diff --git "a/9tE5T4oBgHgl3EQfRQ5h/content/tmp_files/2301.05519v1.pdf.txt" "b/9tE5T4oBgHgl3EQfRQ5h/content/tmp_files/2301.05519v1.pdf.txt" new file mode 100644--- /dev/null +++ "b/9tE5T4oBgHgl3EQfRQ5h/content/tmp_files/2301.05519v1.pdf.txt" @@ -0,0 +1,3293 @@ +Second sound resonators and tweezers as vorticity or velocity +probes : fabrication, model and method +Eric Woillez∗, Jérôme Valentin†and Philippe-E. Roche‡ +Univ. Grenoble Alpes, CNRS, Institut NEEL, F-38042 Grenoble, France +Distributed under a Creative Commons Attribution. +CC-BY | 4.0 International licence +Abstract +An analytical model of second-sound resonators with open-cavity is presented and validated against +simulations and experiments in superfluid helium using a new design of resonators reaching unprecedented +resolution. The model accounts for diffraction, geometrical misalignments and flow through the cavity. It is +validated against simulations and experiments using cavities of aspect ratio of the order of unity operated +up to their 20th resonance in superfluid helium. An important result is that resonators can be optimized +to selectively sense the quantum vortex density carried by the throughflow -as customarily done in the +literature- or alternatively to sense the mean velocity of this throughflow. Two velocity probing methods are +proposed, one taking advantage of geometrical misalignements between the tweezers plates, and another one +by driving the resonator non-linearly, beyond a threshold entailing the self-sustainment of a vortex tangle +within the cavity. +After reviewing several methods, a new mathematical treatment of the resonant signal is proposed, to +properly separate the quantum vorticity from the parasitic signals arising for instance from temperature and +pressure drift. This so-called elliptic method consists in a geometrical projection of the resonance in the +inverse complex plane. Its strength is illustrated over a broad range of operating conditions. +The resonator model and the elliptic method are applied to characterize a new design of second-sound +resonator of high resolution thanks to miniaturization and design optimization. When immersed in a su- +perfluid flow, these so-called +second-sound tweezers provide time-space resolved information like classical +local probes in turbulence, here down to sub-millimeter and sub-millisecond scales. The principle, design +and micro-fabrication of second sound tweezers are detailed, as well as their potential for the exploration of +quantum turbulence +Contents +1 +Introduction to second sound resonators +2 +1.1 +Quantum fluids and second sound +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +2 +1.2 +Generation and detection of second sound waves +. . . . . . . . . . . . . . . . . . . . . . . . . . . +2 +1.3 +From macroscopic second sound sensors to microscopic tweezers . . . . . . . . . . . . . . . . . . . +3 +1.4 +Overview of the manuscript . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +2 +Design, fabrication and mode of operation of second sound tweezers +4 +2.1 +Mechanical design +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +2.2 +Second sound detection and generation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +7 +2.2.1 +Thermometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +7 +2.2.2 +Heating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +8 +2.2.3 +Digression on the operation in the non-linear heating regime +. . . . . . . . . . . . . . . . +9 +2.3 +Microfabrication and assembling +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +10 +2.4 +Electric circuit +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +13 +∗present affiliation: CEA-Liten, Grenoble +†present affiliation: Observatoire de Paris - PSL, CNRS, LERMA, F-75014, Paris, France +‡Corresponding author +1 +arXiv:2301.05519v1 [cond-mat.other] 13 Jan 2023 + +3 +Models of second sound resonators +14 +3.1 +Resonant spectrum of second sound resonator: phenomenological aspects +. . . . . . . . . . . . . +15 +3.2 +Analytical approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +15 +3.3 +Numeric algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +19 +3.3.1 +For a backgroud medium at rest +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +19 +3.3.2 +In the presence of a turbulent flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +21 +3.4 +Quantitative predictions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +22 +3.4.1 +Spectral response of second sound resonators . . . . . . . . . . . . . . . . . . . . . . . . . +23 +3.4.2 +Response with a flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +23 +3.4.3 +Effect of lateral shift of the emitter and receiver plates . . . . . . . . . . . . . . . . . . . . +26 +3.4.4 +Limits of the model +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +27 +3.5 +Quantum vortex or velocity measurements ? +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . +28 +4 +Measurements with second sound tweezers +29 +4.1 +The vortex line density from the attenuation coefficient +. . . . . . . . . . . . . . . . . . . . . . . +30 +4.2 +Analytical method in an idealized case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +32 +4.3 +The elliptic method +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +33 +4.4 +Applications of the elliptic method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +35 +4.4.1 +Suppression of temperature and pressure drifts . . . . . . . . . . . . . . . . . . . . . . . . +35 +4.4.2 +Measure of vortex line density fluctuations +. . . . . . . . . . . . . . . . . . . . . . . . . . +36 +4.4.3 +Filtering the vibration of the plates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +38 +4.5 +Velocity measurements +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +38 +5 +Summary and Perspectives +40 +1 +Introduction to second sound resonators +1.1 +Quantum fluids and second sound +Below the so-called lambda transition, liquid 4He enters a quantum state named He-II. This liquid transition +occurs around Tλ ≃ 2.18 K at saturated vapor conditions. According to Tisza and Landau two-fluid model, +the hydrodynamics of He-II can be described as the hydrodynamics of two interpenetrating fluids, called the +superfluid component and the normal fluid component [Bal07, Gri09]. The superfluid density ρs is vanishingly +small right below the transition, and it increases as temperature decreases. The opposite dependence occurs +for the normal fluid density ρn, which becomes vanishingly small in the zero temperature limit. The properties +of both fluids strikingly differ. The superfluid has zero viscosity and zero entropy. Besides, the circulation of +its velocity field is quantized in units of κ = h/m ≃ 0.997.10−7m2/s, where h is Planck constant and m the +atomic mass of 4He. This quantization constrain results in the existence of filamentary vortices of Ångströmic +diameter, later referred to as the superfluid or quantum vortices[Don91]. Contrariwise, the normal fluid follows +a classical viscous dynamics and carries all the entropy of the He-II[Put74, Kha00]. +The existence of distinct velocity fields vs and vn for the superfluid and normal fluid results in the existence +of two independent sound modes in He-II, as can be shown by linearizing the equations of motion [Put74, +Kha00, Don09]. The so-called “first sound” corresponds to a standard acoustic wave : both fluids are oscillating +in phase (vs = vn), producing oscillations of the local pressure and density ρ = ρs + ρn. The “second sound” +corresponds to both fluids oscillating in antiphase without net mass flow (ρsvs = −ρnvn). Hence, the relative +densities of superfluid and normal fluid locally oscillates, as well as entropy and temperature. +1.2 +Generation and detection of second sound waves +Experimentally, two techniques are mostly used to generate and detect second sound: one mechanical and +the other thermal. Alternative techniques not discussed here have been occasionally used, including optical +scattering (e.g. [PGB70]) and acoustic detection above the liquid-vapor interface[LFF47] or in the flow itself +(e.g. [HR76]). +The mechanical technique consists in exciting and sensing a single component of He-II, either its superfluid +one or the normal fluid component. Indeed, a single fluid motion can be viewed as a superposition of a second +sound and a first sound (i.e. an acoustic wave), with an exact compensation of motion for one of the two fluids +at the location of the transducer. Since the first sound velocity is typically one decade larger than the second +second velocity[DB98], most second sound resonances don’t coincide with acoustic resonances. +In practice, +a selective displacement of the superfluid component is achieved in Peshkov transducers. They are made of +a standard acoustic transducer side-by-side with a fixed porous membrane-filter which tiny pores are viscous +dampers for the normal fluid but are transparent (“superleaks”) for the superfluid [Pes48, HL88]. Alternatively, +2 + +Emitter +Receiver +Emitter +Receiver +Flow +Flow +Figure 1: +Left: A macroscopic resonator for second sound, embedded in the sidewall, is used to sense +the averaged flow properties in the shaded region. +Right: Second sound tweezers. In contrast with the +macroscopic design of the left schematics, this miniaturized resonator, positioned within the flow, allows space +and time resolution of the flow variations. +a selective displacement of the normal fluid component is achieved in oscillating superleak transducers. They +are based on a vibrating porous membrane that is coupled to the motion of normal fluid by viscous forces, and +uncoupled to the inviscid superfluid. They can be manufactured by replacing the membrane of a loudspeaker +or microphone by a millipore or a nucleopore sheet [WBF+69, SE70, DLL80]. +The thermal technique to generate and detect second sound consists in forcing second sound by Joule +effect, and detecting it with a thermometer. Depending on the operating range of temperature and practical +considerations, several types of thermometers can be suitable to detect second sound waves. The literature +being vast, we only list a few thermistor materials and bibliographic entry points. Materials with a negative +temperature coefficients1 include carbon in various forms (aquadag paint, fiber, pencil graphite,..) [HVS01], +doped Ge[Sny62], RuOx [YI18], ZrNx/Cernox [YYK97, FS04] and Ge-on-GaAs [MMP+07]. Transition edge +superconductor thermometers are often preferred when large sensitivity or low resistivity is important, for +instance Au2Bi [Not64], PbSn [CR83, RR01], granular Al [CA68, MSS76] and AuSn [Not64, Lag76, BSS83]. +More information on AuSn is provided in section 2.2.1. The second sound tweezers presented in the present +study resorts to this thermal technique, both for generation and detection of second sound2. +1.3 +From macroscopic second sound sensors to microscopic tweezers +In the presence of superfluid vortices, the superfluid and normal fluid experience a viscous mutual coupling +[Don91], which entails a damping of the second sound waves. This attenuation of second sound by vortices +have been extensively used as a tool to explore the properties of He-II flows over the last 60 years[Vin57], in +particular to explore the properties of quantum turbulence (e.g. see [VJSS19]), a field of applications which +as motivated the development of second sound tweezers. For example, mechanical second sound transducers +were successfully used to study the turbulence of He-II in the wake of a grid by groups in Eugene, Prague and +Tallahassee (e.g. see [SNVD02, BVS+14, MG18]). Examples of thermal second sound transducers successfully +used to study turbulence of He-II flows are described in studies by groups from Paris, Tallahassee, Grenoble +and Gainesville (e.g. see [WPHE81, HVS92, RDD+07, YI18]). +A specific type of probe allows very sensitive probing of the density of quantum vortices in He-II flow : +standing-wave second sound resonators. Such a resonator consists in two parallel plates facing each other, one +functionalized with a second sound emitter and the other with a receiver. The emitter excites the cavity at +resonance to benefit from the amplification of the cavity. The characteristics of the standing wave between the +plates provides information on the properties of the fluid and flow between the plates, in particular the density +of vortex lines, which impacts the amplitude of the standing wave. Aside from vortex density measurements, the +second sound can also provide information on the fluid temperature -since the second sound velocity depends +on it- and on the velocity of the background He-II flow when it induces a phase shift or Doppler effect on the +second sound (e.g. see [DL77, WPHE81, WVR21]). +The characteristics listed above for the standard (macroscopic) second sound resonators remain relevant for +their miniaturized version : second sound tweezers. Beside miniaturization, a key specificity of tweezers is their +1Phosphor bronze wire, a positive temperature coefficient thermometer has also been used in the early days [Pes46]. +2In principle, mechanical and thermal techniques could be combined to generate and detect, although we are not aware of any +composite configuration reported in the literature. +3 + +low footprint on the streamlines when positioned in the core of a flow. This key differences between standard and +tweezers resonators are sketched in figure 1. Consequently, standard resonators provide information on averaged +properties of the flow, while tweezers give access to space and time resolved information. Thus, tweezers are local +probes in the same ways as the hot-wire anemometers or cold-wire thermometers used in turbulence studies. +In the present design for tweezers, the thermal actuation technique is preferred to the mechanical actuation +because it makes it easier to respect the constrains of miniaturization and reduced flow blockage. +1.4 +Overview of the manuscript +The following sections cover independent topics, +Section 3 presents a comprehensive modelling of second-sound resonators accounting for plate misalignment, +advection, finite size and near field diffraction. Diffraction, which has been neglected in previous quantitative +models, turns out to be a dominant source of degradation of the quality factor in our case studies. Applications +to the measurement of vortex concentration or velocity are considered. +Section 4 presents existing methods to process the signal from second sound resonators, and their limits. To +circumvent them, we introduce a new general approach, named the elliptic method, based on a mathematical +properties of resonance. This method allows to dynamically separate the amplitude variations of the standing +wave due to variations of vortex density or variations of velocity, from the phase variations (more precisely, from +the acoustical path variations), for instance due to variations of the second sound velocity themselves resulting +from a temperature drift. +Section 2 reports the design, clean-room fabrication and operation of miniaturized second-sound resonators, +named second-sound tweezers. These tweezers allow to probe the throughflow of helium with an unprecedented +spatial and time resolution. +For better clarity, we first present the second-sound tweezers, which allows to illustrate the topics on mod- +elling and method with a challenging practical case. Though we emphasize that the modelling and methods +introduced in this article are general and relevant to second-sound resonator regardless of their size, including +the macroscopic sensors embedded in parallel walls encountered in the literature. +2 +Design, fabrication and mode of operation of second sound tweezers +The core part of second sound tweezers is a stack composed of a heating cantilever and a thermometer cantilever, +separated by a spacer (see Figs. 2 and 3). Heaters and thermometers are cantilevers composed of a baseplate, +an elongated arm and a tip. The baseplate is the thickest part while the tip is the thinnest one. The active +areas, the emitter and receiver plates, are located on the tips. For a given device, heater and thermometer have +strictly the same mechanical structure, the only difference between them being the chemical elements used in +the serpentine electrical path deposited on the tip. Three cantilever types were fabricated in order to allow +assembling of resonators with three different tip sizes (see Fig.4). The tip widths are 1000µm, 500µm and +250µm. +Next subsection 2.1 presents considerations which prevailed in the mechanical design of the second sound +tweezers. The following one (section 2.2) discusses the detection (thermometry) and generation (heating) of +second sound by the tweezers. The last two subsections present the microfabrication techniques (section 2.3) +and the electrical circuitry used to operate the probes (section 2.4). +2.1 +Mechanical design +Resolution. The tweezers space-resolution Lres is set by the largest dimension of its cavity, which can be +either the inter-plates distance D, also called the “gap”, or the side length L of the plates here assumed to be +squared shaped. The present study mostly focuses on cavities with an aspect ratio of order 1, to benefit from +optimal space averaging of the signal at given space-resolution. The tweezers time-resolution τres is set by the +decay-time of a wave bouncing between the plates. In section 3.2, we introduce and validate a simple model +accounting for dissipation in the cavity due to diffraction loss and residual inclination of the plates. An upper +bound for τres is obtained from the diffraction loss term: τres ≃ L2f/bc2 +2, where b ≈ 0.38, c2 is the second sound +velocity and f is the wave frequency which can be approximated as nc2/2D for the nth mode of resonance (see +eq.6). Thus, the tweezers time-resolution due to diffraction loss can be estimated as +τres. ≃ L2f +bc2 +2 +≃ n L2 +Dc2 +The ratio Lres/τres of the space resolution and time resolution defines a characteristic velocity for which +the probe optimally averages the space-time fluctuations. For instance, in cavities of aspect ratio one (D = L) +operated on its nth resonance, the nominal velocity is estimated as c2/n. These estimates show that cavities +4 + +Figure 2: Second sound tweezers, pictured from two angular perspectives. (Left) Probe as seen in the direction +of the mean flow (or nearly so). The second sound cavity is localized at the upper tip of the probe, in the +encircled area. The bend copper wire through the probe is a temporary joystick used for a fine-alignment of +cavity plates under the microscope. The two coaxial cables for heating and thermometry are visible in the +lower left corner of the picture. (Right) The picture insert shows the same probe after some rotation, and after +removal of the joystick, making visible the through-hole across the silicon stack. The two staggered notches +used for thermal confinement of the standing wave in the cavity are clearly apparent. +Tip +Arm +Baseplate +Cantilever with heater + Kapton / Tracks +spacer +Cantilever with thermometer +Tracks / Kapton +Second sound standing wave +Figure 3: Schematic side view of the constitutive stack of second sound tweezers (the pieces are shown sepa- +rated for illustration, their thicknesses are exaggerated). The active areas, the emitter and receiver plates, are +constituents of the tip. +250 µm +500 µm +1000 µm +Figure 4: Top view of the 3 cantilever types. The tips widths are respectively 1000µm, 500µm and 250µm. +Left: Mechanical structures, all parts are silicon made, different thicknesses are represented by different colors. +The baseplate width is 2.5mm for all types. +Right: Electrical path on each tip type. +Yellow areas are +a deposition of TiPt for heaters and AuSn for thermometers. Orange areas are a thick AuPt deposition for +current leads. +5 + +2.5mm +Baseplate +Arm +Tip +10mm +12.5mm +2.5mmwith aspect ratio of order unity are fittingly sized for second-sound-subsonic flow, that is flows with a mean +velocity of few m/s. +Blocking effect. The above considerations on space resolution are relevant if the measured flow is not +altered by the probe support. The present design conforms to the empirical ×10 rule stating that components +of the support that obstruct the flow on a length scale X should be positioned at least 10X away from the +measurement zone. Accordingly, the cavity is at the end of elongated arms and the cantilevers are 25 mm in +length of decreasing thicknesses and widths, as illustrated by the picture of Fig.2 and by Figure 3. The thickness +successive values are around 520µm, 170µm and 20µm while the width decreases from 2.5mm to 145 µm in the +narrowest zone (resp. 275 µm and 500 µm) for cavities with L = 250 µm (resp. L = 500 µm and L = 1000 µm). +Wave confinement. The spatial resolution of tweezers would be degraded if the second sound standing +wave was spreading out of the L × L × D cavity, by reflection between the supporting arms. A design trick was +implemented to confine the standing wave in the cavity region by breaking the mirror symmetry between the +two cantilevers. Thus, as shown in Fig.2, anti-symmetric notches in the tips prevent the second sound to escape +by bouncing away from the cavity, at least in the geometric-optic approximation where diffraction is neglected. +Mechanical resonances. Besides the ×10 rule consideration, these dimensions are chosen such that the +mechanical vibrations of the arm are pushed up to ∼ 1 kHz or above. The fundamental resonance frequency of +the trapezoidal-shaped arm in vacuum was estimated from the analytical formula in [Lob07] (section 1.3.1.1). +f0 = 8.367 +2π +e +t2 +� +ESi(3w2 + w1) +ρSi(49w2 + 215w1) +We find f0 = 2195 Hz (resp. 1889 Hz and 1569 Hz) using the material properties ESi = 140 GPa, ρSi = +2330 kg/m3 and the dimensions of the intermediate section of the arm having thickness e = 172 µm, length +t = 12.5 mm, and width decreasing from w2 = 1.5 mm to w1 = 250 µm (resp. to 500 µm and 1000 µm). An +experimental validation was done at room temperature in air with an arm with w1 = 1000 µm. Its mechanical +vibration frequency spectrum was measured from a photoreceptor detecting a laser beam reflecting of the +arm. The mechanical excitation was provided either by hitting the table supporting the set-up with a small +hammer or by pointing a jet of compressed air toward the arm. In both cases, the fundamental mechanical +resonance frequency was found to be 1215 Hz, in reasonable agreement with the 1569 Hz prediction given the +uncertainty on the Young modulus and deviations from the trapezoidal shape. As discussed in 4.4.3, indirect +measurements of the resonance frequency were done in 1.2 m/s superfluid flow and gave f ≈ 825 Hz, f ≈ 1050 +Hz and an amplitude of vibration smaller than 1 µm. The decrease in frequency compared to room-temperature +measurement is interpreted as mostly due to a fluidic added mass effect[Sad98]. +Deflection of the tips’ ends. The thicknesses of the tweezers parts are such that the mechanical deflection +at the tip endpoint remains significantly lower than the inter-plate distance under typical operating conditions. +The deflection at the tip endpoint can be estimated by considering separately the arm deflection (with +thickness 172µm) and the tip deflection (with thickness 20µm). As a first approximation, both arm and tip +are considered as cantilever beams of uniform width submitted to a uniformly distributed load, and having one +embedded end and one free end. This geometrical approximation overestimates the deflection of the arm, as its +endpoint is narrower than its base, and it underestimates the deflection of the tip, as the notch is ignored. Still, +this is enough to get order-of-magnitude estimates. The load is estimated as the dynamic pressure of a liquid +helium flow impinging the tweezers in transverse direction at a velocity U=0.1 m/s, that is 10% of the typical +longitudinal flow velocity of 1 m/s. The dynamic pressure P is taken as: +P = 1 +2ρU 2 +where the liquid helium density is ρ ≃ 140 kg/m3. According to Euler-Bernouilli beam theory, the free end +deflection δmax of the cantilever is: +δmax = 3 +2 +Pt4 +ESi.e3 +The total deflection (arm and tip) is upper-bounded by considering the sum of the deflection of a 2.5mm +long tip and a 15mm (not 12.5mm) long arm. This way, the small angle generated on the tip by the arm +deflection is taken into account. Using the values t = 15mm (length), e = 172 µm (thickness) and E = 140GPa +(Young modulus), the deflection of the arm endpoint is found to be 75nm. The deflection of the tip endpoint +with t = 2.5mm and e = 20 µm gives a 37nm deflection Thus, the total mechanical deflection of the tweezers +tip due to a steady lateral flow of 0.1 m/s is a fraction of a micron, that is decades smaller than the inter-plate +distance. +The mechanical resonance of the tweezers arm and tip, discussed above, could lead to deflections larger than +the one due to a steady forcing. The amplitude of those mechanical oscillations was measured in a turbulent +He-II flow up to velocities exceeding 1 m/s, taking advantage of the dependence of the second sound resonance +6 + +with respect to the cavity gap. The measured signal will be presented to illustrate the efficiency of the elliptic +projection method in separating the fluctuations of the acoustical path of the cavity and fluctuations of the +bulk attenuation of second sound between plates. The mechanical oscillations of the cavity gap are found to be +typically 0.5 µm (around 1 kHz). As expected, such a deflection is decades lower than the interplate distance +(1.3 mm in this case) and than the second sound wavelength. +Boundary layer. In the presence of a mean flow through the cavity, a velocity boundary layer will develop +along each tweezers plate. In principle, this boundary layer could contribute to the measured signal and therefore +alter the measurement of the incoming flow, for instance by increasing the density of superfluid vortices in the +cavity and therefore second sound attenuation. As illustrated later, the second sound standing wave that settles +between the plates have nodes of velocity near the plates (e.g. see fig. 10 and fig. 16) while the sensitivity +of second sound to vortices arises in antinodal regions of velocity. As long as the boundary layer thickness is +thin enough, say within a fraction of λ/4 (λ = c2/f is the second sound wavelength), it is not expected to alter +significantly the measured signal. +A first requirement for this condition is that the mean flow direction is parallel to the plates so that the flow +penetrates through the cavity with minimal deflection. A consequence is that plates should be widely separated +when operated in flows with undefined or zero mean velocity, such as the core of a mixing layer. +A second requirement is that the plate thickness is much thinner than λ/4. Present plates are 20 microns +thin, to be compared with λ/4 ≃ D/2n for the nth mode of resonance. For instance, with D = 500 µm and +n = 3, the condition 20 ≪ λ/4 ≃ 83 µm is indeed well satisfied. +A third condition regards the downstream development of the boundary layer thickness, which should also +stay well within λ/4. The physics of boundary layers in He-II is ill-understood[SPB17] but existing experiments +(e.g. +see [SHVS99]) suggests that classical hydrodynamics phenomenology could remain valid in the high +temperature limit. +In classical hydrodynamics, the so-called displacement thickness of a laminar “Blasius” +boundary layer at distance L from its origin is given by +δbl = 1.73 +� +Lν +U +where U is the mean velocity far from the boundary layer and ν is the kinematic viscosity of the fluid. In +He-II, several diffusive coefficients could arguably play the role of ν, in particular the quantum of circulation +around a quantum vortex and the kinematic viscosity associated with the dynamics viscosity of the normal fluid +normalized either by the normal fluid density or by the total density. In the temperature range of interest, all +these diffusive coefficients are within one order of magnitude, typ. 10−8 − 10−7m2/s. Taking ν = 3.10−8 m2/s, +L = 1000 µm and U = 0.5 m/s, one finds δbl = 13 µm, and a boundary layer Reynolds number δblU/ν = 217 +consistent with the laminar picture. This thickness estimate, similar in magnitude with the plate thickness, +satisfies the third requirement δbl ≪ λ/4. +2.2 +Second sound detection and generation +2.2.1 +Thermometry +The temperature-sensitive material used in the present study is AuSn, which fulfills two requirements: (1) it +is compatible with the microfabrication process and (2) it can be tuned to become temperature-sensitive over +a range of special interest to quantum turbulence studies[WVR21], from 1.5 K up to the superfluid transition +temperature Tλ ≃ 2.18K in saturated vapor conditions. Surely, other materials would be more appropriate in +other conditions, e.g. Al has been used previously for the tweezers operated around 1.5 K in [RDD+07]. +The gold-tin AuSn thermometer is a metal-superconductor composite material, with superconducting Sn +islands electrically connected by a gold layer. This granular structure is imaged by electronic microscopy in +Fig.5 (right). The temperature dependence phenomenology can be interpreted in a simple way. Indeed, by +proximity effect, the gold in contact with tin behaves as a superconductor over a spatial extent which depends +on temperature : by adjusting the characteristic length scales and thicknesses of the granular pattern, the +temperature response of the material can be tuned. +As a preliminary study, the temperature of the resistance of a 100 squares long AuSn track was tested for +three different tin thicknesses, as illustrated on the left plot of Fig.6. A description of the conduction mechanism +in AuSn is presented in [BSS83]. +In the present study, AuSn layers are deposited by evaporation with successively a small erosion of the +substrate by argon ion bombardment during 20s then the deposition of a 25nm gold layer and a 100nm tin +layer (hypothetic thickness for a planar - not granular - layer). The thermometer is shaped into a meander +deposited by lithographic technique on the tip of tweezers arm, as pictured in Fig.5 (right plot). The total +resistance at superfluid temperatures doesn’t exceed a few hundreds of ohm, a value chosen to be much larger +than the resistance of the leads but small enough to prevent parasitic effect from the leads’ capacitance (typ. +few hundreds of picoFarads) up to the highest frequencies of operation. +7 + +Figure 5: +Left: Tip of tweezers arm as seen from the facing arm. +The thin meander on the top is the +active heater (Pt) or thermometer (AuSn). +The overlap with gold tracks is apparent on the lower part of +the picture Right: Close up picture by electronic microscopy of the AuSn thermometer showing its granular +aspect. Depending on tweezers models (see fig. 8), the width of the AuSn track is 4, 11 or 24 µm +In order to obtain resistance values in this range, the meander length was fixed close to 700 squares for all +tip sizes. According to the tip size, the track width was adapted so as the serpentine shape occupy the whole +available area on the tip. At ambient temperature, the AuSn layer resistance was found to drift from low values +to their final values during a few days (less than one week) after deposition. After this period, the resistance +was found to be stable at least over 6 months. +Figure 6 (right plot) shows a typical AuSn thermometer resistance R(T0) versus temperature T0 for different +direct current I. Regarding the temperature dependence of resistance, the current density is a more determining +parameter than the total current. Thus, the comparison between left and right plots of figure 6 should be done +at constant values of the ratio of current over track width. At low current density (I � 10µA), the sensitivity +exceeds 1 Ω.mK−1. +At larger current densities, the current-induced magnetic field significantly shifts the +superconducting-metal transition to lower temperature and broaden it, allowing to extend the measurement +range down to 1.6 K and below. +In the range of currents explored in Fig. +6, the reduction of sensitivity +in Ω.K−1 at larger current I is more than compensated by the larger sensitivity in V.K−1 units across the +thermistor. Most measurements presented hereafter are done with a polarization of I ≃ 27 µA. +2.2.2 +Heating +The heater consists in a meander of metal deposited by lithographic technique on the tip of a cantilever, alike +the thermometer (see fig. 5 (left)). The same lithographic mask was used, and therefore the meander length is +also close to 700 squares for all the tip sizes. +As can be seen on figures 4 and 5, a buffer zone was designed between the gold tracks and the meander. Into +this zone, the electrical path is wide but the material is the same as in the meander (platinum for heater). The +buffer zone length is approximately 20 squares. This design aims at providing some thermal isolation between +the meander and the gold track. +Numerous resistive materials are suitable, e.g. chrome was used for the tweezers in [RDD+07] and platinum +in [WVR21]. Present data have been obtained with platinum to benefit from the temperature-independence of +its resistivity at superfluid temperatures [PK82], and nevertheless to allow re-use of these miniature heaters as +miniature thermometers or hot-film anemometers in experiments conducted at higher temperatures where Pt +regains temperature dependence [Kem91]. A 5nm titanium layer was deposited prior to platinum as an adhesion +layer. +The thickness of the Pt layer, around 80 nm, was chosen such that the electrical resistance of the heater at +superfluid temperature is around a few hundreds of ohms, like for the thermometer maximum resistance and +for the same reasons. +The heater is driven with a sinusoidal current at frequency f/2. The resulting Joule effect can be decomposed +into a constant mean heating and the sinusoidal heat flux at the frequency f that drives the second sound +resonance. A benefit of this f/2 excitation is that the signal monitored by the thermometer - centered around f +- is not spoiled by spurious sub-harmonic electromagnetic coupling at f/2 from the excitation circuitry. Thus, +no special care is needed to minimize the electromagnetic cross-talk between the electrical tracks of the heater +and the electrical tracks of the thermometer, despite their proximity. +The non-zero mean heating results is a steady thermal flux in He-II, the corresponding entropy being carried +8 + +10μm +2 + +2.2 + +2.4 + +2.6 + +0 + +0.2 + +0.4 + +0.6 + +0.8 + +1 + +Au 25nm + Sn 90nm + +Au 25nm + Sn 100nm + +Au 25nm + Sn 110nm + +T +λ + transition under saturated vapor +R0 / max(R0) +T0 (K) + +1.7 + +1.8 + +1.9 + +2 + +2.1 + +2.2 + +0 + +100 + +200 + +300 + +1 +μ +A +d +c + + 1 +μ +A +a +c + +3 +μ +A +d +c + + 1 +μ +A +a +c + +10 +μ +A +d +c + + 1 +μ +A +a +c + +20 +μ +A +d +c + + 1 +μ +A +a +c + +30 +μ +A +d +c + + 1 +μ +A +a +c +R0 (Ω) +T0 (K) +Figure 6: +Left: Example of temperature and current dependence of AuSn layers with different Sn thicknesses +deposited on a track of width 50µm . The current polarization spans from 1µA up to 1mA. +Right: Tempera- +ture response of the AuSn track of small tweezers for different electrical currents. The thickness of the tin layer +is 100 nm of Tin, and the width of the track is 4µm. The small difference between both plots -when compared +at similar current densities- is compatible with the uncertainty on the layer thicknesses. This good agreement +indicates that AuSn properties are robust to the full fabrication process of the tweezers. No significant aging of +AuSn properties has been noticed over a few years period. +away from the heater in the form of steady normal fluid flow. This outgoing normal flow is balanced by an +opposite steady mass flow of superfluid towards the heater. Such cross-flows are referred to as counterflows in +the quantum fluid literature[Tou82, NF95]. This steady counterflow adds up to a pure second sound generated +by the heater, but -contrary to it- its effects are not amplified by resonance in the cavity. +Quasi-linear vs non-linear regimes. The second sound resonators are operated with standing waves +of low amplitude, say ∼ 100 µK. +In this limit, the amplitude T of the temperature standing wave nearly +responds linearly to the heating power P. For larger heating power, the ratio T/P decreases with P, which is +interpreted as the result of a turbulent transition within the tweezers which fuels a dense tangle of quantum +vortices dissipating the second sound wave3. The crossover from the quasi-linear to non-linear response of T(P) +is illustrated by figure 7 (left plot) for tweezers at 1.6 K in the absence of external flow. In these conditions and +for these tweezers, the transition occurs around P ≃ 1 W/cm2, where P is the total Joule power normalized +by the heating surface. In other conditions, this transition was observed at smaller power densities, but no +systematic study was carried on the threshold value. +2.2.3 +Digression on the operation in the non-linear heating regime +The present study focuses on the linear regime of heating, but a short digression on higher powers allows +uncovering a noteworthy property of non-linear operation and incidentally backing-up the above interpretations +of the nature of the non-linear regime. Figure 7-right displays the amplitude of the (normalized) temperature +standing wave T/P versus P in flows of different mean velocities U and turbulence intensity of few percents. In +the linear regime (P � 1W/cm2 in the conditions of Fig. 7), the plateaus of T/P decreases when U increases. +Following the classical interpretation (see later), the standing wave T is damped by the vortices present in the +external flow, which concentration increases with U. Interestingly, in the non-linear regime (say for P � 2W/cm2 +in Fig. 7), the dependence of T/P versus U is opposite. The interpretation is that the extra damping of the +3In one dataset, a close look at the quasi-linear region in quiescent superfluid around 1.5 K evidenced a small discontinuity of +T/P versus P dependence around P ⋆ ≃ 10−2W/cm2 (not shown), suggesting another flow transition, but this small effect was +not detectable in other datasets. A Reynolds number of this possible transition can be defined with the transverse characteristic +length scale L ≃ 1 mm, the quantum of circulation κ ≃ 0.997.10−7m2/s and the counterflow superfluid velocity towards the +heater vs ≃ 0.3 mm/s (amplification of velocity by the quality factor of the cavity has not been taken into account). One finds +Res = vsL/κ ≃ 3. Critical Reynolds numbers Res of a few units have already been reported to characterize the threshold of +appearance of a few superfluid vortices across the section of pipes that are closed at one of their ends with a heating plug (see +fig. 3 in [BLR17]), a transition referred as the T1-transition in the counterflow literature [Tou82, NF95]. By analogy, this could +suggest that the discontinuity at P ⋆ might be associated with the appearance of a sparse tangle of the quantum vortices near +the heater, which density is expected to increase for larger P. Such vortices would damp the standing wave, but no such effect +has been detected. Indeed, first the observed damping of the standing wave in quiescent He-II can be accounted for by the sole +effect of diffraction (as shown later), which indicates that all the other sources of loss are comparatively small. Second, loss due to +such “counterflow” vortices would make the T(P) dependence sub-linear rather than linear, which is not clearly observed. Since no +quantitative evidence of these vortices could be clearly identified, this effect was not further explored. +9 + +10-1 +100 +P (W/cm2) +0.8 +1 +1.2 +1.4 +1.6 +1.8 +2 +2.2 +T/P (a.u) +0.5 +1 +1.5 +2 +2.5 +3 +3.5 +P (W/cm2) +-1 +0 +1 +2 +3 +X (mK) +-2 +-1 +0 +1 +2 +3 +Y (mK) +10-1 +100 +P (W/cm2) +0.8 +1 +1.2 +1.4 +1.6 +1.8 +2 +2.2 +T/P (a.u) +Overheating +0 +0.2 +0.4 +0.6 +0.8 +1 +U (m/s) +Figure 7: +Left: Normalized amplitude T of the temperature standing wave versus heating power P for +second sound tweezers in a quiescent He-II around 1.6 K. The transition around P = 1 W/cm2 is interpreted +by the development of a self-sustained vortex tangle within the probe. The insert shows the amplitude of the +temperature standing waves in the complex plane for a subset of frequencies belonging to the same second sound +resonance. In this representation, the extra-dissipation associated with the self-sustained vortex tangle results +in a curvature of the iso-frequency radial “lines” revealing the broadening of the resonance. +Right: Same +quantity for second sound tweezers swept by turbulent flows of different mean velocities but similar turbulence +intensity. In the linear regime P � 1 W/cm2, the velocity dependence is opposite to the one in the non-linear +regime, P � 2 W/cm2, demonstrating respectively vortex and velocity sensing by the probe. +standing wave T now mostly results from the vortices generated within the tweezers by the heating itself. This +vortex density decreases at larger U because vortices are more efficiently swept out of the tweezers. In the +non-linear regime, the second-sound tweezers thus behave as a local anemometer. In the linear regime, we will +show that second-sound tweezers can not only behave as vortex probes -as illustrated by Fig.7-right but also as +anemometers through a mechanism discussed later. +2.3 +Microfabrication and assembling +Mechanical and electrical assembly. +The distance between the plates is set by the spacer, composed of one or several micro-machined silicon +elements. Additionally, two Kapton films with golden copper tracks are inserted in contact with the gold tracks +of the heater and thermometer. The cantilevers, Kapton films and spacer elements are stacked on each other +as shown in Fig.3. The resulting assembly is clamped with a standard picture clip, downsized by electro-wire +erosion, and soldered to the head of a mounting screw. An improvement compared to the clamping technique +introduced in [RDD+07] is the possibility to insert a temporary “joystick” through the whole assembly to allow +precise alignment (or offsetting) of the cavity plates under microscope (see Fig.2-a). +All the mechanical structures of cantilevers and spacers are made of silicon. The cantilevers are fabricated +by processing SOI (Silicon On Insulator) substrates by microelectronic techniques. The substrates diameter +is 100mm, the silicon substrate, oxide and device thicknesses are respectively 500µm, 1µm and 20µm. The +substrates are double side polished. The wafers are oxidized so as to form a 100nm thick SiO2 layer on both +sides. Distinct wafers are used to fabricate heaters and thermometers but the process differs only in the metals +used for tips electrical paths. By using a circular symmetry design, 46 cantilevers are made per wafer. +The cantilevers’ fabrication process is presented in table 1. The serpentine electrical path (red color) is +deposited first on the SOI wafer frontside. The deposition is done using standard photolithography, evaporation +and lift-off sequence. The photoresist used is AZ 5214E from Microchemicals GmbH, processed as a negative +photoresist. Depending on the cantilever type, heater or thermometer, two different evaporation sequences are +used: Ti 5nm + Pt 80nm or Au 25nm + Sn 100nm. Evaporation is preceded by in situ wafer surface cleaning by +an argon ions bombardment during 20s. Lift-off is initiated in an acetone bath during 5min and then completed +by ultrasounds during a few tens of seconds. The current leads (orange color) are deposited during a second +photolithography, evaporation and lift-off sequence. The evaporation sequence is Ti 5nm + Au 200nm + Ti +5nm + Pt 50nm. The usage of a platinum layer was found to facilitate lift-off and may also be useful for brazing +purpose. A thin protective resist layer is deposited on frontside in order to protect it during all subsequent +operations on the backside. +10 + +Figure 8: Overview of mask design (the disk diameter is 100mm). +Side +Step +Surface material +Design +Frontside +1. Electrical circuitry fabrication. +TiAuTiPt (orange) +AuSn/TiPt (red) +Backside +2. Aluminum mask fabrication. +Aluminum +Backside +3. Resist mask fabrication. +4. Etching of surface oxide and +silicon. +5. Resist removal. +Photoresist +Backside +6. Etching of surface oxide and +silicon until buried oxide is reached. +7. Etching of buried oxide. +8. Aluminum removal. +Aluminum +Frontside +9. Resist mask fabrication. +10. Etching of surface oxide and +silicon until opening. +Photoresist +Table 1: Cantilevers fabrication process. +11 + +Phase +Deposition +Etch +Sub-phase +Main +Boost +Main +Gas +C4F8: 250 sccm +SF6: 250 sccm +O2: 45 sccm +SF6: 450 sccm +O2: 45 sccm +Duration +2.2 s +2.0 s +5.5 s +Pressure +14 mTor +20 mTor +75 mTor +Coil power +1200 W +1780 W +1780 W +Platen power +20 W +110 W +50 W +Electromagnet current +0 A +0 A +2 A +Platen frequency +RF 13.56 MHz +He backside pressure +10 Tor +Table 2: Bosch process recipe used during deep silicon etching on backside. +The cantilevers 3D structuration starts with a backside deep etching of silicon. Two superimposed etch +masks are fabricated first, one aluminum mask and one photoresist mask deposited over the aluminum one. +The aluminum mask is made in the same way as the frontside electrical paths. A backside alignment is necessary +during photolithography. The aluminum thickness is 120nm. The protective resist layer on frontside had to be +deposited again after lift-off. The resist mask is made by photolithography on the positive AZ4562 photoresist +spin-coated at 4000rpm. The aluminum mask is identical to the resist mask except within the arm area, as +shown in table 1. This area is covered by resist but not by aluminum. Thus, the resist fully covers aluminum. +After the fabrication of both masks, the deep etching is started, with the resist mask protecting both silicon +oxide and aluminum surfaces. The surface oxide is etched first then the solid silicon. The thin oxide layer is +etched by reactive ion etching (RIE) based on SF6 gas. The silicon was etched in a STS HRM deep reactive +ion etching (DRIE) equipment using a recipe based on Bosch process. The recipe is presented on table 2, the +etch duration is 120 cycles. As shown on table 1, a 200µm wide trench is dug from the backside around the +baseplate and arm areas. The tip area is fully exposed to etching as this part was extracted from the device +layer of SOI only. The longer this phase, the thicker the cantilever arm at the end of process. +Following this first etch phase, the resist is removed by an oxygen plasma and the backside surface oxide is +etched into the freshly uncovered arm area. Then, another silicon etch sequence is applied with the remaining +aluminum mask. Its objective was to thin the arm and to reach the buried oxide of SOI in the trench, at the +same time. The same recipe is used, 200 cycles are applied. The Bosch process is interrupted 3 times, every 50 +cycles, in order to apply a 1min oxygen plasma followed by 20s of an isotropic silicon etching recipe (plasma +pressure 75mTor, SF6 flow 450sccm, O2 flow 45sccm, coil power 1780W, platen power 50W). The objective is +to reduce the parasitic effect generated by the passivation layer deposited on arm sidewalls during the first deep +silicon etching sequence. As the arm area is masked during the first silicon etch sequence and unmasked during +the second one, the passivation layer located along the arm edges is released and could generate locally some +irregular micromasking effect. The oxygen plasma is intended to help remove the floating passivation films. The +isotropic silicon etching is intended to cut the silicon pillars generated by micromasking. The final 50 cycles +of the Bosch process recipe end up reaching the SOI buried oxide layer, at the trench bottom, all around the +cantilever (however the oxide should not be reached into the arm area). It is necessary at this step to check that +the buried oxide is fully uncovered by silicon everywhere in the trench bottom and in the tip area. However, +etching cycles should not be applied in excess so as to avoid some mechanical weakening of the wafer. At this +step, the arm thickness has its final value. The buried oxide is then removed by plasma etching. This oxide +is fully removed at the trench bottom and in the tip area. If this layer is not removed, some bending may +occur on the tip at the end of process, due to mechanical stress of oxide. The aluminum mask is removed in +an aluminum etchant solution at 50°C during a few minutes. The frontside protective resist layer is removed +during an acetone cleaning bath. +The 3D cantilever structuration is ended by a third silicon etching made from the frontside. The etch mask +is formed by photolithography on AZ1512HS resist, deposited at 4000rpm. As shown on table 1, the mask +design includes two bridges on the baseplate sides in order to maintain the cantilever after having opened the +trench that surrounds it. The design also includes the tip contour. The frontside thin surface oxide is etched +first, then the silicon of the device layer. The silicon etching is done with specific conditions. Due to the wafer +mechanical weakness at this step, caused by the multiple deep trenches made on the backside, the processed +wafer is layed on and attached to a blank silicon wafer by Kapton tape. This ensemble is loaded into the etching +chamber. As no thermal bridge is present between the two wafers, the recipe is adapted: low RF powers are +used and a 22s idle time is added after each etching cycle (see table 3). The objective is to avoid overheating +during etching. The silicon etch duration is 50 cycles. +After this sequence, the trench around cantilever is fully opened. The resist is removed by a low power oxygen +plasma. Some spacers are fabricated together with the cantilevers but most of them are made separately from +12 + +Phase +Deposition +Etch +Sub-phase +Main +Delay +Boost +Main +Gas +C4F8: 250 sccm +SF6: 250 sccm +O2: 10 sccm +Duration +3 s +2.0 s +5.5 s +22.5 s +Pressure +14 mTor +20 mTor +40 mTor +40 mTor +Coil power +300 W +300 W +300 W +1 W +Platen power +20 W +100 W +50 W +1 W +Electromagnet current +0 A +0 A +2 A +0 A +Platen frequency +RF 13.56 MHz +He backside pressure +10 Tor +Table 3: Bosch process recipe used during deep silicon etching on frontside. +two silicon wafers with thicknesses 300µm and 525µm. These wafers are covered by thin dielectric layers on +both sides. +2.4 +Electric circuit +Figure 9 shows a circuit used for time-resolved measurements with second-sound tweezers, with example values +of resistances and gain. The time-resolved data presented in this paper have been obtained with such a circuit +and using the following equipment. The front-end of the preamplifier is the EPC1-B model by Celian or an SA- +400F3 model by NF when frequencies above 100 kHz are explored. The lock-in amplifier is a LI-5640 by NF or +Model 7280 by SignalRecovery above 100 kHz. In most conditions, its built-in internal generator provides both +the drive of the tweezers heater (at frequency f/2) and the reference frequency to detect the temperature signal +(at frequency f). The acquisition system is built around PXI-4462 analog input cards by National Instrument, +and it records both the in-phase (X) and quadrature (Y ) signal from the analog outputs of the lock-in amplifier. +In occasional conditions, the temperature signal at the lock-in input is buried in a much larger electro- +magnetic parasitic signal at f/2, and it cannot be properly resolved by the limited voltage dynamic range of +the lock-in amplifier. This situation can occur when the tweezers are operated far from a resonance, where +the second sound signal is small, or when the tweezers are operated at very high frequency (say > 100 kHz), +as electromagnetic coupling increases with frequency. The magnitude of this parasitic coupling depends on +the geometrical and electrical specificities of the tweezers and cables. In the range of parameters explored in +the present study, the order of magnitude of the parasitic voltage induced across the thermometer resistor, +normalized by the voltage applied across the heating resistor, is +0.5% × +f/2 +100kHz +Such situations are handle thanks to the differential input of the lock-in amplifier, by removing a signal +mimicking the parasitic one. In such cases, a two-channel waveform generator is used: one channel driving the +heater (at frequency f/2), another channel mimicking the parasitic signal (at frequency f/2, with manually +tuned amplitude and phase shift) and the "sync" output of the generator synchronizing the lock-in demodulation +(at frequency f). The 33612A generator by Agilent is used for this purpose. Alternatively, the compensation +signal can be generated directly from the lock-in internal generator, completed with a simple RC phase shifter +and eventually ratio transformer. +In principle, any positive temperature coefficient thermistor -like AuSn thermometer- that is not well ther- +malized with the fluid can become unstable when driven by a current source. Indeed, an infinitesimal thermistor +fluctuation from T0 to T0 + δT0 entails a resistance variation of δR = ∂R/∂T0.δT0 > 0, leading to an excess of +Joule dissipation δR.I2 for a constant current drive I. Calling Rth is the thermal resistance of the thermistor- +fluid interface, this extra Joule dissipation results in an overheating Rth × δR.I2 which could lead to a thermal +instability. The stability condition is difficult to predict for spatially distributed thermistor deposited on a +Si crystal and immersed in superfluid. Thus, first tests have been done with a voltage polarization, before +validating empirically the stability of our current polarization. +The frequency bandwidth of the measurements is arbitrarily set by the integration time constant of the +lock-in amplifier. In practice, the performance of the circuit are limited by the 0.65 nV/ +√ +Hz input voltage +noise of the EPC1-B pre-amplifier, all other sources of noise being smaller. For a 27 µA current polarization, a +thermometer sensitivity of 0.5 Ω.mK−1, and a 10 Hz or 1000 Hz bandwidth of demodulation, the temperature +resolution Trms is : +Trms = +√ +10.0.65 +0.5 × 27 µK ≃ 150nK for a 10 Hz measurement bandwidth +13 + +300 kΩ +acquisition +system +10 kΩ +400 Ω +Tweezers heater +Reference +clock +9V batteries +quadrature +phase +Optional noise compensation +Low noise AC preamp +(40 - 50 dB) +Lock-in amplifier +(at freq. f ) ++ +- +Freq. f/2, phase φ +r1 +r2 +Freq. f/2 +Tweezers thermometer +(0 - 300 Ω) +R(T0) +Current source (typ 30 µADC) +Figure 9: Example of circuitry for measurements with a high dynamical reserve. +or +Trms = +√ +1000.0.65 +0.5 × 27 +µK ≃ 1.5µK for a 1 kHz measurement bandwidth +These resolutions are sufficient in standard conditions. Indeed, they are respectively 3 and 2 decades smaller +than the typical amplitude of a second sound at resonance, and reaching the same temperature resolution at +significantly larger bandwidth would be useless given the space-time resolution of the probe itself. If needed, +better resolution could nevertheless be achieved with a larger polarization across the thermometer or using a +cryogenic amplifier (e.g. see [DLF+14] and http://cryohemt.com) before being limited by the thermal noise +floor of the thermistor (typ. 0.15 nV/ +√ +Hz for 200 Ω at 2 K). +3 +Models of second sound resonators +The second sound equations within the linear approximation can be written in terms of the temperature fluc- +tuations T and the velocity of the normal component vn as +∂tvn + σρs +ρn +∇T = 0, +(1) +∂tT + σT0 +cp +∇.vn = 0. +with σ the entropy per unit of mass, cp the heat capacity, and ρs , ρn are the densities of the superfluid and +normal components respectively. All along the present section, T0 is the notation for the bath mean temperature +whereas T denotes the temperature fluctuations, with ⟨T⟩ = 0. +We introduce the second sound velocity c2 defined by the relation +c2 +2 = ρs +ρn +σ2T0 +cp +. +(2) +It can be deduced from Eqs. (1) that both the temperature T and the normal velocity vn follow the wave +equation +∂2 +t T − c2 +2∆T = 0. +(3) +We explain in the present section how Eqs. (1-2-3) can be used to build quantitative models of second sound +resonators. We first focus on phenomenological aspects in sec. 3.1. Then, we give analytical approximations in +sec. 3.2 and an accurate numerical model in sec. 3.3. Finally, we discuss the model quantitative predictions in +secs. 3.4 and 3.5. +14 + +Heater +Thermometer +z=0 +z=D +Figure 10: Schematic representation of the second resonant mode of a second sound cavity. The red curve +displays the temperature field at time t = 0, and the blue curve displays the normal fluid velocity vn at time +t = +1 +4f , where f is the excitation frequency. +3.1 +Resonant spectrum of second sound resonator: phenomenological aspects +The basic idea of second sound resonators is to create a second sound resonance between two parallel plates +facing each other. A second sound wave is excited with a first plate, while the magnitude and phase of the +temperature oscillation is recorded with the second plate used as a thermometer (see Fig. 10). For simplicity, +we assume from now on that the second sound wave is excited by a heating, but the whole discussion can be +straightforward extended to nucleopore mechanized resonators. The temperature oscillations within the cavity +are coupled to normal fluid velocity oscillations according to the second sound equations (1). Both components +oscillate in quadrature, which means that they reach their maximal amplitude with a +1 +4f time shift (Fig. 10). +We note jQ = j0e2iπft the periodic component of the heat flux emitted from the heater. +We assume +throughout the present article perfectly insulating plates, which means that the boundary conditions for the +second sound wave are +vn.n = +� +0 +for z = D +jQ +ρσT0 +for z = 0 , +(4) +where n is the unit vector directed inward the cavity and normal to the plates. The second equation in (4) +reflects the fact that the normal component carries all the entropy in the fluid. As illustrated in Fig. 10, +the thermometer plate is a node of the normal velocity oscillation, whereas the normal velocity amplitude only +vanishes on the heater plate in quadrature (t = +1 +4f + n +2f ). According to the first relation in Eq. (1), the boundary +conditions (4) for the normal velocity translate into the following boundary conditions for the temperature field +∇T.n = +� +0 +for z = D +− ρn∂tjQ +ρρsσ2T0 +for z = 0 . +(5) +In particular, the thermometer plate is an antinode for the temperature. +We display in Fig. 11 a typical experimental spectrum of second sound tweezers, that is, the temperature +magnitude averaged over the thermometer plate, as a function of the heating frequency f. The spectrum is +reminiscent of a Fabry–Perot resonator (described in sec. 3.2): it displays very clear resonant peaks almost +equally separated, and a stable non-zero minimum at the non-resonant frequencies. However, the spectrum of +Fig. 11 displays three major characteristics that can be observed for every tweezers spectrum. First, the location +of the resonant frequencies are slightly shifted compared to the standard values fn given by 2πfnD +c2 += nπ, (n ∈ N) +expected for an ideal Fabry–Perot resonator. Only for large mode numbers do the resonant peaks again coincide +with the expected values. Second, the temperature magnitude vanishes in the zero frequency limit, and the +first modes of the spectrum grow linearly with f. In between, the resonant amplitudes saturate and then slowly +decrease at high frequency. +These latter peculiarities of the frequency response were not described in previous references about second +sound resonators. This prompted us to study different models for second sound resonators, including the finite +size effects and near field diffraction phenomena. +We first describe analytical approximations in sec. +3.2, +then we develop in sec. 3.3 a numerical algorithm based on the exact solution of the wave equation (3). The +numerical scheme can be adapted for various types of planar second sound resonators. We then give quantitative +predictions specifically for the response of second sound tweezers without and in the presence of a flow in sec. +3.4 and a summary of the main results in sec. 3.5. +3.2 +Analytical approximations +The starting point to build our model of second sound tweezers is to assume that all zeroth order physical +effects observed with the tweezers are geometrical effects of diffraction. +This means in particular that we +assume perfectly reflecting resonator plates, and we also neglect bulk attenuation of second sound waves when +15 + +0 +10 +20 +30 +40 +50 +60 +70 +f (kHz) +0 +0.2 +0.4 +0.6 +0.8 +1 +T (K) +10-4 +Linear growth of +the first modes +Stable baseline +Decrease at high +frequency +Figure 11: Experimental spectrum of second sound tweezers of lateral size L = 1 mm and gap D ≈ 1.435 mm. +f is the heating frequency, and T is the thermal wave magnitude. The figure displays the main characteristics +of tweezers typical spectrum: first, the resonant frequencies are not located at the values fn given by 2πfnD +c2 += +nπ, (n ∈ N), displayed by the gray vertical lines. The amplitudes of the resonant modes first increases linearly +with f until they saturate and eventually decrease at high frequency. The baseline level only weakly depends +on f. +the fluid is at rest [CR83, RG84]. These assumptions turn to be self-consistent, because the predictions of +the model developed in sec. 3.3 reproduce the main features observed in experiments. We thus start with +the simplest model of a resonant cavity for planar waves, namely the Fabry–Perot model. We then propose +variations of the Fabry–Perot model, taking progressively into account the particular resonator geometry. We +discuss the predictions of these models and their relevance for our second sound tweezers. +The standard Fabry–Perot model +The Fabry–Perot model corresponds to a one-dimensional resonator +composed of two infinite parallel plates separated by a gap D. In that case, the wave Eq. (3) together with the +boundary conditions Eqs. (5) can be solved exactly, for a periodic heating jQ = j0e2iπft . However, as there is +no energy loss between two perfectly insulating and infinite plates, a bulk dissipation has to be introduced by +hand in the model, to balance energy injection from the heater. This can be done with an additional dissipation +coefficient ξ (in m−1). The temperature at the thermometer plate is T(t) = Re +� +Te2iπft� +(where Re is the real +part operator) with the complex wave amplitude T given by +T = +A +sinh +� +i 2πfD +c2 ++ ξD +�, +(6) +with A = − +j0 +ρcpc2 . An illustration of a Fabry–Perot spectrum is displayed in grey in Fig. 14, with ξD = 0.15 +and A = 1. We introduce the wave number k = 2πf +c2 . It can be proved from Eq. (6) that the spectrum maxima +are reached for the values knD = nπ, (n ∈ N), and correspond to constructive interferences in the cavity. The +baseline level is set by the minima reached for the values knD = nπ + π +2 , (n ∈ N), and correspond to destructive +interferences in the cavity. For the simple Fabry–Perot model of Eq. (6), all the resonant peaks have equal +height and are uniformly separated. Therefore, some main features of experimental spectra are missing, an +indication that important other physical effects have to be included in the model. +Second sound resonators embedded in infinite walls +A possible modification of the Fabry–Perot res- +onator is to consider finite-size heater and thermometer of size L embedded in two parallel and infinite walls +facing each other. This geometry is most commonly encountered in the literature. With such a configuration, +16 + +the thermal wave is not a plane wave any more because it is emitted by a finite size heater. The model thus +contains diffraction effects, that the simplest Fabry–Perot resonator do not display. An illustration of the model +setup is displayed in Fig. 12. An exact solution of the wave equation Eq. (3) can be found using the technique +of image source points. Let Σ1 be the heating plate and Σ2 be the thermometer plate, and we assume that the +thermometer is sensitive to the average temperature over Σ2. Then the response of the tweezers is given by +T(t) = Re +� +Te2iπft� +with +T = +ikj0 +2πρcpc2 +1 +L2 +� +Σ2 +d2r2 +� +Σ1 +d2r1G (r2 − r1) , +with the Green function G(r) defined for every vector r in the (x, y) plane +G(r) = 2 ++∞ +� +n=0 +1 +|(2n + 1)Dez + r|e−ik|(2n+1)Dez+r|. +Such a model correctly predicts that the tweezers spectrum vanishes when the heating frequency f goes to zero. +Yet, it does not reproduce the linear increase of the resonant magnitude of the first modes, neither the decrease +of the resonant peaks at large frequency observed in experiments with second sound tweezers. This means that +other effects have to be taken into account to model a fully-immersed open resonant cavity such as non-perfect +plates alignment and energy loss by diffraction outside the cavity when the latter is not embedded in infinite +walls. +Empirically modified Fabry–Perot model +Contrary to a Fabry–Perot resonator composed of infinite +plates, second-sound resonators are built with plates of finite size L, approximately of the same order as the +gap D between them. Those finite size effects are important as they introduce a frequency-dependent energy +diffracted outside the cavity. This mechanism is sketched in Fig. 12. According to standard diffraction theory, +a finite wave initially of size L with a wavelength λ = c2 +f spreads with a typical opening angle given by λ +L. By +this geometrical effect, a part of the wave energy is lost as the wave reaches the other side of the cavity. The +energy loss is roughly proportional to the surface of the wave cross-section that “misses” the reflector (see the +right panel of Fig. 12). Therefore, the fraction of energy lost at the wave reflection is controlled by the ratio +� +L + 2λD +L +�2 − L2 +� +L + 2λD +L +�2 +≈ 4λD +L2 , +(7) +≈ +4 +NF +, +where we have introduced the Fresnel number NF = L2 +λD. +The tweezers plates are mounted at the top of arms of a few millimeters. The perfect parallelism of the plates +is usually not reached for our tweezers, but a small inclination γ of the order of a few degrees can be observed +instead. A relative inclination γ -even small- of both plates creates an additional energy loss mechanism (see +Fig. 13). Intuitively, this second mechanism is controlled by the non-dimensional number +Ni = λ +γL. +(8) +We assume that the Fabry–Perot model (6) can be corrected using the two non-dimensional numbers NF = +L2f +c2D in Eq. +(7) and Ni = +c2 +γfL in Eq. +(8). +More precisely, based on empirical observations, we find that +second-sound tweezers spectra can be accurately represented by the formula +T = +A +sinh +� +i +� +2πfD +c2 +− a c2D +L2f +� ++ b c2D +L2f + c +� +γfL +c2 +�2�, +(9) +where a, b and c are fitting coefficients. As there is no other small parameter in the problem, a reasonable +assumption is to look for coefficients of order one. We find that the values a ≈ 0.95, b ≈ 0.38 and c ≈ 1.3 give +accurate spectra predictions. An illustration of a modified Fabry–Perot spectrum with Eq. (9) is given in Fig. +14. The linear amplitude growth of the first resonant peaks can be interpreted as a progressive focalization of +the wave, and is thus controlled by the Fresnel diffraction number NF in Eq. (12). The shift proportional to +1 +f in peaks frequency positions, observed in the experimental spectra, is also controlled by NF . The decrease +in resonant magnitude for large mode numbers can be interpreted as a wave deflection outside the cavity, after +back and forth propagation between the plates. This latter effect is controlled by the second non-dimensional +number Ni in Eq. (8). +17 + +L +L +D +L +Thermal wave +Figure 12: Representation of the wave dispersion. The energy loss is controlled by the non-dimensional number +λ +L, according to standard diffraction theory. In this section, we discuss both the case of tweezers embedded in +walls, and the case of free tweezers in open space. +Figure 13: Effect of the plates’ inclination γ. Inclination creates an additional energy loss mechanism controlled +by the second non-dimensional number +λ +γL. +18 + +20 +2 +4 +6 +8 +10 +12 +14 +kD +0 +1 +2 +3 +4 +5 +6 +7 +T +inclination +effect +destructive interferences +focalization +of the wave +Figure 14: Analytical models of second-sound tweezers. The grey curve represents the standard Fabry-Perot +spectrum. The blue curve represents the empirical correction of the Fabry-Perot formula using the two non- +dimensional numbers +λ +L and +λ +γL. +The modified spectrum displays the characteristic features of a tweezers +experimental spectrum, as displayed in Fig. 11. +The major interest of the Fabry–Perot model is to offer an analytical expression to fit locally a resonant peak +of second sound resonator spectrum. The local fit of a peak is of particular interest to interpret the experimental +data, as will be explained in sec. 4. Based on Eq. (9), given a measured resonant frequency f0, we will look for +a fitting expression +T = +A +sinh +� +i 2π(f−f0)D +c2 ++ ξ0D +�, +(10) +valid for second sound frequencies f close to f0. In that expression, ξ0 encapsulates the different geometrical +mechanisms responsible for energy loss when the fluid is at rest. A and ξ0 are thus fitting parameters that can +be found easily with the experimental data obtained by varying f in the vicinity of f0. +3.3 +Numeric algorithm +Sec. 3.2 presents a class of models of increasing complexity, still with an analytical solution. Those models +show how all zeroth-order effects observed with second sound resonators can be recovered from geometrical +diffraction effects. However, they do not allow for quantitative predictions of the resonator spectra, nor do +they allow including the effects of a flow. We develop in the present section a numerical algorithm, based on +the exact resolution of the wave equations with the particular tweezers geometry, with and without flow. The +algorithm could be extended to any second sound resonator with a planar geometry. As will become clear in +the following, this numerical model allows going far beyond the approximate models of sec. 3.2. +3.3.1 +For a backgroud medium at rest +The aim of the present section is to build a numerical algorithm to solve the wave equation (3) for a periodic +heating jQ = j0e2iπft. We look for a solution with the ansatz T(r, t) = Re +� +T(r)e2iπft� +. Then, the wave +equation for T is +∆T + k2T = 0, +where we have introduced the wave number k = 2πf +c2 . The boundary conditions are +� +∇T(r).n1 = − ikj0 +ρcpc2 +for r ∈ Σ1 +∇T(r).n2 = 0 +for r ∈ Σ2 +, +(11) +where Σ1 is the heater plate and Σ2 the thermometer plate. The notations are given in Fig. 15. We propose +the method described below, based on the Huyggens–Fresnel principle. The principle states that every point +19 + +of the wave emitter can be considered as a point source. The linearity of the wave equation can then be used +to reconstruct the entire wave by summation of all point source contributions. The Huyggens–Fresnel principle +has been widely used in the context of electromagnetism, for example to compute diffraction patterns produced +by small apertures, or interference patterns... The major difficulty in the context of second sound tweezers is +that none of the standard approximations of electromagnetism can be done, neither the far-field approximation +nor the small wavelength approximation. This explains why numerical resolution is very useful in this context. +We neglect the tweezers arms, which means that both plates are considered as freestanding, infinitely thin +and perfectly insulating plates. We allow a relative inclination γ around the x-axis and a possible relative lateral +shift Xsh of one plate with respect to the other along the x-axis. We assume that the thermometer is sensitive +to the temperature averaged over Σ2. +Let us introduce the Green function +G(r) = 1 +|r|e−ik|r|, +(12) +which is the fundamental solution of the wave equation +∆G + k2G = 4πδ(r). +(13) +Let Σ be one of our two square plates, and U(r′) be a smooth function defined over Σ. We introduce the wave +defined by +T(r) = −1 +2π +� +Σ +G(r − r′)U(r′) d2r′. +(14) +By linearity, T is a solution of Eq. (3), for all r /∈ Σ, because G is a solution. A asymptotic calculation in the +vicinity of Σ then shows that T satisfies the boundary condition +∇T(r).n +−→ +r→r0∈Σ U(r0), +(15) +where n is the unit vector normal to Σ and directed inward the cavity (see Fig. 15). We are going to use Eqs. +(14) and (15) as the two fundamental relations to build our algorithm. We will compute the solution of the +wave equation as an infinite summation of all the emitted and reflected waves in the cavity. +The first wave T 1 is emitted by the heating plate Σ1 and satisfies the first relation in Eq. (11) +∇T 1(r).n1 = − ikj0 +ρcpc2 +for r ∈ Σ1. +Given Eq. (14) and (15), it is clear that the first wave is given by +T 1(r) = +ikj0 +2πρcpc2 +� +Σ1 +G(r − r′) d2r1. +(16) +Then each time a wave denoted T n hits a plate Σ (Σ1 or Σ2), it produces a reflected wave T n+1 to satisfy the +boundary condition +∇ +� +T n(r) + T n+1(r) +� +.n = 0. +(17) +The situation is sketched in the left panel of Fig. (15). If we choose for T n+1 the expression +T n+1(r) = 1 +2π +� +Σ +G(r − r′) +� +∇T n(r′).n +� +d2r′, +(18) +then Eq. (15) shows that Tn+1 satisfies the boundary condition +∇T n+1(r).n +−→ +r→r0∈Σ −∇T n(r0).n, +which is exactly Eq. (17). Eqs. (16) and (18) define our recursive algorithm. Eq. (18) shows that the reflected +wave is generated by the gradient of the incident wave. Practically, the recursive computation of all forth and +back reflected waves thus requires at each step n the computation of ∇T n only on the plates, rather than T n. +For a reflection at (say) Σ1, we have +∇T n+1(r).n2 = −1 +2π +� +Σ1 +G(r − r1) +� +1 +|r − r1| + ik +� +n2. r − r1 +|r − r1| +� +∇T n(r1).n1 +� +d2r1, +(19) +20 + +D +L +Thermometer +L +n2 +Heater +heat flux +n1 +x +y +z +Xsh/2 +Xsh/2 +n +Figure 15: Left: Geometrical setup of the numerical algorithm and notations. Right: Representation of an +incoming and outcoming wave at the nth reflection. +The solution of the wave equation is finally given by the superposition of all waves T n, that is +T(r) = ++∞ +� +n=1 +T n(r), += 1 +2π +� +Σ1 +G(r − r1) ++∞ +� +n=0 +� +∇T 2n+1(r1).n1 +� +d2r1 ++ 1 +2π +� +Σ2 +G(r − r2) ++∞ +� +n=1 +� +∇T 2n(r2).n2 +� +d2r2. +and the thermometer response is given by +� +T +� +Σ2 = 1 +L2 +� +Σ2 +T(r) d2r. +A simulation of the temperature field at t = 0 of the 5th resonant mode of second sound tweezers with aspect +ratio L +D = 0.4, without lateral shift nor inclination of the plates, is displayed in Fig. 16. It can be clearly seen +in particular that the amplitude of the temperature field decreases along the z-axis contrary to a Fabry–Perot +resonator. This symmetry breaking is due to the diffraction effects associated with the finite size of the plates. +A bulk dissipation can be included in the algorithm, for example, to account for quantum vortex lines inside +the cavity. In that case, let ξ be the second-sound attenuation coefficient (in m−1), the wave number k = 2πf +c2 +of the Green function (12) should be replaced by +k = 2πf +c2 +− iξ. +(20) +3.3.2 +In the presence of a turbulent flow +One of the aims of second-sound resonator modelling is to understand their response in the present of a flow +U sweeping the cavity. One effect of the flow is to advect the second sound wave. In the present section, we +explain how the algorithm of sec. 3.3.1 should be modified to account for this effect. We assume in the following +that the inequality |U| < c2 is strictly satisfied, which means that the flow is not supersonic for second sound +waves. +In the presence of a non-zero flow U, the Green function (12) becomes +G(r, t) = +e−2iπft∗ +|r − Ut∗| +� +1 + U +c2 . r−Ut∗ +|r−Ut∗| +�, +(21) +21 + +2mTn+1Figure 16: +Left: Temperature field fluctuations of the 5th resonant mode of second sound tweezers with aspect +ratio L +D = 0.4, and heating power jQ = 0.185 W/cm2, at the bath temperature T0 = 2K. Right: Temperature +field fluctuations for the same conditions with an additional flow of velocity U +c2 = 0.18 directed upward . The +nodes of the temperature standing wave correspond to antinodes of sound sound velocity, and vice-versa. +where t∗ is the time shift corresponding to the signal propagation from the source +|r − Ut∗| = c2t∗. +(22) +In practice, the flow velocity range reached in quantum turbulence experiments is most often much lower than +the second sound velocity, with |U| hardly reaching a few m/s. Most experiments are done in the temperature +range where 10 < c2 < 20 m/s. We thus introduce the small parameter β = |U| +c2 ≪ 1. Similarly to the standard +approximations of electromagnetism, we assume that the effect of β is mostly concentrated in the phase shift +e−2iπft∗ of Eq. (21). We use the approximation |r − Ut∗| +� +1 + U +c2 . r−Ut∗ +|r−Ut∗| +� +≈ |r|, and we solve Eq. (22) to +obtain t∗ to leading order in β. The Green function then becomes +G(r, t) = e−ik|r|Γ(r,U) +|r| +, +(23) +where as previously k = 2πf +c2 +and +Γ (r, U) = 1 − U +c2 +. r +|r|. +The algorithm detailed in sec. 3.3.1 can be applied straightforward with the Green function Eq. (23). In +particular, Eq. (19) becomes +∇T n+1(r).n2 = −1 +2π +� +Σ1 +G (r − r1, U) +�� +1 +|r − r1| + ik +� r − r1 +|r − r1|.n2 − ik U +c2 +.n2 +� � +∇T n.n1 +� +(r1) d2r1. +(24) +A simulation of the temperature field at t = 0 of the 5th resonant mode of second sound tweezers with aspect +ratio +L +D = 0.4, without lateral shift nor inclination, and with a flow of velocity +U +c2 = 0.18, is displayed in the +right panel of Fig. 16. The effect of the flow can be clearly seen with the upward distortion of the antinodes of +the wave, compared to the reference temperature profile without flow displayed in the left panel. +3.4 +Quantitative predictions +We present in this section the quantitative results obtained with the algorithm of sec. 3.3. The algorithm is +specifically run in the configuration of second sound tweezers, but most predictions are relevant for other types +22 + +Thermometer +Thermometer +Heater +Heater +-0.15-0.1 +-0.05 +-0.15 -0.1 -0.05 +0.05 +0.15 +0.05 +0.1 +0.15 +0 +T (mK) +T (mK)of second sound resonators. We first show that the algorithm can quantitatively account for the experimental +spectra. We then use it to predict the response in the presence of a flow and a bulk dissipation in the cavity. +The predictions are systematically compared to experimental results for second sound tweezers. We eventually +display some experimental observations that illustrate the limits of our model. +3.4.1 +Spectral response of second sound resonators +Given a resonator lateral size L, the model of sec. 3.3 has three geometrical parameters : the gap D, the +inclination γ and the lateral shift Xsh (see notations in Fig. 15). We first sketch qualitatively the importance +of those three parameters. +The gap D is the main parameter: it sets the location of the resonant frequencies, and the quality factor of +the resonances at low mode numbers. For second sound tweezers, the value of D can be usually obtained within +a precision of a few micrometers (D is of the order of 1 millimeter). The relative inclination of the plates γ is +responsible for the saturation of the resonant magnitude and its decrease at large mode numbers. It is typically +smaller than a few degrees. Contrary to the gap, only the order of magnitude of γ, not its precise value, can +be determined from the tweezers spectrum. The lateral shift Xsh has very little impact on the spectrum if the +value Xsh +L +remains small enough (we can typically reach Xsh +L +< 0.1 in the tweezers fabrication). However, the +effect of this parameter is of paramount importance to understand open cavity resonators response in a flow +(such as second sound tweezers), and will be investigated in sec. 3.4.2. We consider the case Xsh = 0 in the +present section. The tweezers size L is known from the probe fabrication process. +The method goes as follows: we first find a gap rough estimation �D, for example from the average spacing +between the experimental resonant peaks. Then we can run a simulation for parallel plates (γ = 0), unit gap +D = 1, and aspect ratio L +� +D, in the range 0 < k∗ < nπ (where n is the number of modes to be fitted, and k∗ = kD +is the non-dimensional wave number). This gives a function fL/ � +D(k∗). The experimental spectrum can then +be fitted with the function T(f) = AfL/ � +D( 2πfD +c2 +), where A and D are the two free parameters to be fitted, +provided the experimental value of c2 is known. The high sensitivity of the location of the resonant frequencies +makes this method very accurate to obtain the gap D. +Once D has been found, new simulations have to be run to find the order of magnitude of γ. +As was +previously said, γ controls the saturation and the decrease of the resonant magnitudes for large mode numbers. +Its value can thus be approximated from a fit of the resonant modes with the largest magnitude. A fit of an +experimental tweezers spectrum is displayed in Fig. 17. The values of the fitting parameters for this spectrum +are D = 1.435 ± 0.003 mm and γ = 4.2 ± 0.5 deg. Given the simplicity of the model assumptions, in particular +the assumptions of perfectly insulating and infinitely thin plates without support arms, the agreement with +experimental results is very good. +Interestingly, the resonators can also be used in some conditions as thermometers. Once the gap D is known +with high enough precision, the spectrum can be fitted using c2 as a fitting parameter instead of D. Away from +the second sound plateau of the curve c2(T0) located around 1.65 K, the value of c2 obtained from the spectrum +gives access to the average temperature with a typical accuracy of one mK, simply by inverting the function +c2(T0). +3.4.2 +Response with a flow +Once the characteristics of the resonator have been determined with a background medium at rest, their response +in a flow can be studied using the modified algorithm presented in sec. 3.3.2. We experimentally observe that +the tweezers response is attenuated in the presence of a superfluid helium flow. This attenuation is related to +two physical mechanisms, illustrated in Fig. 18: first, the thermal wave crossing the cavity is damped by the +quantum vortices carried by the flow. This type of damping is usually considered as being proportional to the +density of quantum vortex lines between the plates. Second, the flow mean velocity is responsible for a ballistic +advection of the thermal wave outside the cavity. The thermal wave emitted by the heater partly “misses” the +thermometer plate, and, even if the wave is not attenuated, a decrease of the tweezers’ response will be observed. +Both mechanisms described above exist in experimental superfluid flows, and cannot be observed independently: +once there is a superfluid flow, quantum vortices are created. One key objective is to be able to separate the +attenuation of the experimental signal due to bulk attenuation inside the cavity, from the attenuation due to +ballistic advection of the wave outside the cavity. We will introduce a mathematical procedure to perform such +a separation on a fluctuating signal. +What cannot be experimentally achieved can be simulated with the tweezers model developed in sec. 3.3. +The bulk dissipation can be implemented in the algorithm with a wave number complex part ξ (see Eq. (20)), +and the flow ballistic deflection can be implemented with a non-zero velocity U (see Eqs. (23-24)). Both effects +can be independently studied, setting alternatively ξ or U to zero. We first detail below the respective effects +23 + +0 +10 +20 +30 +40 +50 +60 +70 +f (kHz) +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +T (K) +10-5 +Experimental data +Numerical simulation +Figure 17: Prediction of the second sound tweezers experimental spectrum of Fig. +11 using the numerical +algorithm. The fitting parameters are the gap D, the inclination γ and the total heating power. +Heater +Thermometer +Bulk +dissipation +Heater +Thermometer +Flow +Figure 18: Schematic representation of the two attenuation mechanisms. +The top panel illustrates a bulk +dissipation of the wave, due for example to the presence of quantum vortices. The bottom panel illustrates the +ballistic deflection of the wave by a flow directed parallel to the plates. +24 + +2 +2.2 +2.4 +2.6 +kD +1 +1.5 +2 +2.5 +3 +T +0 +1 +2 +X +-1.5 +-1 +-0.5 +0 +0.5 +1 +1.5 +Y +0 +0.05 +0.1 +0.15 +0.2 +Figure 19: Numerical simulation: collapse of a resonance due to increasing values of the bulk attenuation ξ. +The left panel display the magnitude of the thermal wave as a function of the wavevector k, and the right panel +display the same resonance in the phase-quadrature plane. It can be seen that the bulk attenuation results in +an homothetic collapse of the resonance, that means, without global phase shift. For a given value of k, the +model predicts that attenuation is directed toward the center of the resonant Kennelly circle (red curves of right +panel). +of ξ and U for perfectly aligned plates (Xsh = 0). +Fig. 19 display the result of a numerical simulation for second sound tweezers of aspect ratio L/D = 1, γ = 0 +and increasing values of bulk dissipation in the range 0 < ξD < 0.2. The left panel display the magnitude of the +second resonant mode as a function of the wave number, and the right panel display the same resonant mode in +the phase-quadrature plane. More precisely, if we call T(k) the thermal wave magnitude recorded by the ther- +mometer, and ϕ(k) its phase, the right panel display the curve Y (k) = T(k) sin(ϕ(k)), X(k) = T(k) cos(ϕ(k)). +The resonant curve (Y (k), X(k)) is called in the following the resonant “Kennelly circle”, because the curve is +very close to a circle crossing the origin. It can even be shown that the resonant curve becomes closer to a +perfect circle for increasing resonant quality factors. The major characteristic to be observed in Fig. 19 is that +the collapse of the resonant Kennelly circle due to bulk attenuation is homothetic. It means that the different +curves have no relative phase shift between each other, when the bulk attenuation increases. The red curves +in the right panel display the displacement in the phase-quadrature plane for a fixed value of the wavevector. +The model predicts that the displacement is directed toward the Kennelly circle center, which implies that +the path at fixed wavevector approximately follows a straight line. By comparison, the left panel of Fig. 21 +display an experimental resonance in the phase-quadrature plane, for second sound tweezers of size L = 1 mm in +superfluid Helium at 1.65 K. The global orientation of the resonant Kennelly circles is simply due to a uniform +phase shift introduced by the measurement devices, and should be overlooked. It can be seen that the resonance +collapse with increasing values of the flow velocity follows the predictions of Fig. 19: it is homothetic. The +red paths correspond to the tweezers signal at fixed heating frequency. Those paths follow approximately a +straight line directed to the Kennelly circle center. The slight deviation in the path orientation compared to +the predictions of Fig.19 can be explained by a second sound velocity reduction and will be discussed in sec. 3.4.4. +Fig. 20 display the result of a numerical simulation for second sound tweezers of aspect ratio L/D = 1, +γ = 0, with ξ = 0 and a flow mean velocity 0 < +U +c2 < 0.2. As there is no tweezers lateral shift Xsh = 0, +negative velocities would lead to the same result from symmetry considerations. The figure illustrates the effect +of pure ballistic advection on a resonance in the phase-quadrature plane. First, it can be seen that the collapse +of the resonant Kennelly circle is accompanied by a relative anti-clockwise phase shift of the curves when the +velocity increases. Also, the displacement of the tweezers signal at fixed wavenumber follow the red straight +paths directed anti-clockwise. This type of signal strongly contrasts with the one of Fig. 19 obtained for a +pure bulk attenuation. The prediction of the left panel in Fig. 20 cannot be directly compared to experiments +because, as stated before, a superfluid flow always carries quantum vortices that overwhelm the tweezers signal +for tweezers satisfying Xsh ≈ 0. +25 + +0 +1 +2 +X +-1.5 +-1 +-0.5 +0 +0.5 +1 +1.5 +Y +0 +0.05 +0.1 +0.15 +0.2 +-0.5 +0 +0.5 +1 +1.5 +X +-1 +-0.5 +0 +0.5 +1 +Y +-0.2 +-0.15 +-0.1 +-0.05 +0 +0.05 +0.1 +0.15 +0.2 +U/c2 +Increasing U +-0.2 +0 +0.2 +U/c2 +-0.5 +0 +0.5 +s +s +Figure 20: Numerical simulation: collapse of a resonance due to ballistic deflection of the thermal wave +in the presence of a flow of velocity U, without bulk attenuation (ξ=0). The left panel display the result for +tweezers without lateral shift (Xsh = 0). Contrary to the results of Fig. 19, it can be seen in the present case +that the collapse is associated with a global anti-clockwise phase shift of the resonant Kennelly circle. Each +red curve represents the attenuation at a given value of the wavevector k. The right panel display the result +for tweezers with a strong lateral shift Xsh = 0.5 × L (where L is the tweezers size). Such tweezers are very +sensitive to the velocity U, with both an attenuation of the resonance and a strong clockwise angular shift of +the Kennelly circle. We note s the curvilinear abscissa of the curve obtained at a given value of k (red curve). +The inset displays the function s(U). This shows that, once calibrated, second-sound tweezers can be used as +anemometers. +3.4.3 +Effect of lateral shift of the emitter and receiver plates +We discuss in this section the consequences of a lateral shift, that is Xsh ̸= 0 with the notations of Fig. 15. +Contrary to the previous sections, the present discussion is restricted to second sound tweezers, for which a +lateral shift has major quantitative effects. A lateral shift would not be as important, for example in the case +of wall embedded resonators. +The lateral shift has a marginal effect on the tweezers spectrum when the background fluid is at rest. An +effect only appears in the presence of a nonzero velocity specifically oriented in the shifting direction U = Uex, +because of the mechanism of ballistic advection of the thermal wave by the flow (see the representation of the +mechanism in Fig. 18). The importance of this effect depends on the tweezers aspect ratio, on the reduced +velocity β = U +c2 , and on the lateral shift Xsh. The lateral shift in the plates’ positioning magnifies the signal +component related to ballistic advection. This property opens the opportunity to build second sound tweezers +for which ballistic advection of the wave completely overwhelms bulk attenuation from the quantum vortices, +which means that the tweezers signal is in fact a measure of the velocity component in the shifting direction. +We illustrate this mechanism in Fig. 20. +The right panel displays a numerical simulation of a tweezers resonant mode in the phase-quadrature plane, +for the parameters L +D = 1, γ = 0 and Xsh = 0.5, for positive and negative values of the flow velocity in the range +−0.2 < U +c2 < 0.2. As can be seen at the first sight, the deformation of the resonant curve - that we equivalently +call the “Kennelly circle” - is very different from a deformation due to a bulk attenuation (see Fig. 19). First, +we observe that the deformation can result in an increase of the magnitude of the thermometer signal, when +the velocity is negative. This can be explained in this configuration, because the thermal wave emitted by the +heating plate is redirected toward the thermometer plate: less energy is scattered outside the cavity when the +wave is first emitted by the heater, and the signal magnitude increases. On contrary, the signal magnitude +decreases when the velocity is positive because the flow advects the emitted thermal wave further away from the +thermometer plate and more energy is scattered outside the cavity. Second, the deformation of the Kennelly +circle is associated to a global clockwise rotation, a phenomenon that is not observed for bulk attenuation in Fig. +19. Coming back to Fig. 20, the red curve displays the displacement in the phase-quadrature plane for a fixed +wave frequency value. The displacement follows a very characteristic curved path always directed clockwise. +Let s(U) be the curvilinear abscissa of the red path. Once calibrated, the value of s can be used as a measure +of the flow velocity component in the ex direction. +The right panel of Fig. 21 displays the experimental signal observed with second sound tweezers of size +26 + +0 +0.05 +0.1 +0.15 +X (mK) +-0.05 +0 +0.05 +0.1 +0.15 +Y (mK) +0 +0.2 +0.4 +0.6 +0.8 +1 +U (m/s) +-0.4 +-0.2 +0 +0.2 +X (mK) +-0.1 +0 +0.1 +0.2 +0.3 +0.4 +Y (mK) +0 +0.2 +0.4 +0.6 +0.8 +1 +U (m/s) +Figure 21: Experiment: Collapse of a second sound resonance for increasing values of the flow mean velocity +U, in superfluid Helium at T0 ≈ 1.65 K. The right panel display the result for tweezers of size L = 1 mm and +minor lateral shift Xsh < 0.1×L. The figure shows a homothetic collapse of the resonant Kennelly circle without +global phase shift, as predicted by the model of Fig. 19. The red curves display the displacement in the phase- +quadrature plane at a fixed value of the second sound frequency f. The right panel displays the experimental +data obtained with shifted second sound tweezers with parameters L = 250 µm and Xsh = 0.5 × L. The +figure qualitatively confirms the clockwise angular shift with increasing values of U, predicted by the numerical +simulations of Fig. 20. +L = 250 µm, D = 431 µm and Xsh ≈ 125 µm, for a positive velocity range 0 < U < 1 m/s. The main +characteristics of a ballistic advection signal can be observed: the Kennelly circle are attenuated with a clear +clockwise rotation, and the signal at fixed frequency follows a curved path in the clockwise direction. This is a +strong indication that those type of tweezers can be used as anemometers. The signal fluctuations of those type +of tweezers were recently characterized in a turbulent flow of superfluid helium [WVR21]. It has been shown in +particular that both the signal spectra and its probability distributions indeed display all the characteristics of +that of turbulent velocity fluctuations. +3.4.4 +Limits of the model +Although the model of sec. +3.3 gives excellent experimental predictions, we still observe some unexpected +phenomena with real second sound tweezers. We discuss two of them in this section. +We have seen in secs. 3.4.2 that the thermal wave complex amplitude T(f) can be represented in the phase- +quadrature plane by a curve (X(f), Y (f)) very close to a circle crossing the origin. This osculating circle will +be called in the following the resonant “Kennelly circle”. The wave is damped in the presence of a superfluid +flow, which can be seen in the phase-quadrature plane as a homothetic shrink of the Kennelly circle toward the +origin. Fig. 22 displays an experimental resonance in the phase-quadrature plane, for U = 0 m/s and U = 0.7 +m/s, together with the fitted Kennelly circles. As can be seen in the figure, the resonant curve at U = 0 has +periodic oscillations in and out the Kennelly circle. We call this phenomenon the “daisy effect”. The daisy +effect progressively disappears for increasing values of U, and cannot be seen any more on the resonant curve at +U = 0.7 m/s. We interpret the daisy effect as a secondary resonance in the experimental setup with a typical +acoustic path of a few centimeters. We assume that the flow kicks out the thermal wave from this secondary +resonant path when U is increased. The daisy effect alters the attenuation measurements close to U = 0, and +should be considered with care before assessing the vortex line densities for low mean velocities. +It has been shown in sec. 3.4.2 that the displacement of the tweezers signal in the phase quadrature plane +for a fixed wave frequency, follows a straight line. +We call “attenuation axis” the direction of this straight +path. The model predicts that the attenuation axis should always be directed toward the center of the resonant +Kennelly circle. Fig. 23 displays a zoom on a part of the Kennelly circle at U = 0, together with the signal +displacement at fixed frequency and for increasing flow velocity. It can be seen that the displacement is indeed +a straight line, but not exactly directed toward the Kennelly circle center. An angle between 20° and 30° is +systematically observed between the attenuation axis and the circle center direction (see Fig. 23). Moreover, +the angle is always positive (with the figure convention) and cannot be interpreted as a ballistic advection, +27 + +-8 +-7 +-6 +-5 +-4 +-3 +-2 +-1 +0 +1 +X +10-7 +-3 +-2 +-1 +0 +1 +2 +3 +4 +Y +10-7 +U=0 m/s +U=0.7 m/s +Figure 22: Experimental resonance obtained with a second sound tweezers at 1.98 K, for two values of the He +flow mean velocity U. The blue curve displays a periodic perturbation of the resonance that we refer to as the +“daisy effect”. The circle is a fit of the Kennelly osculating circle for this resonance. This effect is not predicted +by our model, and we interpret it as a secondary resonance in the experimental setup. The daisy effect perturbs +the measurements at low values of U, but it can be seen on the red curve that the effect disappears for higher +values of U. +that would give a negative angle instead. This effect is thus very likely been attributed to a decrease of the +second sound velocity in the presence of the quantum vortices. Whereas a second sound velocity reduction has +previously been observed in the presence of quantum vortices [LV74, Meh74, MLM78], the exact value of this +reduction turns to be difficult to assess in particular experimental conditions. We therefore keep the second +sound velocity reduction as a qualitative explanation, and we do not try to assess quantitative result from the +attenuation axis angle. +3.5 +Quantum vortex or velocity measurements ? +Let us summarize the discussion of sec 3.4. We have shown that second sound resonators are sensitive to two +physical mechanisms. The first one is the thermal wave bulk attenuation inside the tweezers cavity, due to +the quantum vortices. The second one is thermal wave ballistic advection perpendicular to the plates4. Both +mechanisms exist for all the second sound resonators, but depending on their geometry, they can preferentially +be sensitive to the one or the other mechanism. We call selectivity the fraction of the signal due to quantum +vortices or to ballistic advection. Let T (ξ, U) be the probe signal as a function of the bulk attenuation coefficient +ξ (m−1) and flow velocity U(m/s), we define the vortex selectivity as +Rξ = +��T (ξ, 0) − T (0, 0) +�� +��T (ξ, U) − T (0, 0) +��. +(25) +and by symmetry we define the velocity selectivity as +RU = +��T (0, U) − T (0, 0) +�� +��T (ξ, U) − T (0, 0) +��. +(26) +Further investigations in second sound tweezers experiments have shown that the velocity/vortex selectivity +process only weakly depends on the aspect ratio +L +D. +Indeed, for a given resonator lateral size L, ballistic +advection of the wave outside the cavity increases when the gap D increases, but the number of quantum vortex +lines inside the cavity also increases linearly with D. Altogether, both the ballistic advection and the bulk +attenuation due to the quantum vortices have similar dependence with D, that’s why changing the gap has +no significant effect on selectivity. For second sound tweezers, we observe that the selectivity neither depends +strongly on the mean temperature (that controls the superfluid fraction and the second sound velocity). +4Advection of second sound by velocity is illustrated e.g. in [DL77]. +28 + +-3.5 +-3 +-2.5 +-2 +-1.5 +X +10-4 +-2.5 +-2 +-1.5 +Y +10-4 +U=0 m/s +0 90% for a small shift and low mode number, which means that they can be +used for direct quantum vortex measurements. On contrary, small tweezers (L = 250 µm) can reach a velocity +selectivity RU > 90% for large shift or high mode number, and can thus be used as anemometers, as confirmed +by the experiments reported in [WVR21]. +4 +Measurements with second sound tweezers +Second sound tweezers are singular sensors in the sense that they can measure two degrees of freedom at the +same time, whereas most of hydrodynamics sensors only measure one (e.g. Pitot tubes, Cantilevers, Hot wires). +The tweezers record the magnitude and phase of the thermal wave averaged over the thermometer plate. Both +quantities contain physical information about the system. To summarize it shortly, magnitude variations give +information about quantum vortices in the cavity, whereas phase variations give information about the local +mean temperature and pressure. The local mean velocity has an impact on both magnitude and phase, and +will be specifically treated in sec. 4.5. The aim of the following sections is to explain how properly separate +quantum vortices signal from other signal components. +In the following, we call L⊥ the density of projected quantum vortex lines density (projected VLD) +L⊥ = 1 +V +� +V +sin2 θ(l)dl, +(27) +where V is the tweezers cavity volume, l is the curvilinear abscissa along the vortex lines inside the cavity , +θ(l) is the angle between the quantum vortex line and the direction perpendicular to the plates (vector ez). +29 + +Figure 24: Selectivity to quantum vortices or velocity advection for two second sound tweezers, obtained with +numerical simulations. The color code indicates the fraction of the signal due to bulk attenuation by quantum +vortices (see Fig. 18). The selectivity of the tweezers mainly depends on the lateral shift of both plates one +from another, and the resonant mode number excited in the cavity. The left panel shows that large tweezers +(L = 1mm) are mainly sensitive to quantum vortices. Almost pure quantum vortex signal can be achieved +with carefully aligned tweezers excited at low mode numbers (Rξ > 90%). On the reverse, small tweezers +(L = 250µm) are mainly sensitive to the velocity. Almost pure velocity signal can be achieved by shifting the +heater and the thermometer plates and by exciting the cavity at large mode numbers (RU > 90%). The present +simulation was run with a vortex line density L = 2 × 1010 m−2 and U = 1 m/s, in accordance with the typical +values observed in our experiments. +Assuming isotropy of the vortex tangle, the total quantum vortex lines density (VLD) is +L = 3 +2L⊥. +(28) +A second sound wave is damped in the presence of a tangle of quantum vortices. Let ξV LD (in m−1) be +the bulk attenuation coefficient of second sound waves, it has been found[HV56a, HV56b, Tsa62, SP66, MPS84] +that ξV LD is proportional to L⊥ according to the relation +ξV LD = BκL⊥ +4c2 +, +(29) +where B is the first Vinen coefficient and κ ≈ 9.98 × 10−8 m2/s (for 4He) is the quantum of circulation around +one vortex. +Therefore, Eq. (29) shows that a measure of the bulk attenuation coefficient gives access to the projected +VLD defined by Eq. (27). We recall in sec. 4.1 the standard methods to measure the bulk attenuation coefficient +from a second sound resonance, and we propose in sec. 4.3 a new method called “the elliptic method”. We give +in sec. 4.4 some examples to apply the elliptic method to the experimental data. +4.1 +The vortex line density from the attenuation coefficient +We assume that a single second sound resonance can be accurately represented by the following expression (see +Eq. (10)) +T(f) = T 0 +sinh (ξ0D) +sinh +� +i 2π(f−f0)D +c2 ++ (ξ0 + ξV LD)D +�, +(30) +where f0 is the second sound frequency of the local amplitude maximum, D is the resonator gap, c2 is the +second sound velocity, ξ0 is the attenuation coefficient without flow and ξV LD is the additional bulk attenuation +in the presence of quantum vortices given by Eq. (29). ξV LD = 0 without flow. +A standard method to measure ξV LD goes as follows: we fix the second sound frequency at the resonant +value f0, and we measure the thermal wave amplitude with, and without flow. Eq. (30) then shows that ξV LD +is given by +ξV LD = 1 +Dasinh +� T 0 +T(f0) sinh (ξ0D) +� +− ξ0. +(31) +30 + +26p- +0.5 +0.45 +0.4 +0.35 +0.3 +shift +0.25ity / vortex sensitivity -→0.2 +0.15 +Vortex se +0.1 +0.05 +0 +1234567 +modeK velocity sensitiv +ectivity > 90% +89101112131415 +number0.5 +0.45 +Velocity se +0.4 +0.35 +0.3 +shift +0.25ty / vortex sensitivity → +electivity > 90%0.2 +0.15 +0.1 +0.05 +0 +2345678 +modeK velocity sensitiv +910111213141516 +numberFigure 25: Spectral response of tweezers in the SHREK facility, right below the superfluid transition temperature +Tλ, where second sound velocity c2 is very sensitive to temperature (here D ≃ 500 µm and c2 drifts around a +mean value of 5.4 m/s). The frequency axis is adimensionalize using the second sound velocity c2 calculated +from the temperature recorded near the sidewall of the flow. While the temperature of the bath is regulated, +frequency sweeps are repeated half a dozen of times for different turbulent flow conditions flagged by colors. +A systematic drift of resonance frequencies versus flow conditions is observed ; it is interpreted as an under- +estimation of temperature, and therefore over-estimate of c2, due to turbulent dissipation in the core of the flow. +At a given mean flow, some scatter of the resonance frequencies is apparent ; it is interpreted as noise from +the temperature regulation. The elliptic method introduced in section 4.3 allows separating such temperature +artifacts from the attenuation due to second sound attenuation by quantum vortices. +Eq. (31) shows that beside the value of D , that can be determined from a fit of the tweezers spectrum (sec. +3.4), the value of ξ0 has to be accurately measured. This is usually done by the measurement of the resonant +half width. With some algebra manipulations, it can be found from Eq. (30) that the resonant magnitude +satisfies +��T +��2 = +��T 0 +��2 +sinh2 (ξ0D) +sinh2 (ξ0D) + sin2 � +2π(f−f0)D +c2 +�. +(32) +Let ∆f be the frequency half-width defined by the relation +���T(f0 ± ∆f +2 ) +��� +2 += 1 +2 +��T 0 +��2, it can be shown from Eq. +(32) that ξ0 and ∆f are related by +sin +�π∆fD +c2 +� += sinh (ξ0D) . +(33) +We note in particular that the relation (33) can be used to find ξ0 as long as the resonance quality factor is +high enough, that means, for sinh (ξ0D) < 1. The linear approximations of Eqs. (31) and (33) are usually used +when ξ0D ≪ 1, and they give the well-know approximation: +L⊥ ≃ 4π∆f +Bκ +� T 0 +T(f0) − 1 +� +, +(34) +For low quality factor resonances, another method should be used instead of the resonant half width. The +elliptic method presented in sec. 4.3 allows determining ξ0 for resonances of any quality factors. +The main problem of the method presented above is that it implicitly assumes that there is no variation +of the acoustic path value 2πf0D +c2 +during the measurement. In particular, as c2 depends on temperature and +pressure, it means that the experiment should have an excellent temperature and pressure regulation. This can +become increasingly difficult when the second sound derivatives become steep, close to the superfluid transition. +Moreover, measurements in the presence of a flow are necessary done out of equilibrium as the flow dissipates +energy. +As an example, measurements in such conditions are illustrated by figure 25, which shows second +sound resonances measured close to the superfluid transition in the turbulent Von Karman experiment SHREK +[RBD+14]. Furthermore, we observe that a measurement with a non-zero value of ξV LD can be associated with +an acoustic path shift (i.e. a variation of the factor 2πfD +c2 +). The situation is illustrated in the left panel of Fig. +26, where the acoustic path shift leads to an overestimation of the attenuation and an important error on ξV LD. +31 + +0.08 +0.5 +0.4 +0.07 +0.3 +0.06 +0.2 +0.05 +0.1 +/w) +U +-0.1 +0.03 +-0.2 +0.02 +-0.3 +0.01 +-0.4 +0 +-0.5 +2 元 +4元 +6元 +8元 +10元 +12元 +14元 +2元fD/C 21.6 +1.8 +2 +2.2 +2.4 +kD +0 +0.2 +0.4 +0.6 +0.8 +1 +T +First measurement +VLD=0 +Second measurement +VLD=1e-4 +Acoustic +path shift +Attenuation +error +Without +phase shift +With +phase shift +-0.2 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +X +-0.6 +-0.4 +-0.2 +0 +0.2 +0.4 +0.6 +Y +R +r +Figure 26: An illustration of an attenuation measurement. Left: the resonant mode without (blue curve), +and with quantum vortex attenuation (red curve). The figure shows that an acoustic path shift can lead to +an important error in the attenuation measurement. Right: the resonant mode represented in the phase- +quadrature plane together with the fitted Kennelly circle. The figure shows that the acoustic path shift creates +a phase shift θ. Using the phase measurement θ, the maximal magnitude can be recovered using Eq. (37). +Passive and active approaches have been reported in the literature to handle the most common cause of +acoustical path shift during second sound measurement: the temperature drift and its resulting shift of the +resonance frequency. +A passive approach consists in performing a sweep of the second sound frequency, across the resonance curve. +Afterward, with proper modelling of the resonance, the attenuation and the phase shift can be fitted separately, +e.g. as done in [MSS76]. A limit of this approach is its time resolution, that is restricted by the duration of +frequency scan. Another passive approach consists in performing systematic calibration of the full frequency +responses of the resonator in various conditions, and subsequently interpolating measurements obtained at a +fixed working frequency onto this mapping [VBL+17]. +A standard example of active approaches consist in controlling the helium bath temperature. An alternative +or complementary approach consists in controlling the second-sound frequency so that it always matches the +resonance peak, despite possible drift of the temperature. The resonator itself can provide the feedback signal +of these control loops, for example by monitoring the thermometer or locking the phase of the second sound +signal. An even more direct approach has been recently proposed: the resonator is driven by a self-oscillating +circuit, which frequency adapts dynamically to the drift of the second sound velocity [YIE17]. +Below, we introduce two analytical methods to separate phase shift from attenuation. The first method is +relevant for simple cases (sec. 4.2), while the second one has a broader range of validity (sec. 4.3). +4.2 +Analytical method in an idealized case +The acoustic path shift can be corrected using the resonant representation in the phase-quadrature plane. Let +X(f) and Y (f) be respectively the real and imaginary parts of T(f), the curve (X(f), Y (f)) in the phase- +quadrature plane is very close to a circle crossing the origin. It can even be proved (see sec. 4.3) that the +resonant curve converges to a circle when the quality factor increases, or equivalently when ξ0 decreases5. All +along the present article, this limit circle is called the “Kennelly circle”. An illustration of two resonant curves +with their osculating Kennelly circles is displayed in the right panel of Fig. 26. The acoustic path shift translates +in a phase shift θ in the phase-quadrature plane, such that the amplitude of the second measurement (with +ξV LD > 0) does not correspond to the maximal amplitude R of the attenuated resonant peak (see Fig. 26 for +5In this case, the complex amplitude of the nth temperature resonance is often approximated by the Lorentzian formula [VS71, +DLL80] +T(f) ≃ +Tn +1 + iQn f−fn +fn +(35) +where Tn, Qn and fn are the amplitude at resonance, the quality factor and resonance frequency of the mode of interest. To +highlight that this Lorentzian approximation describes a circle in the complex plane X-Y , it can be written as: +T(f) +T n/2 += 1 + eiφ(f) +(36) +where tan �� = 2F/ +� +F 2 − 1 +� +and F = Qn (f − fn) /fn. +32 + +Figure 27: Sequence of five resonances of second sound tweezers at 1.6K. The flow is weakly turbulent: the +velocity standard deviation is a few percents of the mean velocity displayed on the colorbar. The right side plot +illustrates that each resonance can be approximated by a circle in the complex plane. The global phase shift of +the last resonance (lower circle) compared to the others is attributed to a cut-off of the measurement electronics +at high frequency. +the notations). Using the geometric properties of the Kennelly circle, R can be approximately recovered from +the measured amplitude r with +R = +r +cos θ. +(37) +Thus, a modified version of Eq. (31) can be written to find the VLD attenuation coefficient in the presence +of a phase shift +ξV LD = 1 +Dasinh +� T 0 cos θ +T(f0)e−iθ sinh (ξ0D) +� +− ξ0. +(38) +4.3 +The elliptic method +We present in this section an original method to obtain the values of the acoustic path shift and the attenuation +coefficient, from experimental data. The method, that we call the “elliptic method”, is much simpler to imple- +ment, and much more reliable, than the fit of the Kennelly resonant circle and the use of Eq. (38). Besides, the +method can be used for resonances with very low quality factors. +The method comes from the observation that a pair of two ideal consecutive resonances is transformed into +an ellipse with the complex inversion z → 1 +z in the phase-quadrature plane. The inversion is represented in Fig. +28. To prove this assertion, consider the inversion of the classical Fabry–Perot expression Eq. (6) +1 +T = +sinh +� +i 2πfD +c2 ++ ξD +� +A +. +(39) +Expanding the sinh in the previous expression gives +1 +T = cos +�2πfD +c2 +� sinh (ξD) +A ++ i sin +�2πfD +c2 +� cosh (ξD) +A +. +(40) +Finally, let Xl = Re +� +1 +T +� +and Yl = Im +� +1 +T +� +be respectively the real and imaginary parts of Eq. (40), the +coordinates (Xl, Yl) satisfy the equation +� +Xl +sinh (ξD) /A +�2 ++ +� +Yl +cosh (ξD) /A +�2 += 1 +(41) +which is exactly the cartesian equation of an ellipse with semi-major axis a = cosh(ξD) +A +, and semi-minor axis +b = sinh(ξD) +A +. In particular, we note that the attenuation coefficient ξ can be recovered from the ratio of the +semi-major and semi-minor elliptic axes using the formula +ξ = 1 +Datanh +� b +a +� +. +33 + +0.15 +(yu) +0.1 +0.05 +20 +40 +60 +80 +100 +f (kHz)0.1 +2 +0.05 +1 +(mK) +(s/w) +0 +0 +> -0.05 +U +-1 +-0.1 +-2 +-0.15 +-0.1 +0 +0.1 +X (mK)-1 +-0.5 +0 +0.5 +1 +X +-0.5 +0 +0.5 +Y +-2 +0 +2 +X +-4 +-2 +0 +2 +4 +Y +1/z +Figure 28: Transformation of a pair of consecutive resonances to an ellipse using the inversion of the complex +plane z → 1/z. +0.4 +0.6 +0.8 +1 +1.2 +X +-0.2 +0 +0.2 +Y +1/z +acoustic path +axis +acoustic path +axis +attenuation +axis +u +v +attenuation +axis +ul +vl +Figure 29: A resonant mode in the phase-quadrature plane and its elliptic transform, for an ideal Fabry–perot +resonance with ξ0 = 0.2 and 1.95π < kD < 2.05π. +When the quality factor increases (equivalently when ξ decreases) the ellipse is flattened. The limit of infinite +quality factor (ξ → 0) corresponds to two parallel straight lines in the complex plane. +Second sound tweezers resonances are not ideal Fabry–Perot resonances. Yet, we have argued in sec. 3.2 +that a single second sound resonance can be locally fitted by the following Fabry–Perot equation (see also Eq. +(10)) +T = +A +sinh +� +i 2π(f−f0)D +c2 ++ (ξ0 + ξV LD)D +�. +(42) +This in particular means that the resonant curve in the vicinity of its maximal amplitude is transformed into a +part of an ellipse with the complex inversion z → 1 +z . The curve (Xl(f), Yl(f)) is very close to a straight line, for +frequencies f close to the resonant frequency f0. The situation is illustrated in Fig. 29. The figure shows a part +of ideal Fabry–Perot resonances given by Eq. (42), in the range 1.95π < kD < 2.05π (where k = 2πf +c2 ), and for +increasing values of the VLD attenuation coefficient ξV LD. The left panel displays the different resonant curves +close to their maximal amplitudes, in the phase-quadrature plane. Using the complex inversion, those curves +become almost parallel straight lines, as can be seen in the right panel. The transformation of the resonant +Kennelly circles into parallel straight lines has very nice applications that we explain in the following. +In the phase-quadrature plane, a variation of the acoustic path value 2πfD +c2 +corresponds to a displacement +along the Kennelly circle, whereas a variation of the bulk attenuation ξ corresponds to a displacement orthogonal +to the Kennelly circle. The acoustic path direction and the attenuation direction thus form a local orthogonal +basis. Such a basis is displayed by the red arrows in the left panel of Fig. 29. When the reference point where +the basis is defined moves along the Kennelly circle, the basis rotates and the acoustic path and attenuation +axes have to be redefined. This can become a tiresome task while analyzing the experimental data. Fortunately, +the complex inversion is a conformal mapping, which means that it preserves locally the angles: the local basis +composed of the acoustic path and attenuation axes is transformed into an orthogonal basis (see the red arrows +in the right panel of Fig. 29). More precisely, let z0 be the complex position in the phase-quadrature plane, +34 + +and (u, v) be the two complex (unit) vectors defining the local basis at z0, then the local basis (ul, vl) at the +point +1 +z0 is given by +� +� +� +ul += −u |z0|2 +z2 +0 +vl += −v |z0|2 +z2 +0 +. +(43) +The major advantage of defining the local basis (ul, vl) with the elliptic transform, is that it becomes a global +basis: when the reference point +1 +z0 moves because of a change in the acoustic path value or the attenuation +value, the basis is simply translated in the plane but the vectors (ul, vl) do not change. One can find the global +basis (ul, vl) for a given resonance and use it to find the local basis (u, v) at every point in the phase-quadrature +plane. We will see in sec. 4.4.1 how the global basis (ul, vl) can be easily used to suppress temperature and +pressure drifts during second sound attenuation measurements. +We finally explain how the elliptic method can be used to measure the bulk attenuation coefficient ξ0. We +have seen in sec. 4.1 that the standard methods to find ξ0 are based on the measure of the half-width ∆f +and on Eq. (33). As was said previously, the classical method can only be applied to resonances satisfying +(ξ0D) < 1. It is a global method, in the sense that one has to sweep the frequency to measure a large part of +the resonant curve. The method is only accurate provided the resonance does not deviate too much from an +ideal Fabry–Perot resonance, which is often not satisfied for the first modes of second sound tweezers (see for +example the first mode of Fig. 17). The alternative method consists in expanding (Xl, Yl) given by Eq. (41) to +leading order in f − f0 +� +Xl +∼ sinh(ξ0D) +A +, +Yl +∼ 2π(f−f0)D +c2 +cosh(ξ0D) +A +, +(44) +where f0 is the resonant frequency. Using Eq. (44), we get +Yl(f) +Xl(f) ∼ +2πD +tanh (ξ0D) c2 +(f − f0). +(45) +Yl(f) +Xl(f) is proportional to f in the vicinity of f0, with the proportionality factor +2πD +tanh(ξ0D)c2 . The attenuation +coefficient ξ0 can be found by a linear fit of the function Yl +Xl provided D and c2 are known. +4.4 +Applications of the elliptic method +The motivation to develop and use the elliptic method has come from experimental constrains: in a large super- +fluid experiment, it can be very difficult to control the values of mean temperature and pressure. This is even +more the case if the superfluid experiment dissipates energy. One then expects a drift of the thermodynamics +conditions during the measurement. Regarding second sound resonators, the critical parameter is the second +sound velocity c2, because variations lead to uncontrolled acoustic path shifts. The elliptic method has been +designed to easily filter those variations from experimental data. This includes filtering temperature and pres- +sure drifts (sec. 4.4.1) and the vibration of the tweezers arms (sec. 4.4.3), and properly extract the quantum +vortex lines fluctuations (sec. 4.4.2). +4.4.1 +Suppression of temperature and pressure drifts +This section presents an example of the elliptic method implementation for second sound tweezers, to find the +relation between ⟨ξV LD⟩ and the mean velocity U in the presence of a superfluid flow. +As before, we note (X, Y ) the temperature signal obtained from the second sound tweezers in the phase- +quadrature plane, and (Xl, Yl) the cartesian coordinates obtained by the complex inversion given by +� +� +� +Xl += Re +� +1 +X+iY +� +, +Yl += Im +� +1 +X+iY +� +. +(46) +The coordinates (Xl, Yl) will be called “elliptic coordinates” for convenience. Fig. 30 displays experimental data +obtained from second sound tweezers of size L = 1 mm, in a saturated bath at mean temperature T0 ≈ 2.14 K, +and for different mean flow velocities 0 < U < 1.2 m/s. At such a temperature close to the superfluid transition, +it was difficult to regulate the mean temperature and therefore the second sound velocity. Uncontrolled acoustic +path variations can be observed, for example in the red points of Fig. 30. +The first step consists in sweeping the second sound frequency f in the vicinity of the resonant frequency f0. +The data (X, Y ) obtained, displayed by the black curve of the left panel of Fig. 30, form a part of the Kennelly +circle. As explained in sec. 4.3, the elliptic coordinates (Xl, Yl) given by Eq. (46) form a straight line (see the +35 + +0 +0.1 +0.2 +0.3 +X (mK) +-0.05 +0 +0.05 +0.1 +0.15 +0.2 +Y (mK) +2 +4 +6 +8 +-4 +-3 +-2 +-1 +0 +1 +0 +0.2 +0.4 +0.6 +0.8 +1 +1.2 +U (m/s) +Attenuation +axis +ul +vl +Acoustic path +axis +Figure 30: Experiment: measurement of a resonance with second sound tweezers at T0 ≈ 2.14 K, for different +values of the mean flow velocity U. The left panel presents the data in the phase-quadrature plane, and the +right panel presents the same data after the complex inversion. The red points are obtained for a fixed value +f = 6.35 kHz, and sweeping the flow mean velocity between 0 and 1.2 m/s. +right panel of Fig. 30). Using a linear fit, it is then straightforward to obtain the unit vector vl parallel to the +line defining the acoustic path axis, and the orthogonal vector ul defining the attenuation axis. We then call +Zl = (Xl, Yl) the vector of the elliptic coordinates. The attenuation coefficient ξ0 can be found by the relation +(see Eq. (45)) +Zl(f).vl +Zl(f).ul +∼ +2πD +tanh (ξ0D) c2 +(f − f0). +Fig. 30 displays experimental resonant curves obtained for non-zero mean velocities U > 0 , to illustrate the +robustness of the elliptic method. However, we emphasize that only the resonant curve with U = 0 is necessary +to find the global basis (ul, vl) in the plane of elliptic coordinates. +The second step consists in fixing the second sound frequency to f0 and vary the mean velocity U to look +at the resonance attenuation. The experimental data are displayed by the red points in Fig. 30. In can be +seen in the figure that the bulk attenuation is accompanied by a systematic acoustic path deviation as the +mean velocity increases. For mean velocities U ≈ 1 m/s, energy dissipation in the experiment leads to a data +dispersion along the acoustic path direction. To properly recover the mean VLD attenuation coefficient ⟨ξV LD⟩ +, we use the elliptic coordinates Zl = (Xl, Yl) , and we project it on the attenuation axis ul. We get from Eq. +(44) +Zl(U).ul +Zl(0).ul += sinh ((ξ0 + ⟨ξV LD⟩) D) +sinh (ξ0D) +. +And ⟨ξV LD⟩ is then given by +⟨ξV LD⟩ = 1 +Dasinh +�Zl(U).ul +Zl(0).ul +sinh (ξ0D) +� +− ξ0. +(47) +We note that the previous expression remains accurate even if the second sound frequency chosen for the +measurement is close but not exactly equal to the resonant frequency f0. The average VLD attenuation can +then be converted to the average projected vortex line density ⟨L⊥⟩ using Eq. (29). +4.4.2 +Measure of vortex line density fluctuations +Second sound tweezers are designed to directly probe the small scale vortex line density fluctuations, not only +its average value. The method to probe fluctuations slightly differs from the method used to probe the average +value explained in sec. 4.4.1. The average VLD value can be directly computed using the complex inversion of +experimental data, but this is no longer possible for its fluctuations. Indeed, the tweezers signal have different +sources of noise, like e.g. thermal white noise, interfering frequencies, electromagnetic bursts, etc... Those +signals can usually be considered as independent additive noises in the signal data, and easily filtered out or +attenuated by an appropriate post-processing. On the contrary, the complex inversion is a non-linear transfor- +mation. Using the latter on noisy data can lead to an overestimation of the signal fluctuations closest to zero, +36 + +1/z-2 +-1.5 +-1 +-0.5 +X (V) +x 10-6 +-16 +-12 +-8 +-4 +Y (V) +x 10-7 +0 m/s +1.20 m/s +Elliptic transformation +acoustic path axes +attenuation axes +Figure 31: +perturb the additivity of noise sources and make them much more difficult to filter out. We thus choose to +compute the VLD fluctuations only using linear transformations. +The first step is similar to that of sec. 4.4.1. We sweep the second sound frequency f close to the resonant +frequency f0, in order to measure a part of the Kennelly circle. We then transform this Kennelly circle into a +straight line using the complex inversion, and we find the global basis (ul, vl) in the plane of elliptic coordinates. +A fit of the Kennelly circle, and its transformation into a straight line can be seen in Fig. 31. This experimental +step has to be done just before the fluctuations measurement. +We then record the signal fluctuations (X(t), Y (t)), for different values of the flow mean velocity U. Fig. +31 displays the fluctuating signal for U = 0 and U = 1.2 m/s in the form of clouds of data points. It can be in +particular observed that the U = 1.2 m/s data are shifted compared to the U = 0 m/s data because of both +an average attenuation and an acoustic path shift (see sec. 4.4.1). Let us define ⟨Z(t)⟩ = ⟨X(t)⟩ + i ⟨Y (t)⟩ the +average complex position in the phase-quadrature plane, for a given value of U. Following Eq. (43), the local +basis (u, v) of the attenuation and acoustic path axes can be computed from the global elliptic basis (ul, vl) by +� +� +� +u += −ul +⟨Z⟩2 +|⟨Z⟩|2 +v += −vl +⟨Z⟩2 +|⟨Z⟩|2 +. +(48) +Fig. +31 shows that the local basis (u, v) depends on ⟨Z⟩ and thus on the value of U: for different mean +velocities, the bases are rotated from one another. If there is a significant drift of the mean signal value during +the measurement (as can e.g. be observed in Fig. 22), the local basis will also depend on time, with a typical +timescale that should be much larger than the fluctuation timescale. +The acoustic path fluctuations and the attenuation fluctuations can then be recovered using a projection on +the (u, v) basis. More precisely, let x be the average acoustic path value and δξ = ξ − ⟨ξ⟩ be a small fluctuation +of the attenuation coefficient, a leading order expansion of expression Eq. (42) shows that +T ≈ +A +sinh (ix + ⟨ξ⟩ D) − δξ +AD +sinh (ix + ⟨ξ⟩ D) tanh (ix + ⟨ξ⟩ D). +(49) +We then do the approximation ⟨Z⟩ ≈ A/ sinh (ix + (ξ0 + ⟨ξV LD⟩) D) which is equivalent to neglecting non-linear +corrections in Eq. (49). We finally get +δξ(t) = 1 +Du. (Z(t) − ⟨Z⟩) × +���� +tanh (ix + (ξ0 + ⟨ξV LD⟩) D) +⟨Z⟩ +���� , +(50) +where the value of ξ0 can be found with Eq. (45) and ⟨ξV LD⟩ with Eq. (47). +The use of the elliptic method is illustrated by Figure 32, which reports the probability density function of +the fluctuations of the quantum vortex density in a nearly isotropic superfluid turbulent flow The data of this +37 + +0.5 +1 +1.5 +2 +p (arb. shift) +s / +vortex line density fluct. +velocity fluctuations +Figure 32: Example of the probability density function of vortex line density (dashed line) and velocity (contin- +uous line) measured by second sound tweezers in a superfluid turbulent flow [WVR21]. In abscissa, each signal +s is normalized by its mean value < s >. The nearly Gaussian velocity statistics and skewed vorticity statistics +are reminiscent of those in classical turbulence. +plot were reported together with spectra of vorticity fluctuations. For details about the setup and analysis of +these results, see [WVR21]. +4.4.3 +Filtering the vibration of the plates +One possible source of noise for second sound resonator measurement is the vibration of the plates arms whenever +U ̸= 0. The signature of those vibrations can be very clearly identified in the form of two thin peaks in the +fluctuations power spectrum. Those two peaks are located at the two arms resonant frequencies: their exact +values can vary for different tweezers, but we always observe them around f ≈ 1 kHz (see sec. 4.4). +Fortunately, the tweezers arms vibrations correspond to a variation of the gap D, and thus to acoustic path +fluctuations. Fig. 33 display a part of the tweezers fluctuations power spectrum. The fluctuations are projected +along the attenuation axis (blue curve) and along the acoustic path axis (red curve), following the method +presented in sec. 4.4.2. The two peaks located at f ≈ 825 Hz and f ≈ 1050 Hz are identified on the acoustic +path axis fluctuations power spectrum, whereas the same peaks are damped by many orders of magnitude on +the attenuation axis fluctuations. Using the power spectrum of Fig. 33, we can estimate the order of magnitude +of the gap standard deviation. We find +�� +(δD)2� +≈ 0.5 µm, and +� +⟨(δD)2⟩ +D +≈ 4 × 10−4. This confirms that the +arms vibrations have a negligible impact on the measurement. +4.5 +Velocity measurements +As shown in Sec. 3.5, the second sound tweezers geometry can be optimized to sense specifically velocity rather +than vortex density. For this purpose, one trick consists in shifting one plate with respect to the other in the flow +direction. Figure 34 shows three second sound tweezers and one anemometer that is based on the same principle +as Pitot tubes. All sensors are positioned across a nearly isotropic superfluid flow bounded by a cylindrical pipe +-not shown here- (for details on this set-up, see [RCSR17a]). The figure insert is a close view of the tip of the +left-side tweezers, which is dedicated to velocity measurements. This shift of one plate versus the other in the +downstream direction is clearly visible. +The projection of the anemometer-tweezers signal in the complex plane is not performed along orthogonal +axes. These axes are determined with an in-situ calibration, ramping the mean velocity. The complementary +“Pitot tube” signal is used to calibrate the axis in units of m/s. The duration of velocity ramp is chosen to be +much larger than the time resolution of the Pitot tube. +As an illustration, Figure 32 presents the probability density function of the velocity fluctuations measured +by the anemometer tweezers, together with vortex line density fluctuations recorded by the other tweezers during +the same experimental run (see Fig. 34). As expected in nearly homogeneous and isotropic quantum turbulence +[SCLR12], the velocity statistics are close to a Gaussian when probed at scales significantly larger than the +intervortex distance, which is the case here. Velocity spectra derived from the same dataset are reported in +[WVR21]. +38 + +700 +800 +900 +1000 +1100 +f (Hz) +10-18 +10-17 +10-16 +P(f) +attenuation axis fluctuations +acoustic path axis fluctuations +Figure 33: Experiment: a part of second sound tweezers fluctuation power spectrum, with T0 = 1.65 K, +U = 1.2 m/s and for a tweezers gap D = 1.320 mm. The tweezers arms’ resonances can be clearly identified in +the acoustic path fluctuations. +Figure 34: Example of an arrangement of three second sound tweezers dedicated to velocity (x1, left side) and +vorticity (x2, right side) time series acquisitions, together with a miniature total head pressure tube (loosely +labelled "Pitot tube" on the bottom left side of the picture) used for velocity calibration. +All probes are +mounted on a ring connecting two 76mm-inner-diameter coaxial pipes (see Fig.1 of ref. [WVR21]). The insert +is a close-up view of the shifted plates of the anemometer tweezers. +39 + +Thermometer +Flow +Heater +VLD probing +second sound +tweezers +Second sound +tweezers +anemometer +Miniature Pitot +tube5 +Summary and Perspectives +This study has covered three independent topics : the comprehensive analytical modelling of second-sound +resonators with a cavity allowing a throughflow of superfluid (section 3), new mathematical methods to process +the signal provided by such resonators (section 4) and the miniaturization of immersed second-sound resonators +allowing time and/or space resolved flow sensing (section 2). These so-called second-sound tweezers have been +used throughout the manuscript to demonstrate the strength and limits of the modelling and analysis methods. +Two observations remains unexplained: the origin of some noise on the acoustic path length and a second +order oscillations of the resonator spectral response in quiescent 4He, named the “daisy effect” (section 3.4.4). +Fortunately, both effects don’t impair the measurement of the flow vorticity or velocity. +Some results that were not anticipated when this study was initiated, are worth recalling and discussing: +• the possibility to operate tweezers by over-driving the second-sound standing wave beyond an intrinsic +turbulent transition. In this non-linear mode, the probe becomes sensitive to velocity, which is interpreted +as a signature of the sweeping of the local vortex tangle by the outer flow (section 2.2.3). This operating +mode is somehow analogous of hot film and hot wire anemometry in classical fluid, where sensitivity +to velocity is due to the more-or-less pronounced sweeping of the thermal boundary layer around an +overheated thermometer. +• in the linear regime (standing wave of small amplitude), the probe can be sized and operated to be either +mostly sensitive to quantum vortices or mostly sensitive to the velocity of the throughflow (section 3.5). +This prediction is verified experimentally by comparing statistics in turbulent flows (section 4.5). The +spatial and time resolution of tweezers operated as anemometers -both in non-linear and linear mode- +is close or better than the alternative miniaturized anemometers working in He-II, that is hot-wires +[DBM+15, DRS+21], cantilevers[SMR12, RCSR17b], Pitot tubes [SMR12, RCSR17b] and total head- +pressure probes[MT98, WVR21]. +• in the absence of throughflow, the full spectral response of the resonators, and in particular its quality +factors, can be accurately determined simply taking into account the loss by diffraction and misalignment +of the reflecting plates of the cavity. +In other words, the other sources of dissipation have negligible +contributions in the range of conditions explored here. +This is no longer the case in a presence of a +throughflow carrying quantum vortices since the latter can significantly contribute to the total dissipation. +We have not explored experimentally the production and detection of second-sound by mechanical means, +which implies cavity with rough surfaces (e.g. millipore or nucleopore membranes). In this case, the effect +of vortices pinned on the surface may no longer be negligible ; for a discussion see [DLL80]. +• the possibility to sense the variations of the vortex line density or velocity without knowledge on variations +of the second sound velocity (or acoustical path). More generally, the elliptic method allows a mathematical +decoupling of both effects by a projection method in the (inverse) complex plane. This results is of major +practical interest in flows where the second-sound velocity is not accurately controlled due to residual +temperature variations or thermal gradients. This situation can occur for instance in flows sequentially +driven at various levels of forcing (e.g. to explore a Reynolds number dependence), in inhomogeneous +dissipative flows and in flows close to the lambda superfluid transitions where the second-sound velocities +strongly depends of temperature. +Two applications of second-sound tweezers have been illustrated. Measurement within a turbulent boundary +layer (Fig. 25) are possible thanks to the small size of probe, and measurements of time series in the bulk +of quantum turbulent flow (Fig. 32) are possible thanks to both the time and space resolutions of the probe. +Among other applications the probe can map the velocity or the vorticity field of an inhomogeneous flow. A +mapping of vorticity in a counterflow jet has been recently done and will be reported elsewhere. +Another application of tweezers would be to probe simultaneously the temperature fluctuations and those of +either velocity or vorticity, for instance to explore their correlations in turbulent counterflows or even co-flows. +Indeed, the tweezers thermometer provides a direct measurement of temperature in a bandwidth spanning from +zero frequency up to a fraction of the frequency of the second sound standing wave, and this signal could be +acquired without impairing the measurement of the second sound standing wave. Alternating measurements in +the linear and non-linear modes is also interesting to explore locally both velocity and vorticity in a given flow. +A last example of application is to operate a double-tweezers made of a heating plate between two thermometer +plates, or vice-versa. Such a stack can be used to probe joint statistics of vorticity on one side, and velocity on +the other side, or this arrangement can be used alternatively to probe transverse gradient of either vorticity or +velocity. +40 + +Acknowledgements +We warmly thank our colleague Benoît Chabaud for support during cryogenic tests and operation of the TOUPIE +wind-tunnel. We acknowledge and thank the staff of the PTA and Nanofab clean-rooms in Grenoble, where +microfabrication was done. Data of figure 25 have been acquired in the SHREK facility. We thank all the +members of the SHREK collaboration, in particular Michel Bon Mardion and Bernard Rousset for the specific +operation very near the superfluid transition. We acknowledge the indirect but key contribution of A. Elbakyan +regarding the comprehensiveness of the bibliography. +The probe design, fabrication, cryogenic operation and theoretical analysis took place over 7 years, and was +possible thanks to following research grants: ANR Ecouturb grant (ANR-16-CE30-0016), ANR QUTE-HPC +grant (18-CE46-0013- 03) and EU Horizon 2020 Research and Innovation Program "the European Microkelvin +Platform (EMP)" (824109). +This research was funded in part by the Agence nationale de la recherche (ANR). A CC-BY public copyright +license has been applied by the authors to the present document and will be applied to all subsequent versions +up to the Author Accepted Manuscript arising from this submission, in accordance with the grant’s open access +conditions. +References +[Bal07] +S Balibar. The discovery of superfluidity. Journal of Low Temperature Physics, 146(5-6):441–470, +2007. +[BLR17] +J. Bertolaccini, E. Lévèque, and P.-E. Roche. Disproportionate entrance length in superfluid flows +and the puzzle of counterflow instabilities. Phys. Rev. Fluids, 2:123902, 2017. +[BSS83] +H Borner, T Schmeling, and DW Schmidt. Experimental investigations on fast gold-tin metal film +second-sound detectors and their application. J. Low Temp. Phys., 50(5):405–426, 1983. +[BVS+14] +S Babuin, E Varga, L Skrbek, E Lévêque, and P-E Roche. Effective viscosity in quantum turbulence: +A steady-state approach. EPL, 106(2):24006, 2014. +[CA68] +R W. Cohen and B Abeles. Superconductivity in granular aluminum films. Phys. Rev., 168:444–450, +1968. +[CR83] +M J Crooks and B J Robinson. +Technique for determining second sound attenuation near the +superfluid transition in 4he. Review of Scientific Instruments, 54(1):12–15, 1983. +[DB98] +R J Donnelly and C F Barenghi. The observed properties of liquid helium at the saturated vapor +pressure. Journal of physical and chemical reference data, 27(6):1217–1274, 1998. +[DBM+15] D. Durì, C. Baudet, J.-P. Moro, P.-E. Roche, and P. Diribarne. Hot-wire anemometry for superfluid +turbulent coflows. Review of Scientific Instruments, 86(2):025007, 2015. +[DL77] +P E Dimotakis and G A Laguna. Investigations of turbulence in a liquid helium ii counterflow jet. +Physical Review B, 15:5240, Jun 1977. +[DLF+14] +Q. Dong, Y. X. Liang, D. Ferry, A. Cavanna, U. Gennser, L. Couraud, and Y. Jin. Ultra-low noise +high electron mobility transistors for high-impedance and low-frequency deep cryogenic readout +electronics. Applied Physics Letters, 105(1):013504, 2014. +[DLL80] +D D’Humières, A Launay, and Albert Libchaber. +Pinning of vortices in nucleopores. effect on +second-sound resonators. Journal of Low Temperature Physics, 38:207, Jan 1980. +[Don91] +R. J. Donnelly. Quantized Vortices in Helium-II. Cambridge Studies in Low Temperature Physics. +Cambridge University Press, Cambridge, 1991. +[Don09] +Russell J Donnelly. The two-fluid theory and second sound in liquid helium. Phys. Today, 62(10):34– +39, 2009. +[DRS+21] +P. Diribarne, B. Rousset, Y. A. Sergeev, J. Valentin, and P.-E. Roche. Cooling with a subsonic flow +of quantum fluid. Physical Review B, 103(14):144509, 2021. +[FS04] +S Fuzier and Steven W Van Sciver. Use of the bare chip cernox(tm) thermometer for the detection +of second sound in superfluid helium. Cryogenics, 44:211, Mar 2004. +41 + +[Gri09] +A Griffin. New light on the intriguing history of superfluidity in liquid4he. Journal of Physics: +Condensed Matter, 21(16):164220, 2009. +[HL88] +K Henjes and M Liu. Peshkov transducers of second sound. Journal of Low Temperature Physics, +71(1):97–117, 1988. +[HR76] +J Heiserman and I Rudnick. The acoustic modes of superfluid helium in a waveguide partially +packed with superleak. Journal of Low Temperature Physics, 22(5):481–499, 1976. +[HV56a] +H. E Hall and W. F Vinen. The rotation of liquid Helium II. i. experiments on the propagation of +second sound in uniformly rotating Helium II. Proceedings of the Royal Society of London. Series +A, 238:204, Dec 1956. +[HV56b] +H. E Hall and W. F Vinen. The rotation of liquid Helium II. II. the theory of mutual friction in +uniformly rotating helium II. Proceedings of the Royal Society of London. Series A, 238:215, Dec +1956. +[HVS92] +D. S. Holmes and S. W. Van Sciver. Attenuation of second sound in bulk flowing He II. J. Low +Temp. Phys., 87:73–93, 1992. +[HVS01] +D.K. Hilton and S.W. Van Sciver. Techniques for the detection of second sound shock pulses and +induced quantum turbulence in He II. Cryogenics, 41:347–53, 2001. +[Kem91] +R. C. Kemp. The reference function for platinum resistance thermometer interpolation between +13,8033 k and 273,16 k in the international temperature scale of 1990. Metrologia, 28(4):327, 1991. +[Kha00] +I M Khalatnikov. An Introduction to the Theory of Superfluidity. CRC Press, 2000. +[Lag76] +G. Laguna. Photolithographic fabrication of high frequency second sound detectors. Cryogenics, +16(4):241–243, 1976. +[LFF47] +C. T. Lane, Henry A. Fairbank, and William M. Fairbank. Second sound in liquid Helium II. Phys. +Rev., 71:600–605, May 1947. +[Lob07] +N Lobontiu. Dynamics of microelectromechanical systems. Springer, Boston, MA, 2007. +[LV74] +D Lhuillier and F Vidal. Temperature dependence of the second sound velocity reduction in rotating +liquid helium. Journal of Physics C: Solid State Physics, 7(14):L254, 1974. +[Meh74] +J. B Mehl. New effects in the interaction of second sound with superfluid vortex lines. Physical +Review A, 10(2):601, 1974. +[MG18] +B Mastracci and W Guo. An apparatus for generation and quantitative measurement of homoge- +neous isotropic turbulence in he ii. Review of Scientific Instruments, 89(1):015107, 2018. +[MLM78] +RJ Miller, IH Lynall, and JB Mehl. Velocity of second sound and mutual friction in rotating helium +ii. Physical Review B, 17(3):1035, 1978. +[MMP+07] V. F Mitin, PC McDonald, F Pavese, NS Boltovets, VV Kholevchuk, IY Nemish, VV Basanets, +VK Dugaev, PV Sorokin, and RV Konakova. Ge-on-gaas film resistance thermometers for cryogenic +applications. Cryogenics, 47(9-10):474–482, 2007. +[MPS84] +P Mathieu, B Plaçais, and Y Simon. Spatial distribution of vortices and anisotropy of mutual +friction in rotating He II. Physical Review B, 29:2489, Mar 1984. +[MSS76] +P. Mathieu, A. Serra, and Y. Simon. Critical-region measurements of the mutual-friction parameters +in rotating He II. Phys. Rev. B, 14:3753–3761, 1976. +[MT98] +J. Maurer and P. Tabeling. Local investigation of superfluid turbulence. Europhys. Lett., 43(1):29– +34, 1998. +[NF95] +S. K. Nemirovskii and W. Fiszdon. +Chaotic quantized vortices and hydrodynamic processes in +superfluid helium. Rev. Modern Phys., 67:37–84, 1995. +[Not64] +H A Notarys. Megacycle frequency second sound. PhD thesis, California Institute of Technology, +1964. +[Pes46] +V. P. Peshkov. +Determination of the velocity of propagation of the second sound in helium ii. +Journal of Physics (Moscow), 10:389, 1946. +42 + +[Pes48] +VP Peshkov. Propagation of second sound in helium ii. exp. Zh. Eksp. Teor. Fiz., 18:857, 1948. +[PGB70] +R. L St Peters, T. J. Greytak, and G.B. Benedek. Brillouin scattering measurements of the velocity +and attenuation of high frequency sound waves in superfluid helium. +Optics Communications, +1(9):412–416, 1970. +[PK82] +DB Poker and CE Klabunde. Temperature dependence of electrical resistivity of vanadium, plat- +inum, and copper. Physical Review B, 26(12):7012, 1982. +[Put74] +S J Putterman. Superfluid hydrodynamics. Series in low temperature physics, 3, 1974. +[RBD+14] +B. Rousset, P. Bonnay, P. Diribarne, A. Girard, J.M. Poncet, E. Herbert, J. Salort, C Baudet, +B Castaing, L. Chevillard, F. Daviaud, B. Dubrulle, Y. Gagne, M. Gibert, B. Hébral, T. Lehner, P.- +E. Roche, B. Saint-Michel, and M Bon Mardion. Superfluid high Reynolds von Kármán experiment. +Rev. Sci. Instrum., 85:103908, 2014. +[RCSR17a] E Rusaouen, B Chabaud, J Salort, and P-E Roche. Intermittency of quantum turbulence with +superfluid fractions from 0% to 96%. Phys. Fluids, 29(10):105108, 2017. +[RCSR17b] E. Rusaouen, B. Chabaud, J. Salort, and P.-E. Roche. Intermittency of quantum turbulence with +superfluid fractions from 0% to 96%. Phys. Fluids, 29(10):105108, 2017. +[RDD+07] +P-E Roche, Pantxo Diribarne, Thomas Didelot, Olivier Français, Lionel Rousseau, and Hervé +Willaime. Vortex density spectrum of quantum turbulence. EPL (Europhysics Letters), 77(6):66002, +2007. +[RG84] +M. Raui and A. Guenter. Damping of second sound near the superfluid transition of 4he as a +function of pressure. Physical Review B, 30:5116, Nov 1984. +[RR01] +D Rinberg and M L Rappaport. Fiber bolometer and emitter with negligible reflection for second +sound measurements near the lambda-point. Cryogenics, 41(8):557–561, 2001. +[Sad98] +J E Sader. Frequency response of cantilever beams immersed in viscous fluids with applications to +the atomic force microscope. Journal of Applied Physics, 84(1):64–76, 1998. +[SCLR12] +J. Salort, B. Chabaud, E. Lévêque, and P.-E. Roche. Energy cascade and the four-fifths law in +superfluid turbulence. EPL, 97:34006, 2012. +[SE70] +R. A Sherlock and D. O Edwards. +Oscillating superleak second sound transducers. +Rev. Sci. +Instrum., 41:1603, Jan 1970. +[SHVS99] +M.R. Smith, D.K. Hilton, and S. W. Van Sciver. Observed drag crisis on a sphere in flowing He I +and He II. Phys. Fluids, 11:751–3, 1999. +[SMR12] +J. Salort, A. Monfardini, and P.-E. Roche. Cantilever anemometer based on a superconducting +micro-resonator: Application to superfluid turbulence. Rev. Sci. Instrum., 83:125002, 2012. +[SNVD02] +S.R. Stalp, J. J Niemela, W. J. Vinen, and R. J. Donnelly. Dissipation of grid turbulence in helium +II. Phys. Fluids, 14:1377–9, 2002. +[Sny62] +H. A. Snyder. Use of germanium as a second sound receiver. Review of Scientific Instruments, +33(4):467–469, 1962. +[SP66] +H. A. Snyder and Zimri Putney. Angular dependence of mutual friction in rotating Helium II. Phys. +Rev., 150:110, 1966. +[SPB17] +GW Stagg, NG Parker, and CF Barenghi. Superfluid boundary layer. Physical Review Letters, +118(13):135301, 2017. +[Tou82] +J. T. Tough. Superfluid Turbulence, volume 8, chapter 3, pages 133–219. North-Holland Publishing +Company, Amsterdam, 1982. +[Tsa62] +D.S. Tsakadze. (sov. phys. jetp 15 (1962) 681). Zh. Eksp. Teor. Fiz., 42:985, 1962. +[VBL+17] +E Varga, S Babuin, VS L’vov, Anna Pomyalov, and L Skrbek. Transition to quantum turbulence +and streamwise inhomogeneity of vortex tangle in thermal counterflow. Journal of Low Temperature +Physics, 187(5-6):531–537, 2017. +43 + +[Vin57] +W. F Vinen. Mutual friction in a heat current in liquid helium ii. iii. theory of the mutual friction. +Proceedings of the Royal Society of London. Series A, 242:493, Nov 1957. +[VJSS19] +E Varga, MJ Jackson, D Schmoranzer, and L Skrbek. The use of second sound in investigations of +quantum turbulence in He II. J. Low Temp. Phys., 197(3):130–148, 2019. +[VS71] +F Vidal and Y Simon. Ideal second sound resonance in he-ii. Physics Letters A, 36(3):165–166, +1971. +[WBF+69] R. Williams, S.E.A. Beaver, J.C. Fraser, R.S. Kagiwada, and I. Rudnick. The velocity of second +sound near tlambda. Physics Letters A, 29(5):279–280, 1969. +[WPHE81] P. E. Wolf, B. Perrin, J. P. Hulin, and P. Elleaume. Rotating Couette flow of helium II. J. Low +Temp. Phys., 44:569–593, September 1981. +[WVR21] +E Woillez, J Valentin, and P-E Roche. Local measurement of vortex statistics in quantum turbu- +lence. EPL (Europhysics Letters), 134(7):46002, 2021. +[YI18] +J Yang and GG Ihas. Decay of grid turbulence in superfluid helium-4: Mesh dependence. In Journal +of Physics: Conference Series, volume 969, page 012004. IOP Publishing, 2018. +[YIE17] +J. Yang, G. G Ihas, and D. Ekdahl. Second sound tracking system. Review of Scientific Instruments, +88(10):104705, 2017. +[YYK97] +T Yotsuya, M Yoshitake, and T Kodama. Low-temperature thermometer using sputtered zrnx thin +film. Cryogenics, 37(12):817–822, 1997. +44 +