Damping of a micro-electromechanical oscillator in turbulent superfluid 4 He: A novel probe of quantized vorticity in the ultra-low temperature regime
C. S. Barquist, W. G. Jiang, K. Gunther, N. Eng, Y. Lee, H. B. Chan
DDamping of a micro-electromechanical oscillator in turbulent superfluid He: A novelprobe of quantized vorticity in the ultra-low temperature regime
C. S. Barquist, ∗ W. G. Jiang, K. Gunther, N. Eng, and Y. Lee † Department of Physics, University of Florida, Gainesville, Florida, 32611, USA
H. B. Chan
Department of Physics, Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong (Dated: May 15, 2020)We report a comprehensive investigation of the effects of quantum turbulence and quantizedvorticity in superfluid He on the motion of a micro-electromechanical systems (MEMS) resonator.We find that the MEMS is uniquely sensitive to quantum turbulence present in the fluid. To generateturbulence in the fluid, a quartz tuning fork (TF) is placed in proximity to the MEMS and driven atlarge amplitude. We observe that at low velocity, the MEMS is damped by the turbulence, and thatabove a critical velocity, v c (cid:39) − , the turbulent damping is greatly reduced. We find thatabove v c , the damping of the MEMS is reduced further for increasing velocity, indicating a velocitydependent coupling between the surface of the MEMS and the quantized vortices constituting theturbulence. We propose a model of the interaction between vortices in the fluid and the surface ofthe MEMS. The sensitivity of these devices to a small number of vortices and the almost unlimitedcustomization of MEMS open the door to a more complete understanding of the interaction betweenquantized vortices and oscillating structures, which in turn provides a new route for the investigationof the dynamics of single vortices. I. INTRODUCTION
Similar to classical fluids, superfluids, such as He, He, and Bose-Einstein condensates of ultra cold gases,can also become turbulent [1–5]. This phenomenon goesby the name quantum turbulence (QT). Quantum tur-bulence differs from classical turbulence in at least twokey ways. Firstly, in QT, all of the circulation in thesuperfluid component is due to quantized vortices. Anal-ogous to vortices in type II superconductors, quantizedvortices in a superfluid consist of circulating superfluidaround a normal core of diameter a . The quantum ofcirculation is κ = h/m [6], where h is Planck’s constantand m is the mass of the boson constituting the super-fluid. For He, the vortex core diameter estimated fromthe nonlinear Schr¨odinger equation is a (cid:39) − m [6]and κ = 9 . × − m s − . Because the energy of aquantized vortex is proportional to the square of the cir-culation [6], doubly quantized vortices are unstable, withthe fluid preferring two singly quantized vortices instead.Therefore, all of the circulation is due to singly quan-tized vortices, and QT may be understood as a tangle ofthese identical vortices. Secondly, the superfluid differsfrom the classical fluid in that it has zero viscosity, andviscous dissipation is absent.Despite these differences, in some ways QT is remark-ably similar to classical turbulence. In both He [7, 8]and He [9–12], the decay rate of turbulent energy wasfound to have the same time dependence as predicted bythe classical Kolmogorov-Ohbukov theory. In He, the ∗ cbarquist@ufl.edu † ysl@ufl.edu quintessential k − / law for turbulent fluctuations and in-termittency has also been observed [13]. Remarkably, thesimilarities between quantum and classical turbulence, donot exist only at high temperatures, where the superfluidand normal fluid are coupled by mutual friction. In thisregime, it might be expected that the turbulence in thesuperfluid inherits classical characteristics through its in-teraction with the normal fluid. However, even in theultra-low temperature regime, where the normal fluid iseffectively absent, the superfluid is able to mimic classi-cal turbulent flow on scales larger than the average intervortex distance, (cid:96) = L − / , where L measures vortex linelength per unit volume. By polarizing and forming bun-dles, quantized vortices are able to generate flow on allscales between (cid:96) and the system size [14]. In this way,pure superfluid turbulence can behave quasi-classically[15]. Although, at scales similar to and smaller than (cid:96) ,the individual nature of the quantized vortices becomesapparent and the flow loses any classical character.In the ultra-low temperature regime, in the absence ofviscous damping, how is the energy in quantum turbu-lence dissipated? It is well established through experi-ment [7–12, 16] and simulation [14, 17–21] that at scalesmuch larger (cid:96) , turbulent energy is transferred to smallerscales in a manner similar to the Richardson cascade [22]of classical turbulence. However, at scales similar to (cid:96) this process must stop, and a new process must take over.It is thought that the energy is carried to smaller scalesby the Kelvin wave cascade on individual vortices beforeit is radiated away as phonons. However, direct exper-imental evidence of this process has not been observed,due to the lack of an appropriate experimental probe.When the normal fluid is absent, powerful probes, suchas second sound and tracer particles, cease to function.There has been significant success with injected electrons a r X i v : . [ c ond - m a t . o t h e r] M a y [9–12]. Currently, this method has only been able tomeasure average properties about large scale flow, suchas the average L . In the past two decades, small oscillat-ing objects, such as vibrating wires [7, 8, 23–46]; tuningforks [47–60]; micro-spheres [61–67]; and vibrating grids[40, 68–73], have successfully investigated many proper-ties of QT in He and He over a large range of tempera-tures, including the ultra-low temperature regime. Theyhave been particularly useful in understanding the gener-ation of QT and the crossover from laminar to turbulentflow. However, in He, none are continuously sensitiveto externally applied turbulent flow, as they begin togenerate their own turbulent flow after being exposed tovortices from the turbulent flow. This precludes themfrom being able to measure important quantities, suchas the fluctuations in turbulent energy or in line density, L .In this work, we investigate the effects of QTand quantized vorticity on the motion of a micro-electromechanical systems (MEMS) resonator in theultra-low temperature limit of He. Previously, similarMEMS resonators have been used to study superfluid[74–76] and normal fluid [77] He. To generate QT, aquartz turning fork (TF) is placed in proximity to theMEMS and driven with large amplitude. We find thatthe MEMS is uniquely sensitive to vortices and is ableto continuously monitor the turbulent flow. We alsofind that the coupling between the MEMS and quan-tized vortices is velocity dependent, with a critical veloc-ity of 5 mm s − separating two distinct regimes of cou-pling. While these are not observations of the elusiveKelvin wave cascade, the sensitivity of these devices to asmall number of vortices and the almost unlimited cus-tomization of MEMS open the door to a more completeunderstanding of the interaction between quantized vor-tices and oscillating structures, which in turn provides anew route for the investigation of the dynamics of sin-gle vortices and turbulent fluctuations in the ultra-lowtemperature regime. II. EXPERIMENTALA. Devices
The MEMS device used for this study is 2 µ m thick andconsists of a 125 × µ m square plate with two rowsof capacitively coupled comb electrodes on two oppositesides of the plate. The device is suspended 2 µ m above asubstrate by four springs, which allows for a fluid film tobe formed beneath the device. A diagram of the deviceis shown in Fig. 1(a) along with a cartoon of the crosssection of the device in Fig. 1(b). Due to the geometry ofthe MEMS device, there are several modes of oscillation,which are illustrated in Refs. [78, 79]. In this work, weonly study the behavior of the shear mode, which has itsmotion directed in the plane of the device, as shown inFig. 1(b). In contradistinction to most of the resonators mentioned previously, when oscillating in the shear mode,the whole device is displaced equally, and the velocity isuniform. This is an advantage of our device, as non-uniform velocity profiles have the tendency to blur themeasurements of velocity dependent phenomena.The device is asymmetrically driven and detected viathe comb electrodes. An oscillating voltage, V ( t ) = V f cos(2 πf t/ F = V dCdx , where C is the capacitance betweenthe movable electrode and the fixed electrode, and x isthe displacement of the device from its equilibrium posi-tion. The driving force is then F d = 14 βV f cos(2 πf t ) , (1)where β = dCdx | x =0 = 1 .
44 nF m − (in helium) is calledthe transduction factor [80]. Because the force dependson the square of the voltage, the force is transduced attwice the excitation frequency, and also at f = 0. TheDC component of the force, βV f , negligibly shifts theequilibrium position, and may be ignored.The displacement is detected by using the electrodeson the other side of the device. A DC bias voltage, V b ,applied to the electrodes induces a charge on the elec-trodes. As the device is displaced the capacitance variesby δC , which varies the charge by δq = V b δC = V b βx .The change in the charge is measured by a charge sen-sitive amplifier with amplification α = 0 .
67 pF − . Thedisplacement can be calculated as x = V o αβV b , (2)where V o = αδq is the voltage measured at the output ofthe charge sensitive amplifier.At the low temperatures used in this work, damping ofthe device is minimal and the quality factor, Q , exceeds10 . Because of this, the nonlinear nature of the deviceis apparent, see Fig. 4. The nonlinearity observed in thisdevice is attributed to the nonlinear variation of the ca-pacitance. After a carefully accounting for the nonlinearvariation in capacitance [80], it is found that two addi-tional forces need to be considered: a modification to thespring constant, V b c x , and a nonlinear spring restor-ing force, V b c x , where c and c are constants thatdepend on the geometry of the electrodes. Additionalforces also arise which are proportional to V f ; however,( V f /V b ) < .
002 for all measurements, so they may beignored. Including these nonlinearities the equation ofmotion for the MEMS is¨ x + 2(Γ + Γ x ) ˙ x + ω x + α x = g cos(2 πf t ) , (3)where 2Γ = ∆ ω is the full width at half max (FWHM)of the resonance at low amplitude, 2Γ characterizes thenonlinear damping intrinsically present in the silicon, ω = ( k − V b c ) /m , k is the mechanical spring constant, m is the mass, α = − V b c /m , and g = βV f / m . SpringAnchorFixed Electrodes Movable Electrodes
Movable Plate
Si Substrate
Fixed Electrode Fixed Electrode 𝑑 = 2 𝜇𝑚 𝑡 = 2 𝜇𝑚
LTW v (a)(b) (c) (d) FIG. 1. Schematic diagram of the oscillators used in this experiment (a) A scale diagram of the MEMS. The MEMS consistsof a 125 × µ m square plate, and is suspended above the substrate by springs which are anchored to the substrate at theanchor points. The device is actuated and detected through the capacitively coupled electrodes on either side. (b) A cartoon ofthe cross section of the MEMS (not to scale). The MEMS has a uniform 2 µ m thickness, and is suspended above the substrateby 2 µ m. The arrows indicate the direction of motion for the primary resonance mode discussed. (c) Cartoon of the TF usedto drive turbulence in the experiment. The dimensions of the fork are W = 0 .
10 mm, T = 0 .
23 mm, and L = 2 .
36 mm. Thetines oscillate in the plane of the fork and in antiphase to one another. (d) Schematic of the experimental setup. The TF islocated (cid:39) cos( t ) ω d sin( t ) ω d x ( t ) y ( t ) Memory CPU Data I/O
Zurich Instruments MFLI
FIG. 2. The MEMS and TF are measured through a lock-inreferenced at ω d = 2 πf d , where f d is the driving frequencyfor each respective device. This setup is employed for bothtime and frequency domain measurements, and allows for thecollection of both phase and amplitude information. The TF used in this work is the commercially avail-able Epson C-002RX. These forks are typically used astiming devices for integrated circuits, and they are de-signed to have a frequency of 2 = 32768 Hz. They comepackaged in a hermetically sealed vacuum can, which islathed off exposing the fork. A 3D rendering of the TFis shown in Fig. 1(c). The TF is made of single crys-tal quartz with metal electrodes patterned on the tines(not shown). The fork consists of two large aspect ra-tio tines of length L = 2 .
36 mm, width W = 0 .
10 mm,and thickness T = 0 .
23 mm. When oscillating in its fun-damental mode (the only mode used in this work) thetines oscillate in anti-phase to one another in the plane of the fork. It is actuated and its motion is detected bytaking advantage of the piezoelectirc property of quartz.The electrical properties of the TF are characterized bya single quantity, the fork constant , a [81]. Similar tothe MEMS, the fork is driven by applying an alternat-ing voltage, V ( t ) = V f cos(2 πf t ), directly to one of theelectrodes on the fork. This applies a force on the forkproportional to V ( t ): F d = 12 aV f cos(2 πf t ) . (4)In contradistinction to the MEMS, the force on the TFis at the same frequency as the applied excitation. Asthe fork oscillates, a current is generated proportional tothe velocity of the fork, v ( t ): I ( t ) = av ( t ) . (5)It should be noted that the fork tines do not move withuniform velocity, i.e., the tines have the maximum ve-locity v at the tip and zero velocity at the base. Thecurrent is then amplified using a transimpedance ampli-fier with amplification α = −
10 k Ω. The output voltage, V o = αI , can then be directly related to the velocity us-ing Eq. 5. Calibration of the fork and determination ofthe fork constant are discussed further in the supplemen-tary information. For the fork used in this work it wasfound that a = 2 . µ C m − [80]. Tip Velocity, v (mm s ) T o t a l D a m p i n g F o r c e ( n N ) f A m p li t u d e ( A . U . ) F I = γv F T
19 nN1.4 nN
FIG. 3. Damping force on resonance for various velocitiesat 14 mK in He-II for increasing (blue) and decreasing (or-ange) velocities. The change of slope of the TF’s velocitydependence on force is associated with the generation of tur-bulence. The dashed line is an extrapolation of the potentialflow regime to higher velocities. The excess damping forcedue to turbulence is calculated by subtracting off the extrap-olated damping force from the potential flow regime. Inset:Frequency sweeps of the TF in He-II at 14 mK for drivingforces in the range 1.4 – 19 nN. Here, the responses of the de-vice have been normalized by the driving excitation. The datacollapse to a universal curve on the low velocity tails, but failto collapse on resonance. This indicates that at low velocitiesthe damping is linear and becomes nonlinear beyond v c , dueto the generation of turbulence. B. Measurement Technique
To study the behavior of the MEMS in the presenceof turbulence, the MEMS and the TF are situated inclose proximity, with the TF 3 mm above the MEMS, asdepicted in Fig. 1(d). The devices are located inside ofa copper cell with a cylindrical volume of about 2 cm .The cell is affixed just below the mixing chamber stage ofa dilution refrigerator. To generate turbulence, the TFis driven with a large amplitude. For TFs, the transi-tion to turbulent flow and their behavior in the turbulentregime has been studied extensively over the past decade,in both He and He-B [47–60]. Because the MEMS ismost sensitive to the effects of vortices when the dampingis smallest, all of the measurements made in the turbulentregime are made at the lowest attainable temperature of14 mK to avoid excess damping due to the presence ofnormal fluid.To characterize the generation of turbulence by theTF, we measure the velocity of the TF on resonance as the driving force is varied. This allows us to measurethe velocity dependent damping force experienced by theTF. When the TF is on resonance, the force is in phasewith the velocity, and we may equate the driving forcewith the damping force. In order to remain on reso-nance, a feedback loop was employed, taking advantageof the property that the quadrature component of the TFsignal passes through zero on resonance. The feedbackloop is implemented in a LabVIEW program and adjuststhe frequency of excitation until the quadrature compo-nent is within some specified distance from zero. Figure 3shows the results of this measurement for increasing anddecreasing driving force. At low velocity, the dampingforce on the TF is roughly proportional to the velocity, F ∝ v . This is identified as the laminar regime. Moreaccurately, the flow around the TF in this regime is po-tential, because the viscous normal fluid is absent at thistemperature. As the velocity is increased beyond about140 mm s − , the velocity jumps to a lower value, indicat-ing a sudden increase in the damping, and the dampingforce is no longer proportional to the velocity. This isidentified as the turbulent regime. As the velocity is re-duced in the turbulent regime, the damping force con-tinues to follow the new power law until it eventuallycrosses over into the laminar regime. The velocity wherethe turbulent and potential regimes merge is identifiedas the critical velocity, v c . For our TF, v c = 90 mm s − .This behavior is different from the generation of tur-bulence in classical fluids. In a classical fluid, there is nocritical velocity and the onset of turbulence is continu-ous. Here, in pure superfluid, there is no turbulent flow oremission of vorticity below v c [35, 51, 61]. The differencearises because vorticity in superfluid He is nucleated ex-trinsically from preexisting remnant vortices pinned tothe surfaces of the oscillating structures. The growthof these remnant vortices occurs through the Glaberson-Donnely instability [82], which only happens above a crit-ical velocity determined by the size of the largest vortexpinned to the device.To further illustrate the effects of turbulence genera-tion on the TF, a set of frequency sweeps of the TF areshown in the inset of Fig. 3. There, the sweeps are scaledby the driving excitation. Collapse of the data onto a sin-gle universal curve after scaling indicates that the fork isin the linear regime. Here, it is clearly seen that the dataon resonance do not overlap, and that the data measuredwith larger driving force are situated toward the bottom,which indicates increased damping at higher velocities.However, the tails of the resonance still collapse to a sin-gle curve, confirming that the excess damping is onlypresent above v c .The effects of turbulence on the MEMS device was in-vestigated by performing measurements in both the fre-quency domain and time domain. For both types of mea-surement, the output of the MEMS, after the preamp,was fed into a lock-in amplifier referenced at the fre-quency of the driving force, f d (twice the excitation volt-age frequency). This allows us to collect amplitude andphase information by measuring the components of thesignal in and out of phase with the driving force, seeFig. 2.The frequency response of the MEMS is measured bydriving the MEMS with a fixed amplitude while vary-ing the driving frequency through the resonance. In theabsence of turbulence, the frequency response of the de-vice is modeled by Eq. 3 [83, 84]. The displacement of theMEMS has the following form x ( t ) = A ( ω ) cos( ωt + φ ( ω ))with A ( ω ) = g (cid:113) ( ω − ω + 2Π Aω ) + ω (2Γ + Γ A ) (6)and tan( φ ( ω )) = − ω (Γ + Γ A ) ω − ω + 2Π A ω . (7)Here, Π = α ω . Above a critical amplitude, a c , whichdepends on all of the resonance parameters, a hysteresisappears between sweeping through the resonance withincreasing and decreasing frequency [83]. Figure 4 showsan upward and downward frequency sweep through theresonance of the MEMS, with the directions of the sweepsindicated by the arrows. A clear hysteresis is observed,which is characteristic of the Duffing nonlinearity. Theintrinsic ( i.e. not due to the fluid) nonlinear nature ofthe damping can be seen in Fig. 6(a), where the intrinsicdamping measured in vacuum is shown as the solid greencurve.For the measurements that follow, only downwardsweeps through the resonance (down sweeps) are con-sidered. This is because we are primarily focused on thedamping force experienced by the MEMS. Because ourdevice possess a spring softening nonlinearity, the downsweeps contain the resonance peak (maximum displace-ment). As is true for the linear resonator, at the peakof the nonlinear resonance, the force and velocity are inphase, i.e., φ ( ω peak ) = π/
2. Therefore, by measuring thepeak velocity we may equate the driving force with thedamping force and map the relationship between MEMSvelocity and damping force.The time domain response of the MEMS is measuredby observing its free decay (ringdown). To do this, theMEMS is first energized by driving it at a frequency closeto resonance. For these measurements, we wish to startat large amplitude on the upper branch of the resonance(blue curve of Fig. 3.). To accomplish this, the frequencymust be set above the hysteretic region and slowly re-duced to the desired value. In the absence of turbulence,the behavior of the MEMS under free decay is determinedby solving Eq. 3 for g = 0 [85]. Here the response of theMEMS is given by x ( t ) = A ( t ) cos( ω t + φ ( t )) with A ( t ) = A e − Γ t (cid:113)
14 Γ Γ A (1 − e − t ) (8) Driving Frequency, f D i s p l a c e m e n t ( m ) FIG. 4. Frequency response of the device in vacuum at 6 mK,demonstrating the hysteresis between upward and downwardfrequency sweeps through the resonance. and ˙ φ ( t ) = Π A ( t ) , (9)where A is the initial displacement amplitude. The de-vice parameters can then be determined by fits to eitherthe frequency response or the free decay.In presenting Eqs. 6 - 9, we stated that they only holdtrue in the absence of turbulence. This is because thefunctional form of the damping and frequency shifts onthe device due to the vorticity is not known a priori andis not included in Eq. 3. The response of the device pre-sented above describes the intrinsic behavior of the de-vice, and any deviations may be attributed to the effectof turbulence in the fluid. III. RESULTSA. Frequency Domain
The effect of turbulence on the device can be clearlyseen by comparing two sweeps made with the same driv-ing force. Figure 5(a) shows two down sweeps of theMEMS shear mode, both made with 400 mV p excitation.One sweep was made in the presence of turbulence, la-beled “Turbulent”, and the other in its absence, labeled“Quiescent”. For the turbulent sweep, the velocity of theTF was 126 mm s − . There are several features that dis-tinguish the turbulent sweep from the quiescent sweep.The turbulent sweep transition between bi-stable statesof the Duffing oscillator (the big jump around 23621 Hz)occurs at a higher frequency and lower amplitude, andalso has its phase shifted relative to the other sweep.Extra noise is also observed in the quadrature channels,but not in the amplitude, which can be interpreted asphase noise. Measurements of the phase noise spectraand discussion of the origin of the noise are to be pre-sented in a forthcoming publication. Because the peakoccurs when the driving force is equal to the dampingforce, and because both sweeps were performed with thesame excitation, the lower peak of the turbulent sweepindicates increased damping due to the presence of tur-bulence. The overall shift in phase and the increase infrequency of the bi-stable transition are consequences ofthe peak occurring at a lower amplitude.Figure 5(b) shows a series of frequency sweeps madewhile the TF was generating turbulence at 126 mm s − .The frequency sweeps were made with excitations from100–420 mV p with 20 mV p steps. It can be seen that forthe lowest few excitations the damping is significantlyhigher compared to the other sweeps (labeled “HighDamping”), and on this scale the signal is indistinguish-able from the noise floor. To characterize the velocitydependence of the damping, we record the velocity of theMEMS at the peak for various driving forces. Figure 6shows the result of this measurement. Figure 6(b) depictsthe data shown in Fig. 6(a) with the vacuum damping(solid green curve in Fig. 6(a)) subtracted. Therefore, thedamping presented in Fig. 6(b) may be identified as thedamping from the presence of vortices and turbulence.A preliminary version of the results shown in Fig. 5 andFig. 6 were discussed previously in Refs. [86, 87]In Fig. 6, alongside the measurements made in TF gen-erated turbulence, are measurements of the response ofthe device due to remnant vortices pinned to the sur-face, which are shown as open and closed squares. Thesevortices became pinned to the surface after some turbu-lent event, such as driving the TF or cooling throughthe superfluid transition [88]. They remain attached tothe surface after the turbulence has dissipated. Thesemeasurements were made directly after cooling to basetemperature through the superfluid transition, so thatmany remnant vortices were present, and the MEMS washighly damped. They were also made in the absence ofturbulence generated by the TF. The measurements werefirst made for increasing velocity, then decreasing veloc-ity. A large hysteresis between these measurements canbe seen and is identified with the removal of some vor-tices pinned to the device. We identify the change indamping around v (cid:39) − with the onset of vortexremoval. To observe the hysteresis again, more remnantvortices must be generated. Otherwise, upon increasingthe velocity again, the damping closely follows the lowercurve. We term the process of vortex removal as anneal-ing . While the MEMS may be annealed by use of theshear mode, as seen in Fig. 6, it may be annealed to agreater degree by driving other modes of the device. Theeffects of remnant vortices and the annealing process arediscussed further in Refs. [79, 86, 87].In the presence of turbulence, hysteresis is no longerobserved. This is consistent with the interpretation thatpinned vortices are being removed when the velocity D i s p l a c e m e n t ( m ) Frequency, f D i s p l a c e m e n t ( m ) (a)(b) TurbulentQuiescentHigh Damping
FIG. 5. Frequency response of the MEMS in the presence ofturbulence generated by the TF measured at 14 mK. (a) Twodownward frequency sweeps of the MEMS with (Turbulent)and without (Quiescent) the TF generating turbulence. Themain figure shows the quadrature components of the signaland the inset shows the amplitude. In the turbulent state thepeak amplitude is smaller due to increased damping. Excessnoise in the phase also appears caused by fluctuations of thedamping [79, 86]. (b) A set of frequency sweeps made withexcitations between 100 – 420 mV p in steps of 20 mV p while theTF was generating turbulence ( v TF = 126 mm s − ). For thelowest several excitations, the corresponding sweeps (labeled“High Damping”) experience significant damping, which canbe seen by the relatively small amplitudes. This behavior canbe seen more clearly in Fig. 6. of the device exceeds v (cid:39) − . In the turbulentflow, vortices are continually colliding with the MEMSand becoming pinned; any vortices removed by drivingthe MEMS are quickly replenished. For velocities below5 mm s − , the MEMS experiences significant drag. Thiscorresponds to the “High Damping” regime referenced in T o t a l D a m p i n g F o r c e ( p N ) v TF =189mm s v TF =126mm s v TF =119mm s Remnant Response s )051015202530 T u r b u l e n t D a m p i n g F o r c e ( p N ) Increasing VelocityDecreasing Velocity (a)(b)
FIG. 6. Damping force on the MEMS in the presence of turbu-lence generated by the TF measured by performing frequencysweeps. Also shown is the damping from remnant vortices,which is measured while the TF is at rest. (a) The totaldamping force experienced by the MEMS as a function of ve-locity. The colored and open symbols represent measurementsmade for increasing and decreasing MEMS velocity, respec-tively. (b) The damping force experienced by the MEMS dueto turbulence for various TF velocities. The turbulent damp-ing force is calculated by subtracting the intrinsic dampingof the device, which is shown as the solid curve in (a). Theintrinsic damping was measured in vacuum at 6 mK.
Fig. 5. Upon exceeding this critical velocity, the dampingof the MEMS is reduced, whereupon the velocity of theMEMS jumps to (cid:39)
70 mm s − . Similar to the remnantvortex response, in the high velocity regime the turbulentdamping force is reduced as the velocity increases.To understand the high damping regime it is helpful toconsider how much extra vortex line length would need to be generated to account for the observed damping. Forthis, we consider how much energy is dissipated duringeach cycle of oscillation. We simplify the argument byassuming that the force and velocity are sinusoidal, andthat they are exactly in phase. This last assumption isjustified because we are only considering the damping onresonance. In this case, the energy dissipated per cycleis E = F v / f , where F and v are the amplitudes ofthe force and velocity, respectively. For F = 10 − Nand v = 10 mm s − , the energy dissipated each cycle is E = 2 . × − J. The linear energy density of a vortexis
E/l = ( ρκ / π ) ln( (cid:96)/a ) [6], where ρ is the density ofhelium, (cid:96) is the inter vortex spacing around the device,and a (cid:39) − m is the size of the vortex core. Forour device, a reasonable guess for the vortex spacing isin the range (cid:96) ∼ − µ m. This then yields a linearenergy density E/l = 1 . − . × − J µ m − , whichcorresponds to ∼ µ m/cycle of total increased length.We can estimate the number of vortices pinned to thedevice by constructing a simple model for how the vor-tices are pinned, see Fig. 7(a) – (d). The movable portionof the MEMS is suspended d = 2 µ m above a substrate.Consider a straight vortex bridging the moving plate andthe substrate such that its length is d . If the plate is dis-placed horizontally by a distance x , the vortex length isincreased by δl (cid:39) x / d . For a complete cycle of oscil-lation the increased length is twice that amount. Thedisplacement amplitude of the MEMS near the criticalvelocity is roughly x (cid:39) . µ m. This yields an increaseof length per vortex of δl (cid:39) × − µ m, which suggeststhere are about 400 vortices pinned to the device in thehigh damping regime. This excess vortex length wouldthen be carried away from the device by vortex rings thatare created when the length of an individual vortex islarge enough to intersect itself and cause a reconnectionevent [89, 90]. It is not possible to know the exact distri-bution of vortices on the device; however, they are mostlikely concentrated around the perimeter of the device.Including the electrodes, the perimeter of the MEMS isquite large. For 400 vortices the inter-vortex distance is (cid:96) ∼ − µ m, consistent with our initial guess above.This simple model is consistent with the linear scalingof the force with velocity in the high damping regime:the energy lost per cycle is proportional to the veloc-ity squared, E ∝ v , and the energy transferred to thevortices is proportional to the increased length which isproportional to the velocity squared, E ∝ δl ∝ x ∝ v .To effectively transfer energy from the device to the vor-tex line, the frequency of the device should be matchedto the frequency of a standing mode of the vortex [68].If the frequency of the device is too low, the vortex linewill respond adiabatically and will be in its instantaneousequilibrium position determined by the flow around thedevice. In this limit, there is no accumulation of excesslength through one period of motion. If the frequency ofthe device is much larger than the standing mode, thecoupling of motion is greatly reduced. The frequency ofstanding wave modes for a quantized vortex is given by f ( k ) = 12 π κk π ln (cid:18) ka (cid:19) , (10)where k = nπ/l is the wave number [6]. Taking therange of lengths, l = 2 . . µ m, we find that the fun-damental frequency of the standing mode is in the range27.3–22.9 kHz, which is consistent with the frequency ofthe SH mode (23.6 kHz), indicating that the device canefficiently transfer energy to the vortices.Another possible mechanism for the damping of theMEMS that might be proposed is the removal of energythrough a Kelvin wave cascade. The initial stages of thisprocess are similar to what is described above, i.e., themotion of the device causes motion on the vortices pinnedto the device. Except, instead of the energy being car-ried away from the device as vortex rings, it is ferried tosmaller scales of wave motion on the attached vorticesthrough the Kelvin wave cascade until the vortex can ef-ficiently radiate phonons. However, this process predictsthat the power dissipated should scale as the tenth powerof the displacement amplitude, A [91], which is clearlynot observed, so this process can not fully explain thedamping observed by the device.Because the TF is situated above the MEMS, it is rea-sonable to expect that some vortices will become pinnedto the top of the moving plate ( i.e. the side in contactwith the bulk). However, these vortices should not signif-icantly contribute to the damping because it is unlikelythat their lengths correspond to a Kelvin wave resonanceat the frequency of the device. It may be expected thatvortices pinned to the top of the device could cause theMEMS to transition to turbulent flow in a manner sim-ilar to the TF, but this is not observed. Because of thegeometry of the device, the backflow around the deviceis minimal, and it is the backflow which provides the su-perflow necessary for the transition to turbulence. Forthese reasons, we do not believe that vortices pinned tothe top of the MEMS significantly contribute to the ob-served results.The reduction of damping with increasing velocity, ob-served above the critical velocity, can be understood asthe depinning of vortices from pinning sites on the sur-face. Figures 7(e) – (f) illustrate this process. When thevelocity of the plate is below the depinning velocity, v pin ,the vortex remains attached to the pinning site. Becausethe vortex is pinned, its length is increased as the plateis displaced, which leads to damping as discussed above.When the velocity exceeds v pin the vortex is no longerpinned to a specific place on the surface and is free tomove relative to the surface. Because of this, the motionof the vortex is no longer coupled to the motion of theplate and no longer contributes to the damping. The de-pinning velocity for a vortex pinned between two parallelplates was considered by Schwarz [90], and was found tobe v pin = κ πd ln (cid:18) ba (cid:19) . (11) 𝑣 < 𝑣 𝑝𝑖𝑛 𝑣 > 𝑣 𝑝𝑖𝑛 d = 2𝜇𝑚 𝑏 = 10 −7 − 10 −8 m (a) (b)(c) (d)(e) (f) FIG. 7. A cartoon of the interaction between the MEMSand quantized vortices. (a) – (d) A mechanism for dampingthe MEMS. (a) A quantized vortex is pinned between thedevice and the substrate. (b) Because of the pinning, thevortex is elongated when the device is displaced. (c) Theextra length after displacement is accumulated. Some numberof cycles later the vortex line is long enough to reconnectwith itself. (d) The reconnection ejects a vortex ring, whichremoves length from the pinned vortex, and the process beginsagain. (e) – (f) A mechanism for the decoupling of the motionof the MEMS from the motion of vortices. (e) At velocitieslower than the velocity required to depin a vortex, v pin , themotion of the MEMS is coupled to the motion of the vortex.(f) When the MEMS exceeds v pin the vortex is free to slidealong the surface and the motion is no longer coupled. Adistribution in depinning velocities would cause the dampingto be gradually reduced as the velocity is increased, as seenin Fig. 5(b). Here d = 2 µ m is the MEMS gap size and b = 10 − –10 − m is the size of the pinning site, measured by atomicforce microscopy [78]. However, the argument presentedin Ref. [90] is for uniform superflow between the plates,which is not the case for the MEMS. A vortex pinned be-tween the moving plate and the substrate will experiencesome velocity gradient as the plate is displaced. Despitethis, we use Eq. 11 for an order of magnitude compari-son. For our MEMS, v pin (cid:39)
50 mm s − , which is closeto the region where the damping is observed to decrease,70–200 mm s − . The observed velocity dependence of thedamping may be due to a distribution in the size of pin-ning sites. The surface of the MEMS is rough on thescale of 100 nm, and there is a distribution of bump sizesthat make up this roughness. A detailed discussion ofthe surface characteristics of these MEMS devices is pro-vided in Refs. [78, 92]. Because of the distribution in thebumps there is a distribution in the depinning velocities.As the velocity is increased, more vortices decouple andthe damping is reduced. However, the velocity dependsonly on the logarithm of the bump size, and this is un-likely able to explain this behavior over the whole veloc-ity range. A complete explanation of this phenomena isnot possible because there is still much unknown abouthow the MEMS interacts with the turbulence from theTF. We currently do not know how the MEMS capturesand removes vortices. It is also unknown how the flowfield induced by the turbulent vortices around the deviceaffects its motion. B. Time Domain
We further investigate the effects of turbulence on theMEMS motion by studying the properties of the free de-cay of the MEMS in the turbulent and quiescent regime.In the quiescent regime we investigate the effect of rem-nant vortices. For all measurements, the MEMS wastuned to f d = 23 , .
72 Hz and was driven at 400 mV p .The lock-in was referenced at f d , and the time constantwas chosen such that 1 /τ > ω d − ω ( t ) for all time. Here, ω ( t ) = ω + Π A ( t ) (Eq. 9) depends on the amplitudeand changes during the ringdown measurement due tothe nonlinear restoring force. Note that the measure-ments do not begin on resonance. This is done becausethe noise due to turbulence will readily cause the MEMSto transition from the high amplitude state into the lowamplitude state if the frequency is tuned too close to thetransition frequency [93].From top to bottom, Fig. 8 shows the ringdown re-sponse of the MEMS in the quiescent regime after an-nealing, in the quiescent regime before annealing, andin the turbulent regime. For each different measurementshown, ten individual decays were recorded and averaged.The black curve is the amplitude of motion, A ( t ), and theoscillating curve is the in-phase component of the motion, A ( t ) cos[( ω ( t ) − ω d ) t ]. The out-of-phase component is alsocollected, but it is not plotted for clarity. The frequencyof oscillation at any given time is | f ( t ) − f d | . This shiftin frequency is due to demodulation occurring within thelock-in. Relative to the motion in the quiescent annealedstate, the motion in the presence of remnant vortices andturbulence is more damped, which is consistent with thefrequency domain measurements. The decay in all threecases begins roughly the same, and only deviates as theamplitude is decreased. In the turbulent state, the decayclearly deviates from an exponential time dependence atlower amplitudes. Several more measurements of the freedecay of the MEMS were made for various TF velocitiesin the range of 126 – 183 mm s − . Again, for each TFvelocity ten decays were measured and averaged. The D i s p l a c e m e n t ( m ) Time (s)
AnnealedRemnantTurbulent
FIG. 8. From top to bottom, the response of the MEMS inthe quiescent state after being annealed (see text), in the qui-escent state before annealing, and with turbulence. The solidblack curves represent the amplitude of the device, A ( t ), whilethe oscillating curves represent the component of the signalin-phase with the original driving signal, A ( t ) cos[( ω ( t ) − ω d ) t ](see Eq. 9). At large amplitude, ω ( t ) (cid:39) ω d , and the oscillationis slow. As the amplitude is decreased ˙ φ tends towards zeroand ω ( t ) tends toward ω . At low amplitude, the signal thenoscillates with frequency | ω − ω d | . The out-of-phase signalis also recorded, but is not plotted for clarity. The envelopesof these ringdowns are shown in Fig. 9b in log-linear scale,where the departure from a pure exponential decay can beeasily noticed. results of these measurements are shown in Fig. 9.Visually, it is not obvious from Fig. 8 what effect theturbulence has on the nonlinear frequency shift. To seehow the turbulence affects the frequency shift we firstneed to calculate the frequency of oscillation as the am-plitude decays. To do this, we first calculate the phaseof the oscillator as a function of time from the measuredquadrature components as φ ( t ) = tan − (cid:18) A ( t ) sin[( ω ( t ) − ω ) t ] A ( t ) cos[( ω ( t ) − ω ) t ] (cid:19) . (12)To obtain the total accumulated phase, π is added ev-ery time the argument of Eq. 12 changed signs from − to +. The frequency shift was then calculated by nu-merically differentiating the phase with respect to time.At low amplitude, the noise would cause the sign of thephase to fluctuate and spoil the process described above.A moving average including the 40 nearest points wasperformed to reduce the noise.The amplitude dependent frequency shift for theMEMS measured in the quiescent annealed state is shown0 Displacement ( m ) / ( H z ) Time (S) R i n gd o w n E n v e l o p e , a ( t ) ( m ) AnnTurb [ ]
Rem
Velocity (mm s ) I n s t a n t a n e o u s D a m p i n g R a t e , / ( H z ) Velocity (mms ) D a m p i n g F o r c e , m v ( n N ) (a) (b) (c) FIG. 9. Ringdown response of the MEMS for several different TF velocities between 126-183 mm s − . (a) Amplitude dependentfrequency shift calculated from the time dependent phase (see Eq. 12). The main figure shows the response in the quiescentstate after annealing, and the solid line is a fit to Eq. 9 with Π / π = 2 . µ m − . The inset shows the frequency response for alldifferent TF velocities, including the response shown in the main figure. Within the precision of the measurement, the presenceof turbulence does not alter the linear or nonlinear restoring force. (b) Ringdown envelopes in log-linear scale for various TFvelocities. The measurements are grouped into three categories: “Turb” for measurements made in the turbulent state, “Rem”for measurements made in the quiescent state before annealing, and “Ann” for measurements made in the quiescent state afterannealing. The solid red line is a fit of the “Ann” data to Eq. 8 with Γ / π = 0 .
037 Hz and Γ / π = 0 .
18 Hz µ m − . The insetshows the short time behavior at high velocity before the nonlinearities significantly affect the response of the device. (c) Thevelocity dependence of the instantaneous damping rate, Γ, calculated from Eq. 13. Γ can be understood as the instantaneousslope of an envelope shown in (b). Inset: Damping force experienced by the MEMS calculated as m v , where m is the massof the MEMS, v is the velocity, and Γ is the value displayed in the main figure. in Fig. 9(a). The solid line is a fit to the data using Eq. 9.Around A = 1 . µ m the frequency shift deviates from thefit. This anomaly is due to the ringdown beginning offresonance. The fit yields a nonlinear frequency pullingof Π / π = − . µ m − . In the inset of Fig. 9(a), thenonlinear frequency shifts of the MEMS in both the tur-bulent and quiescent state are shown. Within the pre-cision of the measurement, all of the data lie on top ofeach other. This implies that the presence of turbulencedoes not significantly affect the resonance frequency orthe nonlinear frequency shift of the MEMS.The ringdown envelopes for all measurements areshown in Fig. 9(b) in log-linear scale. When plotted thisway, the slope at any point is the instantaneous damp-ing rate, and a straight line in the plot corresponds toa pure exponential decay. The envelope labeled “Ann”was made in the quiescent state after annealing. Thesolid red line is a fit to the data using Eq. 8. From the fitwe obtain Γ / π = 0 .
037 Hz and Γ / π = 0 .
18 Hz µ m − .Because the normal fluid is absent and the remnant vor-tices have been removed, the measured linear and non-liner damping may be attributed to intrinsic processespolysilicon. Two measurements were made in the quies-cent state directly after turbulence was present, and arelabeled “Rem”. These measurements were made beforeannealing, so the device is still influenced by remnant vor-tices. The envelopes labeled “Turb” were made while theTF was continuously generating turbulence, with veloc-ities in the range 126–183 mm s − . Even in the absence of turbulence, the remnant envelope differs significantlyfrom the annealed envelope. Directly after initiating theringdown, the damping rate begins to decrease. How-ever, as the velocity is reduced further, the damping ratebegins to increase for the remnant case, while it remainsconstant after annealing. The increased damping rate atlow amplitude is more pronounced for the measurementsmade in turbulence. These observations are consistentwith our previous measurements: there is large dampingat low velocities, and at large velocities the damping isreduced.The decay of the MEMS in the presence of vorticitycannot be described by Eq. 8, as the functional form ofthe damping is unknown. However, we can extract thelocal damping rate, Γ, at a given time by computing theslope of ln( A ( t )) at that time, that isΓ = − d ln( A ( t ) /A ) dt . (13)The damping rates extracted this way are shown inFig. 9(c) as a function of the velocity of the resonator.Because of the noise, the data was first smoothed usinga local weighted regression and the numerical derivativewas averaged over the 100 nearest points. The damp-ing rates are peaked at low velocity, and as the velocityincreases the damping falls off like v − until it beginsto plateau around 100 mm s − . The inset of Fig. 9(c)shows the damping force, m v , calculated from the datashown in the main figure. This can be directly compared1 TF Turbulent Power, P (pW ) T u r b / ( m H z ) Γ T urb = Γ R + ζ √ P Γ R / π = 6 .
5m Hz ζ/ π = 0 .
65 mHz/ √ pW FIG. 10. The increase of the MEMS damping rate at highvelocity as a function of the square root of the TF powerinput into turbulence. The damping rate is calculated byfitting a line to the data shown in the inset of Fig. 9(b) fortimes less than 0.6 s. The increase in the damping rate due toturbulence, Γ
Turb is calculated by subtracting the dampingrate measured without turbulence or remnant vortices, Γ A , i.e. Γ Turb = Γ − Γ A . with Fig. 6(a), where it can be seen that the same broadfeatures are displayed. The critical velocity observed inFig. 6(a) can be understood as the velocity at which thedamping rate becomes inversely proportional to v , i.e. Γ ∝ v − . The large jump in velocity above the criticalvelocity seen in Fig. 6, can also be understood from thevelocity dependence of Γ. When Γ ∝ v − , the dampingforce experienced by the MEMS, F ∝ Γ v , is constantand independent of velocity. Therefore, any incrementalincrease in the force will cause the velocity to grow un-til some new process alters the velocity dependence of Γcausing the damping force to once again equal the drivingforce.For the data presented in Fig. 9, it is important to re-member that the data are collected for decreasing veloc-ity. That is, the damping starts off small and is increasedas the velocity is reduced. This increase in damping mustcome from an increased coupling of the device motion tothe motion of vortices surrounding the device. This in-creased coupling can be understood as the same processdescribed in Fig. 7(e)–(f). Also, as the MEMS slows downit is more likely for a vortex line impingent on the deviceto pin to the surface upon collision. The increased like-lihood of vortex capture is due to the increased numberof stable pinning sites at lower velocities.For short times after the beginning of the ringdown,the damping can be considered to be linear, which is demonstrated in the inset of Fig. 9(b). By fitting thedata in the first 0.6 s to a line, the damping rate, Γ canbe extracted. The increase in damping rate arising fromthe turbulence, Γ T urb , can be determined by subtractingthe damping rate of the device in the annealed state, Γ A , i.e. Γ T urb = Γ − Γ A . It is found that Γ T urb grows inproportion the the square root of the turbulent power.This is demonstrated in Fig. 10, where Γ
T urb is plottedagainst the square root of the turbulent power. The solidline is a fit to the data, and the fitting function with thefit values are shown in the figure. The turbulent poweris calculated from the excess force experienced by theTF when it is driving turbulence, F T = F − γv , where γ is determined by fitting in the potential flow regime,see Fig, 3. The power input to turbulent flow is then P = F T v .The dependence of Γ T urb on turbulent power maybe understood by assuming that we are in the ultra-quantum regime of turbulence [15, 16, 20]. This regime isdistinct from the quasi-classical regime discussed in theintroduction, and is characterized by a lack of large scaleflow. Here, the turbulent vortex tangle is random andlacks any significant polarization. In other words, mostof the turbulent energy is contained at the scale of theinter vortex spacing, (cid:96) . With (cid:96) being the characteristiclength scale of the ultra-quantum turbulence, it may beshown by dimensional analysis, that the decay rate of thevortex line density, L , is proportional to κL [94]. To de-termine the steady state value of L we must add a termto account for the creation of line length coming from theTF. Because the energy of a vortex line is proportionalto its length, the rate of increase of line length should beproportional to the power P . Including this we obtain dLdt = χ P P − χ κL , (14)where χ P is a proportionality constant, which can beunderstood at the inverse of the average linear energydensity of a vortex in the turbulent flow. The steadystate value of L is found when dL/dt = 0, yielding L = (cid:115) χ P Pκχ . (15)It is natural to assume that the damping on the de-vice is proportional to the number of vortices interact-ing with the device, and in turn proportional to L . As-suming this, we arrive at Γ T urb ∝ L ∝ √ P . Unfortu-nately, we are unable to make a quantitative calculationof L , because we lack a precise theory for how L corre-sponds to Γ. However, by making a similar measurementfor a greater range of TF velocities, we may be able toobserve the crossover from the ultra-quantum to quasi-classical regime of turbulent flows. In the fully devel-oped quasi-classical regime dL/dt ∝ − L / , which yieldsΓ T urb ∝ P / .The offset of Γ at P = 0, Γ R , is likely due to a semi-permanent background of remnant vortices. The data2point at (0 pW, 6.5 mHz) corresponds to the damping ofboth remnant ringdown measurements, which were madeat different times. One was made after driving the TF at125 mm s − and the other after driving it at 183 mm s − .Although they were made at separate times, their ring-down envelopes overlap almost perfectly. This indicatesthat there are some long lived remnant vortices that arenot removed by simply driving the shear mode to highvelocity. The extra damping of these vortices is thenpresent for all of the ringdown measurements made inturbulence, which is seen as a constant shift of Γ T urb .If there really is a semi-permanent background of rem-nant vortices, then why do the remnant ringdowns dif-fer from the annealed ringdown (see Fig. 9(b))? Beforemaking the annealed measurement, the device was thor-oughly annealed using a combination of different reso-nance modes of the device, and was annealed for a longertime. The extra care in annealing seems to be responsiblefor the removal of these semi-permanent vortices.
IV. CONCLUSION
We have presented measurements of our MEMS de-vice in the presence of turbulence generated by a sec-ondary structure (tuning fork) in the ultra-low tempera-ture regime (14 mK) and demonstrated that our device isuniquely sensitive to turbulence. The uniqueness of thisdevice is its ability to continuously measure the turbulentflow. Until now, all other oscillators measured in super-fluid He begin to generate their own turbulence immedi-ately after being exposed to vorticity in the fluid. Whilethis has enabled many rich experiments [33–37, 95], itprecludes the use of these devices to continuously sensequantum turbulence.To demonstrate the sensitivity of the MEMS, we havepresented measurements of the device in both the fre-quency and time domain. It was observed that below a critical velocity of about 5 mm s − , the damping ofthe MEMS is greatly enhanced relative to its intrinsicdamping. Above the critical velocity, the damping isgreatly reduced. From the time domain measurements,it was observed that this critical velocity corresponds toa change in the velocity dependence of the damping, withthe damping rate changing inversely proportional to thevelocity, Γ ∝ v − . To explain the damping at low veloci-ties and the change in damping above the critical velocitywe propose a model of vortices pinned between the sub-strate and the moving part of the MEMS. The modelaccounts for the reduction in damping above the criticalvelocity by supposing there is a distribution in depinningvelocities, and that when a vortex become depinned itsmotion is decoupled from the motion of the device. How-ever, from the frequency domain measurements of thedevice under the influence of remnant vortices, it is clearthat complete removal of some fraction of the vorticesis also happening above the critical velocity. Throughthe time domain measurements, it was also found thathigh velocity damping of the MEMS scales in proportionto √ P . We interpret this result as an indication thatthe MEMS is sensing the average vortex line density, L ,which scales as √ P in the ultra quantum regime.While there is still much to learn about the interactionof vortices with the MEMS device before more quantita-tive statements can be made, it is clear that there is awealth of information that can be extracted from thesedevices. ACKNOWLEDGMENTS
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