Cavity Optomechanics: – Interactions Between Light and Nanomechanical Motion

Cavity optomechanics: – interactions between light and nanomechanical motion

Florian Marquardt

University of Erlangen-Nuremberg, Germany, and Max-Planck Institute for the Physics of Light (Erlangen) Radiation pressure

Johannes Kepler De Cometis, 1619

(Comet Hale-Bopp; by Robert Allevo) Radiation pressure

Johannes Kepler De Cometis, 1619 Radiation pressure

Nichols and Hull, 1901 Lebedev, 1901

Nichols and Hull, Physical Review 13, 307 (1901) Radiation forces

Trapping and cooling •Optical tweezers •Optical lattices

...but usually no back-action from motion onto light! Optomechanics on different length scales

4 km Mirror on cantilever – Bouwmeester lab, Santa Barbara (2006)

LIGO – Laser Interferometer 14 Gravitational 10 atoms Wave Observatory Optomechanical Hamiltonian

optical laser mechanical cavity mode Optomechanical Hamiltonian

Review “Cavity Optomechanics”: M. Aspelmeyer, T. Kippenberg, FM arXiv:1303.0733

optical laser cavity mechanical mode

laser detuning optomech. coupling Optomechanical Hamiltonian Optomechanical Hamiltonian Optomechanical Hamiltonian

...any dielectric moving inside a cavity generates an optomechanical interaction! The zoo of optomechanical (and analogous) systems

Karrai Bouwmeester Vahala (Caltech) Harris (Yale) (Munich) (Santa Barbara) Kippenberg (EPFL), Carmon, ...

LKB group Mavalvala Teufel, Lehnert (Boulder) (Paris) (MIT) Painter (Caltech)

Stamper-Kurn (Berkeley) Schwab (Cornell) cold atoms

Aspelmeyer (Vienna) The zoo of optomechanical (and analogous) systems The zoo of optomechanical (and analogous) systems Optomechanics: general outlook

Fundamental tests of quantum mechanics in a new regime: entanglement with ‘macroscopic’ objects, unconventional decoherence? [e.g.: gravitationally induced?] 2

Mechanics as a ‘bus’ for connecting hybrid components: superconducting qubits, spins, photons, cold atoms, .... Precision measurements small displacements, masses, forces, and accelerations Painter lab Optomechanical circuits & arrays Exploit nonlinearities for classical and quantum information processing, storage, and ampliﬁcation; study collective Tang lab (Yale) dynamics in arrays

FIG. 1: Overview of the accelerometer design. a, Canonical example of an accelerometer. When the device (blue frame) experiences a constant acceleration a, a test mass m undergoes a displacement of x = ma/k. b, Frequency response χ(ω) of an accelerometer on a log-log | | plot featuring a resonance at fm = k/m/2π with Qm = 10. c, False-colored SEM-image of a typical optomechanical accelerometer. A test mass of size 150 µm 60 µm 400 nm (green) is suspended on highly stressed 150 nm wide and 560 µm long SiN nano-tethers, which allow for high oscillator frequencies× × (> 27 kHz) and high mechanical Q-factors (> 106). On the upper edge of the test mass, we implement a zipper photonic crystal nanocavity (pink). The cross-shaped cuts on the test mass facilitate undercutting the device. d, Zoom-in of the optical cavity region showing the magnitude of the electric ﬁeld E(r) for the fundamental bonded mode of the zipper cavity. The top beam is mechanically | | anchored to the bulk SiN and the bottom beam is attached to the test mass. e, Schematic displacement proﬁle (not to scale) of the fundamental in-plane mechanical mode used for acceleration sensing. f, SEM-image of an array of devices with different test mass sizes.

(1 2 106), and strong thermo-optomechanical back-action Fig. 2a and appendix E). A balanced detection scheme allows to− damp× and cool the thermal motion of the test mass. for efficient rejection of laser amplitude noise, yielding shot- noise limited detection for frequencies above 1 kHz. Figure 1c shows a scanning-electron microscope image ∼ of the device studied here, with the test mass structure and Figure 2b shows the electronic power spectral density nano-tethers highlighted in green. The fundamental in-plane (PSD) of the optically transduced signal obtained from the mechanical mode of this structure is depicted in Fig. 1e device in Fig. 1c. The cavity was driven with an incident = and is measured to have a frequency of f = 27.5 kHz, in laser power of Pin 116 µW, yielding an intracavity photon- m number of 430. The two peaks around 27.5 kHz arise from good agreement with finite-element-method simulations from ≈ 12 thermal Brownian motion of the fundamental in- and out- which we also extract a motional mass of m = 10 10− kg. The measured mechanical Q-factor is Q = 1.4 ×106 in vac- of-plane mechanical eigenmodes of the suspended test mass. m × The transduced signal level of the fundamental in-plane reso- uum (see appendix G), which results in an estimated ath = 1.4 µg/√Hz. The region highlighted in pink corresponds to nance, the mode used for acceleration sensing, is consistent = the zipper optical cavity used for monitoring test mass mo- with an optomechanical coupling constant of gOM 2π 5.5 GHz/nm, where g ∂ω /∂x is defined as the optical× tion, a zoom-in of which can be seen in Figure 1d. The cav- OM ≡ o ity consists of two patterned photonic crystal nanobeams, one cavity frequency shift per unit displacement. The dotted green attached to the test mass (bottom) and one anchored to the line depicts the theoretical thermal noise background of this bulk (top). The device in Fig. 1c is designed to operate in mode. The series of sharp features between zero frequency the telecom band, with a measured optical mode resonance at (DC) and 15 kHz are due to mechanical resonances of the anchored fiber-taper. The noise background level of Fig. 2b is λo = 1537 nm and an optical Q-factor of Qo = 9,500. With the optical cavity field being largely confined to the slot be- dominated by photon shot-noise, an estimate of which is indi- tween the nanobeams, the optical resonance frequency is sen- cated by the red dotted line. The cyan dotted line in Fig. 2b sitively coupled to relative motion of the nanobeams in the corresponds to the electronic photodetector noise, and the pur- plane of the device (thex ˆ-direction in Fig. 1c). A displace- ple dashed line represents the sum of all noise terms. The ment of the test mass caused by an in-plane acceleration of the broad noise at lower frequencies arises from fiber taper mo- supporting microchip can then be read-out optically using the tion and acoustic pick-up from the environment. The right- setup shown in Fig. 2a, where the optical transmission through hand axis in Fig. 2b quantifies the optically transduced PSD the photonic crystal cavity is monitored via an evanescently- in units of an equivalent transduced displacement amplitude coupled fiber taper waveguide [25] anchored to the rigid side of the fundamental in-plane mode of the test mass, showing a of the cavity. Utilizing a narrow bandwidth (< 300 kHz) laser measured shot-noise-dominated displacement imprecision of /√ source, with laser frequency detuned to the red side of the cav- 4 fm Hz (the estimated on-resonance quantum-back-action √ ity resonance, fluctuations of the resonance frequency due to displacement noise is 23 fm/ Hz, and the corresponding on- motion of the test mass are translated linearly into amplitude- resonance SQL is 2.8 fm/√Hz; see appendix I 4). fluctuations of the transmitted laser light field (see inset in At this optical power the observed linewidth of the mechan- 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 Laser-cooling towards the ground state

1x1010 1x109 1x108 x 7 MIT 1 10 Yale x 6

1 10 LKB 100000 10000 1000 IQOQI MPQ JILA 100 John Teufel 10 CaltechOskar 2011 Painter Boulder 2011 phonon number phonon 1 0.1 minimum possible 0.01 phonon number 0.001 ground state 0.0001 0.00001 0.01 0.1 1 10

analogy to (cavity-assisted) FM et al., PRL 93, 093902 (2007) laser cooling of atoms Wilson-Rae et al., PRL 99, 093901 (2007) Optomechanics (Outline)

Displacement detection Linear optomechanics

-1 0 1 100 2 Nonlinear dynamics 3

System Bath Quantum theory of cooling Interesting quantum states Towards Fock state detection Hybrid systems: coupling to atoms Optomechanical crystals & arrays Optical displacement detection input laser optical cantilever cavity

reﬂection phase shift Thermal ﬂuctuations of a harmonic oscillator

Classical equipartition theorem:

extract Possibilities: temperature! •Direct time-resolved detection •Analyze ﬂuctuation spectrum of x Fluctuation spectrum Fluctuation spectrum Fluctuation-dissipation theorem

General relation between noise spectrum and linear response susceptibility

susceptibility

(classical limit) Fluctuation-dissipation theorem

General relation between noise spectrum and linear response susceptibility

susceptibility

(classical limit) for the damped oscillator: Fluctuation-dissipation theorem

General relation between noise spectrum and linear response susceptibility

susceptibility

(classical limit) for the damped oscillator:

area yields ...yields variance of x: temperature! 8

Displacement spectrum

-13 a b 10 n =0.93 n d =4,500 m -29 nd =18 n m =27 10 4 n =11,000 n =0.55 nd =71 n m =22 d m 2 ) ) z z -30 H

H 10 / / nd =280 n m =8.5 -14 2 1 n =28,000 nm =0.36 ( 10 d m

( 8 o

x 6 S n =1,100 n =2.9 S/P -31 d m 4 10 n =0.34 nd =89,000 m

2 g/S nd =4,500 n m =0.93 n =0.42 nd =180,000 m -15 -32 10 10 10.556 10.558 10.3 10.4 10.5 10.6 10.7 Frequency (M H z) Frequency (M H z) c 100 Teufel et al., Nature 2011

10 n

y m c n n c a p u c c

O 1

0.1

0 1 2 3 4 5 10 10 10 10 10 10

D rive Photons, nd

FIG. 3. Sideband cooling the mechanical mode to the ground state. a, The displacement noise spectra and Lorentzian fits (shaded region) for five different drive powers. With higher power, the mechanical mode is both damped (larger linewidth) and cooled (smaller area) by the radiation pressure forces. b, Over a broader frequency span, the normalized sideband noise spectra clearly show both the narrow mechanical peak and a broader cavity peak due to finite occupancy of the mechanical and electrical modes, respectively. A small, but resolvable, thermal population of the cavity appears as the drive power increases, setting the limit for the final occupancy of the coupled optomechanical system. At the highest drive power, the coupling rate between the mechanical oscillator and the microwave cavity exceeds the intrinsic dissipation of either mode, and the system hybridizes into optomechanical normal modes. c, Starting in thermal equilibrium with the cryostat at T =20mK, sideband cooling reduces the thermal occupancy of the mechanical mode from nm =40into the quantum regime, reaching a minimum of nm =0.34 0.05. These data demonstrate that the parametric interaction between photons and phonons can initialize the strongly coupled, electromechanical± system in its quantum ground state. Measurement noise 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 “Standard Quantum Limit”

imprecision noise full noise (including back-action)

back-action noise SQL zero-point imprecision noise ﬂuctuations

intrinsic (thermal & scale) (log noise measured laser power (log scale) quantum) noise

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” 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: ! Optomechanics (Outline)

Displacement detection Linear optomechanics

-1 0 1 100 2 Nonlinear dynamics 3

System Bath Quantum theory of cooling Interesting quantum states Towards Fock state detection Hybrid systems: coupling to atoms Optomechanical crystals & arrays Static behaviour

oscillating mirror Basic physics: Statics radiation x cantilever input laser pressure ﬁxed mirror Frad

Experimental proof of static bistability: Frad(x)=2I(x)/c A. Dorsel, J. D. McCullen, P. Meystre, λ E. Vignes and H. Walther: 2 λ/2 Phys. Rev. Lett. 51, 1550 (1983) F

Vrad(x)

Veﬀ = Vrad + VHO x hysteresis Dynamics: Delayed lightBasic response physics: dynamics

quasistatic F finite sweep!rate

sweep x ﬁniteﬁnite cavity optical ring-down ringdown rate γ time – delayed response to cantilever motion ⇒delayed response to cantilever motion

F Fdx ! < 0 > 0 x cooling 0 heating Höhberger-Metzger and Karrai, (ampliﬁcation) Nature 432, 1002 (2004): 300K to 17K [photothermalC. Höhberger force]!Metzger and K. Karrai, Nature 432, 1002 (2004) (with photothermal force instead of radiation pressure) Equations of motion

cantilever input laser optical cavity Equations of motion

cantilever input laser optical cavity

mechanical mechanical frequency damping radiation pressure equilibrium position

laser detuning cavity amplitude from resonance decay rate Linearized optomechanics

(solve for arbitrary )

Optomechanical frequency shift (“optical spring”) Effective optomechanical damping rate Linearized dynamics

Effective Optomechanical optomechanical frequency shift damping rate (“optical spring”)

laser detuning cooling heating/ softer stiffer ampliﬁcation Linearized dynamics

Effective Optomechanical optomechanical frequency shift damping rate (“optical spring”)

laser detuning cooling heating/ softer stiffer ampliﬁcation Optomechanical Hamiltonian

optical laser cavity mechanical mode

laser detuning optomech. coupling Quantum optomechanics: Linearized Hamiltonian

large amplitude quantum ﬂuctuations (laser drive)

bilinear interaction tunable coupling!

Sufﬁcient to explain (almost) all current optomechanical experiments in the quantum regime Mechanics & Optics

After linearization: two linearly coupled harmonic oscillators!

mechanical oscillator driven optical cavity Different regimes red-detuned blue-detuned cavity pump

beam-splitter squeezer (cooling) QND (entanglement) Optomechanics (Outline)

Displacement detection Linear optomechanics

-1 0 1 100 2 Nonlinear dynamics 3

System Bath Quantum theory of cooling Interesting quantum states Towards Fock state detection Hybrid systems: coupling to atoms Optomechanical crystals & arrays Dynamics: Delayed light response

quasistatic F finite sweep!rate

sweep x ﬁnite cavity ring-down rate γ delayed response to cantilever motion ⇒ Self-induced oscillations

F Fdx ! < 0 > 0 x cooling 0 heating total

C. Höhberger!Metzger and K. Karrai, Nature 432, 1002 (2004) (with photothermal force instead of radiation pressure) Dynamics: Delayed light response

quasistatic F finite sweep!rate

sweep x ﬁnite cavity ring-down rate γ delayed response to cantilever motion ⇒ Self-induced oscillations

F Fdx ! < 0 > 0 x cooling 0 heating total

C. Höhberger!Metzger and K. Karrai, Nature 432, 1002 (2004) (with photothermal force instead of radiation pressure) Beyond some laser input power threshold: instability Cantilever displacement x Amplitude A Time t An optomechanical cell as a Hopf oscillator

amplitude laser power

phase

bifurcation

Amplitude ﬁxed, phase undetermined! Attractor diagram

power fed into the cantilever

0 cantilever -1 energy 0 FM, Harris, Girvin, 1 PRL 96, 103901 (2006) 2 detuning 100 3 Ludwig, Kubala, FM, NJP 2008 Attractor diagram

power fed into the cantilever

1 power balance

0 cantilever -1 energy 0 FM, Harris, Girvin, 1 PRL 96, 103901 (2006) 2 detuning 100 3 Ludwig, Kubala, FM, NJP 2008 Attractor diagram

power fed into the cantilever

1 power balance

0 cantilever -1 energy 0 FM, Harris, Girvin, 1 PRL 96, 103901 (2006) 2 detuning 100 3 Ludwig, Kubala, FM, NJP 2008 Attractor diagram

10 Höhberger, Karrai, IEEE proceedings 2004 Carmon, Rokhsari, Yang, Kippenberg, Vahala, PRL 2005 5 + FM, Harris, Girvin, transition PRL 2006

Oscillation amplitude Oscillation - Metzger et al., PRL 2008 0 ï 0 detuning Attractor diagram

10 Höhberger, Karrai, IEEE proceedings 2004 Carmon, Rokhsari, Yang, Kippenberg, Vahala, PRL 2005 5 + FM, Harris, Girvin, PRL 2006

Oscillation amplitude Oscillation - Metzger et al., PRL 2008 0 ï 0 detuning Attractor diagram

10 Höhberger, Karrai, IEEE proceedings 2004 Carmon, Rokhsari, Yang, Kippenberg, Vahala, PRL 2005 5 + FM, Harris, Girvin, PRL 2006

Oscillation amplitude Oscillation - Metzger et al., PRL 2008 0 ï 0 detuning Coupled oscillators? Coupled oscillators?

...

Collective dynamics in an array of coupled cells? Phase-locking: synchronization! Synchronization: Huygens’ observation

(Huygens’ original drawing!) Coupled pendula synchronize...... even though frequencies slightly different Classical nonlinear collective dynamics: Synchronization in an optomechanical array

0.06

2 1 two coupled Ω cells

0.05 k/m 0 0.04 synchronized →

0.03 π → 0.02 unsynchronized mechanical coupling mechanical 0.01 -0.91 0.0 0.61

0.01 0.02 0.03 0.04 frequency difference δΩ/Ω1 G. Heinrich et al., Phys. Rev. Lett. 107, 043603 (2011) ## Experiments (two cells, joint optical mode) # Michal Lipson lab, Cornell Hong Tang lab, Yale

sync sync laser detuning

mechanical frequency ## (Zhang et al., PRL 2012) (Bagheri, Poot, FM, Tang; PRL 2013) Figure 2. Synchronized motion of two optomechanical oscillators. a#$%&#&'()*+,(-#

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# Effective Kuramoto model eff. Kuramoto model optomech. model Hopf model

G. Heinrich et al., Phys. Rev. Lett. 107, 043603 (2011) Effective Kuramoto model eff. Kuramoto model optomech. model Hopf model

Effective Kuramoto-type model for coupled Hopf oscillators:

G. Heinrich et al., Phys. Rev. Lett. 107, 043603 (2011) Effective Kuramoto model eff. Kuramoto model optomech. model Hopf model

Effective Kuramoto-type model for coupled Hopf oscillators: sin(δϕ) 1

-1 0 1 2 3 4 5 6 0 1 2 3 4 5 6 time tδΩ/2π time tδΩ/2π

G. Heinrich et al., Phys. Rev. Lett. 107, 043603 (2011) Synchronization in optomechanical arrays Optomechanical arrays Optomechanical array: Many coupled optomechanical cells optical mode mechanical mode

laser drive

Possible design based on “snowﬂake” 2D optomechanical crystal (Painter group), here: with suitable defects forming a superlattice (array of cells) Pattern formation in optomechanical arrays

6.28319 work with Christian Brendel, Roland Lauter, 60 Steve Habrake, Max Ludwig

20 initial state initial mechanical phase (B,A can be calculated from

Phase vortex 0 0 microscopic parameters) 0 20 40 60

6.28319 6.28319 6.28319

60 60 60

40 40 40

20 20 20 transient state

0 0 0 0 0 0 0 20 40 60 0 20 40 60 0 20 40 60 6.28319 6.28319 6.28319

60 60 60

40 40 40

20 20 20

steady state Spatiotemporal Spiral patterns Chaos 0 0 0 0 0 0 0 20 40 60 0 20 40 60 0 20 40 60 2

ˆ The mechanical mode (bj) is characterized by a frequency photon correlations Ω. The cavity mode (âj) is transformed into the frame rotating at the laser frequency (∆ = ω ω ) and 3 laser − cav driven at the rate αL. In the most general case, both 1 photons and phonons can tunnel between neighboring 1 sites ij at rates J/z and K/z,wherez denotes the coordination number. The full Hamiltonian of the array 0.001 0.01 0.1 1 is given by Hˆ = Hôm,j + Hînt,with 0.5 j 4 3 2 ˆ J K ˆ ˆ ˆ ˆ Hint = aî†aˆj +âiaˆj† bi†bj + bibj† .(2) 1

− z − z optical coupling strength 0 i,j i,j -1.5 -1 -0.5 0 detuning To bring this many-body problem into a treatable form, ˆ ˆ ˆ ˆ we apply the Gutzwiller ansatz Ai†Aj Ai† Aj + Figure 2. Loss of photon blockade for increasing optical cou- ≈ pling in an array of optomechanical cavities. The equal time Aˆ† Aˆj Aˆ† Aˆj to Eq. (2). The accuracy of this ap- i − i photon correlation function shows anti-bunching (g(2)(0) < 1) proximation improves if the number of neighboring sites z (2) increases. For identical cells, the index j can be dropped and bunching (g (0) > 1) as a function of detuning ∆ and optical coupling strength J.Thesmallestvaluesofg(2)(0) and the Hamiltonian reduces to a sum of independent 2 are found for a detuning ∆0 = g0 /Ω. When increasing the contributions, each of which is described by coupling J while keeping the intracavity− photon number constant, i.e. along the dashed line, photon blockade is lost (inset, g(2)(0) as black solid line). For a smaller driving power (inset, ˆ ˆ ˆ ˆ ˆ ˆ 5 Hmf = Hom J aˆ† aˆ +â aˆ† K b† b + b b† .(3) blue solid line, αL =5 10− κ), anti-bunching is more pro- − − nounced, and the behavior· is comparable to that of a nonlin- Hence, a Lindblad master equation for the single cell den- ear cavity (inset, dashed line). The hatched area in the main figure outlines a region where a transition towards coherent sity matrixρ ˆ, dρˆ/dt = i[Hˆmf , ρˆ]+κ [â]ˆρ + Γ [ˆb]ˆρ can − D D mechanical oscillations has set in (see main text and further be employed. The Lindblad terms [Aˆ]ˆρ = AˆρÂˆ† 1 ˆ ˆ 1 ˆ ˆ D − figures). κ =0.3 Ω, αL =0.65 κ, g0 =0.5 Ω, Γ =0.074 Ω. 2 A†Aρˆ 2 ρÂ†A take into account photon decay at a rate κ and− mechanical dissipation (here assumed due to a zero temperature bath) at a rate Γ. large couplings. Similar physics has recently been ana- Photon statistics. - Recently, it was shown that the ef- lyzed for coupled qubit-cavity arrays, [30]. For very large fect of photon blockade [7] can appear in a single optome- coupling strengths, though, the density plot of Fig. 2 re- chanical cell: The interaction with the mechanical mode veals signs of the collective mechanical motion (hatched induces an effective nonlinearity for the photon field of area). There we observe the correlation function to os- 2 strength g0/Ω [7, 27]. Hence, the presence of a single cillate (at the mechanical frequency) and to show strong photon can hinder other photons from entering the cav- bunching. We will now investigate this effect. ity. To observe this effect, the nonlinearity must be com- Collective mechanical quantum effects. - To describe 2 parable to the cavity decay rate, i.e. g0/Ω κ, and the the collective mechanical motion of the array, we focus laser drive weak (α κ) [7, 28]. L on the case of purely mechanical intercellular coupling To study nonclassical effects in the photon statistics, (K>0, J = 0) for simplicity. Note, though, that the we analyze the steady-state photon correlation function effect is also observable for optically coupled arrays, as (2) 2 g (τ)= aˆ†(t)â†(t + τ)â(t + τ)â(t) / aˆ(t)†aˆ(t) [29] at discussed above. (2) equal times (τ = 0), with g (0) = 1 for a coherent state, As our main result, Figs. 3(a)and 4(a) show the (2) and g (0) < 1(> 1) indicating anti-bunching (bunch- sharp transition between incoherent self-oscillations and ing). Here (Fig. 2), we probe the influence of the collec- a phase-coherent collective mechanical state as a function tive dynamics by varying the optical coupling strength J, of both laser detuning ∆ and coupling strength K:Inthe while keeping the mechanical coupling K zero for clarity. regime of self-induced oscillations, the phonon number We note that, when increasing J, the optical resonance ˆb†ˆb reaches a finite value. Yet, the expectation value effectively shifts: ∆ ∆ + J . To keep the photon ˆTransitionb remains small towards and constant coherent in time. When mechanical increas- → oscillations number fixed while increasing J, the detuning has to be ing the intercellular coupling, though, ˆb suddenly starts adapted [30]. In this setting, we observe that the inter- Observationoscillating: for strong inter-cell coupling: action between the cells suppresses anti-bunching (inset spontaneous time-dependence of phonon field of Fig. 2). Photon blockade is lost if the intercellular “order parameter” iΩ t (“mechanical coherence”) coupling becomes larger than the effective nonlinearity, ˆb (t)=¯b + re− eff . (4) 2 2J g0/Ω. Above this value, the photon statistics shows bunching, and ultimately reaches Poissonian statistics for Our more detailed analysis (see below) indicates that 1

coherence 0.5 optical coherence

mechanical 0 0 0.25 0.5 quantum mean-ﬁeld inter-cell coupling result Mechanical quantum states

Incoherent mechan. oscillations (weak inter-cell coupling) 15 5 Mechanical Wigner density shows incoherent mixture of 0 all possible oscillation phases 0 -5 0 15 -5 0 5 Coherent mechan. oscillations (strong inter-cell coupling) 15 Mechanical Wigner density 5 shows preferred phase (coherent state) – 0 spontaneous symmetry breaking! 0 -5 0 15 -5 0 5 2

− z − z optical coupling strength 0 i,j i,j -1.5 -1 -0.5 0 detuning To bring this many-body problem into a treatable form, ˆ ˆ ˆ ˆ we apply the Gutzwiller ansatz Ai†Aj Ai† Aj + Figure 2. Loss of photon blockade for increasing optical cou- ≈ pling in an array of optomechanical cavities. The equal time Aˆ† Aˆj Aˆ† Aˆj to Eq. (2). The accuracy of this ap- i − i photon correlation function shows anti-bunching (g(2)(0) < 1) proximation improves if the number of neighboring sites z (2) increases. For identical cells, the index j can be dropped and bunching (g (0) > 1) as a function of detuning ∆ and optical coupling strength J.Thesmallestvaluesofg(2)(0) and the Hamiltonian reduces to a sum of independent 2 are found for a detuning ∆0 = g0 /Ω. When increasing the contributions, each of which is described by coupling J while keeping the intracavity− photon number constant, i.e. along the dashed line, photon blockade is lost (inset, g(2)(0) as black solid line). For a smaller driving power (inset, ˆ ˆ ˆ ˆ ˆ ˆ 5 Hmf = Hom J aˆ† aˆ +â aˆ† K b† b + b b† .(3) blue solid line, αL =5 10− κ), anti-bunching is more pro- − − nounced, and the behavior· is comparable to that of a nonlin- Hence, a Lindblad master equation for the single cell den- ear cavity (inset, dashed line). The hatched area in the main figure outlines a region where a transition towards coherent sity matrixρ ˆ, dρˆ/dt = i[Hˆmf , ρˆ]+κ [â]ˆρ + Γ [ˆb]ˆρ can − D D mechanical oscillations has set in (see main text and further be employed. The Lindblad terms [Aˆ]ˆρ = AˆρÂˆ† 1 ˆ ˆ 1 ˆ ˆ D − figures). κ =0.3 Ω, αL =0.65 κ, g0 =0.5 Ω, Γ =0.074 Ω. 2 A†Aρˆ 2 ρÂ†A take into account photon decay at a rate κ and− mechanical dissipation (here assumed due to a zero temperature bath) at a rate Γ. large couplings. Similar physics has recently been ana- Photon statistics. - Recently, it was shown that the ef- lyzed for coupled qubit-cavity arrays, [30]. For very large fect of photon blockade [7] can appear in a single optome- coupling strengths, though, the density plot of Fig. 2 re- chanical cell: The interaction with the mechanical mode veals signs of the collective mechanical motion (hatched induces an effective nonlinearity for the photon field of area). There we observe the correlation function to os- 2 strength g0/Ω [7, 27]. Hence, the presence of a single cillate (at the mechanical frequency) and to show strong photon can hinder other photons from entering the cav- bunching. We will now investigate this effect. ity. To observe this effect, the nonlinearity must be com- Collective mechanical quantum effects. - To describe 2 parable to the cavity decay rate, i.e. g0/Ω κ, and the the collective mechanical motion of the array, we focus laser drive weak (α κ) [7, 28]. L on the case of purely mechanical intercellular coupling To study nonclassical effects in the photon statistics, (K>0, J = 0) for simplicity. Note, though, that the we analyze the steady-state photon correlation function effect is also observable for optically coupled arrays, as (2) 2 g (τ)= aˆ†(t)â†(t + τ)â(t + τ)â(t) / aˆ(t)†aˆ(t) [29] at discussed above. (2) equal times (τ = 0), with g (0) = 1 for a coherent state, As our main result, Figs. 3(a)and 4(a) show the (2) and g (0) < 1(> 1) indicating anti-bunching (bunch- sharp transition between incoherent self-oscillations and ing). Here (Fig. 2), we probe the influence of the collec- a phase-coherent collective mechanical state as a function tive dynamics by varying the optical coupling strength J, of both laser detuning ∆ and coupling strength K:Inthe while keeping the mechanical coupling K zero for clarity. regime of self-induced oscillations, the phonon number We note that, when increasing J, the optical resonance ˆb†ˆb reaches a finite value. Yet, the expectation value effectively shifts: ∆ ∆ + J . To keep the photon ˆb remains small and constant in time. When increas- → number fixed while increasing J, the detuning has to be ing theTransition intercellular coupling, towards though, coherentˆb suddenly mechanical starts adapted [30]. In this setting, we observe that the inter- oscillating: oscillations action between the cells suppresses anti-bunching (inset of Fig. 2). Photon blockade is lost if the intercellular “order parameter” iΩ t (“mechanical coherence”) coupling becomes larger than the effective nonlinearity, ˆb (t)=¯b + re− eff . (4) 2 2J g0/Ω. Above this value, the photon statistics shows bunching, and ultimately reaches Poissonian statistics for Our more detailedmechanical analysis coherence (see below)(color-coded) indicates that 0.5 coherent

0.25 1.5 incoherent 1 0.5 onset of self-oscillations 0 0 -1 mechancial inter-cell coupling -1.5 -0.5 0 0.5 1 laser detuning

Max Ludwig, FM, PRL 111, 073603 (2013):Quantum many-body dynamics in optomechanical arrays Optomechanics (Outline)

Displacement detection Linear optomechanics

-1 0 1 100 2 Nonlinear dynamics 3

System Bath Quantum theory of cooling Interesting quantum states Towards Fock state detection Hybrid systems: coupling to atoms Optomechanical crystals & arrays Cooling with light

optical cavity cantilever input laser

Current goal in the ﬁeld: ground state of mechanical motion of a macroscopic cantilever

Classical theory:

Pioneering theory and optomechanical damping rate experiments: Braginsky (since 1960s ) Cooling with light

optical cavity cantilever input laser

Current goal in the ﬁeld: ground state of mechanical motion of a macroscopic cantilever

Classical theory: quantum limit?

shot noise! Pioneering theory and optomechanical damping rate experiments: Braginsky (since 1960s ) Cooling with light

optical cavity cantilever input laser

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 Quantum noise approach

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

bath provides energy bath absorbs energy transition rate Quantum theory of optomechanical cooling Spectrum of radiation pressure ﬂuctuations

radiation pressure force photon number

photon shot noise spectrum Radiation pressure noise spectrum

Photon number: nˆ =â†â ¯hωR Radiation pressure force: Fˆ = nˆ Quantum! L theory" of Noise spectrum for photonoptomechanical number: cooling + ∞ iωt 2 κ Snn(ω)= dt e ( nˆ(t)ˆn(0) n¯ )=¯n Spectrum of radiation# $−pressure (fluctuationsω + ∆)2 +(κ/2)2 #−∞ Detuning laser/cavity resonance: ∆ = ω ω Cavity decay rate: κ L − R

∆ detuning : − ) [norm.] ω ( κ

FF cavity linewidth S

cavity emits energy / absorbs energy Radiation pressure noise spectrum

Photon number: nˆ =ˆa†ˆa ¯hωR Radiation pressure force: Fˆ = nˆ L Noise spectrum for photon number:! " + ∞ iωt 2 κ Snn(ω)= dt e ( nˆ(t)ˆn(0) n¯ )=¯n # $− Quantum(ω + ∆)2 +( theoryκ/2)2 of #−∞ Detuning laser/cavity resonance: ∆ = ω ω Cavity decay rate: κ L −optomechanicalR cooling Cooling rate ∆ − ) [norm.] ω ( κ Quantum limit for FF

S cantilever phonon number

cavity emits energy / absorbs energy

FM, Chen, Clerk, Girvin, PRL 93, 093902 (2007) Ground-state cooling also: Wilson-Rae, Nooshi, Zwerger, Kippenberg, PRL 99, 093901 (2007); needs: high optical ﬁnesse / Genes et al, PRA 2008 large mechanical frequency experiment with Kippenberg group 2007 Laser-cooling towards the ground state

1x1010 1x109 1x108 x 7 MIT 1 10 Yale x 6

1 10 LKB 100000 10000 1000 IQOQI MPQ JILA 100 10 Caltech 2011 Boulder 2011 phonon number phonon 1 0.1 minimum possible 0.01 phonon number 0.001 ground state 0.0001 0.00001 0.01 0.1 1 10

analogy to (cavity-assisted) FM et al., PRL 93, 093902 (2007) laser cooling of atoms Wilson-Rae et al., PRL 99, 093901 (2007) Optomechanics (Outline)

Displacement detection Linear optomechanics

-1 0 1 100 2 Nonlinear dynamics 3

System Bath Quantum theory of cooling Interesting quantum states Towards Fock state detection Hybrid systems: coupling to atoms Optomechanical crystals & arrays Squeezed states

Squeezing the mechanical oscillator state Squeezed states

Squeezing the mechanical oscillator state

x Squeezed states

Squeezing the mechanical oscillator state

Periodic modulation of spring constant: Parametric ampliﬁcation Squeezed states

Squeezing the mechanical oscillator state

Periodic modulation of spring constant: Parametric ampliﬁcation Squeezed states

Squeezing the mechanical oscillator state

Periodic modulation of spring constant: Parametric ampliﬁcation Squeezed states

Squeezing the mechanical oscillator state

Periodic modulation of spring constant: Parametric ampliﬁcation Measuring quadratures (“beating the SQL”) Clerk, Marquardt, Jacobs; NJP 10, 095010 (2008) amplitude-modulated input ﬁeld (similar to stroboscopic measurement)

measure only one quadrature, back-action noise affects only the other one....need: Measuring quadratures (“beating the SQL”) Clerk, Marquardt, Jacobs; NJP 10, 095010 (2008) amplitude-modulated input ﬁeld (similar to stroboscopic measurement)

measure only one quadrature, back-action noise affects only the other one....need: p reconstruct mechanical Wigner density (quantum state tomography) x Measuring quadratures (“beating the SQL”) Clerk, Marquardt, Jacobs; NJP 10, 095010 (2008) amplitude-modulated input ﬁeld (similar to stroboscopic measurement)

measure only one quadrature, back-action noise affects only the other one....need: p reconstruct mechanical Wigner density (quantum state tomography) x Optomechanical entanglement

Bose, Jacobs, Knight 1997; Mancini et al. 1997 Optomechanical entanglement

coherent mechanical state entangled state

n=0 n=1 (light ﬁeld/mechanics) n=2 n=3

Bose, Jacobs, Knight 1997; Mancini et al. 1997 Proposed optomechanical which-path experiment and quantum eraser

Recover photon coherence if interaction time equals a multiple of x the mechanical period! cf. Haroche experiments in 90s Marshall, Simon, Penrose, Bouwmeester, PRL 91, 130401 (2003) Optomechanics (Outline)

Displacement detection Linear optomechanics

-1 0 1 100 2 Nonlinear dynamics 3

System Bath Quantum theory of cooling Interesting quantum states Towards Fock state detection Hybrid systems: coupling to atoms Optomechanical crystals & arrays “Membrane in the middle” setup

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

Thompson, Zwickl, Jayich, Marquardt, Girvin, Harris, Nature 72, 452 (2008) optical frequency

membrane transmission frequency

membrane displacement Experiment (Harris group, Yale)

Mechanical frequency: ωM=2π⋅134 kHz 50 nm SiN membrane Mechanical quality factor: Q=106÷107

Optomechanical cooling from 300K to 7mK

Thompson, Zwickl, Jayich, Marquardt, Girvin, Harris, Nature 72, 452 (2008) Towards Fock state detection of a macroscopic object Detection of displacement x: not what we need! Towards Fock state detection of a macroscopic object Detection of displacement x: not what we need!

membrane transmission frequency

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

membrane transmission frequency

membrane displacement Towards Fock state detection of a macroscopic object

membrane transmission frequency

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

membrane transmission frequency

membrane displacement phase shift of measurement beam:

(Time-average over QND measurement cavity ring-down time) of phonon number! Towards Fock state detection of a macroscopic object (phonon number) simulated phase shift

Time Towards Fock state detection of a macroscopic object (phonon number) simulated phase shift

Time Towards Fock state detection of a macroscopic object Signal-to-noise ratio:

Optical freq. shift per phonon:

Noise power of freq. measurement: (phonon number) simulated phase shift Ground state lifetime: Time Towards Fock state detection of a macroscopic object Signal-to-noise ratio:

Optical freq. shift per phonon:

very challenging! Noise power of freq. measurement: (phonon number) simulated phase shift Ground state lifetime: Time Towards Fock state detection of a macroscopic object Ideal single-sided cavity: Can observe only phase of reﬂected light, i.e. x2: good Two-sided cavity: Can also observe transmitted vs. reﬂected intensity: linear in x!

- need to go back to two-mode Hamiltonian! - transitions between Fock states! Single-sided cavity, but with losses: same story Detailed analysis (Yanbei Chen’s group, PRL 2009) shows: need absorptive part of photon decay (or 2nd mirror) Miao, H., S. Danilishin, T. Corbitt, and Y. Chen, 2009, PRL 103, 100402 Optomechanics (Outline)

Displacement detection Linear optomechanics

-1 0 1 100 2 Nonlinear dynamics 3

System Bath Quantum theory of cooling Interesting quantum states Towards Fock state detection Hybrid systems: coupling to atoms Optomechanical crystals & arrays 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, Stamper-Kurn (Berkeley) Nature Phys. 4, 561 (2008). cold atoms F. Brennecke, S. Ritter, T. Donner, and T. Esslinger, Science 322, 235 (2008).

...proﬁt from small mass of atomic cloud

Here: Coupling a single atom to a macroscopic mechanical object Challenge: huge mass ratio 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 Optomechanics (Outline)

Displacement detection Linear optomechanics

-1 0 1 100 2 Nonlinear dynamics 3

System Bath Quantum theory of cooling Interesting quantum states Towards Fock state detection Hybrid systems: coupling to atoms Optomechanical crystals & arrays Many modes 8

mechanical mode

optical mode Scaling down

cm usual optical cavities 10 µm

LKB, Aspelmeyer, Harris, Bouwmeester, .... Scaling down

cm usual optical cavities 10 µm

50 µm LKB, Aspelmeyer, Harris, Bouwmeester, .... microtoroids

Vahala, Kippenberg, Carmon, ... Scaling down

cm usual optical cavities 10 µm

50 µm LKB, Aspelmeyer, Harris, Bouwmeester, ....

microtoroids

Vahala, Kippenberg, Carmon, ... optomechanics in photonic circuits optomechanical crystals

5 µm O. Painter et al., Caltech H. Tang et al., Yale 10 µm Optomechanical crystals

free-standing photonic crystal structures optical modes advantages: tight vibrational conﬁnement: high frequencies, small mass (stronger quantum effects) tight optical conﬁnement: vibrational modes large optomechanical coupling (100 GHz/nm) integrated on a chip

from: M. Eichenﬁeld et al., Optics Express 17, 20078 (2009), Painter group, Caltech Optomechanical arrays

laser drive

cell 1

optical mode mechanical mode cell 2 ... collective nonlinear dynamics: classical / quantum cf. Josephson arrays Dynamics in optomechanical arrays

Outlook •2D geometries •Quantum or classical information processing and storage (continuous variables) •Dissipative quantum many-body dynamics (quantum simulations) •Hybrid devices: interfacing GHz qubits with light Photon-phonon translator

(concept: Painter group, Caltech) strong optical “pump”

1 phonon photon 2 Superconducting qubit coupled to nanomechanical resonator

2010: Celand & Martinis labs Josephson phase qubit

piezoelectric nanomechanical resonator (GHz @ 20 mK: ground state!) swap excitation between qubit and mechanical resonator in a few ns! Conversion of quantum information

phonon-photon translation

Josephson nano- phase mechanical light qubit resonator Recent trends

•Ground-state cooling: success! (spring 2011) [Teufel et al. in microwave circuit; Painter group in optical regime] •Optomechanical (photonic) crystals •Multiple mechanical/optical modes •Option: build arrays or ‘optomechanical circuits’ •Strong improvements in coupling •Possibly soon: ultrastrong coupling (resolve single photon- phonon coupling) •Hybrid systems: Convert GHz quantum information (superconducting qubit) to photons •Hybrid systems: atom/mechanics [e.g. Treutlein group] •Levitating spheres: weak decoherence! [Barker/ Chang et al./ Romero-Isart et al.] 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, ....

Precision measurements [e.g. testing deviations from Newtonian Kapitulnik lab (Stanford) gravity due to extra dimensions] Optomechanical circuits & arrays Exploit nonlinearities for classical and quantum information processing, storage, and ampliﬁcation; study collective Tang lab (Yale) dynamics in arrays Parameters of Optomechanical Systems Mechanical damping rate

rate of energy loss, linewidth in mechanical spectrum

rate of re-thermalization, ground state decoherence rate

Mechanical quality factor, number of oscillations during damping time

Optomechanical coupling strength

bilinear interaction tunable coupling! Single-photon optomechanical coupling rate nonclassical mechanical quantum states, ....

Linearized (driving-enhanced) optomechanical coupling rate optomechanical damping rate, state transfer rate, ...

1 3

6 6 10 1 4 2 9 2 8 [Hz] S

4 1 1 /

0 10 1 7 1 6 1 8 1 5 2 0 1 2 7 5

= 7 2 N 2 3 / 9 10 g 0 2 8 2 5 2 1 microwave -2 2 2 photonic crystals 0 2 6 =1 nanoobjects 1 9 0 /N 4 0 microresonators 10 g 2 4 -4 31 mirrors 10 cold atoms = 3 -2 /N 1 0 1 10 g 0 2 1

-4 single-photon coupling rate g 10 101 103 105 107 109 1011 cavity decay rate NS [Hz] Cooperativities

Linearized (driving-enhanced) cooperativity

Optomechanically induced transparency, instability towards optomech. oscillations Linearized (driving-enhanced) quantum cooperativity ground state cooling, state transfer, entanglement, squeezing of light, ...

Single-photon cooperativity

Single-photon “quantum” cooperativity

4 T T 10 2 9 =3 =3 2 8 0mK K 102 0 100 2 5 8 -2 6

10 2 3 1 6 1 1 5 1 7 9 1 1 3 -4 1 4 2 7

10 2 4 2 2 1 8 -6 1 5 10 4 2 1

2 2 0 2 6 31 10-8 1 9

-10 3 7 10 microwave photonic crystals -12 1 0 10 nanoobjects -14 microresonators 10 mirrors 1 2 single-photon cooperativity C 10-16 cold atoms 10-18 101 103 105 107 109 1011 mechanical frequency : S [Hz] m Photon interaction

photon blockade, photon QND measurement, ...

2 2 9

10 2 8

10-2

-6 2 7 1 3 6

10 1 1 2 5 8 2 3 1 7 1 6

1 4 5 1 5 1 8 9 2 0 2 4 -10 2 1

2 6 10 2 2 1 2 1 9 4 7 -14 31 10 microwave 3 photonic crystals nanoobjects -18 microresonators 10 1 0 mirrors photon blockade parameter D

cold atoms 1 2 10-22 10-6 10-4 10-2 100 102 sideband resolution : N m Text Linear Optomechanics Nonlinear Optomechanics Displacement detection Self-induced mechanical oscillations Optical Spring Synchronization of oscillations Cooling & Ampliﬁcation Chaos Two-tone drive: “Optomechanically induced transparency” Nonlinear Quantum State transfer, pulsed operation Optomechanics Wavelength conversion Phonon number detection Radiation Pressure Shot Noise Phonon shot noise Squeezing of Light Photon blockade Squeezing of Mechanics Optomechanical “which-way” Entanglement experiment Precision measurements Nonclassical mechanical q. states Nonlinear OMIT Optomechanical Circuits Noncl. via Conditional Detection Bandstructure in arrays Single-photon sources Synchronization/patterns in arrays Coupling to other two-level Transport & pulses in arrays systems Note:

Yesterday’s red is today’s green! Linear Optomechanics Nonlinear Optomechanics Displacement detection Self-induced mechanical oscillations Optical Spring Attractor diagram? Cooling & Ampliﬁcation Synchronization of oscillations Two-tone drive: “Optomechanically induced Chaos transparency” Ground state cooling State transfer, pulsed operation Wavelength conversion Radiation Pressure Shot Noise Squeezing of Light Squeezing of Mechanics Light-Mechanics Entanglement White: yet unknown challenges/goals Accelerometers Single-quadrature detection, Wigner density Nonlinear Quantum Optomechanics with an active medium Optomechanics Measure gravity or other small forces Mechanics-Mechanics entanglement QND Phonon number detection Pulsed measurement Phonon shot noise Quantum Feedback Photon blockade Rotational Optomechanics Optomechanical “which-way” experiment Nonclassical mechanical q. states Multimode Nonlinear OMIT Mechanical information processing Noncl. via Conditional Detection Bandstructure in arrays Single-photon sources Synchronization/patterns in arrays Coupling to other two-level systems Transport & pulses in arrays Optomechanical Matter-Wave Interferometry Optomechanics

Review “Cavity Optomechanics”: M.Aspelmeyer, T. Kippenberg, FM arXiv:1303.0733