Thermally Induced Parametric Instability in a Back-Action Evading

Nov 7, 2012 - ABSTRACT: We report the results of back-action evading experiments utilizing a tightly coupled electro-mechanical system formed by a rad...
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Thermally Induced Parametric Instability in a Back-Action Evading Measurement of a Micromechanical Quadrature near the Zero-Point Level J. Suh,† M. D. Shaw,†,‡ H. G. LeDuc,‡ A. J. Weinstein,† and K. C. Schwab*,† †

Applied Physics, California Institute of Technology, Pasadena, California 91125, United States Jet Propulsion Laboratory, California Institute of Technology, Pasadena, California 91109, United States

‡

ABSTRACT: We report the results of back-action evading experiments utilizing a tightly coupled electro-mechanical system formed by a radio frequency micromechanical resonator parametrically coupled to a NbTiN superconducting microwave resonator. Due to excess dissipation in the microwave resonator, we observe a parametric instability induced by a thermal shift of the mechanical resonance frequency. In light of these measurements, we discuss the constraints on microwave dissipation needed to perform BAE measurements far below the zero-point level. KEYWORDS: Back-action evasion, micromechanical resonator, superconducting resonator, parametric instability

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nonlinearity in the capacitive coupling between the mechanical resonator and the electrical resonator.11 This nonlinearity arises due to the non-negligible value of ∂2C/∂x2 that produces an electrostatic spring constant, ke = (1/2)(∂2C/∂x2)V2, and a frequency shift of the mechanical device, Δωm ≈ (1/2)ωmke/ km, where V is the instantaneous voltage oscillating across the coupling capacitance and km is the effective spring constant. In this work, we identify another source of parametric instability that is driven by thermally induced shifts of the mechanical resonance frequency. In the two-tone BAE scheme,2 the energy stored in the microwave resonator oscillates at a frequency of 2ωm. Dissipation in the microwave resonator generates heat, and due to the short thermal relaxation time of the micrometer-scale superconducting device, the temperature of the mechanical resonator follows the modulated power at 2ωm. The mechanical resonance frequency ωm depends on the temperature, and therefore ωm is modulated at 2ωm. Once the amplitude of this modulation exceeds the mechanical damping rate, a parametric instability arises.12 A somewhat similar thermally induced parametric instability has been observed in an opto-mechanical structure13 at room temperature, where significant heating reaching ∼60 μW was required for instability. In our measurement, the high quality factor of the mechanical resonator combined with low thermal capacitance and conductance makes it more susceptible to the parametric effect and lowers the level of dissipation required at the onset of instability. As a result, we observe that less than 100 pW of dissipation is enough to initiate the oscillation, significantly lower than reported in ref 13. This instability limits

hen performing continuous linear position measurements of a harmonic oscillator, quantum mechanics places a lower bound on the total measurement uncertainty, xSQL = (ℏ/2mωm)1/2, called the standard quantum limit (SQL),1,2 where ℏ is the reduced Planck constant, m is the mass, and ωm is the resonance frequency. This limit is the result of the Heisenberg uncertainty principle: a position measurement generates quantum back-action on the oscillator momentum, and under free evolution this back-action increases the uncertainty in subsequent position measurements. However, since the 1970s it has been understood how to avoid this back-action limitation if one is willing to give-up complete knowledge of the oscillator’s motion. Coupling exclusively to one motional quadrature3,4 forms a back-action evading (BAE) measurement that is an elementary example of a quantum nondemolition measurement.5 In this case, quantum backaction must perturb the uncoupled quadrature, but this perturbation does not evolve in time to affect the coupled quadrature;1 the two quadratures are decoupled and are constants of the motion. Furthermore, starting from a thermal state, it should be possible to prepare a squeezed state of motion by implementing feedback in combination with a BAE measurement with measurement imprecision sufficiently far below the SQL.2 Although BAE measurements have been performed on optical fields6 and mechanical resonators,7−11 a BAE measurement of mechanical motion with imprecision below the SQL has not been demonstrated yet. The work presented here investigates the experimental challenges to realize this elementary technique at such an extreme sensitivity. Recent attempts have achieved an imprecision of 4·xzp, where the zeropoint of motion xzp = xSQL. In these measurements, the sensitivity was limited by a parametric instability caused by a © 2012 American Chemical Society

Received: September 7, 2012 Revised: October 30, 2012 Published: November 7, 2012 6260

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etched using a BCl3/Cl2 plasma to form the capacitor counterelectrode and ground plane. Next, a 230 nm sacrificial layer of SiO2 is DC-sputtered under RF bias to give a smooth step edge over the metal layer. To allow the top layer of metal to make electrical contact with the bottom layer, and to provide mechanical anchoring for the freestanding capacitor membrane, a via is patterned and etched into the sacrificial layer. To allow a smooth step edge over the sacrificial layer, we developed a tapered SiO2 plasma etch process based on reflow of PMMA. A bilayer of PMMA and UVN-30 resists are spun on the wafer, the top UVN-30 layer is patterned with photolithography, and the pattern is transferred into the PMMA with an O2 RIE plasma etch. The thicknesses of the layers are chosen such that the UVN-30 has been completely removed at the point that the plasma etches completely through the PMMA. The PMMA is then reflowed for 5 min at 200 °C, creating a smooth tapered edge which is transferred into the SiO2 with a CHF3/O2 ICP plasma etch. Next, a 300 nm layer of tensile NbTiN is sputtered onto the wafer and patterned and etched to form the top layer of the membrane and the spiral inductor. As tensile refractory metal films are known to have degraded microwave properties,18 careful optimization of the sputtering parameters was required to maintain film quality. Compressive NbTiN films sputtered under identical conditions to those used in this sample had a transition temperature Tc = 14.2 K and a resistivity ρ ∼ 100 μΩ·cm, while the optimized tensile films were measured to have Tc = 13.7 K and ρ ∼ 200 μΩ·cm. Finally, the suspended capacitor membrane and vacuum-gap crossovers for the spiral inductor are released in a 90 min exposure to 10:1 BOE followed by critical point drying. To minimize losses due to dielectrics,19 none of the sacrificial SiO2 remained on the finished device after the HF etch to release the structure. The cryogenic and electronic measurement setup is similar to that reported in ref 11 and includes cryogenic attenuators to absorb thermal noise at microwave frequencies, superconducting coaxial cables, cryogenic circulators at the mixing chamber of the dilution refrigerator, and a cryogenic HEMT amplifier at 4 K. We employ room temperature, tunable sapphire whispering gallery mode filters20 to attenuate the phase noise of the microwave sources used to generate the two-tone BAE pumps. We utilize a filter mode with a quality factor of ∼20 000 which shows a nearby antiresonance which can be tuned to be coincident with the LC resonance. We achieve phase noise rejection of >20 dB at the microwave resonance frequency which should produce less than 0.1 quanta in the microwave resonator as a result of the phase noise of the source (3 = −140 dBc at 10 MHz from a carrier). The NbTiN membrane shows a fundamental mechanical resonance at 10.01 MHz (= ωm/2π) which is close to the expected value from finite element simulations assuming a restoring force from Young’s modulus of 105 GPa and a density of 5500 kg/m3. The mechanical resonance has a line width of 200 Hz (= γm/2π) at 100 mK. The microwave LC resonance frequency is 7.07 GHz (= ωLC/2π) with a total line width κtot = 2π·(672 ± 3) kHz. κtot is dominated by a very large and unexpected internal loss, κint = 2π·(609 ± 16)kHz, whose origin has not yet been identified. It is this large electrical dissipation which results in the parametric instability observed in this work. The input and output coupling strength are κin = 2π·(23 ± 5) kHz and κout = 2π·(40 ± 8) kHz. The coupling between the electrical and the mechanical resonators is given by the modulation of capacitance due to mechanical motion:21 g =

the maximum measurement strength and ultimately bounds minimum quadrature imprecision at 2.5xzp. Based upon these observations, a thermal model for the heat flow in the system is presented to elucidate the experimental challenges for reaching subzero-point motion sensitivity in the two-tone BAE scheme. Figure 1 shows scanning electron micrograph of the device and an outline of the fabrication steps. Our mechanical

Figure 1. (a, left) Scanning electron micrograph of the mechanical resonator which serves as capacitance in the microwave resonator circuit. (right) Device micrograph showing microwave resonator circuit composed of spiral inductor and vacuum gap capacitor. The image is taken with 45° of tilt. (b) Outline of device fabrication process. (1) A 40 nm layer of compressively strained NbTiN is sputtered onto a Si wafer, patterned with photolithography, and dry etched to form the ground plane, CPW microwave lines, membrane capacitor counterelectrode, and ground contact for the spiral inductor. The diagram on the left is a side-view layer stack, while the diagram on the right is a top-down schematic. Relative feature sizes and thicknesses are exaggerated for clarity. The inset gives a color-coded legend for the different material layers. The red line through the membrane indicates the plane through which the material stack is shown. (2) A 230 nm sacrificial layer of SiO2 is sputtered under RF bias to give a rounded step edge profile over the NbTiN layer. A tapered dry etch process is used to etch a via “window’’ through the SiO2 to allow the two metal layers to contact and to provide mechanical anchoring for the membrane capacitor and inductor crossovers. (3) A 300 nm layer of tensile NbTiN is sputtered onto the structure, lithographically patterned, and dry etched to form the top of the membrane capacitor and the body of the spiral inductor. (4) The sacrificial layer is removed with a 90 min exposure to buffered oxide etch to form a self-supporting membrane capacitor and vacuum-gap crossover structures in the spiral inductor.

resonator is a 15 μm diameter NbTiN membrane, 300 nm thick, separated by a gap of 230 nm from a NbTiN base electrode (Figure 1a), forming the capacitive element of a microwave LC resonator. NbTiN is chosen due to the low dissipation observed in microwave kinetic inductance detectors,14 which is a result of the low density of two-level systems (TLS).15−17 We begin our fabrication process with a ⟨100⟩ oriented highresistivity (10 kΩ·cm) single-crystal Si substrate (Figure 1b). After removing the native Si oxide with buffered oxide etch (BOE), we deposit 40 nm of compressively stressed NbTiN by DC magnetron sputtering at a base pressure of 1 × 10−9 Torr. This material is then patterned with photolithography and ICP 6261

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∂ωLC/∂x. Compared to nanowires used by Hertzberg et.al.,11 the planar geometry provides 3 orders of magnitude tighter coupling (g ≃ 2π·3.8 MHz/nm) due to the larger fraction of its capacitance which is sensitive to mechanical motion21 (g0 = g·xzp ≈ 2π·5.8 Hz). Since ∂2C/∂x2 is proportional to C, whereas the measurement rate 4g20np/κ scales as C2, a larger coupling capacitance also increases the threshold for the parametric instability due to ∂2C/∂x2 observed with the wire resonators;11 np (= CV2/(2ℏωLC)) is the number of pump photons, for the instantaneous voltage V oscillating across the microwave resonator capacitance C. We estimate this type of instability to occur at a pump strength of np ∼ 1010 photons where we predict the measurement imprecision on one quadrature will be ∼10−1·xzp. The LC resonator loss and coupling rates are determined by calibrated S21 measurements, performed independently at 300 mK. The measured LC coupling rates relate number of photons to incident power. By fitting optical damping21 versus number of photons, we determine g0; g is then calculated using xzp = 1.5 fm which is estimated from the size and density of the NbTiN membrane. The parametric coupling g leads to up and down-conversion of microwave photons, and the mechanical motion is detected by measuring these frequency converted microwave photons as they leave the LC resonator (Figure 2a). Figure 2b is an example of a measured thermomechanical noise spectrum, upconverted from a single microwave tone at ωred = ωLC − ωm. As in ref 22, the mechanical occupation nTm is calibrated by

integrating these mechanical noise peaks at various sample temperatures (Figure 2c). The integrated thermomechanical noise power, Pm, with a correction for detection efficiency22 is linear in temperature in excellent agreement with equipartition theorem keffx2̅ = kBT. This gives the calibration of the average mechanical occupation nm = keffx2̅ /(ℏωm) against the measured noise power Pm. The finite occupation of the microwave resonator nLC is more challenging to determine accurately. Measurements of the microwave noise spectrum radiated from the LC resonator, combined with an estimated κout leads to an estimate of nLC. However, uncertainties in the performance of the microwave circuit, such as attenuations and gains, lead to an uncertainty of no less than a few dB. Ideally, for temperature far below the freeze-out temperature of the LC resonator, T < ℏωLC/kB ≈ 340 mK, nLC is expected to be far less than 1. However, as we see in this experiment and in others in the field,11,21,22 nLC becomes significant when pumping the device at a sufficiently large pump strength. This finite occupation is responsible for limiting the sideband resolved cooling process and for the production of a squashing effect of the observed mechanical occupation:22 nm = nTm − 2nLC where nm is the observed mechanical occupation. To accurately determine nLC, we utilize the following technique. We stabilize the dilution refrigerator at 100 mK and apply a single, red-detuned pump tone at ωred = ωLC − ωm. The pump tone is relatively weak and produces negligible cooling at np = 104, where the optical damping23 is less than 10% of γm. The LC resonator is then driven through the input port with varying levels of white noise generated through the amplification of 300 K thermal noise: in this way, we can control nLC. The apparent mechanical occupation is then measured versus the observed power in the microwave resonance which radiates out the output port as shown in Figure 2d. The squashing relationship is then used to calibrate nLC. In this way, we transfer the thermal calibration from the mechanical resonator, based on the equipartition theorem, onto the LC resonator. By combining two equal-intensity pump tones with frequencies ωLC ± ωm, the internal cavity field is amplitudemodulated by ωm, and BAE detection of a single motional quadrature is realized.2 It is essential to generate tones of equal power at the device, since an imbalance results in back-action on the observed quadrature X. Thus for each BAE measurement, at each pump power, we perform a separate detuned twotone measurement (DTT) with tones at ωLC ± (ωm + Δ), where Δ = 2π·2kHz ≪ κtot. This results in two mechanical noise peaks, offset by 2Δ, each paired with one pump tone. The frequency difference between the up- and down-converted thermomechanical peaks is small enough that the difference in microwave transmission is negligible. From the slope of the transmission curve at ωLC, we estimate this difference to be less than 10−4. As a result, we can carefully compare the integrated noise power under each peak and adjust the microwave pump strength to balance the power in each tone. With a measurement and averaging time of a few minutes, this comparison and balance can be done to an accuracy of approximately 0.1%. After balancing the pump powers for each data point, we set Δ = 0 to prepare BAE tones (see inset of Figure 3b). The measured mechanical occupation nm for each number of photons np is shown in Figure 3b. In the BAE configuration, the mechanical resonator becomes unstable and exhibits an abrupt increase in nm above np ∼ 2 × 107. The instability is

Figure 2. (a) The detector microwave resonator is modeled as parallel RLC circuit. Its capacitance C is modulated by small mechanical motion x as shown up to second order in x. Applied microwave tones (Pin) experience frequency conversion by this modulation leave the resonator (Pout). (b) A measured output spectrum showing thermomechanical noise at 50 mK. A Lorentzian is fitted to the data (red) which gives the total power (Pm), resonance frequency (ωm), and line width (γm). (c) Measured thermomechanical noise vs sample temperature. A correction factor (κ/κ0)2 on Pm is given to account for the small change of line width of microwave resonance(κ) over the sample temperature. The linear relation demonstrates the equipartition theorem, and it calibrates the mechanical quanta against Pm for the rest of the measurements. (d) Calibration of the occupation of microwave resonator nLC. We use a noise squashing effect22 to convert calibrated nm to nLC. 6262

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Figure 4. (a, green) Mechanical resonance frequency shifts up as more pump photons are applied. Δωm reaches 2γm at np ∼ 2 × 107 which satisfies the criterion of parametric instability, agreeing with the observation in 2b. (blue) The expected ∂2C/∂x2 contribution to Δωm. (b, inset) Mechanical resonance frequency shift (Δωm) vs sample temperature. The temperature deduced from Δωm, T(ωm), is plotted together with the mode temperature measured by thermomechanical noise power, T(Pm). Their agreement indicates the observed mechanical resonance shift, and the parametric instability is the thermal effect.

Parametric instability was also observed in the recent work of Hertzberg et al.,11 where it was attributed to the nonlinearity in coupling capacitance with respect to motion(∂2C/∂x2). However, the frequency pulling due to this effect in our system is estimated to have the opposite sign and to be 3 orders of magnitude smaller than what is observed here (see blue line in Figure 4a). We have also considered a possible optical spring effect due to the cavity shift caused by driven two-level systems,16 but it predicts a negative frequency pulling which is contrary to our observations. By contrast, we identify this instability as a thermal effect. As the temperature of the mechanical resonator increases with pump power due to dissipation in the microwave resonator, its resonance frequency shifts. In the inset of Figure 4b, we measure ωm at different sample temperatures and find a monotonic increase. The magnitude of ∂ωm/∂T is found to be 30 times larger than we observe in other measurements.25,26 Using ωm as our thermometer, we deduce the temperature of the mechanical mode and compare it to the thermomechanical noise. Figure 4b shows good agreement in between the two, indicating that the frequency shift Δωm is the thermal effect. Given the high internal loss in the microwave resonator and the strong dependence of device parameters on np, we expect that microwave dissipation is the source of heat in the observed thermal parametric oscillation. In Figure 5b, we plot the microwave resonator mode temperature Tcav versus number of

Figure 3. (a) The position imprecision reaches 2.5·xzp before onset of parametric instability. (red) Expected ximp from measured device parameters. (b) Average mechanical occupation (nm) vs number of pump photons (np). (red, blue) Measured from red or blue-detuned tones in DTT. (green) Measured in BAE. (inset) Back-action evasion (BAE) and detuned two-tone (DTT) measurement. We prepare BAE tones by balancing two mechanical noise powers measured with detuned symmetric tones from BAE configuration. (c) Mechanical damping (γm) vs np. The mechanical resonator becomes unstable at np ∼ 2 × 107 in BAE. This is absent in DTT, indicating the nature of instability is parametric oscillation due to modulation of microwave power at 2ωm with BAE tones. (d) fractional line width narrowing η shows linear relation with np, which is consistent to parametric amplification12 via spring constant modulation Δkm = (∂km/∂np)np.

accompanied by a decrease in mechanical damping which is shown in part c. The measurement imprecision, which is the dominated by the detector amplifier noise floor, reaches 2.5·xzp before reaching instability (Figure 3a). If we utilized a near quantum limited microwave amplifier,24 we expect to have achieved an imprecision of 0.2·xzp. Although it is not detectable in our current realization of this experiment, at this maximum pump strength, we expect the motion of the mechanical resonator to be conditionally squeezed to reach VX/VY ≅ 0.4 where VX and VY are the variances of two quadratures X and Y. We do not observe these effects of line width narrowing and instability in nm when driving the system with two detuned pump tones, which suggests parametric amplification occurring in the BAE configuration. Also, fractional line width narrowing η (= Δγm/γm) in the BAE configuration is proportional to np as shown in Figure 3d, which is consistent with degenerate parametric amplification12 with an effective spring constant proportional to np. The modulation of the microwave field in the BAE configuration implies that the number of pump photons varies in time as np,BAE(t) = np(1 + cos(2ωmt)). If the mechanical resonance frequency depends on np, the parametric amplification becomes possible when the resonance frequency is modulate at 2ωm, that is, Δωm = Δωm0 cos(2ωmt). When Δωm0 is larger than the mechanical line width γm, the threshold for parametric oscillation is reached.12 In Figure 4a, we observe the mechanical resonance frequency shifts in the BAE configuration as a function of np. It shows that Δωm reaches 480 Hz at np = 2 × 107 and satisfies Δωm0 > γm.

Figure 5. (a) An example of LC resonator noise spectrum. (b) The temperature of microwave resonator (Tcav) increases as more photons are applied. Tcav is measured from total noise power under Lorentzian (K). fit in a. (red) Power law fit yields T = (0.021 ± 0.01)n0.28±0.03 p 6263

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capacity of superconducting niobium titanium.32 The thermal time constant of the mechanical resonator is RmCm ≃ 0.2 ns ≪ 1/(2ωm) (= 8 ns), and the thermal response of the mechanical resonator is expected to be fast enough to support parametric excitation. Also, having measured the empirical power law of Tcav as a function of pump power, we estimate that the microwave dissipation must be kept below 0.3 pW to keep the microwave resonator close to the ground state (nLC < 1/2). At this level of dissipation, we estimate that Tm increases by ∼50 μK based on our estimate of RT, which is negligible compared to our lowest bath temperature 20 mK. With a relatively straightforward improvement in device design, we expect that subquantum limit quadrature sensitivity is within reach. For instance, with κint/2π ≃ 40kHz,21 κout/2π = 500 kHz, and a more typical ∂ωm/∂T, which is 10 times lower than observed here, the parametric instability is estimated to occur at np ∼ 109. At this np, the measurement imprecision becomes 10−1·xzp, well below the quantum limit. At this point, the mechanical mode is conditionally squeezed, and feedback force can be employed to transform it into an unconditionally squeezed state2 with an optimized Kalman filter.33 In conclusion, a back-action evading measurement of a single quadrature of micromechanical motion showed near-zero point imprecision reaching 2.5xzp. It is limited by a parametric instability driven by a thermal frequency shift of the mechanical mode due to the internal dissipation of the microwave resonator. By minimizing the dissipation, subquantum limit quadrature sensitivity is believed to be within reach, which is the first step toward generating mechanical squeezed states with feedback.

pump photons np. Tcav is derived from Bose-Einstein statistics, nLC = 1/[exp(ℏωLC/kBTcav) − 1], with the occupation nLC measured from the integrated noise under the microwave resonance,22 as shown in Figure 5a, utilizing the squashing calibration described above. The empirical power law fit gives Tcav ∼ n0.28±0.03 . Since the microwave resonator has constant p loss throughout the measured range of np, this indicates the cavity electron temperature Tcav behaves as Tcav ∼ P0.28±0.03 for diss the dissipated power Pdiss, which is close to P1/4 diss. In principle, electron−phonon coupling in metal gives rise to the fifth power 27,28 law, such that Tcav ∼ P1/5 However, impurities in the diss. system can change this power law toward the fourth power,29,30 and the power law close to 1/4 is shown in a previous study on electron heating in normal aluminum.31 These observations support a speculation that the internal loss might be because of the normally conducting region of a size about Ω ∼ (100 nm)3, possibly existing in many joints and crossovers between the top and bottom layers of NbTiN. The volume Ω is estimated based upon a known electron−phonon coupling parameter28 Σ ∼ 2 × 109 W·m−3·K−5 in the fifth power law Pdiss = ΣΩ(T5cav − T50), where T0 (= 100 mK) is the bath temperature. By measuring the mechanical mode temperature simultaneously, we develop a thermal model describing the temperatures of various parts of the system and the heat flow (Figure 6b). The heat Pdiss is generated in the electron system in

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AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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Figure 6. (a) The mechanical mode temperature Tm (= T(Pm)) and microwave resonator temperature Tcav is plotted over power dissipation in the microwave resonator (Pdiss). The sample temperature is 100 mK. (b) The heat is generated in the electron system and flows into the bath through the phonon system. Based upon the thermal model, we estimate thermal time constant of the mechanical resonator to be about 0.2 ns, which is short enough to sustain the parametric oscillation.

ACKNOWLEDGMENTS We would like to acknowledge generous and essential support from DARPA (DARPA-QUANTUM HR0011-10-1-0066) and the National Science Foundation (NSF-DMR 1052647, NSFIQIM 1125565). Fabrication was performed at the Microdevice Laboratory at JPL, and the Kavli Nanoscience Institute at Caltech.

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NbTiN which is at the temperature of Tcav. It flows into the bath through the phonon system, raising its temperature and also the mechanical mode temperature Tm, as shown in Figure 6a. Since the mechanical resonance shift is mostly caused by interactions with two-level systems in the amorphous solid,15 the temperature deduced from the resonance shift, T(ωm), reflects the phonon temperature where two-level systems are thermalized. The fact that we observe a good agreement between T(ωm) and the temperature from thermomechanical noise, T(Pm), indicates the mechanical mode is also tightly thermalized to the phonon system and the other thermal flow channels are negligible. Therefore Pdiss flows mostly through Rm, which is the thermal resistance between the contanttemperature thermal bath and the phonon system. With measured ΔTm ≃ 10 mK at Pdiss ≃ 54 pW, we deduce Rm ≃ 1.8 × 108 K/W. We also estimate the thermal capacitance of the mechanical resonator as Cm ≃ 1 aJ/K from the phonon heat

REFERENCES

(1) Caves, C. M.; Thorne, K. S.; Drever, R. W. P.; Sandberg, V. D.; Zimmermann, M. Rev. Mod. Phys. 1980, 52, 341. (2) Clerk, A. A.; Marquardt, F.; Jacobs, K. New J. Phys. 2008, 10, 095010. (3) Braginskii, V. B.; Vorontsov, Y. I. Sov. Phys. Usp. 1975, 17, 644. (4) Thorne, K. S.; Drever, R. W. P.; Caves, C. M.; Zimmermann, M.; Sandberg, V. D. Phys. Rev. Lett. 1978, 40, 667. (5) Bocko, M. F.; Onofrio, R. Rev. Mod. Phys. 1996, 68, 755. (6) Levenson, M. D.; Shelby, R. M.; Reid, M.; Walls, D. F. Phys. Rev. Lett. 1986, 57, 2473. (7) Bocko, M. F.; Johnson, W. W. Phys. Rev. A 1984, 30, 2135. (8) Speitz, G. W.; Mann, A. G.; Hamilton, W. O.; Oelfke, W. C. Phys. Lett. A 1984, 104, 335. (9) Marchese, L. E.; Bocko, M. F.; Onofrio, R. Phys. Rev. D 1992, 45, 1869. (10) Caniard, T.; Verlot, P.; Briant, T.; Cohadon, P. F.; Heidmann, A. Phys. Rev. Lett. 2007, 99, 110801.

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Letter

(11) Hertzberg, J. B.; Rocheleau, T.; Ndukum, T.; Savva, M.; Clerk, A. A.; Schwab, K. C. Nat. Phys. 2010, 6, 213. (12) Rugar, D.; Grüter, P. Phys. Rev. Lett. 1991, 67, 699. (13) Zalalutdinov, M.; Zehnder, A.; Olkhovets, A.; Turner, S.; Sekaric, L.; Ilic, B.; Czaplewski, D.; Parpia, J. M.; Craighead, H. G. Appl. Phys. Lett. 2001, 79, 695. (14) Barends, R.; Vercruyssen, N.; Endo, A.; de Visser, P. J.; Zijlstra, T.; Klapwijk, T. M.; Diener, P.; Yates, S. J. C.; Baselmans, J. J. A. Appl. Phys. Lett. 2010, 97, 023508. (15) Phillips, W. A. Rep. Prog. Phys. 1987, 50, 1657. (16) Gao, J. The Physics of Superconducting Microwave Resonators. Ph.D. thesis, California Institute of Technology, Pasadena, CA, 2008. (17) Kumar, S.; Gao, J.; Zmuidzinas, J.; Mazin, B. A.; LeDuc, H. G.; Day, P. K. Appl. Phys. Lett. 2008, 92, 123503. (18) Kuroda, K.; Yuda, M. J. Appl. Phys. 1988, 63, 2352. (19) Cicak, K.; Allman, M. S.; Strong, J. A.; Osborn, K. D.; Simmonds, R. W. IEEE Trans. Appl. Supercond. 2009, 19, 948. (20) Dick, G. J.; Santiago, D. G.; Wang, R. T. IEEE Trans. Ultrason., Ferroelectr., Freq. Control 1995, 42, 812. (21) Teufel, J. D.; Li, D.; Allman, M. S.; Cicak, K.; Sirois, A. J.; Whittaker, J. D.; Simmonds, R. W. Nature (London) 2011, 471, 204. (22) Rocheleau, T.; Ndukum, T.; Macklin, C.; Hertzberg, J. B.; Clerk, A. A.; Schwab, K. C. Nature (London) 2010, 463, 72. (23) Marquardt, F.; Chen, J. P.; Clerk, A. A.; Girvin, S. M. Phys. Rev. Lett. 2007, 99, 093902. (24) Ho Eom, B.; Day, P. K.; LeDuc, H. G.; Zmuidzinas, J. Nat. Phys. 2012, 8, 623. (25) LaHaye, M. D.; Buu, O.; Camarota, B.; Schwab, K. C. Science 2004, 304, 74. (26) Suh, J. Coupled dynamics of a nanomechanical resonator and superconducting quantum circuits. Ph.D. thesis, California Institute of Technology, Pasadena, CA, 2011. (27) Roukes, M. L.; Freeman, M. R.; Germain, R. S.; Richardson, R. C.; Ketchen, M. B. Phys. Rev. Lett. 1985, 55, 422. (28) Wellstood, F. C.; Urbina, C.; Clarke, J. Phys. Rev. B 1994, 49, 5942. (29) Sergeev, A.; Mitin, V. Phys. Rev. B 2000, 61, 6041. (30) Karvonen, J. T.; Maasilta, I. J. Phys. Rev. Lett. 2007, 99, 145503. (31) Timofeev, A. V.; García, C. P.; Kopnin, N. B.; Savin, A. M.; Meschke, M.; Giazotto, F.; Pekola, J. P. Phys. Rev. Lett. 2009, 102, 017003. (32) Bishoff, J.; Vassilev, P. G.; Goncharov, I. N. Cryogenics 1982, 22, 131. (33) Bruland, K. J.; Garbini, J. L.; Dougherty, W. M.; Sidles, J. A. J. Appl. Phys. 1998, 83, 3972.

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