Partial Optomechanical Refrigeration via Multimode Cold-Damping Feedback

Christian Sommer1 and Claudiu Genes1, 2

1Max Planck Institute for the Science of Light, Staudtstraße 2, D-91058 Erlangen, Germany2Department of Physics, University of Erlangen-Nuremberg, Staudtstraße 2, D-91058 Erlangen, Germany

(Dated: December 10, 2019)

We provide a fully analytical treatment for the partial refrigeration of the thermal motion ofa quantum mechanical resonator under the action of feedback. As opposed to standard cavityoptomechanics where the aim is to isolate and cool a single mechanical mode, the aim here is toextract the thermal energy from many vibrational modes within a large frequency bandwidth. Weconsider a standard cold-damping technique where homodyne read-out of the cavity output field isfed into a feedback loop that provides a cooling action directly applied on the mechanical resonator.Analytical and numerical results predict that low final occupancies are achievable independentlyof the number of modes addressed by the feedback as long as the cooling rate is smaller than theintermode frequency separation. For resonators exhibiting a few nearly degenerate pairs of modescooling is less efficient and a weak dependence on the number of modes is obtained. These scalingshint towards the design of frequency resolved mechanical resonators where efficient refrigeration ispossible via simultaneous cold-damping feedback.

In recent decades great progress has been accomplishedin laser cooling of microscopic and macroscopic objects,ranging from atoms, ions and molecular systems toselected modes of micro-mechanical oscillators or levi-tated nanoparticles [1, 2]. In standard cavity quantumoptomechanics with macroscopic mechanical resonatorsor levitated nanoparticles a crucial goal is to isolate andcool a given vibrational mode of interest [2–14] close to itsquantum ground state. This can for example be utilizedtowards high-precision sensing applications. Differenttechniques have been employed among which two standout: cavity resolved sideband cooling (or cavity-assistedcooling) [15–19] and feedback-aided cooling (in particularthe cold-damping technique) [20–28]. As mechanicalresonators typically exhibit a large number of vibrations,cooling of a single mode leaves the overall temperatureof the object largely unaltered. For regimes wheremany vibrational modes can be found within a singlecavity resonance it is interesting to ask what is theefficiency of cold-damping in the simultaneous reductionof the occupancy of a few modes. This could lead tothe partial refrigeration of mechanical resonators withenhanced sensing capabilities in a much larger frequencybandwidth.

We provide here a theoretical investigation of colddamping simultaneously applied to N mechanicalresonances [29–31] in order to provide a roadmap forpartial refrigeration. The analytical treatment consists infinding solutions for a set of quantum Langevin equationsdescribing the evolution of N vibrational modes coupledto a single optical mode and to thermal and opticalreservoirs. In this sense our approach is general andcould be tailored to a variety of systems such as a setof nano-particles in dipole traps, or ions in an ion trap,where the thermal excitations is distributed into Ncollective oscillation modes. However, for the simplicityof presentation we will only refer to the system depictedin Fig. 1 consisting of an optomechanical cavity with amovable end-mirror or highly-reflective membrane. The

-

Laser

Feedback

Membrane

w1

w2

w3

Cavity

Figure 1. Multi-mode cold damping. The output of a drivenoptomechanical system is homodyne detected providing in-formation on the collective displacement of many vibrationalmodes (bright mode). A feedback loop is utilized to design adirect force to be applied onto the resonator that can lead thefreezing of its thermal fluctuations (cold-damping effect).

cavity output with frequency components correspondingto the radiation pressure coupling to many vibrationalmodes, is passed through a feedback device that allows forthe choice of a correct back-action onto the mechanicalresonator that can compensate the heating effect of theenvironment. We analyze the final occupancies of allmodes undergoing cold damping and compare this tothe expected results obtained for cooling of independentisolated modes [20]. As the readout field providesinformation solely on a bright mode (the generalizedquadrature that the cavity field couples to) the feedbackonly directly cools this mode. It would then be expectedthat the uncoupled N − 1 orthogonal dark modes wouldconsiderably slow down the overall refrigeration process.This would be indeed the case for completely degeneratemechanical modes where only 1/N of the total thermalenergy can be removed. Instead, for non-degeneratemodes, the main theoretical result of this calculation(backed by numerical simulations) indicates that the

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efficiency of the cooling process is roughly independentof the number of modes undergoing cooling dynamics.This occurs as inter-mode frequency mismatches offerdark to bright mode couplings leading to an efficientsympathetic cooling mechanism for all modes involved.A detrimental aspect of increasing the number of modesis the enhanced probability of finding consecutive modesof nearly degenerate frequency. The result previouslypointed out in the case of resolved sideband cooling of twomechanical modes, holds here in the cold-damping caseas well, which is that the cooling efficiency is degraded asmodes approach the point of degeneracy [32, 33]. Thisindicates a roadmap to efficient partial refrigerationconsisting of the design of mechanical resonators withclose to linearly spaced normal mode frequencies wherethe optical cooling rate is only limited by the minimalinter-mode frequency separation.

Linearized Langevin equations — We consider an op-tomechanical cavity with a movable end-mirror or highly-reflective membrane exhibiting N independent modes ofvibrations each of effective mass mj and frequency ωj(as depicted in Fig. 1). The quantum motion of the me-chanical modes is described by the displacement Qj andmomentum Pj quadrature operators with standard com-mutations [Qj , Pj′ ] = i~δjj′ . A single cavity mode atfrequency ω and loss rate κ is described by a bosonicoperator A with [A,A†] = 1. The optomechanical in-teraction is a standard radiation pressure Hamiltonian∑j ~g

(j)OMA

†AQj where g(j)OM is the single photon single

phonon coupling rate. Driving is executed with a laserof power P and frequency ω` via the non-moving mirrorat rate ε =

√2Pκ/~ω` (where κ is the photon loss rate).

We follow a standard quantum Langevin treatment ofoptomechanics [20] (see Appendix) where the operatorsare split into classical averages plus zero-average quan-tum fluctuations: A = 〈A〉 + a, Qj = 〈Qj〉 + qj andPj = 〈Pj〉 + pj . The problem then becomes linear inthe limit | 〈A〉 | 1 and the following set of Langevinequations can be written for the fluctuations:

qj = ωjpj , (1a)

pj = −ωjqj − γjpj +Gjx− gj ∗ yest + ξj , (1b)

x = −κx+√

2κxin, (1c)

y = −κy +∑Nj=1Gjqj +

√2κyin, (1d)

where x = (a + a†)/√

2 and y = i(a† − a)/√

2 are thequadratures of the cavity field fluctuations and xin, yin

are the corresponding optical input noise terms similarlydefined from the input optical noise operator ain. Thezero-average noise terms are delta-correlated in time〈ain(t)ain†(t′)〉 = δ(t− t′). The effective optomechanical

couplings Gj =√

2g(j)OM 〈A〉 are enhanced by the large

cavity field amplitude. We set the condition that the

effective cavity detuning ∆ = ω − ω` −∑j g

(j)OM 〈Qj〉,

containing a collective mechanically-induced frequencyshift is kept at zero value (see Appendix). For each

mode j thermalization with the environment is describedby a zero-averaged Gaussian stochastic noise term ξjand a rate γj . The fluctuation-dissipation relation isfulfilled as described by the two-time correlation function

〈ξj(t)ξj′(t′)〉 = γj/ωj∫ Ω

0dω/2πe−iω(t−t′)Sth(ω)δjj′

where Ω is the frequency cutoff of the reservoir andSth(ω) = ω[coth (~ω/2kBT ) + 1] is the thermal noisespectrum. The correlation function becomes a stan-dard white noise input with delta correlations bothin frequency and time for sufficiently high tempera-tures kBT ~ωj . This results in the approximateform 〈ξj(t)ξj′(t′)〉 ≈ (2nj + 1)γjδ(t − t′)δjj′ , wherethe occupancy of each vibrational mode is given bynj = 1/ [exp(~ωj/kBT )− 1] ≈ kBT/~ωj .Notice that the cavity field only provides information ona generalized mode which we dub bright mode obtainedas a linear combination of individual modes quadratures.All other collective modes orthogonal to this one couldthen be defined as dark modes and apparently donot participate in the cooling dynamics. However,as we will show, intermodal correlations are built-upduring the dynamics and all individual modes are affected.

Feedback loop — The feedback force on the j’s modegiven by the convolution term (gj ∗ y)(t) =

∫∞−∞ dsgj(t−

s)y(s) depends on past dynamics of the detected quadra-ture y that is driven by the weighted sum of the oscillatorfluctuations qj . Here, the causal kernel

gj(t) = g(j)cd ∂t

[θ(t)ωfbe

−ωfbt]

(2)

contains the feedback gain terms g(j)cd and feedback band-

width ωfb. Notice that in the limit ωfb →∞ the feedback

becomes gj(t) = g(j)cd δ′(t). The component injected into

the feedback loop yest is the estimated intracavity phasequadrature. This results from a measurement of the out-put quadrature yout =

√2κy(t)− yin(t) and additionally

considering a detector with quantum efficiency η (mod-eled by an ideal detector preceded by a beam splitter withtransmissivity

√η, which mixes the input field with an

uncorrelated vacuum field yv(t)). The estimated signal isthen written as

yest(t) = y(t)− yin(t) +√η−1 − 1yv(t)√2κ

. (3)

Multi-mode cold damping — We proceed by firstformally eliminating the dynamics of the cavity quadra-tures to find a set on 2N nonlinear differential equa-tions for the dynamics of mechanical modes. Underthe conditions of fast feedback and fast cavity dynam-ics, we then can simplify this to a set of linear differ-ential equations with analytical solutions. We startby writing a formal solution for the y-quadrature as

y(t) =∫ t−∞ dse−κ(t−s)

(∑Nj=1Gjqj(s) + yin(s)

)and es-

timate the effect of the cold-damping convolution termappearing as a drive term for the membrane’s momentum

3

in Eq. 1b (see Appendix)

(gj ∗ y)(t) = g(j)cd ωfb

∑Nk=1Gkωk(h ∗ pk)(t) + ξyin .

The right-hand side term contains the added cold-dampingdecay rate acting on the j’s mode as well as some cross-terms which dissipatively couple distinct modes. In thelossy cavity limit κ ωj and fast feedback ωfb ωj thefirst term of the convolution above can be approximatedby an instantaneous value pj/(κωfb). This allows one toturn Eqs. 1 into a set of linear differential equations:

qj = ωjpj , (4a)

pj = −ωjqj − Γjjpj −∑k 6=j Γjkpk + ξj + ξopt,j . (4b)

The added optical noise ξopt,j is a sum of a feedbackinduced noise term as well as the direct radiation pressurenoise. The diagonal term contains the cold-dampingoptical loss

Γjj = γj +g

(j)cd Gjωjκ

, (5)

expected for a single isolated uncoupled mode [20]. Thepresence of adjacent modes leads to a dissipative cross-talk

at similar rates Γjk = g(j)cd Gkωk/κ. The back-action both

due to feedback as well as to radiation pressure effects iscontained within the term ξopt which adds to the thermal

noise. In a simplified regime where all Γjk = Γ the gen-

eralized quadrature∑j pj/

√N defines the bright mode

which is directly addressed by the feedback loop. Fordegenerate frequencies this is the only mode effectivelycooled at a damping rate NΓ. All other N − 1 collectivemodes orthogonal to the bright one become effectivelydark - not directly addressed by the feedback loop. As thethermal energy is equally spread in all collective modesonly 1/N of total energy of the resonator is extracted. Inthe non-degenerate case, frequency mismatches lead tocoupling of dark modes to the bright mode and thus to amechanism of efficient sympathetic cooling of all degreesof freedom present (see Appendix for the derivation ofdark-bright mode couplings similar to the procedure ap-plied in Ref. [34]).The 2N equations of motion can be cast in acompact form given by v = Mv + nin wherev = (q1, p1, q2, p2, . . . , qN , pN )> and nin = (0, ξ1 +ξopt,1, . . . , 0, ξN + ξopt,N )>. In the case that the solutionis stable and all the eigenvalues of the matrix M have neg-ative real parts, the system achieves a steady state fullycharacterized by the covariance matrix V = 〈v(t)v>(t)〉.Under the assumption that the cavity is lossy and thefeedback is fast one can then show that the diffusion ma-trix is delta-correlated 〈nin(t)n>in(t′)〉 = Dinδ(t− t′) andthe covariance matrix is computed by solving a steadystate Lyapunov equation MV + VM> = −Din. For

Γjj ≈ (g(j)cd Gjωj)/κ one can then analytically estimate

the momentum and displacement fluctuation variancesas:

〈p2i 〉 =

(ni +

1

2

)γiΓii

+G2i

2Γiiκ+∑j 6=i

Γij2Γii

(ω2i Γjj + ω2

jΓii)

Λij(ω2i − ω2

j

)2 +∑k 6=i,j

1(ω2i − ω2

j

) ( ω2i ΓjkΛik

(ω2i − ω2

k)−ω2jΓikΛjk(ω2j − ω2

k

)) ,

(6a)

〈q2i 〉 = 〈p2

i 〉+∑j 6=i

ΓijΛij

2(ω2i − ω2

j

) , (6b)

where we have defined the following quantity

Λij :=g

(j)cd

g(i)cd

(2ni + 1)γi +g

(i)cd

g(j)cd

(2nj + 1)γj +(g

(j)cd Gi − g

(i)cdGj)

2

κg(i)cd g

(j)cd

. (7)

Notice that the apparent divergence in the denominatorscomes from neglecting the bare mechanical damping rateγi with respect to any of the diagonal or mutual dampingrates Γij . The regime of interest where we compare

analytical results with numerical simulations assumesnon-degenerate modes where the frequency separationis much larger than any γi. Exact results (a bit morecumbersome) can however be obtained for any case as

4

(a) (b) (c) (e)

gcd

G

(f)Dw

1 3 5 7

(d)

1 3 5 7

Figure 2. Multimode cold damping. Comparison between cooling efficiency in the case of linear dispersion in (a) versus dispersionrelation showing two quasi-degenerate modes in (b) for a resonator with eight independent vibrational modes. The initial(orange disks) and final occupancies (black disks), obtained from a Monte Carlo simulation, are displayed and compared to thepredictions of the Lyapunov equation (red stars). Results obtained from a simplification assuming independent damping ofeach oscillator are indicated by gray stars. For nearly-degenerate modes cooling is strongly inhibited while the other modes arevirtually unaffected. In (c),(d) the average energies are presented as a function of time for case (a) in (c) and case (b) in (d). In(e) dependence of the occupation number normalized final energy with respect to the frequency difference between two modes isobtained from a Monte Carlo simulation (orange solid line) and via solving the Lyapunov equation (black dashed line). In (f)the final occupancy of an individual mode is plotted versus variations of the intra-cavity light amplitude (affecting all the Gj)and feedback gain.

shown in the Appendix.

Discussions — A few observations can immediately bemade on the expressions above. First, one notices that theequipartition theorem is generally not fulfilled signifyingthat the final state cannot be described by an effectivetemperature. Then, in the absence of mode-mode cou-pling (where all Γij are set to zero) the result is theexpected one [20] where the final state fulfills the equipar-tition theorem and achieves an occupancy of roughlyniγi/Γii. Notice that there is also a residual occupancycoming from the noise in the x quadrature; this can beneglected under the assumption that the cavity-assistedcooling rate G2

i /2κ is much smaller than the feedbackdamping rate Γii easily fulfilled when the gain is large

enough such that g(i)cd ωi Gi. Extra terms arise from

inter-mode momentum-momentum 〈pipj〉 and momentum-displacement 〈piqj〉 quantum correlations due to the factthat the feedback force term contains a sum of all mo-mentum quadratures.In the limit of linear dispersion relation where ωj ≈ω + j∆ω and ω ∆ω the final occupancy can be easilysimplified to (see Appendix)

Ei ≈ niγiΓii

+1

4∆ω2

∑〈i,j〉j 6=i

Γij2

(1 +

ΓjjΓii

)Λij

+∑〈i,j,k〉k 6=i,j

1

Γii

(ΓjkΛik

(i− j)(i− k)− ΓikΛjk

(i− j)(j − k)

),

(8)

where we have defined the energy per mode as

Ei = (〈p2i 〉 + 〈q2

i 〉)/2 and where we can approximateΛij ≈ (2ni + 1)γi + (2nj + 1)γj . Notice that the sumsare only performed on the nearest neighbors in frequencyspace. This is a crucial aspect as one sees that the heatingeffects stemming from the coupling to the neighbormodes do not scale with the number of modes N butonly depend on the relative inter-mode minimal distance.For the case that Γii,Γij ∆ω the expression aboveindicates that the final occupancies are close to whatwould have been expected from N independent feedbackloops, each specialized to a given mode.The comparison between numerical and analyticalresults in Fig. 2a,b indicate that nearly-degeneratemodes are cooled with a lower efficiency. In fact, theexpression above reproduces well this effect showingthat the maximum optical cooling rate cannot exceedthe inter-mode frequency separation ∆ω. To furtherelucidate this aspect, we focus in Fig. 2e on a targetmode at frequency ω and plot its final occupancy in thepresence of an adjacent mode at frequency ω + ∆ω asa function of the frequency separation between the twomodes. The final occupancy of mode ω is unaffectedfor ∆ω > Γ while at near degeneracy the cooling iscompletely inefficient. In the intermediate regime, thescaling with (∆ω)−2 predicted by Eq. 8 holds.While the damping rate includes a product of the cavityintra-cavity field amplitude (via the coefficients Gj) with

the feedback gain (via the terms g(j)cd ), back-action noise

limits the final achievable occupancy. Figure. 2f showsa density plot of the final occupancy of a target modeas a function of these two parameters. We have kept

the relative ratios of all the parameters Gj and g(j)cd and

5

simultaneously varied them. The result shows that pastthe optimal regime for cooling the radiation pressure andfeedback noise add to the achievable final occupancy.

Conclusions—We have theoretically analyzed prospectsfor using a single cold-damping feedback loop for thepartial refrigeration of a multi-mode mechanical resonatorwithin a large frequency window. As a main resultwe have derived an analytical expression for the finaloccupancy of the modes undergoing cooling dynamics.The expression predicts a generalization of a previouslyknown result obtained in the case of two mechanicalmodes, i.e. that efficient refrigeration requires theabsence of nearly-degenerate vibrations in the mechanicalspectrum of the resonator. This can be understood as asympathetic cooling mechanism where a bright collectivemode is directly damped by the feedback loop andprovides sympathetic cooling of dark modes to which iscoupled via frequency disorder. Under these conditions,simultaneous cooling can achieve final temperatures closeto the case of isolated mode addressing.

While we have illustrated our approach on the manymodes of a macroscopic mechanical resonator, ouranalysis is quite general as it can also be applied to thecooling of collective vibrational modes of ions in iontraps, of atoms in dipole traps, levitated nanoparticles inoptical tweezers etc. For interacting systems held in thesame externally designed potential (as is the case of iontraps), the frequency spectrum of the collective modes

can be tailored to eliminate near degeneracy pointsand single particle addressing ensures a simultaneousdriving of all collective modes. For mesoscopic systemsexhibiting a limited number of mechanical resonancesthe mechanism can lead to the full refrigeration of theobject’s motion. This could be for example achieved intwo-dimensional atomic array optomechanics [35, 36] orin ensembles of interacting levitated nanoparticles inneighboring optical tweezer traps.

Finally, let us comment on the assumed form ofcold-damping feedback: in Eq. (2) we have assumed thatthe electronic loop can provide an instantaneous feedbackonto the system. This assumption is contained in theargument of the Heaviside function θ(t). This assumesfast electronics which can respond much quicker thanthe oscillation time of the system. For some currentexperiments [23] feedback delay is an issue as it can leadto inadvertent heating of the target mode instead of theenvisioned cooling [37]. The effect of a time delay τfbcan be included in the argument θ(t − τfb) and will beaddressed in the future following a similar formalism.

Acknowledgments - We acknowledge financial supportfrom the Max Planck Society and from the German Fed-eral Ministry of Education and Research, co-funded bythe European Commission (project RouTe), project num-ber 13N14839 within the research program ”PhotonikForschung Deutschland”. We acknowledge initial discus-sions with Muhammad Asjad and very useful commentson the manuscript from Andre Xuereb and David Vitali.

[1] H. J. Metcalf and P. van der Straten, Laser Cooling andTrapping (Springer, New York, 1999).

[2] M. Aspelmeyer, T. J. Kippenberg, and F. Marquardt,“Cavity optomechanics,” Rev. Mod. Phys. 86, 1391–1452(2014).

[3] D. Windey, C. Gonzalez-Ballestero, P. Maurer,L. Novotny, O. Romero-Isart, and R. Reimann, “Cavity-based 3d cooling of a levitated nanoparticle via coherentscattering,” Phys. Rev. Lett. 122, 123601 (2019).

[4] U. Delic, M. Reisenbauer, D. Grass, N. Kiesel, V. Vuletic,and M. Aspelmeyer, “Cavity cooling of a levitatednanosphere by coherent scattering,” Phys. Rev. Lett. 122,123602 (2019).

[5] M. Rossi, N. Kralj, S. Zippilli, R. Natali, A. Borrielli,G. Pandraud, E. Serra, G. Di Giuseppe, and D. Vitali,“Enhancing sideband cooling by feedback-controlled light,”Phys. Rev. Lett. 119, 123603 (2017).

[6] J. B. Clark, F. Lecocq, R. W. Simmonds, J. Aumentado,and J. D. Teufel, “Sideband cooling beyond the quantumbackaction limit with squeezed light,” Nature 541, 191EP – (2017).

[7] L. Qiu, I. Shomroni, S. P, and T. J. Kippenberg, “High-fidelity laser cooling to the quantum ground state ofa silicon nanomechanical oscillator,” arXiv:1903.10242(2019).

[8] P. Asenbaum, S. Kuhn, S. Nimmrichter, U. Sezer, andM. Arndt, “Cavity cooling of free silicon nanoparticles

in high vacuum,” Nature Communications 4, 2743 EP –(2013).

[9] S. Mancini, D. Vitali, and P. Tombesi, “Optomechanicalcooling of a macroscopic oscillator by homodyne feedback,”Phys. Rev. Lett. 80, 688–691 (1998).

[10] C. Schafermeier, H. Kerdoncuff, U. B. Hoff, H. Fu,A. Huck, J. Bilek, G. I. Harris, W. P. Bowen, T. Gehring,and U. L. Andersen, “Quantum enhanced feedback cool-ing of a mechanical oscillator using nonclassical light,”Nature Communications 7, 13628 EP – (2016).

[11] N. Kiesel, F. Blaser, U. Delic, D. Grass, R. Kaltenbaek,and M. Aspelmeyer, “Cavity cooling of an opticallylevitated submicron particle,” Proceedings of the Na-tional Academy of Sciences 110, 14180–14185 (2013),https://www.pnas.org/content/110/35/14180.full.pdf.

[12] J. Millen, P. Z. G. Fonseca, T. Mavrogordatos, T. S.Monteiro, and P. F. Barker, “Cavity cooling a singlecharged levitated nanosphere,” Phys. Rev. Lett. 114,123602 (2015).

[13] B. Rodenburg, L. P. Neukirch, A. N. Vamivakas, andM. Bhattacharya, “Quantum model of cooling and forcesensing with an optically trapped nanoparticle,” Optica3, 318–323 (2016).

[14] K. E. Khosla, G. A. Brawley, M. R. Vanner, and W. P.Bowen, “Quantum optomechanics beyond the quantumcoherent oscillation regime,” Optica 4, 1382–1387 (2017).

[15] S. Gigan, H. R. Bohm, M. Paternostro, F. Blaser,

6

G. Langer, J. B. Hertzberg, K. C. Schwab, D. Bauerle,M. Aspelmeyer, and A. Zeilinger, “Self-cooling of a mi-cromirror by radiation pressure,” Nature 444, 67–70(2006).

[16] V. B. Braginsky, S. E. Strigin, and S. P. Vyatchanin,“Parametric oscillatory instability in Fabry–Perot interfer-ometer,” Phys. Lett. A 287, 331 (2001).

[17] F. Marquardt, J. P. Chen, A. A. Clerk, and S. M. Girvin,“Quantum theory of cavity-assisted sideband cooling ofmechanical motion,” Phys. Rev. Lett. 99, 093902 (2007).

[18] I. Wilson-Rae, N. Nooshi, W. Zwerger, and T. J. Kippen-berg, “Theory of ground state cooling of a mechanicaloscillator using dynamical backaction,” Phys. Rev. Lett.99, 093901 (2007).

[19] J. D. Teufel, T. Donner, D. Li, J. W. Harlow, M. S.Allman, K. Cicak, A. J. Sirois, J. D. Whittaker, K. W.Lehnert, and R. W. Simmonds, “Sideband cooling ofmicromechanical motion to the quantum ground state,”Nature 475, 359–363 (2011).

[20] C. Genes, D. Vitali, P. Tombesi, S. Gigan, and M. As-pelmeyer, “Ground-state cooling of a micromechanicaloscillator: Comparing cold damping and cavity-assistedcooling schemes,” Phys. Rev. A 77, 033804 (2008).

[21] V. Steixner, P. Rabl, and P. Zoller, “Quantum feedbackcooling of a single trapped ion in front of a mirror,” Phys.Rev. A 72, 043826 (2005).

[22] P. Bushev, D. Rotter, A. Wilson, F. m. c. Dubin,C. Becher, J. Eschner, R. Blatt, V. Steixner, P. Rabl,and P. Zoller, “Feedback cooling of a single trapped ion,”Phys. Rev. Lett. 96, 043003 (2006).

[23] M. Rossi, D. Mason, J. Chen, Y. Tsaturyan, andA. Schliesser, “Measurement-based quantum control ofmechanical motion,” Nature 563, 53–58 (2018).

[24] P. F. Cohadon, A. Heidmann, and M. Pinard, “Coolingof a mirror by radiation pressure,” Phys. Rev. Lett. 83,3174–3177 (1999).

[25] M. Poggio, C. L. Degen, H. J. Mamin, and D. Rugar,“Feedback cooling of a cantilever’s fundamental mode be-low 5 mk,” Phys. Rev. Lett. 99, 017201 (2007).

[26] D. J. Wilson, V. Sudhir, N. Piro, R. Schilling, A. Ghadimi,and T. J. Kippenberg, “Measurement-based control ofa mechanical oscillator at its thermal decoherence rate,”Nature 524, 325 EP – (2015).

[27] G. P. Conangla, F. Ricci, M. T. Cuairan, A. W. Schell,N. Meyer, and R. Quidant, “Optimal feedback cooling ofa charged levitated nanoparticle with adaptive control,”Phys. Rev. Lett. 122, 223602 (2019).

[28] F. Tebbenjohanns, M. Frimmer, A. Militaru, V. Jain,and L. Novotny, “Cold damping of an optically levitatednanoparticle to microkelvin temperatures,” Phys. Rev.Lett. 122, 223601 (2019).

[29] W. H. P. Nielsen, Y. Tsaturyan, C. B. Møller, E. S.Polzik, and A. Schliesser, “Multimode optomechani-cal system in the quantum regime,” Proceedings ofthe National Academy of Sciences 114, 62–66 (2017),https://www.pnas.org/content/114/1/62.full.pdf.

[30] P. Piergentili, L. Catalini, M. Bawaj, S. Zippilli, N. Mal-ossi, R. Natali, D. Vitali, and G. D. Giuseppe, “Two-membrane cavity optomechanics,” New Journal of Physics20, 083024 (2018).

[31] X. Wei, J. Sheng, C. Yang, Y. Wu, and H. Wu, “Control-lable two-membrane-in-the-middle cavity optomechanicalsystem,” Phys. Rev. A 99, 023851 (2019).

[32] C. Genes, D. Vitali, and P. Tombesi, “Simultaneous cool-

ing and entanglement of mechanical modes of a micromir-ror in an optical cavity,” New Journal of Physics 10,095009 (2008).

[33] C. F. Ockeloen-Korppi, M. F. Gely, E. Damskagg,M. Jenkins, G. A. Steele, and M. A. Sillanpaa, “Sidebandcooling of nearly degenerate micromechanical oscillatorsin a multimode optomechanical system,” Phys. Rev. A99, 023826 (2019).

[34] J. del Pino, F. A. Y. N. Schroder, A. W. Chin, J. Feist,and F. J. Garcia-Vidal, “Tensor network simulation ofnon-markovian dynamics in organic polaritons,” Phys.Rev. Lett. 121, 227401 (2018).

[35] E. Shahmoon, D. M. Lukin, and F. S. Yelin, “Quan-tum optomechanics of a two-dimensional atomic array,”arXiv:1810.01052 (2018).

[36] E. Shahmoon, M. D. Lukin, and S. F. Yelin, “Chapterone - collective motion of an atom array under laserillumination,” (Academic Press, 2019) pp. 1 – 38.

[37] S. Zippilli, N. Kralj, M. Rossi, G. Di Giuseppe, andD. Vitali, “Cavity optomechanics with feedback-controlledin-loop light,” Phys. Rev. A 98, 023828 (2018).

[38] K. Jacobs, Stochastic Processes for Physicists: Under-standing Noisy Systems (Cambridge University Press,New York, 2010).

7

Appendix A: Linearized quantum Langevin equations in optomechanics

We start with a quantum formulation of the system’s dynamics of a few independent oscillation modes withfrequencies ωj of a membrane resonator (where j = 1, ...N). The equations of motion for the collection of modes arewritten as

Qj = ωjPj , (A1a)

Pj = −ωjQj − γjPj + g(j)OMA

†A+ ξj , (A1b)

A = −(κ+ i∆0)A+ i∑Nj=1 g

(j)OMAQj + ε+

√2κain, (A1c)

in terms of dimensionless position and momentum quadratures Qj and Pj for each of the N independent membraneoscillation modes. The term ∆0 = ωc − ω` describes the detuning of the cavity resonance frequency ωc from the laserfrequency ωl and κ its decay rate. The oscillator frequencies are given by ωj . The radiation pressure coupling is given

by g(j)OM for the j’s mode and the input laser power by ε =

√2Pκ/~ω`. The damping of the j’s resonator mode is

described by the parameter γj and is with the associated zero-averaged Gaussian stochastic noise term leading tothermalization with the environment. The noise term can be fully described by the two-time correlation function:

〈ξj(t)ξj′(t′)〉 =γjωj

∫ Ω

0

dω

2πe−iω(t−t′)Sth(ω)δjj′ , (A2)

where Ω is the frequency cutoff of the reservoir and Sth(ω) = ω[coth (~ω/2kBT ) + 1] is the thermal noise spectrum.The correlation function becomes a standard white noise input with delta correlations both in frequency and time forsufficiently high temperatures kBT ~ωj . This results in the approximate form 〈ξj(t)ξj′(t′)〉 ≈ (2nj + 1)γjδ(t− t′)δjj′ ,where the occupancy of each vibrational mode is given by nj = (exp(~ωj/kBT )− 1)−1 ≈ kBT/~ωj . The cavity inputnoise is described by ain and follows the the correlation function 〈ain(t)a†in(t′)〉 = δ(t− t′). For an intense cavity fieldand rewriting the operators A = 〈A〉+ a, Qj = 〈Qj〉+ qj and Pj = 〈Pj〉+ pj as a sum of their expectation value and afluctuation term we can simplify the equations of motion in Eq. A1a and obtain for the equations of motion for theexpectation values

〈Qj〉 = ωj〈Pj〉, (A3a)

〈Pj〉 = −ωj〈Qj〉 − γj〈Pj〉+ g(j)OM|〈A〉|

2, (A3b)

〈A〉 = −(κ+ i∆0)〈A〉+ i

N∑j=1

g(j)OM〈A〉〈Qj〉+ ε, (A3c)

which at steady (〈Qj〉 = 〈Pj〉 = 〈A〉 = 0) state results in

〈A〉 =ε[

κ+ i

(∆0 −

∑j

(g(j)OM

)2

ωj|〈A〉|2

)] , (A4)

where ∆ = ∆0 −∑j

(g(j)OM

)2

ωj|〈A〉|2 is the effective cavity detuning including radiation pressure effects and 〈Qj〉 =

(g(j)OM/ωj)|〈A〉|2.

For the fluctuations where we can omit all nonlinear terms a†a and aqj since |〈A〉| 1, we obtain the linearizedequations of motion

qj = ωjpj , (A5a)

pj = −ωjqj − γjpj + ξj +Gjx, (A5b)

x = −κx+ ∆y +√

2κxin, (A5c)

y = −κy −∆x+∑Nj=1Gjqj +

√2κyin, (A5d)

where x = (1/√

2)(a+a†) and y = (i/√

2)(a†−a) are the quadratures of the cavity field and xin and yin are formulated

correspondingly. The effective optomechanical coupling terms are given by Gj =√

2g(j)OM〈A〉.

8

Appendix B: Multimode cold damping

For cold damping with many resonator modes in the quantum mechanical treatment we start with the equations ofmotion given by

qj = ωjpj , (B1a)

pj = −ωjqj − γjpj +Gjx+ ξj −∫ ∞−∞

dsgj(t− s)yest(s), (B1b)

x = −κx+√

2κxin, (B1c)

y = −κy +∑Nj=1Gjqj +

√2κyin, (B1d)

where the effective cavity detuning is kept at zero ∆ = 0.

Feedback details

The quadrature component that is injected into the feedback mechanism yest is the estimated intracavity phasequadrature. This results from a measurement of the output quadrature yout =

√2κy(t)−yin(t) additionally considering a

detector with quantum efficiency η which is modeled by an ideal detector preceded by a beam splitter with transmissivity√η, which mixes the input field with an uncorrelated vacuum field yv(t). The estimated phase quadrature is decribed

by

yest(t) = y(t)− yin(t) +√η−1 − 1yv(t)√2κ

. (B2a)

Eliminating the cavity quadratures

We can eliminate the cavity field quadratures by formally integrating their equations of motion to obtain:

x(t) = Gj

∫ t

−∞dse−κ(t−s)xin(s), (B3a)

y(t) =

∫ t

−∞dse−κ(t−s)

N∑j=1

Gjqj(s) +√

2κ

∫ t

−∞dse−κ(t−s)yin(s). (B3b)

The yest(s) term introduces both terms proportional to the qj as well as noise terms stemming from the cavity inputnoise yin(s) as well as from the vacuum filled port noise yv. We can first work out the terms coming from y as:

(gj ∗ y) = g(j)cd ωfb

∫ ∞−∞

dse−ωfb(t−s)δ(t− s)y(s)− g(j)cd ω

2fb

∫ ∞−∞

dsθ(t− s)e−ωfb(t−s)y(s)

=

∫ t

−∞dsκe−κ(t−s) − ωfbe

−ωfb(t−s)

(κ− ωfb)

[∑Nk=1 g

(j)cd ωfbGkqk(s) +

√2κg

(j)cd ωfby

in(s)]. (B4a)

To obtain a dependence with respect to pj we apply integration by parts noticing that the convolution above containsa derivative of the following function:

h(t− s) =e−κ(t−s) − e−ωfb(t−s)

ωfb − κ(B5)

and the relation qj = ωjpj and we obtain

(gj ∗ y) =

N∑k=1

g(j)cd ωfbGkωk

∫ t

−∞dsh(t− s)pk(s) +

√2κg

(j)cd ωfb

∫ t

−∞ds∂sh(t− s)yin(s) (B6)

9

We can now write in simplified notation the reduced set of equations of motion for the 2N resonator modes quadratures

qj = ωjpj , (B7a)

pj = −ωjqj −∫ ∞−∞

ds(γjδ(t− s) + g

(j)cd ωfbGjωjθ(t− s)h(t− s)

)pj(s)−

∑k 6=j

g(j)cd ωfbGkωk

∫ t

−∞dsh(t− s)pk(s)

+ ξj + ξfb + ξvac + ξrp. (B7b)

The three sources of noise are owed to the direct feedback action, to the feedback filtered vacuum action in the lossport and to the intra-cavity radiation pressure effect:

ξfb = −g

(j)cd ωfb√

2κ

∫ ∞−∞

dsφ1(t− s)yin(s), (B8a)

ξvac = −g

(j)cd ωfb√

2κ

√η−1 − 1

∫ ∞−∞

dsφ2(t− s)yv(s), (B8b)

ξrp =√

2κGj

∫ ∞−∞

dsφ3(t− s)xin(s), (B8c)

with the following definitions

φ1(t) = θ(t)(ωfb(ωfb + κ)e−ωfbt − 2κ2e−κt)/(ωfb − κ)− δ(t), (B9a)

φ2(t) = θ(t)ωfbe−ωfbt − δ(t), (B9b)

φ3(t) = θ(t)e−κt, (B9c)

for the convolution kernels.

Performing the fast-feedback fast-cavity approximation

In the limit of lossy cavity and fast feedback where κ, ωfb ωj we can estimate∫ t

−∞dsh(t− s)pj(s) ≈ pj(t)

∫ t

−∞dsh(t− s) =

pj(t)

ωfbκ(B10)

and we end up with a set of coupled linear differential equations for the mechanical mode quadratures:

qj = ωjpj , (B11a)

pj = −ωjqj − Γjjpj −∑k 6=j

Γjkpk + ξj + ξfb + ξvac + ξrp. (B11b)

The off-diagonal rates are defined as

Γjk = (g(j)cd Gkωk)/κ (B12)

while the diagonal damping rates are the expected independent cooling rates

Γjj = γj + (g(j)cd Gjωj)/κ (B13)

Solving the Lyapunov equation

The set of differential equations presented in Eq. B11a can be cast into the form

v = Mv + nin (B14a)

10

with v = (δq1, δp1, . . . δqN , δpN )> and nin = (0, η1, . . . , 0, ηN ) where all noise terms have been gathered into a singleterm ηj = ξj + ξfb + ξvac + ξrp. From the general solution

v(t) = eM(t−t0)v(t0) +

∫ t

t0

dseM(t−s)nin(s) (B15)

we obtain the correlation matrix

V = 〈v(t)v>(t)〉 =

∫ t

t0

ds

∫ t

t0

ds′eM(t−s)〈nin(s)n>in(s′)〉eM>(t−s′), (B16a)

where we have ignored the transient solution which will decay strongly for large times t. Regarding the noise correlationterm 〈nin(s)n>in(s′)〉 component wise we obtain 〈nin,i(s)nin,j(s

′)〉 6= 0 if i and j are both even numbers, which isresulting in

〈nin,2i(s)nin,2j(s′)〉 = 〈ηi(s)ηj(s′)〉

= 〈ξi(s)ξj(s′)〉+g

(i)cd g

(j)cd ω

2fb

2κ

[〈(φ1 ∗ yin)(s)(φ1 ∗ yin)(s′)〉+ (η−1 − 1)〈(φ2 ∗ yv)(s)(φ2 ∗ yv)(s′)〉

]+ 2κGiGj〈(φ3 ∗ xin)(s)(φ3 ∗ xin)(s′)〉 − g(i)

cd ωfbGj〈(φ1 ∗ yin)(s)(φ3 ∗ xin)(s′)〉

− g(j)cd ωfbGi〈(φ3 ∗ xin)(s)(φ1 ∗ yin)(s′)〉

= (2ni + 1)γiδijδ(s− s′) +g

(i)cd g

(j)cd ω

2fb

4κη

(δ(s− s′)− ωfb

2e−ωfb|s−s′|

)+GiGjκ

κ

2e−κ|s−s

′|

+ i

(ωfbe

−ωfb|s−s′| − κe−κ|s−s′|

2(ωfb − κ)

)ωfb

(g

(i)cdGjθ(s− s

′)− g(j)cd Giθ(s

′ − s)), (B17a)

for i, j ∈ 1, . . . , N. For ωfb, κ ωj ,Γj we can approximate δ(t) ≈ (ωfb/2)e−ωfb|t| as well as δ(t) ≈ (κ/2)e−κ|t|

resulting in

〈ηi(s)ηj(s′)〉 ≈(

(2ni + 1)γiδij +GiGjκ

)δ(s− s′). (B18a)

For δ-correlated noise we can simplify the correlation matrix to

V =

∫ t

t0

dseM(t−s)DineM>(t−s), (B19a)

where Din,2i,2j = (2ni + 1)γiδij +GiGj/κ for even index numbers and is zero otherwise. The Lyapunov equation forthe N -oscillator system which determines the steady solution of the correlation matrix is given by

MV + VM> = −Din, (B20)

and can be solved exactly. Evaluating the individual components we obtain the set of equations

Yii = 0, (B21a)

ωjYij + ωiYji = 0, (B21b)

ΓiiXii +∑j 6=i

ΓijXij − (2ni + 1)γi −G2i

κ= 0, (B21c)

ωi (Xii − Zii)−∑j 6=i

ΓijYij = 0, (B21d)

(ω2i − ω2

j )Xij −(ω2i Γjj + ω2

jΓii)

ωiYij −

∑k 6=i,j

(ωiΓjkYik − ωjΓikYjk) = 0, (B21e)

−(ω2i − ω2

j

)ωi

Yij − (Γii + Γjj)Xij − ΓjiXii − ΓijXjj −∑k 6=i,j

(ΓikXjk + ΓjkXik) +2GiGjκ

= 0, (B21f)

11

with Xij = 〈pipj + pjpi〉, Yij = 〈qipj + pjqi〉 and Zij = 〈qiqj + qjqi〉. In the case that γj (g(j)cd Gjωj)/κ where

Γjj ≈ (g(j)cd Gjωj)/κ and Γij = (g

(i)cd /g

(j)cd )Γjj , we can simplify the expression in Eq. B21f and we obtain

−(ω2i − ω2

j

)ωi

Yij −

(g

(j)cd

g(i)cd

)(2ni + 1)γi −

(g

(i)cd

g(j)cd

)(2nj + 1)γj −

(g

(j)cd Gi − g

(i)cdGj

)2

κg(i)cd gcd,j

= 0. (B22a)

Here, we define

Λij :=

((g

(j)cd

g(i)cd

)(2ni + 1)γi +

(g

(i)cd

g(j)cd

)(2nj + 1)γj +

(g(j)cd Gi − g

(i)cdGj)

2

κg(i)cd g

(j)cd

), (B23a)

and we obtain

〈p2i 〉 =

(ni +

1

2

)γiΓii

+G2i

2Γiiκ+∑j 6=i

Γij2Γii

(ω2i Γjj + ω2

jΓii)

Λij(ω2i − ω2

j

)2 +∑k 6=i,j

1(ω2i − ω2

j

) ( ω2i ΓjkΛik

(ω2i − ω2

k)−ω2jΓikΛjk(ω2j − ω2

k

)) ,

(B24a)

〈q2i 〉 = 〈p2

i 〉+∑j 6=i

ΓijΛij

2(ω2i − ω2

j

) . (B24b)

Therefore, the energy of the j’s mode is given by

1

2

(〈p2i 〉+ 〈q2

i 〉)

=

(ni +

1

2

)γiΓii

+G2i

2Γiiκ

+∑j 6=i

Γij2Γii

(ω2i Γjj + ω2

jΓii)

Λij(ω2i − ω2

j

)2 +∑k 6=i,j

1(ω2i − ω2

j

) ( ω2i ΓjkΛik

(ω2i − ω2

k)−ω2jΓikΛjk(ω2j − ω2

k

))+

ΓijΛij

4(ω2i − ω2

j

)

(B25a)

≈ niγiΓii

+G2i

2Γiiκ+∑j 6=i

ΓijΓii

(ω2i Γjj + ω2

jΓii)((

g(j)cd

)2

niγi +(g

(i)cd

)2

njγj

)(g

(i)cd g

(j)cd )

(ω2i − ω2

j

)2

+∑k 6=i,j

1(ω2i − ω2

j

)ω

2i Γjk

((g

(k)cd

)2

niγi +(g

(i)cd

)2

nkγk

)(g

(i)cd g

(k)cd ) (ω2

i − ω2k)

−ω2jΓik

((g

(k)cd

)2

njγj +(g

(j)cd

)2

nkγk

)(g

(j)cd g

(k)cd )

(ω2j − ω2

k

)

+

Γij

((g

(j)cd

)2

niγi +(g

(i)cd

)2

njγj

)2(g

(i)cd g

(j)cd )

(ω2i − ω2

j

) . (B25b)

The term G2i /(2Γiiκ) = Gi/

(2g

(i)cd ωfbωi

)is in general smaller than one and can be mostly ignored if niγi/Γii > 1. For

a sequence of frequencies with ωj ≈ ω + j∆ω and ω ∆ω we obtain for ω2i − ω2

j ≈ 2ω(i− j)∆ω. By considering onlynearest neighbors in frequencies since the terms decay quadratically with distance we obtain

1

2

(〈p2i 〉+ 〈q2

i 〉)≈ ni

γiΓii

+1

4∆ω2

∑〈i,j〉j 6=i

Γij

(

1 +ΓjjΓii

)(niγi + njγj) +

∑〈i,j,k〉k 6=i,j

(Γjk(niγi + nkγk)

Γii(i− j)(i− k)− Γik(njγj + nkγk)

Γii(i− j)(j − k)

) ,

(B26a)

showing that the lower bound of the energy niγi/Γii can be reached when ∆ω is much larger than Γij . Here, we have

considered that g(i)cd ≈ gcd for all i ∈ 1, . . . , N.

12

(a) (b)

(c)

( )f( )e( )d

Figure 3. Cold damping for two modes. (a) Full solutions for cold damping using convolutions for two modes with ω1,2 = (1, 0.9)and in the degenerate case with ω1,2 = (1, 1). In (b) and (c) the corresponding phase space trajectories are presented.The magnified signal presented in (d) and (e) show a comparison between the full solutions (solid lines) and the ones withapproximated damping rates (dashed lines) at the beginning and a later stage of the evolution, respectively. In the degeneratecase the effect of the feedback stops when both modes have a relative phase shift of π. In (f) the average energy is presentedas a function of time. The dashed green line shows the final energy at steady state obtained by the Lyapunov equation fortwo oscillators. The simulation parameters are given by γ1,2 = (4, 3) × 10−5ω1, G1,2 = (0.16, 0.1) × ω1, κ = 3ω1, ωfb = 3.5ω1,gcd,1,2 = (0.8, 0.8) and τ = 0.05ω−1.

Appendix C: Cooling of two adjacent modes

To visualize the feedback cooling process we perform classical simulations of the stochastic differential equations. InFig. 3 we show the results for cooling two modes close to and at frequency degeneracy. The simulation is performedusing the full convolutional description of the feedback process (solid lines) and is compared to the approximatedform (dashed lines), which show good agreement. Especially in the degenerate case at a large time duration frominitialization as presented in Fig. 3e it is visible that the feedback stops when both modes acquire a phase shift of πwith respect to each other. Fig. 3f shows the average energy over many trajectories of the two mode system. In thecase of frequency degeneracy only up to half of the initial energy of the system is removed since only the bright modecan be accessed by the method.

Appendix D: Numerical integration of Langevin equations

To test the results derived by solving the Lyapunov equation, we perform numerical Monte-Carlo simulations forthe equations of motion. Here, the initial conditions are obtained from a Boltzmann distribution representing theinitial thermal state. The numerical integration can be obtained from the differential form of the stochastic differentialequations of motion

dqj = ωjpjdt, (D1a)

dpj = −ωjqjdt− γjpjdt− (gj ∗ y)dt+√

(2nj + 1)γjdW (t), (D1b)

dy = −κydt+

N∑j=1

Gjqjdt, (D1c)

where dW (t) describes an infinitesimal Wiener increment (dW 2 = dt) that guarantees that the fluctuation dissipationtheorem is fulfilled [38]. In our case we use the Runge-Kutta fourth-order method (RK4) that guarantees numericalstability for the integration.

Appendix E: Numerical simulations for many modes

The analytical versus numerical comparison in the main text is based on a small number of resonances such that theresults are easily comprehensible and easy to visualize graphically. However, cooling of many more modes is possibleusing the same technique as we numerically prove in Fig. 4. A limitation for extending numerics to even more modescomes from the limited available computational resources.

13

(a) (b) (c)

Figure 4. Cold damping of many modes . (a) Linear dispersion for N = 50 oscillator modes. (b) Initial (orange disks) andsteady state (black disks) occupations of the 50-modes obtained from the dynamical simulations are presented and compared tothe results from the Lyapunov equation (red stars) and in the case of independent damping (gray stars). In (c) the dynamicalevolution of the average energies reaching convergence are shown for all modes.

Appendix F: Bright and dark mode cooling dynamics

The cavity output fed into the cold-damping loop contains information solely on a linear combination of individualmodes momentum quadratures. Moving to generalized quadratures can provide insight into the efficiency of coolingversus frequency degeneracy (or level of disorder) in the system. Analytical considerations on the simplified system ofequations Eqs. B11a can already shed light onto the bright versus dark modes relevance in the cooling process. Let usassume Γjk = Γ for set of indexes (we have checked that the predictions of the full equations of motion and of thesimplified version of Eqs. B11a indeed agree and show the relevant dynamics around degeneracy points). An analysisof the simplified equations:

qj = ωjpj , (F1a)

pj = −ωjqj − (Γ + γ)pj − Γ∑k 6=j pk + ξj + ξopt,j , (F1b)

shows that a generalized momentum quadrature can be defined as P1 = 1/√N∑j pj (with the corresponding position

quadrature Q1 = 1/√N∑j qj) corresponding to the bright mode. Notice that the commutation relation for these

quadratures are fulfilled. Dark modes (not directly addressed by the feedback loop) can be defined via a Gram-Schmidtprocedure as Pk =

∑j αkjpj and Qk =

∑j αkjqj such that commutations are insured. Notice also that orthogonality

to each other and to the bright mode is fulfilled meaning that∑j αjk = 0 and

∑j α∗jkαjk′ = δkk′ . Conservation of

energy requires that∑j ωj(p

2j + q2

j ) =∑k Ωk(P2

k +Q2k) where the generalized quadratures eigen-frequencies depend

on the coefficients αjk. With notations v = (q1, p1, ...qN , pN ) and u = (Q1,P1, ...QN ,PN ) we can write a generaltransformation u = Tv and an equation of motion for the vector u of generalized quadratures

u = T (Mω,γ −MΓ)T> + uin. (F2)

with

Mω,γ =

0 ω1 0 0 ··· 0 0−ω1 −γ 0 0 ··· 0 0

0 0 0 ω2 ··· 0 00 0 −ω2 −γ ··· 0 0

......

......

. . ....

...0 0 0 0 ··· 0 ωN0 0 0 0 ··· −ωN −γ

and MΓ =

0 0 0 0 ··· 0 00 −Γ 0 −Γ ··· 0 −Γ0 0 0 0 ··· 0 00 −Γ 0 −Γ ··· 0 −Γ

......

......

. . ....

...0 0 0 0 ··· 0 00 −Γ 0 −Γ ··· 0 −Γ

, (F3a)

The transformation is defined such that it diagonalizes the damping matrix and can be written as

T =

α11 0 α12 0 ··· α1N 00 α11 0 α12 ··· 0 α1Nα21 0 α22 0 ··· α2N 00 α21 0 α22 ··· 0 α2N

......

......

. . ....

...αN1 0 αN2 0 ··· αNN 0

0 αN1 0 αN2 ··· 0 αNN

, (F4a)

14

where the vector (α11, α12, · · · , α1N ) = 1/√N(1, 1, · · · , 1) describes the bright symmetric mode. The diagonalized

matrix has the form

TMΓT> =

0 0 0 0 · · · 0 00 −NΓ 0 0 · · · 0 00 0 0 0 · · · 0 00 0 0 0 · · · 0 0...

......

.... . .

......

0 0 0 0 · · · 0 00 0 0 0 · · · 0 0

. (F5a)

showing that feedback damping only affects the bright mode. However, sympathetic cooling of all other collectivemodes appears from off-diagonal coupling terms of bright to dark modes in the transformed matrix

TMω,γT> =

0∑α2

1jωj 0∑α1jα2jωj ··· 0

∑α1jαNjωj

−∑α2

1jωj 0 −∑α1jα2jωj 0 ··· −

∑α1jαNjωj 0

0∑α2jα1jωj 0

∑α2

2jωj ··· 0∑α2jαNjωj

−∑α2jα1jωj 0 −

∑α2

2jωj 0 ··· −∑α2jαNjωj 0

......

......

. . ....

...0

∑αNjα1jωj 0

∑αNjα2jωj ··· 0

∑α2

Njωj

−∑αNjα1jωj 0 −

∑αNjα2jωj 0 ··· −

∑α2

Njωj 0

(F6a)

The frequencies of the generalized quadratures can be determined as Ωk =∑j α

2kjωj and the intermode couplings are

defined as gkk′ =∑j αkjαk′jωj . The bright mode always exhibits a frequency as a harmonic average of all individual

mode frequencies: Ω1 =∑j ωj/N . In these simplified notations one can then write the equations of motion:

Q1 = Ω1P1 +

N∑k′=2

g1k′P ′k, (F7a)

P1 = −Ω1Q1 − (NΓ + γ)P1 −N∑k′=2

g1k′Q′k + Ξk, (F7b)

Qk = ΩkPk +

N∑k′=2

gkk′P ′k, (F7c)

Pk = −ΩkQk − γPk −N∑k′=2

gkk′P ′k + Ξk. (F7d)

Note that only the bright mode is directly driven by the feedback and exhibits an enhanced damping rate NΓ. In thenext step, the coupling of the dark modes to the damped bright mode stemming from off-diagonal terms proportionalto frequency mismatches gkk′ can then lead to sympathetic cooling of all degrees of freedom.

1. The fully degenerate case

In the case of full degeneracy where ωi = ω for all i ∈ 1, . . . , N the intermode couplings vanish altogether and theequations of motion for collective modes show full separation:

Q1 = ωP1, (F8a)

P1 = −ωQ1 − (NΓ + γ)P1 + Ξj , (F8b)

Qj = ωPj , (F8c)

Pj = −ωQj − γPj + Ξj . (F8d)

The bright mode the is cooled at NΓ while all the dark modes are left completely unaffected (as there is no dampingterm except for the γ associated with the thermalization to the environmental temperature). The efficiency of thecooling process is then minimal as only 1/N of the total energy is removed.

15

2. Illustration: two and three modes

Let us illustrate the role of frequency disorder in coupling bright to dark modes in the case on only two or threedistinct modes present. For two modes the dark combination is defined by the coefficients (1/

√2,−1/

√2). The

equations of motion become

Q1 =ω1 + ω2

2P1 +

ω1 − ω2

2P2, (F9a)

P1 = −ω1 + ω2

2Q1 − (2Γ + γ)P1 −

ω1 − ω2

2Q2 + Ξ1, (F9b)

Q2 =ω1 + ω2

2P2 +

ω1 − ω2

2P1, (F9c)

P2 = −ω1 + ω2

2Q2 − γP2 −

ω1 − ω2

2Q1 + Ξ2. (F9d)

Sympathetic cooling of the dark mode is driven by the indirect coupling to the feedback loop via the direct coupling tothe position quadrature of the bright mode proportional to the frequency difference.For three modes we obtain a more complicated coupling map where as expected the bright mode is damped quickly at3Γ

Q1 =ω1 + ω2 + ω3

3P1 +

ω1 − 2ω2 + ω3

3√

2P2 +

ω1 − ω3√6P3, (F10a)

P1 = −ω1 + ω2 + ω3

3Q1 − (3Γ + γ)P1 −

ω1 − 2ω2 + ω3

3√

2Q2 −+

ω1 − ω3√6Q3 + Ξ1, (F10b)

Q2 =ω1 + 4ω2 + ω3

6P2 +

ω1 − 2ω2 + ω3

3√

2P1 +

ω1 − ω3

2√

3P3, (F10c)

P2 = −ω1 + 4ω2 + ω3

6Q2 − γP2 −

ω1 − 2ω2 + ω3

3√

2Q1 −

ω1 − ω3

2√

3Q3 + Ξ2, (F10d)

Q3 =ω1 + ω3

2P3 +

ω1 − ω3√6P1 +

ω1 − ω3

2√

3P2, (F10e)

P3 = −ω1 + ω3

2Q3 − γP3 −

ω1 − ω3√6Q1 −

ω1 − ω3

2√

3Q2 + Ξ2. (F10f)

The bright mode then in turn couples to both dark modes which are also coupled among each other.

g1,2

g1,3

g1,4B

D1

D2

D3

D9

g1,10

Feedback

(a) (b)

Co

up

ling

Str

en

gth

-g1k

d

Mode-Number k

Figure 5. Mode coupling . (a) The bright mode B experiences directly the action of the feedback loop and sympathetically coolsthe rest of the modes Dj (depicted is the case of 10 independent modes). (b) Plot of the coupling strength between the brightmode and dark modes (in units of the frequency difference δ) in the linear dispersion case showing a linear decrease.

16

3. Coupling rates for the linear dispersion case

For a sequence of frequencies ωj = ω + (j − 1)δ we obtain for the matrix elements

(TMω,γT>)(2i−1)(2k) =

N∑j=1

αijαkj(ω + (j − 1)δ) = ωδik + δ

N∑j=1

(j − 1)αijαkj (F11a)

= (ω − δ)δik + δ

N∑j=1

jαijαkj , (F11b)

where the other non zero entries are following (TMT>)(2i)(2k−1) = −(TMT>)(2i−1)(2k). This results in the followingcoupling matrix

TMω,γT> =

0 ω+(N−1)δ/2 0 δ

∑jα1jα2j ··· 0 δ

∑jα1jαNj

−(ω+(N−1)δ/2) 0 −δ∑jα1jα2j 0 ··· −δ

∑jα1jαNj 0

0 δ∑jα2jα1j 0 ω+(N−1)δ/2 ··· 0 δ

∑jα2jαNj

−δ∑jα2jα1j 0 −(ω+(N−1)δ/2) 0 ··· −δ

∑jα2jαNj 0

......

......

. . ....

...0 δ

∑jαNjα1j 0 δ

∑jαNjα2j ··· 0 ω+(N−1)δ/2

−δ∑jαNjα1j 0 −δ

∑jαNjα2j 0 ··· −(ω+(N−1)δ/2) 0

. (F12a)

The common frequency of all collective modes is ω + (N − 1)δ/2 and the off diagonal elements show the bright-dark aswell as the dark-dark couplings. Notice that the bright to dark rates diminish progressively when scanning through althe collective modes. This is illustrated in Fig. 5 for a case including 10 independent vibrational resonances.