6 GHz hyperfast rotation of an optically levitated nanoparticle in vacuum
Yuanbin Jin, Jiangwei Yan, Shah Jee Rahman, Jie Li, Xudong Yu, Jing Zhang
66 GHz hyperfast rotation of an optically levitated nanosphere in vacuum
Yuanbin Jin , , Jiangwei Yan , , Shah Jee Rahman , , Jie Li , Xudong Yu , ∗ , Jing Zhang † The State Key Laboratory of Quantum Optics and Quantum Optics Devices,Institute of Opto-Electronics, Shanxi University, Taiyuan 030006, China Collaborative Innovation Center of Extreme Optics,Shanxi University, Taiyuan, Shanxi 030006, China Zhejiang Province Key Laboratory of Quantum Technology and Device,Department of Physics and State Key Laboratory of Modern OpticalInstrumentation, Zhejiang University, Hangzhou 310027, China
We report an experimental observation of a record-breaking ultra-high rotation frequency about6 GHz in an optically levitated nanosphere system. We optically trap a nanosphere in the gravitydirection with a high numerical aperture objective lens, which shows significant advantages in com-pensating the influences of the scattering force and the photophoretic force on the trap, especiallyat intermediate pressures (about 100 Pa). This allows us to trap a nanoparticle from atmosphericto low pressure (10 − Pa) without using feedback cooling. We measure a highest rotation frequencyabout 4.3 GHz of the trapped nanosphere without feedback cooling and a 6 GHz rotation withfeedback cooling, which is the fastest mechanical rotation ever reported to date. Our work providesuseful guides for efficiently observing hyperfast rotation in the optical levitation system, and mayfind various applications such as in ultrasensitive torque detection, probing vacuum friction, andtesting unconventional decoherence theories.
In recent years, levitated nanoparticles in vacuum haveattracted considerable interests and become an impor-tant platform for ultrasensitive force detection [1, 2],the study of macroscopic quantum phenomena [3–5],and nonequilibrium thermodynamics [6–9], among manyothers. Over the past decade, significant progress hasbeen made in the experimental realization of coolingthe motion of trapped nanoparticles [10–17] and themotional quantum ground state has been achieved [5].Such a system has also been employed for the funda-mental test of unconventional decoherence theories at themacro scale [18–24]. In Refs. [19–24] the relevant degreeof freedom of motion is the center-of-mass (CoM) mo-tion. Other degrees of freedom of motion of the levitatednanoparticle, such as the torsional vibration [25], the pre-cession motion [26], and rotation [27–32], provide alsorich physics to explore. Recent theoretical work [33, 34]show that the rotational degree of freedom may of-fer considerable advantages in testing the continuous-spontaneous-localization collapse theory. Furthermore,hyperfast rotation [30–32] has many important applica-tions, such as in testing material properties in extremeconditions [35] and detecting the quantum form of rota-tional friction [36]. Recently, a hyperfast rotation of fre-quency about 1 GHz (5.2 GHz) of a trapped nanosphere(nanodumbbell) has been reported [30, 32]. The rota-tion of a nanodumbbell is much faster than that of ananosphere in the same size because it receives a muchlarger optical torque under the same trap and air pres-sure.Stable optical levitation at low and high vacuum canbe achieved without feedback cooling of the micro andnano-particle’s motion. However, feedback cooling of theCoM motion is typically required to prevent particle lossfrom the trap at intermediate pressures (around 100 Pa), where photophoretic forces, sphere de-gassing, and othersources of noise not present in high vacuum may playsignificant roles [37]. In this work, we show that, byadopting a vertical-up layout of the trapping light, we canstably trap a nanosphere from an atmospheric pressureto high vacuum (10 − Pa) without using feedback cool-ing. Therefore, our work could enable feedback-free opti-cal trapping over almost full ranges of vacuum pressures.Consequently, we measure a fastest 4.3 GHz rotation ofthe trapped nanosphere without feedback cooling. Dueto the coupling between the rotation and the CoM mo-tion, at high rotation frequency the nanoparticle is easilylost from the trap in high vacuum. We thus apply thefeedback cooling to the CoM motion, which improves thestability of the trap and makes it possible to reach evenhigher vacuum, and consequently, we measure a highestrotation frequency about 6 GHz at 8 × − Pa. Thisis, to our knowledge, the highest rotation frequency everreported for a mechanical object.
Result
Trap nanoparticle from atmospheric to lowpressure without using feedback cooling.
The ex-perimental setup is depicted in Fig. 1. We opticallytrap a silica nanoparticle in vacuum using a 1064 nmlaser in gravity direction. The laser first passes throughan acousto-optic modulator (AOM) for shifting the fre-quency and controlling the power. The frequency-shiftedlaser is then coupled into a single-mode polarizationmaintaining fibre, of which the output beam passes suc-cessively through a quarter and a half-wave plate. Thevertically propagating 1064 nm laser is strongly focusedby a high numerical aperture (NA=0.95) objective lens a r X i v : . [ phy s i c s . op ti c s ] D ec Fiber detector Spectrum analyzer X&Y detection Z detection
Vacuum chamber
PBS1 PBS2 PBS3 BDDM PBS4λ/2 λ/4λ/2 λ/2 λ/2 λ/2
CoM motion detectionRotation detection
AOM
CoM motion feedback signals z xy
FIG. 1. (Color online). Schematic diagram of the experimen-tal setup, which includes four parts: vacuum system, rota-tion detection, CoM motion detection, and feedback system. λ/
4: quarter-wave plate; λ/
2: half-wave plate; BS: dichroicbeam splitter; PBS(1-4): polarized beam splitter; DM: D-shape mirror; BD: beam dump. in a vacuum chamber for trapping the particles. The po-larization of the light can be adjusted precisely by thecombination of the two wave plates. The power of thelaser before entering the chamber is 300 mW, and thetotal transmission of the chamber window and the objec-tive lens is about 52%, leading to an effective trappingpower about 156 mW in our experiment. The diameter ofthe trapping laser is 3.2 mm before entering the objectivelens and about 1.1 µ m at the focus point. The intensitydistribution in the x – y plane ( z in the axial direction)at the focus region is slightly asymmetric because of thevector diffraction of the light. The trapping light afterthe focus point is collimated by another high numericalaperture lens (NA=0.68) and then divided into two partsby a polarized beam splitter (PBS). One part is used tomeasure the nanosphere’s rotation signal using a detectorwith a flat gain of about 10 V/A in a broad band rangeof DC-12 GHz, and the power input into the detector is 1mW. The other part is used to measure the CoM motionin three directions. The rotation signal is analyzed by aspectrum analyzer.A small dielectric particle in a strongly focused lightbeam feels a three-dimensional gradient force. In this sit-uation, two relevant effects must be considered. First, fora single trapping beam configuration, the axial trappingforce is crucial because the axial gradient force is smallcompared to the radial direction. Besides, in the axial di- rection, the particle also feels a scattering force from thelight, which tends to push the particle out of the trap.Consequently, the equilibrium position of the particle ismoved away from the focus point along the propagat-ing direction of the trapping light, which decreases thewell depth in this direction. Second, in high vacuum thethermal transfer between the particle and the backgroundgases is restrained. Therefore, the particle is heated toa high and uniform internal temperature. In parallel,in low vacuum the particle has a low and also uniforminternal temperature due to a quick heat exchange be-tween the nanoparticle and the air molecules. However,there are internal temperature gradients induced by thetrapping laser at intermediate pressures, leading to a non-uniform distribution of temperature on the nanoparticle’ssurface. When air molecules hit the nanoparticle, thoserebounding from the warmer side will have higher en-ergy than those rebounding from the cooler side. Thisimparts a net force (i.e., the photophoretic force) on theparticle, which is in the vertical-up direction in our sys-tem. This force can easily kick the particle out of thetrap, especially in medium vacuum. To restrain thesedetrimental effects, we implement a vertical-up layoutfor the trapping light, which can compensate the influ-ences of the scattering and photophoretic forces using itsown gravity of the particle. As a result, we can stablytrap a nanosphere from an atmospheric pressure to highvacuum without using feedback cooling. This results inabout 50% success probability of trapping a nanoparti-cle below an intermediate pressure 100 Pa to lower pres-sures. Furthermore, we can monitor the intensity of thescattering light from the trapping laser by imaging thenanoparticle via CCD. By further selecting the nanopar-ticles at atmospheric pressure with an intermediate scat-tering intensity, we can increase the success probabilityto more than 90% below an intermediate pressure. Thosenanoparticles with much higher or lower intensity of thescattering light cannot reach high vacuum in our experi-ment.The CoM motion of the particle in a strongly focusedlaser generally has three eigen frequencies in three direc-tions due to the vector diffraction of the light [38]. Inour experiment, the eigen frequency in x , y , z directionsare about 210 kHz, 220 kHz, and 90 kHz, respectively.The damping rate γ of the CoM motion is proportionalto the product of the radius of the nanosphere R and theair pressure p in a certain air pressure regime, accordingto the kinetic theory γ = αRp + β , where α is a constantand β is a high-order term of Rp . Hence, by measuringthe damping rates at different pressures, the radius of thenanosphere can be inferred. It is about 95 ± Rotation without using feedback cooling.
Theangular momentum of the trapping light can be trans-ferred to the nanoparticle due to the absorption, bire-fringence, and asymmetric shape of the particle [30]. Thetransferred angular momentum provides a torque, whichdrives the particle to rotate. We denote the total driv-ing torque the particle receives as M o . Meanwhile, theinteraction with the gas molecules in the vacuum cham-ber damps the rotation of the particle, which causes adrag torque M d . Under the driving and drag torques,the rotational motion equation of the particle is [30]:2 πI df r dt = M o + M d , (1)where I = 0 . mR is the moment of inertia of thenanoparticle and m is its mass. The drag torque M d is proportional to the frequency of the rotation undera certain air pressure, M d = − πIf r γ d [39], where γ d = pR / ( ηmv ) is the damping rate of the rotation mo-tion, with v the mean molecular velocity, and η the ac-commodation factor accounting for the efficiency of theangular momentum transferred to the particle via colli-sions with gas molecules. According to this equation, inthe beginning as the rotation gets faster under the driv-ing torque, the drag torque increases accordingly. Even-tually, the rotation speed increases to a certain pointand remains constant at a certain air pressure as a resultof the balance between the driving torque and the dragtorque. The rotation frequency in the steady state canbe solved, which is f r = πγ d M o I . In order to measure therotation frequency, the light after trapping the nanopar-ticle is split by a PBS and detected by a fast detector.Intuitively, a nanoparticle acts as a half-wave plate, andthe rotated nanoparticle is like a polarization modulator.One period (2 π ) of the rotation of the nanoparticle willgenerate a 4 π modulation in the polarization of the trap-ping light. Thus, a frequency shift arises for the photonsafter interacting with the particle and the shift amountis 2 f r , with f r the rotation frequency of the nanoparticle.Consequently, we obtain the 2 f r signal in the spectrumanalyzer.Considering the circularly polarized trapping laser, thetotal driving torque is proportional to the light intensity: M o ∝ I e ( I e is the intensity of the trapping light at theequilibrium point of the particle). Hence, the rotationfrequency shows a linear dependence upon the trappinglaser power. Moreover, the rotation direction can be al-tered by changing the chirality of the light. For the el-liptical polarization, the light can be decomposed into acircular and a linear polarization component. The bire-fringence and asymmetric shape of the particle aligns theparticle along the linear polarization, while the circularpolarization component drives the particle to rotate [40].Therefore, the weights of these two components deter-mine the motion of the particle: If the effect of the circu-lar polarization component is stronger than that of thelinear polarization, the particle starts to rotate; if the op-posite, the rotation would not occur. Here, the ellipticityof polarization is controlled by adjusting the angle of thefast axis of the quarter-wave plate. Rotation motion dis- appears at the angle ranging from − ◦ to 18 ◦ as shownin Fig. 2. As we change the chirality of the polarization,the rotation direction of the nanoparticle is changed.In order to observe the rotation of the nanoparticle, wefirst trap the nanoparticle below an intermediate pressure100 Pa to lower pressures with success probability morethan 90%, and we then can observe the rotation in highvacuum with probability about 90%. In Fig. 3, we mea-sure the rotation frequency of three trapped nanospheresversus the air pressure for a fixed laser power 300 mWwithout feedback cooling. We use two vacuum gauges, aresistance gauge with measurement range from 5 × − Pa to 10 Pa, and a hot cathode ionization gauge withmeasurement range from 10 − Pa to 0.2 Pa. This resultsin a slight mismatch between the two traces measured bythe two vacuum gauges for the same nanoparticle at pres-sure around 0.2 Pa. We observe a beat signal of about8.6 GHz, corresponding to a rotation frequency about4.3 GHz, at 0.01 Pa, as shown in the top-right inset ofFig. 3. In the bottom-left inset, we show the fluctuationof the rotation frequency for one of the nanospheres. Thefrequency uncertainty becomes larger as the pressure re-duces.
Rotation with feedback cooling.
At low pressure,the rotation of the nanoparticle is very fast, which re-sults in the coupling between the rotation and the CoMmotion [27]. This coupling can cause instability of thetrap. In order to reduce this deleterious coupling, weimplement feedback controls to cool the CoM motion ofthe nanoparticle in three directions (see Fig. 1). The dis-placement signals in three directions are sent into broad-bandwidth lock-in amplifiers for generating the corre-sponding double-frequency signals, which are then inputinto a function generator of AOM for modulating thepower of the trapping laser and cooling the CoM motions[41]. This parametric feedback cooling results in signif- - 4 5 - 3 0 - 1 5 1 5 3 0 4 51 0 - 2 - 1 Rotation Frequency (GHz)
A n g l e o f (cid:1) / 4 ( D e g )
FIG. 2. (Color online). Measured rotation frequency versusthe angle of the fast axis of the quarter-wave plate at differentpressures: 5 Pa, 0.5 Pa and 0.1 Pa (from bottom to top). R o t a t i o n F re qu e n c y ( GH z ) Pressure (Pa)
Frequency (GHz) P S D ( a . u . ) s ( GH z ) Frequency (GHz) -2 -1 -4 -3 -2 -1 R o t a t i o n F re qu e n c y ( GH z ) Pressure (Pa) -2 -5 -3 -1 Pressure (Pa)
FIG. 3. (Color online). Measured rotation frequency ofthree trapped nanospheres (red, blue, green traces) ver-sus air pressure without feedback cooling. Top-right inset:Power spectrum density of a rotation signal of 8.6 GHz at0.01 Pa. Bottom-left inset: Standard deviation of the ro-tation frequency versus air pressure measured for one of thenanospheres. At each pressure, we perform 120 measurementsto obtain the standard deviation. icantly improved stability of the trap. Figure 4(a) and(b) illustrate the fluctuation of the rotation frequencybefore and after the feedback cooling at 0.16 Pa, respec-tively. Figure 4(c) shows the rotation frequencies versusthe air pressure with feedback cooling for three differentnanospheres. The highest rotation frequency observedis about 6 GHz at 8 × − Pa and the correspondingbeat signal is 12.17 GHz, as shown in the top-right in-set of Fig. 4(c). At the rotation frequency higher than 4GHz, the linear dependence of γ d on the pressure is nolonger valid, which results in steeper slopes of the traces.Therefore, we must carefully control the evacuating speedof the vacuum pump to obtain the highest rotation fre-quency in this region. The bottom-left inset shows thefluctuation of the rotation frequency measured for one ofthe nanospheres with feedback cooling. The fluctuationof the rotation frequency is significantly reduced by thefeedback cooling, comparing with the inset of Fig. 3. Discussion
In conclusion, we have adopted a vertical-up layout ofthe trapping light in an optical levitation system, whichallows us to trap a nanosphere from an atmosphericpressure to high vacuum without using feedback cool-ing. Once the nanosphere is trapped in high vacuum,by further including feedback cooling, we have measureda record high rotation frequency about 6 GHz. In ourexperiment, the rotation is hyperfast and close to theregime where the internal forces generated were strongenough to break up the material. Our work thus provides
Time (s) R o t a t i o n F re qu e n c y ( M H z ) R o t a t i o n F re qu e n c y ( GH z ) Pressure (Pa)
Frequency (GHz) P S D ( a . u . ) (c) (a) (b) s ( GH z ) -2 -1 -3 -2 -1 R o t a t i o n F re qu e n c y ( GH z ) Pressure (Pa) -2 -5 -3 Pressure (Pa)
FIG. 4. (Color online). The fluctuation of the rotation fre-quency (a) without feedback cooling and (b) with feedbackcooling at 0.16 Pa. We sample the rotation frequency 30times per second. (c) Measured rotation frequency of threetrapped nanospheres versus air pressure with feedback cool-ing. Top-right inset: Power spectrum density of a rotationsignal of 12.17 GHz at 8 × − Pa. Bottom-left inset: Stan-dard deviation of the rotation frequency versus air pressuremeasured for one of the nanospheres. an important platform for studying vacuum friction andthe material properties under extreme conditions. Thesystem can also be used for ultrasensitive torque detec-tion [32] and micron-scale pressure gauges [42]. Further-more, our work sheds light on the test of the continuous-spontaneous-localization collapse theory by using the ro-tational degrees of freedom [33, 34].
Methods
Experimental setup.
A 1064 nm laser beam ofTEM00 Gaussian mode, emitted from a diode-pumpedsingle-frequency laser, passes through an acousto-opticmodulator (AOM) for controlling its power used for cool-ing the CoM motion of nanoparticle. The 1064 nm laserbeam is strongly focused in the anti-gravity direction bya high NA objective lens (Nikon CF IC EPI Plan 100X,NA=0.95), of which the working distance (WD) is 0.3mm. The strongly focused beam is then collimated byanother high NA aspheric lens (Thorlabs C330TMD-C,NA=0.68) with WD of 1.8 mm. These two lenses areplaced in the vacuum chamber.The nanosphere’s rotation signal is measured by a fastdetector (New focus 1554-A) with a flat gain of about 10 V/A in a broad band range of DC-12 GHz. The CoMmotion of the nanosphere in the x and y directions canbe detected by using a D-shape reflective mirror, whichsplits the laser beam into two equal parts in space. Thenthe two parts are focused respectively by two short focuslenses ( f = 30 mm) and detected by a pair of photodiodesin a current-subtraction detector. When the nanosphereslightly leaves its equilibrium position in the radial direc-tion, the light intensities detected by the two photodiodesare slightly different and the intensity difference is pro-portional to the displacement of the nanosphere. In orderto detect the motion in the z direction, the beam is sepa-rated by a beam splitter into two parts with imbalancedintensity (1:2). One part is completely detected by thephotodiode, while the other part is partially detected,but they are balanced in a current-subtraction detector.A slight change of the nanosphere’s position in the axialdirection leads to a slight move of the beam focus in the z detection, and thus a slight change of the light intensityon the photodiode. Through this way, the nanosphere’sposition in the axial direction can be measured. Thecurrent-subtraction detectors have a high common moderejection ratio (CMRR) and the measured value is largerthan 60 dB @1 MHz. The conversion gain of the current-voltage is 10 V /A .To load the nanoparticle, we tried silica nanoparticlesproduced by different manufacturers, and finally selectedthe non-functionalized silica nanosphere (Bangs Labora-tories, Inc.), which gave us the best result. Its nomi-nal diameter is about 170 nm with specification rangeof 20%. The hydro-soluble silica nanospheres are firstdiluted in the high-purity ethanol with concentration ofabout 1 . × /ml and are then sonicated for 30 min-utes. The dilution solution is poured into an ultrasonicnebulizer (OMRON NE-U22). The droplets containingthe nanospheres are dispersed by the ultrasonic nebu-lizer and guided through a thin tube near the focus ofthe objective lens in the vacuum chamber. Once a par-ticle is trapped in the focused beam, the vacuum pumpthen starts to evacuate the chamber.For parametric feedback cooling of the nanoparticle’sCoM motion, we first use the phase-locked loop tech-nology to map the CoM motion signals in the three di-rections to three sine signals via two lock-in amplifiers(Zurich Instruments HF2LI 50 MHz). The lock-in am-plifiers generate the corresponding double-frequency sig-nals, of which the amplitude and phase can be easily con-trolled. Finally, we send these signals with appropriateamplitude and phase into the driver of AOM to modulatethe trapping laser power. In this way, the nanoparticle’sCoM motion in the three directions can be cooled. Corresponding authors. ∗ jiance [email protected]; † [email protected], [email protected]. DATA AVAILABILITY
All data generated or analysed during this study areincluded in this published article. Additional data arealso available from the corresponding authors upon rea-sonable request.This research is supported by the MOST (GrantNo. 2016YFA0301602), NSFC (Grant No. 11234008,11474188, 11704234) and the support from theXPLORER PRIZE.
AUTHOR CONTRIBUTIONS
J.Z. designed research. J.Z. and X.Y. supervised re-search. Y.J., J.Y., S.R., X.Y., and J.Z. performed theexperiments. J.Z., X.Y., and J.L. wrote the manuscript.All authors interpreted the results and reviewed themanuscript.
COMPETING INTERESTS
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