This is the submitted version of the article: Urgell C., Yang W., De Bonis S.L., Samanta C., Esplandiu M.J., Dong Q., Jin Y., Bachtold A.. Cooling and self-oscillation in a nanotube electromechanical resonator. Nature Physics, (2020). 16. : 32 - . 10.1038/s41567-019-0682-6. Available at: https://dx.doi.org/10.1038/s41567-019-0682-6 Cooling and Self-Oscillation in a Nanotube Electro-Mechanical Resonator C. Urgell,1, ∗ W. Yang,1, ∗ S. L. de Bonis,1 C. Samanta,1 M. J. Esplandiu,2 Q. Dong,3 Y. Jin,3 and A. Bachtold1 1 ICFO - Institut De Ciencies Fotoniques, The Barcelona Institute of Science and Technology, Mediterranean Technology Park, 08860 Castelldefels (Barcelona), Spain 2 Catalan Institute of Nanoscience and Nanotechnology (ICN2), CSIC and BIST, Campus UAB, Bellaterra, 08193 Barcelona, Spain 3 Centre de Nanosciences et de Nanotechnologies, CNRS, Univ. Paris-Sud, Univ. Paris-Saclay, C2N Marcoussis, 91460 Marcoussis, France Abstract Nanomechanical resonators are used with great success to couple mechanical motion to other degrees of freedom, such as photons, spins, and electrons [1, 2]. Mechanical vibrations can be efficiently cooled and amplified using photons, but not with other degrees of freedom. Here, we demonstrate a simple yet powerful method for cooling, amplification, and self-oscillation using electrons. This is achieved by applying a constant (DC) current of electrons through a suspended nanotube in a dilution fridge. We demonstrate cooling down to 4.6 ± 2.0 quanta of vibrations. We also observe self- oscillation, which can lead to prominent instabilities in the electron transport through the nanotube. We attribute the origin of the observed cooling and self-oscillation to an electrothermal effect. This work shows that electrons may become a useful resource for quantum manipulation of mechanical resonators. ∗ These authors contributed equally to this work. 1 The vibrations of mechanical resonators have been coupled to electrons in different trans- port regimes, such as single-electron tunneling [3–9], Kondo [10], and quantum Hall ef- fect [11, 12]. Because mechanical resonators are excellent force sensors, a small electrostatic force created by electrons generates a large displacement of the resonator. Conversely, the displacement reacts back on the electrons by a sizeable amount. This backaction of elec- trons on the resonator has been frequently studied by measuring the change in resonance frequency and in energy decay rate of vibrations [3–12]. In principle, the backaction of elec- trons can also be used to cool and amplify thermal vibrational fluctuations and to generate self-oscillation by applying a DC electron current [13, 14]. Signatures of a modest cooling down to ∼ 200 quanta as well as self-oscillation were observed over a decade ago in a pio- neer work [15] where a resonator is coupled to a superconducting single-electron transistor, but a quantitative understanding of the backaction has not been reported yet. Meanwhile, many theoretical schemes have been proposed to cool mechanical vibrations using electrons in different electron transport regimes; see for instance Refs. [16–20]. However, these cooling schemes could not be implemented due to experimental difficulties. In this Letter, we show efficient backaction cooling in a current-biased suspended nanotube precooled in a dilution fridge. Carbon nanotubes are a versatile system for the study of both electron transport and nanomechanics. Different electron transport regimes can be reached by tuning the transmis- sion of electrons between the nanotube and electrodes [21]. Interaction can lead to electron attraction, Kondo behaviours, and Wigner states [21–23]. On the other hand, carbon nan- otubes are so small that they make the lightest mechanical resonators fabricated thus far. Cooling a nanotube resonator in a dilution fridge leads to high quality factors [24, 25]. As a result, the force sensitivity of the resonator is record high [26], and the effect of the electron-vibration coupling is expected to be especially large. Suspending a carbon nanotube between two metal electrodes is key to form a nanome- chanical resonator and to carry out state-of-the-art electron transport measurements. This suppresses the electron backscattering in the nanotube due to the charge impurities and the rugosity of the substrate. We grow the carbon nanotube between two metal electrodes in the last step of the fabrication process using chemical vapour deposition in order to minimise residual contamination [24]. Measurements are carried out by applying a DC voltage to the source electrode (Vsd ) and the gate electrode (Vg ) patterned at the bottom of the trench 2 (Fig. 1a). We detect the electrical current from the drain electrode using a RLC resonator with frequency ωRLC = 2π · 1.27 MHz and a high-electron-mobility-transistor amplifier [26]. We record the differential conductance Gdiff of the device by applying an oscillating voltage Vsdac to the source electrode with the frequency set at ωRLC . Using a capacitive transduction scheme [26], we measure thermal vibrations with resonance frequency ω0 by applying Vsdac at the frequency ω0 − ωRLC . In order to avoid perturbations from the measurement, we keep the amplitude of Vsdac much smaller than kB T /e, where kB is the Boltzmann constant, T the temperature of the cryostat, and e the electron charge. All the measurements presented here are carried out at the base temperature of the fridge except when stated differently. Electron transport measurements indicate that electrons are in the Kondo regime [21]. A regular shell filling with Kondo ridges at zero source-drain bias is observed upon sweeping Vg (Figs. 1b,c). Unlike normal Coulomb blockade, Gdiff increases in every second conductance valleys when decreasing temperature. This shows the SU(2) nature of the Kondo effect in this device. Energy decay measurements of thermal vibrations reveal that the quality factor Q = 6.8 · 106 is remarkably high when compared to previous works [24–26]. This is also higher than the quality factor inferred from the spectral resonance linewidth, since the energy decay rate is smaller than the spectral resonance linewidth (Figs. 1d-f). The difference is attributed to dephasing. The resonance frequency can be tuned by sweeping both Vg and Vsd (Figs. 1g,h). The slopes ∂ω0 /∂Vg and −∂ω0 /∂Vsd are often rather similar, suggesting that they are related to the same origin, that is, the mechanical tension induced by the static displacement of the nanotube. Thermal vibrations can be cooled with the cryostat down to ∼ 70 mK (Fig. 1i). We attribute the saturation of the displacement variance at low temperature to the electric noise in the circuit. We observe instabilities in the conductance arising periodically in Vg (see arrows in Fig. 1b). They emerge at finite source-drain bias in charge stability diagram measure- ments. In these instability regions, the peaks in conductance are truncated (Figs. 2a-c) and conductance traces as a function of Vg appear noisy (Fig. 2d). While similar conductance instabilities were previously reported [6], we will show below that these instabilities are related to large-amplitude vibrations. The measured conductance instabilities originate from the switching of the mechanical motion between thermal noise and self-oscillation (Fig. 3). Upon sweeping Vsd through the 3 instability region, the variance of the displacement δz 2 dramatically increases (Fig. 3a), and the decay rate gets suppressed towards zero near the border of the instability region (shaded in yellow in Fig. 3b). These two experimental facts point to the development of self-oscillation. The phase-space and the histogram of the two quadratures of the motion can be described by the superposition of the distributions of a donut and a Gaussian-like peak (Figs. 3h,i), suggesting that the motion switches back and forth between self-oscillation with high δz 2 and thermal noise with low δz 2 (Sec. III in Supplementary Information). This switching is further supported by the fact that the resonance lineshape is unusually broad (Fig. 3g); indeed, the large linewidth is then related to the large fluctuations of δz 2 and the resonator nonlinearity. Pure self-oscillation can also be observed without any switches to thermal vibrations; this often happens at higher Vsd values. The shift in resonance frequency due to electron backaction (Fig. 3c) is difficult to quantify, since the resonance frequency also depends on the mechanical tension induced by Vsd , the temperature rise of the nanotube lattice due to Joule heating, and the variance of the displacement through the mechanical nonlinearity. Mechanical vibrations are cooled down to 4.6 ± 2.0 quanta at Vg = −943 mV upon increasing the source-drain bias to Vsd ' 0.565 mV (Figs. 4a-c). Cooling is accompanied with a strong increase of the decay rate (Fig. 4d), indicating a backaction effect. This efficient cooling occurs when the transconductance is negative and large (Fig. 4e). We observe cooling at other Vg values when the transconductance is negative as well (Sec. II in Supplementary Information). The determination of the number of quanta is robust against the hypothetical miscalibration of the amplification chain and of the attenuation along the coaxial cables (Sec. I in Supplementary Information). The uncertainty in the transconductance, which enters into the transduction of the displacement, is 5.7 % of its value at Vg = −943 mV and Vsd ' 0.565 mV. As explained above, the frequency shift due to backaction cannot be quantified, since the frequency shift depends on various other effects that are difficult to disentangle experimentally. The nanotube experiences Joule heating due to the current flowing through the resonator. Backaction cooling predicts that the phonon occupation at finite bias is given by n(Vsd ) = Γbath nbath /Γdecay (Vsd ) where nbath = kB Tbath /h̄ω0 is the thermal phonon number and Γbath is the coupling to the thermal bath. We would achieve much lower phonon occupation, if the bath temperature Tbath was given by the cryostat temperature, while setting Γbath 4 to the measured decay rate at zero-bias. This indicates that Joule heating is sizable. We deduce the bath temperature from the measured values of n and Γdecay (Fig. 4f) [27, 28]. The temperature rise can be well described by Joule heating for different Vg values using the 0 phenomenological relation Tbath = Tvib +ηGVsd2 (Fig. 4f), where Tvib 0 is the measured vibration temperature at zero-bias, G is the conductance, and η is the same constant for all the Vg values. The temperature rise is not accounted for by the electrostatic force associated to the electron shot noise of the nanotube, since the temperature rise does not depend linearly on Vsd and it is independent of Vg to a first approximation. The shot noise of the suspended nanotube behaves in the usual way with a Fano factor between 0.2 and 0.3 (Sec. IV of Supplementary Information). Here, we discuss the possible origins of the observed backaction. It could be related to the usual backaction in electro-mechanical resonators [3–10], where conducting electrons generate an electrostatic force on the nanotube and the retardation of the force is given by the transmission of electrons between the nanotube and electrodes. However, we do not observe resonance frequency dips when sweeping Vg (Fig. 1g), showing that the strength of this backaction is weak. Moreover, this backaction predicts cooling at the conductance peaks [13, 14], which is the opposite of what is observed in Fig. 2, that is, self-oscillation near conductance peaks. This shows that the backaction measured in electro-mechanical resonators at zero source-drain bias cannot describe our results at finite bias. Another possible mechanism could be related to the retardation created by the circuit, where the vibration-induced current noise of the nanotube generates a retarded electrostatic force due to the capacitance of the circuit. However, the predicted decay rate is too weak to produce the cooling observed in Figs. 4a,c. We conclude that backaction with electrostatic origins cannot account for our findings. We attribute the origin of the backaction to an electrothermal effect [29], which is an analogue of the photothermal backaction often observed in opto-mechanical resonators [30]. The power GVsd2 of Joule heating modifies the mechanical tension in the nanotube through the effective thermal expansion coefficient of the device. This results in a net displacement δz of the resonator when the nanotube is bent by e.g. the static electrostatic force associated dG with Vg . This displacement reacts back on the dissipated power via δG = dz δz with a delay given by both the capacitance of the circuit and the thermalization time of the device [29]. 5 This electrothermal effect modifies the decay rate by dGdiff Cg0 ∆Γback = −α Vg zs Vsd2 . (1) dVg Cg Here, Cg is the capacitance between the nanotube and the gate electrode, Cg0 is its derivative with respect to z, and zs is the static displacement. We use α as a free parameter in our analysis, since α depends on various quantities that are difficult to quantify. These include the thermalisation time, the effective thermal expansion coefficient, and the three- dimensional profile of the static bending of the nanotube. dGdiff The electron transport in the device controls the electrothermal backaction through dVg . dGdiff When dVg is positive and large, the total decay rate of the resonator can become effectively dGdiff negative, leading to self-oscillation. When dVg is strongly negative so that ∆Γback >> Γbath , the vibrations are efficiently cooled. Equation 22 reproduces qualitatively the decay rate measured when increasing Vsd towards the self-oscillation regime shaded in yellow in Fig. 3b and to the strong cooling regime in Fig. 4d (see pink lines). See Sec. V of the Supplementary Information for more discussion on the different backactions. This cooling method with electrons becomes efficient by precooling the resonator in a dilution fridge, so that the quality factor is high and the motion is sensitive to backaction. Electrothermal backaction is effective for small resonators, since the low mass increases the backaction strength, and the small heat capacity enhances the heating effect. Future studies may enable ground-state cooling. This may be achieved by enhancing the backaction rate (Eq. 22) using devices with higher transconductance and stronger coupling to the gate (to increase the Cg0 /Cg ratio). Cooling mechanical vibrations with electrons may become a useful resource for quantum manipulation of mechanical resonators. Acknowledgments This work is supported by the ERC advanced grant 692876, the Foundation Cellex, the CERCA Programme, AGAUR, Severo Ochoa (SEV-2015-0522), the grant FIS2015-69831-P of MINECO, and the Fondo Europeo de Desarrollo Regional (FEDER). We wish to thank Brian Thibeault for help in fabrication (UCSB). Author contributions W.Y. fabricated the devices with the support of C.U. and M.J.E. in the growth. C.U. and W.Y. carried out the measurements. C.U., W.Y., S.B., C.S., Q.D., and Y.J developed the detection circuit. C.U., W.Y., and A.B. did the analysis of the data and wrote the manuscript. A.B. supervised the work. 6 G diff (e 2/h) 0.5 1.0 1.5 a b 1.5 1 Vsd (mV) 0.5 0 -0.25 -720 -680 -640 -600 -560 -520 -480 Vg (mV) c 1.6 Gdiff (e2/h) 1.0 15mK 500mK 1500mK 0.4 -720 -680 -640 -600 -560 -520 -480 Vg (mV) d e f Energy autocorr. (a.u.) 1.0 width 4 V = -492 mV 90.58 MHz Vg = -492 mV g Sz (pm/Hz)  (Hz) 3 Vsd = 0 mV Vsd = 0 mV 100 2 0.5 decay 1 10 0 0.0 -2 -1 0 1 2 0 50 100 -1000 -800 -600 /2 (kHz) t (ms) Vg (mV) g h i 93 1 quanta number  (MHz) 1 z2 (10-20m2) 0/2 (kHz) 92 100 0 91 -1 90 0 0 -700 -600 -500 -0.2 -0.1 0.0 0.1 0.2 0.0 0.2 0.4 0.6 Vg (mV) Vsd (mV) T (K) FIG. 1: Characterization of the nanotube electro-mechanical resonator. (a) Measurement schematic and scanning electron microscopy image of the suspended nanotube. DC voltages Vsd and Vg are applied to electrodes S and G, respectively. (b) Differential conductance as a function of Vsd and Vg . Yellow arrows indicate regions of conductance instabilities. (c) Differential conductance as a function of Vg with Vsd = 0 mV for different temperatures. (d) Displacement spectral density. (e) Energy decay obtained from the autocorrelation of the time trace of X 2 + Y 2 , where X and Y are the two quadratures of thermal vibrations. The pink line indicates an exponential decay. (f ) Decay rate and spectral resonance width as a function of Vg with Vsd = 0 mV. (g,h) Resonance frequency as a function of Vg and Vsd . We set Vsd = 0 mV in g and Vg = −560 mV in h. (i) Variance of the displacement of thermal vibrations as a function of temperature at Vg = −185 mV. 7 G diff (e 2 /h) 0.5 1.0 1.5 c a 0.6 b no instability 1.4 1.4 0.4 Gdiff (e2/h) 1.2 Vsd (mV) Gdiff (e2/h) 1.2 0.2 1.0 1.0 Vg=-614mV Vg = -616 mV 0.8 0 -620 -616 -612 0.0 0.2 0.4 0.6 0.0 0.2 0.4 0.6 0.8 Vg (mV) Vsd (mV) Vsd (mV) d 1.6 Vsd = 0.4 mV Gdiff (e2/h) 1.2 0.8 0.4 -720 -680 -640 -600 -560 -520 -480 Vg (mV) FIG. 2: Conductance instabilities. (a) Differential conductance as a function of Vsd and Vg . Yellow arrows in the different panels indicate regions of conductance instabilities. (b,c) Differential conductance as a function of Vsd for two different Vg values. (d) Differential conductance as a function of Vg . 8 a b width c 91.82 1000 decay  (MHz) z2 (nm) 1 91.81 / (Hz) 100 91.80 0.1 10 91.79 1 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 0.0 0.2 0.4 0.6 0.8 Vsd (mV) Vsd (mV) Vsd (mV) d 10 e f 1 Vsd = 0.15 mV 0.2 Counts (a.u.) Sz (pm/Hz ) 0.1 Y (nm) 0 1 0.01 -0.2 -3 -2 -1 0 1 2 3 -0.2 0 0.2 -0.2 0.0 0.2 (kHz) X (nm) X (nm) g h 4 i 1 Vsd = 0.25 mV Counts (a.u.) 2 Sz (pm/Hz) 0.1 Y (nm) 10 0 -2 0.01 1 -4 -16 -8 0 8 16 -4 -2 0 2 4 -4 -2 0 2 4 X (nm) X (nm) (kHz) FIG. 3: Self-oscillation at Vg = −616 mV. (a) Variance of the displacement as a function of Vsd . The yellow shaded area represents the region with self-oscillation. (b) Decay rate and spectral resonance width as a function of Vsd . The pink line is the expected decay rate using Eq. 22. (c) Resonance frequency as a function of Vsd . (d-f ) Displacement spectral density, the phase-space of the two quadratures of the motion, and the associated histogram at Vsd = 0.15 mV. (g-i) Same as d-f but at Vsd = 0.25 mV. The arrows in (i) indicate the donut distribution. 9 a b c 4 14 0.1 n = 131 n = 4.6 quanta number 100 Vsd = 0.315mV Vsd = 0.565mV Sz (pm/Hz) Sz (pm/Hz) z2 (nm) 12 10 10 3 0.01 8 0.0 0.2 0.4 0.6 -6 -4 -2 0 2 4 6 -6 -4 -2 0 2 4 6 Vsd (mV)  (kHz) / (kHz) d e f 10 width 2 Vg = -472 mV 1000 Vg = -662 mV dGdiff /dVg (mS/V) decay 1 1 Vg = -943 mV Tbath (K) / (Hz) 100 0 -1 0.1 10 -2 0.01 0.0 0.2 0.4 0.6 0.0 0.2 0.4 0.6 0.01 0.1 1 Vsd (mV) Vsd (mV) Vsd (mV) FIG. 4: Cooling at Vg = −943 mV. (a) Variance of the displacement as a function of Vsd . The error bars arise from the uncertainty in the fitting of the resonance lineshape to a Lorentzian. (b,c) Displacement spectral density at two different Vsd . (d) Decay rate and spectral resonance width as a function of Vsd . The pink line is the expected decay rate using Eq. 22. (e) Transconductance as a function of Vsd . (f ) Bath temperature of mechanical vibrations as a function of Vsd for three different Vg values. The solid lines indicate the dependence expected from Joule heating; see text. 10 I. DETECTION OF MECHANICAL VIBRATIONS Mechanical vibrations are electrically detected using a RLC resonator and a HEMT am- plifier cooled at liquid-helium temperature (Fig. 1a of the main text) [26]. Displacement modulation is transduced capacitively into current modulation by applying an input oscil- lating voltage Vsdac across the nanotube. The frequency ωsd /2π of the oscillating voltage is set to match ωsd = ω0 ± ωRLC , where ω0 /2π is the resonance frequency of the nanotube resonator and ωRLC /2π the resonance frequency of the RLC resonator. Thermal vibrations are measured by recording the current noise at ∼ ωRLC . The current δI is related to the displacement of the nanotube δz by δI = βδz, (2) 1 dGdiff C0 ac g β= Vg Vsd . (3) 2 dVg Cg dGdiff Here, dVg is the transconductance, Vg is the static gate voltage, Cg is the capacitance between the nanotube and the gate electrode, and Cg0 is the derivative of Cg with respect to z. The spectral density Szz of the displacement noise in the main text is obtained from the measured spectral density of the current noise using Eqs. 2 and 3. The calibration of the number of quanta is obtained in a reliable way thanks to the equipartition theorem mω02 δz 2 = kB T, (4) where m is the effective mass, kB the Boltzmann constant, and T the temperature. In practice, we measure the spectral density of the current noise to quantify the variance 2 of the current δIres associated to the mechanical resonance of thermal vibrations. The 2 measurement of δIres as a function of temperature in Fig. 1i of the main text determines  2 Cg m C 0 = 2.5 × 10−33 kg · m2 using Eqs. 2-4. This allows us to quantify the effective g temperature Tvib of the thermal vibrations at any Vg and DC voltage Vsd applied to the 2 dGdiff source electrode by measuring δIres and dVg and using 2 4ω02  Cg 2 Tvib = m 2 δIres . (5) Cg0  dGdiff kB dVg Vg Vsdac Importantly, the determination of Tvib does not depend on the hypothetical inaccurate cal- ibration of the attenuation along the coaxial cables created by thermal contraction and of 11 the amplification chain. Indeed, such inaccurate calibration, if sizeable, would have an effect  2 on Vsdac and m CC0 g , but it would be canceled out when determining Tvib . The number of g kB Tvib 1 quanta of vibrations is obtained using n = h̄ω0 − 2 with h̄ the reduced Planck constant. In order to quantify Szz and δz 2 , we estimate the capacitance Cg from the separation ∆Vg = 23.1 mV between two conductance peaks in the Coulomb blockade regime at large positive Vg values (Fig. S1). We obtain Cg = e/∆Vg = 6.94 × 10−18 F. We get Cg0 = 7 × 10−12 F/m from the measurement of the variance of the displacement as a function of temperature using m = 2.7 ag. 0.03 0.02 Gdiff (e2/h) 0.01 0.00 1.7 1.8 1.9 2.0 Vg (V) FIG. 1: Coulomb blockade measurements. Differential conductance Gdif f as a function of gate voltage at 10 K and zero source-drain bias. The energy decay rate Γdecay is estimated by measuring the time trace of the two quadra- tures of thermal vibrations and by quantifying the autocorrelation of the amplitude squared. From these time trace measurements, we also obtain the phase-space of the two quadratures and the associated histogram (Fig. 3 of main text). II. RELATION BETWEEN ELECTRON TRANSPORT AND VIBRATION COOL- ING dGdiff Figures S2a,b show that the measurements of the differential conductance Gdif f and dVg as functions of Vg and Vsd are remarkably regular over a large range of gate voltage. This reflects the high quality of the nanotube. The shell filling with Kondo ridges at zero source- 12 Gdiff (e2/h) 0.5 1.0 1.5 a 1.5 V ds(mV) 0 -1.5 -1000 -900 -800 -700 -600 -500 -400 Vg (mV) dGdiff /dVg (mS/V) -5 0 5 b 1.5 V ds(mV) 0 -1.5 -1000 -900 -800 -700 -600 -500 -400 Vg (mV) c 2 dGdiff/dVg (mS/V) 0 -2 -4 -1000 -900 -800 -700 -600 -500 -400 Vg (mV) FIG. 2: Electron transport measurements. (a,b) Differential conductance and transconduc- tance as a function of Vsd and Vg measured at the base temperature of the fridge. (c) Transcon- ductance as a function of Vg measured at Vsd = 0.7 mV. drain bias is observed over the full range of Vg . The instability in the conductance discussed in Figs. 1b and 2d of the main text appears periodically in gate voltage over the full Vg range as well. dGdiff Figures S2b,c show that regions with strongly negative dVg emerge periodically in Vg at finite source-drain voltage. This occurs for Vsd in the range between 0.4 mV and 1.1 mV. dGdiff We observe efficient cooling in these strongly negative dVg regions, as demonstrated by the measured spectra of thermal vibrations in Figs. S3a-c. 13 a 5 b 4 c 5 Vg = -472mV n = 14.5±1.6 Vg = -568mV n = 13.5±1.8 Vg = -662mV n = 16.2±2.7 Vsd = 0.74mV Vsd = 0.75mV Vsd = 0.85mV Sz (pm/Hz) Sz (pm/Hz) Sz (pm/Hz) 4 3 4 3 2 3 -5 0 5 -5 0 5 -5 0 5  (kHz)  (kHz)  (kHz) FIG. 3: Cooling at different Vg values. (a-c) Displacement power spectral density of thermal dGdiff vibrations showing low occupation number in regions where dVg is strongly negative. III. SELF-OSCILLATION The vibrations of the nanotube in the instability region switch back and forth between thermal motion and self-oscillation, as it can be seen in the time traces of one of the two quadratures (X) and of the amplitude (R) in Fig. S4. In these traces, the amplitude of thermal vibrations is low, whereas the amplitude in self-oscillation is high. These switches between thermal motion and self-oscillation occur randomly in time. Pure self-oscillation can also be observed without any switches to thermal vibrations. See for instance Fig. S5. This often happens at high Vsd values. IV. SHOT NOISE MEASUREMENT Here we describe how we measure the shot noise of the nanotube device. The spectral density of the current noise SII is transformed into spectral density of voltage noise SVV −1 −1 −1 through the total impedance Ztot = (Rdiff + ZRLC ) , where Rdiff is the nanotube differential resistance and ZRLC is the effective impedance of the RLC circuit. The voltage fluctua- tions, which are amplified by the high-electron-mobility-transistor amplifier (HEMT), are measured at the frequency ωRLC = 2π · 1.27 MHz over ∼ 80kHz bandwidth. Our noise mea- surement contains the background contribution SIIbg related to the Johnson-Nyquist noise of the circuit and of the HEMT noise. This background contribution is independent of the source-drain voltage Vsd , so that it can be quantified from the current noise measured at Vsd = 0 mV. After the substraction of this background contribution, we determine the 14 a 4 2 X, R (nm) 0 -2 -4 3.4 3.6 3.8 4 4.2 4.4 Time (s) b 4 2 X, R (nm) 0 -2 -4 7.4 7.6 7.8 8 8.2 8.4 8.6 Time (s) c 4 2 X, R (nm) 0 -2 -4 10.4 10.6 10.8 11 11.2 11.4 11.6 Time (s) FIG. 4: Time domain measurements. (a-c) Three different time traces of one quadrature X (blue) and corresponding amplitude R (orange) at Vsd = 0.25mV for V g = −616mV, plotted from the Fig. 3h in the main text. Fano factor F of the nanotube device at finite Vsd from the measured current noise using F = SII (Vsd )/(2eIsd ), where e is the electron charge, and Isd is the DC current at a given source-drain bias Vds . The Fano factor in Fig. S6 is smaller than one because of the suppres- sion of the electron transmission below one and perhaps because of the electron correlation in the device. 15 a b c 1 Vsd = 1.20 mV 1 Counts (a.u.) Sz (pm/Hz) 0.1 Y (nm) 0 10 0.01 -1 -2 -1 0 1 2 -1 0 1 -1 0 1 (kHz) X (nm) X (nm) FIG. 5: Pure self-oscillation. (a) Displacement spectral density at Vsd = 1.2 mV and Vg = −616 mV. (b) The phase-space of the two quadratures of the motion. (c) Histogram associated to the phase-space in b. a b c 40 Vg = -941 mV 0.3 Vg = -620 mV SII (10-26A2/Hz) 35 1.0 1.0 SII (10 A /Hz) Fano Factor 20 Rdiff (k) Rdiff (k) 0.2 -26 2 30 0.5 -492mV 25 18 0.5 -568mV 0.1 -620mV 20 -941mV 0.0 -999mV 16 0.0 0.0 0.0 0.5 1.0 1.5 2.0 0.0 0.5 1.0 1.5 2.0 0.5 1.0 1.5 2.0 Vsd (mV) Vsd (mV) Vsd (mV) FIG. 6: Shot noise measurements. (a,b) Differential resistance Rdiff (grey) and current noise density SII (blue) as a function of Vsd for two different gate voltage values. (c) Fano factor as a function of Vsd for different Vg values. V. BACKACTION A. Retardation time due to the circuit Figure S7 shows the simplified electrical circuit used to evaluate the electrostatic and the electrothermal backactions when the retardation is given by the circuit. We consider the impedances relevant at the resonance frequency of the resonator. The nanotube with conductance G is connected on the source electrode to the capacitance CRC ' 60 pF of the coaxial cable and the resistance R50 = 50 Ω of an attenuator, which form the impedance of the circuit −1 −1 ZT = R50 + iωCRC . (6) 16 The mechanical vibrations modulate the nanotube conductance by the amount δG. When a DC voltage Vsd is applied to the source electrode nanotube, the conductance modulation generates an oscillating current δiac at the frequency close to ω0 . The current flowing through ZT creates an oscillating voltage δvac on the source electrode, so that δvac = −δiac ZT , (7) δiac = δGVsd + Gδvac . (8) Reference [29] made a similar analysis as here. The difference in the two analysis comes from the fact that our device is biased with a constant voltage, while the device in Ref. [29] is biased with a constant current. δVac S D G + δG R50 ZC = 1 iωCRC FIG. 7: Simplified electrical circuit. The retardation time τRC of the backaction on the vibrations is of the order of 1/ω0 . The retardation time is related to the delay of the modulation of δvac with respect to δG. We thus express δvac as ZT 1 − iωR50 CRC δvac = − δGVsd ' −R50 2 2 δGVsd , (9) 1 + ZT G 1 + ω 2 R50 CRC where we use R50 G << 1 in the last equality. The argument of the complex number in the numerator is ϕ = −arctan (ωR50 CRC ), so that the retardation time is arctan (ω0 R50 CRC ) τRC = . (10) ω0 From the values of CRC , R50 , and ω0 ' 2π · 92 MHz, we get that ω0 R50 CRC = 1.7. Therefore, the retardation time τRC of the circuit is of the order of 1/ω0 . The estimation ω0 τRC ∼ 1 is relevant, since this enhances cooling [31]. B. Electrostatic backaction with the retardation due to the circuit As described in the last subsection, the modulation of the voltage δvac on the source electrode is due to the vibration-induced modulation of the conductance, when the nanotube 17 is biased with a constant voltage. Assuming symmetric electrical contacts, the voltage modulation on the nanotube is δvNT = 12 δvac . This results in the electrostatic force 1 1 − iωR50 CRC ∂G δF = Cg0 Vg δvNT = − Cg0 Vg R50 2 2 Vsd δz. (11) 2 1 + ω 2 R50 CRC ∂z The real part of this backaction force leads to the shift of the spring constant, and the imaginary part to the shift of the decay rate. Using F = −m∆Γback dz dt and dz dt = iωz, we get dGdiff (Cg0 Vg )2   1 R50 ∆kback = Vsd , (12) 2 1 + (R50 ωCRC )2 dVg Cg 2 dGdiff (Cg0 Vg )2   1 R50 CRC ∆Γback =− Vsd . (13) 2m 1 + (R50 ωCRC )2 dVg Cg The retardation time of the backaction on the vibrations is about 1/ω0 . This backaction cannot account for our data. Equation 13 cannot account for the efficient cooling in Figs. 4a,c of the main text, since the predicted ∆Γ is one order of magnitude smaller than that measured in Fig. 4d of the main text. C. Electrothermal backaction with the retardation due to the circuit The closed loop of the backaction goes as follows. The dissipated power increases the temperature of the device. The effective thermal expansion of the device leads to the dis- placement of the nanotube. This displacement reacts back on the dissipated power via ∂G δG = ∂z δz. The delay of the retardation time is τRC . The dissipated power of the voltage-biased nanotube is P = (G + δG)(Vsd + δvac )2 . The first-order expansion of the power reads ZT G δP1 = Vsd2 δG − 2 V 2 δG. (14) 1 + ZT G sd The first term of this equation leads to backaction when taking into account the thermalisa- tion time of the device, as discussed in the next subsection. The second term results in the change of the decay rate because of the retardation of the circuit. This is what is discussed here. The modulation of the dissipated power leads to the modulation of the mechanical tension in the nanotube. The tension modulation depends on the temperature profile along the 18 nanotube and the electrodes, which is something hard to know precisely especially at low temperature when the electron transport is quasi-coherent [32]. In what follows, we assume for simplicity that the dissipation occurs solely in the nanotube, and that temperature rises by δT = δP τph /Cheat . Here, Cheat is the heat capacity of the nanotube and τph is the thermalisation time of the nanotube. We do an additional simplification using τph ' L/v ' 0.1 ns, where L is the nanotube length and v ' 104 m/s is the phonon velocity in nanotubes [33]. Assuming that the thermal expansion is solely occurring in the nanotube, δL the nanotube expands by L = αTEC δT where αTEC is the thermal expansion coefficient of the nanotube. Using Hook0 s law, the change of the mechanical tension is given by δTmech = 2πrE2d δL L where E2d = 340 N/m is the two-dimensional Young0 s modulus of graphene and r the nanotube radius. Overall, the mechanical tension is related to the dissipated power by αTEC E2d τph δTmech = 2πrδP1 . (15) Cheat We emphasize that we would get a linear relation between the tension and the power as in Eq. 15 albeit with a different ratio δTmech /δP1 , if we were considering dissipation in the electrodes and/or thermal expansion of the electrodes. The modulation of the mechanical tension generates a shift in the spring constant and in the decay rate. For this, we use the EulerBernoulli equation that reads " Z  2 # 2 d2 Z d4 Z ES L dZ dZ ρS 2 = −EI 4 + Tmech + dx (16) dt dx 2L 0 dx dx2 where ρ is the nanotube mass density, S the nanotube cross-sectional area, Z the dis- placement at the coordinate x along the nanotube axis, t the time, E the nanotube three- dimensional Young0 s modulus, and I the moment of inertia. We assume that the restoring force is solely given by the mechanical tension, as it is the case in our experiment, so that 4 EI ddxZ4 → 0. We set Z(x, t) = zs × φs (x) + z1 (t) × φ1 (x). (17) Here, φs (x) and φ1 (x) are the profiles of the static deformation and the measured eigenmode with max(φs (x)) = max(φ1 (x))=1, whereas zs and z1 (t) are the associated time depen- dent displacements. We use φs (x) = φ1 (x) = sin(πx/L), a good approximation since the nanotube is under tensile tension. The equation of motion is obtained by multiplying the EulerBernoulli equation by φ1 (x) and integrating it along x. The mechanical tension is 19 0 0 Tmech = Tmech − δTmech where Tmech is the time-independent tension in the nanotube. The time-dependent tension creates a term proportional to z1 . The real part of this term induces a shift in the spring constant, and the imaginary part leads to a shift in the decay rate, 1 dGdiff Cg0 ∆kback = αm Vg zs Vsd2 , (18) CRC R50 dVg Cg dGdiff Cg0 ∆Γback = −α Vg zs Vsd2 , (19) dVg Cg π 3 r αTEC E2d τph 1 2   2CRC GR50 α= . (20) L Cheat m (ωCRC R50 )2 + 1 The retardation time of the backaction on the vibrations is about 1/ω0 , that is, τ ' 2 ns. We now compare the measurements of the decay rate as a function of Vsd in Fig. 3b and Fig. 4d with Eq. 19 (pink lines). We estimate that the static displacement is zs = −0.97 nm Cg0 Vg2 at Vg = −616 mV and zs = −2.08 nm at Vg = −943 mV using zs = − π4 mω02 from the derivation of the EulerBernoulli equation. We use Cheat = 1.6·10−22 J/K from Ref. [34] where the specific heat capacity of nanotubes is 3 · 10−5 J/gK at 0.1 K. The only free parameter left is the thermal expansion coefficient. From the comparison between the measurements and this model, we get αTEC = 9 · 10−8 1/K. Although we did not find any report on αTEC for nanotubes, graphene, and graphite at such low temperatures, the order of magnitude that we get is rather realistic. To finish this subsection, we discuss the third-order expansion of the power modulation related to Eq. 14, since it is relevant for the self-oscillation regime. The third-order expansion reads  2 ZT δP3 = Vsd2 δG3 . (21) 1 + ZT G Carrying out the same derivation as that described above, we obtain two additional back- action force terms, that is, a Duffing force and a nonlinear decay force of the form z 2 dz dt . dGdiff Depending on the sign of dVg , the nonlinear decay force can be negative, so that this force further increases the amplification, especially when the amplitude of motion is large. The exact derivation of the nonlinear decay force is difficult due to its renormalisation by the other nonlinear forces. The study of this nonlinear force is beyond the scope of this Letter. 20 D. Electrothermal backaction with the retardation due to the thermalisation time of the device In contrast to the backaction discussed in the last subsection, this backaction arises from the modulation of the power δP1 = Vsd2 δG in Eq. 14 associated to the thermalisation time of the device. The derivation of the backaction is similar to that above. 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