Quantum Theory of Cavity-Assisted Sideband Cooling of Mechanical Motion Florian Marquardt,¹ Joe P. Chen,² ,4 A. A. Clerk,³ and S. M. Girvin² ¹ Department of Physics, Arnold-Sommerfeld-Center for Theoretical Physics, and Center for NanoScience, Ludwig-Maximilians-Universität München, Theresienstrasse 37, 80333 Munich, Germany ²Department of Physics, Yale University, P.O. Box 208120, New Haven, Connecticut 06520-8120, USA 3 Department of Physics, McGill University, 3600 rue University, Montreal, QC Canada H3A 2T8 4 Department of Physics, Cornell University, 109 Clark Hall, Ithaca, New York 14853-2501, USA (Received 22 January 2007; published 28 August 2007) We present a quantum-mechanical theory of the cooling of a cantilever coupled via radiation pressure to an illuminated optical cavity. Applying the quantum noise approach to the fluctuations of the radiation pressure force, we derive the optomechanical cooling rate and the minimum achievable phonon number. We find that reaching the quantum limit of arbitrarily small phonon numbers requires going into the good- cavity (resolved phonon sideband) regime where the cavity linewidth is much smaller than the mechanical frequency and the corresponding cavity detuning. This is in contrast to the common assumption that the mechanical frequency and the cavity detuning should be comparable to the cavity damping. DOI: 10.1103/PhysRevLett.99.093902 Recent years have seen significant advances in fabricat- ing, measuring, and controlling mechanical systems on the micro- and nanometer scale [ 1 – 4 ]. A number of experi- ments to explore the quantum regime of these mechanical devices have been proposed, including generation of non- classical states [ 5 ], entanglement [ 6 – 11 ], and quantum- limited measurements [ 12 , 13 ]. Many of these proposals require the system to be cooled to its ground state, i.e., to temperatures below 20 mK even for 1 GHz resonators. This is hard or impossible using bulk refrigeration, but it may be feasible using nonequilibrium cooling techniques analo- gous to the laser-cooling schemes for trapped ions and neutral atoms [ 14 ]. Cooling via an active feedback loop [ 15 – 17 ] has already been considered elsewhere (including its quantum limits [ 18 ]). Here we will focus on passive cooling (without feedback). In this approach the cantilever displacement is coupled parametrically to a driven resonator (or two-level system). When the drive frequency is chosen appropriately, the cantilever is cooled. The lowest achievable temperature is determined by the resonator's quantum fluctuations (photon shot noise). This approach has been considered theoretically for a few specific realizations of the cooling system: a Cooper- pair box [ 19 ], the superconducting single-electron transis- tor [ 4 , 20 , 21 ], quantum dots [ 22 ], and ions [ 23 ]. However, for experimental simplicity one often prefers to get rid of the auxiliary quantum system mediating between cantile- ver and radiation field. This is achieved by coupling di- rectly to the field of an optical cavity via radiation pressure. A number of recent experiments [ 24 – 31 ] have produced very promising results on self-cooling in such systems, using both photothermal and radiation pressure forces. In this Letter, we present a fully quantum-mechanical de- scription, providing the basic theory for future ground-state cooling experiments. Cooling rates and steady-state tem- PACS numbers: 42.79.Gn, 07.10.Cm, 42.50.Lc peratures for such systems (and related ones like a driven LC circuit [ 32 ] ) have been derived only (semi-)classically so far. Specifically, Braginsky and Vyatchanin [ 33 ] con- sidered the optomechanical damping within a semiclassi- cal theory. Here, we use the quantum noise approach to find simple and transparent but fully quantum-mechanical ex- pressions, starting directly from the force fluctuation spec- trum. Our results are valid both for the good-cavity regime (resolved mechanical sidebands) and the bad-cavity regime (unresolved sidebands). Earlier (semi-)classical calcula- tions [ 24 , 32 , 33 ] are reproduced in the appropriate limits. We show, in particular, that it should be possible to cool the cantilever to its quantum-mechanical ground state by choosing the cantilever resonance frequency much larger than the cavity ringdown rate, a regime that has not been considered so far. Ground-state cooling is required for the experiments mentioned above and may permit studies of the optomechanical instability [ 28 , 34 – 36 ] in the quantum regime. Moreover, we present an exact solution of the linearized coupled equations of motion to account for the ''strong cooling'' limit, where the cooling rate exceeds the cavity ringdown rate. Consider a mechanical degree of freedom î coupled parametrically with strength A to the cavity oscillator Â = ñ(ω R − A¥)[â†â − ⟨â†â⟩] + Â M + Âdrive + Â k + Â r,
where ω R is the cavity resonance frequency at the equilib- rium cantilever position x = 0 in the presence of the mean radiation pressure, Â M = ħω M ĉ † ĉ is the mechanical os- cillator, H drive is the optical drive, H x is the cavity damp- ing, and Â r is the mechanical damping. For a cavity of length L , the coupling constant is A = + ω R / L . Opening the port used to supply the classical drive to the cavity also admits vacuum noise. Splitting the cavity field into classical and quantum parts â(t) = e- iωι ′ [ ā + d(t)] yields the photon number autocorrelation function (in the absence of coupling to the cantilever)
where Δ = ω L − ω R is the laser-cavity detuning and ñ is the mean photon number.
The symmetrized spectrum was used in [ 33 ] to estimate the effects of the shot noise (assumed to dominate over tech- nical laser noise). Note, however, that it is the asymmetry in the noise which leads to cooling or heating. The noise of the radiation pressure force ˆ F = ˙ hAˆ n at positive frequency, S FF (+ ω M ), corresponds to the ability of the cavity to absorb a quantum of energy from the cantilever, while noise at negative frequency SFF(-wM) corresponds to emission of energy into the cantilever. The net optical damping rate of the cantilever from Fermi's golden rule is
where xzpf is the cantilever zero-point fluctuation and S FF = h 2 A 2 S nn . This formula requires both Γ opt + Г M ≪ κ and a large mechanical quality factor, Q M = ω M / Γ M  » 1. Essentially, the interaction with î (three wave mixing) tries to Raman scatter the drive photons into the high density of states at the cavity resonance. For negative detuning △ < 0, this is uphill in energy [see Eq. ( 3 ) and Fig. 1(a)], and ¥ is cooled. A somewhat similar mechanism involving an oscillator with nonlinear damping was con- sidered by Dykman [ 37 ]. The nonequilibrium noise can be assigned a unique effective temperature T eff , provided the cantilever is per- fectly harmonic with weak optical and mechanical damp- ing, Γ M + Γ opt ≪ κ , so that it responds only at frequencies ±ω M . Thus, without mechanical damping the mean steady-state number of cantilever phonons ñ M is given by the detailed balance expression (for △ < 0)
Cooling (heating) is indicated by Teff ^ () Fig. 1 ). From Eq. ( 3 ) we obtain
For the special case of detuning Δ = −ω M (which is optimal for ω M ≫ κ ), we have the simple limit
which shows that for ω M ≥ κ , the ground state can be approached (provided Γ opt » Γ M ). For the case Δ = −ω , we obtain from Eq. ( 4 ):
After including the mechanical damping Γ M , a rate equa- tion yields the full expression for the mean steady-state phonon number (Fig. 2 ),
where ñ M is the equilibrium mechanical mode occupation number determined by the mechanical bath temperature. For ñ T ⁄ s 1, Eq. ( 9 ) reduces to the classical expression. Note that ground-state cooling is possible only for initial phonon numbers ñ T , ≪ Q M : We emphasize once more that a large detuning |Δ| = ω M ≫ κ offers the advantage of an arbitrarily small mini- mum phonon number ( 7 ). The only price to pay is to increase the input intensity, in order to keep constant the number of photons inside the cavity, ñ = ñ max /[1 + (2Δ/κ)²], where π max is the photon number at resonance, proportional to the input power. A large detuning increases the cooling efficiency in terms of the circulating power, thus limiting the heating of the cantilever by residual absorption of photons. The optomechanical frequency shift (see, e.g., [30,33] ) can also be derived from the force spectrum:
Equations (4) , (5) , (9) , and (10) are valid for arbitrary spectra S FF . These might result from the contribution of more than one mode, or relate to a different setup, e.g., a driven electrical circuit [ 32 ]. A related discussion [ 20 ] has considered cooling by a single-electron transistor. We now compare with commonly employed simpler models [ 24 , 26 , 32 ], where one postulates the light intensity to relax in an exponential fashion, dn/dt = γ(ñ(x) − n), with ñ(x) denoting the equilibrium photon number as a function of position. After linearization, one obtains an effective optomechanical damping rate [ 24 , 26 , 32 ] Γ opt ＝ (hA 2 /my)[1 + (ω M /y) 2 ]−1 ∂ . In general, for a fixed γ ∞ κ , this differs from the correct result ( 4 ), and it predicts a maximum Γ opt for γ ～ ω M . For radiation pressure cooling, one has to employ a detuning-dependent decay rate, γ ≡ [( $ )² + Δ ²]/κ, to recover Eq. ( 4 ) in the limit ω M ≪ κ. For bolometric forces, where the time lag is due to a finite heat relaxation rate γ ≪ κ, our quantum theory does not apply due to the dissipative nature of the force, while Γ ′ opt is valid in the semiclassical limit. The quantum result ( 4 ) for Γ opt can be reproduced by linearizing the equation for the complex light amplitude a [29,33], da/dt = i(△ + Ax)a −挙(a − ā). This still does not yield the correct steady-state phonon number ( 9 ), un- less zero-point fluctuations are included. To go beyond weak coupling, we now derive an exact solution of the linearized Heisenberg equations of motion for d and ĉ , using the input-output formalism [ 38 ] (see also [27] ):
Here the effective light amplitude has been expressed in terms of a frequency, α = ā(ω RXZPF/L) with |ā|² = ñ. The solution yields the cantilever spectrum S cc (ω) = { dteiω! { t (t) ĉ} :
where
We introduced the response functions of mirror and optical resonator, χ M (ω) = [ L M − i(ω − ω M )]− 1 and χ R (ω) = ''self-energy'' Σ(ω) = − i|α|²[χ R(ω) − χ R(−ω)]. The quantum noise results given previously are valid in the weak-coupling limit Γ M , Γ opt ≪ κ . Then, the optome- chanical damping and the ''optical spring'' frequency shift can be read off the self-energy as Im Σ(ωм) = −Γ opt/2 and Re Σ(ω M ) = δω M , coinciding with the expressions given above [Eqs. ( 4 ) and ( 10 )]. The steady-state average phonon number ñ M = ∫ f f S cc (ω) reproduces Eq. ( 9 ). The optical output spectrum (of ∂ out = ∂ in + √ R∂ ) dis- plays a Stokes peak at ω = − ω M and an anti-Stokes peak at ω = +ω M , with weights given by Γ opt(ñ M + 1)ñ M and Г optñм (ñ M + 1), respectively, which are the rates of pro- cesses leading to heating (cooling) of the cantilever by redshifting (blueshifting) a reflected photon. The ratio of the peak intensities thus provides a measure of the phonon number ñ , and the anti-Stokes peak vanishes upon reach- ing the mechanical ground state. The exact solution allows us also to discuss the regime of strong cooling. In the limit Γ opt ≈ 4 | α | 2 / κ ≫ Γ M , it adds a term ñ (assuming κ, α ≪ ω M , and Γ M ≪ κ ). Then a minimal phonon number ñmin ≥ ñmΓ M/κ is found as a function
of Γ opt at (Γ opt/κ)² = (Γ M/κ)(ñm/ñm)/2. For Γ opt/κ > 1/2, the mirror resonance peak splits into a pair of peaks at −ω M ± α , getting hybridized with the driven cavity mode (see arrow in inset of Fig. 2 ). At even larger photon number or smaller ω M , when α ≈ ω M /2, the static bistability [ 39 ] precludes reaching the desired detuning Δ = − ω M (see inset of Fig. 2 ). Thus, the far-detuned regime ω M ≫ κ has the additional strong advantage of avoiding the bistability, which already interferes with cooling in some current schemes [ 29 ]. Recent experiments [ 26 , 28 – 31 ] combine all the neces- sary ingredients and some of them [ 28 , 29 , 31 ] have already demonstrated ω M / K ～ 1, the threshold of the good-cavity regime. As an example of what might be possible in the near future, we mention estimates for two realistic setups: a microtoroidal  resonator  [ 29 ]  with  ω M / 2 π = 60 MHz, κ/2π = 50 MHz, and Q M = 10 4 , and a mechanical mem- brane  between  mirrors  [ 31 ]  with  ω M / 2π = 100 kHz, κ/2π = 8 kHz, and Q м = 4 · 10 6 . Both of these could be cooled down to ñ M < 1 after precooling to T = 300 mK. This could be achieved using P in ~ 5 mW (or P in ～ 1 nW, respectively), while remaining in the weak- coupling regime Γ opt ≪ κ ～ ω м . We have obtained the full quantum theory of cavity- assisted sideband cooling of a cantilever, based on the quantum noise approach applied to the fluctuations of the radiation pressure. Only in the previously unexplored re- gime of a detuning much larger than the cavity linewidth can the cantilever be cooled to arbitrarily small phonon numbers. The theory analyzed here may form the basis for future experiments that venture into the quantum regime of mechanical motion. Dykman, I. Wilson-Rae, We M. thank and T. Kippenberg for comments, and particularly J. Harris for useful contributions. This work was supported in part by the NSF under Grants No. ITR-0325580, No. DMR- 0342157, and No. DMR-0603369, by the NSERC, and by the SFB 631 and NIM of the DFG. Note added.—Essentially similar conclusions were reached recently in [40] . A. Erbe, C. Weiss, W. Zwerger, and R. H. Blick, Phys. [1] Rev. Lett. 87 , 096106 (2001). X. M. H. Huang, C. A. Zorman, M. Mehregany, and M. L. [2] Roukes, Nature (London) 421, 496 (2003). R. G. Knobel and A. N. Cleland, Nature (London) 424, [3] 291 (2003). [4] A. Naik et al. , Nature (London) 443 , 193 (2006). M. P. Blencowe and M. N. Wybourne, Physica [5] (Amsterdam) 280B , 555 (2000). S. Mancini, V. I. Manko, and P. Tombesi, Phys. Rev. A 55, [6] 3042 (1997). S. Bose, K. Jacobs, and P.L. Knight, Phys. Rev. A 59, [7] 3204 (1999). W. Marshall, C. Simon, R. Penrose, and D. Bouwmeester, [8] Phys. Rev. Lett. 91 , 130401 (2003). A. D. Armour, M. P. Blencowe, and K. C. Schwab, Phys. [9] Rev. Lett. 88 , 148301 (2002). J. Zhang, K. Peng, and S. L. Braunstein, Phys. Rev. A 68, [10] 013808 (2003). [11] M. Pinard, A. Dantan, D. Vitali, O. Arcizet, T. Briant, and A. Heidmann, Europhys. Lett. 72, 747 (2005). [12]  V. B. Braginsky and F. Y. Khalili, Quantum Measurement (Cambridge University Press, Cambridge, England, 1992). [13] M.D. LaHaye, O. Buu, B. Camarota, and K.C. Schwab, Science 304 , 74 (2004). [14] D. J. Wineland and W. M. Itano, Phys. Rev. A 20, 1521 (1979). [15] J. M. W. Milatz and J. J. van Zolingen, Physica (Amsterdam) 19 , 181 (1953). [16] P.F. Cohadon, A. Heidmann, and M. Pinard, Phys. Rev. Lett. 83, 3174 (1999). [17] D. Kleckner and D. Bouwmeester, Nature (London) 444, 75 (2006). [18] J. M. Courty, A. Heidmann, and M. Pinard, Eur. Phys. J. D 17, 399 (2001). [19] I. Martin, A. Shnirman, L. Tian, and P. Zoller, Phys. Rev. B 69, 125339 (2004). [20] A. A. Clerk and S. Bennett, New J. Phys. 7, 238 (2005). [21] M.P. Blencowe, J. Imbers, and A.D. Armour, New J. Phys. 7, 236 (2005). [22] I. Wilson-Rae, P. Zoller, and A. Imamoglu, Phys. Rev. Lett. 92 , 075507 (2004). [23] L. Tian and P. Zoller, Phys. Rev. Lett. 93 , 266403 (2004). [24] C. Höhberger-Metzger and K. Karrai, Nature (London) 432 , 1002 (2004); note that in Eq. (7), the frequency ratio should be dropped [K. Karrai (private communication)]. J. G. E. Harris, B. M. Zwickl, and A. M. Jayich, Rev. Sci. [25] Instrum. 78 , 013107 (2007). [26] S. Gigan et al. , Nature (London) 444 , 67 (2006). M. Paternostro, S. Gigan, M.S. Kim, F. Blaser, H.R. [27] Böhm, and M. Aspelmeyer, New J. Phys. 8, 107 (2006). [28] O. Arcizet, P.F. Cohadon, T. Briant, M. Pinard, and A. Heidmann, Nature (London) 444, 71 (2006). [29] A. Schliesser, P. Del'Haye, N. Nooshi, K. J. Vahala, and T. J. Kippenberg, Phys. Rev. Lett. 97 , 243905 (2006). [30] T. Corbitt, Y. Chen, E. Innerhofer, H. Muller-Ebhardt, D. Ottaway, H. Rehbein, D. Sigg, S. Whitcomb, C. Wipf, and N. Mavalvala, Phys. Rev. Lett. 98 , 150802 (2007). [31] J.D. F. Thompson, B. M. Zwickl, Jayich, A. M. Marquardt, S. M. and J. G. E. Harris, Girvin, arXiv:0707.1724. D. J. Wineland et al. , arXiv:quant-ph/0606180. [32] [33] V.B. Braginsky and S.P. Vyatchanin, Phys. Lett. A 293, 228 (2002). [34] T. Carmon, H. Rokhsari, L. Yang, T. J. Kippenberg, and K. J. Vahala, Phys. Rev. Lett. 94 , 223902 (2005). [35] F. Marquardt, J. G. E. Harris, and S. M. Girvin, Phys. Rev. Lett. 96 , 103901 (2006). [36] T. Corbitt et al. , Phys. Rev. A 74 , 021802(R) (2006). M. Dykman, Sov. Phys. Solid State 20 , 1306 (1978). [37] D. F. Walls and G. J. Milburn, Quantumoptics (Springer, [38] New York, 1995). A. Dorsel, J. D. McCullen, P. Meystre, E. Vignes, and H. [39] Walther, Phys. Rev. Lett. 51 , 1550 (1983). T. J. [40] I. Wilson-Rae, N. Nooshi, W. Zwerger, and Kippenberg, preceding Letter, Phys. Rev. Lett. 99, 093901 (2007).