science.sciencemag.org/cgi/content/full/science.aba3993/DC1

Supplementary Materials for

Cooling of a levitated nanoparticle to the motional quantum ground state

Uroš Delić*, Manuel Reisenbauer, Kahan Dare, David Grass, Vladan Vuletić, Nikolai Kiesel, Markus Aspelmeyer*

*Corresponding author. Email: uros.delic@univie.ac.at (U.D.); markus.aspelmeyer@univie.ac.at (M.A.)

Published 30 January 2020 on Science First Release DOI: 10.1126/science.aba3993

This PDF file includes: Supplementary Text Figs. S1 to S7 References

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Supplementary text

The experimental setupThe optical part of the experiment is based on a Nd:YAG laser (Innolight Mephisto, 2W) witha wavelength of λ = 1064 nm and a frequency ω1 = 2πc/λ (Fig. S1). The laser is split onpolarizing beamsplitters (PBS1 and PBS2) into three laser paths, which are used for stabilizationto the optomechanical cavity (OM), for optical levitation of a silica nanoparticle and as the localoscillator in heterodyne detection (34).

The first beam (blue) is stabilized to a TEM00 mode of the OM cavity with a Pound-Drever-Hall (PDH) lock. We use an electro-optical modulator (EOM) to generate sidebandsat a frequency of ω1 ± ∆ωFSR, where ∆ωFSR ≈ 14.0192 GHz is the free spectral rangeof the OM cavity. The filtering cavity (FC) selects only the upper sideband at a frequencyω2 = ω1 + ∆ωFSR. The error signal from the PDH detector is used to stabilize the laser fre-quency ω2 to the OM cavity resonance ωcav,2 by acting back directly onto the master laserfrequency ω1, thereby automatically stabilizing ω1 to the adjacent longitudinal cavity resonancefrequency ωcav,1 = ωcav,2 −∆ωFSR as well.

The second beam (red) seeds a fiber amplifier (FA, Keopsys, maximal output power 5W) andserves as the trapping laser for the optical tweezer. The optical mode leaving the fiber amplifieris expanded with a telescope, its polarization adjusted with a half-waveplate (λ/2) and tightlyfocused with a high NA microscope objective (MO, NA= 0.8). The microscope objective sitson a triaxial nano-positioner (Mechonics MX35, step size ∼ 8 nm) that is used to position thelevitated nanoparticle within the OM cavity modes. The OM cavity resonance ωcav,1, which isinitially empty, is driven from the inside by anti-Stokes and Stokes tweezer photons scatteredoff the levitated particle at frequencies ω1+Ωm and ω1−Ωm, respectively (Ωm is the mechanicalfrequency of the particle, where m = x, y, z). Note that we can introduce a relative detuning∆ between the tweezer laser ω1 and the cavity resonance ωcav,1 by changing the EOM drivingtone from ∆ωFSR to ∆ωFSR + ∆. The PDH lock will instantaneously stabilize the frequencyω2 = ω1 + ∆ωFSR + ∆ to the cavity resonance ωcav,2, resulting in a detuned tweezer laser witha frequency ω1 = ωcav,1−∆. We use this mechanism to control the optomechanical interactionfor cavity cooling.

The third beam (dark red) is frequency shifted by an acousto-optical modulator (AOM) toω1 + ωhet with ωhet/2π = 10.2MHz and used as a local oscillator for a balanced heterodyneread-out of the particle motion (hetD).

Stokes and anti-Stokes scattered photons that leak out of the OM cavity through both mirrorsare coherently combined on a beamsplitter (BS1) and used as signal beam for the heterodyneread-out (hetD). A movable mirror (MM) in combination with an error signal (sigL) is used tocompensate length fluctuations between the two signal paths (light leaking through the ”left”and ”right” OM cavity mirror) before superimposition on BS1 (35).

The OM cavity itself has a length of L ≈ 1.07 cm and a linewidth of κ/2π ≈ 193 kHz. Amore detailed description about the interface between OM cavity and tweezer, and positioning

2

of the levitated particle with respect to the OM cavity can be found in (18, 34, 36).

Heterodyne detection of the Stokes and anti-Stokes photonsWe monitor the particle motion that is coupled to the cavity (xmotion) with a heterodyne detec-tion, which allows us to simultaneously detect both Stokes (heating) and anti-Stokes (cooling)photons leaking out of the cavity. These photons are differently amplified by the cavity re-sponse, which results in an asymmetry that prefers the anti-Stokes over Stokes photons. Notethat cavity cooling of the x motion by coherent scattering comes with a benefit of strong sup-pression of classical laser intensity and phase noise (19, 34). The local oscillator originatesdirectly from the Mephisto laser such that the added noise by the fiber amplifier is not observedin the fundamental heterodyne spectrum (obtained without the particle signal). We additionallyoperate the heterodyne at a frequency of 10.2 MHz and for low local oscillator powers therebyguaranteeing shot-noise limited detection (Fig. S2), i.e. the classical intensity noise introducedby the laser is orders of magnitude below the shot noise level (34). A heterodyne spectrum isshown in Fig. S3 highlighting the features corresponding to motion along each axis.

The intrinsic Stokes and anti-Stokes scattered rates ΓS ∝ nx + 1 and ΓAS ∝ nx are pro-portional to the phonon occupation nx. The cavity modifies the scattering rates following thecavity response T (∆, ω) at the particle motional frequency Ωx:

T (∆, ω) =

(κ2

)2(κ2

)2+ (∆− ω)2

, (S1)

where κ is the cavity linewidth, ∆ is the detuning of the tweezer laser and ω is the relevantfrequency in a heterodyne spectrum. We observe the modified scattering rates in the heterodynespectrum Shet(ω) with the final ratio of anti-Stokes to Stokes sideband given as (23):

Shet(Ωx)− 1Shet(−Ωx)− 1

=nx

nx + 1

(κ2

)2+ (∆ + Ωx)

2(κ2

)2+ (∆− Ωx)2

. (S2)

Therefore, it is crucial to determine the exact cavity linewidth κ and detuning ∆ of the tweezerlaser in order to correctly extract the phonon occupation nx, following the procedure describedin Ref. (23). After accounting for the cavity response, the remaining asymmetry of the scatteringrates is given only by the different probabilities for transitions between adjacent levels and canbe used to extract the phonon occupation in the steady state (sideband asymmetry thermometry).

In practice, strongly cooled x motion and weakly cooled y motion typically overlap in theheterodyne spectrum. Following one-dimensional analysis of our system, we evaluate the oc-cupation nx via three distinct methods consistent results: from a joint fit of both Stokes andanti-Stokes sidebands of the two peaks, from a fit to only the motional sidebands of the x-motion (obtained by generously deleting points around the y peak) and from integrated areas of

3

the data points accredited to the x motion (37). A joint fit of the two motional sidebands in thisone-dimensional case, normalized to shot noise, is given by:

Shet(ω) = aS

γx2

(ω + Ωx)2 +(γx2

)2+aAS

γx2

(ω − Ωx)2 +(γx2

)2 + 1. (S3)The ratio of fit amplitudes aS and aAS is subsequently used to extract the occupation nx as:aAS/aS = nxT (∆,−Ωx)/((nx + 1)T (∆,Ωx)).

It has been argued that in the strong cooperativity regime mode hybridization may occur,which would introduce correlations between the x- and y-motion (38). As an ultimate worstcase estimate of the phonon occupation, we also integrate the full area of the Stokes and anti-Stokes side in the range±(250−300) kHz, thus integrating over spectral peaks of both motions.For the detuning ∆/2π ≈ 315 kHz we obtain a final phonon occupation of nx = 0.91 ± 0.04,which still demonstrates ground state cooling of the particle motion.

Cavity linewidth

Here we describe how we determine the cavity linewidth κ. The principal locking scheme isdescribed in the text above. Instead of driving the cavity with the tweezer scattered photons, wenow drive the cavity resonance ωcav,1 with the laser at a variable frequency ω1 = ωcav,1 − ∆through one of the OM cavity mirrors. We tune the detuning ∆ over the cavity resonance andmonitor the laser power in transmission of the OM cavity (Fig. S4):

Ptran ∝ T (∆) =

(κ2

)2(κ2

)2+ ∆2

. (S4)

As this procedure is regularly repeated, we combine all measurements to get an average cavitylinewidth κ/2π = (193 ± 4) kHz. We extract the mean free spectral range ∆ωFSR/2π ≈14.0192 MHz from the scan as well, which allows us to calculate the cavity length as L =cπ/∆ωFSR ≈ 1.07 cm.

Detuning of the tweezer laser

We use the cavity induced asymmetry of the residual phase noise and the spectrum of weaklycooled z-motion in the heterodyne detection to extract the detuning of the tweezer laser. Theratio of the spectra on each side of the heterodyne frequency, as its source is purely classical, isgiven by:

Shet(ω)

Shet(−ω)=

(κ2

)2+ (ω + ∆)2(

κ2

)2+ (ω −∆)2

, (S5)

4

where ∆ is the tweezer laser detuning. The typical error of this method is at maximum ±10kHz, which is used in the error estimation of the phonon occupation.

Positioning stabilityThe peak size in the heterodyne measurement at the heterodyne frequency ωhet is given bythe product of the local oscillator power PLO and the optical power leaking out of the cavityPcav ∝ nphot, where nphot is the intracavity photon number. It depends on the nanoparticleposition x0 along the cavity standing wave as nphot ∝ cos2(kx0). Therefore, the height of theheterodyne peak holds information about the positioning stability of the nanoparticle. In orderto calibrate the absolute particle position, we initially place the nanoparticle around the cavityantinode (x0 = 0) and monitor the peak height, with the maximum peak size subsequentlyrescaled to nphot = 1. The minimum intracavity photon number nphot = 0 is reached at a cavitynode (x0 = λ/4 = 256 nm), where the optimal cooling of the x motion takes place. We assumeno changes in the optical power of the local oscillator and the detection efficiency during thismeasurement. Using this calibration we are able to follow the particle position in the vicinityof the cavity node during cavity cooling, which shows that the particle stays within a couple ofnanometers (Fig. S5). Besides long term drifts, the most of the particle motion around the cavitynode comes from the vibrations transferred from the turbo pump, which are at a frequency ofωjitter/2π = 54 Hz (Note that in Figure S5 this jitter is not seen as we average over it to showonly long term positioning stability). This is far away from any relevant mechanical frequenciesand doesn’t induce additional heating of the particle motion.

Noise sourcesLaser phase noise

By using the cavity drive deduced from the cooling performance in Ref. (19) and accountingfor the increased tweezer power to reach higher mechanical frequencies, we calculate the cavitydrive of Ed/2π ≈ 4 × 109 Hz. The resulting intracavity photon number is subsequently givenby:

nphot(x0) =E2d cos

2(kx0)(κ2

)2+ ∆2

, (S6)

where x0 is the particle position along the cavity standing wave. The expected number ofphonons added from classical laser phase noise at the optimal detuning ∆ ∼ Ωx and for aparticle positioned 3 nm away from the cavity node at λ/4 is (39–41):

nphase =nphot(λ/4 + 3nm)

κSφ̇φ̇(Ωx) < 0.025, (S7)

5

where the measured laser frequency noise of a Mephisto laser at the motional frequency Ωx/2π =305 kHz is Sφ̇φ̇(Ωx) < 0.1Hz

2/Hz (42). This corresponds to a phase-quadrature noise contri-bution of maximally Cpp = 2.5× 10−3 in excess to fundamental vacuum fluctuations (25).

Laser intensity noise

Intensity noise of the tweezer laser leads to parametric heating due to the modulation of the trapintensity and manifests itself as a negative damping rate (43):

γmint = −π2Ω2mSRIN(2Ωm). (S8)

Here, SRIN(2Ωm) is the relative intensity noise (RIN) at twice the mechanical frequency Ωm.While the intensity noise of a Mephisto laser is typically around −135 dB/Hz, the fiber am-plifier used to increase the trapping power significantly increases the laser intensity noise atfrequencies below 100 kHz (Fig. S6). We therefore intentionally increase the trapping powersuch that second harmonics of the motional frequencies are in the range of low laser intensitynoise. The calculated damping rates are (γxint, γ

yint, γ

zint)/2π = −(6, 5, 0.5) mHz. Heating is

therefore negligible for the strongly cooled x and y motion with cooling rates of at least 1 kHz(see below). However, for the only sporadically cooled z motion, the heating rate can becomecomparable to the cooling rate.

Laser intensity noise can also add to the phonon occupation through cavity backaction (25,44). For our parameters at optimal detuning, the corresponding amplitude-quadrature noisecontribution yields Cqq = 2× 10−5 in units of tweezer laser shot noise. Therefore, for a particleplaced 4 nm away from the cavity node this adds at maximum nint ∼ 10−4 phonons to theoccupation nx (25). As in the case of phase noise, the strong suppression of intracavity photonsrenders the noise contribution negligible.

Heating ratesIn our system, the heating rate is dominated by gas collisions at the relevant pressures of∼ 10−6mbar. This was studied in detail in Ref. (18) in the same experimental apparatus and usingparticles of the same nominal diameter. This allows us to extrapolate the gas heating rate atany pressure and for any mechanical frequency (or trapping power). For our experimentalparameters at the detuning of ∆/2π ≈ 315 kHz, the gas heating rate is Γgas/2π = (16.1± 1.2)kHz, which we subtract from the inferred heating rate Γx/2π = (20.6 ± 2.3) kHz to estimatethe recoil heating rate Γrec/2π = 4.3± 1 kHz. This is in a good agreement with our calculatedrecoil heating Γrec,calc/2π = 6 kHz, which we obtain from a model of the trapping potentialbased on experimental mechanical frequencies (34, 45) (note that a direct measurement of therecoil heating rate would require operating at much lower pressures (26)). Together with theindependently measured cooling rates γx, which are obtained from the fits to the mechanicalsidebands using Eq. S3, we can estimate the phonon occupation via nx = (Γgas + Γrec,calc)/γx.The thus obtained values are in good agreement with the measurements obtained by sideband

6

asymmetry (Figure S7). We use the full optomechanical theory from Ref. (46) to describe thecooling performance of our system and plot the theoretical green band shown in Figure 2 ofthe main text. In addition to independently measured system parameters, the upper and lowerbound of the band is given by the range of experimental pressures during our measurements.

Cavity cooling of other motional axesIn principle, cavity cooling by coherent scattering enables cooling in all three dimensions (19,47, 48). In our current experiment, however, we optimize cooling only along the cavity (x-)axis, specifically by aligning the tweezer polarization orthogonal to the cavity axis (therebymaximizing the scattering component along the x-axis), and by positioning the particle at thecavity node. We explore here the cooling of the y and z motion.

Cavity cooling of the z motion

There are two distinct mechanisms by which the z motion is coupled to the cavity mode:

1. Cavity cooling by coherent scattering is a genuine three-dimensional cooling process. Theoptimal position for the cooling of the z motion is at the cavity antinode (19), while at thecavity node the z motion is not cooled at all. However, the particle is always positionedin the vicinity of the cavity node and thus experiences some minimal cooling of the zmotion.

2. The tweezer laser is slightly non-orthogonal to the cavity mode, which leads to a projec-tion of the z motion onto the cavity axis. This results in some cooling of the z motioneven at the cavity node.

Harmonics of the z motion in the heterodyne detection are present due to the nature ofcoupling to the cavity mode (34). For a particle at any position x0, the area of the i-th harmonicin the heterodyne spectrum is proportional to the 2i-th order central moment 〈z2i〉 of the zmotion. The ratio of the first and second harmonic after removing the cavity response from theheterodyne detection is:

gquadglin

〈z4〉〈z2〉

=kBTzmΩ2z

(k − 1/zR

2

)2, (S9)

where m = 2.83 fg is the nanoparticle mass (18), k = 2π/λ with a wavelength λ = 1064 nm,zR = WxWyπ/λ is the Rayleigh length with trap waists Wx = 0.67 µm and Wy = 0.77 µm andTz is the temperature of the z motion. For the tweezer detuning ∆/2π = 315 kHz the averageobserved temperature is Tz ≈ 80 K and the respective occupation is nz = kBTzh̄Ωz = 2× 10

7. Thetemperature greatly varies between different measurements, but the particle is cooled enoughalong the z-axis to be stably trapped in high vacuum. Further cooling of the z-motion is possiblewith feedback cooling schemes (11, 49–51).

7

Cavity cooling of the y motion

The optical potential created by the tweezer is, due to the tight focusing, elliptical in the focalplane (45). As a result, the particle mechanical frequencies along the x- and y-axes are non-degenerate. The orientation of the elliptical trap in the focal plane can be aligned by rotatingthe polarization of the tweezer laser. In this experiment, we optimize the polarization such thatthe y motion is orthogonal to the cavity axis. However, due to experimental imperfections,there remains always a small projection of the y motion onto the cavity axis. Since the particleposition is optimized for maximal coupling to the motion along the cavity axis, the y motionis also coupled to the cavity axis and hence is also moderately cooled by coherent scattering.We estimate the occupation number by computing the power spectral density of the heterodynedetection and fitting the motional peak of the y motion to extract the damping rate of at leastγy/2π = 0.5 kHz. Together with the calculated heating rate Γy/2π ≈ 20 kHz, dominated bygas collisions and recoil heating of the trapping laser, we can estimate the occupation of the ymotion to be well below ny < 100 phonons for any detuning.

Decoherence rateWe estimate the decoherence in our experiment after preparing the nanoparticle in its groundstate of motion and during a subsequent hypothetical free fall based on Romero-Isart et al. (28).

The heating rate of Γx/2π = 21±3 kHz is given by the optical recoil heating rate Γrec/2π ≈6 kHz and recoil by background gas collisions Γgas/2π ≈ 15 kHz. It limits the coherence timein the optical trap to Ttrap = 1/Γx = 7.6 ± 1 µs corresponding to approximately 15 coherentoscillations before the ground state is populated with one phonon.

When the optical trap is turned off, the particle is in free fall and coherently expands. Themajor decoherence mechanism is then position localization via background gas collisions. Weare operating in an interesting transitional regime, in which the initial wavepacket size xzpf =σ(t = 0) =

√h̄

2mΩx= 3.1 pm is smaller than the thermal de Broglie wavelength of the scattering

gas molecules λth = 2a = h/√mgaskBT = 19 pm by a factor of 6.2 (T : environment

temperature, mgas ≈ 28u: mass of one gas molecule, u = 1.67 × 10−27 kg: atomic massunit).

In the short-distance regime where σ(0)� 2a the decoherence rate increases quadraticallywith the wavepacket size (and hence with time due to the linear expansion) until it saturates inthe regime σ(t) � 2a (52, 53). To estimate the expected coherence time and maximum ex-pansion of the wavepacket, we can determine the localization parameter from our experimentalheating rate in the optical trap Λ = Γx/x2zpf . This is obtained by comparing the energy in-crease by momentum diffusion (Eq. 14 in (28)) with the heating rate: 2Λh̄

2

2m= Γxh̄Ωx. From

this, we can determine the maximum coherence length ξmax = 10.2 pm and the corresponding

8

expansion time tmax = 1.42 µs in the short-distance approximation (Eq. 19 and 20 in (28)):

tmax =

(3m (2n̄x + 1)

2Λh̄Ωx

)1/3

ξmax =√

2

(2h̄Ωx

3mΛ2(2n̄x + 1)

) 16

(S10)

where n̄x ≈ 0.43 is the inital occupation of the wavepacket. Note that the obtained valuesslightly underestimate the expected values, as the expansion occurs in the transition to satura-tion.

What conditions are required to achieve an expansion of the wavepacket until it reaches thesize of the nanosphere itself? Most of this evolution occurs with a wavepacket size σ(t) that ismuch larger than the deBroglie wavelength of the gas molecules and thus in the long-distanceapproximation. Here, the decoherence rate is given by Γsat = λ2thΛ = Γxλ

2th/x

2zpf ≈ 3.6 MHz

(note that Γsat is denoted γ in (28)) and essentially corresponds to the rate of collision withsingle gas molecules, as in the long-distance approximation already a single collision localizesthe nanoparticle.

Given the expansion of the undisturbed wavepacket via σ(t) = xzpf(1 + Ωxt) we require anexpansion time of τ ≈ r/(xzpfΩx) = 12 ms, demanding a decoherence rate below 84 Hz. Thisis achievable by a reduction of the pressure by at least a factor of 5×104 to below 2×10−11 mbar.However, at these pressures blackbody radiation of the internally hot particle becomes relevant.The total localization parameter due to blackbody radiation consists of three contributions: ab-sorption Λbb,a, emission Λbb,e and scattering Λbb,sc (28):

Λbb = Λbb,sc + Λbb,e + Λbb,a ≈ 2.3× 1020 Hz/m2,with:Λbb,sc ≈ 1015 Hz/m2

Λbb,e(a) ≈ 2.3× 1020 Hz/m2(1.4× 1018 Hz/m2).(S11)

Here we assumed an environment temperature of 300 K, an average refractive index of �bb =2.1 + i × 0.57 and an internal temperature of 700 K, which is a conservative upper boundbased on measured heating rates in Ref. (18). This estimate relies on the model presentedin Ref. (54), where the particle motion is effectively coupled to two baths via the scatteringof gas particles. Briefly summarizing, the momentum distribution of incoming gas particlescorresponds to the environmental temperature of 300 K. The outgoing gas particles, in contrast,have thermalized (to an extent given by the accommodation factor) with the internal temperatureof the nanosphere. This consequently determines their momentum distribution and by recoilthe heating rate of the nanoparticle center-of-mass motion. This model allows to infer theinternal temperature from the heating rate. At our pressure and below we expect the internaltemperature of 700K to be given by equilibrium between absorption from the trapping laserand emission of black-body radiation (55). The expansion time tmax,bb ≈ 0.55 ms and length

9

ξmax,bb ≈ 2 nm is therefore limited by the blackbody radiation. To further reduce decoherence tothe desired level (below the gas scattering contribution) requires cryogenic temperatures (below130K) for both the internal particle temperature and the environment. This could be achievedeither by combining a cryogenic (ultra-high) vacuum environment with laser refrigration of thenanoparticle (56) or with low-absorption materials (57).

10

hetD

Nd:YAG

sig L

AOM

PDH

Laser FC

EOM

BS

2

z

xy

vac

ω₂

ω₁

ω₁

ω₁+ωhet

MM

FA

BS1

PBS1

PBS2

MO

OM

Figure S1: The optical setup. Light from a Nd:YAG master laser (wavelength λ = 1064nm,frequency ω1) is split into three beams with polarizing beam splitters (PBS1 and PBS2). Thefirst beam (blue) is frequency shifted by a combination of an electro-optical modulator (EOM)and a filtering cavity (FC) to ω2 = ω1 + ∆ωFSR, with ∆ωFSR the free spectral range of theoptomechanical cavity (OM). This beam is used for a Pound-Drever-Hall lock (PDH) that sta-bilizes the master laser on a TEM00 mode of the OM cavity. Note that this scheme automaticallystabilizes laser ω1 to the OM cavity, too. The second beam (red) seeds a fiber amplifier (FA)whose output is used for levitation of a silica nanoparticle in the optical tweezer. We achievepolarization control of the tweezer laser by a half-waveplate (λ/2) before the microscope ob-jective (MO). The third laser (dark red) is frequency shifted by an acousto-optical modulator(AOM) to ω1 + ωhet and serves as local oscillator for a balanced heterodyne detection (hetD).The microscope objective (MO) sits on a triaxial nano-positioner (not shown) and is used toposition the nanoparticle with respect to the OM cavity mode. The OM cavity is driven viaphotons scattered off of the particle from the tweezer laser. Light leaking out of both cavitymirrors is combined on a beamsplitter (BS1) and used as signal beam for heterodyne read-out(hetD) of the particle motion. To ensure phase stability at BS1 between both beams leaking outof the cavity, a movable mirror (MM) and an error signal derived from one port of BS1 (detectorsigL) are used to stabilize their relative phase.

11

0 100 200 300 400 500 600 700

0

5

10

15

PLO [ W]

Integratedarea

[a.u.]

Figure S2: Detected sideband power in the relevant frequency band for vacuum signalinput. For different local oscillator powers (LO) and a blocked signal path we integrate thearea in a frequency range (−350,−250) kHz (blue) and (250, 350) kHz (orange) around theheterodyne frequency ωhet, which are relevant to the occupation estimates of the Stokes andanti-Stokes sideband, respectively. The linear scaling of the bandpower with the LO power is anecessary and sufficient signature for shot noise.

12

Cavity

Enve

lope

ω-ωhet

/2 [kHz]

-300 -150 0 150 300

Sh

et [×

sh

ot n

ois

e]

1.5

1.25

1

x

yzzy

x

Figure S3: Heterodyne spectrum of the Stokes and anti-Stokes sidebands. The inelasticallyscattered Stokes (red) and anti-Stokes (blue) sidebands are mixed with a strong local oscillator(PLO ≈ 400µW) which is detuned from the optical tweezer by ωhet/2π = 10.2 MHz. Theresulting spectrum contains information about the motion of the nanoparticle in all 3 orthog-onal directions. Highlighted in red and blue, respectively, are the features corresponding tothe Stokes and anti-Stokes sidebands of the axial motion, labelled according to the axis of mo-tion. The features corresponding to the x motion are significantly smaller in amplitude andmore broad compared to the z and y motion as the motion along the x-axis is already heavilycooled when this trace was taken. The black solid line indicates the cavity transmission func-tion normalized to the Stokes sideband power. The central purple feature corresponds to thecarrier frequency. Unlabelled peaks are attributed to either intensity noise of the tweezer laseror electronic noise imprinted on the local oscillator.

13

-400 -200 0 200 400 6000.0

0.2

0.4

0.6

0.8

1.0

( - FSR)/2 [kHz]

Transmission[a.u.]

Figure S4: Cavity response in transmission. We regularly scan over a cavity resonance bydetuning a laser derived from another cavity mode. Here 26 of these measurements are super-imposed to each other with a final cavity decay rate κ/2π = (193± 4) kHz. The average cavityfree spectral range from the scans is ∆ωFSR/2π = 14.0192 GHz.

14

0 1 2 3 4 5

1

2

3

Time [s]

x0-

/4[nm]

Figure S5: Particle distance from the cavity node. We position the nanoparticle in the vicinityof the cavity node. The absolute position is calibrated from the intracavity photon number fora particle placed at the cavity antinode. The particle is at most ∼ 4 nm away from the cavitynode, which strongly suppresses elastic scattering into the cavity and phase noise, leading tolow phase noise heating.

15

Mephisto

Mephisto & FA

1 10 100 1000-140

-130

-120

-110

-100

-90

Frequency [kHz]

RIN

[dB/Hz]

Figure S6: Laser intensity noise of the tweezer laser. Laser intensity noise at twice the me-chanical frequency leads to parametric heating. We measure the relative intensity noise (RIN)from the Mephisto laser (blue) and the combined noise with the fiber amplifier used for trap-ping (orange). The added intensity noise of the fiber amplifier is due to the limitations of theamplifier’s current locking circuit. However, at the relevant frequency band (shaded gray area)the intensity noise is at the level of the noise intrinsic to the Mephisto laser.

16

100 200 300 400 500 6000

1

2

3

4

5

Detuning Δ 2π [kHz]

nx

Figure S7: Phonon occupation deduced from sideband thermometry (black) and fromheating and cooling rates (red). We measure the phonon occupation from sideband asymme-try, which does not depend on detection calibration, and compare it to the occupation obtainedfrom the independently measured heating and cooling rates. The occupations obtained fromthese two methods are in a good agreement.

17

References

1. A. Ashkin, J. M. Dziedzic, J. E. Bjorkholm, S. Chu, Observation of a single-beam gradient force optical trap for dielectric particles. Opt. Lett. 11, 288 (1986). doi:10.1364/OL.11.000288 Medline

2. D. E. Chang, C. A. Regal, S. B. Papp, D. J. Wilson, J. Ye, O. Painter, H. J. Kimble, P. Zoller, Cavity opto-mechanics using an optically levitated nanosphere. Proc. Natl. Acad. Sci. U.S.A. 107, 1005–1010 (2010). doi:10.1073/pnas.0912969107 Medline

3. O. Romero-Isart, M. L. Juan, R. Quidant, J. I. Cirac, Toward quantum superposition of living organisms. New J. Phys. 12, 033015 (2010). doi:10.1088/1367-2630/12/3/033015

4. P. F. Barker, M. N. Shneider, Cavity cooling of an optically trapped nanoparticle. Phys. Rev. A 81, 023826 (2010). doi:10.1103/PhysRevA.81.023826

5. O. Romero-Isart, A. C. Pflanzer, F. Blaser, R. Kaltenbaek, N. Kiesel, M. Aspelmeyer, J. I. Cirac, Large quantum superpositions and interference of massive nanometer-sized objects. Phys. Rev. Lett. 107, 020405 (2011). doi:10.1103/PhysRevLett.107.020405 Medline

6. A. D. O’Connell, M. Hofheinz, M. Ansmann, R. C. Bialczak, M. Lenander, E. Lucero, M. Neeley, D. Sank, H. Wang, M. Weides, J. Wenner, J. M. Martinis, A. N. Cleland, Quantum ground state and single-phonon control of a mechanical resonator. Nature 464, 697–703 (2010). doi:10.1038/nature08967 Medline

7. Y. Chu, P. Kharel, T. Yoon, L. Frunzio, P. T. Rakich, R. J. Schoelkopf, Creation and control of multi-phonon Fock states in a bulk acoustic-wave resonator. Nature 563, 666–670 (2018). doi:10.1038/s41586-018-0717-7 Medline

8. R. Riedinger, S. Hong, R. A. Norte, J. A. Slater, J. Shang, A. G. Krause, V. Anant, M. Aspelmeyer, S. Gröblacher, Non-classical correlations between single photons and phonons from a mechanical oscillator. Nature 530, 313–316 (2016). doi:10.1038/nature16536 Medline

9. J. I. Cirac, M. Lewenstein, K. Mølmer, P. Zoller, Quantum superposition states of Bose-Einstein condensates. Phys. Rev. A 57, 1208–1218 (1998). doi:10.1103/PhysRevA.57.1208

10. M. Rossi, D. Mason, J. Chen, Y. Tsaturyan, A. Schliesser, Measurement-based quantum control of mechanical motion. Nature 563, 53–58 (2018). doi:10.1038/s41586-018-0643-8 Medline

11. F. Tebbenjohanns, M. Frimmer, V. Jain, D. Windey, L. Novotny, Motional Sideband Asymmetry of a Nanoparticle Optically Levitated in Free Space. Phys. Rev. Lett. 124, 013603 (2020). doi:10.1103/PhysRevLett.124.013603

http://dx.doi.org/10.1364/OL.11.000288http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=19730608&dopt=Abstracthttp://dx.doi.org/10.1073/pnas.0912969107http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=20080573&dopt=Abstracthttp://dx.doi.org/10.1088/1367-2630/12/3/033015http://dx.doi.org/10.1103/PhysRevA.81.023826http://dx.doi.org/10.1103/PhysRevLett.107.020405http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=21797585&dopt=Abstracthttp://dx.doi.org/10.1038/nature08967http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=20237473&dopt=Abstracthttp://dx.doi.org/10.1038/s41586-018-0717-7http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=30464340&dopt=Abstracthttp://dx.doi.org/10.1038/nature16536http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=26779950&dopt=Abstracthttp://dx.doi.org/10.1103/PhysRevA.57.1208http://dx.doi.org/10.1038/s41586-018-0643-8http://dx.doi.org/10.1038/s41586-018-0643-8http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=30382202&dopt=Abstracthttp://dx.doi.org/10.1103/PhysRevLett.124.013603

12. P. Horak, G. Hechenblaikner, K. M. Gheri, H. Stecher, H. Ritsch, Cavity-Induced Atom Cooling in the Strong Coupling Regime. Phys. Rev. Lett. 79, 4974–4977 (1997). doi:10.1103/PhysRevLett.79.4974

13. V. Vuletić, S. Chu, Laser cooling of atoms, ions, or molecules by coherent scattering. Phys. Rev. Lett. 84, 3787–3790 (2000). doi:10.1103/PhysRevLett.84.3787 Medline

14. M. Aspelmeyer, T. J. Kippenberg, F. Marquardt, Cavity optomechanics. Rev. Mod. Phys. 86, 1391–1452 (2014). doi:10.1103/RevModPhys.86.1391

15. N. Kiesel, F. Blaser, U. Delić, D. Grass, R. Kaltenbaek, M. Aspelmeyer, Cavity cooling of an optically levitated submicron particle. Proc. Natl. Acad. Sci. U.S.A. 110, 14180–14185 (2013). doi:10.1073/pnas.1309167110 Medline

16. J. Millen, P. Z. G. Fonseca, T. Mavrogordatos, T. S. Monteiro, P. F. Barker, Cavity cooling a single charged levitated nanosphere. Phys. Rev. Lett. 114, 123602 (2015). doi:10.1103/PhysRevLett.114.123602 Medline

17. N. Meyer, A. L. R. Sommer, P. Mestres, J. Gieseler, V. Jain, L. Novotny, R. Quidant, Resolved-Sideband Cooling of a Levitated Nanoparticle in the Presence of Laser Phase Noise. Phys. Rev. Lett. 123, 153601 (2019). doi:10.1103/PhysRevLett.123.153601 Medline

18. U. Delić et al., arXiv 1902.06605 [physics.optics] (18 February 2019).

19. U. Delić, M. Reisenbauer, D. Grass, N. Kiesel, V. Vuletić, M. Aspelmeyer, Cavity Cooling of a Levitated Nanosphere by Coherent Scattering. Phys. Rev. Lett. 122, 123602 (2019). doi:10.1103/PhysRevLett.122.123602 Medline

20. D. Windey, C. Gonzalez-Ballestero, P. Maurer, L. Novotny, O. Romero-Isart, R. Reimann, Cavity-Based 3D Cooling of a Levitated Nanoparticle via Coherent Scattering. Phys. Rev. Lett. 122, 123601 (2019). doi:10.1103/PhysRevLett.122.123601 Medline

21. S. Nimmrichter, K. Hammerer, P. Asenbaum, H. Ritsch, M. Arndt, Master equation for the motion of a polarizable particle in a multimode cavity. New J. Phys. 12, 083003 (2010). doi:10.1088/1367-2630/12/8/083003

22. A. H. Safavi-Naeini, J. Chan, J. T. Hill, T. P. M. Alegre, A. Krause, O. Painter, Observation of quantum motion of a nanomechanical resonator. Phys. Rev. Lett. 108, 033602 (2012). doi:10.1103/PhysRevLett.108.033602 Medline

23. R. W. Peterson, T. P. Purdy, N. S. Kampel, R. W. Andrews, P.-L. Yu, K. W. Lehnert, C. A. Regal, Laser Cooling of a Micromechanical Membrane to the Quantum Backaction Limit. Phys. Rev. Lett. 116, 063601 (2016). doi:10.1103/PhysRevLett.116.063601 Medline

24. See supplementary materials.

http://dx.doi.org/10.1103/PhysRevLett.79.4974http://dx.doi.org/10.1103/PhysRevLett.84.3787http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=11019206&dopt=Abstracthttp://dx.doi.org/10.1103/RevModPhys.86.1391http://dx.doi.org/10.1073/pnas.1309167110http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=23940352&dopt=Abstracthttp://dx.doi.org/10.1103/PhysRevLett.114.123602http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=25860743&dopt=Abstracthttp://dx.doi.org/10.1103/PhysRevLett.123.153601http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=31702279&dopt=Abstracthttps://arxiv.org/abs/1902.06605http://dx.doi.org/10.1103/PhysRevLett.122.123602http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=30978033&dopt=Abstracthttp://dx.doi.org/10.1103/PhysRevLett.122.123601http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=30978044&dopt=Abstracthttp://dx.doi.org/10.1088/1367-2630/12/8/083003http://dx.doi.org/10.1103/PhysRevLett.108.033602http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=22400740&dopt=Abstracthttp://dx.doi.org/10.1103/PhysRevLett.116.063601http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=26918990&dopt=Abstract

25. V. Sudhir, D. J. Wilson, R. Schilling, H. Schütz, S. A. Fedorov, A. H. Ghadimi, A. Nunnenkamp, T. J. Kippenberg, Appearance and Disappearance of Quantum Correlations in Measurement-Based Feedback Control of a Mechanical Oscillator. Phys. Rev. X 7, 011001 (2017). doi:10.1103/PhysRevX.7.011001

26. V. Jain, J. Gieseler, C. Moritz, C. Dellago, R. Quidant, L. Novotny, Direct Measurement of Photon Recoil from a Levitated Nanoparticle. Phys. Rev. Lett. 116, 243601 (2016). doi:10.1103/PhysRevLett.116.243601 Medline

27. O. Romero-Isart, A. C. Pflanzer, M. L. Juan, R. Quidant, N. Kiesel, M. Aspelmeyer, J. I. Cirac, Optically levitating dielectrics in the quantum regime: Theory and protocols. Phys. Rev. A 83, 013803 (2011). doi:10.1103/PhysRevA.83.013803

28. O. Romero-Isart, Quantum superposition of massive objects and collapse models. Phys. Rev. A 84, 052121 (2011). doi:10.1103/PhysRevA.84.052121

29. A. A. Geraci, S. B. Papp, J. Kitching, Short-range force detection using optically cooled levitated microspheres. Phys. Rev. Lett. 105, 101101 (2010). doi:10.1103/PhysRevLett.105.101101 Medline

30. J. Bateman, I. McHardy, A. Merle, T. R. Morris, H. Ulbricht, On the existence of low-mass dark matter and its direct detection. Sci. Rep. 5, 8058 (2015). doi:10.1038/srep08058 Medline

31. D. Rickles, C. M. DeWitt, The Role of Gravitation in Physics: Report from the 1957 Chapel Hill Conference (2011); https://edition-open-sources.org/media/sources/5/Sources5.pdf.

32. S. Bose, A. Mazumdar, G. W. Morley, H. Ulbricht, M. Toroš, M. Paternostro, A. A. Geraci, P. F. Barker, M. S. Kim, G. Milburn, Spin Entanglement Witness for Quantum Gravity. Phys. Rev. Lett. 119, 240401 (2017). doi:10.1103/PhysRevLett.119.240401 Medline

33. C. Marletto, V. Vedral, Gravitationally Induced Entanglement between Two Massive Particles is Sufficient Evidence of Quantum Effects in Gravity. Phys. Rev. Lett. 119, 240402 (2017). doi:10.1103/PhysRevLett.119.240402 Medline

34. U. Delić, thesis, University of Vienna (2019).

35. R. Uberna, A. Bratcher, B. G. Tiemann, Coherent polarization beam combination. IEEE J. Quantum Electron. 46, 1191–1196 (2010). doi:10.1109/JQE.2010.2044976 Medline

36. D. Grass, thesis, University of Vienna (2018).

37. M. Underwood, D. Mason, D. Lee, H. Xu, L. Jiang, A. B. Shkarin, K. Børkje, S. M. Girvin, J. G. E. Harris, Measurement of the motional sidebands of a nanogram-scale oscillator in the quantum regime. Phys. Rev. A 92, 061801 (2015). doi:10.1103/PhysRevA.92.061801

38. M. Toroš, T. S. Monteiro, arXiv 1909.09555 [quant-ph] (28 November 2019).

http://dx.doi.org/10.1103/PhysRevX.7.011001http://dx.doi.org/10.1103/PhysRevLett.116.243601http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=27367388&dopt=Abstracthttp://dx.doi.org/10.1103/PhysRevA.83.013803http://dx.doi.org/10.1103/PhysRevA.84.052121http://dx.doi.org/10.1103/PhysRevLett.105.101101http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=20867507&dopt=Abstracthttp://dx.doi.org/10.1038/srep08058http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=25622565&dopt=Abstracthttps://edition-open-sources.org/media/sources/5/Sources5.pdfhttp://dx.doi.org/10.1103/PhysRevLett.119.240401http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=29286711&dopt=Abstracthttp://dx.doi.org/10.1103/PhysRevLett.119.240402http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=29286752&dopt=Abstracthttp://dx.doi.org/10.1109/JQE.2010.2044976http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=21151233&dopt=Abstracthttp://dx.doi.org/10.1103/PhysRevA.92.061801https://arxiv.org/abs/1909.09555

39. P. Rabl, C. Genes, K. Hammerer, M. Aspelmeyer, Phase-noise induced limitations on cooling and coherent evolution in optomechanical systems. Phys. Rev. A 80, 063819 (2009). doi:10.1103/PhysRevA.80.063819

40. T. J. Kippenberg, A. Schliesser, M. L. Gorodetsky, Phase noise measurement of external cavity diode lasers and implications for optomechanical sideband cooling of GHz mechanical modes. New J. Phys. 15, 015019 (2013). doi:10.1088/1367-2630/15/1/015019

41. A. H. Safavi-Naeini, J. Chan, J. T. Hill, S. Gröblacher, H. Miao, Y. Chen, M. Aspelmeyer, O. Painter, Laser noise in cavity-optomechanical cooling and thermometry. New J. Phys. 15, 035007 (2013). doi:10.1088/1367-2630/15/3/035007

42. J. Hoelscher-Obermaier, thesis, University of Vienna (2017).

43. T. A. Savard, K. M. O’Hara, J. E. Thomas, Laser-noise-induced heating in far-off resonance optical traps. Phys. Rev. A 56, R1095–R1098 (1997). doi:10.1103/PhysRevA.56.R1095

44. A. M. Jayich, J. C. Sankey, K. Børkje, D. Lee, C. Yang, M. Underwood, L. Childress, A. Petrenko, S. M. Girvin, J. G. E. Harris, Cryogenic optomechanics with a Si 3 N 4 membrane and classical laser noise. New J. Phys. 14, 115018 (2012). doi:10.1088/1367-2630/14/11/115018

45. L. Novotny, B. Hecht, Principles of Nano-Optics (Cambridge Univ. Press, 2012).

46. C. Genes, D. Vitali, P. Tombesi, S. Gigan, M. Aspelmeyer, Ground-state cooling of a micromechanical oscillator: Comparing cold damping and cavity-assisted cooling schemes. Phys. Rev. A 77, 033804 (2008). doi:10.1103/PhysRevA.77.033804

47. V. Vuletić, H. W. Chan, A. T. Black, Three-dimensional cavity Doppler cooling and cavity sideband cooling by coherent scattering. Phys. Rev. A 64, 033405 (2001). doi:10.1103/PhysRevA.64.033405

48. P. Domokos, P. Horak, H. Ritsch, Semiclassical theory of cavity-assisted atom cooling. J. Phys. B 34, 187–198 (2001). doi:10.1088/0953-4075/34/2/306

49. J. Gieseler, B. Deutsch, R. Quidant, L. Novotny, Subkelvin parametric feedback cooling of a laser-trapped nanoparticle. Phys. Rev. Lett. 109, 103603 (2012). doi:10.1103/PhysRevLett.109.103603 Medline

50. T. C. Li, S. Kheifets, M. G. Raizen, Millikelvin cooling of an optically trapped microsphere in vacuum. Nat. Phys. 7, 527–530 (2011). doi:10.1038/nphys1952

51. J. Vovrosh, M. Rashid, D. Hempston, J. Bateman, M. Paternostro, H. Ulbricht, Parametric feedback cooling of levitated optomechanics in a parabolic mirror trap. J. Opt. Soc. Am. B 34, 1421 (2017). doi:10.1364/JOSAB.34.001421

52. W. H. Zurek, Decoherence, einselection, and the quantum origins of the classical. Rev. Mod. Phys. 75, 715–775 (2003). doi:10.1103/RevModPhys.75.715

http://dx.doi.org/10.1103/PhysRevA.80.063819http://dx.doi.org/10.1088/1367-2630/15/1/015019http://dx.doi.org/10.1088/1367-2630/15/3/035007http://dx.doi.org/10.1103/PhysRevA.56.R1095http://dx.doi.org/10.1088/1367-2630/14/11/115018http://dx.doi.org/10.1088/1367-2630/14/11/115018http://dx.doi.org/10.1103/PhysRevA.77.033804http://dx.doi.org/10.1103/PhysRevA.64.033405http://dx.doi.org/10.1088/0953-4075/34/2/306http://dx.doi.org/10.1103/PhysRevLett.109.103603http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=23005289&dopt=Abstracthttp://dx.doi.org/10.1038/nphys1952http://dx.doi.org/10.1364/JOSAB.34.001421http://dx.doi.org/10.1103/RevModPhys.75.715

53. A. O. Caldeira, A. J. Leggett, Influence of damping on quantum interference: An exactly soluble model. Phys. Rev. A 31, 1059–1066 (1985). doi:10.1103/PhysRevA.31.1059 Medline

54. J. Millen, T. Deesuwan, P. Barker, J. Anders, Nanoscale temperature measurements using non-equilibrium Brownian dynamics of a levitated nanosphere. Nat. Nanotechnol. 9, 425–429 (2014). doi:10.1038/nnano.2014.82 Medline

55. E. Hebestreit, R. Reimann, M. Frimmer, L. Novotny, Measuring the internal temperature of a levitated nanoparticle in high vacuum. Phys. Rev. A 97, 043803 (2018). doi:10.1103/PhysRevA.97.043803

56. A. T. M. A. Rahman, P. F. Barker, Laser refrigeration, alignment and rotation of levitated Yb3+:YLF nanocrystals. Nat. Photonics 11, 634–638 (2017). doi:10.1038/s41566-017-0005-3

57. A. C. Frangeskou, A. T. M. A. Rahman, L. Gines, S. Mandal, O. A. Williams, P. F. Barker, G. W. Morley, Pure nanodiamonds for levitated optomechanics in vacuum. New J. Phys. 20, 043016 (2018). doi:10.1088/1367-2630/aab700

http://dx.doi.org/10.1103/PhysRevA.31.1059http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=9895586&dopt=Abstracthttp://dx.doi.org/10.1038/nnano.2014.82http://www.ncbi.nlm.nih.gov/entrez/query.fcgi?cmd=Retrieve&db=PubMed&list_uids=24793558&dopt=Abstracthttp://dx.doi.org/10.1103/PhysRevA.97.043803http://dx.doi.org/10.1038/s41566-017-0005-3http://dx.doi.org/10.1038/s41566-017-0005-3http://dx.doi.org/10.1088/1367-2630/aab700

Delic-SM.cover.page.pdfscience.sciencemag.org/cgi/content/full/science.aba3993/DC1science.sciencemag.org/cgi/content/full/science.aba3993/DC1science.sciencemag.org/cgi/content/full/science.aba3993/DC1Cooling of a levitated nanoparticle to the motional quantum ground stateCooling of a levitated nanoparticle to the motional quantum ground state

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