev. Lett. 101, (2008)

54 Comparison theory/experiment low laser power medium laser power Transmission Displacement x0 I/I0=1.8 % I/I0=8.3 % Amplitude Displacement x0 Metzger et take al., into a PRL 2008 displaceme employ a s 20 8 ẍ where x i d mode with and effecti 60 khz, Γ 1 diation pre the param now couple present set second mo adjustable ulation of two modes experimen between th

55 high laser power Comparison theory/experiment Coupled self-oscillations of two mechanical modes + + I/I0=100 % Displacement x 0 Displacement x 0 Metzger et al., PRL di th no pr se ad ul tw ex be I( no ou va ca at co its

56 Optomechanics (Outline) Introduction Displacement detection Linear optomechanics Nonlinear dynamics System Bath Quantum theory of cooling Interesting quantum states Towards Fock state detection Coupling to the motion of a single atom

57 Cooling with light input laser optical cavity cantilever Current goal in the field: ground state of mechanical motion of a macroscopic cantilever Classical theory: Pioneering theory and experiments: Braginsky (since 1960s ) optomechanical damping rate

58 Cooling with light input laser optical cavity cantilever Current goal in the field: ground state of mechanical motion of a macroscopic cantilever Classical theory: quantum limit? Pioneering theory and experiments: Braginsky (since 1960s ) shot noise! optomechanical damping rate

59 Cooling with light input laser optical cavity cantilever Quantum picture: Raman scattering sideband cooling Original idea: Sideband cooling in ion traps Hänsch, Schawlow / Wineland, Dehmelt 1975 Similar ideas proposed for nanomechanics: cantilever + quantum dot Wilson-Rae, Zoller, Imamoglu 2004 cantilever + Cooper-pair box Martin Shnirman, Tian, Zoller 2004 cantilever + ion Tian, Zoller 2004 cantilever + supercond. SET Clerk, Bennett / Blencowe, Imbers, Armour 2005, Naik et al. (Schwab group) 2006

60 Quantum noise approach System Bath (radiation field in cavity) (cantilever)

61 spectrum Quantum noise approach

62 Quantum noise approach spectrum bath provides energy bath absorbs energy transition rate

63 Quantum theory of optomechanical cooling Spectrum of radiation pressure fluctuations radiation pressure force photon number photon shot noise spectrum

64 Quantum theory of optomechanical cooling Spectrum of radiation pressure fluctuations S FF (ω) [norm.] κ detuning : cavity linewidth cavity emits energy / absorbs energy

65 Quantum theory of optomechanical cooling Cooling rate S FF (ω) [norm.] κ Quantum limit for cantilever phonon number cavity emits energy / absorbs energy FM, Chen, Clerk, Girvin, PRL 93, (2007) also: Wilson-Rae, Nooshi, Zwerger, Kippenberg, PRL 99, (2007); Genes et al, PRA 2008 Ground-state cooling needs: high optical finesse / large mechanical frequency experiment with Kippenberg group 2007

66 Towards the ground state phonon number 1x x10 9 1x10 8 1x10 7 1x MIT LKB Yale minimum possible phonon number IQOQI MPQ JILA Caltech Results from groups at MIT (Mavalvala), LKB (Pinard, Heidmann, Cohadon), Yale (Harris), IQOQI Wien (Aspelmeyer), MPQ Munich (Kippenberg), JILA Boulder (Lehnert), Caltech (Schwab)

67 Strong coupling: resonances of light and mechanics hybridize cantilever spectrum strong coupling splitting frequency driven cavity cantilever detuning FM, Chen, Clerk, Girvin, PRL 93, (2007)

68 Experiment: Optomechanical hybridization (Mechanical spectrum under strong illumination) b mw Theory! +/ /! m Δ=0.92! m Δ=1.02! m Δ=1.13! m c Δ/! m Noise power spectrum (a.u.) Noise power spectrum (a.u.) Frequency (MHz) Frequency (MHz) Aspelmeyer group 2009

69 Experiment: Optomechanically induced transparency (Light field transmission of a second, weak probe beam) Kippenberg group 2010

70 Optomechanics (Outline) Introduction Displacement detection Linear optomechanics Nonlinear dynamics System Bath Quantum theory of cooling Interesting quantum states Towards Fock state detection Coupling to the motion of a single atom

71 Squeezed states Squeezing the mechanical oscillator state

72 Squeezed states Squeezing the mechanical oscillator state

73 Squeezed states Squeezing the mechanical oscillator state

74 Squeezed states Squeezing the mechanical oscillator state t x

75 Squeezed states Squeezing the mechanical oscillator state t x Periodic modulation of spring constant: Parametric amplification

76 Squeezed states Squeezing the mechanical oscillator state t x Periodic modulation of spring constant: Parametric amplification

77 Squeezed states Squeezing the mechanical oscillator state t x Periodic modulation of spring constant: Parametric amplification

78 Squeezed states Squeezing the mechanical oscillator state t x Periodic modulation of spring constant: Parametric amplification

79 Measuring quadratures ( beating the SQL ) amplitude-modulated input field (similar to stroboscopic measurement) Clerk, Marquardt, Jacobs; NJP 10, (2008) measure only one quadrature, back-action noise affects only the other one...need:

80 Measuring quadratures ( beating the SQL ) amplitude-modulated input field (similar to stroboscopic measurement) Clerk, Marquardt, Jacobs; NJP 10, (2008) measure only one quadrature, back-action noise affects only the other one...need: p reconstruct mechanical Wigner density (quantum state tomography) x

81 Measuring quadratures ( beating the SQL ) amplitude-modulated input field (similar to stroboscopic measurement) Clerk, Marquardt, Jacobs; NJP 10, (2008) measure only one quadrature, back-action noise affects only the other one...need: p reconstruct mechanical Wigner density (quantum state tomography) x

82 Measuring quadratures ( beating the SQL ) amplitude-modulated input field (similar to stroboscopic measurement) Clerk, Marquardt, Jacobs; NJP 10, (2008) measure only one quadrature, back-action noise affects only the other one...need: p reconstruct mechanical Wigner density (quantum state tomography) x

83 Optomechanical entanglement

84 Optomechanical entanglement

85 Optomechanical entanglement n=0 n=1 n=2 n=3 entangled state (light field/mechanics) coherent mechanical state

86 Proposed optomechanical which-path experiment and quantum eraser p x Recover photon coherence if interaction time equals a multiple of the mechanical period! cf. Haroche experiments in 90s Marshall, Simon, Penrose, Bouwmeester, PRL 91, (2003)

87 Optomechanics (Outline) Introduction Displacement detection Linear optomechanics Nonlinear dynamics System Bath Quantum theory of cooling Interesting quantum states Towards Fock state detection Coupling to the motion of a single atom

88 Towards Fock state detection of a macroscopic object n=3 n=2 n=1 n=0

89 Towards Fock state detection of a macroscopic object n=3 Electron oscillations in Penning trap: n=2 Peil and Gabrielse, 1999 n=1 n=0

90 Membrane in the middle setup optical frequency Thompson, Zwickl, Jayich, Marquardt, Girvin, Harris, Nature 72, 452 (2008)

91 Membrane in the middle setup optical frequency Thompson, Zwickl, Jayich, Marquardt, Girvin, Harris, Nature 72, 452 (2008) frequency membrane transmission membrane displacement

92 Experiment (Harris group, Yale) Mechanical frequency: ωm=2π 134 khz Mechanical quality factor: Q= nm SiN membrane Current optical finesse: ( ) [almost sideband regime] Optomechanical cooling from 300K to 7mK Thompson, Zwickl, Jayich, Marquardt, Girvin, Harris, Nature 72, 452 (2008)

93 Towards Fock state detection of a macroscopic object Detection of displacement x: not what we need!

94 Towards Fock state detection of a macroscopic object Detection of displacement x: not what we need! frequency membrane transmission membrane displacement

95 Towards Fock state detection of a macroscopic object Detection of displacement x: not what we need! frequency membrane transmission membrane displacement

96 Towards Fock state detection of a macroscopic object frequency membrane transmission membrane displacement phase shift of measurement beam:

97 Towards Fock state detection of a macroscopic object frequency membrane transmission membrane displacement phase shift of measurement beam: (Time-average over cavity ring-down time) QND measurement of phonon number!

98 Towards Fock state detection of a macroscopic object Measured phase shift (phonon number) Time Thompson, Zwickl, Jayich, FM, Girvin, Harris, Nature 72, 452 (2008)

99 Towards Fock state detection of a macroscopic object Measured phase shift (phonon number) Time Thompson, Zwickl, Jayich, FM, Girvin, Harris, Nature 72, 452 (2008)

100 Towards Fock state detection of a macroscopic object Measured phase shift (phonon number) Time Thompson, Zwickl, Jayich, FM, Girvin, Harris, Nature 72, 452 (2008)

101 Towards Fock state detection of a macroscopic object Measured phase shift (phonon number) Time Thompson, Zwickl, Jayich, FM, Girvin, Harris, Nature 72, 452 (2008)

102 Towards Fock state detection of a macroscopic object Signal-to-noise ratio: Measured phase shift (phonon number) Optical freq. shift per phonon: Noise power of freq. measurement: Ground state lifetime: Time Thompson, Zwickl, Jayich, FM, Girvin, Harris, Nature 72, 452 (2008)

103 Towards Fock state detection of a macroscopic object Measured phase shift (phonon number) need: higher finesse smaller mass T=300mK + optomechanical cooling to ground state higher reflectivity of membrane: rc>0.999 Signal-to-noise ratio: Optical freq. shift per phonon: Noise power of freq. measurement: Ground state lifetime: Time Thompson, Zwickl, Jayich, FM, Girvin, Harris, Nature 72, 452 (2008)

104 Towards Fock state detection of a macroscopic object Measured phase shift (phonon number) need: higher finesse smaller mass T=300mK + optomechanical cooling to ground state higher reflectivity of membrane: rc>0.999 Signal-to-noise ratio: Optical freq. shift per phonon: Noise power of freq. measurement: Thompson, Zwickl, Jayich, FM, Girvin, Harris, Nature 72, 452 (2008) Ground state lifetime: TimeAlternative: Phonon shot noise measurement (Clerk, FM, Harris 2010)

105 Optomechanics (Outline) Introduction Displacement detection Linear optomechanics Nonlinear dynamics System Bath Quantum theory of cooling Interesting quantum states Towards Fock state detection Coupling to the motion of a single atom

106 Atom-membrane coupling Note: Existing works simulate optomechanical effects using cold atoms K. W. Murch, K. L. Moore, S. Gupta, and D. M. Stamper-Kurn, Nature Phys. 4, 561 (2008). F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, Science 322, 235 (2008). Stamper-Kurn (Berkeley) cold atoms...profit from small mass of atomic cloud Here: Coupling a single atom to a macroscopic mechanical object Challenge: huge mass ratio

107 Coupling a single atom to a heavy object: Why it is hard atoms 1 atom

108 Coupling a single atom to a heavy object: Why it is hard atoms 1 atom interaction: freq. shift coupling freq. shift coupling term small!

109 Strong atom-membrane coupling via the light field existing experiments on optomechanics with cold atoms : labs of Dan-Stamper Kurn (Berkeley) and Tilman Esslinger (ETH) collaboration: LMU (M. Ludwig, FM, P. Treutlein), Innsbruck (K. Hammerer, C. Genes, M. Wallquist, P. Zoller), Boulder (J. Ye), Caltech (H. J. Kimble) Hammerer et al., PRL 2009 Goal: atom membrane atom-membrane coupling

110 Cavity-mediated coupling (b) cavity response (c) (d) (e) for each cavity mode: atom-membrane coupling via virtual transitions: (likewise for membrane) detuning laser-cavity

111 Decoherence and decay Thermal ground-state decay rate: Note: T limited by light absorption, T~1K Relaxation of atom/membrane motion via driven cavity modes: Choose Atomic momentum diffusion from photon scattering: Parameter optimization for currently achievable setups: can reach strong coupling regime!

112 Example: State transfer (a) Transfer of squeezed state from atom to membrane (b) Squeezing transferred to membrane (maximized over time) Atom Membrane -9 db Transferred Squeezing [db] -3 db -6 db -9 db FIG. 2: (a) Wigner functions of atom and membrane (upper and [ ] Exploit toolbox for single-atom manipulation Creation of arbitrary atom states and transfer to membrane Transfer of membrane states to atom and measurement

113 Optomechanics (Outline) Introduction Displacement detection Nonlinear dynamics System Bath Quantum theory of cooling Interesting quantum states Towards Fock state detection Coupling to the motion of a single atom Optomechanical arrays

114 Scaling down cm usual optical cavities 10 µm LKB, Aspelmeyer, Harris, Bouwmeester,...

115 Scaling down cm usual optical cavities 50 µm 10 µm LKB, Aspelmeyer, Harris, Bouwmeester,... microtoroids Vahala, Kippenberg, Carmon,...

116 Scaling down cm usual optical cavities 50 µm 10 µm LKB, Aspelmeyer, Harris, Bouwmeester,... microtoroids optomechanics in photonic circuits Vahala, Kippenberg, Carmon,... optomechanical crystals 5 µm H. Tang et al., Yale O. Painter et al., Caltech 10 µm

117 Optomechanical crystals free-standing photonic crystal structures optical modes advantages: tight vibrational confinement: high frequencies, small mass (stronger quantum effects) vibrational modes tight optical confinement: large optomechanical coupling (100 GHz/nm) integrated on a chip from: M. Eichenfield et al., Optics Express 17, (2009), Painter group, Caltech

118 Optomechanical arrays laser drive cell 1 cell 2 optical mode mechanical mode... collective nonlinear dynamics: classical / quantum cf. Josephson arrays

119 Nonlinear dynamics of a single optomechanical cell: Self-induced oscillations F Fdx blue-detuned laser: < 0 > 0 anti-damping red-detuned laser: x damping (cooling)

120 Nonlinear dynamics of a single optomechanical cell: Self-induced oscillations F Fdx blue-detuned laser: < 0 > 0 anti-damping red-detuned laser: x damping (cooling) Beyond some laser input power threshold: instability Cantilever displacement x Time t Amplitude A

121 Attractor diagram 15 Oscillation amplitude transition detuning Höhberger, Karrai, IEEE proceedings 2004 Carmon, Rokhsari, Yang, Kippenberg, Vahala, PRL 2005 FM, Harris, Girvin, PRL 2006 Metzger et al., PRL 2008

122 Attractor diagram 15 Oscillation amplitude detuning Höhberger, Karrai, IEEE proceedings 2004 Carmon, Rokhsari, Yang, Kippenberg, Vahala, PRL 2005 FM, Harris, Girvin, PRL 2006 Metzger et al., PRL 2008

123 Attractor diagram 15 Oscillation amplitude detuning Höhberger, Karrai, IEEE proceedings 2004 Carmon, Rokhsari, Yang, Kippenberg, Vahala, PRL 2005 FM, Harris, Girvin, PRL 2006 Metzger et al., PRL 2008

124 An optomechanical cell as a Hopf oscillator amplitude laser power phase bifurcation Amplitude fixed, phase undetermined!

125 An optomechanical cell as a Hopf oscillator amplitude laser power phase bifurcation Amplitude fixed, phase undetermined!... Collective dynamics in an array of coupled cells? Phase-locking: synchronization!

126 Synchronization: Huygens observation Coupled pendula synchronize... (Huygens original drawing!)...even though frequencies slightly different...due to nonlinear effects

127 Fireflies synchronizing (Source: YouTube)

128 Coupled phase oscillators

129 Coupled phase oscillators

130 Coupled phase oscillators

131 Coupled phase oscillators

132 The Kuramoto model Kuramoto model: captures essential features often found as limiting model Kuramoto 1975, 1984 Acebron et al., Rev. Mod. Phys. 77, 137 (2005)

133 The Kuramoto model Synchronization: phase lag

134 The Kuramoto model 1 phase locking 0 coupling K

135 Many phase oscillators

136 Many phase oscillators

137 Many phase oscillators

138 Many phase oscillators

139 Kuramoto model: Phase-locking transition... infinite-range coupling Kuramoto model displays phase transition... coupling strength

140 laser drive Phase locking of two optomechanical cells cell 1 optical mode mechanical mode cell 2... Two optomechanical cells, fixed laser drive, increasing mechanical coupling

141 Phase locking of two optomechanical cells 0.8 synchronization sin(δϕ) time time -0.4 time phase-slip π mechanical coupling k/mω 2 1

142 Phase locking of two optomechanical cells sin(δϕ) time synchronization time time k/mω 2 1 mechanical coupling synchronized 0 π unsynchronized phase-slip π mechanical coupling k/mω 2 1 frequency difference δω/ω 1

143 Effective Kuramoto model eff. Kuramoto model optomech. model Hopf model Standard Kuramoto model: Effective Kuramoto model for coupled Hopf oscillators: 1 0 sin(δϕ) time tδω/2π time tδω/2π

144 frequency detuning Frequency locking 0.01 Mechanical spectrum in light field intensity 0.00 (optom.) fluctuations (Hopf) frequency 1.02

145 Desynchronization for increasing drive laser power G 2 α 2 max/mω smaller effective coupling K for larger drive! Hopf bifurcation 1.00 frequency

146 Desynchronization for increasing drive laser power G 2 α 2 max/mω smaller effective coupling K for larger drive! frequency time

147 Array with common optical mode chaos frequency

148 Dynamics in optomechanical arrays Outlook 2D geometries Information storage and classical computation Dissipative quantum many-body dynamics...

149 Optomechanics: general outlook Bouwmeester (Santa Barbara/Leiden) Fundamental tests of quantum mechanics in a new regime: entanglement with macroscopic objects, unconventional decoherence? [e.g.: gravitationally induced?] Mechanics as a bus for connecting hybrid components: superconducting qubits, spins, photons, cold atoms,... Kapitulnik lab (Stanford) Tang lab (Yale) Precision measurements [e.g. testing deviations from Newtonian gravity due to extra dimensions] Optomechanical circuits & arrays Exploit nonlinearities for classical and quantum information processing, storage, and amplification; study collective dynamics in arrays

150 Optomechanics Recent review on optomechanics: APS Physics 2, 40 (2009) Recent review on quantum limits for detection and amplification: Clerk, Devoret, Girvin, Marquardt, Schoelkopf; RMP 2010