Review of cavity optomechanical cooling
aa r X i v : . [ qu a n t - ph ] N ov Review of cavity optomechanical cooling ∗ Yong-Chun Liu a) † , Yu-Wen Hu a) , Chee Wei Wong b) and Yun-Feng Xiao a) ‡ a) State Key Laboratory for Mesoscopic Physics and School of Physics, Peking University, Beijing 100871,China b) Optical Nanostructures Laboratory, Columbia University, New York, New York 10027, USA
Quantum manipulation of macroscopic mechanical systems is of great interest in both fundamen-tal physics and applications ranging from high-precision metrology to quantum information processing.A crucial goal is to cool the mechanical system to its quantum ground state. In this review, we fo-cus on the cavity optomechanical cooling, which exploits the cavity enhanced interaction betweenoptical field and mechanical motion to reduce the thermal noise. Recent remarkable theoretical andexperimental efforts in this field have taken a major step forward in preparing the motional quantumground state of mesoscopic mechanical systems. This review first describes the quantum theory ofcavity optomechanical cooling, including quantum noise approach and covariance approach; then theup-to-date experimental progresses are introduced. Finally, new cooling approaches are discussedalong the directions of cooling in the strong coupling regime and cooling beyond the resolved sidebandlimit.
Keywords: cavity optomechanics, optomechanical cooling, cavity cooling, groundstate cooling, mechanical resonator
PACS:
Contents
1. Introduction 22. Quantum theory of cavity optomechanical cooling 5 ∗ We thank Yi-Wen Hu and Xingsheng Luan for helpful discussions. Project supported by 973 program (GrantNo. 2013CB328704), National Natural Science Foundation of China (Grant Nos. 11004003, 11222440, and11121091), and RFDPH (No. 20120001110068). Y.-C. Liu acknowledges the support from the ScholarshipAward for Excellent Doctoral Student granted by Ministry of Education. † E-mail:[email protected] ‡ ∼ yfxiao/index.html . Recent experimental progresses 154. New cooling approaches 18
5. Summary and outlook 24
1. Introduction
Optomechanics is an emerging field exploring the interaction between light and mechanicalmotion. Such interaction originates from the mechanical effect of light, i.e., optical force.Radiation pressure force (or scattering force) and optical gradient force (or dipole force) aretwo typical categories of optical forces. The radiation pressure force originates from the factthat light carries momentum. The momentum transfer from light to a mechanical object exertsa pressure force on the object. This was noticed dating back to the 17th century by Kepler,who noted that the dust tails of comets point away from the sun. In the 1970s, H¨ansch andSchawlow [1], Wineland and Dehmelt [2] pointed out the possibility of cooling atoms by usingradiation pressure force of a laser. This was subsequently realized experimentally [3], and it hasnow become an important technique for manipulating atoms. The gradient force stems fromthe electromagnetic field gradient. The nonuniform field polarizes the mechanical object in away that the positively and negatively charged sides of the dipole experience different forces,leading to nonzero net optical force acting on the object. It was first demonstrated by Ashkinthat focused laser beams can be used to trap micro- and nano-scale particles [4]. This hasstimulated the technique of optical tweezers, which are widely used to manipulate living cells,DNA and bacteria. There are other kinds of optical forces, for instance, photothermal force(bolometric force) [5], which results from the thermalelastic effect.The optical forces exerting on macroscopic/mesoscopic mechanical objects are typicallyigure 1: Schematic of a generic optomechanical system, with a laser-driven optical cavity. Theleft mirror is fixed and the right mirror is movable.very weak. To overcome this problem, optical cavities are employed, which resonantly enhancethe intracavity light intensity so that the optical forces become pronounced. For example, in aFabry-P´erot (FP) cavity consisting of a fixed mirror and a movable mirror attached to a spring(Fig. 1), light is reflected multiple times between the two mirrors, and thus the cavity fieldbuilds up, resulting largely enhanced optical force exerting on the movable mirror. The study ofthis field named as cavity optomechanics was pioneered by Braginsky and co-workers [6, 7] withmicrowave cavities. Later, experiments in the optical domain demonstrated the optomechanicalbistability phenomenon [8], where a macroscopic mirror had two stable equilibrium positionsunder the action of cavity-enhanced radiation pressure force. Further experiments observed op-tical feedback cooling of mechanical motion based on precise measurement and active feedback[9, 10], and along this line cooling to much lower temperatures was realized later [11, 12, 13]. Onthe other hand, after the observation of radiation-pressure induced self-oscillations (parametricinstability) in optical microtoroidal cavities [14, 15, 16], passive cooling, which uses purely theintrinsic backaction effect of the cavity optomechanical system, attracted much attention in thepast decade [17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. Moreover, recent theoretical and exper-imental efforts have demonstrated optomechanically induced transparency [28, 29, 30, 31, 32],optomechanical storage [33], normal mode splitting [34, 35, 36, 37, 38, 39], quantum-coherentcoupling between optical modes and mechanical modes [40, 41] and state transfer at differentoptical wavelengths [42, 43, 44, 45, 46, 47, 48]. Various experimental systems are proposed andinvestigated, including FP cavities [17, 18], whispering-gallery microcavities [15, 49, 50, 51], mi-croring cavities [52], photonic crystal cavities [53, 54, 55, 56], membranes [57, 58, 59, 60], nanos-rings [61], nanorods [62, 63, 64], hybrid plasmonic structures [65], optically levitated particles[66, 67, 68, 69, 70, 71, 72], cold atoms [73, 74, 75] and superconducting circuits [76]. Recentlymuch attention is also focused on the studies of single-photon strong optomechanical coupling[77, 78, 79, 80, 81, 82], single-photon transport [83, 84, 85, 86], nonlinear quantum optomechan-ics [87, 88, 89, 90], quadratic coupling [57, 91, 92, 93, 94, 95, 96, 97, 98], quantum superposition[99, 100, 101], entanglement [102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113], squeez-ing [114, 115, 116, 117, 118], decoherence [119], optomechanical arrays [120, 121], quantumhybrid systems [122, 123, 124, 125, 126], Brillouin optomechanics [127, 128], high-precisionmeasurements [129, 130, 131, 132, 133, 134] and so on.The rapidly growing interest in cavity optomechanics is a result of the importance of thissubject in both fundamental physics studies and applied science. On one hand, cavity op-tomechanics provides a unique platform for the study of fundamental quantum physics, forexample, macroscopic quantum phenomena, decoherence and quantum-classical boundary. Onthe other hand, cavity optomechanics is promising for high-precision measurements of smallforces, masses, displacements and accelerations. Furthermore, cavity optomechanics providesmany useful tools for both classical and quantum information processing. For instance, op-tomechanical devices can serve as storages of information, interfaces between visible light andmicrowave. Optomechanical systems also serve as the “bridge” or “bus” in hybrid photonic,electronic and spintronic components, providing a routing for combining different systems toform hybrid quantum devices. A number of excellent reviews covering various topics has beenpublished in the past [135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149,150, 151, 152, 153, 154, 155, 156, 157].As the first crucial step for preparing mechanical quantum states, cooling of mechanicalresonators has been one of the central research interests in the past decade. Currently, itlacks a comprehensive review on most recent theoretical and experimental progresses of cavityoptomechanical cooling, especially new cooling approaches for guiding future experiments. Inthis review we focus on this issue, addressing the quantum theory, recent experiments and newdirections. The rest of this paper is organized as follows. In Sec. 2, we present the basicquantum theory of cavity optomechanical cooling. Starting from the system Hamiltonian, weintroduce the linearization of the interaction. Then the methods for calculating the cooling ratesnd cooling limits are shown, including quantum noise approach and covariance approach. InSec. 3, we review the up-to-date experimental progress towards cooling to the quantum groundstate. Recent theoretical approaches for improving the cooling performance are discussed inSec. 4. A summary is presented in Sec. 5.
2. Quantum theory of cavity optomechanical cooling
The basic idea of cavity optomechanical cooling is that the optical field introduces extradamping for the mechanical mode. Qualitatively, such optical damping is introduced becausethe optical force induced by the cavity field reacts with a finite delay time, corresponding to thephoton lifetime of the cavity. Let us take a FP cavity optomechanical system as an example(Fig. 1). On one hand, when the movable mirror is at different position, the cavity field andthus the optical force. exerting on the mirror are also different. On the other hand, when theposition of the movable mirror changes, the subsequent change of the cavity field requires sometime-lag due to the finite photon lifetime. Therefore, the optical force also depends on thevelocity of the movable mirror. This velocity-dependent optical force is similar to the frictioncaused by the intrinsic mechanical damping, and leads to extra damping (or amplification) ofthe mechanical motion. In the classical picture, for an optical damping rate Γ opt , the resultingeffective temperature of the mechanical mode being cooled is T eff = γT / ( γ + Γ opt ), where γ isthe intrinsic mechanical damping rate and T is the environmental temperature. Note that themechanical mode is selectively cooled, i. e., only the mode of interest is cooled while the bulktemperature of the mechanical object keeps unchanged. For the FP cavity case, the mechanicalmode of interest is the center-of-mass motion of the movable mirror.The above classical description of cavity optomechanical cooling is not accurate in somecases. For example, it does not predict cooling limits [158, 159, 160]. To accurately modelthe cooling process, in the following we provide full quantum theory of cavity optomechanicalcooling. Here both the cavity field and the mechanical oscillation are described as quantizedbosonic fields. Starting from the system Hamiltonian and taking the dissipations into consid-eration, we can write down the quantum Langevin equations and master equation to describethe system dynamics. For different parameter regimes, cooling performance can be obtainedsing different approaches. Let us consider a generic cavity optomechanical system with a single optical cavity modecoupled to a mechanical mode, which is canonically modeled as a FP cavity with one fixedmirror and one movable mirror mounted on a spring (Fig. 1). The system Hamiltonian is givenby H = H free + H int + H drive . (1)The first term ( H free ) is the free Hamiltonian of the optical and mechanical modes, describedby H free = ω c a † a + ω m b † b. (2)Here both of the optical and the mechanical modes are represented by quantum harmonicoscillators, where a ( a † ) is the bosonic annihilation (creation) operator of the optical cavitymode, b ( b † ) is the bosonic annihilation (creation) operator of the mechanical mode, and ω c ( ω m ) is the corresponding angular resonance frequency. The commutation relations satisfy[ a, a † ] = 1 and [ b, b † ] = 1. The displacement operator of the mechanical mode is given by x = x ZPF ( b † + b ), where x ZPF = p ~ / (2 m eff ω m ) is the zero-point fluctuation, with m eff beingthe effective mass of the mechanical mode.The second term of Eq. (1) ( H int ) describes the optomechanical interaction between theoptical mode and the mechanical mode, which is written as H int = ga † a ( b † + b ) , (3)where g = [ ∂ω c ( x ) /∂x ] x ZPF represents the single-photon optomechanical coupling strength.This Hamiltonian can be obtained by simply considering that the cavity resonance frequency ismodulated by the mechanical position and using Taylor expansion ω c ( x ) = ω c + x∂ω c ( x ) /∂x + O ( x ) ≃ ω c + g ( b † + b ). A more rigorous and detailed derivation of this Hamiltonian can befound in Law’s paper [161]. Note that we focus on the radiation pressure force and the opticalgradient force. For the photothermal force, the Hamiltonian can be found in [162].The last term of Eq. (1) ( H drive ) describes the optical driving of the system. Assume thatthe system is excited through a coherent continuous-wave laser, and then the Hamiltonian isiven by H drive = Ω ∗ e iω in t a + Ω e − iω in t a † . (4)Here ω in is the input laser frequency and Ω = p κ ex P/ ( ~ ω ) e iφ denotes the driving strength,where P is the input laser power, φ is the initial phase of the input laser and κ ex is the input-cavity coupling rate.In the frame rotating at the input laser frequency ω in , the system Hamiltonian is transformedto H = − ∆ a † a + ω m b † b + ga † a ( b † + b ) + (Ω ∗ a + Ω a † ) , (5)where ∆ = ω in − ω c is the input-cavity detuning. The quantum Langevin equations are givenby ˙ a = (cid:0) i ∆ − κ (cid:1) a − iga ( b + b † ) − i Ω −√ κ ex a in , ex − √ κ a in , , (6)˙ b = (cid:0) − iω m − γ (cid:1) b − iga † a −√ γb in , (7)where κ is the intrinsic cavity dissipation rate; κ = κ + κ ex is the total cavity dissipationrate; γ is the dissipation rate of the mechanical mode; a in , , a in , ex and b in are the noise oper-ators associated with the intrinsic cavity dissipation, external cavity dissipation (input-cavitycoupling) and mechanical dissipation. The correlations for these noise operators are given by h a in , ( t ) a † in , ( t ′ ) i = h a in , ex ( t ) a † in , ex ( t ′ ) i = δ ( t − t ′ ) , (8) h a † in , ( t ) a in , ( t ′ ) i = h a † in , ex ( t ) a in , ex ( t ′ ) i = 0 , (9) h b in ( t ) b † in ( t ′ ) i = ( n th + 1) δ ( t − t ′ ) , (10) h b † in ( t ) b in ( t ′ ) i = n th δ ( t − t ′ ) . (11)Here n th is the thermal phonon number given by n th = (cid:16) e ~ ω m k B T − (cid:17) − , (12)where T is the environmental temperature and k B is Boltzmann constant. Note that for mi-crowaves the thermal occupations should also be included in Eq. (8) and (9). Here we focuson optical frequencies and thus the thermal photon number is negligible.Coherent laser input results in the displacements of both the optical and mechanical har-monic oscillators. For convenience, a displacement transformation is applied, i. e., a → a + α , -1n m+1, -1n m , n m +1, n m C DE FA B ...... ... ...
Figure 2: Level diagram of the linearized Hamiltonian (17). | n, m i denotes the state of n pho-tons and m phonons in the displaced frame. The solid (dashed) curves with arrows correspondto the cooling (heating) processes. See text for details. Figures reproduced with permissionfrom Ref. [163] c (cid:13) b → b + β , where α and β are c -numbers, denoting the displacements of the optical and me-chanical modes; a and b are the displaced operators, representing the quantum fluctuationsof the optical and mechanical modes around their classical values. By separating the classicaland quantum components, the quantum Langevin equations are rewritten as˙ α = (cid:0) i ∆ ′ − κ (cid:1) α − i Ω , (13)˙ β = (cid:0) − iω m − γ (cid:1) β − ig | α | , (14)˙ a = (cid:0) i ∆ ′ − κ (cid:1) a − igα ( b + b † ) − iga ( b + b † ) −√ κ ex a in , ex − √ κ a in , , (15)˙ b = (cid:0) − iω m − γ (cid:1) b − ig (cid:16) α ∗ a + αa † (cid:17) − iga † a − √ γb in , (16)where the optomechanical-coupling modified detuning ∆ ′ = ∆ − g ( β + β ∗ ). Under strong drivingcondition, the classical components dominate and the nonlinear terms iga ( b + b † ) and iga † a in Eqs. (15) and (16) can be neglected, respectively. Then we obtain the linearized quantumLangevin equations for a and b , and the corresponding Hamiltonian is given by H L = − ∆ ′ a † a + ω m b † b + ( Ga † + G ∗ a )( b + b † ) , (17)where G = αg is the coherent intracavity field enhanced optomechanical coupling strength.Initially the phonons are in a thermal equilibrium state and the thermal phonon numberis n th . Then the interaction between the photons and the phonons, as described by the lasterm in Eq. (17), leads to the modification of the phonon number. Figure 2 displays thelevel diagram and the coupling routes among different states [163], where | n, m i represents thenumber state with n ( m ) being the photon (phonon) number in the displaced frame. Denotedby the dashed curves, there are three kinds of heating processes: swap heating ( B ), quantumbackaction heating ( D ) and thermal heating ( F ). Thermal heating is an incoherent processarising from the interaction between the mechanical object and the environment. Swap heatingand quantum backaction heating are the accompanying effect when radiation pressure is utilizedto cool the mechanical motion, corresponding to the coherent interaction processes a b † and a † b † , respectively. Swap heating emerges when the system is in the strong coupling regimewhich enables reversible energy exchange between photons and phonons. Meanwhile, quantumbackaction heating can pose a fundamental limit for backaction cooling. The solid curves ( A , C and E ) illustrate cooling processes associated with energy swapping, counter-rotating-waveinteraction and cavity dissipation. In the following we derive the cooling rate and cooling limitsby taking all the above processes. In the weak coupling regime, the optomechanical cooling can be analyzed using the per-turbation theory, where optomechanical coupling is regarded as a perturbation to the opticalfield. The power spectrum of the optical force exerting on the mechanical motion S F F ( ω ) iscalculated with the absence of coupling to the mechanical resonator. Then the cooling (heat-ing) rate is proportional to S F F ( ± ω m ), corresponding to the ability for absorbing (emitting) aphonon by the intracavity field.From Eq. (17) we obtain the optical force acting on the mechanical motion, describedby F = − ( G ∗ a + Ga † ) /x ZPF . The quantum noise spectrum of the optical force is given bythe Fourier transformation of the autocorrelation function S F F ( ω ) ≡ R h F ( t ) F (0) i e iωt dt . Thecalculation is best performed in the frequency domain. In the absence of the optomechanicalcoupling, from Eq. (15) we obtain − iω ˜ a ( ω ) = (cid:16) i ∆ ′ − κ (cid:17) ˜ a ( ω ) −√ κ ex ˜ a in , ex ( ω ) −√ κ ˜ a in , ( ω ) , (18)hich yields ˜ a ( ω ) = √ κ ˜ a in ( ω ) i ( ω + ∆ ′ ) − κ , (19)where ˜ a in ( ω ) = p κ ex /κ ˜ a in , ex ( ω )+ p κ /κ ˜ a in , ( ω ). Using F ( ω ) = − [ G ∗ ˜ a ( ω ) + G ˜ a † ( ω )] /x ZPF , thespectral density of the optical force is obtained as S F F ( ω ) = κ | Gχ ( ω ) | x = | G | x κ [ ω + ∆ ′ ] + κ . (20)The rate for absorbing and emitting a phonon by the cavity field are respectively given by A ∓ = S F F ( ± ω m ) x = | G | κ [ ω m ± ∆ ′ ] + κ . (21)We can also derive the spectral density of the mechanical mode S bb ( ω ) by considering thefull equations (Eq. (15) and (16)) − iω ˜ a ( ω ) = (cid:0) i ∆ ′ − κ (cid:1) ˜ a ( ω ) − iG [˜ b † ( ω ) + ˜ b ( ω )] −√ κ ex ˜ a in , ex ( ω ) −√ κ ˜ a in , ( ω ) , (22) − iω ˜ b ( ω ) = (cid:0) − iω m − γ (cid:1) ˜ b ( ω ) − i [ G ∗ ˜ a ( ω ) + G ˜ a † ( ω )] − √ γ ˜ b in ( ω ) , (23)from which we obtain˜ b ( ω ) ≃ √ γ ˜ b in ( ω ) − i √ κ n G ∗ χ ( ω ) ˜ a in ( ω ) + Gχ ∗ ( − ω ) ˜ a † in ( ω ) o iω − i ( ω m +Σ ( ω )) − γ . (24)where Σ ( ω ) = − i | G | [ χ ( ω ) − χ ∗ ( − ω )] , (25) χ ( ω ) = − i ( ω +∆ ′ )+ κ . (26)In the second step of the derivation we have neglected the terms containing ˜ b † ( ω ), which isnegligible near ω = ω m . Here Σ ( ω ) represents the optomechanical self energy and χ ( ω ) isthe response function of the cavity mode. It shows that the optomechanical coupling leadsto the modification of both the mechanical resonance frequency and the mechanical dampingrate, which are termed as optical spring effect and optical damping effect, respectively. Thefrequency shift δω m and the extra damping Γ opt are given by δω m = ℜ Σ ( ω m ) = | G | ℑ h − i ( ω m +∆ ′ )+ κ − − i ( ω m − ∆ ′ )+ κ i , (27)Γ opt = − ℑ Σ ( ω m ) = 2 | G | ℜ h − i ( ω m +∆ ′ )+ κ − − i ( ω m − ∆ ′ )+ κ i . (28) N o r m a li z ed S FF () / m Figure 3: Normalized optical force spectrum S F F ( ω ) for ∆ ′ = − ω m (red solid curve), 0 (blackdashed curve), and ω m (blue dotted curve). The dashed vertical lines denotes ω/ω m = ± κ = 0 . ω m .The spectral density of the mechanical mode is given by S bb ( ω ) = Z ∞−∞ ˜ b † ( ω ) ˜ b ( ω ′ ) dω ′ = γn th + κ | Gχ ( − ω ) | (cid:12)(cid:12) iω − i ( ω m + Σ( ω )) − γ (cid:12)(cid:12) . (29)Figure 3 plots the normalized S F F ( ω ) for different laser detunings. It shows that reddetuning leads to A − > A + , corresponding to cooling. In this case, typically the opticaldamping rate Γ opt = A − − A + is much larger than the intrinsic mechanical damping rate, andthen the cooling limit is obtained as n f = γn th + A + Γ opt . (30)Note that n cf = γn th / Γ opt is the classical cooling limit while n qf = A + / Γ opt corresponds to thefundamental quantum limit, as the heating rate A + originates from the quantum backaction.This fundamental limit can be simplified as n qf = 4 ( ω m + ∆ ′ ) + κ − ω m ∆ ′ . (31)The minimal cooling limit is given by n qf , min = 12 s κ ω − ! , (32)obtained when ∆ ′ = − p ω + κ / ω m ≪ κ ), the quantum limit is n qf , min = κ/ (4 ω m ) for ∆ ′ = − κ/
2. In this case the minimum phonon number cannot reach 1, whichprecludes ground state cooling. In the resolved sideband limit ( ω m ≫ κ ), the quantum limitis simplified as n qf , min = κ / (16 ω ) for ∆ ′ = − ω m . In this limit ground state can be achieved[158, 159]. For the linear regime under strong driving, the mean phonon number can be computedexactly by employing the quantum master equation and solving a linear system of differentialequations involving all the second-order moments. This approach holds for both weak andstrong coupling regimes.With the linearized Hamiltonian Eq. (17), the quantum master equation reads˙ ρ = i [ ρ, H L ] + κ (cid:16) a ρa † − a † a ρ − ρa † a (cid:17) + γ ( n th + 1) (cid:16) b ρb † − b † b ρ − ρb † b (cid:17) + γ n th (cid:16) b † ρb − b b † ρ − ρb b † (cid:17) , (33)To calculate the mean phonon number, we need to determine the mean values of all the second-order moments, ¯ N a = h a † a i , ¯ N b = h b † b i , h a † b i , h a b i , h a i and h b i [163, 164], which aredetermined by a linear system of ordinary differential equations ddt ¯ N a = − i (cid:16) G h a † b i − G ∗ h a † b i ∗ + G h a b i ∗ − G ∗ h a b i (cid:17) − κ ¯ N a , (34) ddt ¯ N b = − i (cid:16) − G h a † b i + G ∗ h a † b i ∗ + G h a b i ∗ − G ∗ h a b i (cid:17) − γ ¯ N b + γn th , (35) ddt h a † b i = (cid:2) − i (∆ ′ + ω m ) − κ + γ (cid:3) h a † b i − i (cid:0) G ∗ ¯ N a − G ∗ ¯ N b + G h a i ∗ − G ∗ h b i (cid:1) , (36) ddt h a b i = (cid:2) i (∆ ′ − ω m ) − κ + γ (cid:3) h a b i − i (cid:0) G ¯ N a + G ¯ N b + G + G ∗ h a i + G h b i (cid:1) , (37) ddt h a i = (2 i ∆ ′ − κ ) h a i − iG (cid:16) h a b i + h a † b i ∗ (cid:17) , (38) ddt h b i = ( − iω m − γ ) h b i − i (cid:16) G ∗ h a b i + G h a † b i (cid:17) . (39)Note that in the above calculation, cut-off of the density matrix is not necessary and thesolutions are exact.In the stable regime, which requires | G | < − (4∆ ′ + κ ) ω m / (16∆ ′ ) for red detuning ∆ ′ < ′ = − ω m and cooperativity C ≡ | G | / ( γκ ) ≫
1, the finalphonon occupancy reads [163]¯ N std ≃ | G | + κ | G | ( κ + γ ) γn th + 4 ω (cid:0) κ + 8 | G | (cid:1) + κ (cid:0) κ − | G | (cid:1) ω (4 ω + κ − | G | ) . (40)Here the first term, being proportional to the environmental thermal phonon number n th , isthe classical cooling limit; the second term, which does not depend on n th , corresponds to thequantum cooling limit. This quantum limit originates from the quantum backaction, consistingof both dissipation quantum backaction related to the cavity dissipation and interaction quan-tum backaction associated with the optomechanical interaction. In the resolved sideband case,Eq. (40) reduces to ¯ N std ≃ γ (4 | G | + κ )4 | G | ( κ + γ ) n th + κ + 8 | G | ω − | G | ) . (41)In the weak coupling regime, it further reduces to ¯ N wkstd ≃ γn th / (Γ + γ ) + κ / (16 ω ) withΓ = 4 | G | /κ . In the strong coupling regime, ¯ N strstd ≃ γn th / ( κ + γ ) + | G | / [2( ω − | G | )]. Inthis case the classical limit is restricted by the cavity dissipation rate κ , while the interactionquantum backaction limit suffers from high coupling rate | G | .To study the cooling dynamics beyond the steady state, the differential equations need tobe solved to obtain the time evolution of the mean phonon number ¯ N b . For weak coupling,we have ¯ N wk b ≃ n th ( γ + Γ e − Γ t ) / ( γ + Γ) + [ κ / (16 ω )](1 − e − Γ t ) , which shows that the meanphonon number decays exponentially with the cooling rate Γ. This cooling rate is limited bythe coupling strength, since in the cooling route A → E as shown in Fig. 2, the energy flowfrom the mechanical mode to the optical mode (process A ) is slower than the cavity dissipation(process E ).In the strong coupling regime, the time evolution of the mean phonon number is describedby [163] ¯ N str b = ¯ N str b, + ¯ N str b, , ¯ N str b, ≃ n th γ + e − κ + γ t [ κ − γ +( κ + γ ) cos( ω + − ω − ) t ] κ + γ , ¯ N str b, ≃ | G | (cid:20) − e − κ + γ t cos( ω + + ω − ) t cos( ω + − ω − ) t (cid:21) ω − | G | ) , (42)
200 400 60010 -1 -1 G m G m G m G m N b (a) N b t ( -1m ) (b) Figure 4: (a) Time evolution of mean phonon number ¯ N b for G/ω m = 0 . .
01, 0 .
02 and 0 . N b for G/ω m = 0 .
005 and 0 .
01 with a wider time interval. The shadowedregion shows the same time interval with (a). Other parameters: n th = 10 , γ/ω m = 10 − , κ/ω m = 0 .
05. The dotted horizontal lines correspond to the steady-state cooling limits, givenby Eq. (41). Figures reproduced with permission from Ref. [163] c (cid:13) ω ± = p ω ± | G | ω m are the normal eigenmode frequencies. The phonon occupancyexhibits oscillation under an exponentially-decaying envelope and can be divided into two dis-tinguishable parts ¯ N str b, and ¯ N str b, , where the first part originates from energy exchange betweenoptical and mechanical modes, and the second part is induced by quantum backaction. ¯ N str b, reveals Rabi oscillation with frequency ∼ | G | , whereas the envelopes have the same exponen-tial decay rate Γ ′ = ( κ + γ ) / | G | . This is because, in thestrong coupling regime, the cooling route A → E is subjected to the cavity dissipation (process E ), which has slower rate than the energy exchange between phonons and photons (process A ). This saturation prevents a higher cooling speed for stronger coupling. Figures 4(a) and(b) plot the numerical results based on the master equation for various G . It shows that forweak coupling the cooling rate increases rapidly as the coupling strength increases, whereasfor strong coupling the envelope decay no longer increases, instead the oscillation frequencybecomes larger.igure 5: (a) Scanning electron microscope (SEM) image of the cantilever, a doubly clampedfree-standing Bragg mirror (520 µ m long, 120 µ m wide and 2.4 µ m thick) that had beenfabricated by using ultraviolet excimer-laser ablation in combination with a dry-etching process.(b) Layout of the micromirror optical cavity. The microresonator mirror is etched upon a 1cm silicon chip. The coupling mirror of the cavity is a standard low-loss silica mirror. (c)SEM image of a deformed silica microsphere. (d) SEM image of the mechanical system formedby a doubly clamped SiN beam. A circular, high-reflectivity Bragg mirror is used as the endmirror of a FP cavity. Figures reproduced with permission from: (a) Ref. [17] c (cid:13) (cid:13) (cid:13) (cid:13)
3. Recent experimental progresses
Pioneering work of cavity optomechanical cooling dates back to 1960th by Braginsky andcoworkers [6, 7], where they demonstrated the modification of mechanical damping rate asa result of the retarded nature of the cavity-enhanced optical force due to the finite cavityphoton lifetime. In 2006, radiation pressure cooling was realized in three groups using differentoptomechanical systems, including suspended micromirrors [17, 18] (Fig. 5 (a) and (b)) andmicrotoroids [19]. Later in 2008, cooling in the resolved sideband regime was achieved [20].Soon afterwards, with environmental pre-cooling under cryogenic condition, cooling to only afew phonons was demonstrated [21, 22, 23] (Fig. 5 (c) and (d)). Recently, several groups havecooled the mechanical motion close to the quantum ground state both in the microwave domain[24, 25] (Fig. 6) and in the optical domain [26, 27, 40] (Fig. 7).igure 6: (a) SEM image of the Nb-Al-SiN sample. The nanomechanical resonator is 30 µ mlong, 170 nm wide and 140 nm thick, and is formed of 60 nm of stoichiometric, high-stress, low-pressure chemical-vapour-deposition SiN and 80 nm of Al. (b) SEM image of the aluminium(grey) electromechanical circuit fabricated on a sapphire (blue) substrate. A 15- µ m-diametermembrane is suspended 50 nm above a lower electrode. Figures reproduced with permissionfrom: (a) Ref. [24] c (cid:13) (cid:13) .
85 quanta was achieved,with the ground state occupancy probability greater than 50% ( P g = 0 . .
68 GHz. Themechanical Q-factor was 10 , corresponding to an intrinsic mechanical damping rate of 35 kHz.The optical Q-factor was 4 × , and thus the optical damping rate was 500 MHz. By fittingthe measured data of mechanical damping effect, the single-photon optomechanical couplingstrength is determined to be 910 kHz. With 2000 intracavity photons, the minimum meanphonon occupancy was observed to be 0 . ± .
08. The cooling was limited for higher drivepowers, which resulted from the increase of the bath temperature due to optical absorption andthe increase of the intrinsic mechanical damping rate induced by the generation of free carriersthrough optical absorption.igure 7: Top panel: Photonic nanocrystal nanobeam cavity with phononic shield. (a) Scanningelectron microscope (SEM) image of the patterned silicon nanobeam and the external phononicbandgap shield. (b) Enlarged image of the central region of the nanobeam. (c) Simulationof the optical and mechanical modes. (d) Enlarged image of the nanobeam-shield interface.(e) Simulation of the localized acoustic resonance at the nanobeamCshield interface. Bottompanel: Spoke-supported microtoroidal cavity. (a) SEM image of the spoke-anchored toroidalresonator. (b) Sketch of an optical whispering gallery mode. (c) Simulation of the fundamentalradial breathing mechanical mode. Figures reproduced with permission from: Top panel, Ref.[26] c (cid:13) (cid:13) n f = 1 . , and thus the cavity decay rate reaches κ/ π <
10 MHz. The mechanical resonancefrequency for such spoke-anchored toroidal resonator with 31 µ m diameter was 78 MHz. In aHe-3 buffer gas cryostat with 650 mK temperature, the mechanical resonator was pre-cooled to ∼
200 quanta. Through optomechanical cooling, the final mean phonon occupancy was reducedto 1 . ± .
1. Further cooling was limited by the laser reheating of the sample and the onset ofnormal modes. For the latter, cooling in the strong coupling regime was limited by the swapeating (see the next section). In this experiment, besides cooling, they also demonstrated thequantum-coherent coupling between the mechanical mode and the optical mode.
4. New cooling approaches
The current cooling approach as shown in Sec. 2 has achieved great successes, while thereare still some major challenges. First, saturation effect appears in the strong optomechanicalcoupling regime as a result of swap heating. Secondly, to achieve ground state cooling, it requiresthe resolve sideband condition, which is stringent for many cavity optomechanical systems. Inthis section we review recent cooling approaches to improve the cooling performance along thesedirections
Recent experiments have reached the regime of strong optomechanical coupling, which iscrucial for coherent quantum optomechanical manipulations. However, as mentioned in Sec.2, strongly-coupled optomechanical cooling has predicted only limited improvement over weakcoupling due to the saturation effect of the steady-state cooling rate. In Ref. [163], Liu et al.proposed to dynamically tailor the cooling and heating processes by exploiting the modulationof cavity dissipation. In this proposal, the internal cavity dissipation is abruptly increased eachtime when the Rabi oscillation reaches a minimum-phonon state. At this time the system hastransited from state | n, m i to state | n + 1 , m − i (Fig. 2). Once a strong dissipation pulse isapplied to the cavity so that the process E dominates, the system will irreversibly transit fromstate | n + 1 , m − i to state | n, m − i . The dissipation pulse has essentially behaves as a switchto halt the reversible Rabi oscillation, resulting in the suppression of the swap heating. Suchdissipative cooling is verified in Figs. 8 (a) and (b), which plot the modulation scheme and thecorresponding time evolution of mean phonon number ¯ N b . At the end of the first half Rabioscillation cycle, t ∼ π/ (2 | G | ), a dissipation pulse is applied. After that, the phonon numberreaches and remains near the steady-state limit. Without modulation (blue dashed curve), thesteady-state cooling limit is reached only after t ≃ /ω m ; while with the modulation (redsolid curve), it only takes t ≃ /ω m to cool below the same limit, corresponding to 50 times
10 20 30 40 350 40010 -1 -1 Steady-state cooling limit N b (b)Single dissipation pulse (a) ( t ) m (c) ( t ) / ( ) Steady-state cooling limitsInstantaneous-state cooling limit ONON (d) N b t ( -1m ) Modulation OFF
Figure 8: (a) Modulation scheme of the cavity dissipation rate κ ( t ) and (b) the correspondingtime evolution of mean phonon number ¯ N b with (red solid curve) and without (blue dashedcurve) modulation for G/ω m = 0 . κ/ω m = 0 .
05. (c) Modulation scheme of κ ( t ) /κ (0)and (d) the corresponding ¯ N b for G/ω m = 0 . κ (0) /ω m = 0 .
01 (red solid curve) and 0 . κ (0); the dash-dottedline denotes the instantaneous-state cooling limit independent of κ (0), given by Eq. (43); the“ON” and “OFF” regions corresponds that the modulation is turned on and off, respectively;the vertical coordinate range from 10 to 10 is not shown. Other parameters: n th = 10 , γ/ω m = 10 − . Figures reproduced with permission from Ref. [163] c (cid:13) N ins ≃ πγn th | G | + π | G | ( ω − | G | )( ω − | G | ) . (43)ere the first term comes from ¯ N str b, for t ≃ π/ (2 | G | ), which shows a πκ/ (4 | G | ) times reductionof classical steady-state cooling limit. The second term of ∼ π | G | /ω , obtained from ¯ N str b, when t ≃ π/ω m , reveals that the second order term of | G | /ω m in quantum backaction has beenremoved, leaving only the higher-order terms. It is also demonstrated in Fig. 8 (c) and (d) thatthe modulation is switchable. If the modulation is turned on (“ON” region), the system willreach the instantaneous-state cooling limit; if the modulation is turned off (“OFF” region), thesystem transits back to the steady-state cooling limit.Under frequency matching condition ( ω + + ω − ) / ( ω + − ω − ) = k ( k = 3 , t ≃ π/ (2 | G | ),the optimized instantaneous-state cooling limit is obtained as [163]¯ N optins ≃ πκ | G | " γn th κ + | G | ω − | G | ) , (44)which reduces both the classical and quantum steady-state cooling limits by a factor of πκ/ (4 | G | ).This reduction is significant when the system is in the deep strong coupling regime. Typically,the cooling limits can be reduced by a few orders of magnitude. For example, when G/ω m = 0 . κ/ω m = 0 . N std = 3 .
4, while ¯ N optins = 0 .
03, corresponding to more than 100times of phonon number suppression.
To loosen the stringent resolved sideband condition, a few approaches have been proposed,which can be divided into the following three directions: novel coupling mechanisms, parametermodulations and hybrid systems.
The first direction is searching for novel optomechanical coupling mechanism, for instance,dissipative coupling, where the mechanical motion couples to the cavity decay rate instead ofthe cavity resonance frequency [166]. In the system described in Fig. 1, the displacement of themechanical object couples to the cavity resonance frequency ω ( x ), which is sometimes termeddispersive coupling. For the dissipative coupling, the displacement of the mechanical objectcouples to the cavity decay rate κ ( x ). It was predicted [166] that this can yield novel coolingbehavior, capable of reaching the quantum ground state without the resolved sideband limit.igure 9: (a) Schematics of the microdisk-waveguide optomechanical system for dissipativecoupling. (b) SEM image of the fabricated device. (c) Schematics of the three-mirror system.The movable mirror with two perfectly reflecting surfaces is placed inside a driven cavity withtwo transmissive fixed mirrors. Figures reproduced with permission from: (a)-(b) Ref. [168] c (cid:13) (cid:13) .2.2. Parameter modulations The second direction is to introduce modulations of the system parameters, such as inputlaser intensity [173, 174, 175], mechanical resonance frequency [176] and other parameters [177].In Ref. [174, 175], optimal control method were introduced, allowing ultra-efficient cooling viapulsed laser inputs. The idea is to use interference between optical pulses incident on thesystem, where a sequence of fast pulses adds a term to the effective optomechanical interactionHamiltonian. In the usual continuous driving case, the interaction term has the form x m x c ,while pulsed laser input generates an effective interaction term with the form p m p c , where x m and p m ( x c and p c ) are the quadrature operators of the mechanical (optical) mode. Byoptimizing the pulse duration time, the total effective interaction is described by the beam-splitter Hamiltonian x m x c + p m p c ∝ ab † + a † b . As a result, the counter-rotating-wave term iseliminated, avoiding the quantum backaction heating.In Ref. [176], Li et al. propose a ground state cooling scheme by taking advantage of amechanical resonator with time-dependent frequency, using a three-mirror system (Fig. 9 (c)).In this scheme, strong laser input is used to generate optical spring effect, where the effectiveresonance frequency of the mechanical mode is determined by the optical driving. Fast groundstate cooling can be achieved by designing the trajectory of the effective frequency from theinitial time to the final time.Liao and Law have investigated the cooling performance using chirped-pulse coupling bymodulating the input laser intensity and phase [178]. In this scheme, owing to the frequencymodulation in chirped pulses, cooling can be realized without the need for high-precision controlof the laser detuning and pulse areas. The third direction is to construct hybrid systems, for example, couple atoms to the opticalcavity. In Ref. [179], Genes et al. have described a hybrid system by coupling a two-levelensemble to the cavity optomechanical system (Fig. 10 (a) and (b)). The two-level ensemblecouples to the cavity mode, creating intracavity narrow bandwidth loss or gain. It inducestailored asymmetric structure of the cavity noise spectrum interacting with the mechanicalmode. As a result, This allows cooling via inhibition of the Stokes-scattering process or en-igure 10: (a) Couple two-level atoms to the optical cavity with a movable mirror. Theatoms are in the ground states. (b) Same as (a) but the atoms are in the excited states.(c) Micromechanical membrane in a cavity coupled to a distant atomic ensemble. Figuresreproduced with permission from: (a)-(b) Ref. [179] c (cid:13) (cid:13)
5. Summary and outlook
In summary, we have reviewed the quantum theory, recent experiments and new directionsof cavity optomechanical cooling. Particularly, we summarize the new cooling scheme alongtwo directions: cooling in the strong coupling regime and cooling beyond the resolved sidebandlimit. Novel cooling schemes for efficiently suppressing the thermal noise are still being explored.These schemes add complexity to the current experimental systems, and thus efforts should betaken to demonstrate these schemes in real experiments, which would be possible in the nearfuture. There are more appealing challenges such as cooling massive mechanical objects up tokilograms and room-temperature ground state cooling of mechanical resonators.With currently rapid experimental and theoretical advances in cavity optomechanics, itopens up new avenues to the foundations of quantum physics and applications. Quantummanipulation of macroscopic/mesoscopic mechanical objects provides a direct method to testfundamental quantum theory in a hitherto unachieved parameter regime. For example, micromechanical structures consist of typically 10 atoms and weigh 10 − kilograms, while grav-itational wave detectors comprise more than 10 atoms and weigh up to several kilogram.For applications, cavity optomechanics provides new aspects for measurement with high pre-cision and for sensing with high sensitivity. Particularly, cavity optomechanics offers a newarchitecture for solid-state realization of quantum information processing. The developmentof cavity optomechanical cooling will enable quantum manipulation of mechanical objects andgeneration of non-classical mechanical states, which will provide unique resources for quantumcommunication and quantum computation. References [1] H¨ansch T W and Schawlow A L 1975
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