Performance and limits of feedback cooling methods for levitated oscillators: a direct comparison
PPerformance and limits of feedback cooling methods for levitated oscillators: a directcomparison
T. W. Penny, A. Pontin, and P. F. Barker ∗ Department of Physics and Astronomy, University College London,Gower St, London WC1E 6BT, United Kingdom (Dated: February 19, 2021)Cooling the centre-of-mass motion is an important tool for levitated optomechanical systems,but it is often not clear which method can practically reach lower temperatures for a particularexperiment. We directly compare the parametric and velocity feedback damping methods, whichare used extensively for cooling the motion of single trapped particles in a range of traps. Byperforming experiments on the same particle, and with the same detection system, we demonstratethat velocity damping cools the oscillator to lower temperatures and is more resilient to imperfectexperimental conditions. We show that these results are consistent with analytical limits as well asnumerical simulations that include experimental noise.
I. INTRODUCTION
Levitated nanoparticles in high vacuum are thermallyand mechanically well isolated from the environment.They can be cooled from room temperature to low occu-pancy, or even to the groundstate, via cavity or feedbackcooling [1–3]. Levitated oscillators are increasingly seenas ideal candidates for tests of fundamental quantum me-chanics with proposed experiments to create large masssuperpositions via matter-wave interferometry [4–6] or tostudy mechanisms of wavefunction collapse [7, 8]. Lev-itated nanoparticles have also been used for weak forcemeasurements [9] with direct measurements of photonrecoil [10], and even for testing the standard model [11].Controlling the motion of levitated nanoparticles hasseen much interest in the last decade, particularly in cool-ing the centre-of-mass (CoM) temperature towards theground state [12, 13]. This is seen as a milestone in gain-ing full quantum control of macroscopic objects and is aninitial step in matter-wave interferometry protocols [14].Cooling the motion of a trapped nanoparticle to evenfairly modest temperatures can be useful for preventingparticle loss in high vacuum [9, 15].Ground state cooling has been achieved using coher-ent scattering from an optical tweezer into a cavity modeto remove energy from the particle motion [1]. Previ-ous attempts to cool trapped nanoparticles have includedother forms of cavity cooling utilising direct trappingin the cavity [16] and hybrid traps [17]. An alterna-tive method is feedback cooling based on measurementsof the trapped particle motion. Modulating the trap-ping potential at twice the particle frequency (parametricfeedback) or applying a linear force proportional to theparticle’s velocity (velocity damping) are two techniquesthat have been used extensively. Average phonon occu-pancies of 62 . .
56 phonons [2] havebeen achieved respectively in optical tweezer set-ups us-ing feedback cooling. ∗ [email protected] As both techniques are commonly used, it is natu-ral to ask which is likely to achieve lower temperaturesfrom both a theoretical and experimental perspective.In this paper we directly compare parametric feedbackcooling and velocity damping for a particle confined ina Paul trap. Parametric feedback is implemented bytracking the instantaneous phase of a trapped particle us-ing a phase-locked loop (PLL) and modulating the trap-ping potential with a frequency-doubled signal phase-locked to the particle motion with an appropriate phaseshift. This implementation is commonly used in opticaltraps [18, 19]. Although parametric feedback has beenimplemented in Paul traps before [20, 21], this is the firsttime it has been realised using a PLL. The feedback signalfor velocity damping is generated by estimating the veloc-ity of the particle from a position measurement. We havetaken common concepts from PLL theory and appliedthese to the optomechanical system to set bounds on theminimum temperatures achievable with parametric cool-ing using a PLL. Both cases of cooling are simulated andexperimentally demonstrated on the same particle underidentical experimental conditions. We consider the mini-mum achievable temperature of each method and discussthe implications of experimental imprecision. Finally, weexamine the energy distributions of the cooled oscillator.
II. FEEDBACK COOLING SCHEMES
A levitated nanoparticle can be considered a thermaloscillator in a 3D harmonic potential. The motion ineach direction x i , where i = { x, y, z } , obeys an equationof motion which is given by:¨ x i + γ ˙ x i + ω i x i = F th,i m , (1)where γ is the gas damping, ω i is the frequency of oscil-lation, m is the mass of the trapped particle and F th,i is arandom Langevin force that satisfies (cid:104) F th,i ( t ) F th,j ( t (cid:48) ) (cid:105) =2 mγ k B T δ ( t − t (cid:48) ) δ i,j where k B is the Boltzmann con-stant and T is the temperature of the surrounding ther- a r X i v : . [ qu a n t - ph ] F e b mal bath. The CoM temperature of the particle canbe estimated using the variance of the motion, T CoM = mω i (cid:104) x i (cid:105) /k B . With no additional forces being appliedto the particle it is equal to the temperature of the sur-rounding thermal bath i.e. T CoM = 293 K.Without loss of generality the equations of motion canbe considered in 1D with similar equations applying toall directions. The effects of the interactions betweenmodes are considered later. Velocity damping cools anoscillator by applying a force proportional to the velocityto increase the damping. However, any noise in the de-tection will also be fed back to the oscillator. Eq. 1 canbe modified to include these effects such that [22]:¨ x + γ ˙ x + ω x = F th m − γ fb ( ˙ x + δ ˙ x ) , (2)where γ fb is the damping due to feedback, ω = ω x , and δ ˙ x is a stochastic additive noise in the feedback signal.Parametric feedback cools a trapped particle by mod-ulating the trapping potential at twice the frequency ofthe particle motion. The modulation can be created bymultiplying the current position and velocity together, x ( t ) ˙ x ( t ) [23]. However, this scheme is often implementedusing a digital PLL to lock a numerically controlled oscil-lator (NCO) to the phase of the particle. The frequencydoubled output from the NCO can then be used as thefeedback signal (after an appropriate phase shift) [18].Although both are considered parametric feedback thesetwo implementations produce different particle dynam-ics because the feedback signal from a PLL has a con-stant amplitude whereas the amplitude of the feedbackfrom multiplying position and velocity is dependent onparticle energy [24, 25]. The equation of motion of theoscillator under parametric feedback with a PLL is:¨ x + γ ˙ x + (1 − G sin(2( ω t + θ o ))) ω x = F th m , (3)where G is the modulation depth and θ is a time de-pendent phase set by the PLL. Fig. 1 shows a schematicof a general digital PLL with a breakdown of the phasedetector to show how it is implemented in the simulationand experiment. Signals from the NCO and the oscilla-tor are fed into a phase detector which produces an out-put proportional to the phase difference between the twosignals with a constant of proportionality, K d . A loopcontroller with transfer function F ( s ), where s is com-plex frequency, is applied to the output then multipliedby a gain constant, K o , to produce the control signal forthe NCO. By modifying the controller transfer functionand gain constant, the PLL can be made to accuratelytrack the phase by minimising the phase detector output.Despite being digitally implemented we will consider alltransfer functions in their analogue equivalent form forthe purposes of analysis. This is valid provided any fea-tures in the transfer function are well below the Nyquistfrequency of the digital system as they are here. III. SIMULATION
The simulations are implemented using the energy con-serving (symplectic) leapfrog method [26]. Eq. 1, with anadditional feedback force, is rewritten as a system of twofirst order equations: ˙ x = v (4)˙ v = − γ v − ω x + 1 m ( F th + F fb ) (5)and each variable is progressed one half-timestep out ofsync: x n +1 = x n + v n + ∆ t (6) v n + = v n − + ˙ v n ∆ t (7)where ∆ t is the timestep size in the simulation. Thermalforce noise and measurement noise are simulated using astring of random numbers with a Gaussian distributionand variance equal to (2 k B T γ ) / ( m ∆ t ) and S nn / ∆ t re-spectively where S nn is the detection noise spectral den-sity which is assumed to be uncorrelated and white. Thegas damping is pressure dependent obeying the equation γ = (1 + π ) π MNR v T m where M is the gas molecularmass, N is the pressure dependent gas particle density, R is the sphere radius and v T = (cid:113) k B T πM is the averagethermal velocity of the air molecules [27].Two methods to generate the feedback signal for ve-locity damping were considered. The first method uses aWiener filter [28, 29]. Wiener filtering allows us to extractthe best estimation of a variable from a noisy measure-ment of a related variable provided we have knowledgeof the transfer function of the system. More explicitlyif we measure position, x , with additional noise in or-der to estimate velocity, v , where x and v are relatedvia S xx ( ω ) = ω S vv ( ω ) then the optimum filter is givenby [28]: W ( ω ) = ω S nn ( ω ) S xx ( ω ) . (8)A caveat is that Wiener filtering only works for sta-tionary processes, which is not strictly true when coolinga harmonic oscillator, since the transfer function is al-tered by the cooling process. However, the process isonly non-stationary during the transient period of ini-tial cooling therefore the transfer function of the steadystate can be calculated using the applied feedback gainand used when computing the Wiener filter. By applyingthe Wiener filter to a time-dependent position measure-ment the velocity can be estimated. Alternatively, the FIG. 1. A basic digital PLL loop consists of a NCO that tracks the oscillator phase through an input that alters the frequency.A phase detector takes both the oscillator and NCO signals as inputs and produces an output proportional to the phasedifference. In the simulation and experiment the phase detector mixes the signals to produce X and Y quadratures of theoscillator signal. Low-pass filters are used to remove noise and the 2 ω component. The X and Y quadratures can then beused to calculate the phase difference between the NCO and oscillator. The digitally implemented loop controller, with an anequivalent analogue transfer function F ( s ), is adjusted such that the NCO tracks the phase of the oscillator. The NCO acts asan integrator and must be considered when calculating the closed-loop transfer function. velocity can be predicted by delaying the measured po-sition signal by π ω seconds. This method is valid underthe high-Q approximation, where ω ≈ ω over the widthof the transfer function. The estimated velocity from ei-ther method can then be multiplied by a gain and usedas the feedback signal.For parametric feedback, a digital PLL was imple-mented in the simulation. A sinusoidal function withdirect access to the phase is used as the NCO. The phaseof the oscillator at each timestep is calculated by de-modulating the current oscillator position (with addedmeasurement noise) with the signal from the NCO toobtain the X and Y quadratures. Filtering the quadra-tures using second-order exponential smoothing, with abandwidth B quad , removes the 2 ω component of the sig-nal allowing the phase difference to be calculated using δθ = − arctan ( Y /X ). We use K d = K o = 1 in the simu-lation. To generate a control signal in the simulation weuse a loop controller with a transfer function of: F ( s ) = − ( τ τ + 1 τ s ) (9)where τ and τ are the two time-constants of the con-troller. This form of the transfer function demonstratesits equivalence to a PI loop. This is one of the mostwidely employed loop controllers in PLLs and providesa balance between narrow bandwidth and loop stabil-ity [30]. Implementing a digital filter of the form G ( s ) = − F ( s ) /s gives the open-loop transfer function. Usingthis we can directly calculate a new phase, θ o , for ourNCO from the phase detector output. It is useful to notethat the closed-loop transfer function becomes [30]: θ o θ i = H ( s ) = 2 ω n ζs + ω n s + 2 ω n ζs + ω n (10)where ω n = (cid:113) K o K d τ and ζ = τ (cid:113) K o K d τ are the naturalfrequency and the damping factor of the PLL. The PLLbandwidth can be defined using the 3dB cut-off of theclosed-loop transfer function [30]: B dB = ω n [2 ζ + 1 + (cid:112) (2 ζ + 1) + 1] . (11)This determines the rate of change of the oscillatorphase that can be tracked by the PLL. IV. THEORETICAL ANALYSIS
Eq. 2 can be solved to find the variance of the oscilla-tor. Making the high-Q approximation, which is valid atlow pressures, the CoM temperature of the oscillator iscalculated as [22, 31]: T CoM = T γ γ + γ fb + 12 mω k B γ fb γ + γ fb S nn (12)where the second term gives the contribution from thedetection noise. Physically this results from noise inthe measurement being fed into the motion of the par-ticle causing heating. This also leads to a phenomenonknown as noise squashing where correlations between de-tection noise and particle motion make the power spectraldensity (PSD) of the particle motion from the detectorbeing used to generate the feedback signal (in-loop de-tector) appear as if it is being cooled below the noisefloor [22, 32]. In the limit γ fb (cid:29) γ the optimum feed-back gain provides an effective damping given by: γ fb = (cid:115) γ k B T S nn mω (13)with a minimum temperature of: T CoM = (cid:115) S nn mω γ T k B . (14)In contrast to velocity damping, a theoretical analy-sis of the PLL is extremely difficult when adding noiseinto the closed-loop due to the non-linear nature of thePLL. A limited analysis can be done using a simplifiedloop controller with only proportional control, F ( s ) = P where P is a constant, and the assumption of small phaseerror, sin( δθ ) = δθ [24, 33]. In this regime the tempera-ture of the oscillator is still decoupled from variations inphase error and we cannot predict the effect of detectionnoise on the oscillator temperature. However, it can beshown that the bandwidth of the PLL limits the modu-lation depth that can be applied whilst still maintainingphase tracking according to: G lim = 2 B dB ω . (15)In Fig. 2a) we show the temperature of a simulated os-cillator being cooled parametrically for a range of ζ and ω n values at a fixed modulation depth. To reflect the ex-perimental conditions in our set-up, a particle radius of R = 193 . ρ = 1850 kg m − is used. Theoscillator frequency is set to be ω = 2 π ×
277 Hz witha pressure of P = 2 . × − mbar giving an intrinsiclinewidth of 780 µ Hz. The detection noise spectral den-sity is S nn = 1 . × − m Hz − . The quadrature filterbandwidth, B quad , is fixed at 2 π ×
400 Hz for all parame-ter values so that it is much larger than any B dB valuesused and does not interfere with the PLL loop controller.Provided ζ is large enough there is an optimum valueof B dB = 2 π ×
104 Hz that is unaffected by individual ζ and ω n values. The large ζ limit is the equivalent oflarge DC loop gain being required for good tracking [30].From now on we can just consider the PLL to containthree parameters B quad , B dB and G . The quadraturefilter bandwidth must be large enough so that it doesnot interfere with the loop controller but it must be suf-ficiently small to eliminate the 2 ω component in the de-modulated signal. We find that keeping B quad = 5 B dB is sufficient. For larger bandwidths this makes it impos-sible to completely remove the 2 ω component from thedemodulated signal, however, the PLL still tracks and cools the oscillator. This leaves only two independentparameters to adjust, G and B dB .We show in Fig. 2b) the temperature of a cooled os-cillator as the modulation depth is adjusted for severaldifferent PLL bandwidths. It can be seen that for eachbandwidth there is an optimum gain that increases as thebandwidth is increased as predicted by Eq. 15. Heuris-tically, this results from the linewidth of the oscillatorincreasing as it is cooled. Once the linewidth is largerthan the PLL bandwidth the particle phase can no longerbe tracker consistently so the phase error increases andthe particle is cooled less effectively. This is confirmedin Fig. 2c) which shows the linewidth of a cooled oscil-lator at optimum gain for several PLL bandwidths. Astraight line fit to the first four points gives a gradientof 0.6 suggesting the PLL struggles to track the oscil-lator even when the linewidth is less than B dB due tothe more complex loop controller and large phase error.Similar trends to those in Fig. 2 are seen for alternativepressures, oscillator frequencies and particle masses. Us-ing Eq. 15 we can calculate an achievable temperature atany particular bandwidth, this is given by: T lim = T γ B dB = T γ G lim ω . (16)These simulations show the modulation depth is not lim-ited to 1 .
5% as previously reported for optical traps [34].We found the modulation depths were also not limited bythis value in the experiment where modulation depths ofup to 5% were used. Fig. 2b) additionally shows thebandwidth cannot be indefinitely increased without in-curring a penalty on the effectiveness of the PLL. Fora sinusoidal input into the PLL with amplitude V s , theloop SNR can be defined as SN R L = V s / B L S nn [30] where B L is the noise bandwidth of the loop (in hertz). Forthe loop controller used in this numerical simulation B L = ω n ( ζ + ζ ). If the SNR drops below ∼ SN R L = (cid:104) x (cid:105) B L S nn . We can then define thelower bound the temperature of the oscillator can reachbefore the PLL unlocks as: T lim = mω k B B L S nn . (17)These two limits allow us to bound the smallest achiev-able temperature of the oscillator during parametric feed-back cooling. Note that unlike Eq. 12 they are not acomplete analytical expression for the temperature butbounds on what can be achieved since they do not in-clude the effect of phase noise on the temperature, i.e.,the model does not include the backaction of the feed-back scheme. Furthermore, in the derivation of Eq. 17 FIG. 2. a) Heatmap showing the CoM temperature for para-metric cooling with different ω n and ζ parameters. The redline shows a constant bandwidth of 104 Hz along which thetemperature is at a minimum. The inset shows the tem-perature variation along the red line. b) Temperature of aparametrically cooled oscillator against modulation depth forseveral different PLL bandwidths. c) The linewidth of thecooled oscillator at the optimum gain against the bandwidth.The orange line shows a straight line fit to the first four datapoints with a gradient of 0.6. we have exchanged a constant amplitude signal for a sig-nal with a varying amplitude and considered only theaverage. In reality, the PLL often tracks the signal on a much shorter timescale than the evolution of the oscilla-tor amplitude. If at any point during the measurementthe instantaneous loop SNR drops below 1 the PLL willunlock and the oscillator temperature will increase. Thismeans that in practice the oscillator will never reach thetemperature given by Eq. 17, however, it can never besignificantly lower than this. We can use these bounds topredict a bandwidth at which the minimum temperaturewill occur. Using the relation B dB ≈ πB L (valid in thelimit ζ (cid:29)
1) we find the optimum bandwidth for coolingis: B dB = (cid:115) πγ k B T S nn mω . (18)Using Eq. 16 the minimum achievable temperature is: T CoM = (cid:115) S nn mω γ T πk B (19)which is lower than the minimum temperature that canbe achieved with velocity damping. This is because themodel for velocity damping includes backaction from thenoise in the feedback whereas the model for the paramet-ric feedback does not. Simulations must be used to fullyinclude the effects of noise from the PLL on the particlemotion as shown below. V. EXPERIMENT
Paul traps utilise an alternating electric field to trapcharged particles since Gauss’ Law forbids a minimumfor three-dimensional static electric fields in free space.For a linear Paul trap the potential is [35]:Φ( x, y, z, t ) = U κz ( − x + y z )+ V cos ( ω rf t )( η x − y r + 1) (20)where U is the DC voltage applied to the endcap elec-trodes, V is the AC voltage applied to the rod electrodesat angular frequency ω rf , and the parabolic coefficients r , z , κ and η are determined by the geometry of thetrap. In the case of no damping the particle motion inone-dimension can be approximated to be [36]: x i ( t ) ≈ AC cos ( ω i t )(1 − q i cos ( ω rf t )) (21)where A is determined by the initial conditions of theparticle, C is a function of particle and trap parame-ters, ω i ≈ ω rf (cid:113) a i + q i is the ’secular frequency’ and a i and q i are known as the stability parameters of the trap. FIG. 3. a) Simplified experimental set-up. A focused 1030 nm laser illuminates the particle. The scattered light can becaptured and focused onto a CMOS camera to track the motion. The forward scattered and unscattered light is also collectedand focused onto balanced photodiodes to generate the signal used in the feedback electronics. b) PSD of the particle used inthis experiment with ω z = 2 π ×
223 Hz taken with CMOS camera at 1 . × − mbar with fit (orange line). The variance ofthe PSD gives a particle mass of 5 . ± . × − kg. c) Spectrum measured on the balanced detection at 2 . × − mbarshowing all three modes of motion during cooling. The modes have frequencies ω x = 2 π ×
482 Hz, ω y = 2 π ×
450 Hz, and ω z = 2 π ×
229 Hz. The spectrum is left uncalibrated since each mode requires a separate calibration.
The stability parameters are given by q x = q y = qV ηmω rf r , q z = 0 and a x = a y = − . a z = qU κmω rf z . For this approx-imation to hold the conditions | a i | , q i (cid:28) ω i and a smaller, driven ’micromotion’ at higherfrequencies, ω rf ± ω i [36].The Paul Trap used in this experiment consisted of fourparallel rods held by printed circuit board (PCB) similarto the trap in reference [37]. The PCB allowed for easyelectrical connections to the rods and had two ring elec-trodes etched into the surface as endcaps to confine theparticle along the trap axis. The PCB was gold coated tominimise charge build up causing stray fields around thetrap. For this trap the geometric factors are r = 1 . z = 3 . κ = 0 .
071 and η = 0 .
82. Typical volt-ages and trap frequencies used were V = 100 −
400 V, U = 50 −
150 V and ω rf = 2 π × − × − mbar using the electrospray tech-nique with a quadrapolar guide [37] and can be pumpeddown to low pressures without feedback. Individualnanospheres could be easily charged to approximately1500 elementary charges with this method. Trapped par-ticles were detected visually on a CMOS camera usingscattered light from a 1030 nm diode laser.The radius of the trapped particle could be determined using the CMOS camera to track the motion of the par-ticle [37, 38]. Fig. 3b) shows a PSD of the particle mo-tion in the z -direction at a pressure of 1 . × − mbar.Assuming a CoM temperature of 293 K and density of1850 kg m − , a radius of 190 ± . ∼ . − correspondingto ∼
421 charges.Real time detection of the particle motion was doneusing balanced photodiodes as shown in Fig. 3a). Allthree modes of motion have a projection perpendicularto the laser beam and therefore motion along all axescan be detected using a single balanced detector (spectrashown in Fig. 3c)). The signal from the balanced photo-diodes can be sent directly to either a PLL or FGPA togenerate the feedback signal. The balanced detector canbe calibrated for the z − mode by acquiring timetraces onboth the balanced detector and the CMOS camera simul-taneously then comparing the variance in the z − mode.This can be done at any pressure so does not require theassumption that the calibration remains constant at allpressures unlike calibration by assuming thermal equi-librium at a high pressure [39]. For balanced detectionthe laser was typically focused onto the particle with anintensity of 1 . × W m − . Increasing the laser inten-sity by a factor of 3 was found to have no effect on thefrequency or position of the particle therefore at theseintensities any effect to the particle motion can be con-sidered negligible.A Red Pitaya FGPA was used to generate the feed-back signal for the velocity damping scheme using theIQ module in the PyRPL software package. A signalproportional to the measured motion of the particle withan arbitrary delay and gain could be produced. Othermodes in the feedback signal were found to couple to theparticle motion and cause heating. To prevent this theinput signal was filtered around the appropriate spec-tral peak. The x - and y -modes were cooled by adding asignal to an appropriate rod of the Paul trap such thatthe force opposes the particle motion. The z -mode wascooled by applying the feedback to one of the endcapsusing electronics built in-house.A Zurich Instruments HF2LI lock-in amplifier was usedas a PLL to generate the feedback signal for parametricfeedback cooling. The loop controller parameters of thePLL were automatically generated by the lock-in ampli-fier based on a user defined bandwidth. The signal from afrequency doubled NCO with continuously tunable phasecould be output as the modulation signal. The z -modewas cooled by modulating both endcaps using electronicsbuilt in-house with a maximum modulation depth of 5%.Although we only consider the temperature of the z -mode, the x - and y -motion of the particle was cooled us-ing velocity damping throughout the experiment. Thisminimises the cross-coupling between modes and im-proves the noise floor of the CMOS camera detection.The feedback on the z -mode could easily be switched be-tween parametric cooling and velocity damping withoutlosing the trapped particle. VI. COOLING
Fig. 4a) shows the CoM temperature against dampingrate for velocity damping in both the experiment andsimulations alongside the analytical results. Experimen-tally, cooling was performed on the z -mode of the oscilla-tor with a frequency of ω z = 2 π ×
277 Hz at a pressure of2 . × − mbar with an expected intrinsic linewidth of780 µ Hz. The simulations were performed with the sameparameters using the experimentally measured detectionnoise spectral density of S nn = 1 . × − m Hz − andnominal particle radius and density of R = 193 . ρ = 1850 kg m − . The black circles show the simulationresults where the Wiener filter described by Eq. 8 is usedto estimate the velocity based on a measurement of theparticle position that includes detection noise. These re-sults agree with the analytical results (dark blue line)across all feedback gains. The result of using a delayedposition signal as a feedback signal are shown by the pur-ple circles. Using an additional bandpass filter to removedetection noise in the feedback signal similar to the ex-periment makes no difference to the CoM temperatures.For low feedback gains the simulation temperatures agree with the analytical results and are lower than when usingan optimum Wiener filter. This is due to phase delays inthe Wiener filter. For large feedback gains the simulationbegins to cool less effectively than predicted. This is be-cause as the effective damping increases ω can no longerbe assumed constant over the oscillator linewidth andthe assumption of high-Q is no longer valid. Simulationswere run for feedback gains higher than 200 Hz but theoscillator temperature was significantly higher than 1 Kso are not shown. Although both schemes for generatinga feedback signal achieve similar minimum temperaturesat these parameters, as the Q-factor of the oscillator de-creases the Wiener filter technique will cool to lower tem-peratures. For example, for a 70 nm particle at the samepressure and oscillation frequency ( γ = 2 . ± B dB as predicted. The simulation de-viates from this trend as B dB increases due to greaterphase noise in the NCO arising from the smaller detec-tion SNR at lower temperatures. It can be seen that thetemperature begins to increase for higher bandwidths asthe PLL begins to unlock and heat the particle due tolow SN R L . The CoM temperature never goes below thebound defined by Eq. 17 (cyan line). The experiment(green circles) shows higher temperatures than the sim-ulation for all bandwidths with a minimum temperatureof 280 ±
20 mK. This is lower than previously achievedby parametric feedback in a Paul trap [20, 21]. Oncethe bandwidth increases above 100 Hz (the grey regionin Fig. 4b)) the PLL begins to lose lock and the oscilla-tor becomes unstable. In the experiment a quadrature
FIG. 4. Analytical, simulation and experimental results ofcooling the axial motion of the particle. Experimentally bothcooling schemes were done on the same particle with the samedetection parameters. The frequency of motion was 277 Hz.a) Cooling with velocity damping. Green circles are experi-mental data, black circles are simulation using a Wiener fil-ter to predict the velocity and magenta circles are simulationwhere a delayed position signal predicts the velocity. Thered and cyan lines show the first and second terms in Eq. 12respectively and the dark blue line shows the total. b) Cool-ing with parametric feedback via a PLL. Green circles areexperimental data, black circles are the ideal simulation andmagenta circles are the improved model simulation. The redline represents Eq. 16 and the cyan line represents Eq. 17. Theshaded region shows when the PLL begins to unlock from theoscillator in the experiment. For both parametric feedbackand velocity damping the modulation was experimentally in-creased to the maximum gain. bandwidth of B quad = 5 B dB was used based on the sim-ulation results.To understand what limited the final temperature ofthe experiment an improved model was designed to morerealistically simulate the experiment. Due to instabili-ties in the amplitude and frequency of the trap poten-tial, the frequency of the particle experiences a smoothdrift [41]. This was approximated in the model by a slowsinusoidal modulation of the oscillator frequency and in-creases the CoM temperature for low bandwidths wherethe modulation is bigger than or comparable to B dB .As seen in Fig. 3c), other modes of motion appear in thedetection signal which the PLL can lock to at high band- widths causing modulation at the wrong frequency andless efficient cooling of the particle. These were addedto the simulation along with second-order harmonics tomatch experimental spectra. In the experiment, the par-ticle equilibrium position can be pushed away from thegeometric centre of the trap due to stray fields. This in-troduces heating when parametric feedback is turned ondue to a shifting equilibrium position [42, 43]. This wasimplemented in the simulation by introducing a constantforce on the particle. The lock-in amplifier used has a’range’ feature that was included in the improved model.This limits the frequency difference between the NCOand the oscillator. Finally, the modulation depth wascapped at 5% to match the experimental limit. The pur-ple circles in Fig. 4b) show the results of this improvedmodel. Much better agreement is now seen between thesimulation and experiment below 100 Hz. Once B dB isincreased above this in the simulation the CoM temper-ature is unlikely to match the experimental results sincethe oscillator becomes unstable similarly to what is ob-served in the experiment.The lowest experimentally achieved temperature wasan order of magnitude lower for velocity damping thanparametric cooling with the PLL. Our simulations showthat this is partly due to other modes in the detectionsignal, a drift of the central frequency of the oscillatorand the particle being offset from the centre of the trapwhich do not affect the velocity damping scheme. Thisis because velocity damping acts on the particle fromone direction therefore any changes in position can becompensated for by a change in the feedback gain. Inaddition, any changes to the central frequency are auto-matically tracked since the position measurement is usedas the feedback signal and other frequencies in the de-tection signal do not couple to the z -mode. Even in thecase with only one mode and white noise in the detec-tion the simulation shows the backaction on the particledue to measurement noise is larger for PLL parametricfeedback than for velocity damping. Similar trends areseen at alternative pressures, particle radii and oscillatorfrequencies. VII. ENERGY DISTRIBUTIONS
A trapped nanoparticle obeying Eq. 1 is expected tohave an energy distribution given by the Boltzmann-Gibbs (thermal) distribution: ρ ( E ) = 1 Z α e − EkBT (22)where Z α is the normalisation constant such that (cid:82) ∞ ρ ( E ) dE = 1. By adding feedback to the oscillatorwe can expect to alter the dynamics and change the en-ergy distribution of the particle.In the case of velocity damping and PLL parametricfeedback we can use the Stratonovitch-Kaminskii Limit FIG. 5. Energy distribution for experimental data in unitsof k B . a) Distributions from a parametrically cooled oscil-lator (with PLL) at two different temperatures (dots) withexpected analytical distributions (lines). The distributionsagree with the analytical prediction. b) Oscillator cooledwith velocity damping. The experimental results agree withthe analytical prediction. All experimental distributions andanalytical predictions include a contribution from detectionnoise [44]. theorem to write a Fokker-Plank equation and calculatethe distributions [24, 45, 46]: ρ ( E ) vd = 1 Z vdα e − E ( γ γfb )2 γ kBT mω Snnγ fb (23) ρ ( E ) P LL = 1 Z P LLα e − EkBT (1+ Gω γ ) (24)where Z vdα and Z P LLα are the normalisation constants.Detection noise in the feedback signal has been includedin the derivation of ρ ( E ) vd . Both distributions still de-scribe a Boltzmann-Gibbs distribution in contrast to anoscillator being cooled parametrically without a PLLwhich produces a highly non-thermal distribution [25].Fig. 5. shows the energy distribution in units of k B forboth velocity damping and parametric feedback at dif-ferent temperatures. These confirm that experimentallythe oscillator is still characterised by a Boltzmann-Gibbsdistribution when cooled parametrically or with velocitydamping. Due to the small SNR at low oscillator tem-peratures the distributions include some detection noisewhich manifests as an exponential distribution for whiteuncorrelated noise. Also shown are the expected distribu-tions based on the measured temperature and detection noise. Our simulations suggest that as the SNR of thePLL becomes low the distribution will begin to deviatefrom the analytic result. This is because the phase errorwill be larger for small SNR, which is proportional to theoscillator energy, and the PLL will not track the phase asaccurately. This will lead to larger energies experiencinggreater damping similar to the case of parametric feed-back without a PLL [25]. However, detection noise in theexperiment will make this deviation hard to measure. VIII. CONCLUSIONS
We have shown velocity damping is a more effectivecooling scheme than parametric feedback using a PLLunder an identical experimental conditions. Our simu-lations have shown that this is fundamentally a resultof the larger backaction in parametric feedback. How-ever, additional signals due to the x - and y -modes andhigher order harmonics, an off-centre particle equilib-rium position, and modulation of the particle frequencydue to instabilities in the trap potential were also shownto heat the particle during parametric feedback with aPLL. These have no effect on the temperature from ve-locity damping since any additional signals from x - and y -modes do not couple to the z -mode, the force is ap-plied in only one direction and therefore independent ofposition, and any modulation of the central frequencyis automatically expressed in the feedback signal. Fur-thermore, it was demonstrated that for low-Q oscillatorsa Wiener filter will produce a better estimate of veloc-ity than the delayed position method leading to lowerCoM temperatures. Practically, parametric feedback iseasier to implement in optical traps since it requires mod-ulation of only the trapping beam. Additionally, as thetrapped particle will always be centred in the x − y planeof the optical potential, it is not affected by heating dueto off-centre trapping in these directions. In a Paul trap,additional electrodes are required to cancel stray electricfields, therefore, velocity damping can be easily imple-mented by applying the feedback signal to these elec-trodes. Kalman filtering could be used for both para-metric feedback and velocity damping to more accuratelypredict the state of the particle. However, previous stud-ies have shown this is unlikely to make a large improve-ment on the minimum achievable temperature of the par-ticle [2, 34]. Lastly, unlike standard parametric coolingwhich leads to non-thermal energy distributions, bothschemes studied here produce cold thermal distributions. ACKNOWLEDGMENTS
The authors acknowledge funding from the EPSRCGrant No. EP/N031105/1 and the H2020-EU.1.2.1 TEQproject Grant agreement ID: 766900. A.P. has re-ceived funding from the European Union’s Horizon 2020research and innovation programme under the Marie0Sklodowska-Curie Grant Agreement ID: 749709 [1] U. Deli´c, M. Reisenbauer, K. Dare, D. Grass, V. Vuleti´c,N. Kiesel, and M. Aspelmeyer, “Cooling of a levitatednanoparticle to the motional quantum ground state,”
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