Full Rotational Control of Levitated Silicon Nanorods
Stefan Kuhn, Alon Kosloff, Benjamin A. Stickler, Fernando Patolsky, Klaus Hornberger, Markus Arndt, James Millen
FFull Rotational Control of Levitated Silicon Nanorods
Stefan Kuhn, ∗ Alon Kosloff, † Benjamin A. Stickler, FernandoPatolsky, Klaus Hornberger, Markus Arndt, and James Millen University of Vienna, Faculty of Physics, VCQ, Boltzmangasse 5, 1090 Vienna, Austria School of Chemistry, Tel-Aviv University, Ramat-Aviv 69978, Israel University of Duisburg-Essen, Lotharstraße 1, 47048 Duisburg, Germany
Optically levitated nano-objects in vacuum are amongst the highest-quality mechanical oscillators,and thus of great interest for force sensing, cavity quantum optomechanics, and nanothermodynamicstudies. These precision applications require exquisite control. Here, we present full control over therotational and translational dynamics of an optically levitated silicon nanorod. We trap its centre-of-mass and align it along the linear polarization of the laser field. The rod can be set into rotation at apredefined frequency by exploiting the radiation pressure exerted by elliptically polarized light. Therotational motion of the rod dynamically modifies the optical potential, which allows tuning of therotational frequency over hundreds of Kilohertz. Through nanofabrication, we can tailor all of thetrapping frequencies and the optical torque, achieving reproducible dynamics which are stable overmonths, and analytically predict the motion with great accuracy. This first demonstration of fullro-translational control of nanoparticles in vacuum opens up the fields of rotational optomechanics,rotational ground state cooling and the study of rotational thermodynamics in the underdampedregime.
Introduction—
Nanofabrication has advanced all areasof science, technology and medicine [1], including the fieldof optomechanics, where the motion of a mechanical os-cillator is controlled by light. The quantum ground stateof motion has been reached in optomechanical crystaldevices [2], and superconducting microwave circuits [3].Ground-state cooling enables the coherent transductionof signals [4], the production of non-classical states oflight and matter [5], and the ultra-sensitive detection ofmotion [6] and forces [7]. Coherent optomechanical tech-nology is limited by the coupling between the mechanicaldevice and its environment, which leads to decoherence ofquantum states, and by a reduction in mechanical qualityfactor due to clamping forces on the oscillator.These limitations can be overcome by optically levi-tating the mechanical system, such that it oscillates ina harmonic trapping potential. Optical trapping is ap-plicable from atoms in vacuum [8, 9], to complex organ-isms in liquid [10]. By optically levitating nanoscale ob-jects in vacuum, ultra-high mechanical quality factors( Q ∼ ) are predicted [11], and it may be possi-ble to generate macroscopic quantum superpositions [12].Such massive quantum systems could test the limits ofquantum physics [13, 14], looking for the existence ofnew mechanisms of wave-function collapse such as spon-taneous localisation [15] or gravitational effects [16–18].These goals require a high degree of control over all ofthe dynamics of the nanoparticle.The field of levitated optomechanics is growing rapidly,with progress including feedback [19–21] and cavity cool-ing [22–25] to the milli-Kelvin level and below, the sens-ing of forces on the zepto-Newton scale [26], and thestudy of Brownian motion [27] and equilibration [28] inthe underdamped regime. Experiments are often limitedby the quality of commercially available nanoparticles. α β y x z a) b)c) α α β FIG. 1: a) Nanofabricated silicon nanorods of length (cid:96) (cid:39) (725 ±
15) nm and diameter d (cid:39) (130 ±
13) nm are optically levitated ina standing laser wave at low pressures. The light they scatter iscollected by a multimode optical fiber placed close to the trapwaist. b) The rods have five degrees of freedom which can becontrolled; three translational ( x, y, z ) and two rotational ( α, β ).c) By monitoring the scattered light, trapping of all five degreesof freedom can be observed in the power spectral density (PSD)when the trap light is linearly polarized. This data was acquiredat a pressure of 4 mbar. The appearance of the various harmonicscan be explained by slight misalignment of the trap as discussedin Supplementary Information 1.
Impurities lead to absorption of the trapping light, caus- a r X i v : . [ qu a n t - ph ] M a r ing loss at low pressures [29], and even graphitization oflevitated diamond [30]. Recently, rotation has been de-tected in levitated particles [1, 31, 32, 34], displaying fargreater rotation rates than experiments in liquid [35–38].In this work, we trap clean, nanofabricated siliconnanorods, and study their center-of-mass and rotationalmotion. Our particles are of uniform, tailored sizeand shape, allowing a high degree of repeatability, pre-dictability and control of the dynamics. We are able totrap the nanorods, trap and control their orientation, andtunably spin them using the radiation pressure exerted bythe light field. While rotational control has been achievedin liquid [35–37, 39, 40], this is the first demonstration oftrapping nanorods in vacuum, and we observe novel fea-tures such as shape enhanced light-matter interactionsand dynamic reshaping of the trapping potential. Suchfull control opens the way to optomechanical rotationalcooling [4, 34, 41], even to the quantum level [4]. EXPERIMENTAL SETUP
A single silicon nanorod is optically trapped in thefocus formed by two counterpropagating laser beamsof wavelength λ = 1550 nm, see Fig. 1a). At thiswavelength, silicon exhibits a high relative permittiv-ity, ε r = 12, and negligible absorption, which is sup-ported by the fact that we see no signature of heatingdue to light absorption (following the method in Ref.[29]). The nanorods are tailored to have a length of (cid:96) = (725 ±
15) nm and a diameter of d = (130 ±
13) nm,corresponding to a mass M = (1 . ± . × amu.They are fabricated onto a silicon chip following themethods described in Ref. [1]. The laser trap is charac-terized by a beam waist radius w ≈ µ m and the totalpower P tot = 1 .
35 W, making a large volume trap to en-hance the rate of capture. The nanorods are trapped in aclean N environment at a pressure of p g = 4 mbar, afterbeing launched by laser desorption from a silicon wafer,see Refs. [1, 43]. Up to 10 nanorods are simultaneouslytrapped, and we perturb the trapping field until a sin-gle nanorod remains. The rods can be stably trappedfor months at any pressure above 1 mbar, below whichthey are lost, as observed in experiments with sphericalnanoparticles [22, 29]. TRAPPING THE NANORODS
The motional state of the nanorod is described by itscenter-of-mass position ( x, y, z ) and by its orientation( α, β ), see Fig. 1b), where x points counter-parallel tothe direction of gravity and z along the beam axis. Theorientation of the rod is parametrized by α , the angle be-tween the x -axis and the projection of the rod onto the x - y -plane, and β the angle between the rod’s symmetry S i gn a l ( m V ) c)b) α z, α , β LinearCircular t ( s) μ LASER POL. CONTROL AMP CIRCULATOR50:50
SCATTERED LIGHT
TRAPVACUUM ← FREE SPACE ← F R EE S P A C E a) PCP PCP PCP
FIG. 2: a) Experimental setup. Light at λ = 1550 nm is producedby a fiber laser (Keysight 81663A), and then goes through anelectro-optical in-fiber polarization controller (EOSPACE),allowing us to realise arbitrary waveplate operations. The light isamplified in a fiber amplifier (Hangzhou Huatai OpticHA5435B-1) and split equally to make the two arms of the trap.Stress induced birefringence in the fibers can be accounted forwith polarization controlling paddles (PCP). The system iscompletely fiber-based until out-coupled to the aspheric trappinglenses (f=20 mm). The inset shows an SEM micrograph of a rodthat was launched and captured on a sample plate. The scatteredlight signal reveals the nanorod dynamics in case of b) co-linearpolarization, and c) the strongly driven rotation of the rod forcircularly polarized trapping light. axis and the beam propagation axis. The motion of thenanorod is measured via the light that the rod scattersout of the trap, which is collected with a 1 mm diametermultimode optical fiber as described in Ref. [1].The polarization of the two trapping beams determinesthe properties of the optical trap. In the case of co-linearpolarization the rod aligns with the field polarization andis thus trapped in all its degrees of freedom. The result-ing trapping frequencies can be measured in the powerspectral density (PSD) of the scattered light signal, asshown in Fig. 1c). Using a LiNb-polarization controller,we can perform arbitrary wave-plate operations on thepolarization of the trapping light [44]. The optical setup(see Fig. S1a) ) is designed such that the rod experiencesthe same polarization from both arms of the counterprop-agating trap. By realising a half-waveplate operation onthe linearly polarized trapping beam we can align the rodalong any direction orthogonal to the trap axis, as hasbeen observed in liquid [36].The trapping frequencies of a harmonically capturedrod can be calculated as [4] f x,y = 12 π (cid:115) P tot χ (cid:107) π(cid:37)cw , f z = 12 π (cid:115) P tot χ (cid:107) k π(cid:37)cw ,f β = 12 π (cid:115) P tot χ (cid:107) π(cid:37)cw (cid:96) (cid:18) ∆ χχ (cid:107) + ( k(cid:96) ) (cid:19) ,f α = 12 π (cid:115) P tot ∆ χπ(cid:37)cw (cid:96) , (1)where k = 2 π/λ , (cid:37) = 2330 kg m − is the density of sil-icon, χ (cid:107) = ε r − χ = ( ε r − / ( ε r + 1) is the sus-ceptibility anisotropy [45]. At the maximum input powerwe measure f x,y = (1 . ± .
2) kHz, f z = (124 ±
1) kHz, f α = (134 ±
1) kHz and f β = (175 . ± .
5) kHz, seeFig. 1c). For comparison, a silicon nanosphere of thesame volume under the same experimental conditionswould have f z = 58 kHz, and a silica sphere would have f z = 47 kHz, illustrating the great potential for siliconnanorods in cavity cooling experiments [4]. We can usethe measured frequencies to deduce the trapping waistradius w = (27 ± µ m, which is the only free experi-mental parameter. The measured frequencies agree wellwith the theoretical expectations, as shown in Fig. 3c).The slight ( < f β is attributed to the fact that therods have finite diameter and the generalized Rayleigh-Gans approximation [4] is not strictly valid. SPINNING THE NANORODS
When the trapping light is circularly polarized, thetrapping potential for the α motion vanishes whilst thestanding wave structure along z is retained. The radia-tion pressure of the laser field exerts a constant torque N α acting on α . Adapting the theory presented in Ref.[4], the resulting torque is obtained as N α = P tot ∆ χ(cid:96) d k cw [∆ χη ( k(cid:96) ) + χ ⊥ η ( k(cid:96) )] , (2)where the two functions η , ( k(cid:96) ) are given by η ( k(cid:96) ) = 34 (cid:90) − d ξ (1 − ξ )sinc (cid:18) k(cid:96)ξ (cid:19) ,η ( k(cid:96) ) = 38 (cid:90) − d ξ (1 − ξ )sinc (cid:18) k(cid:96)ξ (cid:19) . (3)For short rods, k(cid:96) (cid:28)
1, one has η (cid:39) η (cid:39) m g takes theform [47] Γ = d(cid:96)p g M (cid:114) πm g k B T (cid:18)
32 + π (cid:19) , (4)where T is the gas temperature.The maximum steady-state rotation frequency is ob-tained by balancing the torque Eq. 2 with the dampingEq. 4, f α, max = N α πI Γ , (5)with I = M (cid:96) /
12 the rod’s moment of inertia. This ex-pression agrees well with the measured value of the rota-tion frequency f α, rot as a function of power and pressure,as shown in Figs. 3d) and e), respectively.A comparison of the PSD for the co-linear and the cir-cular polarization traps is shown in Fig. 3c). The peakrelated to the trapping frequency at f α vanishes and apronounced peak at 2 f α, rot arises. We are only sensitiveto 2 f α, rot due to the symmetry of the rod. The rotationof the rod in the circularly polarized field results in a re-duced average susceptibility and thus a weaker trappingpotential, which shifts the axial trapping frequency to f z, rot = 94 kHz as discussed in Supplementary Informa-tion 1. The rapid rotation in α leads to a stabilization in β and hence the complete suppression of the peak at f β in Fig. 3c). A similar effect has also been observed forspinning microspheres [32].The broad distribution of frequencies about 2 f α, rot isdue to perturbations temporarily decreasing the rotationrate, which then takes time to spin back up to the max-imum value. For example, irregular excursions in theradial x, y directions lead to variations in the instanta-neous rotation frequency via variation in the local lightintensity, as shown in Fig. 3f), with the correlation clearlyshown in Fig. 3g). The maximum rotation rate is limitedby pressure in this set-up, with an ultimate limit pre-sumably set by material properties. In previous work,rotation rates of 50 MHz were observed for free nanorodsin UHV [1]. TUNING THE ROTATIONAL FREQUENCY
To study the effect of driven rotation in more detailwe use the polarization controller to perform a quarter-waveplate (QWP) operation on the trapping light andtrack the motion of the rod at each setting, see Fig. 4.Starting from a linear polarization along x and increasingits ellipticity at first leads to a shift of all trapping fre-quencies to lower values due to a reduced trap depth, asshown in Fig. 4a). At a QWP setting of 30 ◦ the radiationpressure induced torque starts driving the rod into rota-tion over the trapping potential in the direction of α , the f z z f z, , z, , - a)b) c)d) e) f)g) Linear , trappedCircular , rotating 100 200 300 Frequency (kHz) -5 -4 -3 -2 PS D ( m V H z - ) LinearCircular2f , ,rot , ,rot Frequency (kHz) -4 -3 -2 -1 Power (W) f , , r o t ( M H z ) Pressure (mbar) f , , r o t ( M H z ) Radial Motion
Time (ms) S i g . ( m V ) Rotational Frequency f , , r o t ( M H z )
380 400 420
Radial position signal (mV) f , , r o t ( M H z ) FIG. 3:
Comparing the dynamics when the nanorod is a) trapped in all degrees of freedom by linearly polarized light and b) driven torotate in the α direction by circularly polarized light. c) The PSD for circularly (red) and linearly (blue) polarized light. For circularpolarization, the trapped frequency f α vanishes, and the rotational frequency f α, rot appears. The peak at f β vanishes since the motionin β is stabilized when the rod is spinning. Markers indicate predicted trapping frequencies. The rotational frequency scales d) linearlywith power, and e) decreases with increasing pressure, as predicted by Eq. (5). Markers represent the mean value of f α, rot , the shadedareas represent the full range of f α, rot , and solid lines are the theoretically expected maximal value of f α, rot . The broad frequencydistribution of f α, rot is due to coupling between the motion in α and x, y (radial). f) Perturbations from the equilibrium position (lowerpanel) are reflected in instantaneous frequency fluctuations (top panel). g) The correlation between the radial position and f α, rot . frequencies f α,β vanish, and f z drops to a steady valueof f z, rot = (94 ±
1) kHz, as also seen in Fig. 3c). Rotat-ing beyond 45 ◦ , one may expect the nanorod to becometrapped again at 60 ◦ , however the rod is not trapped un-til 85 ◦ . When starting at 90 ◦ and decreasing the QWPangle, the rod spins at 60 ◦ , and is not trapped until 5 ◦ ,showing a symmetric hysteresis, see Fig 4b).This effect is due to the anisotropy of the suscepti-bility tensor: A trapped rod experiences the full trapdepth related to χ (cid:107) whereas the trapping potential fora spinning rod is proportional to the susceptibility av-eraged over rotations in the 2D plane orthogonal to thebeam axis ( χ (cid:107) + χ ⊥ ) /
2, which is smaller by a factor of1.7. Thus, it requires a greater torque to spin a trappedrod than to maintain the rotation of an already spinningrod. The value of f α, rot varies with the ellipticity of thelight, as shown in Fig. 4b). By exploiting the dynamicalmodification of the trap depth we can extend the rangeover which the rotation frequency can be tuned to manyhundreds of Kilohertz. CONCLUSIONS
In summary, we present a method to capture and lev-itate nanofabricated silicon nanorods at low pressures,working with telecoms wavelengths in a fibre-based setup.We can precisely control the length and diameter of ournanorods, meaning we can tailor rods to attain particu-lar trapping and rotational frequencies. We are able totrap all relevant degrees of freedom, and control the ori-entation of the rods via the polarization of the trappingbeams. By using circularly polarized light we can spin thenanorods at more than 1 MHz, and tune this frequencyover hundreds of Kilohertz by introducing ellipticity intothe field polarization and through a dynamic modifica-tion of the trapping potential. When the rod is spinningwe notice a stabilization of the tilt angle β and a cou-pling to the radial motion x, y . The system is very wellunderstood as shown by the excellent agreement betweenexperiment and theory. The high degree of control opensthe way to study rotational optomechanics [48–50], ori-entational decoherence [51, 52], rotational underdampedBrownian motion and stochastic thermodynamics, andsynchronisation of multiple rotors due to optical binding[53]. This is the first use of silicon in an optical trap invacuum, and its high susceptibility and low absorption in f z f , f - a) b) QWP angle (degrees) F r e qu e n c y ( k H z ) QWP angle (degrees) f , , r o t ( k H z ) FIG. 4:
Effect of performing a quarter-waveplate (QWP)operation on the trapping light at 5 mbar, either increasing fromfrom 0 ◦ (crosses) or decreasing from 90 ◦ (circles). At 0 ◦ and 90 ◦ the trap is linearly polarized along the y -axis. At 45 ◦ thepolarization is circular. a) Shift of the trapping frequencies fordifferent QWP settings. For small deviations from linearpolarization the trapping frequencies decrease due to a lowertrapping potential. At 30 ◦ from the starting linear polarization,the light is circularly polarized enough to drive f α, rot , at whichpoint f α,β vanish, and f z drops. At 85 ◦ from the starting linearpolarization, the motion becomes trapped again. b) Due to thishysteresis the driven rotational frequency f α, rot can be tuned overseveral hundred Kilohertz via the ellipticity of the trapping field.The markers indicate the mean value of f α, rot , and the shadedregion represents the range of measured frequencies. this frequency band, combined with the shape-enhancedsusceptibility of rods, will enable rotational cavity cool-ing to the quantum level [4, 34]. Such deeply trapped,cooled particles may be used as point sources for orien-tation dependent interference experiments [54, 55]. Funding Information
We are grateful for financial support by the AustrianScience Funds (FWF) in the project P27297 and DK-CoQuS (W1210-3). J.M. acknowledges funding fromthe European Unions Horizon 2020 research and innova-tion programme under the Marie Sk(cid:32)lodowska-Curie grantagreement No 654532. F.P. acknowledges the LegacyProgram (Israel Science Foundation) for its support.
Acknowledgments
We acknowledge support by S. Puchegger and the fac-ulty center for nanostructure research at the Universityof Vienna in imaging the nanorods. We thank Frank forhis dedication to the project.See Supplement 1 for supporting content. ∗ Electronic address: [email protected] † Co-first Author[1] B. Bhushan,
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DISTINGUISHING DIFFERENT TRANSLATIONAL AND ROTATIONAL MOTIONS
All trapping frequencies of the nanorods can be calculated, based on their geometry as measured through scanningelectron microscopy. The scattered light signal contains information about all degrees of freedom, with a high signal-to-noise ratio (SNR) [S1]. In the manuscript, all data for ( x, y, z, α ) is extracted from the scattered light detector D4,and the motion in β is extracted from a polarization dependent measurement on the scattered light D5.We independently check all motional degrees of freedom to confirm our assignation of frequencies, using otherdetectors. This information is not used in the manuscript. The full optical layout is shown in Fig. S1, and eachdetector and its uses are outlined in table I. Motion in z and α : We expect the trapping frequencies for z and α to be f z = 124 kHz and f α = 134 kHzrespectively. In the data we see a double-peak in the Power Spectral Density (PSD) of the scattered light signal D4around these frequencies, as shown in Fig. S2a). To confirm experimentally which peak is which we implement anadditional detection scheme. The nanorod rotates the polarization of the trapping light, depending on the angle itmakes to the polarization axis (i.e. by an amount proportional to α ).We monitor the trapping light that has interacted with the nanorod by collecting some of the light coupled backinto the fiber outcouplers, splitting it off with a 99:1 fiber beamsplitter and turning its linear polarization by 45 ◦ .This light then goes through a fiber polarizing beamsplitter (PBS), and each arm of the PBS is coupled onto a fastfiber-coupled balanced detector D1 (Thorlabs PDB420C). By this method we measure just the rotation in α , as shownin Fig. S2a). This confirms that the higher frequency peak is due to motion in α .We also monitor the intensity fluctuations of this picked-off light on detector D2, which yields information aboutthe motion in z , as described in Ref. [S2]. The quality of this signal is poor, due to the large beam waist of our trap,and could be improved through a difference measurement. It reconfirms that the peak at 124 kHz is due to motion in LASER POL. CONTROL AMP CIRCULATOR50:5050:50PBS 99:1
TRAP ← FREE SPACE ← F R EE S P A C E PBS
D1 D2D3 D4D5
PCPPCP PCP PCP
FIG. S1:
The full optical layout, including mechanisms for independently measuring different trapping frequencies. A small portion ofthe free-space light is split on a D-shaped mirror, yielding the radial motion x on detector D3. A portion of the light that is coupled backinto the fibers is picked off, and split equally to be sent to two detectors. D2 monitors the motion in z , as described in Ref. [S2], and D1the motion in α . We collect the light which the nanorod scatters using a multimode fiber, and monitor its intensity on detector D4, anduse a lens to collect the scattered light to perform a polarization sensitive measurement in the 45 ◦ basis on detector D5. Detector Model Motional sensitivityD1 Fiber coupled differencing photodetector, Thorlabs PDB420C α D2 Fiber coupled photodetector Bookham PP-10GC58L x, y, z
D3 Homebuilt differencing photodetector x or y D4 Multimode fiber coupled Hamamatsu photodiode G12180-005A withFemto HCA-100M-50K-C amplifier All degrees of freedomD5 Free space photodetecotr Thorlabs DET10C β TABLE I:
Detectors used to measure the motion of a levitated silicon nanorod. Only detectors D4 and D5 are used in the manuscript,D1-3 are used for confirming our assignation of motional degrees of freedom. z . Motion in x, y : We pick off a small amount of the light that has interacted with the nanorod using a free-space97:3 beamsplitter. This light is incident on a D-shaped mirror to cut the beam in half, and the resulting two beamsare measured on a differencing photodiode D3. This yields the motion in y , as described in Ref. [S2]. By performing ahalf-waveplate operation on the trapping light we can rotate the nanorod to measure x . Due to the large beam waistof our trap, the SNR is poor for this signal. Motion in β : Using a lens we collect the scattered light emitted in the opposite direction to that collected by themultimode fiber. This light is sent through a polarizing beamsplitter cube, which is rotated by 45 ◦ with respect to the x -axis, and monitored with a photodiode D5. This yields information about the motion in β , as shown in Fig. S2b). Further notes on detection
The SNR from the scattered light detector D4 is significantly better than that from detectors D1,2,3. Because ofthis, even though the z and α peaks overlap, fitting the PSD of D4 is the most accurate method for monitoring thedynamics of the nanorod. However, we can always use the other detectors to confirm our findings. The SNR of thescattered light signal is so good because we only collect light that is scattered by the nanorod, with virtually zero a) b)
50 100 150 200
Frequency (kHz) -10 -9 -8 PS D ( V H z - ) Scattered light detector , detection
50 100 150 200 250 300
Frequency (kHz) -14 -12 -10 -8 -6 PS D ( V H z - ) Scattered light detector - detection FIG. S2: a) A comparison of the PSD of the signal from the scattered light detector D4 (red), which is sensitive to motion in alldirections, and the polarization sensitive detector D1 (blue), which is only sensitive to α . This confirms that the higher frequency peakcorresponds to the α motion, which agrees with our calculations. b) A comparison of the PSD from the scattered light detectors D4 (red)and D5 (blue, polarization sensitive). D5 is sensitive to the frequency f β , which isn’t visible on detector D4. background.We expect the scattered light signal D4 to be only sensitive to the first harmonic of all motions 2 f x,y,z,α,β , since itdepends on position squared. However, Fig. S2a) shows that we measure the fundamental motions f z,α on D4. Thisis due to a slight misalignment between the trapping beams, and because the polarization in each arm of the trapis not perfectly identical. We confirm this through calculation, which also confirms that D4 is not sensitive to thefundamental frequency f β .The polarization (rather than intensity) sensitive detectors D1,5 are sensitive to the sign of the motion in α, β respectively, and so the PSDs in Fig. S2 show the frequency f α,β and not its harmonic. FITTING DATA
To extract trapping frequencies we fit the PSD of the scattered light time series with the functionPSD( ω ) = C d k B TM d Γ d ( ω d − ω ) + ω Γ d , (S1)where d labels the degrees of freedom (i.e. x, y, z, α, β ), Γ d is the (angular) momentum damping rate, ω d is thetrapping frequency, M d is the particle’s mass or moment of inertia and C d is the calibration between our measuredsignal and absolute motional information. The derivation of this PSD is standard, e.g. [S2, S3]. When recording thetime series of the particle’s motion, the PSD can usually be calibrated to extract C and convert from units of V Hz − to m Hz − . However, our signal contains both translational ( x, y, z ) and rotational ( α, β ) information, so such aglobal calibration isn’t possible. As an indication of sensitivity, the peak sensitivity for motion in the z direction fromthe PSD from detector D4 is 3 µ m / Hz − .Each degree of freedom can be fit with this expression. Figure S3 shows an example of fitting the PSD. Due to theproximity of the z and α peaks in the scattered light PSD, we fit the data with a sum of two of the functions definedin Eq. S1 to extract the parameters for both the z and α motions. TRAPPING FREQUENCY POWER SCALING
We expect all trapping frequencies to scale with the square root of power, see Eq. (1) of the manuscript. Figure S4shows the measured trapping frequencies as a function of power in comparison to the theoretical expectation. The
50 100 150 200 250 300 350 400
Frequency (kHz) -4 -2 PS D ( m V H z - / ) -3 -2 PS D f z f , z , - f x,y FIG. S3:
The PSD of the scattered light signal showing all motional degrees of freedom. Solid red lines are fits to the data using Eq. S1. f z f , f - f z,rot Power (W) F r e qu e n c y ( k H z ) FIG. S4:
Variation in the trapping frequencies with total trap power, markers are data, solid lines are the theoretical predictions. Thefrequency f z, rot is the trap frequency in the z direction when the nanorod is rotating in the α direction. Experimental uncertainties aresmaller than the data markers. only unknown experimental parameter is the laser waist w , which we extract from the ratio of f z to f x . The ratioof f z to f α confirms the length of our nanorod, which agrees with scanning electron microcope images. We observeexcellent agreement between theory and f α,z , and also for the axial frequency when the rod is rotating f z, rot . Thediscrepancy ( < f β is attributed to the fact that the rods have finite diameter( d = 130 nm) and, the generalized Rayleigh-Gans approximation [S4] is not strictly valid. REDUCTION OF TRAPPING POTENTIAL WHEN THE PARTICLE IS ROTATING
Due to the rotation of the rod in the plane orthogonal to the trap axis, only the averaged susceptibility ( χ (cid:107) + χ ⊥ ) / z -direction. This means that in Eq. (1) of the manuscript, χ (cid:107) has to be replaced by ( χ (cid:107) + χ ⊥ ) /
2. This agrees well with the measured reduction of the trapping frequency, seeFig. S4.
EXTRACTING INFORMATION FROM THE ROTATIONAL MOTION
When the trapping light is circularly polarized the nanorod rotates in the plane orthogonal to the trap axis ( α direction) with the frequency f α, rot . As shown in the paper, f α, rot has a broad distribution. To analyze this motionwe extract the instantaneous frequency, using time bins 100 times longer than the mean rotational period. In thepaper we present the mean value of f α, rot and display shaded regions around the data points representing the minimumand maximum values of f α, rot . The theoretical analysis predicts the maximum rotation frequency f α, max , which isdetermined by the balance between the torque exerted by the light field and the rotational friction due to collisionswith gas molecules. ∗ Electronic address: [email protected] † Co-first Author[S1] S. Kuhn, P. Asenbaum, A. Kosloff, M. Sclafani, B. A. Stickler, S. Nimmrichter, K. Hornberger, O. Cheshnovsky, F. Patolsky,and M. Arndt, “Cavity-Assisted Manipulation of Freely Rotating Silicon Nanorods in High Vacuum.” Nano Lett. , 5604–5608 (2015).[S2] Gieseler, J., Deutsch, B., Quidant, R. and Novotny, L., “Subkelvin Parametric Feedback Cooling of a Laser-TrappedNanoparticle.” Phys. Rev. Lett. , 103603 (2012).[S3] Millen, J., Deesuwan, T., Barker, P. F. and Anders, J., “Nanoscale temperature measurements using non-equilibriumBrownian dynamics of a levitated nanosphere.” Nature Nano. , 425 (2014).[S4] Stickler, B. A., Nimmrichter, S., Martinetz, L., Kuhn, S., Arndt, M. and Hornberger, K., “Rotranslational Cavity Coolingof Dielectric Rods and Disks.” Phys. Rev. A94