Cavity-Assisted Back Action Cooling of Mechanical Resonators

We analyze the quantum regime of the dynamical backaction cooling of a mechanical resonator assisted by a driven harmonic oscillator (cavity). Our treatment applies to both optomechanical and electromechanical realizations and includes the effect of thermal noise in the driven oscillator. In the perturbative case, we derive the corresponding motional master equation using the Nakajima-Zwanzig formalism and calculate the corresponding output spectrum for the optomechanical case. Then we analyze the strong optomechanical coupling regime in the limit of small cavity linewidth. Finally we consider the steady state covariance matrix of the two coupled oscillators for arbitrary input power and obtain an analytical expression for the final mechanical occupancy. This is used to optimize the drive's detuning and input power for an experimentally relevant range of parameters that includes the"ground state cooling"regime.


Introduction
Recent progress in the emerging fields of nanoelectromechanical [1,2] and optomechanical systems [3,4] promises to enable quantum limited control of a single macroscopic mechanical degree of freedom [5,6]. This is relevant in the context of high precision measurements [7]- [10] like single spin magnetic resonance force microscopy [11,12] (MRFM) and for fundamental studies of the quantum to classical transition [13,14,15]. A paradigmatic goal which has triggered a surge of activity is to prepare the eingenmode associated to an ultra long lived mechanical resonance (angular frequency ω m and Q-value Q m ) in its quantum ground state with high fidelity [16]- [28]. The considerable difficulty to achieve the desired combination k B T ω m , Q m 1 with state of the art micro-fabrication and cryogenic techniques [2] has naturally motivated ideas to use cooling schemes analogous to the laser-cooling of atoms [29]- [32].
In these schemes the mechanical resonator's displacement is coupled parametrically to an auxiliary high frequency bosonic or fermionic resonator (pseudospin) that can act as a "cooler" [33]. To drive the latter while monitoring its output allows detection of the mechanical displacement. Naturally there is a back-action force associated to this measurement process [34,35,36]. Due to the dissipative dynamics of the cooler for a negative detuning of the drive this force becomes anti-correlated with the Brownian motion resulting in net cooling. In turn the quantum fluctuations of the cooler -which in the atomic laser cooling manifest in the inherent stochastic nature of the spontaneous photon emissions -result in a quantum noise spectrum for this backaction force that sets a fundamental lower bound for the final temperature [30,31]. Thus the structured reservoir afforded by the driven cooler and its environment provides an effective thermal bath for the mechanical resonator. The concomitant absorptions of motional quanta (cooling) correspond to Raman scattering processes in which a drive quanta is up converted, while emission events (heating) are associated to Raman processes in which a drive quanta is down converted [cf. Fig. 1

(b)].
A host of concrete realizations of the above generic scenario have been discussed in the literature. These range from electronic or electrical devices where the cooler is provided by a (superconducting) single electron transistor [19], a Cooper-pair box [37], an LC-circuit [3], or a quantum dot [38]; to optomechanical systems where this role is played by an optical cavity mode [4]. In the latter systems, which are equivalent to a Fabry-Perot with a moving mirror [cf. Fig. 1(a)], the optical field couples parametrically to the mechanical motion via radiation pressure. This effect has already been thoroughly demonstrated experimentally [39] and harnessed to provide appreciable cooling, albeit in the classical regime [20,21,23,26,27]. Recent experiments involve both optical and microwave cavities. In turn completely analogous effects have been shown using a capacitively coupled radio frequency LC-circuit [24] and a superconducting microwave cavity [28]. These systems share the common feature that the cooler is well approximated by a single harmonic oscillator which henceforth will be referred to as the "cavity". While for optical cavities the vacuum constitutes an  excellent approximation for the input when the drive is switched off, in the case of radio and microwave frequencies thermal noise in the cavity needs to be taken into account. A quantum treatment of the corresponding temperature limits has already been given which predicts that ground state cooling is possible when the mechanical oscillation frequency is larger than the cavity's linewidth κ [40,41,42]. Here we provide a rigorous analysis of the cooling dynamics that drives the mechanical resonator mode to a thermal state with a well defined effective final temperature that for a finite Q m is imprinted on the cavity's output. This is done in Sec. 3 where we obtain the motional master equation and the corresponding output spectrum. The above picture is only valid for perturbative optomechanical coupling and thus breaks down when the resulting cooling rate becomes comparable to the cavity linewidth κ or to the mechanical oscillation frequency [41] ω m . In the Doppler limit ω m κ the system then approaches the parametric instability or settles into a regime where backaction effects are purely diffusive with no net cooling. On the other hand in the resolved sideband limit κ ω m as the optomechanical coupling exceeds the cavity linewidth the system enters into a strong coupling regime in which the motion hybridizes with the cavity fluctuations. The ensuing optomechanical normal modes are then cooled simultaneously. This phenomenon is analyzed in Sec. 4 in the limit of small cavity linewidth where we find that the dynamics of each of the normal modes can be described by a master equation analogous to the one valid in the perturbative regime with a cooling rate given by half the cavity linewidth. Finally, in the same Section we derive an analytical expression for the final (steady state) average mechanical occupancy (phonon number) valid for arbitrary optomechanical coupling and use it to optimize the parameters of the drive.

Optomechanical Master Equation
In optomechanical systems (and their electromechanical analogues) the cavity resonant frequency ω p depends inversely on a characteristic length that is modified by the mechanical resonator's displacement. The fact that ω p ω m allows for an adiabatic treatment of this effect which results in the aforementioned parametric coupling. In general the leading contribution to the latter is linear in the mechanical displacement -but situations in which it is instead quadratic can be readily engineered [42,26]. The desired cooling dynamics is induced by a slightly detuned electromagnetic drive (angular frequency ω L ), which for optomechanical systems corresponds to an incident laser and in the electromechanical case is afforded by a suitable external AC voltage. Thus the Hamiltonian describing the coupled system (in a rotating frame at ω L ) is given by [43] Here a p (a m ) is the annihilation operator for the electromagnetic (mechanical) oscillator and ∆ L is the detuning of the drive from ω p . Here we have defined Ω ≡ 2 P κ ex / ω L , where P is the input power of the drive and κ ex is the cavity decay rate into the associated outgoing electromagnetic modes. The dimensionless parameter η, characterizing the non-linear coupling between the cavity and the mechanical resonator, is given by η = (ω p /ω m )(x 0 /L) where x 0 is the zero point motion of the mechanical resonator mode and the characteristic length L depends on the physical realization. In the optomechanical case it corresponds to an effective optical cavity length while for electromechanical realizations [24] L = 2dC tot /C c , where C c ∝ 1/d is the dynamical capacitance, d is the distance between the corresponding electrodes, and C tot is the total capacitance. We treat the losses induced by the electromagnetic and mechanical baths within the rotating wave Born-Markov approximation using the standard Lindblad form Liouvillians [44]. It is important to note that the validity of a rotating wave approximation (RWA) in the environmental coupling responsible for the mechanical losses only amounts to Q m 1 provided the optomechanical coupling is weak enough that there is no appreciable mixing between the annihilation and creation operators of the modes (cf. Sec. 4). Clearly if the latter is not satisfied the usual RWA will result in the unwarranted neglection of resonant terms. This can be borne out by comparing the corresponding displacement spectra and results in the condition η|α|ω m max{ √ ω m κ, ω m }. Henceforth we focus on parameters that satisfy it which include the most relevant regimes for cooling and ensure that the system doe not approach the instability for any detuning. Thus the evolution for the density matrix of the resonator-cavity system readṡ The total cavity decay rate κ has two contributions: (i) the rate at which photons are lost from the open port (where the driving field comes in) given by κ ex and (ii) the "internal loss" rate κ − κ ex due to the other losses of the electromagnetic resonator (i.e absorption inside the dielectric, scattering into other modes, etc.). Naturally the thermal noise is determined by the Bose number n(ω). At room temperature and for optical frequencies n(ω p ) is negligible, however for the much lower radio and microwave frequencies characterizing electromechanical setups this quantity can be comparable to the final mechanical occupancies achieved. Similarly γ m = ω m /Q m is the mechanical resonator's natural linewidth and n(ω m ) its mean occupation number at thermal equilibrium (i.e. in the absence of the drive).
To study the cooling process it proves useful to apply a canonical transformation of the form a p → a p + α, a m → a m + β with the amplitudes α, β chosen so that the linear terms in the transformed Liouvillian cancel out. This condition leads to the following coupled equations for the amplitudes We assume η 1 and that the mechanical dissipation rate γ m is much smaller than ω m . To lowest order in the small parameters η and 1/Q m we obtain α ≈ Ω/(2∆ L + iκ) and β ≈ −η|α| 2 . Here |α| 2 is the steady state occupancy of the cavity in the absence of optomechanical coupling and β is the static shift of the mechanical amplitude due to the radiation pressure. The normal coordinates after the transformation are shifted so that the new amplitudes correspond to the deviation from the steady state equilibrium position. This transformation leaves the dissipative part of the Liouvillian invariant and transforms the Hamiltonian into: where we have introduced the effective detuning ∆ L + 2η 2 |α| 2 ω m → ∆ L . It is interesting to note that if the bosonic cooler is replaced by a fermionic one so that a p → σ − we obtain a Hamiltonian that resembles the one describing a trapped ion in the Lamb-Dicke regime. As a result for perturbative η (as analyzed in the next Section) the cooling cycle [cf. Fig. 1(b)] becomes analogous to the Lamb-Dicke regime of atomic laser-cooling [40].

Master equation for mechanical motion
We first focus on the regime in which the input laser power P is low enough so that the time scales over which the populations of the mechanical resonator's Fock states evolve (leading to cooling or heating) are much slower than those associated to the losses of the cavity and to the free mechanical frequency. As will become clear below this requires Here we also assume [n(ω m ) + 1]γ m κ, ω m which must hold to allow for appreciable cooling, and η 2 1 (in optomechanical realizations η 10 −4 ). Hence the electromagnetic degrees of freedom can be treated as a structured environment that affects the mechanical motion perturbatively. Along these lines the latter can be described by a Markovian master equation for its reduced density matrix [44].
To derive it we take the optomechanical master equation in the shifted representation, transform to an interaction picture for the resonator mode and adiabatically eliminate the cavity using the Nakajima-Zwanzig formalism [45,44,46]. The optomechanical coupling and the mechanical losses are treated perturbatively. We define the projection with ρ (th) where n p ≡ n(ω p ) is the Bose number, and introduce the formal parameter ζ such that with We note that Pρ is a stationary state of L 0 for any ρ implying that L 0 P = 0 while PL 0 = 0 follows from trace preservation, so that we have As will emerge from our derivation the basic idea is that the rates for cooling and heating set the relevant time scale (zeroth order in 1/ζ) which is widely separated from the mechanical oscillation period 2π/ω m and the cavity ring down 1/κ (order 1/ζ 2 ). In fact, the asymptotic expansion for ζ → ∞ pursued below amounts to a controlled expansion in the ratio between the fast and the slow timescales. Given that we are interested in the behaviour as t → ∞, the initial condition is immaterial and we choose for simplicity one in the P-manifold so that Qρ 0 = 0. Subsequently by explicitly integrating the differential equation for the irrelevant part Qρ, we obtain a closed equation for the relevant part Pρ: where T + (T − ) is the time-ordering (anti-time-ordering) operator. For the purpose of analyzing the asymptotic limit ζ → ∞ of Eq. (14) we have which can be understood considering the corresponding Laplace transforms. Substitution of Eqs. (13), (15) into Eq. (14) and the change of variables τ = ζ 2 (t − τ ) then yields Here we have also used Q 2 = Q, P 2 = P, and PL 2 = L 2 P. The leading order in 1/ζ of the evolution over the aforementioned relevant time scale will be determined by the limit as ζ → ∞ of the Laplace transform of the above. In the time domain all the fast rotating terms (frequencies of order ζ 2 ω m ) drop out and Eq. (16) reduces to We note that as P = lim t→∞ e L 0 t we have It follows that the restriction of L 0 to the Q-manifold has only eigenvalues with negative real parts which allows us to establish We focus on the parameter regime where |α| 2 n p . The behavior of the cavity correlations that determine the second term in Eq. (17) implies that in this regime contributions arising from the cubic term in L (±) 1 are negligible compared to those generated by the quadratic term -for n p = 0 the contributions of the former are higher order in η. The conditions that warrant this linearization of L (±) 1 for the case n p = 0 will be discussed further in the next Subsection. The restriction of the quadratic term to the P-manifold vanishes and we obtain Here we have used Eqs. (5), (10), (11), and introduced the optomechanical coupling the reduced density matrix for the mechanical mode µ ≡ Tr p {ρ}, and the cavityquadratures' correlations with X p (0) ≡ (α * a p + αa † p )/ √ 2|α|. If we substitute Eqs. (5), (12), (20) into (17), trace out the cavity, and rearrange we finally obtain the following master equatioṅ where the cooling (heating) rate A − (n p ) [A + (n p )] and the mechanical frequency shift ∆ m induced by the optomechanical coupling are given by It is straightforward to calculate the necessary two-time correlations using the quantum regression theorem given the steady state moments: and that the evolution of the mean amplitude reads Thus we obtain with which substituted into Eqs. (24), (25) leads to where the corresponding rates in the absence of thermal noise in the driven cavity are given by: The above can be related to the input power and the frequency of the drive via: The master Eq. (23) generalizes the one obtained in Ref. [40] by including thermal noise in the cavity input. As shown below, this effect is significant for determining the ultimate limit to which the mechanical resonator can be cooled for ratios ω p /ω m like the ones that characterize electromechanical realizations. Equation (23) has as its steady state a thermal state which defines the effective final temperature to which the mechanical resonator is cooled. The corresponding final occupancy reads whereñ f is the quantum backaction contribution derived in Refs. [40,41], namelỹ We note that in the "unshifted" representation there is in addition a coherent shift of the resonator's normal coordinate so that If we now consider the appreciable cooling limit Γ γ m and minimize with respect to the detuning we obtain for the optimal detuning ∆ opt L = ω 2 m + κ 2 /4. The first term corresponds to the "linear cooling" limited by thermal noise. In turn, the second term shows that the final occupancy is necessarily bounded by the equilibrium thermal occupancy of the cooler. Finally, the last term for n p = 0 corresponds to the fundamental temperature limit imposed by the quantum backaction which in the Doppler regime κ ω m reduces to κ/4ω m , precluding ground state cooling, while in the RSB regime it yields κ 2 /16ω 2 m corresponding to occupancies well below unity [40,41]. We note that for a given κ/ω m < √ 32 occupancies below unity are only attainable within a finite detuning window |∆ L + 3ω m | ≤ 8ω 2 m − κ 2 /4, as illustrated in Figure 2.

Output power spectrum and temperature measurement
The spectrum of the cavity output constitutes a crucial observable to understand the backaction cooling. It allows to visualize the cooling cycle as a frequency up-conversion process and for optomechanical realizations it provides an efficient way to measure the final temperature [40]. We focus on the latter for which n p = 0 and consider the experimentally relevant case in which the output is measured in the same modes in which the coherent laser drive is fed [23]. To calculate its spectrum we apply the standard input-output formalism [44] and treat the parameters η|α|, η perturbatively along the lines of the previous Subsection. For this purpose it proves useful to consider the quantum Langevin equation for a p associated to the Liouvillian (7), namelẏ where δa in and b in correspond to the vacuum noise associated, respectively, to the laser mode and to the other cavity losses. Here we use the shifted representation [cf. (4)] and also take into account that Hamiltonian (1) presupposes the standard time-dependent canonical transformation that maps the coherent input state associated to the laser into a classical field so that a in (t) → δa in (t) + a in (t) . If we assume that the term linear in η is a specified function of time and that the solution a This integral equation can be iterated to generate the following Dyson series type result a p (t) = a (0) p (t) + In turn, the output spectrum is given by which in the shifted representation reads We now seek the lowest non-trivial order in η. The output a out has a c number part arising from the classical drive and the cavity steady state amplitude and an operator part corresponding to the fluctuations. Equation (43) directly implies that terms involving the c numbers only contribute to the "main line" ∝ δ(ω − ω L ). Furthermore, as in the shifted representation the steady state of the electromagnetic modes is the vacuum |0 it follows from Eqs. (41), (43) that the contribution of the cross term with the operator part -which vanishes if the cubic term in Eq. (4) is omitted -is at most higher order in η and can be neglected relative to the contribution bilinear in the c numbers. The latter corresponds to the classical reflection coefficient that is straightforward to obtain considering the near-resonant scattering into modes (real or fictitious) responsible for the other losses κ − κ ex . Thus we arrive at where we have used a  44) we adopt (as in the previous Subsection) an interaction picture for the resonator mode [i.e. a m (t) → e −iωmt a m (t)] and make the substitutions t + τ − τ 1 → τ 1 , t + τ 1 → τ 1 in the time integrals. In this representation the mechanical mode operators evolve slowly compared with 1/κ so that in line with our derivation of a Markovian master equation for the mechanical motion we can factor them out of the time integrals whose upper limit can be extended to infinity. This Markovian approximation yields where we have used Eq. (28) with n p = 0. The two time averages of the mechanical mode operators can now be calculated from the master equation (23) using the quantum regression theorem. The needed one time averages satisfy Finally, a straightforward calculation leads from Eqs. (44), (45), (46), and (29) to the spectrum already given in Ref. [40], namely where we have introduced γ eff ≡ γ m + Γ which is the total dissipation rate for the mechanical mode in the presence of the drive that determines the linewidths of the motional sidebands peaked at ω L ± ω m . One should note that the above corresponds to photon rate per unit frequency and that the weight of the motional sidebands relative to the main line is of order η 2 (2n f + 1) [η 2 10 −8 for typical systems]. Here unlike the case of atomic laser cooling [30,31] the mechanical dissipation induces an asymmetry in the weights of the sidebands that allows to retrieve the final temperature directly from the steady state. The "blue" sideband weighted by N − = κex κ A − n f corresponds to the up-converted photons (anti-Stokes scattering) responsible for the cooling while the "red" sideband weighted by N + = κex κ A + (n f + 1) corresponds to the down-converted photons (Stokes scattering) that result in heating.
It is interesting to note that the formalism of this Section does not presuppose linearizing around the steady state and neglecting the cubic term in Hamiltonian (4) accordingly, but rather the validity of such treatment emerges from a controlled procedure that would allow to incorporate the necessary corrections if the intrinsic nonlinearity η were larger. A straightforward self-consistency criterion is to compare the steady state fluctuation of the cubic term with that of the quadratic one. An heuristic estimate of their ratio can be extracted from the total weight of the motional sidebands in Eq. (47) which together with the analysis in Subsec. 3.1 implies that the cubic term can be neglected provided the conditions η 2 (2n f + 1)ω 2 κ 2 and |α| 2 η 2 are satisfied -for the typical parameters in cooling experiments these are always met.

Small cavity linewidth limit κ g m
The treatment in the previous section is only applicable when the cooling rate A − given by Eqs. (30), (31), (32) is much smaller than κ (note that the heating rate A + is always bounded by A − for negative detuning). This condition for arbitrary negative detuning results in g 2 m κ 2 so that it follows (as expected) that the motional master equation is only warranted for small enough optomechanical coupling. In the Doppler regime when this is violated the aforementioned condition g m max{ √ ω m κ, ω m } underpinning the RWA for the mechanical losses will also fail. In contrast in the resolved sideband regime (RSB) relevant for ground state cooling κ ω m implies that there is a wide parameter range of interest in which Eq. (2) remains valid while Eq. (23) fails. Here we consider this RSB regime beyond perturbative optomechanical coupling. As g m becomes comparable to κ it becomes necessary to follow the coupled dynamics of both modes as described by Eq. (2) which for g m > κ/2 exhibits normal mode splitting. Though for a Gaussian initial condition the approach to the steady state is always amenable to a straightforward description, in the intermediate regime g m ∼ κ there will be no simple analog of Eq. (23) that allows to visualize the cooling process in terms of phonon jumps. In turn, deep in the strong coupling regime but away from the instability, i.e. for κ g m ω m , the dynamics can be described by two decoupled master equations for the optomechanical normal modes analogous to Eq. (23).
Within the latter parameter range it is permissible to start from a Hamiltonian description including the optomechanical coupling and treat the losses (as described by γ m , κ) perturbatively. We focus on the "resonant case" −∆ L = ω m -which in the RSB regime can be shown to be optimal for minimizing the final occupancy -and consider the canonical transformation that diagonalizes Hamiltonian (4) for η = 0 with g m = 0 [after performing the convenient rotation a p → (α/|α|)a p ]. The latter is given by where the eigenfrequencies of the normal modes read If we now consider the expansion of the above in the small parameter g m /ω m the zeroth order of Eq. (49) yields a splitting given by g m while Eq. (48) reduces to the transformation that diagonalizes the rotating wave part of Hamiltonian (4) -that results from neglecting the terms that involve a m a p , a † m a † p . The latter transformation does not mix the annihilation and creation operators. For small but finite g m /ω m there will be small admixtures that for the purpose of analyzing the cooling dynamics will only be relevant insofar as they give rise to qualitatively new terms in the dissipative part of the Liouvillian -otherwise they can be shown to result in contributions of relative order (g m /ω m ) 2 for all values of the other parameters. Hence we have To proceed we: (i) apply the transformation given by Eq. (48) to the master equation (2), (ii) transform the result to an interaction picture with respect to the (now diagonal) Hamiltonian, (iii) neglect all the resulting fast rotating terms which are ∝ e ±2iωmt or ∝ e ±igmt (up to corrections higher order in g m /ω m ) and (iv) expand to lowest order in g m /ω m following the aforementioned "qualitative" criterion. Naturally (iii) relies on the small cavity linewidth condition κ g m . Thus we obtaiṅ The only corrections in (g m /ω m ) 2 appear in the first term and correspond to the "high power limit" of the heating induced by the quantum backaction of the cavity. Analogous heating terms ∝ γ m are neglected given that they are comparable to corrections to the RWA treatment of the mechanical dissipation. It follows from Eq. (51) that the losses do not couple the normal modes (annihilation operators a ± ) so that the steady state is given by the tensor product of thermal states for each of them characterized by average occupancies (where we neglect higher order terms in g m /κ) to which the average phonon number converges with a cooling rate (κ + γ m )/2. Hence Eqs. (50), (52) finally yield where the corrections are higher order in the small parameters (g m /ω m ) 2 , γ m /κ. The first term corresponds to the heating associated to the mechanical dissipation and can be identified with the corresponding term in Eq. (35) showing a saturation of the usual linear cooling law. Similarly, the second and third terms can be identified with the corresponding contributions in Eq. (35) arising from the thermal noise in the cavity and the quantum backaction. Thus ground state cooling requires k B T / Q m κ ω m and ω p < k B T . We note that Eqs. (50), (52) imply that a 2 m SS = 0 so that the reduced state for the motion is also thermal in the small κ limit.
If we now compare Eqs. (38) and (53) and consider minimizing them with respect to g m within their respective ranges of validity, heuristic considerations imply that the following formula is expected to always constitute a lower bound for the final mechanical occupancy optimized with respect to the parameters of the drive

Final occupancy for arbitrary ratio g m /κ and optimal parameters
The approximate expressions (35), (53) for the final mechanical occupancy that we have derived in the limits g m κ and g m κ provide a basic understanding of the requirements for ground state cooling and the expected order of magnitude for the optimum. Notwithstanding they have the drawback that they fail to settle which is the optimal input power as minimization with respect to g m shifts this variable away from the domain where they are valid. In addition given the experimental progress towards achieving ultra-cold states in these systems it is clear that precise quantitative predictions of the steady state for a given input are highly desirable. To this effect we complement the above analysis by deriving an analytical expression for n f valid for with It is straightforward to show that in the limits g m κ with ∆ L < 0, and κ g m ω m with ∆ L = −ω m we recover, respectively, the approximate expressions (35), (53) [to lowest order in γ m /Γ].
The analytical formulas Eq. (61), (62) provide a quantitative basis for a precise analysis of the final temperatures that can be attained for arbitrary values of g m /κ. In particular they can be readily minimized numerically with respect to the drive's detuning and input power subject to the constraint imposed by the stability condition R −1 > 0 that emerges from the Routh-Hurwitz criteria. This is the natural optimization problem that is posed by an experimental realization in which the natural linewidth of the cavity is fixed or hard to modify (admittedly sweeping this parameter within some range should be straightforward in electromechanical setups). We have performed this optimization using instead the exact expression that follows from Eq. (55), i.e. without the small γ m approximation so that the transition to the trivial regime min{n f } = n(ω m ) is captured. Figure 3(a) compares this optimum (solid lines) as a function of the thermal equilibrium occupancy n(ω m ) with the lower bound n TL (dashed lines) furnished by Eq. (54) for representative values of the other parameters. The blue lines correspond to n p = 0 (i.e. ω p /ω m → ∞) for values of κ/ω m in the Doppler regime (light blue) and in the RSB regime (dark blue). The red lines show the latter with ω p /ω m = 10 3 instead. There are three distinct regimes: (i) n(ω m ) < n TL so that the optomechanical coupling only raises the temperature and g opt m = 0, (ii) the minimum occupancy is determined by the fundamental quantum backaction limit and is thus independent of the ambient temperature [i.e. constant as a function of n(ω m )], and (iii) the minimum occupancy is determined by classical noise and is thus linear in the ambient temperature. In this latter regime for finite ω p /ω m there is a noticeable transition at which the minimum deviates from the n p = 0 result as the cavity becomes thermally activated. One can also note that in this linear regime n TL is only reachable in the RSB limit. In turn Fig. 3(b) shows the corresponding optimal values for g m . The saturation at high temperatures just reflects the fact that (n m f + n p f )/n(ω m ) becomes temperature independent in that limit and attains its minimum at a finite g m [c.f. (62)]. Finally, Fig. 4(a) and (b) illustrate, respectively, the dependence of the optimum with the ratios κ/ω m and ω p /ω m as the ambient temperature is varied in the RSB regime. The fundamental quantum backaction limit results in a distinct shoulder. In Fig. 4(a) its lower edge traces a line with a slope consistent with the quadratic dependence on κ/ω m . It is clear from Fig. 4(b) that for ratios ω p /ω m 10 4 and n(ω m ) 10 3 (parameters that are relevant for electromechanical setups [24]) thermal noise in the cavity needs to be taken into account.

Conclusion
In summary, we have analyzed the cavity-assisted backaction cooling of a mechanical resonator in the quantum regime by deriving effective master equations for the relevant degrees of freedom both in the perturbative and strong coupling regimes. These provide a description of the cooling dynamics that allows to establish a simple lower bound for the final occupancy [Eq. (54)] that can only be reached in the RSB regime. This bound implies that ground state cooling is only possible when the cavity linewidth is much larger than the heating rate induced by the mechanical dissipation but much smaller than the mechanical oscillation frequency, and the equilibrium thermal occupancy of the cavity is well below unity. In addition we give an analytical expression for the final occupancy valid in all regimes of interest that allows for a straightforward optimization of the parameters of the drive. Finally we analyze the dependence of this optimum on the ambient temperature, the cavity linewidth, and the ratio of the cavity's frequency to the mechanical frequency.