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 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 Optomechanics on different length scales

4 km

LIGO – Laser Interferometer Gravitational Wave Observatory Optomechanics on different length scales

4 km Mirror on cantilever – Bouwmeester lab, Santa Barbara

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

optical cantilever input laser cavity Optomechanical systems

radiation pressure force

optical cantilever input laser cavity 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) The zoo of optomechanical (and analogous) systems

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

LKB group Mavalvala Lehnert (NIST) (Paris) (MIT) Wineland (NIST)

Stamper-Kurn (Berkeley) Schwab (Cornell) cold atoms ... Aspelmeyer (Vienna) 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 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 Optomechanics (Outline)

Introduction Displacement detection Linear optomechanics

0

-1 0 1 100 2 Nonlinear dynamics 3

System Bath Quantum theory of cooling Interesting quantum states Towards Fock state detection Coupling to the motion of a single atom 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 spectrum Fluctuation spectrum

“Wiener-Khinchin theorem” Fluctuation spectrum

“Wiener-Khinchin theorem”

area yields variance of x: 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: Displacement spectrum

T=300 K Experimental curve: Gigan et al., Nature 2006 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”

(measured) + backaction noise

+ imprecision noise

true spectrum “Standard Quantum Limit”

(measured) (measured) + backaction noise imprecisionfull noise noise backactionnoise + imprecision noise intrinsic fluctuations true spectrum coupling to detector (intensity of measurement beam) “Standard Quantum Limit”

(measured) (measured) + backaction noise imprecisionfull noise noise backactionnoise + imprecision noise intrinsic fluctuations 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” 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: ! Enforcing the SQL (Heisenberg) in a weak optical measurement

reflection phase shift: (here: free space) N photons arrive in time t Poisson distribution for fluctuations: a coherent laser beam 1. Uncertainty in phase estimation:

2. Fluctuating force: momentum transfer

Uncertainty product: Heisenberg Optomechanics (Outline)

Introduction Displacement detection Linear optomechanics

0

-1 0 1 100 2 Nonlinear dynamics 3

System Bath Quantum theory of cooling Interesting quantum states Towards Fock state detection Coupling to the motion of a single atom 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 Cooling by damping

T=300 K

Thompson, Zwickl, Jayich, Marquardt, Girvin, Harris, Nature 72, 452 (2008) Optomechanics (Outline)

Introduction Displacement detection Linear optomechanics

0

-1 0 1 100 2 Nonlinear dynamics 3

System Bath Quantum theory of cooling Interesting quantum states Towards Fock state detection Coupling to the motion of a single atom 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 Self-induced oscillations

PHYSICAL REVIEW LETTERS week ending PRL 96, 103901 (2006) 17 MARCH 2006

Cantilever motion Γ 0.5 tric mirrors as part of a high-finesse optical cavity at low x(t) a x0 0 0 temperatures. With modest adjustments to the dimensions A 0 of [22], it should be possible to realize cantilevers with b 0.1 Cavity intensity 10 -0.2 2 spring constant 1N=m, mass m 10ÿ kg, and me- | α chanical Q 105supporting a mirror capable of achieving | 0.05 c  5 a cavity finesse of = 10 (with l 10 cm), and use 0 T 2 | Output intensity -0.4

  α 2 Time d them at an input power of Pin 20 nW at T 0:3K. For a   1 |1-2 b c d FM, Harris, Girvin, Dynamical potential U(A) the Yale (Caltech [25,26]) setups, our dimensionless units 0 1 PRL 96, 1039010 (2006) 0.5 1 0 5 10 15 20 25 30 are scaled by x 3 pm (3.5 pm), t 10 s Oscillation amplitude A ÿ Time ω0t/2π (19 ns), and we have ÿ 10 5 (5:4 10ˆ 4) and !  ÿ  ÿ 0  1 0:65 . For Pin 20 nW (5 mW), we find P 1 9:5 , FIG. 2 (color online). Left: Oscillations of the light energy †2   † 2 and 1 corresponds to an energy of 0.8 pJ (0.4 nJ) or stored inside the cavity, @!Lnmax , and the output light j j6 ˆ 9 2 j j 2 10 (3 10 ) photons inside the cavity. An energy intensity Pout Pin 1 2 , for given sinusoidal cantilever  ~ 5 motion (with !ˆ j1,ÿx xj 5, A 20). See text for real U 1 or T 1 (see below) equals 0.6 K (10 K). Note 0 ˆ ˆ 0 ˆ ˆ thatˆ in the Munichˆ experiment [9] the optical resonances units. Right: Dynamical potential U A , for P 1 and ÿ 4 3 2 1 † ˆ ˆ overlap and P 10 18 ÿ precludes the multistability. 10ÿ ; 10ÿ ; 10ÿ ; 10ÿ (bottom to top curve). Insets show the  ÿ  time evolution of the cavity intensity 2 (thick line) and the The cavity resonance peak x 1= 1 2ix gives j j †ˆ ÿ † output intensity 1 2 2 (thin line, scaled down by a factor rise to a barrier in the effective static cantilever potential 0.1), obtained forj oscillationsÿ j corresponding to the dynamical obtained by integrating the right-hand-side (rhs) of Eq. (2), 2 attractors a,b,c,d [minima in U A ]. Note the additional oscil- !0 2 P 2 † Veff x x x0 arctan 2x . There can be two lation in for each subsequent attractor. †ˆ 2 ÿ † ÿ 2 † j j local minima of Veff, leading to static bistability [6,7]. However, it is known [5] that the time lag generated by 2 2 1 P t !0 x x0 ; (4) the finite cavity ring-down time ÿ introduces additional hj †j i ˆ ÿ † damping or antidamping when x is to the left or right of the P P t 2x_ P ÿ x_ 2 : (5) barrier, respectively. Here we will focus on the regime rad ˆ hj †j i ˆ fric ˆ h i where the antidamping leads to an instability discovered The dependence on x; A, and !0 follows from the coeffi- previously [2,16,17,27], which destroys the stable solution cients of Eq. (3): x_ 0. Then the system settles into self-sustained oscilla- 2 2 ˆ n ; (6) tions, whose full nonlinear dynamics we explore here. hj j i ˆ n j j Dynamics in the unstable regime.—For the parameters X ~ 2 from above, the effects of radiation during one cycle are P rad x_ A!0 Im n n 1: (7) ˆ hj j i ˆ n ‡ weak, such that x t carries out approximately sinusoidal X oscillations at the † unperturbed frequency: x t x We note that a force following the light intensity with a Acos ! t . This fact is the basis of our analytical † theory.‡ time lag  (e.g., photothermal forces [9]) would enter 0 Eq. (2) in the form  1 t dt 2 t exp t For very † high P this approximation breaks down and P ÿ 0 0 t = , and leads to a factor 1ÿ1 i!j j 1,† inside‰ÿ theÿ chaotic motion may result (as was observed in [25] for 0 R 0 ÿ imaginary† Š part on the rhs of Eq. (7).‡ † P 6000), which will not be analyzed here. The dynam- ics of the light amplitude t resembles that of a driven The power balance equation can be recast into the form † damped oscillator which is swept through resonance non- P~ x; A ÿ rad † ; (8) adiabatically (see Fig. 2). The exact solution for a given P~ A ˆ P sinusoidal x t can be written as a Fourier series, t fric † † †ˆ ~ 2 2 i’ t in!0t with Pfric A ! A =2. Stable attractors are those where e † n ne , with 0 the ratio decreases †ˆ for increasing A. A P 1 Jn After solving the force balance Eq. (4) for x x x ;A , ÿ !0† ˆ 0 † n 1 ; (3) we can plot the contour lines of the left-hand side of Eq. (8) ˆ 2 in!0 ix ‡ 2 ÿ in the x ;A plane. These yield the possible equilibrium 0 † and a global phase ’ t A=!0 sin !0t , where Jn is the values of the oscillation amplitude A as a function of x0, Bessel function of the †ˆ first kind. The† output† light intensity see Fig. 3. The red dots show the results of a simulation of 2 is [28] given by Pin 1 2 , and the Fourier transform of the initial equations of motion, Eqs. (1) and (2). t directly yieldsj theÿ sidebandj spectrum. This diagram may be observed experimentally by † Dynamical multistability.—The possible attractors (x; A) sweeping x0, either via tuning the laser frequency or by have to fulfill two conditions resulting from Eq. (2) for any applying some force to the cantilever. After initial tran- periodic motion. The total time-averaged force x has to sients, the system settles into one of the attractors (specific vanish, and the net power input via the radiationh pressurei amplitude A), which may be identified either via its distinct force (due to the Doppler shift caused by the moving pattern of light emission from the cavity (see Fig. 2) or by mirror) must equal the power dissipated through friction, the total power loss due to friction, Pin Pout x x_ 0: P ÿA2!2=2, or by obtaining x t from interferometryÿ ˆ h i ˆ in 0 † 103901-2 Self-induced oscillations

cavity intensity oscillations: PHYSICAL REVIEW LETTERS week ending PRL 96, 103901 (2006) 17 MARCH 2006

Cantilever motion Γ 0.5 tric mirrors as part of a high-finesse optical cavity at low x(t) a x0 0 [analytical 0 temperatures. With modest adjustments to the dimensions A 0 solution] of [22], it should be possible to realize cantilevers with b 0.1 Cavity intensity 10 -0.2 2 spring constant 1N=m, mass m 10ÿ kg, and me- | 1. force balance: α chanical Q 105supporting a mirror capable of achieving | 0.05 c  5 a cavity finesse of = 10 (with l 10 cm), and use 0 T 2 | Output intensity -0.4

  α 2 Time d them at an input power of Pin 20 nW at T 0:3K. For 2. powera balance:   1 |1-2 b c d FM, Harris, Girvin, Dynamical potential U(A) the Yale (Caltech [25,26]) setups, our dimensionless units 0 1 PRL 96, 1039010 (2006) 0.5 1 0 5 10 15 20 25 30 are scaled by x 3 pm (3.5 pm), t 10 s Oscillation amplitude A ÿ Time ω0t/2π (19 ns), and we have ÿ 10 5 (5:4 10ˆ 4) and ! Two equations for two unknowns:  ÿ  ÿ 0  1 0:65 . For Pin 20 nW (5 mW), we find P 1 9:5 , FIG. 2 (color online). Left: Oscillations of the light energy †2   † 2 and 1 corresponds to an energy of 0.8 pJ (0.4 nJ) or stored inside the cavity, @!Lnmax , and the output light j j6 ˆ 9 2 j j 2 10 (3 10 ) photons inside the cavity. An energy intensity Pout Pin 1 2 , for given sinusoidal cantilever  ~ 5 motion (with !ˆ j1,ÿx xj 5, A 20). See text for real U 1 or T 1 (see below) equals 0.6 K (10 K). Note 0 ˆ ˆ 0 ˆ ˆ thatˆ in the Munichˆ experiment [9] the optical resonances units. Right: Dynamical potential U A , for P 1 and ÿ 4 3 2 1 † ˆ ˆ overlap and P 10 18 ÿ precludes the multistability. 10ÿ ; 10ÿ ; 10ÿ ; 10ÿ (bottom to top curve). Insets show the  ÿ  time evolution of the cavity intensity 2 (thick line) and the The cavity resonance peak x 1= 1 2ix gives j j †ˆ ÿ † output intensity 1 2 2 (thin line, scaled down by a factor rise to a barrier in the effective static cantilever potential 0.1), obtained forj oscillationsÿ j corresponding to the dynamical obtained by integrating the right-hand-side (rhs) of Eq. (2), 2 attractors a,b,c,d [minima in U A ]. Note the additional oscil- !0 2 P 2 † Veff x x x0 arctan 2x . There can be two lation in for each subsequent attractor. †ˆ 2 ÿ † ÿ 2 † j j local minima of Veff, leading to static bistability [6,7]. However, it is known [5] that the time lag generated by 2 2 1 P t !0 x x0 ; (4) the finite cavity ring-down time ÿ introduces additional hj †j i ˆ ÿ † damping or antidamping when x is to the left or right of the P P t 2x_ P ÿ x_ 2 : (5) barrier, respectively. Here we will focus on the regime rad ˆ hj †j i ˆ fric ˆ h i where the antidamping leads to an instability discovered The dependence on x; A, and !0 follows from the coeffi- previously [2,16,17,27], which destroys the stable solution cients of Eq. (3): x_ 0. Then the system settles into self-sustained oscilla- 2 2 ˆ n ; (6) tions, whose full nonlinear dynamics we explore here. hj j i ˆ n j j Dynamics in the unstable regime.—For the parameters X ~ 2 from above, the effects of radiation during one cycle are P rad x_ A!0 Im n n 1: (7) ˆ hj j i ˆ n ‡ weak, such that x t carries out approximately sinusoidal X oscillations at the † unperturbed frequency: x t x We note that a force following the light intensity with a Acos ! t . This fact is the basis of our analytical † theory.‡ time lag  (e.g., photothermal forces [9]) would enter 0 Eq. (2) in the form  1 t dt 2 t exp t For very † high P this approximation breaks down and P ÿ 0 0 t = , and leads to a factor 1ÿ1 i!j j 1,† inside‰ÿ theÿ chaotic motion may result (as was observed in [25] for 0 R 0 ÿ imaginary† Š part on the rhs of Eq. (7).‡ † P 6000), which will not be analyzed here. The dynam- ics of the light amplitude t resembles that of a driven The power balance equation can be recast into the form † damped oscillator which is swept through resonance non- P~ x; A ÿ rad † ; (8) adiabatically (see Fig. 2). The exact solution for a given P~ A ˆ P sinusoidal x t can be written as a Fourier series, t fric † † †ˆ ~ 2 2 i’ t in!0t with Pfric A ! A =2. Stable attractors are those where e † n ne , with 0 the ratio decreases †ˆ for increasing A. A P 1 Jn After solving the force balance Eq. (4) for x x x ;A , ÿ !0† ˆ 0 † n 1 ; (3) we can plot the contour lines of the left-hand side of Eq. (8) ˆ 2 in!0 ix ‡ 2 ÿ in the x ;A plane. These yield the possible equilibrium 0 † and a global phase ’ t A=!0 sin !0t , where Jn is the values of the oscillation amplitude A as a function of x0, Bessel function of the †ˆ first kind. The† output† light intensity see Fig. 3. The red dots show the results of a simulation of 2 is [28] given by Pin 1 2 , and the Fourier transform of the initial equations of motion, Eqs. (1) and (2). t directly yieldsj theÿ sidebandj spectrum. This diagram may be observed experimentally by † Dynamical multistability.—The possible attractors (x; A) sweeping x0, either via tuning the laser frequency or by have to fulfill two conditions resulting from Eq. (2) for any applying some force to the cantilever. After initial tran- periodic motion. The total time-averaged force x has to sients, the system settles into one of the attractors (specific vanish, and the net power input via the radiationh pressurei amplitude A), which may be identified either via its distinct force (due to the Doppler shift caused by the moving pattern of light emission from the cavity (see Fig. 2) or by mirror) must equal the power dissipated through friction, the total power loss due to friction, Pin Pout x x_ 0: P ÿA2!2=2, or by obtaining x t from interferometryÿ ˆ h i ˆ in 0 † 103901-2 Attractor diagram

power fed into the cantilever

1

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

15 Color scale: power fed into cantilever

10

5 + FM, Harris, Girvin, PRL 96, 103901 (2006) -

Cantilever oscillation amplitude 0 ï 0  detuning Attractor diagram

15 Color scale: power fed into cantilever

10

5 + FM, Harris, Girvin, PRL 96, 103901 (2006) - Multistability

Cantilever oscillation amplitude 0 ï 0  detuning Attractor diagram

15 Color scale: power fed into cantilever

10

5 + FM, Harris, Girvin, PRL 96, 103901 (2006) -

Cantilever oscillation amplitude 0 ï 0  detuning Attractor diagram

15 Color scale: power fed into cantilever

10

5 + FM, Harris, Girvin, PRL 96, 103901 (2006) -

Cantilever oscillation amplitude 0 ï 0  detuning Attractor diagram

15 Color scale: power fed into cantilever

10

5 + FM, Harris, Girvin, PRL 96, 103901 (2006) -

Cantilever oscillation amplitude 0 ï 0  detuning 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, 133903 (2008) Comparison theory/experiment 4

Transmission Amplitude Metzger ettake al., into account the second mode as well. The total PRL 2008 1 displacement is x(t)=x0 + x1(t)+x2(t), and we have to employ a set of equations: low laser I/I0=1.8 % 20 power x¨ = ω2x Γ x˙ + F bol[x(t)]/m , (9) i − i i − i i i i 0.5 where xi denotes the coordinate of the i-th mechanical mode with frequency ωi, mechanical damping rate Γi, and effective mass mi (where ω1/2π =8.7 kHz, ω2/2π = 60 kHz, Γ1 = 30.0 Hz, Γ2 = 150 Hz). We neglected the ra- medium diation pressure force, as this is much smaller anyway for 4 laser power I/I0=8.3 % the parameters of this setup. The mechanical modes are now coupled indirectly by the bolometric force. For the 8 present setup, this force changes sign when going to the 2 bol bol second mode. Choosing F2 m1/F1 m2 = 28.8 as an adjustable parameter, we have found the numerical− sim- ulation of these coupled nonlinear equations for the first two modes to be in surprisingly good agreement with the experiment (Fig. 3). We have to note that the relation Displacement x0 Displacement x0 between the measured “amplitude”, i.e. first harmonic of 40 + I(t) at frequency ω1, and the actual amplitude A1 does 4 not hold exactly if both modes are excited simultane- ously. 20 At maximum laser power, there are actually two inter- + vals with simultaneous excitation of both modes (indi- cated in Fig. 3). Specifically, the onset of such a regime 440 4 0 4 at x0 λ/8 can be interpreted as follows: Taking into ac- count≈F bol/F bol < 0, we see that the second mode gains Displacement x0 Displacement x0 2 1 its energy from dipping into the resonance at x = λ/2, while the first is still provided with energy due to the Figure 3: (Color online) Experiment vs. theory. The trans- mitted light intensity (left) and the amplitude of self-induced resonance at x = 0. cantilever oscillations (right), from a simulation of the theo- Numerical evidence shows that the steady-state mo- retical model (red full curves) and from the experiment (blue tion consists of sinusoidal oscillations in x1,2 at the re- data points), at increasing input power levels (top to bottom). spective eigenfrequencies, of nearly constant amplitudes Theoretical “transmission” curves display the (rescaled) time- and without phase locking (for the parameters explored averaged circulating light intensity from the simulation. “Am- 2 here). Thus x(t) x + [A + δA (t)] cos(ω t + φ ), plitude” curves are obtained from the power in the intensity ≈ 0 i=1 i i i i where δAi(t)/Ai 1, and the φi are arbitrary phases. sidebands at the fundamental mechanical mode of the can- ￿ ￿ tilever (see main text). For clarity, the hysteresis observed Higher input powers will lead to excitations of additional upon sweeping x0 up or down has been shown only in the mid- modes, and the system might go into a chaotic regime of dle panel. The region of instability (shaded interval) grows motion. with increasing input power. Simultaneous self-induced oscil- Conclusions. - We have analyzed the nonlinear dy- lations of the first two mechanical modes set in at the highest namics of an optomechanical system, by measuring and power displayed (in the two intervals indicated in the plot). The calculated amplitude of the second mode is shown as a explaining its attractor diagram. The comparison of dashed line. data and theoretical predictions have revealed the on- set of multi-mode dynamics at large optical power, with two mechanical modes of the cantilever participating in down. the radiation-driven self-sustained oscillations. These ef- Beginning at Iin =0.57 I0, a second interval of self- fects could find applications in highly sensitive force or oscillatory behaviour appears to the left of the resonance, displacement detection [23]. In the future, it would be growing stronger and wider with increasing laser power. interesting to observe the attractor diagram in systems This initially completely unexpected result may be ex- of a high optical finesse [21, 22] (with delayed radiation plained by invoking the influence of higher mechanical dynamics), or the self-excitation of multiple mechanical modes. These may be excited by the radiation as well, modes of sub-wavelength mechanical resonators interact- leading to coupled (multimode) nonlinear dynamics with ing with the radiation field inside a cavity [14, 29]. The richer features than discussed up to now. whole field of quantum nonlinear dynamics in systems of In order to describe the behaviour in that case, we now this kind also remains to be explored. 4

Transmission Amplitude take into account the second mode as well. The total 1 displacement is x(t)=x0 + x1(t)+x2(t), and we have to employ a set of equations: 20 x¨ = ω2x Γ x˙ + F bol[x(t)]/m , (9) i − i i − i i i i 0.5 where xi denotes the coordinate of the i-th mechanical mode with frequency ωi, mechanical damping rate Γi, and effective mass mi (where ω1/2π =8.7 kHz, ω2/2π = 60 kHz, Γ1 = 30.0 Hz, Γ2 = 150 Hz). We neglected the ra- diation pressure force, as this is much smaller anyway for 4 Comparison theory/experiment the parameters of this setup. The mechanical modes are now coupled indirectly by the bolometric force. For the Metzger8 et al., present setup, this force changes sign when going to the 2 PRL 2008 second mode. Choosing F bolm /F bolm = 28.8 as an 2 1 1 2 − Coupled self-oscillations adjustable parameter, we have found the numerical sim- ulation of these coupled nonlinear equations for the first high laser of two mechanical modes power two modes to be in surprisingly good agreement with the experiment (Fig. 3). We have to note that the relation I/I0=100 % between the measured “amplitude”, i.e. first harmonic of 40 + I(t) at frequency ω1, and the actual amplitude A1 does 4 not hold exactly if both modes are excited simultane- ously. 20 At maximum laser power, there are actually two inter- + vals with simultaneous excitation of both modes (indi- cated in Fig. 3). Specifically, the onset of such a regime 440 4 0 4 at x0 λ/8 can be interpreted as follows: Taking into ac- count≈F bol/F bol < 0, we see that the second mode gains Displacement x0 Displacement x0 2 1 its energy from dipping into the resonance at x = λ/2, while the first is still provided with energy due to the Figure 3: (Color online) Experiment vs. theory. The trans- mitted light intensity (left) and the amplitude of self-induced resonance at x = 0. cantilever oscillations (right), from a simulation of the theo- Numerical evidence shows that the steady-state mo- retical model (red full curves) and from the experiment (blue tion consists of sinusoidal oscillations in x1,2 at the re- data points), at increasing input power levels (top to bottom). spective eigenfrequencies, of nearly constant amplitudes Theoretical “transmission” curves display the (rescaled) time- and without phase locking (for the parameters explored averaged circulating light intensity from the simulation. “Am- 2 here). Thus x(t) x + [A + δA (t)] cos(ω t + φ ), plitude” curves are obtained from the power in the intensity ≈ 0 i=1 i i i i where δAi(t)/Ai 1, and the φi are arbitrary phases. sidebands at the fundamental mechanical mode of the can- ￿ ￿ tilever (see main text). For clarity, the hysteresis observed Higher input powers will lead to excitations of additional upon sweeping x0 up or down has been shown only in the mid- modes, and the system might go into a chaotic regime of dle panel. The region of instability (shaded interval) grows motion. with increasing input power. Simultaneous self-induced oscil- Conclusions. - We have analyzed the nonlinear dy- lations of the first two mechanical modes set in at the highest namics of an optomechanical system, by measuring and power displayed (in the two intervals indicated in the plot). The calculated amplitude of the second mode is shown as a explaining its attractor diagram. The comparison of dashed line. data and theoretical predictions have revealed the on- set of multi-mode dynamics at large optical power, with two mechanical modes of the cantilever participating in down. the radiation-driven self-sustained oscillations. These ef- Beginning at Iin =0.57 I0, a second interval of self- fects could find applications in highly sensitive force or oscillatory behaviour appears to the left of the resonance, displacement detection [23]. In the future, it would be growing stronger and wider with increasing laser power. interesting to observe the attractor diagram in systems This initially completely unexpected result may be ex- of a high optical finesse [21, 22] (with delayed radiation plained by invoking the influence of higher mechanical dynamics), or the self-excitation of multiple mechanical modes. These may be excited by the radiation as well, modes of sub-wavelength mechanical resonators interact- leading to coupled (multimode) nonlinear dynamics with ing with the radiation field inside a cavity [14, 29]. The richer features than discussed up to now. whole field of quantum nonlinear dynamics in systems of In order to describe the behaviour in that case, we now this kind also remains to be explored. Optomechanics (Outline)

Introduction Displacement detection Linear optomechanics

0

-1 0 1 100 2 Nonlinear dynamics 3

System Bath Quantum theory of cooling Interesting quantum states Towards Fock state detection Coupling to the motion of a single atom 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 =ˆa†ˆa ¯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 Towards the ground state

1x1010 1x109 1x108

7 MIT 1x10 Yale 1x106 100000 LKB 10000 IQOQI 1000 MPQ 100 10 1 minimum possible JILA

phonon number phonon 0.1 phonon number Caltech 0.01 0.001 0.0001 0.00001 0.01 0.1 1 10

Results from groups at MIT (Mavalvala), LKB (Pinard, Heidmann, Cohadon), Yale (Harris), IQOQI Wien (Aspelmeyer), MPQ Munich (Kippenberg), JILA Boulder (Lehnert), Caltech (Schwab) Strong coupling: resonances of light and mechanics hybridize 20 40

strong 0 coupling 1.51.5 1.5 1.0 0.5 cantilever spectrum

splitting

cantilever 1.01.0 frequency driven cavity

0.50.5 0.50.5 1.0 1.01.5 1.5 detuning FM, Chen, Clerk, Girvin, PRL 93, 093902 (2007) LETTERS NATURE | Vol 460 | 6 August 2009

have to lie at sidebands equal to the dressed state frequencies v6 coupling gives rise to the two side-peaks in the so-called Mollow relative to the incoming laser photons of frequency vL, that is, they triplet. It is interesting to note that the analogy even holds for the have to be emitted at sideband frequencies vL 6 v1 or vL 6 v2. single-photon regime, in which both systems are close to their Homodyne detection provides us with direct access to the optical quantum ground state. For both cases (that is, the atom–cavity sys- sideband spectrum, which is presented in Fig. 2a for the resonant case tem and the cavity–optomechanical system), a sufficiently strong D < vm. For small optical pump power, that is, in the regime of weak single-photon interaction g0 would allow one to obtain the well- coupling, the splitting cannot be resolved and one obtains the well- known vacuum Rabi splitting as well as state-dependent level spa- known situation of resolved sideband laser cooling, in which Stokes cing, which is due to intrinsic nonlinearities in the coupling. and anti-Stokes photons are emitted at one specific sideband fre- We should stress that normal mode splitting alone does not establish quency. The splitting becomes clearly visible at larger pump powers, a proof for coherent dynamics, that is, for quantum interference effects. which is unambiguous evidence for entering the strong coupling With the present experimental parameters, such effects are washed out regime. Indeed, at a maximum optical driving power of ,11 mW, by thermal decoherence and normal mode splitting has a classical we obtain a coupling strength g 5 2p 3 325 kHz, which is larger than explanation in the framework of linear dispersion theory30. Still, the both k 5 2p 3 215 kHz and cm 5 2p 3 140 Hz and which corre- demonstration of normal mode splitting is a necessary condition for sponds to the magnitude of the level crossing shown in Fig. 2b. As future quantum experiments. is expected, for detunings D off resonance, the normal mode frequen- We finally comment on the prospects for mechanical quantum cies approach the values of the uncoupled system. state manipulation in the regime of strong coupling. One important These characteristics of our strongly driven optomechanical sys- additional requirement in most proposed schemes is the initializa- tem are reminiscent of a strongly driven two-level atom, and indeed a tion of the mechanical device close to its quantum ground state. strong and instructive analogy exists. If an atom is pumped by a Recent theoretical results show that both ground state laser cooling strong laser field, optical transitions can only occur between dressed and strong coupling can be achieved simultaneously, provided that kBT 20,22 atomic states, that is, atomic states ‘dressed’ by the interaction with the conditions BQ =k=vm are fulfilled . Thus, in addition to the laser field. For strong driving, any Rabi splitting that is induced by operating in the resolved sideband regime, a thermal decoherence strong coupling is effectively of order G n (n , mean number of rate that is small compared to the cavity decay rate is required. 0 hiL L laser photons; G0, electric dipole coupling) and one therefore obtains Cryogenic experiments have demonstrated thermal decoherence an equally spaced level splitting, fully analogouspffiffiffiffiffiffiffiffiffi to the coupled opto- rates as low as 20 kHz for nanomechanical resonators for a 20 mK mechanical spectrum. From this point of view, the optomechanical environment temperature9. For our experiment, temperatures below modes can be interpreted in a dressed state approach as excitations of 300 mK would be sufficient to combine strong coupling with ground mechanical states that are dressed by the cavity radiation field. The state cooling. origin of the sideband doublet as observed in the output field of the We have demonstrated strong coupling of a micromechanical strongly driven optomechanical cavity correspondsExperiment: to the resonance resonatorOptomechanical to an optical cavity field. Thishybridization regime is a necessary pre- fluorescence spectrum of a strongly driven atom, in which strong condition to obtaining quantum control of mechanical systems. (Mechanical spectrum under strong illumination) ab1×10–10 1.4 6.9 mW 10.7 mW Δ =1.02!m Theory 0.1×10–10 1.2 m ! / 1.0 –10 1×10 +/–

3.8 mW !

0.5×10–10

Δ Δ

0.8 Δ

=0.92 =1.02 =1.13 ! ! !

0.6 m m m

6.9 mW 0.6 0.8 1.0 1.2 1.4 1.6 –10 1.5×10 Δ/!m c

5×10–10 1×10–10 3×10–10 4×10–10 Noise power spectrum (arbitrary units) 2.7×10–10 10.7 mW 3×10–10 (a.u.) (a.u.) 3×10–10 2.4×10–10

–10 2×10–10

Noise power spectrum 2.1×10 Noise power spectrum 0.80.7 0.9 1.0 1.1 0.80.70.9 1.0 1.1 Frequency (MHz) Frequency (MHz) 2×10–10 0.7 0.8 0.9 1.0 1.1 Frequency (MHz) Aspelmeyer group 2009 Figure 2 | Optomechanical normal mode splitting and avoided crossing in assuming two independent Lorentzian curves, red solid lines are the sum the normal-mode frequency spectrum. a, Emission spectra of the driven signal of these two fits. b, Normal mode frequencies obtained from the fits to optomechanical cavity, obtained from sideband homodyne detection on the the spectra as a function of detuning D. For far off-resonant driving, the strong driving field after its interaction with the optomechanical system (see normal modes approach the limiting case of two uncoupled systems. Dashed Supplementary Information). The power levels from top to bottom (0.6, 3.8, lines indicate the frequencies of the uncoupled optical (diagonal) and 6.9, 10.7 mW) correspond to an increasing coupling strength of g 5 78, 192, mechanical (horizontal) resonator, respectively. At resonance, normal mode 260 and 325 kHz (g 5 0.4, 0.9, 1.2, 1.5 k). All measurements are performed splitting prevents a frequency degeneracy, which results in the shown close to resonance (D 5 1.02 vm). For strong driving powers a splitting of the avoided level crossing. Error bars, s.d. Solid lines are simulations (see cavity emission occurs, corresponding to the normal mode frequencies of Supplementary Information). For larger detuning values, the second normal true hybrid optomechanical degrees of freedom. This normal mode splitting mode peak could no longer be fitted owing to a nearby torsional mechanical is an unambiguous signature of the strong coupling regime. All plots are mode. c, Normal mode spectra measured off resonance. shown on a logarithmic scale. Green dashed lines are fits to the data 726 ©2009 Macmillan Publishers Limited. All rights reserved Experiment: Optomechanically induced transparency 9 (Light field transmission of a second, weak probe beam)

Kippenberg2 group 2010 FIG. 3: Observation of OMIT. a) Theoretically expected intracavity probe power A− ,oscillationamplitudeX,probepower 2 2 | | transmission tp and the homodyne signal thom as a function of the modulation frequency Ω/2π (top to bottom panels). The first two panels have| | additionally been normalized| to unity.| When the two-photon resonance condition is met, the mechanical oscillator is excited ∆￿ =0,givingrisetodestructiveinterferenceofexcitationpathwaysforanintracavityprobefield.Theprobetransmission therefore exhibits an inverted dip, which can be easily identified in the homodyne signal. b) Experimentally observed normalized homodyne traces when the probe frequency is scanned by sweeping the phase modulator frequency Ω for different values of control beam detuning ∆¯ . While the center of the response of the bare optical cavity shifts correspondingly, the sharp dip characteristic of OMIT occurs always for ∆￿ =0.Thepowerofthecontrolbeamsenttothecavityis0.5mWinthesemeasurementsandtheHe-3buffer gas has a pressure of 155 mbar at a temperature of 3.8K. Themiddlepanelshowstheoperatingconditionswherethecontrolbeamistuned to the lower motional sideband ∆¯ Ωm = 2π 51.8MHz. Theregionaroundthecentraldip(orangebackground)isstudiedinmore ≈− detail− in· a dedicated experimental series (cf. figure 4). Optomechanics (Outline)

Introduction Displacement detection Linear optomechanics

0

-1 0 1 100 2 Nonlinear dynamics 3

System Bath Quantum theory of cooling Interesting quantum states Towards Fock state detection Coupling to the motion of a single atom Squeezed states

Squeezing the mechanical oscillator state Squeezed states

Squeezing the mechanical oscillator state Squeezed states

Squeezing the mechanical oscillator state Squeezed states

Squeezing the mechanical oscillator state

t

x Squeezed states

Squeezing the mechanical oscillator state

t

x

Periodic modulation of spring constant: Parametric amplification Squeezed states

Squeezing the mechanical oscillator state

t

x

Periodic modulation of spring constant: Parametric amplification Squeezed states

Squeezing the mechanical oscillator state

t

x

Periodic modulation of spring constant: Parametric amplification Squeezed states

Squeezing the mechanical oscillator state

t

x

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 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 Optomechanical entanglement Optomechanical entanglement

coherent mechanical state entangled state

n=0 n=1 (light field/mechanics) n=2 n=3 Proposed optomechanical which-path experiment and quantum eraser

p

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)

Introduction Displacement detection Linear optomechanics

0

-1 0 1 100 2 Nonlinear dynamics 3

System Bath Quantum theory of cooling Interesting quantum states Towards Fock state detection Coupling to the motion of a single atom Towards Fock state detection of a macroscopic object

n=3

n=2

n=1

n=0 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 “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 Mechanical quality factor: Q=106÷107 50 nm SiN membrane Current optical finesse: 7000÷15000 (5⋅105) [almost sideband regime]

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) Measured phase shift

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

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

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

Time Thompson, Zwickl, Jayich, FM, Girvin, Harris, Nature 72, 452 (2008) Towards Fock state detection of a macroscopic object Signal-to-noise ratio:

Optical freq. shift per phonon:

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

Introduction Displacement detection Linear optomechanics

0

-1 0 1 100 2 Nonlinear dynamics 3

System Bath Quantum theory of cooling Interesting quantum states Towards Fock state detection Coupling to the motion of a single atom 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 Coupling a single atom to a heavy object: Why it is hard

1014 atoms

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

1014 atoms

1 atom

interaction:

freq. shift coupling freq. shift coupling term small! 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 Strong coupling of a mechanical oscillator and a single atom

K. Hammerer1,5, M. Wallquist1,5, C. Genes1, M. Ludwig2, F. Marquardt2, P. Treutlein3, P. Zoller1,5, J. Ye4,5, H. J. Kimble5 1 Institute for Theoretical Physics, University of Innsbruck, and Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, Technikerstrasse 25, 6020 Innsbruck, Austria 2 Department of Physics, Center for NanoScience, and Arnold Sommerfeld Center for Theoretical Physics, Ludwig-Maximilians-Universitat¨ Munchen,¨ Theresienstr. 37, D-80333 Munich, Germany 3 Max-Planck-Institute of Quantum Optics and Department of Physics, Ludwig-Maximilians-Universitat¨ Munchen,¨ Schellingstr. 4, D-80799 Munich, Germany 4 JILA, National Institute of Standards and Technology and University of Colorado, Boulder, CO 80309-0440 USA 5 Norman Bridge Laboratory of Physics 12-33, California Institute of Technology, Pasadena, CA 91125 USA (Dated: May 7, 2009) We propose and analyze a setup to achieve strong coupling between a single trapped atom and a mechanical oscillator. The interaction between the motion of the atom and the mechanical oscillator is mediated by a quan- tized light field in a laser driven high-finesse cavity. In particular, we show that high fidelity transfer of quantum states between the atom and the mechanical oscillator is in reach for existing or near future experimental param- eters. Our setup provides the basic toolbox for coherent manipulation, preparation and measurement of micro- and nanomechanical oscillators via the tools of atomic physics.

Recent experiments with micro- and nanomechanical oscil- (a) lators coupled to the optical field in a cavity are approach- ing the regime where quantum effects dominate [1, 2, 3]. Cavity-mediated coupling

In light of this progress, the question arises to what extent (b) (c) e s n the quantized motion of a mesoscopic mechanical system o p s e r

can be coherently coupled to a microscopic quantum object y t i v a [4, 5, 6, 7, 8, 9], the ultimate challenge being strong cou- c pling to the motion of a single atom. For a direct mechani- (d) cal coupling the interaction involves scale factors m/M 7 4 ∼ 10− 10− depending on the ratio of the mass of the atom − ￿ m to the mass of the mechanical oscillator M [4]. It is there- (e) fore difficult to achieve a coherent coupling for exchange of a single vibrational quantum that is much larger than relevant for each cavity mode: dissipation rates. FIG. 1: (a) Strong coupling of the motion of a single atom to a vibrational degree of freedom of a micron-sized membrane can be In this Letter we show, however, that strong coupling (likewise for membrane) achieved in a two mode cavity (for details see text). (b) Cavity re- atom-membrane coupling via virtual transitions: can be realized between a single trapped atom and an opto- sponse as a function of frequency. Two cavity modes are driven mechanical oscillator. The coupling between the motion of by two lasers of frequencies ω1 and ω2, with red and blue detun- a membrane [10] – representing the mechanical oscillator – ing respectively. (c) The two frequenciesdetuning drivelaser-cavity two atomic transi- and the atom is mediated by the quantized light field in a laser tions, e.g. the D1,2 lines of Cs, both with red detuning, causing AC driven high-finesse cavity. Remarkably, in this setup a co- Stark shift of the ground state. (d) (left side) The atom is trapped in the potential from the two optical lattices (red and blue curves) herent coupling for single-atom and membrane exceeding the 2 u1,2(x) = sin (k1,2x) with wave vectors k1 = k2. (right side) dissipative rates by a factor of ten is within reach for present or The membrane is placed at a point of steepest slope￿ of the intensity near future experimental parameters [11]. Entering the strong profiles u1,2(x) where opto-mechanical coupling is maximal. (e) A coupling regime provides a quantum interface allowing the small displacement of the membrane will shift the cavity resonances coherent transfer of quantum states between the mechanical [cf. dashed line in (b)] resulting in a spatial shift of the trap potential oscillator and atoms, opening the door to coherent manipu- for the atom, and thus an effective linear atom-membrane coupling lation, preparation and measurement of micromechanical ob- as in Eq. (1). (Displacements and frequency shifts are not to scale.) jects via the well-developed tools of atomic physics, and per- arXiv:0905.1015v1 [quant-ph] 7 May 2009 translates into a variation of the intensity of the intracavity haps the birth of quantum phononics. light field. In addition, we assume that this intracavity field We propose and analyze a setup which combines the recent provides an optical lattice as a trap for a single atom. Thus advances of micromechanics with membranes in optical cavi- for the setup of Fig. 1a the motion of the membrane will be ties [10] and cavity QED with single trapped atoms [11] (see coupled via the dynamics of the optical trap to the motion of Fig. 1a). We consider a membrane placed in a laser driven the atom, and vice versa. This coupling is strongly enhanced high-finesse cavity representing the opto-mechanical system by the cavity finesse which is a key ingredient in achieving the with radiation pressure coupling. In this setup the motion of strong coupling regime. the membrane manifests itself as a dynamic detuning of cav- In the following we are interested in a configuration which ity modes. For a cavity mode driven by a detuned laser this - after integrating out the internal cavity dynamics - realizes 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! Example: State transfer 4 Transfer of squeezed state Squeezing transferred to membrane from atom to membrane (maximized over time) (a) (b) applied in Eq. (1), the effective dynamics is described by ]

B HI G(ama† +h.c.) in the interaction picture. This in-

d at m

[ -3 dB

o ￿ t g teraction swaps the state of the atom and the membrane after n A i

z -6 dB π e a time Gt = . Thus, states which are easily created on the -9 dB e 2 u

q side of the atom (e.g., squeezed or Fock states) can be trans- S

e d n e a

r ferred to the membrane. In Fig. 2 we study such a transfer of a r -9 dB r b e f m s squeezed state based on the exact solution of the master equa- e n a M r tion in Eq. (2). The figure also illustrates the importance of T [ ] limiting the loss in order to achieve quantum state transfer or readout. The general analysis provided here shows that con- FIG. 2:Exploit (a) Wigner toolbox functions for single-atom of atom and membrane manipulation (upper and dition (9) is the principal bottleneck for a reduction of losses. lower panels, respectively). At t =0(left panels) the atom is in 2 •Creation of arbitrary atom states and transfer to membrane κth Especially the ratio F might be further increased by im- a squeezed•Transfer state of (9 membrane dB) and the states membrane to atom in and a thermal measurement state with a γmM mean number of phonons n¯ =5. An exact solution of the equation of proving material properties and nanostructuring, though there motion (2) with losses Γ =Γ c, Γm, Γat at rate Γ =0 .1 G shows will always be an apparent tradeoff between good mechani- π × that after a time Gt = 2 (right panels) the states are exchanged, cal isolation and a large thermal link. Another rather obvious up to a trivial rotation in phase space by 90◦. (b) Squeezing trans- route for improvement is to use a small ensemble of N atoms Γ ferred to membrane (maximized over time), versus loss rate , for trapped inside the cavity [14, 15, 16], resulting in a √N en- the indicated values of initial atomic position fluctuations. hancement of the atom-cavity coupling. However, our main point here is to identify the general conditions for achieving α and therefore the absolute timescale of the system are not strong coupling of a single atom to a massive mechanical os- fixed by Eqs. (6) to (9). These equations actually impose con- cillator, and to demonstrate that it is possible to meet these ditions on the properties of the system at the single photon conditions with state of the art systems. level. The necessary cavity amplitude α, and with it the ab- Support by the Austrian Science Fund through SFB FO- solute timescale of the dynamics, will finally follow from the QUS, by the Institute for Quantum Optics and Quantum In- resonance condition ωat = ωm. formation, by the European Union through project EuroSQIP, Example: We will show now that the interaction between a by NIST and NSF, and by the DFG through NIM, SFB631 and single Cs atom and a SiN membrane of small effective mass the Emmy-Noether program is acknowledged. MW, KH, PZ M =0.4 ng mediated by a high-finesse optical micro–cavity and JY thank HJK for hospitality at Caltech. can enter the strong coupling regime. Firstly, we assume a large cavity finesse of 2 105 which is consistent with a [1] A. Schliesser et al., arXiv:0901.1456v1. F ￿ × 5 measured value of Im(n) 1.5 10− for the absorption in [2] S. Groeblacher et al., arXiv:0901.1801v1. ￿ × a SiN membrane inside a cavity [12]. A small cavity waist of [3] For a recent review and further references see F. Marquardt, S. M. Girvin, arXiv:09050566 (2009). w0 = 10 µm results in a cooperativity parameter of C = 140. ∆ [4] L. Tian and P. Zoller, Phys. Rev. Lett. 93, 266403 (2004). A ratio of κ 18 satisfies Eqs. (6) and (8). Secondly, for the [5] P. Treutlein et al., Phys. Rev. Lett. , 140403 (2007). ￿m 13 99 mass ratio of =6 10− and an amplitude reflectivity M × [6] N. Lambert et al., Phys. Rev. Lett. 100, 136802 (2008). r =0.45 we choose a ratio δ 450 in order to approximately [7] P. Rabl et al., Phys. Rev. B 79, 041302 (2009). γ ￿ satisfy condition (7) and at the same time to ease requirements [8] C. Genes et al., Phys. Rev. A 77, 050307 (2008). for condition (9). Thirdly, from the data measured in [13] we [9] S. Singh et al., Phys. Rev. Lett. 101, 263603 (2008). infer a value of k κ 10 nW/K for the dimensions of [10] J. D. Thompson et al., Nature 72, 452 (2008). B th ￿ [11] R. Miller et al., J. Phys. B 38, S551 (2005). the membrane (100 µm 100 µm 50 nm) = (l l d) [12] D. J. Wilson, C. A. Regal, S. B. Papp and H. J. Kimble, (in × × × × 7 required here [20]. A mechanical quality factor of Q = 10 preparation, 2009). and a resonance frequency ωm =2π 1.3 MHz set the left [13] B. L. Zink, F. Hellman, Solid State Comm. 129, 199 (2004). × hand side of Eq. (9) to 45. Finally, the resonance condition [14] Y. Colombe et al., Nature 450, 272 (2007). ∼ ωat = ωm demands a circulating power Pc 850 µW which [15] K. W. Murch et al., Nat. Phys. 4, 561 (2008). ￿ [16] F. Brennecke et al., Science , 235 (2008). will cause heating of ￿ 2.5 K for the given thermal link. In 322 order to make a statement about the absolute timescales, we [17] We assume that losses to other levels can be excluded. [18] We require ξ(¯xat), θ(¯xat) 1. For the two cavity modes with still need to fix the cavity length. For L = 50 µm we find a 1 ￿ wave numbers k1,2 = 2 (k δk) the intensity extrema fulfil cavity mediated coupling G 2π 45 kHz and decoherence k tan(kx)= δk tan(δkx±). The potential minima close to ￿ × − rates Γc, Γm, Γat 0.1 G. It it thus indeed possible to enter points where δkx nπ have the desired properties. ￿ × ￿ the strong coupling regime with state of the art experimental [19] κth is chosen here such as to have dimensions of Hz. parameters. [20] In [13] a thermal link of kB κth￿ =0.1 µW/K was measured While being a surprising result on its own, entering the for a thin, square membrane (d￿ = 200 nm,l￿ = 5 mm) with power dissipated in a central square area (l1￿ =2.5 mm). From regime of strong coupling holds promise for diverse appli- the solution of the Laplace equation we estimate the thermal cations, including for preparation and readout of quantum κth d l￿ l link to scale like κ d ln l / ln 2w 0.1. This is states of mesoscopic massive oscillators. In the regime ω = th￿ ￿ ￿ 1￿ 0 ￿ m confirmed by a finite-element simulation. ω G, where the rotating wave approximation can be ` ´ ` ´ at ￿ Optomechanics (Outline)

Introduction Displacement detection

0

-1 0 1 100 2 Nonlinear dynamics 3

System Bath Quantum theory of cooling Interesting quantum states Towards Fock state detection Coupling to the motion of a single atom Optomechanical arrays 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: Delayed light response

quasistatic F finite sweep!rate

sweep x finite cavity ring-down rate γ delayed response to cantileverNonlinear motion dynamics of a single ⇒ optomechanical cell: Self-induced oscillations

F Fdx blue-detuned laser: ! < 0 > 0 anti-damping x red-detunedcooling laser: 0 heating damping (cooling) 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 cantileverNonlinear motion dynamics of a single ⇒ optomechanical cell: Self-induced oscillations

F Fdx blue-detuned laser: ! < 0 > 0 anti-damping x red-detunedcooling laser: 0 heating damping (cooling) 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 Attractor diagram

15

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

15

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

15

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 An optomechanical cell as a Hopf oscillator

amplitude laser power

phase

bifurcation

Amplitude fixed, phase undetermined! 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! Synchronization: Huygens’ observation

(Huygens’ original drawing!) Coupled pendula synchronize...... even though frequencies slightly different ...due to nonlinear effects Fireflies synchronizing (Source: YouTube) Coupled phase oscillators Coupled phase oscillators Coupled phase oscillators Coupled phase oscillators 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) The Kuramoto model

Synchronization: phase lag The Kuramoto model

1 phase locking

0 coupling K Many phase oscillators Many phase oscillators Many phase oscillators Many phase oscillators Kuramoto model: Phase-locking transition ... infinite-range coupling Kuramoto model displays phase transition ...

coupling strength Phase locking of two optomechanical cells

laser drive

cell 1

optical mode mechanical mode cell 2 ...

Two optomechanical cells, fixed laser drive, increasing mechanical coupling sin(δϕ) ￿ ￿ -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 phase-slip 0.005 time mechanicalcoupling 0.010 time → 0.015 synchronization optomechanical cells Phase locking of Phaselocking two π 0.020 k/m → time Ω 0.025 0 1 2 sin(δϕ) ￿ ￿ -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8 phase-slip 0.005 time mechanicalcoupling 0.010 time → 0.015 synchronization optomechanical cells Phase locking of Phaselocking two π 0.020 k/m → time Ω 0.025 0 1 2

mechanical coupling 2 k/mΩ1 0.01 0.02 0.03 0.04 0.05 0.06

synchronized frequencydifference 0.01

→ → π

-0.91 0 0.02

unsynchronized 0.0 δ 0.03 Ω / Ω 1 0.61 0.04 Effective Kuramoto model eff. Kuramoto model

optomech. model Hopf model Standard Kuramoto model:

Effective Kuramoto model sin(δϕ) 1 for coupled Hopf oscillators: 0

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

0.01

Mechanical spectrum 0.00 (optom.) in light field intensity fluctuations -0.01

1.0 1.02

0.01

0.00 (Hopf) -0.01 frequency detuning frequency

0.98 1.00 1.02 frequency Desynchronization for increasing drive

0.2 3 1 smaller effective Ω 0.2 coupling K for /m larger drive! 2 max α 2 G ￿ 0.1 0.1

0.8 0.8 0.4 0.4 Hopf laser power laser 0 0.0 0.0 bifurcation 0.0 1.00 1.02 1.04 1.00 1.02 1.04 frequency Desynchronization for increasing drive

0.2 3 1 smaller effective Ω 0.2 coupling K for /m larger drive! 2 max α 2 G ￿ 0.1 0.1

0.8 0.8

0.4 0.4 laser power laser 0 0.0 0.0 0.0 1.00 1.02 1.04 1.00 1.02 1.04 frequency

2.0

-2.0

time Array with common optical mode

chaos

2.0 2

1 1.0 0.6 0.6

0.3 0.3

0 0.0

0.8 1.0 1.2 0.8 1.0 1.2 frequency Dynamics in optomechanical arrays

Outlook •2D geometries •Information storage and classical computation •Dissipative quantum many-body dynamics •... 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 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