Cavity Optomechanics
Abstract
We review the field of cavity optomechanics, which explores the interaction between electromagnetic radiation and nano or micromechanical motion. This review covers the basics of optical cavities and mechanical resonators, their mutual optomechanical interaction mediated by the radiation pressure force, the large variety of experimental systems which exhibit this interaction, optical measurements of mechanical motion, dynamical backaction amplification and cooling, nonlinear dynamics, multimode optomechanics, and proposals for future cavity quantum optomechanics experiments. In addition, we describe the perspectives for fundamental quantum physics and for possible applications of optomechanical devices.
Contents
 I Introduction
 II Optical cavities and mechanical resonators
 III Principles of optomechanical coupling
 IV Experimental realizations and optomechanical parameters
 V Basic consequences of the optomechanical interaction
 VI Quantum optical measurements of mechanical motion
 VII Optomechanical cooling
 VIII Classical Nonlinear Dynamics
 IX Multimode optomechanics
 X Quantum Optomechanics
 XI Future Perspectives
 XII Acknowledgements
 XIII Appendix: Experimental Challenges
I Introduction
Light carries momentum which gives rise to radiation pressure forces. These forces were already postulated in the 17 century by Kepler, who noted that the dust tails of comets point away from the sun during a comet transit Kepler (1619). The first unambiguous experimental demonstrations of the radiation pressure force predicted by Maxwell were performed using a light mill configuration Nichols and Hull (1901); Lebedew (1901). A careful analysis of these experiments was required to distinguish the phenomenon from thermal effects that had dominated earlier observations. As early as 1909, Einstein derived the statistics of the radiation pressure force fluctuations acting on a moveable mirror Einstein (1909), including the frictional effects of the radiation force, and this analysis allowed him to reveal the dual waveparticle nature of blackbody radiation. In pioneering experiments, both the linear and angular momentum transfer of photons to atoms and macroscopic objects were demonstrated by Frisch Frisch (1933) and by Beth Beth (1936), respectively.
In the 1970s Arthur Ashkin demonstrated that focused lasers beams can be used to trap and control dielectric particles, which also included feedback cooling Ashkin (1978, 2006). The nonconservative nature of the radiation pressure force and the resulting possibility to use it for cooling atomic motion was first pointed out by Hänsch and Schawlow and by Dehmelt and Wineland Hänsch and Schawlow (1975); Wineland and Dehmelt (1975). Laser cooling was subsequently realized experimentally in the 1980s and has become since then an extraordinarily important technique Stenholm (1986). It has, for example, allowed cooling of ions to their motional ground state and it is the underlying resource for ultracold atom experiments. Many applications have been enabled by laser cooling Metcalf and van der Straten (1999), including optical atomic clocks, precision measurements of the gravitational field, and systematic studies of quantum manybody physics in trapped clouds of atoms Bloch and Zwerger (2008).
The role of radiation pressure and its ability to provide cooling for larger objects was already investigated earlier by Braginsky in the context of interferometers. Braginsky considered the dynamical influence of radiation pressure on a harmonically suspended endmirror of a cavity. His analysis revealed that the retarded nature of the force, due to the finite cavity lifetime, provides either damping or antidamping of mechanical motion, two effects that he was able to demonstrate in pioneering experiments using a microwave cavity Braginsky and Manukin (1967); Braginsky et al. (1970). In later experiments, these phenomena were also observed in microwavecoupled scale mechanical resonators Cuthbertson et al. (1996). Independently, similar physics was explored theoretically for solidstate vibrations Dykman (1978). In the optical domain, the first cavity optomechanical experiment Dorsel et al. (1983) demonstrated bistability of the radiation pressure force acting on a macroscopic endmirror.
Braginsky also addressed the fundamental consequences of the quantum fluctuations of radiation pressure and demonstrated that they impose a limit on how accurately the position of a free test mass (e.g. a mirror) can be measured Braginsky and Manukin (1977); Braginsky and Khalili (1995). A detailed analysis by Caves clarified the role of this ponderomotive quantum noise in interferometers Caves (1980). These works established the standard quantum limit for continuous position detection, which is essential for gravitational wave detectors such as LIGO or VIRGO.
During the 1990s, several aspects of quantum cavity optomechanical systems started to be explored theoretically. These include squeezing of light Fabre et al. (1994); Mancini and Tombesi (1994) and quantum nondemolition (QND) detection of the light intensity Jacobs et al. (1994); Pinard et al. (1995), which exploit the effective Kerr nonlinearity generated by the optomechanical interaction. It was also pointed out that for extremely strong optomechanical coupling the resulting quantum nonlinearities could give rise to nonclassical and entangled states of the light field and the mechanics Mancini et al. (1997); Bose et al. (1997). Furthermore, feedback cooling by radiation pressure was suggested Mancini et al. (1998). Around the same time, in a parallel development, cavityassisted lasercooling was proposed as a method to cool the motion of atoms and molecules that lack closed internal transitions Hechenblaikner et al. (1998); Vuletic and Chu (2000).
On the experimental side, optical feedback cooling based on the radiation pressure force was first demonstrated in Cohadon et al. (1999) for the vibrational modes of a macroscopic endmirror. This approach was later taken to much lower temperatures Kleckner and Bouwmeester (2006); Poggio et al. (2007). At the same time, there was a trend to miniaturize the mechanical element. For example, the thermal motion of a scale mirror was monitored in a cryogenic optical cavity Tittonen et al. (1999). Producing highquality optical FabryPerot cavities below that scale, however, turned out to be very challenging. In spite of this, it was still possible to observe optomechanical effects of retarded radiation forces in microscale setups where the forces were of photothermal origin, effectively replacing the cavity lifetime with a thermal time constant. Examples include demonstration of the optical spring effect Vogel et al. (2003), feedback damping Mertz et al. (1993), selfinduced oscillations Zalalutdinov et al. (2001); Höhberger and Karrai (2004), and cavity cooling due to the dynamical backaction of retarded photothermal light forces Höhberger Metzger and Karrai (2004).
Yet, for future applications in quantum coherent optomechanics it is highly desirable to be able to exploit the nondissipative radiation pressure force. Both the advent of optical microcavities and of advanced nanofabrication techniques eventually allowed to enter this regime. In 2005 it was discovered that optical microtoroid resonators with their high optical finesse at the same time contain mechanical modes and thus are able to display optomechanical effects, in particular radiationpressure induced selfoscillations Rokhsari et al. (2005); Carmon et al. (2005); Kippenberg et al. (2005) (i.e. the effect Braginsky termed ‘‘parametric instability’’^{1}^{1}1Braginsky called the process of dynamical backaction amplification and the concomitant selfinduced coherent oscillations “parametric oscillatory instability”, as this effect is undesirable in gravitational wave interferometers which were the basis of his analysis.). In 2006 three different teams demonstrated radiationpressure cavity cooling, for suspended micromirrors Gigan et al. (2006); Arcizet et al. (2006a) and for microtoroids Schliesser et al. (2006). Since then, cavity optomechanics has advanced rapidly and optomechanical coupling has been reported in numerous novel systems. These include membranes Thompson et al. (2008) and nanorods Favero et al. (2009) inside FabryPerot resonators, whispering gallery microdisks Wiederhecker et al. (2009); Jiang et al. (2009) and microspheres Ma et al. (2007); Park and Wang (2009); Tomes and Carmon (2009), photonic crystals Eichenfield et al. (2009a, b), and evanescently coupled nanobeams Anetsberger et al. (2009). In addition, cavity optomechanics has been demonstrated for the mechanical excitations of cold atom clouds Murch et al. (2008); Brennecke et al. (2008). Optomechanical interactions are also present in optical waveguides  as first studied and observed in the context of squeezing, where the confined mechanical modes of fibers lead to Guided Acoustic Wave scattering Shelby et al. (1985). Nowadays there are a number of systems where such optomechanical interactions are explored in the absence of a cavity, such as waveguides in photonic circuits or photonic crystal fibres, see e.g. Li et al. (2008); Kang et al. (2009). These setups lie somewhat outside the scope of the concepts presented in this review, but we emphasize that they are very promising for applications due to their large bandwidth.
Optomechanical coupling has also been realized using microfabricated superconducting resonators, by embedding a nanomechanical beam inside a superconducting transmission line microwave cavity Regal et al. (2008) or by incorporating a flexible aluminum membrane into a lumped element superconducting resonator Teufel et al. (2011b). In these systems the mechanical motion capacitively couples to the microwave cavity. This approach ties cavity optomechanics to an independent development that has been garnering momentum since the late 1990s, which is concerned with measuring and controlling the motion of nano and micromechanical oscillators using electrical and other nonoptical coupling techniques. Examples include coupling of mechanical oscillators to single electron transistors Cleland et al. (2002); LaHaye et al. (2004a); Naik et al. (2006) or a quantum point contact Cleland et al. (2002); FlowersJacobs et al. (2007). Besides a wealth of possible applications for such devices in sensitive detection Cleland and Roukes (1998); Rugar et al. (2004); LaHaye et al. (2004b), these methods provide the possiblility of realizing mechanical quantum devices Schwab and Roukes (2005); Blencowe (2005); Ekinci and Roukes (2005) by direct interaction with twolevel quantum systems Cleland and Geller (2004); Rugar et al. (2004); WilsonRae et al. (2004); LaHaye et al. (2009); O’Connell et al. (2010); Arcizet et al. (2011); Kolkowitz et al. (2012). For recent comprehensive general reviews of nanomechanical systems (in particular electromechanical devices), we refer the reader to Blencowe (2005); Poot and van der Zant (2012); Greenberg et al. (2012).
It should be noted that in atomic systems quantum coherent control of mechanical motion is state of the art since early pioneering experiments with trapped ions – for reviews see Leibfried et al. (2003); Blatt and Wineland (2008); Jost et al. (2009). In fact, quantum information processing in these systems relies on using the quantum states of motion to mediate interactions between distant atomic spins Cirac and Zoller (1995). In contrast, the fabricated nano and micromechanical structures that form the subject of this review will extend this level of control to a different realm, of objects with large masses and of devices with a great flexibility in design and the possibility to integrate them in onchip architectures.
There are several different motivations that drive the rapidly growing interest into cavity optomechanics. On the one side, there is the highly sensitive optical detection of small forces, displacements, masses, and accelerations. On the other hand, cavity quantum optomechanics promises to manipulate and detect mechanical motion in the quantum regime using light, creating nonclassical states of light and mechanical motion. These tools will form the basis for applications in quantum information processing, where optomechanical devices could serve as coherent lightmatter interfaces, for example to interconvert information stored in solidstate qubits into flying photonic qubits. Another example is the ability to build hybrid quantum devices that combine otherwise incompatible degrees of freedoms of different physical systems. At the same time, it offers a route towards fundamental tests of quantum mechanics in an hitherto unaccessible parameter regime of size and mass.
A number of reviews Kippenberg and Vahala (2007, 2008); Marquardt and Girvin (2009); Favero and Karrai (2009); Genes et al. (2009a); Aspelmeyer et al. (2010); Schliesser and Kippenberg (2010); Cole and Aspelmeyer (2012); Aspelmeyer et al. (2012); Meystre (2012) and brief commentary papers Karrai (2006); Cleland (2009); Marquardt (2011); Cole and Aspelmeyer (2011) on cavity optomechanics have been published during the past few years, and the topic has also been discussed as part of a larger reviews on nanomechanical systems Poot and van der Zant (2012); Greenberg et al. (2012). Here we aim for a comprehensive treatment that incorporates the most recent advances and points the way towards future challenges.
This review is organized follows: We first discuss optical cavities, mechanical resonators, the basic optomechanical interaction between them and the large range of experimental setups and parameters that are now available. We then go on to derive the basic consequences of the interaction (such as optomechanical damping and the optical spring effect), describe various measurement schemes, and present the quantum theory of optomechanical cooling. After studying nonlinear effects in the classical regime, we address multimode setups and the wide field of proposed applications in the quantum domain, before concluding with an outlook.
Symbol  Meaning 

Mechanical frequency  
Mechanical damping rate  
Mechanical quality factor,  
Laser detuning from the cavity resonance,  
Overall cavity intensity decay rate, from input coupling and intrinsic losses, .  
Optomechanical singlephoton coupling strength, in  
Lightenhanced optomechanical coupling for the linearized regime,  
Optical frequency shift per displacement,  
Mechanical zeropoint fluctuation amplitude,  
Photon annihilation operator, with the circulating photon number  
Phonon annihilation operator, with the phonon number  
Average number of phonons stored in the mechanical resonator,  
Average phonon number in thermal equilibrium,  
Photon number circulating inside the cavity,  
Optical susceptibility of the cavity,  
Mechanical susceptibility,  
Quantum noise spectrum, (Sec. II.2.3)  
(Symmetrized) mechanical zeropoint fluctuations,  
: standard quantum limit result for added noise in displacement measurement (Sec. VI.1)  
Optomechanical damping rate (Sec. V.2.2): max. for  
Optical spring (mechanical frequency shift, Sec. V.2.1): for  
Minimum reachable phonon number in lasercooling, for 
Ii Optical cavities and mechanical resonators
In this section we recall the basic aspects of optical cavities and of mechanical resonators, as needed to describe cavity optomechanical systems. Much more about these topics can be found in standard textbooks on quantum optics, e.g. Walls and Milburn (1994), and on nanomechanical systems Cleland (2003).
ii.1 Optical resonators
Optical resonators can be realized experimentally in a multitude of forms of which several types will be discussed later in the review. Here we give a unifying description of the optical properties and provide the mathematical description of a cavity that is pumped with a single monochromatic laser source.
ii.1.1 Basic properties
We first consider the classical response of a simple FabryPerot resonator, which will allow to introduce the relevant parameters to characterize an optical cavity. A FabryPerot resonator or etalon consisting of two highly reflective mirrors, separated by a distance , contains a series of resonances which are given by the angular frequency Here is the integer mode number. The separation of two longitudinal resonances is denoted as the free spectral range (FSR) of the cavity:
(1) 
In the following, we will almost always focus on a single optical mode, whose frequency we will denote .
Both the finite mirror transparencies and the internal absorption or scattering out of the cavity lead to a finite photon (intensity) cavity decay rate^{2}^{2}2In this review we shall use for the photon (energy) decay rate, such that the amplitude decay rate is given by . In some papers the latter is denoted as . .
A further useful quantity is the optical finesse, , which gives the average number of roundtrips before a photon leaves the cavity:
(2) 
The optical finesse is a useful parameter as it gives the enhancement of the circulating power over the power that is coupled into the resonator. Alternatively, we can introduce the quality factor of the optical resonator,
(3) 
where is the photon lifetime. Note that the quality factor is also used to characterize the damping rate of mechanical resonators (see below). Generally speaking, the cavity decay rate can have two contributions, one from losses that are associated with the (useful) input (and output) coupling and a second contribution from the internal losses. It is useful to differentiate these two contributions. For the case of a highQ cavity, the total cavity loss rate can be written as the sum of the individual contributions:
Here, refers to the loss rate associated with the input coupling, and refers to the remaining loss rate. For example, in the case of a waveguide coupled to a microtoroidal or microsphere resonator, is the loss rate associated with the waveguideresonator interface and describes the light absorption inside the resonator. For the case of a FabryPerot cavity, is the loss rate at the input cavity mirror and summarizes the loss rate inside the cavity, including transmission losses at the second cavity mirror as well as all scattering and absorption losses behind the first mirror. Note that by splitting the total decay rate into these two contributions, we are assuming that the photons going into the decay channel will not be recorded. More generally, one could distinguish between more decay channels (e.g. input mirror, output mirror, absorption).
ii.1.2 Inputoutput formalism for an optical cavity
A quantum mechanical description of a cavity that is coupled to the outside electromagnetic environment can be given either via master equations (if only the internal dynamics is of interest) or via a framework known as inputoutput theory, if one also wants to access the light field being emitted by (or reflected from) the cavity. Inputoutput theory allows us to directly model the quantum fluctuations injected from any coupling port (such as the input mirror) into the cavity. In addition, it takes into account any coherent laser drive that may be present. For more details beyond the brief discussion provided below, see e.g. Gardiner and Zoller (2004); Clerk et al. (2010a).
Inputoutput theory is formulated on the level of Heisenberg equations of motion, describing the timeevolution of the field amplitude inside the cavity. One finds that the amplitude experiences decay at a rate . At the same time, its fluctuations are constantly replenished via the quantum noise entering through the various ports of the cavity. In the present case, we distinguish between the channels associated with the input coupling (decay rate ) and the other loss processes (overall decay rate , including loss through the second mirror). The equation of motion reads:
(4) 
In the classical case, would be replaced by a properly normalized complex amplitude of the electric field of the cavity mode under consideration. Indeed, the classical version of this equation (and the following ones) can be obtained by simply taking the average, such that . We have chosen a frame rotating with the laser frequency , i.e. and have introduced the laser detuning with respect to the cavity mode (see also Sec. III.2). Note that a similar equation can also be written down for the mechanical oscillator in order to describe its dissipation and the associated noise force, comprising quantum and thermal contributions (see Sec. III.3).
The input field should be thought of as a stochastic quantum field. In the simplest case, it represents the fluctuating vacuum electric field coupling to the cavity at time , plus a coherent laser drive. However, the same formalism can also be used to describe squeezed states and other more complex field states. The field is normalized in such a way that
is the input power launched into the cavity, i. e. is the rate of photons arriving at the cavity. The same kind of description holds for the “unwanted” channel associated with .
According to the inputoutput theory of open quantum systems, the field that is reflected from the Fabry Perot resonator (or coupled back into the coupling waveguide) is given by:
(5) 
Note that this inputoutput relation describes correctly the field reflected from the input mirror of a FabryPerot resonator. The above equation describes also the transmitted pump field of an evanescently coupled unidirectional waveguide resonator system, such as a whispering gallery mode resonator coupled to a waveguide Cai et al. (2000). In this case the above expression would yield the transmitted pump field.
We still have to consider the case of a twosided cavity, e.g. a twosided Fabry Perot cavity. Other examples in this review include a waveguide coupled to superconducting stripline cavities or fibertaper coupled photonic crystal defect cavities. In these cases there are both transmitted and reflected fields. In all of these cases there are two options for the description. If the field transmitted through the second mirror is not of interest to the analysis, one may lump the effects of that mirror into the decay rate , which now represents both internal losses and output coupling through the second mirror. If, however, the field is important, it should be represented by an additional term of the type in Eq. (4). Then an equation analogous to Eq. (5) will hold for the output field at that second mirror.
In the following, we will not be concerned with noise properties, but focus instead on classical average quantities (for a singlesided cavity), taking the average of Eqs. (4) and (5).
We can solve the equation (4) first for the steadystate amplitude in the presence of a monochromatic laser drive whose amplitude is given by . Noting that we obtain:
(6) 
The expression linking the input field to the intracavity field will be referred to as the optical susceptibility,
Thus, the steadystate cavity population , i.e the average number of photons circulating inside the cavity, is given by:
(7) 
were is the input power launched into the cavity. The reflection or transmission amplitude (for the case of a FabryPerot cavity or a waveguidecoupled resonator, respectively) can be calculated by inserting Eq. (6) into Eq. (5). Using the symbol for the reflection amplitude in the sense of figure 2 case (b), we obtain:
(8) 
The square of this amplitude gives the probability of reflection from the cavity (for FabryPerot) or transmission in the case of a unidirectional waveguide resonator system. From this expression, several regimes can be differentiated. If the external coupling dominates the cavity losses (, the cavity is called “overcoupled”. In that case and the pump photons emerge from the cavity without having been absorbed or lost via the second mirror (a property that is important as discussed below in the context of quantum limited detection). The case where refers to the situation of “critical coupling”. In this case, on resonance. This implies the input power is either fully dissipated within the resonator or fully transmitted through the second mirror (in the case of a FabryPerot cavity with denoting the decay through the second mirror). The situation is referred to as “undercoupling” and is associated with cavity losses dominated by intrinsic losses. For many experiments this coupling condition is not advantageous, as it leads to an effective loss of information.
The physical meaning of reflection (or transmission) depends sensitively on the experimental realization under consideration. One can distinguish four scenarios, which are outlined in the figure 2.
ii.2 Mechanical resonators
ii.2.1 Mechanical normal modes
The vibrational modes of any object can be calculated by solving the equations of the linear theory of elasticity under the appropriate boundary conditions that are determined by the geometry^{3}^{3}3A powerful simulation approach in this context are finite element (FEM) simulations. Cleland (2003). This eigenvalue problem yields a set of normal modes and corresponding eigenfrequencies . The mechanical displacement patterns associated with mechanical motion are given by the strain field where designates the normal mode.
For the purposes of this review, we will mostly focus on a single normal mode of vibration of frequency (where ’’ stands for ’mechanical’), assuming that the mode spectrum is sufficiently sparse such that there is no spectral overlap with other mechanical modes. The loss of mechanical energy is described by the (energy) damping rate which is related to the mechanical quality factor^{4}^{4}4In the context of mechanical dissipation often the loss tangent is quoted, its relation to the quality factor being . by If one is interested in the equation of motion for the global amplitude of the motion, one can utilize a suitably normalized (see below) dimensionless mode function , such that the displacement field would be . Then the temporal evolution of can be described by the canonical simple equation of motion of a harmonic oscillator of effective mass :
(9) 
Here denotes the sum of all forces that are acting on the mechanical oscillator. In the absence of any external forces, it is given by the thermal Langevin force (see Sec. II.2.3). In the above equation the (energy) damping rate has been assumed to be frequency independent. Deviations of this model are treated for example in Saulson (1990).
A brief remark about the effective mass is necessary at this point Cleland (2003); Pinard et al. (1999). The normalization that has been chosen for the mode function affects the normalization of . However, it will always be true that the potential energy is given by . This value can then be compared to the expression for the potential energy that arises from a calculation according to the theory of elasticity. Demanding them to be equal yields the correct value for the effective mass (which therefore is seen to depend on the normalization that was chosen for the mode function). Of course, for the simple case of a centerofmass oscillation of a solid object, a natural definition of is the center of mass displacement in which case the effective mass will be on the order of the total mass of the object. A treatment of effective mass in optomechanical experiments is found in Pinard et al. (1999).
Eq. (9) can be solved easily, which is best done in frequency space. We introduce the Fourier transform via . Then defines the susceptibility , connecting the external force to the response of the coordinate:
(10) 
The low frequency response is given by where is the spring constant^{5}^{5}5To describe the response of a high Q oscillator near resonance one can approximate by a Lorentzian, i.e. using yields .
The quantum mechanical treatment of the mechanical harmonic oscillator leads to the Hamiltonian
Here the phonon creation () and annihilation () operators have been introduced, with
where
is the zeropoint fluctuation amplitude of the mechanical oscillator, i.e. the spread of the coordinate in the groundstate: , and where denotes the mechanical vacuum state. The position and momentum satisfy the commutator relation . The quantity is the phonon number operator, whose average is denoted by . In the following, we will typically not display explicitly the contribution of the zeropoint energy to the energy of the oscillator.
We briefly discuss the effect of dissipation. If the mechanical oscillator is coupled to a high temperature bath, the average phonon number will evolve according to the expression:
For an oscillator which is initially in the ground state, this implies a simple time dependence of the occupation according to , where is the average phonon number of the environment. Consequently, the rate at which the mechanical oscillator heats out of the ground state is given by:
The latter is often referred to as the thermal decoherence rate, and given by the inverse time it takes for one quantum to enter from the environment. In the above expression the high temperature limit has been taken, i.e. This expression shows that to attain low decoherence a high mechanical Q factor and a low temperature bath are important. The change of population of a certain Fock state can be described within the framework of the Master equation approach. This approach allows to calculate the decoherence rate of other quantum states such as a Fock state . The latter is given by (see e.g.Gardiner and Zoller (2004)):
revealing that higher Fock states exhibit a progressively higher rate of decoherence.
ii.2.2 Mechanical dissipation
The loss of mechanical excitations, i.e. phonons, is quantified by the energy dissipation rate . The origins of mechanical dissipation have been intensively studied over the last decades and comprehensive reviews are found for example in Cleland (2003); Ekinci and Roukes (2005). The most relevant loss mechanisms include:

clamping losses, which are due to the radiation of elastic waves into the substrate through the supports of the oscillator Wang et al. (2000); Cross and Lifshitz (2001); Mattila (2002); Park and Park (2004); Photiadis and Judge (2004); Clark et al. (2005); Bindel and Govindjee (2005); Judge et al. (2007); WilsonRae (2008); Anetsberger et al. (2008); Eichenfield et al. (2009b); Cole et al. (2011); Jöckel et al. (2011);

materialsinduced losses, which are caused by the relaxation of intrinsic or extrinsic defect states in the bulk or surface of the resonator Yasumura et al. (2000); Mohanty et al. (2002); Southworth et al. (2009); Venkatesan et al. (2010); Unterreithmeier et al. (2010). Such losses have been successfully described by a phenomenological model involving two level defect states, which are coupled to the strain via the deformation potential Anderson et al. (1972); Phillips (1987); Tielbürger et al. (1992); Hunklinger et al. (1973); Seoánez et al. (2008); Remus et al. (2009). In the context of nano and micromechanical oscillators the two level fluctuator damping has been revisited Seoánez et al. (2008); Remus et al. (2009).
The various dissipation processes contribute independently to the overall mechanical losses and hence add up incoherently. The resulting mechanical quality factor is given by labels the different loss mechanisms. , where
Another helpful quantity is the socalled “” product, which plays an important role in the phase noise performance of oscillators. In the context of optomechanics, it quantifies the decoupling of the mechanical resonator from a thermal environment. Specifically,
ii.2.3 Susceptibility, noise spectra and fluctuation dissipation theorem
If one measures the motion of a single harmonic oscillator in thermal equilibrium, one will observe a trajectory oscillating at the eigenfrequency . However, due to the influence of both mechanical damping and the fluctuating thermal Langevin force, these oscillations will have a randomly timevarying amplitude and phase. Both amplitude and phase change on the time scale given by the damping time . Such realtime measurements have been performed in optomechanical systems Hadjar et al. (1999) (see Fig. 4).
In experiments, the mechanical motion is often not analyzed in realtime but instead as a noise spectrum in frequency space. This allows to easily separate the contributions from different normal modes. We briefly recapitulate the relevant concepts. Given one particular realization of the trajectory obtained during a measurement time , we define the gated Fourier transform over a finite time interval :
(11) 
Averaging over independent experimental runs, we obtain the spectral density . In the limit , the WienerKhinchin theorem connects this to the Fourier transform of the autocorrelation function, also called the noise power spectral density:
(12) 
Here we have defined:
(13) 
The only assumption which has been made is that is a stationary random process. From Eqs. (12,13), we immediately obtain the important result that the area under the experimentally measured mechanical noise spectrum yields the variance of the mechanical displacement, :
(14) 
Furthermore, in thermal equilibrium, the fluctuationdissipation theorem (FDT) relates the noise to the dissipative part of the linear response,
(15) 
where denotes the mechanical susceptibility introduced above and we have treated the hightemperature (classical) case. For weak damping (), this gives rise to Lorentzian peaks of width in the noise spectrum, located at (see Fig. 5). Integration of according to Eq. (14) yields the variance, which for weak damping is set by the equipartition theorem: .
In the quantum regime, the natural generalization of Eq. (13) contains the product of Heisenberg timeevolved operators, , which do not commute. As a consequence, the spectrum is asymmetric in frequency. The quantum FDT
(16) 
implies that for at . Our discussion of dynamical backaction cooling will mention that this means the bath is not able to supply energy, as there are no thermal excitations. In this review we will also consider the symmetrized noise spectrum, . For more on noise spectra, we refer to Clerk et al. (2010a).
Iii Principles of optomechanical coupling
iii.1 The radiation pressure force and optomechanical coupling
In our discussion the fundamental mechanism that couples the properties of the cavity radiation field to the mechanical motion is the momentum transfer of photons, i.e. radiation pressure. The simplest form of radiation pressure coupling is the momentum transfer due to reflection that occurs in a Fabry Perot cavity. A single photon transfers the momentum (: photon wavelength). As a consequence the radiation pressure force is given by
Here denotes the cavity round trip time. Therefore, describes the radiation pressure force caused by one intracavity photon. The parameter which appears in this expression describes also the change of cavity resonance frequency with position, i.e. the frequency pull parameter. In the next section, which introduces a Hamiltonian description of the interaction between a movable mirror and optical cavity, this relation will be derived in its full generality.
More generally, the optomechanical coupling can arise for example by direct momentum transfer via reflection (FabryPerot type cavities with a moveable endmirror, microtoroids), by coupling via a dispersive shift of the cavity frequency (membrane in the middle, levitated nanoobjects trapped inside the cavity) or by optical nearfield effects (e.g. nanoobjects in the evanescent field of a resonator or a waveguide just above a substrate).
Various radiation pressure forces have been investigated in the pioneering work of Ashkin, who first demonstrated that small dielectric particles can be trapped in laser light Ashkin (2006). The relevant forces are generally referred to as gradient (or dipole) forces, as the force arises from the gradient of the laser field intensity. The particle is attracted to the center of the Gaussian trapping laser beam. If denotes the laser electric field distribution, the timeaveraged dielectric energy of the particle in the field is given by the polarizability) which correspondingly yields a force In addition to the gradient force, scattering forces occur for a traveling wave. These forces scale with , i.e. the wavenumber of the electromagnetic radiation, in contrast to the gradient forces. In addition there is also a contribution from the strainoptical effect, i.e. the straindependent polarizability. The strainoptical coupling is the dominant coupling mechanism in guided acoustic wave scattering Shelby et al. (1985); Locke et al. (1998). Independent of the physical interpretation of the force, however, the optomechanical interaction in an optomechanical system can always be derived by considering the cavity resonance frequency shift as a function of displacement (i.e. the “dispersive” shift). This will be the basis for our Hamiltonian description adopted in the next section. (with
It is important to note at this point that a significant difference between the trapping of particles in free space and micromechanical systems is the fact that the latter are also subject to radiation forces based on thermal effects. Absorption of light can heat a structure and deform it, which corresponds to the action of a force (e.g. in an asymmetric, bimorph structure, including materials of different thermal expansion). These photothermal forces can in many ways lead to effects similar to retarded radiation pressure forces, with the thermal relaxation time of the structure replacing the cavity photon lifetime. However, since such forces are based on absorption of light, they cannot form the basis for future fully coherent quantum optomechanical setups, since at least the coherence of the light field is thereby irretrievably lost.
iii.2 Hamiltonian formulation
The starting point of all our subsequent discussions will be the Hamiltonian describing the coupled system of a radiation mode interacting with a vibrational mode (Fig. 1). For brevity we will refer to the radiation field as “optical”, even though the important case of microwave setups is included here as well.
We will focus here on the simplest possible model system in cavity optomechanics, which has been used to successfully describe most of the experiments to date. In this model, we restrict our attention to one of the many optical modes, i.e. the one closest to resonance with the driving laser. Moreover, we also describe only one of the many mechanical normal modes. This is mostly arbitrary, as the displacement frequency spectrum will show peaks at any of the mechanical resonances. Still, as long as the dynamics is linear with independently evolving normal modes, the model will provide a valid approximation. In some cases, like sidebandresolved cooling, it may be possible to experimentally select a particular mechanical mode by adjusting the laser detuning, whereas in other cases, like nonlinear dynamics, an extended description involving several mechanical modes may become crucial.
The uncoupled optical () and mechanical () mode are represented by two harmonic oscillators, which is typically an excellent approximation at the displacements generated in the experiments:
(17) 
In the case of a cavity with a movable end mirror the coupling of optical and mechanical mode is parametric, i.e. the cavity resonance frequency is modulated by the mechanical amplitude^{6}^{6}6Note that such a setup is also considered for discussions of the dynamical Casimir effect, where cavity photons are created by the mechanical modulation of the boundaries. In the optomechanical scenarios considered here, however, the mechanical frequencies are too small for this effect to play a role.:
For most experimental realizations discussed in this review, it suffices to keep the linear term, where we define the optical frequency shift per displacement as (but see Sec. VI.2.2 for another example). A more detailed derivation of the optomechanical Hamiltonian can be found in an early paper Law (1995).
We mention in passing that other coupling mechanisms have been discussed. For example, the transparency of a moving Bragg mirror, and hence , can depend on its velocity Karrai et al. (2008). More generally, the displacement may couple to the cavity decay rate, yielding . This case (sometimes termed “dissipative coupling”), which is of practical relevance in some setups Li et al. (2009c), can lead to novel physical effects, e.g. in cooling Elste et al. (2009).
For a simple cavity of length , we have . The sign reflects the fact that we take to indicate an increase in cavity length, leading to a decrease in if . In general, expanding to leading order in the displacement, we have:
(18) 
Here , as defined before. Thus, the interaction part of the Hamiltonian can be written
(19) 
where
(20) 
is the vacuum optomechanical coupling strength, expressed as a frequency. It quantifies the interaction between a single phonon and a single photon. We stress that, generally speaking, is more fundamental than , since is affected by the definition of the displacement that is to some extent arbitrary for more complicated mechanical normal modes (see the discussion in Sec. II.2.1). Therefore, in the following we will almost always refer to . Further below, we will also mention , which is an oftenused measure for the effective optomechanical coupling in the linearized regime. It will be enhanced compared to by the amplitude of the photon field. The Hamiltonian reveals that the interaction of a movable mirror with the radiation field is fundamentally a nonlinear process, involving three operators (three wave mixing).
The radiation pressure force is simply the derivative of with respect to displacement:
(21) 
The full Hamiltonian will also include terms that describe dissipation (photon decay and mechanical friction), fluctuations (influx of thermal phonons), and driving by an external laser. These effects are formulated most efficiently using the equations of motion and the inputoutput formalism (see Sec. II.1.2, and also the next section). Here, we just remark that it is convenient to change the description of the optical mode by switching to a frame rotating at the laser frequency . Applying the unitary transformation makes the driving terms timeindependent^{7}^{7}7, and generates a new Hamiltonian of the form
(22) 
where
(23) 
is the laser detuning introduced already in Sec. II.1.2, and where we have omitted () driving, decay, and fluctuation terms, which will be discussed below. Eq. (22) is the frequently used starting point in cavity optomechanics.
We now introduce the socalled “linearized” approximate description of cavity optomechanics. To this end, we split the cavity field into an average coherent amplitude and a fluctuating term:
(24) 
Then, the interaction part of the Hamiltonian
(25) 
may be expanded in powers of . The first term, , indicates the presence of an average radiation pressure force . It may be omitted after implementing an appropriate shift of the displacement’s origin by . The second term, of order , is the one we keep:
(26) 
The third term, , is omitted as being smaller by a factor . Without loss of generality, we will now assume realvalued. Thus, the Hamiltonian in the rotating frame reads
(27) 
where the quadratic interaction part
(28) 
is referred to as “linearized”, since the resulting coupled equations of motion will be linear in this approximation. Note that the remaining terms in Eq. (27) no longer contain the laser driving, as that has already been taken care of by the shift implemented in Eq. (24). In the literature up to now, the combination
(29) 
is often referred to as “the optomechanical coupling strength”. Obviously, it depends on the laser intensity and is thus less fundamental than the singlephoton coupling (obtained for ). In the linearized regime described here, the optomechanical system can be viewed in analogy to a linear amplifier Botter et al. (2012) that receives optical and mechanical input fields.
The linearized description can be good even if the average photon number circulating inside the cavity is not large. This is because the mechanical system may not be able to resolve the individual photons if the decay rate is sufficiently large. The detailed conditions for the linearized approximation to be valid may depend on the questions that are asked. We will return to this question in the section on nonlinear quantum optomechanics (Sec. X.6).
We briefly note that is one neccessary condition for the socalled “strong coupling” regime of cavity optomechanics, where the mechanical oscillator and the driven optical mode hybridize (Sec. VII.2). A much more challenging condition is to have , i.e. the singlephoton optomechanical coupling rate exceeding the cavity decay rate. In the latter regime, nonlinear quantum effects will become observable (see Sec. X.6).
Depending on the detuning, three different regimes can be distinguished with respect to the interaction (28), especially in the sidebandresolved regime (, which we assume in the remainder of this section). For , we have two harmonic oscillators of (nearly) equal frequency that can interchange quanta: the mechanical oscillator and the driven cavity mode. Within the rotatingwave approximation (RWA) we thus can write the interaction as
(30) 
This is the case relevant for cooling (transferring all thermal phonons into the cold photon mode; Sec. VII.1) and for quantum state transfer between light and mechanics (Sec. X). In the quantumoptical domain, it is referred to as a “beamsplitter” interaction.
For , the dominant terms in RWA
(31) 
represent a “twomode squeezing” interaction that lies at the heart of parametric amplification Clerk et al. (2010a). In the absence of dissipation, this would lead to an exponential growth of the energies stored both in the vibrational mode and the driven optical mode, with strong quantum correlations between the two. Thus, it may be used for efficiently entangling both modes (Sec. X). Focussing on the mechanical mode alone, the growth of energy can be interpreted as “antidamping” or amplification (Sec. V.2.2). If the intrinsic dissipation is low enough, this behaviour may trigger a dynamical instability that leads to selfinduced mechanical oscillations. The resulting features will be discussed in Sec. VIII.
Finally, when , the interaction
(32) 
means that the mechanical position leads to a phase shift of the light field, which is the situation encountered in optomechanical displacement detection (Sec. VI). In addition, this interaction Hamiltonian can be viewed as implementing QND detection of the optical amplitude quadrature , since that operator commutes with the full Hamiltonian in this case.
iii.3 Optomechanical equations of motion
The mechanical motion induces a shift of the optical resonance frequency, which in turn results in a change of circulating light intensity and, therefore, of the radiation pressure force acting on the motion. This kind of feedback loop is known as optomechanical “backaction”. The finite cavity decay rate introduces some retardation between the motion and the resulting changes of the force, hence the term “dynamical” backaction.
A convenient starting point for the analytical treatment of the radiationpressure dynamical backaction phenomena (Sec. V.2 and Sec. VII) is the inputoutput formalism. This formalism (briefly introduced in Sec. II.1.2) provides us with equations of motion for the cavity field amplitude and, analogously, for the mechanical amplitude . These equations have the form of Quantum Langevin equations^{8}^{8}8In the standard approximation adopted here, these equations are Markoffian, i.e. without memory and with deltacorrelated noise., since both the light amplitude and the mechanical motion are driven by noise terms that comprise the vacuum noise and any thermal noise entering the system:
(33)  
(34) 
Please see Sec. II.1.2 for remarks on the inputoutput treatment and the optical decay rates . With regard to the damping term for the mechanical dissipation, we note that this treatment is correct as long as . Otherwise the equations would have to be formulated on the level of the displacement , with a damping force .
The noise correlators associated with the input fluctuations are given by:
(35)  
(36)  
(37)  
(38) 
Here we have assumed that the optical field has zero thermal occupation (), which is an approximation that is valid for optical fields at room temperature, although it may fail for the case of microwave fields, unless the setup is cooled to sufficiently low temperatures. In contrast, the mechanical degree of freedom is typically coupled to a hot environment, with an average number of quanta given by . Together with these correlators, the quantum Langevin equations describe the evolution of the optical cavity field and the mechanical oscillator, including all fluctuation effects.
It is equally useful to give the classical, averaged version of these equations that will be valid for sufficiently large photon and phonon numbers, in the semiclassical limit. Then, we can write down the equations for the complex light amplitude and the oscillator position :
(39)  
(40) 
Here we have neglected all fluctuations, although these could be added to describe thermal and even, in a semiclassical approximation, quantum noise forces. The term represents the laser drive. Note that we have also chosen to write the mechanical equation of motion in terms of the displacement, where . This becomes equivalent to the equation given above only for weak damping, . These fully nonlinear coupled differential equations will be the basis for our discussion of nonlinear phenomena, in particular the optomechanical parametric instability (also called “selfinduced oscillations” or “mechanical lasing”, Sec. VIII).
The equations of motion Eq. (33), (34) (and likewise their classical versions) are inherently nonlinear as they contain the product of the mechanical oscillator amplitude and the cavity field (first line) or the radiation pressure force that is quadratic in photon operators (second line). While they can still be solved numerically in the classical case, for the quantum regime they are of purely formal use and in practice cannot be solved exactly, neither analytically nor numerically. However, in many situations that we will encounter it is permissible to linearize this set of equations around some steadystate solution: . Using and keeping only the linear terms, we find the following set of coupled linear equations of motion:
(41)  
(42)  
(43) 
These correspond to what one would have obtained alternatively by employing the “linearized” coupling Hamiltonian of Eq. (28) and then applying inputoutput theory. Here we have (as is common practice) redefined the origin of the mechanical oscillations to take into account the constant displacement that is induced by the average radiation pressure force. It is evident that now the mutual coupling terms between the optical and mechanical degrees of freedom are linear in the field operators, and that the strength is set by the fieldenhanced coupling rate .
As shown in the later sections, these linearized equations are able to fully describe several phenomena, including optomechanical cooling, amplification, and parametric normal mode splitting (i.e. strong, coherent coupling). They can be solved analytically, which is best performed in the frequency domain (see Sec. V.2).
For completeness, we display the linearized quantum equations in frequency space:
(44)  
(45) 
Here is the Fourier transform of . Note the important relation , which has to be taken care of while solving the equations.
It is equally useful to consider the linearized version of the classical equations of motion for the light amplitude and the displacement, Eqs. (39) and (40):
(46)  
(47) 
Finally, we display them in frequency space, in the form that we will employ in Sec. V.2.
(48)  
(49)  
Again, note .
Iv Experimental realizations and optomechanical parameters
The increasing availability of highquality optomechanical devices, i.e. highQ mechanical resonators that are efficiently coupled to highQ optical cavities, has been driving a plethora of experiments during the last years that are successfully demonstrating the working principles of cavity optomechanics. We now discuss some of the most frequently used architectures.
iv.1 Optomechanical parameters
The following table summarizes the relevant optomechanical parameters for some typical current experimental implementations. These are: the mechanical resonator frequency and mass ; the fundamental mechanical (phonon) and optical (photon) dissipation rates and , respectively; the “” product, which is a direct measure for the degree of decoupling from the thermal environment (specifically, is the condition for neglecting thermal decoherence over one mechanical period); the sideband suppression factor that determines the ability to realize groundstate cooling (see Sec.VII); and finally the bare optomechanical coupling rate , which corresponds to the cavity frequency shift upon excitation of a single phonon.