An Introduction to Cavity Optomechanics

Home , Cavity optomechanics, Laser detuning

An introduction to cavity optomechanics

Andreas Nunnenkamp University of Basel

1. What? Why? How? 2. Sideband cooling 3. Strong-coupling regime 4. Dissipative coupling Introduction Mechanical effects of light

Dipole or gradient force Scattering force

• ac-Stark shift • radiation pressure • optical lattices • laser cooling

Loudon, The Quantum Theory of Light (OSP) Foot, Atomic Physics (OUP) Radiation-pressure force

2E k ∆p =2k = c k ∆p∆N 2Pin − F = = ∆t c

Loudon, The Quantum Theory of Light (OSP) Foot, Atomic Physics (OUP) Cavity optomechanics

radiation pressure force mechanical cavity drive cavity mode oscillator

Marquardt & Girvin, Physics 2, 40 (2009) Kippenberg & Vahala, Science 321, 1172 (2008) REVIEW sitivity by increasing optical power in interfer- scopic and which is harmonically suspended. This Many more structures exist that should ometers. On the other hand, a red-detuned pump end mirror has been realized in multiple ways, also realize an optomechanical interaction in wave can create a radiation component of mechan- such as from an etched, high-reflectivity mirror an efficient manner. In particular, nanopho- ical damping that leads to cooling of the mechan- substrate (14, 15), a miniaturized and harmonical- tonic devices such as photonic crystal mem- ical mode; i.e., a reduction of the mechanical ly suspended gram-scale mirror (28), or an atomic brane cavities or silicon ring resonators might mode’sBrownianmotion(9, 26). force cantilever on which a high-reflectivity and be ideal candidates owing to their small mode One description of this process is given in micron-sized mirror coating has been transferred volume, high-Finesse, and finite rigidity. Owing Fig. 2B, wherein a feedback loop that is inherent (29). A natural optomechanical coupling can occur to their small length scale, these devices ex- to the cavity optomechanical system is described. in optical microcavities, such as microtoroidal hibit fundamental flexural frequencies well The elements of this loop include the mechanical cavities (13)ormicrospheres,whichcontainco- into the gigahertz regime, but their mechanical and optical oscillators coupled through two dis- existingExperiments high-Q,opticalwhisperinggallerymodes, & qualityMotivation factors have so far not been studied, tinct paths. Along the upper path, a force acting and radio-frequency mechanical modes. This cou- nor has optomechanical coupling been ob- on the mechanical oscillator (for instance, the pling can also be optimized for high optical and served. As described in the next section, such thermal Langevin force or a signal force) causes a mechanical Q (30). In the case of hybrid systems, high frequencies are interesting in the context mechanical displacement, which (for a detuned of regenerative oscillation and ground laser) changes the cavity field due to the opto- state cooling. mechanical coupling (the interferometric mea- kg Hz 1. Mechanical sensing 2. Quantum network Gravity wave Cooling and Amplification Using surement process). However, the amplitude detectors fluctuations, which contain information on the (LIGO, Virgo, GEO,..) Dynamical Back-Action mirror position, are also coupled back to the The cooling of atoms or ions using radi- mechanical oscillator via radiation pressure Harmonically ation pressure has received substantial (lower path), resulting in a back-action. A blue- suspended attention and has been a successful tool detuned pump wave sets up positive feedback gram-scale mirrors in atomic and molecular physics. Dy- (the instability), whereas red detuning introduces namical back-action allows laser cooling negative feedback. Resonant optical probing of mechanical oscillators in a similar man- (where the excitation frequency equals the cavity Mirror coated ner. The resemblance between atomic on April 22, 2011 AFM-cantilevers Mechanical frequency resonance frequency, w = w0) interrupts the laser cooling and the cooling of a mechan- feedback loop because changes in position only icalRugar oscillator et al. coupled, Nature to 430 an optical, 329 (or(2004) Rabl et al., Nature Physics 6, 602 (2010) change the phase, not the amplitude, of the field. electronic) resonator is a rigorous one Micromirrors

As described below, this feedback circuit also Mass (34). In both cases, the motion (of the clarifies the relation between “feedback cooling” ion, atom, or mirror) induces a change in and cooling by dynamic back-action. the resonance frequency, thereby coupling the motion to the optical3. (or cavity)Quantum-classical transition SiN membranes Experimental Systems 3 resonance (Fig. 2C). Indeed, early work Systems that exhibit radiation-pressure dynamic has exploited this coupling to sense the back-action must address a range of design con- atomic trajectories of single atoms in www.sciencemag.org siderations, including physical size as well as Fabry-Perot cavities (35, 36)and,more dissipation. DynamicRealization back-action of an optomechanical relies on opti- interface between ultracold atoms and a membraneOptical microcavities recently, in the context of collective atomic 1, 2, 1, 2, 3, 1, 2, 3 Stephan Camerer, ∗ Maria Korppi, ∗ Andreas Jöckel, 1, 2 1, 2 1, 2, 3, cal retardation; i.e., is most prominentDavid Hunger, for photonTheodor W. Hänsch, and Philipp Treutlein † motion (37, 38). This coupling is not only 1Fakultät für Physik, Ludwig-Maximilians-Universität, Schellingstraße 4, 80799 München, Germany lifetimes comparable to or2Max-Planck-Institut exceeding für the Quantenoptik, me- Hans-Kopfermann-Str. 1, 85748 Garching, Germany restricted to atoms or cavities but also has 3Departement Physik, Universität Basel, Klingelbergstrasse 82, 4056 Basel, Switzerland chanical oscillation period. Very low optical dis-(Dated: July 20, 2011) CPW-resonators MHz been predicted for a variety of other We have realized a hybrid optomechanical system bypg coupling ultracold atoms to a micromechan- coupled to nano- sipation also means that photonsical membrane. The are atoms recycled are trapped in an optical lattice, which is formed by retro-reflection of a systems. For example, the cooling of a laser beam from the membrane surface. In this setup, the lattice laser light mediates an optomechan- resonators ical coupling between membrane vibrations and atomic center-of-mass motion. We observe both the many times, thereby enhancingeffect of the the membrane weak vibrations photon onto the atoms as well as the backaction of the atomic motion mechanical oscillator can be achieved Downloaded from onto the membrane. By coupling the membrane to laser-cooled atoms, we engineer the dissipation Penrose & Bouwmeester et al., PRL 91,130401 (2003) pressure on the mirror. Onrate the of the membrane. other Our hand, observations the agree quantitatively with a simple model. using coupling to a quantum dot (39), a mechanical dissipation rate governs the rate of trapped ion (40), a Cooper pair box (41), Laser light can excert a force on material objects Fig. 3. Experimental cavity optomechanical systems. heating of the mechanicalthrough radiation mirror pressure and mode through the by optical the dipole an LC circuit (5, 32), or a microwave strip- force [1]. In the very active field of optomechanics [2], (Top to Bottom) Gravitational waveultracold detectors atoms in [photo environment, limitingsuch the light forces effectiveness are exploited for cooling of opto- and control line cavity (33). Although the feedback of the vibrations of mechanical oscillators, with possi- credit LIGO Laboratory], harmonicallyoptical suspended resonators gram- Marquardt & Girvin, Physics 2, 40 (2009) mechanical cooling.ble It applications also sets in precision the forcerequired sensing and am- studies of loop of Fig. 2B explains how damping quantum physics at macroscopic scales. This has many scale mirrors (28), coated atomic force microscopy canti- similarities with the field of ultracold atoms [3], where plification level necessary to induce regenerative FIG. 1: Optomechanical coupling of atoms and membrane. and instability can be introduced intoKippenberg the & Vahala, Science 321, 1172 (2008) radiation pressure forces are routinely used for laser cool- leversAlaserbeamofpower (29), coatedP is partially micromirrors reflected at a SiN mem- (14, 15), SiN3 membranes ing [1] and optical dipole forces are used for trapping and brane of reflectivity r and forms a 1D optical lattice for an ul- oscillations. Thesequantum considerations manipulation of atomic illustrate motion, most the notably dispersivelytracold atomic ensemble. coupled Motion of the membrane to an displaces optical cavity (31), optical cavity optomechanical system, the ori- in optical lattices [4, 5]. the lattice and thus couples to atomic motion. Conversely, importance of high opticalIn a number Finesse of recent theoretical and mechan- papers it has been microcavitiesatomic motion is imprinted ( as13 a power, 16 modulation), and∆P onto superconducting microwave gins of cooling and mechanical amplifi- proposed that light forces could also be used to couple the laser, thus modulating the radiation pressure force on the ical Q in system design.the motion of atoms in a trap to the vibrations of a single membrane. t is the transmittivity of the optics between atoms cation are better understood with the use resonatorsand membrane. Arrows coupled illustrate the todirection a of nanomechanical forces and beam (33). The mode of a mechanical oscillator [6–16]. In the resulting displacements at a specific point in time. In the main text, all It is only in thehybrid past optomechanical 3 years system that the a atoms series could be of used massesforces and displacements range are from positive if kilograms pointing to the right. to picograms, whereas fre- of a motional sideband approach, as de- to read out the motion of the oscillator, to engineer its innovative geometriesdissipation, (shown and ultimately in to perform Fig. quantum 3) has informa- quencies range from tens of megahertz down to the hertz scribed in Fig. 4 (13). tion tasks such as coherently exchanging the quantum Fig. 1, see also [14]. A laser beam of power P , whose fre- reached a regime wherestate the of the observation two systems. In recent of radiation- experiments using level.quency ω CPW,is red detuned coplanar with respect waveguide.to an atomic transi- Cooling has been first demonstrated for magnetic [17] or surface-force coupling [18], atoms were tion, impinges from the right onto a SiN membrane oscil- pressure dynamic back-actionused to detect vibrations could of micromechanical be observed. oscillators. lator and is partially retroreflected. The reflected beam micromechanical oscillators coupled to However, the backaction of the atoms onto the oscillator is overlapped with the incoming beam such that a 1D These advances havevibrations, relied which on is the required availability for cooling and manipulat- and yetoptical another lattice potential approach for ultracold atoms has is generated separated optical and optical cavities (14–16)and,usinganelectrome- ing the oscillator with the atoms, could not be observed. [4]. A displacement of the membrane xm displaces the 2 improvements in high-FinessearXiv:1107.3650v1 [quant-ph] 19 Jul 2011 Here we report mirror the experimental coatings implementation (as of a mechanicallattice potential, resulting degrees in a force ofF = freedommωatxm onto by using a min- chanical analog, for a Cooper pair box coupled hybrid optomechanical system in which an optical lat- each atom, where m is the atomic mass and ωat the trap used in gravity wavetice detectors) mediates a long-distance and also coupling onmicro- between ultra- iaturefrequency in high-Finesse a harmonic approximation optical to the lattice cavity po- and a separate to a nanomechanical beam (41). Because the me- cold atoms and a micromechanical membrane oscillator tential well. The membrane motion thus couples through [14]. If the trap frequency of the atoms in the lattice is Fcom = NF to the center of mass (c.o.m.) motion of and nanofabricationmatched techniques to the eigenfrequency [which of the membrane, are the the cou- nanometrican ensemble of N atoms membrane trapped in the (lattice.31). Con- Whereas the afore- chanical modes in experiments are high Q (and pling leads to resonant energy transfer between the two versely, an atom displaced by xat from the bottom of its underlying enabling technologysystems. We observe for both nano- the eff andect of micro- the membrane mentionedpotential well experiences embodiments a restoring optical dipole have force been in the optical are thus very well isolated from the reservoir), they vibrations onto the atoms as well as the backaction of F = mω2 x in the lattice. On a microscopic level, d − at at electromechanical systemsthe atomic motion (27 onto)]. the A membrane. commonly We demonstrate domain,Fd is due to absorption devices and stimulated in the emission, micro- leading and radiowave do- are easily resolved in the spectra of detected probe that the dissipation rate of the membrane can be engi- to a redistribution of photons between the two running used hybrid systemneered consists by coupling of it to a laser-cooled conventional- atoms, as predicted mainwave components have forming also thebeen lattice fabricated [19]. Each redis- (22, 32), such as a light reflected from the optical cavity (Fig. 1B). by recent theoretical work [14]. tribution event results in a momentum transfer of 2¯hk input mirror made withThe a coupling high-reflectivity scheme we investigate coating is illustrated in nanomechanicalto the atom, where k = ω/c. The resonator photon redistribution coupled± to a super- Furthermore, their effective temperature can be and an end mirror whose dimensions are meso- conducting microwave resonator (33). inferred from the thermal energy kBT (where kB is

1174 29 AUGUST 2008 VOL 321 SCIENCE www.sciencemag.org Cavity optomechanics

We start with Hˆ = ω (ˆx)â aˆ + ω ˆb ˆb C † M † L where the position of the mechanical oscillator 15 xˆ = x (ˆb + ˆb ) with xZPF = 10− m ZPF † 2mω M ∼ is parametrically coupled to the cavity mode L x ω (x)= ω 1 ω C L + x R ≈ − L R We obtain the “standard model of optomechanics” =1 radiation-pressure force ˆ xˆ ˆ ˆ H = ωR 1 aˆ†aˆ + ωM b†b ∂Hînt ωR − L Fˆ = = aˆ†aˆ − ∂xˆ L + optical drive/decay “three-wave mixing” + thermal fluctuations Marquardt & Girvin, Physics 2, 40 (2009) N.B. We neglect the dynamical Casimir effect. Kippenberg & Vahala, Science 321, 1172 (2008) Cavity optomechanics

The Hamiltonian

ˆ ˆ ˆ ˆ ˆ H = (ωR + g0xˆ)ˆa†aˆ + ωM b†b + Hdrive + Henv ˆ ˆ xˆ = xzpf b + b† xzpf = g = g0xzpf 2mωM Position operator Zero point motion Coupling rate Three-wave mixing Far from equilibrium Radiation pressure force with the dimensionless parameters Ω/κ ∆/ωm drive strength & detuning

ωm/κ good/bad-cavity limit ωm/γ mechanical quality factor cavity shift per phonon oscillator displacement per g/κ in units of line width g/ωm photon in units of its ZPF Marquardt & Girvin, Physics 2, 40 (2009) Kippenberg & Vahala, Science 321, 1172 (2008) T =0

Damped & driven optical resonator

Damped & driven harmonic oscillator Damped & driven optical resonator ω ω R ωR L κ κ F (t)=F0 cos ωLt Coherent drive E(t)=E0 cos ωLt iω t x(t)=x0 cos(ωLt ϕ) Coherent state aˆ α = αe− L α − →number| ﬂuctuations (shot noise)!|

x0 ϕ α 2 arg α | | π κ

ω ωR L ωR ωL Displacement readout of the mechanical oscillator

ϕ arg α General relation between noise spectrum and linear response susceptibility ˆ ˆ ˆ H = (ωR + g0xˆ)ˆa†aˆ + ωM b†b ϕ x ∼ susceptibility ωR ωL

A single experimental run might look like...... and calculating(classical the noise limit) power spectrum for the damped oscillator:

figure by F. Marquardt Classical equipartition theorem: Classicalwe can find equipartitionthe temperature from thetheorem: area extract Possibilities: temperature! •Direct time-resolved detection •Analyze fluctuation spectrum of x extractfigure by F. Marquardt Possibilities: temperature! •Direct time-resolved detection •Analyze fluctuation spectrum of x Sideband cooling Sideband cooling: the cavity-enhanced scattering picture !"#$%&'(()*+,&-.$&/$/0%"+$ 3"4*-56$+."+'$7&8"/"+&#'"--$%*+,&9*'-:%$; cantilevers, membranes, wires, carbon nanotubes need additional cooling Γopt Γopt ↑ ↓ ωL ωM = kHz GHz M+9&>6&19-&6( − -DZG )$N,1J ωM L,''*'(K,)K%(08*1*- kBT ωM κ 2'&&(%0$)& A8*1*-(K,)K%(3,''*' 1GHz 50mK ωR Z ∼ figure by J. Harris ωM ˆ H = (ωR + g0xˆ)â†aˆ

:&&/(;<(=,+%*->?$&@(&1($+<@(A?#(11@(BC!CB"(DEBBFG< Red-sideband cooling at $-6/(((((((((((H<(I8&-(&1($+<@((A?#(Ground-state112 BC!CBE(DEBBFG(2*'(29++(18&*'J cooling is ωL = ωR ωM − #$%&'()**+,-./(*01,)$+(2*')&%(,-)'&$%possible&(3&34'$-&5%(6$30,-.(7,18*91($66,-.( in the sideband- !" ωout = ωin +2+9)19$1,*-%ωM resolved regime ω κ M Marquardt et al., PRL 99, 093902 (2007) see also: Laser cooling of atoms & ions Wilson-Rae et al., PRL 99, 093901 (2007) Sideband cooling: the classical picture

Fˆ aˆ†aˆ ∝

∆ = ω ω x L − R ∝

dxFopt < 0 Damping is due to the ﬁnite time lag between mirror position and radiation-pressure force: ωM κ Marquardt, Clerk & Girvin, J. Mod. Opt. 55, 3329 (2008) 2 2 ω mx[ω]= mω x[ω]+imωΓ x[ω]+ (¯x)x[ω]/(1 iωτ) (3) − − M M F − Comparing the last two terms (the intrinsic damping with the imaginary part of the optomechanical term), we ﬁnd that the optomechanical damping rate is given by

(¯x) ωM τ Γopt = F 2 , (4) mωM 1+(ωM τ) for the simple ansatz of Eq. (1). According to this analysis, one would expect the maximum effect to occur when ωM τ = 1, i.e. when the time-delay matches the period of the cantilever motion. As we will see further below, this conclusion is not upheld by the full quantum-mechanical analysis for the case of radiation pressure. Equation (1) and the subsequent analysis holds exactly for the bolometric force. In that case, τ is the finite time of thermal conductance and (x)= F maxI(x)/Imax is the displacement-dependent bolometric force, where I(x)= F 2 Imax/(1+(2∆(x)/κ) ) is the intensity profile and ∆(x)=xωR/L is the position- dependent detuning betweenSideband incoming lasercooling: radiation the classical and optical picture resonance at x = 0. For the radiation-pressure force, we can use the present analysis only in the regime of κ ωM , and only if we allow for a position dependent relaxation rate 1/τ (see [32]). In both cases, however, the shape of Γopt as a function of cantilever position x¯ is determined by the slope of the intensity profile, i.e. in particular by the mechanics sign of . To the left of the resonance, where > 0, we indeed obtain extra F ΓFM Γopt damping: Γopt > 0. As long as there are no extra fluctuations introduced by the light-induced force (i.e. if we may disregard shot noise), the effective temperature of the mechanical degree of freedomT is therefore reducedT according=0 to the ratio of intrinsic and optomechanical damping rates:

Γ ΓM opt →∞ The coupling to the cold optical Teff = T . 0 (5) ΓM + Γopt → bath leads to increased damping →neglects number fluctuations (shot noise)! without additional fluctuations. This can be obtained, for example, by solving the Langevin equation that in- Marquardt, Clerk & Girvin, J. Mod. Opt. 55, 3329 (2008) cludes the thermal fluctuations of the mechanical heat bath (whose strength is set by ΓM according to the fluctuation-dissipation theorem). Then the effective temperature may be defined according to the equipartition theorem: 2 2 mω˜M (x x¯) = kBTeff ,where˜ωM contains the frequency renormalization due to the− real part of the optomechanical term in Eq. (3). 3 Quantum noise approach

In the quantum regime, we have to take into account the shot noise that tends to heat the cantilever and enforces a ﬁnite quantum limit for the cantilever phonon number. The quantum picture can also be understood as Raman scattering:

4 Incoming photons, red-detuned with respect to the optical resonance, absorb a phonon from the cantilever, thereby cooling it. However, there is also a finite probability for phonon emission, and thus heating. The purpose of a quantum theory is to discuss the balance of these effects. Incoming photons,The idea red-detuned behind the with quantum respect noise to the approach optical to resonance, quantum-dissipative absorb a systems phononis from to describe the cantilever, the environment thereby cooling fully it. by However, the correlator there is of also the a finite fluctuating force cantileverprobability for phonon emission, and thus heating. The purpose of a quantum phonon Incomingthat couples photons, to red-detuned the quantum with100 system respect of to interest. the optical If resonance, the coupling absorb is weak a enough, numbertheory isknowledge to discuss of the the balance correlator of these is e suffects.fficient to fully describesideband- the influence of the (log.)Thephonon idea behind from the the cantilever, quantum thereby noise10 approach cooling it. to quantum-dissipative However, thereresolved is also systems a finite probabilityenvironment. for phonon In our emission, case, this and thus means heating. looking The at purpose the spectrum of a quantum of the radiation is to describe the environment fully by1 the correlator of the fluctuatingregime force theorypressure is1.0 to discuss force fluctuations, the balance of these which eff areects. produced by the shot noise of photons that couples to the quantum system of0.1 interest. If the coupling is weak enough, circulating radiation power insideThe idea the behind driven the optical quantum cavity noise approach mode, i.e. to quantum-dissipative a nonequilibrium systems environment. In knowledgeis to describe of the correlator the environment is suffi fullycient0.01 by to the fully correlator describe of the the influence fluctuating of force the 0 environment.other In0.1 applications, our case, this we means might looking be dealing at the with spectrum the electrical of the radiation field fluctuations thatproduced, couples to e.g., the quantum by a driven system0.001 electronic of interest. circuit If the coupling (superconducting is weak enough, single electron pressureknowledge force fluctuations, of the correlator which is su arefficient produced to fully by describe the shot the noise influence of photons of the transistor, quantum pointminimum phonon number 0.0001 contact, LC circuit) capacitively coupled to some insideenvironment. the driven optical In our2 case, cavity this mode, means i.e. looking a nonequilibrium at the spectrum environment. of the radiation In 0.001 0.01 0.1 1 10 100 other100 pressure applications,nanobeam. force1 fluctuations, we The might general be which dealing formulas are produced with remain the by electrical the the same shot field noise for fluctuations all of of photons these cases, and produced,insideonly e.g., the the driven by noise a driven optical spectrum electronic cavity changes. mode, circuitmechanical i.e. (superconducting a nonequilibrium frequency vs. ring-down environment. single rate electron In (a) 0 detuning (b) transistor,other applications, quantumThe Fourier point we transform might contact, be of LC dealing the circuit) force with correlator capacitively the electrical defines coupled field the fluctuations to spectral some noise den- nanobeam.produced,sity: TheIncoming e.g., general by a photons, driven formulas electronic red-detuned remain circuit the with same (superconducting respect for all to of the these single optical cases, electron resonance, and absorb a Figure 2: (a)only Cantilever thetransistor, noisephonon spectrum phonon quantum from number changes. point the contact, cantilever, as a function LC thereby circuit) of coolingcapacitively circulating it. However, coupled radiation to there some is also a finite nanobeam. The general formulas remain the same for all of these cases, and power inside theThe cavity Fourier andprobability transform as a function for of the phonon force of detuning emission, correlator between and definesiω thust ˆ laser the heating.ˆ spectral and The optical noise purpose den- of a quantum only the noise spectrum changes.SFF(ω)= dt e F (t)F (0) . (6) sity: theory is to discuss the balance of these effects. resonance. The meanThe phonon Fourier number transform in of steady the force state correlator is plotted defines on the a logarithmicspectral noise den- cantilever The idea behind the quantumphonon noise approach to quantum-dissipative100 systems scale. Contour linessity:The indicate noise spectrumn ¯ = 1 andSFFn ¯ is=0 real-valued.1. Fornumber this and plot, non-negative. the following However,sideband- in contrast iωt (log.) 10 is to describeS (ω the)= environmentdt e 3Fˆ fully(t)Fˆ(0) by the. correlator6 of the(6) fluctuatingresolved force FF 1 ˆ ˆregime parameters have beento the used: classicalωM /κ case,=0 it.3, isn¯ asymmetricth = 10 ,Q in= frequency,ωM1.0/ΓM = since 10 ,F and(t) and F (0) do not that couples to the quantumicirculating systemωt radiationˆ power ofˆ interest. If the0.1 coupling is weak enough, (ωR/ωM )(xZPF/L)commute.0.012. (b) This The asymmetrySFF minimum(ω)= hasdt phonon e an importantF number(t)F (0) physical. as a function meaning:0.01 of (6) Contributions 0 0.1 The noise≈ spectrumknowledgeSFF ofis real-valued the correlator and is non-negative. sufficient to However, fully describe0.001 in contrast the influence of the the ratio between theat positive mechanical frequencies frequency indicateω and the the possibility optical of cavity’s the environment ring- to absorb M ˆ minimum phonon number 0.0001ˆ to the classicalenvironment. case, it is asymmetric In our case, in this frequency, means since lookingF (2t at) and theF spectrum(0) do not of the radiation The noise spectrum SFF is real-valued and non-negative. However,0.001 in0.01 contrast0.1 1 10 100 down rate κ, accordingenergy, to while Eq. those(15). at Ground-state negative frequencies cooling100 is imply1 possible its ability in the to release energy commute.to the This classicalpressure asymmetry case, force it is has fluctuations, asymmetric an important in which frequency, physical are produced since meaning:Fˆ(t) by and Contributions theFmechanicalˆ(0) shot dofrequency not noise vs. ring-down of rate photons regime ω κ, i.e.(to the the “good cantilever). cavity” All or the “resolved optomechanical(a) sideband”0 detuning eff limit.ects can(b) be described in terms of M at positivecommute. frequenciesinside This asymmetry the indicate driven has optical the an possibility important cavity mode, physicalof the i.e. environment meaning: a nonequilibrium Contributions to absorb environment. In SFF, as long as the coupling isFigure weak. 2: (a) Cantilever phonon number as a function of circulating radiation energy,at while positiveother those frequencies applications, at negative indicate frequencies we the might possibilitypower implybe inside dealing the of cavity its the ability and with as environment a function the to of release electrical detuning to between energy absorb field laser and fluctuations optical (to theenergy, cantilever).The while optomechanical those All the at negative optomechanical damping frequenciesresonance. rate effects imply The is mean can given its phonon be ability by described number the in to steady di releaseff stateinerence terms is plotted energy of on of noise a logarithmic spectra produced, e.g., by a drivenscale. electronic Contour lines circuit indicaten ¯ = (superconducting 1 andn ¯ =0.1. For this plot, the single following electron Here the equation has been written in the interaction picture (disregarding3 the 6 SFF,(to asat long the positive cantilever). astransistor, the and coupling All negative quantum the is optomechanical weak. frequencies, pointparameters contact, eff haveects LC been can used: circuit) beωM described/κ =0 capacitively.3, n¯th = in 10 ,Q terms= ωM coupled/Γ ofM = 10 , and to some (ωR/ωM )(xZPF/L) 0.012. (b) The minimum phonon number as a function of oscillations atω ˜MSFF),, and as long as the coupling is weak. ≈ The optomechanicalnanobeam. damping The general rate is formulas giventhe ratio by between remain the the di mechanicalfference the frequency same of noiseω forM and all thespectra of optical these cavity’s cases, ring- and 2 at positiveThe and optomechanical negative frequencies, damping ratex isdown given rate κ, by according the di toff Eq.erence (15). Ground-state of noise cooling spectra is possible in the Sidebandonly the noise cooling: spectrum the quantum changes.ZPFregime ωM κnoise, i.e. the “good approach cavity” or “resolved sideband” limit. at positive and negative1 frequencies,Γopt = 2 [SFF(ωM ) SFF( ωM )] . (7) [Aˆ]ˆρThe= Fourier(2AˆρÂˆ transform† Aˆ†Aˆρˆ of theρÂˆ† forceAˆ) correlator− − defines(10) the spectral noise den- D 2 x2 − −Here the equation has been written in the interaction picture (disregarding the sity: ZPF2 oscillations atω ˜ ), and This formulaΓopt is obtained= xZPF[S byFF( applyingωM ) SFF Fermi’s(M ωM )] Golden. Rule to derive(7) the transi- At weak couplingΓopt =all2 you need[SFF to(ω knowM−) isS FFthe−( forceωM )] spectrum. (7) 2 ˆ ˆ 1 ˆ ˆ ˆˆˆ ˆ ˆ is the standard Lindbladtion rates operator arising for from downward the coupling (A =ˆ− ofa the) or− cantilever[A upward]ˆρ = (2AρÂ† (toAA†A theρˆ=ˆρÂa light††A) field, i.e.(10) from This formula is obtainedˆ byˆ applying Fermi’s GoldenDi Ruleωt 2ˆ to deriveˆ− the− transi- transitions in theThis oscillator.the formula term H isint Restricting obtained= F byxˆ in applying ourselves theSFF Hamiltonian. Fermi’s(isω the)= to standard the Goldendt Lindblad populations e These Rule operatorF ( to aret for)F derive downward(0)ρnn the(,we.Aˆ =â transi-) or upward (Aˆ =â†) (6) tion rates arising from the−ω couplingR of thetransitions cantilever in the oscillator. to the Restricting light ourselves field, to i.e. the populationsfrom ρnn,we tion rates arisingwith fromFˆ = theaˆ†aˆ coupling of the cantilever to the light field, i.e. from obtain the equation for the phononL numbern ¯ = obtainnˆ = the equationn forρ thenn phonon: numbern ¯ = nˆ = nρnn: ˆ ˆ 2 n 2 n the termthe termHint TheH=înt F noise=xˆ inFˆx spectrumˆ thein the Hamiltonian.x Hamiltonian.SFF is real-valued These These are are and non-negative.x However, in contrast Obtain− the ratesopt withZPF Fermi’s Golden Rule opt˙ ZPFopt −Γ = SFF(ωM ) , Γ n¯ = Γ=M n¯th + Γ S(ΓFFM + Γ(opt)¯n,ωM ). (11) (8) to the classicalopt case,2 it is asymmetric in frequency,2↑ − since Fˆ(t) and Fˆ(0) do not n¯˙ = Γ n¯ 2 ↓+2Γ (Γ +whichΓ yields)¯n, the2 steady-state2 ↑ phonon number in the(11)− presence of optomechanical optMoptxthZPFx M optoptopt xZPFx Γcommute.= ThisZPFS ↑ asymmetry(ω− ) , Γcooling: has= an importantZPFS ( physicalω ). meaning:(8) Contributions Γ =2 2FFSFFM(ωM ) , Γ = 2 2 FFSFF( ωMM ). (8) These↓ rates↓ enter the complete master↑ ↑ equation− for− the densityO matrixρ ˆ of the ΓM n¯th + Γoptn¯M which yields the steady-stateat positive phonon frequencies number in indicate the presence the possibility ofn¯ optomechanicalM = of the environment. to(12) absorb cantilever in the presence of the equilibrium heatΓ bathM + Γopt (that would lead to a cooling: TheseThese rates rates enterAndenergy, entercalculate the while complete the the complete thosesteady-state master at master negative equationphonon equation frequencies number for for the the density density imply matrix matrix its abilityρ ˆρ ôfof the to the release energy cantileverthermal in population the presencen ¯ of) the and equilibrium theThis radiation is the weighted heat bath average field: of (that the thermal would and the lead optomechanical to a phonon cantilever in the(to thepresence cantilever). of theth All equilibrium thenumbers. optomechanical heat It represents bath the correct (that effects generalization would can beof lead the described classical to formula a for in the terms of effective temperature,opt Eq. (5).opt Here thermal populationn ¯th) and the radiationO field:Γopt = Γ Γ optical damping thermal populationS , asn ¯th long) and as the the radiation coupling field: is weak. ↓ − ↑ FF ΓMoptn¯th + Γoptn¯M opt opt 1 ˙ O Γ 1 SFF(ωM ) − ↑ n¯Mρˆ = (Γ + ΓM (¯nth +. 1)) n¯M[â=]+(=Γoptminimalopt+ Γphonon=M n¯ numberth(12)) [â1†] ρˆ (13) (9) The optomechanicalopt damping rateopt isΓopt givenΓ /Γ by the1 S diFFff( erenceωM ) − of noise spectra ρˆ˙ =opt(Γ + ↓ΓΓM(¯n+ Γ+opt 1)) [â]+(optDΓ + Γ n¯↑ ) [â ] ρˆ − D (9) ρˆ˙ = (Γ + Γ (¯nM +th 1)) [â]+(Γ + Γ Mn¯ ↓ th) ↑ [â− ]† ρˆ (9) at positive↓ andM negativeth frequencies,D ↑ M th D † This is the weighted average↓ of the thermalD and the↑ optomechanicalMarquardt et al., PRLD 99, 093902 phonon (2007) Clerk et al., RMP 82, 1155 (2010) Wilson-Rae et al., PRL 996 , 093901 (2007) x2 numbers. It represents the correct generalizationΓ = ofZPF the[S classical(ω ) formulaS ( forω )] the. (7) opt 2 FF M − FF − M effective temperature, Eq. (5). Here 5 5 This formula is obtained5 by applying Fermi’s Golden Rule to derive the transi- opt 1 O Γ tion rates arising1 from theSFF coupling(ωM ) of the− cantilever to the light field, i.e. from n¯ = ↑ = = 1 (13) M the termopt Hîntopt= Fˆxˆ in the Hamiltonian. These are Γopt Γ /Γ −1 SFF( ωM ) − − − ↓ ↑ 2 2 opt xZPF opt xZPF Γ = 2 SFF(ωM ) , Γ = 2 SFF( ωM ). (8) ↓ ↑ − These rates enter6 the complete master equation for the density matrixρ ˆ of the cantilever in the presence of the equilibrium heat bath (that would lead to a thermal populationn ¯th) and the radiation field:

opt opt ρˆ˙ = (Γ + ΓM (¯nth + 1)) [ˆa]+(Γ + ΓM n¯th) [ˆa†] ρˆ (9) ↓ D ↑ D

5 Incoming photons, red-detuned with respect to the optical resonance, absorb a phonon from the cantilever, thereby cooling it. However, there is also a finite probability for phonon emission, and thus heating. The purpose of a quantum theory is to discuss the balance of these effects. The idea behind the quantum noise approach to quantum-dissipative systems is to describe the environment fully by the correlator of the fluctuating force that couples to the quantum system of interest. If the coupling is weak enough, knowledge of the correlator is sufficient to fully describe the influence of the environment. In our case, this means looking at the spectrum of the radiation pressure force fluctuations, which are produced by the shot noise of photons inside the driven optical cavity mode, i.e. a nonequilibrium environment. In other applications, we might be dealing with the electrical field fluctuations produced, e.g., by a driven electronic circuit (superconducting single electron transistor,is the minimal quantum phonon point number contact, reachable LC circuit) by optomechanical capacitively coupled cooling. to This some quantum limit is reached when Γ Γ . Then, the cooling effect due to extra nanobeam. The general formulasopt remain M the same for all of these cases, and only thedamping noise is spectrum balanced changes. by the shot noise in the cavity, which leads to heating. Sideband cooling: the quantum noise approachˆ The FourierThe radiation transform pressure of the force force is correlator proportional defines to the the photon spectral number: noise den-F = ( ω /L)â aˆ. A brief calculation for the photon number correlator inside a sity: R † cantilever driven cavity [32]Calculate yields itsradiation spectrumthe force in(i.e. the shot-noise) form of a Lorentzianspectrum that is shifted pressure ∆ = ωM force − by the detuning ∆ = ωL ωR betweeniωt ˆ laserˆ and optical resonanceωR frequency laserSFF(ω)=− dt e F (t)F (0) .with Fˆ = aˆ†aˆ (6) ωR: L optical cavity The noise spectrum SFF is real-valued and2 non-negative. However, in contrast (a) ωR (b) κ (c) to the classical case, itS isFF asymmetric(ω)= in frequency,n¯p since Fˆ(t) and, Fˆ(0) do not(14) L (ω + ∆)2 +(κ/2)2 figure by F. Marquardt commute. This asymmetry has an important physical meaning: Contributions cantilever atphonon positive wheren frequencies¯pFigureis the 1: photon (a) indicate The100 number standard the possibility circulating optomechanical of inside the environment the setup cavity. treatedRed-sideband A to in plot absorbthe cooling of text: the at A driven number sideband- (log.) resulting steady-state phonon10 number is shown in Fig. 2a. energy, whileoptical those at cavity negative with a frequencies movable mirror. implyresolved (b) its Moving ability the to mirror releaseωL = inω energyR a cycleωM can result Inserting this spectrum1 into Eqs. (7) andregime (13) yields the optomechanical (to the cantilever).1.0 in work All extracted the optomechanical by the light-field, effects due can to be the described finite cavity in terms ring-down− of rate. (c) ωout = ωin + ωM circulating radiation power cooling rate and the minimum0.1 phonon number as a function of detuning ∆. SFF, as long asRadiation the coupling pressure is weak. force noise spectrum. O 0.01 2 2 0 The minimum ofn ¯ is reached at a detuning ∆ = ω +(κ/2) , and it is The optomechanical0.1 M damping rate is given by the differenceGround-stateM of noise spectra cooling is (see Fig. 2b): 0.001 − at positive and negative frequencies,minimum phonon number 0.0001 possible in the sideband- have2 been observed in radiation-pressure driven microtoroidal optical resonators 0.001 0.01 0.1 1 10 100 ω κ 100 1[14, 15] and other2 setups [16, 17]. For a recent2 reviewresolved see [18]. regime The studyM of these xZPFOmechanical1 frequency vs. ring-downκ rate (a) 0 detuningsystemsΓopt has(b)=min beenn ¯ made[=SFF even(ωM1+ more) SFF fruitful( ωM by)]1. the. realizationMarquardt et al., that PRL(7) 99 the(15), 093902 same (2007) (or 2M 2  figure− by2 F. ωMarquardt− −  essentiallyClerk et similar) al., RMP 82 physics, 1155 (2010) may be observedM in systemsWilson-Rae ranging et al., PRL from 99, 093901 driven (2007) LC Figure 2:This (a) Cantilever formula phonon iscircuits obtained number coupled by as a applying to function cantilevers of Fermi’s circulating [19] Golden over radiation superconducting Rule to derive single the transi- electron transis- For slow cantilevers, ω κ,wehavemin¯nO = κ/(4ω ) 1. Ground-state power insidetion the rates cavity arising andtors as a from and function microwave the of couplingM detuning cavities between of the coupled laser cantilever and toM optical nanobeams to the lightM [20, field, 21, i.e. 22, 23, from 24, 25, 26] to resonance. The meancooling phononˆ becomes numberˆ possible in steady for state high-frequency is plotted on a logarithmic cantilevers (and/or high-finesse cav- the term Hintclouds= F xˆ ofin cold the atoms Hamiltonian. in an optical These lattice, are whose oscillations couple to the light scale. Contour linesities), indicate whenn ¯−κ= 1ω and.Then,wefindn ¯ =0.1. For this plot, the following M 3 6 parameters have been used:fieldωM / [27,κ =02 28]..3, n¯ Coolingth = 10 ,Q to= theωM ground-state/ΓM = 102 , and may open the door to various quan- opt x opt x (ωR/ωM )(xZPF/L) 0.012.tum (b) The effects minimumZPF in these phonon systems, number including as a functionZPF2 “cat” of states [29], entanglement [30, 31] ≈ Γ = 2 SFF(ωM ) , Γ =κ 2 SFF( ωM ). (8) the ratio between the mechanical↓ frequency ω and the opticalO ↑ cavity’s ring- − and Fock state detectionM minn ¯ [12].M . (16) down rateTheseκ, according rates enterto Eq.All the (15). the complete Ground-state intrinsic master optomechanical cooling equation is≈ possible4ωM for cooling in the the density experiments matrix areρ ˆ basedof the on the fact regime ω κ, i.e. the “good cavity” or “resolved sideband” limit. Mcantilever As explained in thethat presence thein Ref. radiation [33], of the these field equilibrium introducestwo regimes heat extra can bath damping be brought (that for would directly the lead cantilever. into to a cor- In such a thermalrespondence populationclassical withn ¯th picture, the) and known the the radiation e regimesffective temperature for field: laser-cooling of the of single harmonically mechanical bound mode of in- Here the equation has been written in the interaction picture (disregarding the atoms, namelyterest is the related Doppler to the limit bath for temperatureωM κ andT theby resolvedTeff /T = sidebandΓM /(Γopt regime+ΓM ), where oscillations atω ˜M ), and opt opt for ωMρˆ˙ =ΓMκ(.Γand +ΓoptΓMare(¯nth the+ intrinsic 1)) [â]+( mechanicalΓ + Γ dampingM n¯th) [â rate†] ρˆ and the optomechanical(9) cooling1 ↓ rate, respectively.D Thus there↑ is no limitD to cooling in this regime, pro- [Aˆ]ˆρ = (2AˆρÂˆ† Aˆ†Aˆρˆ ρÂˆ†Aˆ) (10) D 2 − − vided the laser power (and thus the cooling rate Γopt) can be increased without ˆ ˆ is the standard4 Lindblad Strong operatorany deleterious for coupling downward effects (A =ˆ such ea)ff orects as upward unwanted (A =â† heating) by absorption, and provided transitions in the oscillator.the Restricting cooling rate ourselves remains to the su populationsfficiently smallerρnn,we than the mechanical frequency and obtain the equationUp for to the now, phonon we have number assumedn ¯ = nˆ = that5n theρnn: coupling between light and mechanical the cavity ring-down rate.n However, at sufficiently low temperatures, the un- degree of freedom is sufficiently weak to allow for a solution in terms of a mas- avoidableopt photon shot noise inside the cavity counteracts cooling. To study n¯˙ = ΓM n¯th + Γ (ΓM + Γopt)¯n, (11) ter equation,the resulting employing↑ − quantum the rates limits obtained to cooling, from the a fully quantum quantum-mechanical noise approach. theory is which yields theHowever, steady-state as phonon the coupling number in becomes the presence stronger of optomechanical (e.g. by increasing the laser input cooling: called for, which we provided in Ref. [32], based on the general quantum noise power), Γapproach.opt may reach Independently, the cavity a decay derivation rate κ emphasizing. Then, the the spectrum analogy of to force ion sideband fluctuations is itself modifiedO by the presence of the cantilever. It becomes nec- coolingΓM wasn¯th + developedΓoptn¯M in Ref. [33]. In the present paper, we will review and n¯M = . (12) essary toillustrate solve forΓM our the+ Γ theory. coupledopt Wedynamics start by of the outlining light field the basic and the classical mechanical picture, then This is the weightedmotion. average Thispresent of has the the thermal been quantum done and thein noise Ref. optomechanical [32], approach by writing phonon that provides down the a Heisenberg transparent equa- and straight- numbers. It representstions of the motionforward correct generalization for way the to cantilever derive of the cooling classical and the rates formula optical and for quantum the mode, and limits solving for the them phonon after number. effective temperature, Eq. (5). Here linearization.Finally, Here we we illustrate will only the discuss strong the coupling result. regime that was first predicted in our opt 1 O Γ Ref. [32].1 SFF(ωM ) − ↑ n¯M = = opt opt = 1 (13) Γopt Γ /Γ 1 SFF( ωM ) − ↓ ↑ − − 7

6 2 Sideband cooling: the experimental results

displacement spectrum a Sxx(ω) n = 27 nd = 18 m 10–29

nd = 71 nm = 22 ˆ ˆ mean phonon number ) b†b c 2 –1 10 10–30 n = 8.5 nd = 280 m Hz 2

(m 101 x n nd = 1,100 nm = 2.9 m S n n¯ 1 c 10–31 0

Occupancy 10

nd = 4,500 nm = 0.93

10–1 10–32 10.556 10.558 ω 100 101 102 103 104 105 Drive photons, n Frequency (MHz) cavity photons d aˆ†aˆ c Mechanical systems are now ﬁrmly in the quantum regime. → atoms, superconducting circuits, and mechanics Teufel et al., Nature 475, 359 (2011) Strong-coupling regime Nonlinear Optomechanics: Two-phonon cooling, mechanical squeezing, and optical output spectra

A. Nunnenkamp, K. Børkje, J. G. E. Harris, and S. M. Girvin Departments of Physics and Applied Physics, Yale University, New Haven, Connecticut 06520, USA (Dated: February 19, 2010) Motivated by recent optomechanics experiments using the membrane-in-the-middle geometry and ultracold atoms in optical resonators we explore the physics of optomechanical systems where the mechanical oscillator is coupled quadratically rather than linearly to one mode of the optical cavity field. Two-phonon cooling can occur if the cavity width is larger than the mechanical frequency (good-cavity limit) and larger than the heating rate due to the thermal bath. Using Fermi’s Golden rule we write down an effective master equation for the mechanical oscillator which can be solved exactly. In the classical limit the added nonlinear damping changes the steady-state phonon number distribution from a Planck to a Gaussian distribution. In the quantum limit we find a non-zero minimal phonon number even for infinitely strong coupling. Finally, we show how to achieve mechanical squeezing by driving the cavity on both sidebands and calculate cavity output and squeezing spectra.

PACS numbers:

Introduction. Whereas initial work on optomechanical sys- Hamiltonian. We start from the following Hamiltonian tems was motivated by building sensitive measurement de- 2 vices to detect gravitational waves, the field has now become Hˆ = ωR + gxˆ aˆ†aˆ + ωMˆb†ˆb + Hˆdrive + Hˆdecay (1) an active area of research in its own right investigating macroscopic quantum coherence in solid-state devices. As of today where ωR is the cavity resonance frequency, g = ωcav (x)/2 there are many different systems in which one mode of the is the curvature of the cavity dispersion relation ωcav(x), optical cavity field couples linearly to one mode of the me- xˆ = xZPF(ˆb + ˆb†) is the position of the mechanical oscil- 1/2 chanical oscillator. Consequentially, many properties of this lator with zero-point fluctuations xZPF =(2mωM )− , fre- standard setup have been discussed, including red-sideband quency ωM and mass m. aˆ and ˆb are annihilation operators cooling in the resolved-sideband limit, normal-mode splitting, obeying bosonic commutation relations. It appropriately de- optical Kerr-like squeezing, backaction evasion by driving on scribes systems with membrane-in-the-middleLinear optomechanics geometry [1], both sidebands, mechanical squeezing either using squeezed ultracold atoms in optical resonators [2] as well as nano- input light or by driving on the sidebands, entanglement be- electromechanical structures [3]. iω t tween light and one or multiple membranes. Expressing the optical fieldThe as aˆ Hamiltonian= e− L (¯a+dˆ), we obtain Recently, experiments with the membrane-in-the-middle the following quantum masterˆ equation ˆ ˆ ˆ ˆ geometry [1], ultracold atomic gases in optical resonators [2] H = ωRaˆ†aˆ + ωM b†b + g(b + b†)â†aˆ as well as nano-electromechanical structures [3] have realized ˙ = i Hˆ0, + κ [dˆ] + γ(1 +n ¯th) [ˆb] + γn¯th [ˆb†] a nonlinear optomechanical coupling where the mechanical − D D D (2) iωLt ˆ ˆb = ¯b +ˆc mode is coupled quadratically rather than linearly to the cavity with the system HamiltonianFor drive and decay we write aˆ = e − a¯ + d and mode. It enables Fock state detection [1, 4–6], but otherwise → bistability its properties remain largely unknown [7]. 2 2 Hˆ = ∆dˆ†dˆ+ ω ˆb†ˆb + gax¯ (ˆb + ˆb†) (dˆ† + dˆ) (3) 0 M Hˆ ZPF= ∆dˆ†dˆ+ ω cˆ†cˆ + ga¯(ˆc +ˆc†)(dˆ+ dˆ†)+gdˆ†dˆ(ˆc +ˆc†) In this paper we take the first step to change this situa- − M − 2 tion and explore the physics of nonlinear optomechanics. Fo- where we have introduced the detuningdetuning ∆ = ω ω , the # of photons a¯ L − R cussing on the good-cavity limit and the case where the de- cavity damping rate κ, the mechanical damping rateEffectiveγ as well coupling rate Bilinear coupling | | tuning between drive and cavity matches twice the mechani- as the thermal phonon number n¯ . [ô] =ôoˆ† (ô†oˆ + opticalth bathD − mechanical bath cal frequency, we first solve the full quantum master equation oˆ†oˆ)/2 denotes the standard dissipator in Lindblad form. numerically. We find that two-phonon cooling occurs as long Experiments with the membrane-in-the-middle geometry optical cavity mechanical mode as the cavity width is larger than the heating rate due to the are in the regime of weak coupling and large thermal phonon or 5 T =06 T nth thermal bath, while the reverse effect, i.e. heating of the op- number: κ = 10 Hz, ωM = 10 Hz, ΓM = 1Hz, g = 6 2 15 5 ga¯ tical field, becomes important in the opposite limit. Using 10 Hz/nm , xZPF = 10− m, a¯ = 10 ,κ so that g¯ = γ 2 4 ˆ cˆ Fermi’s Golden rule we then write down an effective master gax¯ ZPF =0.1Hz, while n¯th = 10 at T = 300mKdand 7 equation for the mechanical oscillator which can be solved ex- n¯th = 10 at T = 300K [1]. Experiments with ultracold actly. In the classical limit two-phonon cooling can be under- atoms in optical resonators, in contrast, show strong optome-∆ ωM stood as added nonlinear damping which changes the steady- chanical coupling while the effect of the thermal phonon− bath 6 5 2 3 state phonon number distribution from a Planck to a Gaussian is negligible: κ = 10 Hz, ωM = 10 Hz, gxZPF = 10 Hz [2]. distribution. In the quantum limit we find a non-zero minimal Current nano-electromechanical structures have only a small phonon number even for infinitely strong coupling since two- quadratic optomechanical coupling so that nonlinear effects phonon cooling processes preserve the photon-number parity. are expected to be small corrections and will not be discussed 6 7 Finally, we point out that if the cavity is driven on both side- in this paper: κ = 10 Hz, ωM = 10 Hz, ΓM = 10Hz, 3 2 14 4 2 bands the system maps onto a degenerate parametric oscillator g = 10 Hz/nm , xZPF = 10− m, a¯ = 10 and n¯th = 10 which opens up the possibility of mechanical squeezing. at T = 150mK [3]. StaticStatic part behaviour of radiation-pressure force oscillating mirror x Veff = Vho + Vrad radiation input laser cantileverκ effective mechanical potential pressure0=i∆a¯ a¯ iΩ ig(¯b + ¯b∗)¯a fixed mirror − 2 − − F Frad γ 2 0= iωM¯b ¯b ig a¯ x − − 2 − | | increasing 2 laser power Experimental proof of staticΩ bistability: ( )=2 ( ) n¯A. Dorsel,= J. D. McCullen, P. Meystre, FradV (xx)= F (Ix)x /c cav 2 2 − E. Vignes andκ H. Walther: λ 2 +(∆ + χn¯cav) 2 λ/2 Phys. Rev. Lett. 51, 1550 (1983) F 2g2 bistability with χ ≈ ωM Vrad(x) position = + Veff circulatingVrad intensityVHO x laser detuning figure adapted from F. Marquardt hysteresis Experiment: Dorsel et al., PRL 51, 1550 (1983) Nonlinear Optomechanics: Two-phonon cooling, mechanical squeezing, and optical output spectra

PACS numbers:

Introduction. Whereas initial work on optomechanical sys- Hamiltonian. We start from the following Hamiltonian tems was motivated by building sensitive measurement de- 2 vices to detect gravitational waves, the field has now become Hˆ = ωR + gxˆ aˆ†aˆ + ωMˆb†ˆb + Hˆdrive + Hˆdecay (1) an active area of research in its own right investigating macroscopic quantum coherence in solid-state devices. As of today where ωR is the cavity resonance frequency, g = ωcav (x)/2 there are many different systems in which one mode of the is the curvature of the cavity dispersion relation ωcav(x), optical cavity field couples linearly to one mode of the me- xˆ = xZPF(ˆb + ˆb†) is the position of the mechanical oscil- 1/2 chanical oscillator. Consequentially, many properties of this lator with zero-point fluctuations xZPF =(2mωM )− , fre- standard setup have been discussed, including red-sideband quency ωM and mass m. aˆ and ˆb are annihilation operators cooling in the resolved-sideband limit, normal-mode splitting, obeying bosonic commutation relations. It appropriately de- optical Kerr-like squeezing, backaction evasion by driving on scribes systems with membrane-in-the-middle geometry [1], both sidebands, mechanical squeezing either using squeezed ultracold atoms in optical resonators [2] as well as nano- input light or by driving on the sidebands, entanglement be- electromechanical structures [3]. iω t tween light and one or multiple membranes. Expressing the optical field as aˆ = e− L (¯a+dˆ), we obtain Recently, experiments with the membrane-in-the-middle the following quantum master equation geometry [1], ultracold atomic gases in optical resonators [2] as well as nano-electromechanical structures [3] have realized ˙ = i Hˆ0, + κ [dˆLinear] + γ(1 + n ¯optomechanicsth) [ˆb] + γn¯th [ˆb†] a nonlinear optomechanical coupling where the mechanical − D D D (2) mode is coupled quadratically rather than linearly to the cavity with the system Hamiltonian mode. It enables Fock state detection [1, 4–6], but otherwise The Hamiltonian its properties remain largely unknown [7]. 2 2 Hˆ = ∆dˆ†dˆ+ ω ˆb†ˆb + gax¯ (ˆb + ˆb†) (dˆ† + dˆ) (3) 0 M Hˆ ZPF= ∆dˆ†dˆ+ ω cˆ†cˆ + ga¯(ˆc +ˆc†)(dˆ+ dˆ†)+gdˆ†dˆ(ˆc +ˆc†) In this paper we take the first step to change this situa- − M − 2 tion and explore the physics of nonlinear optomechanics. Fo- where we have introduced the detuningdetuning ∆ = ω ω , the # of photons a¯ L − R cussing on the good-cavity limit and the case where the de- cavity damping rate κ, the mechanical damping rateEffectiveγ as well coupling rate Bilinear coupling | | tuning between drive and cavity matches twice the mechani- as the thermal phonon number n¯ . [ô] =ôoˆ† (ô†oˆ + The thequationsD of −motion cal frequency, we first solve the full quantum master equation oˆ†oˆ)/2 denotes the standard dissipator in Lindblad form. numerically. We find that two-phonon cooling occurs as long Experiments with the membrane-in-the-middle˙ κ geometry dˆ= i∆dˆ dˆ √κdˆ iga¯(ˆc +ˆc†) as the cavity width is larger than the heating rate due to the are in the regime of weak coupling and large thermal phonon in 5 6 − 2 − − thermal bath, while the reverse effect, i.e. heating of the op- number: κ = 10 Hz, ωM = 10 Hz, ΓM = 1Hzγ , g = 6 2 15˙ 5 ˆ ˆ tical field, becomes important in the opposite limit. Using 10 Hz/nm , xZPF = 10− cˆm=, a¯ =iω 10M cˆ, so thatcˆ g¯ =√γcîn iga¯(d + d†) 2 −4 − 2 − − Fermi’s Golden rule we then write down an effective master gax¯ ZPF =0.1Hz, while n¯th = 10 at T = 300mK and 7 equation for the mechanical oscillator which can be solved ex- n¯th = 10 at T = 300KExact[1]. Experiments solution with ultracold actly. In the classical limit two-phonon cooling can be under- atoms in optical resonators, in contrast, show strong optome- cˆ(ω)=...cˆ + ...cˆ† + ...dˆ + ...dˆ† cˆ† (ω)ˆc (ω) = n δ(ω + ω) stood as added nonlinear damping which changes the steady- chanical coupling while the effect of thein thermal phononin bathin in in in th state phonon number distribution from a Planck to a Gaussian is negligible: κ = 106Hz, ωˆ = 105Hz, gx2 = 103Hz [2].ˆ ˆ dM(ω)=...cîn +ZPF...cîn† + ...din + ...din† cîn(ω)ˆcin† (ω) =(nth + 1)δ(ω + ω) distribution. In the quantum limit we find a non-zero minimal Current nano-electromechanical structures have only a small cˆ† (ω)ˆc† (ω) = cˆ (ω)ˆc (ω) =0 phonon number even for infinitely strong coupling since two- quadratic optomechanical coupling so that nonlinear effects in in in in phonon cooling processes preserve the photon-number parity. are expected to be small correctionsWe can and calculate will not be all discussed observables! 6 7 Finally, we point out that if the cavity is driven on both side- in this paper: κ = 10 Hz, ωM = 10 Hz, ΓM = 10Hz, Walls and Milburn, Quantum Optics (OUP) 3 2 14 4 2 bands the system maps onto a degenerate parametric oscillator g = 10 Hz/nm , xZPF = 10− m, a¯ = 10 and n¯th = 10 which opens up the possibility of mechanical squeezing. at T = 150mK [3]. Normal-mode splitting in the strong-coupling limit

iωt iωt S (ω)= dt e cˆ†(t)ˆc(0) S (ω)= dt e dˆ† (t)dˆ (0) cc out out out with dˆout = dˆ+ √κdˆin

1.5 1.5 Sout(ω)=SFF( ω)Sxx(ω) cantilever spectrum − b 10–13 nd = 4,500 nm = 0.93 S ( ω) cc n = 11,000 n = 0.55 − Sxx(ω) d m

cantilever ) 1.0 1.0 –1 n = 28,000 n = 0.36 –14 d m

(Hz 10 o S/P n = 0.34 nd = 89,000 m frequency driven cavity g/π n = 0.42 nd = 180,000 m 0.50.5 0.50.5 1.01.0 1.51.5 10–15 10.3 10.4 10.5 10.6 10.7 detuning Frequency (MHz) ﬁgure by F. Marquardt In the strong-coupling limit cavity and mechanics hybridize. Gröblacher el al., Nature 460, 724 (2009) Teufel et al., Nature 471, 204 (2011) Theory: Dobrindt et al., PRL 101, 263602 (2008) REPORTS

pressure force oscillating at the frequency dif- where dsin is the amplitude of the probe field cently in the mechanical domain (14), in which the ference W. If this driving force oscillates close drive, and we abbreviate D′ ≡ W − Wm.Wehave optical and mechanical systems are hybridized to to the mechanical resonance frequency Wm, the assumed a high-quality factor of the mechanical dressed states that differ by ℏWc in their energy. mechanical mode starts to oscillate coherently, oscillator (Gm << Wm)andthecontrolbeamde- OMIT is realized using toroidal whispering- −iWt dx(t)=2Re[Xe ]. This in turn gives rise to tuning D −Wm.Thesolutionfortheintracavity gallery-mode microresonators (Fig. 2A) (10, 17). Stokes- and anti-Stokes scattering of light from probe field¼ amplitude reads The cavity is operated in the undercoupled the strong intracavity control field. If the system regime (hc <1/2),whichtogetherwithmodal resides deep enough in the resolved-sideband coupling between counterpropagating modes − phck (RSB) regime with k << Wm,Stokesscattering A 2 dsin 5 (SOM Sec. 7) leads to a nonzero probe (ampli- ¼ ′ Wc =4 ð Þ (to the optical frequency wl − W)isstrongly −iD k=2 ffiffiffiffiffiffiffi ′ tude) transmission tr = tp(D′ =0,Wc =0)at ð þ Þþ−iD Gm=2 suppressed because it is highly off-resonant with þ resonance (Fig. 2B), even in the absence of the optical cavity. We can therefore assume that This solution has a form well known from the the control beam. In the case of the present device, 2 2 only an anti-Stokes field builds up inside the response of an EIT medium to a probe field (1). |tr| ≈ 0.5. [Note, however, that |tr| <0.01canbe cavity, da(t) ≈ A− e−iWt.However,thisfieldof The coherence between the two ground states of achieved with silica toroids (24).] To separate the frequency wp = wl + W is degenerate with the an atomic L system, and the coherence between effects of this residual transmission from OMIT, near-resonant probe field sent to the cavity. De- the levels probed by the probe laser undergo the we introduce the normalized transmission of the ′ structive interference of these two driving waves same evolution as do the mechanical oscillation probe tp tp − tr = 1 − tr . can suppress the build-up of an intracavity probe amplitude and the intracavity probe field in the The mechanical¼ð Þ motionð Þ was detected using field. These processes are captured by the Langevin case of optomechanically induced transparency a balanced homodyne detection scheme (fig. equations of motion for the complex amplitudes (OMIT). The role of the control laser’sRabi S1) measuring the phase quadrature of the field A− and X,whichrequireinthesteadystate(SOM frequency in an atomic system is taken by the op- emerging from the cavity (25). This allows ex- Eqs. S26 and S27) tomechanical coupling rate Wc 2aGxzpf ,where tracting the parameters of the device used in these ¼ ′ − xzpf ℏ=2meff Wm designates the spread of experiments, which are given by (meff, G/2p, −iD k=2 A −iGaX phck dsin 3 ¼ ð þ Þ ¼ þ ð Þ the ground-state wave function of the mechanical Gm/2 p, Wm/2 p, k/2 p) ≈ (20 ng, −12 GHz/nm, pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi oscillator. For Wc > Gm, k the system enters the 41 kHz, 51.8 MHz, 15 MHz), placing it well into ′ ffiffiffiffiffiffiffi − 2meff Wm −iD Gm=2 X −iℏGaA 4 the resolved sideband regime (25). To probe the ð þ Þ ¼ ð Þ strong coupling regime (22, 23)investigatedre- cavity transmission spectrum in the presence of on August 19, 2011 LETTER RESEARCH a control beam, the Ti:sapphire control laser is A C frequency modulated at frequency W using a

a 1 Control C broadband phase modulator, creating two side- on field trol Probe in 2 |T|

G/2π = –360 kHz G/2π = 0 G/2π = 680 kHz Probe Control field Probe out 0 Probe /8 Probe π field field Probe in Optomechanically-inducedRESEARCHPhase LETTER transparency (OMIT) –π/8 www.sciencemag.org

7.4730 7.4735 7.4740 7.4730 7.4735 7.47402 7.4730 7.4735 7.4740 x(t) | p

/2 (GHz) /2 (GHz) t /2 (GHz) Zp π Zp π 1 Zp π the number of drive photons in the cavity. If the drive is detuned so that a 1 B b c D b –1 its upper mechanical sidebandCavity is near the cavity resonance, ∆ = ωM nd = 5 × 10 7.4740 mode − d~ vdzVm {vc = Vm (Fig. 2c), the modified mechanical res- 0 x(t) ) ðÞ) 7.4735 7.4735 n = 5 × 100 onance frequency V’m and damping rate C’m closely follow the ima- d 7.4735 Probe out (GHz (GHz π π (GHz)

ginary and real parts of the cavity response./2 In the resolved sideband /2 p p π Z Z 7,8,24 regime these quantities are well approximated by : n = 5 × 101 /2 Probe d p 2 Downloaded from

| 7.4730 Z T laser 2 0.98 1.00 1.02 Control laser 7.4730 4g d 7.4730

probe frequency probe 1 V V z 0 2 ’m< m 2 1 nd = 5 × 10 2 0.90.80.7 ð Þ 1.0 1.1 1.2 1.3 2 7.4725 7.4624 7.4629k z4d 7.4624 | transmission Probe power 7.4629 |tr| /2 (GHz /2 (GHz Zd π ) ProbeZd π laser) offset frequency Ω/Ω Optical frequency m 2 drive frequency 3 c 106 d 4g k nd = 5 × 10 Fig. 1. Optomechanically inducedC’m

(GHz ∝ 4 coupledon the drumto an external asπ a function propagating of d. mode The incident at the rate microwavekex.Throughtheexternalmode,theresonatoris power Pin is 10 /2 p n = 5 105 populatedheld constant with a control atZ 107.4730 pW. field (onlyAs this intracavity poweri( fieldω is applied isω shown).)t very The farrespo+ from+nsei(ω of the thisω driven)t optomechanical Fig.d 2. ×Optomechanical system. (Top)Atoroidal coherent signal dˆ(t) = A−e− P − L + A e P − L systemcavity is resonance, probed by a it weak results probe in field a greatly sent toward reduced the cavity, number the transmission of photons of which (i.e., the returned microcavity is used to10 demonstrate3 OMIT: The res- 7.4725 n =

2 2 d Coupling rate, field “Probe out”)isanalyzedhere.(B)ThefrequencyofthecontrolfieldisdetunedbyD from the cavityg / onator is coupled to the control and probe fields in the cavity, given by nd 5 (2Pinkex/Bvd)/(k 1 4D ), where π 5 × 106 resonanceD 5 v 2 frequency,v . Even where for a this detuning moderate-power close to the lower microwave mechanical drive sideband, withD ≈−Wm,ischosen.The using a tapered fiber.102 The optical mode couples probe laserd ’sfrequencyisoffsetbythetunableradiofrequencyc 7.4620 1 7.4625 7.4630W from the control laser.7.4725 The dynamics7.4730 of 7.4735 7.4740 100 102 104 106 Drive frequency, /2 (GHz) through radiation pressure force to the mechanical # Zd π ndA−800, the effects on the mechanical oscillator2 are2 quite striking; Drive photons, nd interest occur whenκ the probe laser is tuned over the opgticala¯ resonance of the cavity, which hasProbe a linewidth frequency, Zpradial/2π (GHz) breathing mode of the structure. In this ring Figure 4 | Spectroscopythe intrinsic∝ in the strong-coupled mechanical regime. damping a, The normalized is dominatedγin the transmission by the at damping a frequency vdz fromV’m. c, When the drive amplitude is of k = k0 + kex.(C)Levelschemeoftheoptomechanicalsystem.Thecontrolfieldistunedclosetored-i(Ω + ∆)+ 4 geometry, the cavity transmission, defined by the magnitude and phase of the cavity2 transmission in the presence of a strong 2increasedi( (Ωnd < 10ω), theM two) resonances show an avoided crossing between the microwavethe drive. microwaveThe data shown in red, photons.− green and blue Knowing are for three differentnd andeigenmodes estimating of the coupledxzp system.fromd, For geo- the largestFigure amplitude 3 drive| Demonstration of the strong-coupling regime. a, Normalized sideband transitions, in which a mechanical excita−tion quantum6 − is annihilatedWeis (mechanical et al., occupationScience 330ratio, 1520 of the (2010) returned probe-field amplitude divided detuningsndmetry,with→ fitsn to we− equation1) can when (3) fit (black). a these photonb, For data relatively is added to low equations to drive the power cavity (1)(nd (optical< and5 3 10 (2)), occupation the (shown driven transmissionn in→ blackn spectra+1),thereforecouplingthe are splitmicrowave by much more cavity than k transmissionby the incoming in the presence probe field of a is microwave simply given drive by applied the (nd < 10), them mechanicalm sideband appears as a sharp dip or peak in the probe or Cm2for1 a broad range of detunings.p p Teufel et al., Nature 471, 204 (2011) correspondingin Fig. 2d) towith energy extract eigenstates.ΩG/2=p 5ω56 The6 probe7 MHzω field nm probes. transitions in which themechanicaloscillatorwith D 52Vm, with successivetransmission plots through for increasing the tapered drive fiber. amplitude (Bottom)Undernd.At from resonance, it inherits the cavity’s larger dampingP rate as itLquantum memory,Theory: a quantum Agarwal bus architecture and for quantumHuang, informa- PRA 81, 041803 (2010) moderate drive amplitude (nd < 10), the interference between the drive and approachesoccupationvJustc. At the as largest the is unchanged. microwave coupling, the (D normal-mode photons)Transmissionoftheprobelaserpower splitting strongly− is affecttion processing the mechanical analogousthrough to that mode, achieved the withoptomechanical ion traps, and a way system to the chosen waveguide-toroid coupling conditions, there appreciable for a broad range of drive and probe frequencies (Fig. 4d). investigate decoherence for large, spatially separatedprobe photons superposition results in a narrow peak in the cavity spectrum, whose2 width is inthe the symmetry case of a critically of the coupled parametric cavity k0 = interactionkex as a function suggests of normalized that probe the laser frequency offset, is a nonzero probe power transmission |tr| at res- states27. This experimentwhen the demonstrates control field that is by off engineering (blue lines) a microwave and on (green lines). Dashed and full linesgiven correspond by the mechanical to the onance. linewidth TheC control’m. This field represents induces an the additional onset of trans- resonant circuitmechanics with a free-standing should micromechanical influence the membrane cavity or mode.Last, To we consider investigate ground-state this, cooling, we which is the next step for- 11 electromechanically induced transparency . When C’m becomes comparable2 drum, a largemodelsapply coupling both based is possible a on microwave the without full (Eq. degradation drive 1) and tone of approxima eitherv theandtiveward. a (Eq. Assuming second 5) calculat thermal probeions, equilibrium tone respectively.v at a. base cryostat temperature of parency window with a contrast up to 1 − |tr| . mechanical or electromagnetic quality factors. Our measurementsd 20 mK, we expect the cavity to be in itsp groundto statek, the and the cavity thermal resonance splits into normal modes. The eigenmodes of the have shownHere, clear quantitative the drive agreement tone will with induce theoretical an predictions interactionoccupancy between of the mechanical the mechanics mode to be nsystemm < 40 quanta. are no Strong longer purely electrical or mechanical, but are a pair of hybrid for dynamicaland back-action the cavity, and normal-mode while the splitting, probe with tone the latter will monitorcoupling implies the that response resolved sideband of the coolingelectromechanical can be used to reduce resonances. b, The transmission as a function of probe being exceptionally large, namely 10% of Vm. Simple refinements—www.sciencemag.orgthe occupancy of the mechanicalSCIENCE mode byVOL a factor 330 of k/Cm 10< 5,000 DECEMBER 2010 1521 such as reducingcavity.d, This using thinner technique membranes provides or exploring a way higher- to measure(refs 7–9), placing the spectroscopy both (coupled) resonators of frequency in their ground and state. number of drive photons shows that the driven system enters the order modes—could further enhance the coupling strength and relax Once this is achieved, the thermal lifetime of a single phonon would the ‘dressed’ cavity states in the presence of the electromechanical strong-coupling regime (g $ k, Cm). Here, the logarithmic colour scale show cooling requirements. This new system shows the performance neces- be 1/Cth < 130 ms, which is orders of magnitude longer than typical sary to notinteraction. only observe quantum Figure effects 3a shows but also a (with series additional of cavitysuperconducting probe spectra qubit lifetimes. taken Verifying with thisthe achievement transmission would as it varies from 213 dB (dark red) to 0 dB (blue). c,The circuitry) enable the realization of a number of important goals. For require either a better52 cryogenic amplifier21,28measured(for continuous coupling mea- rate g (red) follows the expected dependence on the number continuoussuccessively measurements, higher these include microwave detection of power zero-point appliedsurements) with or aD superconductingVm. Once qubit4 (for discrete measurements). 1 22 of drive photons, with a fit to pnd shown in black. motion , evadingthe drive quantum power back-action is largeor Heisenberg-limited enough that detec-g

Ground-state cooling Normal-mode splitting

c 102

101 nm n n¯ 1 c b

Occupancy 10

10–1

100 101 102 103 104 105

Drive photons, nd

2 µm Nature 475, 359 (2011) Nature 471, 204 (2011)

Coherent coupling Sideband thermometry signal Homodyne

Ω displac. Mechanical

Nature 482, 63 (2012) PRL 108, 033602 (2012) Dissipative coupling Dispersive vs. dissipative coupling

Dispersive coupling Dissipative coupling

∂κ ∂ω κ(ˆx)=κ + xˆ ω (ˆx)=ω + R xˆ ∂x R R ∂x Hˆ Hˆ xˆ int ∝ damp Hînt aˆ†aˆxˆ with ∝ Hˆ √κ (â†aˆ +â†aˆ) damp ∝ q q q

Elste et al., PRL 102, 207209 (2009) Cooling with dissipative coupling

1 ∆ =+κ/2 0.8

) 0.6 ω ( FF

S 0.4 Hînt Hˆdampxˆ with ∝ 0.2 Hˆ √κ (â†aˆ +â†aˆ) damp ∝ q q q 0 ï4 ï2 0 2 4 ω/κ Two noise sources! Fˆ = Fˆ1 + Fˆ2 Zero temperature bath! Fˆ1 dîn + dˆ† Fˆ dˆ+ dˆ† Γ↑ SFF( ωM ) ∝ in 2 ∝ opt ∝ − They are random, but not independent! dˆ(ω) χ (ω)dˆ This enables ground ∝ R in κ 2 state cooling outside the S (ω) 1 + i∆ χ (ω) FF ∝ − 2 R resolved-sideband limit! 1 with χR(ω)=[κ/2 i(ω + ∆)]− →Fano line shape Elste et al., PRL 102, 207209 (2009) − Conclusions

xˆ Hˆ = ω 1 aˆ†aˆ + ω ˆb†ˆb R − L M radiation-pressure force

• Applications: sensors, transducers, decoherence • Damping is due to the ﬁnite time lag between mirror and radiation-pressure force: ωM κ • With red-sideband cooling ω L = ω R ω M the ground state was reached in experiments:− n¯ 1 • At strong coupling optics and mechanics hybridize. http://www.physinfo.fr/houches/pdf/Marquardt.pdf