1 A coherent nanotube oscillator driven by electromechanical backaction Edward Laird A coherent nanotube oscillator driven by electromechanical backaction

Yutian Wen Natalia Ares Tian Pei Felix Schupp Andrew Briggs

(at Oxford)

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The limits of displacement sensing Suspended carbon nanotube ~106 nucleons

Diameter ~5 nm

Length ~800 nm

• High mechanical quality factor • High frequency (200 MHz ≡ 10 mK) • Mechanically compliant • Can be integrated into electronic circuits 3 The limits of displacement sensing: Magnetic resonance microscopy

Force per proton: ~10-20 N

Review: Poggio & Harzheim (2018)

4 A coherent nanotube oscillator driven by electromechanical backaction

1. Measuring vibrations with a single-electron transistor

2. Creating a nanomechanical oscillator

3. Characterizing the oscillator

5 Question 1: How precisely can we measure nanomechanical vibrations?

Standard quantum limit for continuous measurements (m/√Hz):

Quality factor

Mechanical frequency

Theory: Caves et al. (RMP 1982) Mass Experiment: LaHaye et al. (Science 2004) Review: Clerk et al. (RMP 2010) 6 Question 2: Can we make a laser for sound?

Microwave amplification by stimulated emission of radiation (Townes, 1955)

7 A vibrating carbon nanotube device

Laird et al (2011)

~2 nm

1 µm

8 The single-electron transistor Energy levels nA ) Current ( Current Gate voltage (V) Hüttel et al (van der Zant group, 2009) 9 Sensing vibrations

Motion Current

Displacement

Hüttel et al (2009) 10 Fast readout for nanotube vibrations

SiGe amplifier

Details: Wen et al. (2018) Schupp et al. (2018) 11 How precisely can we measure nanomechanical vibrations?

12 Is this a laser analogue?

13 REPORTS tions do not simply smooth the stepwise (G ~ f0), damping of the mechanical motion is addition to the overall softening of k yielding the transition from the static single-electron tuning minimized, and we observe the highest value of frequency dips of Fig. 2, the fluctuating charge on shifts. Instead, fluctuations caused dips in the Q.OnaCoulombpeak,chargefluctuationsare the dot also changes a,givingasofteningspring resonant frequency up to one order of magnitude maximized (G >> f0), and Q decreases to a few (a <0)outsideofthefrequencydip(Coulomb greater than the single-electron tuning shifts. As thousand. These results explicitly show that detec- valleys) and a hardening spring (a >0)insidethe shown in (13, 22)anddiscussedindetailinthe tor back-action can cause substantial mechanical frequency dip (Coulomb peaks), as shown in Fig. supporting online material (SOM) (20), the dy- damping. The underlying mechanism for the damp- 3. The sign of a follows the curvature of f0(Vg) namic force modifies the nanotube’sspringcon- ing is an energy transfer occasionally occurring induced by the fluctuating electron force, giving a stant (k)resultinginasofteningofthemechanical when a current-carrying electron is pushed up to change in sign at the inflection point of the fre- resonance. The shape of the frequency dip can be ahigher(electrostatic)energybythenanotube quency dip. Nonlinearity from the single-electron altered by applying a finite bias (Vsd)acrossthe motion before tunneling out of the dot. This gain force in our device dominates and is much stronger nanotube. Starting from deep and narrow at small in potential energy is later dissipated in the drain than that from the mechanical deformation (20). Vsd =0.5mV,thedipbecomesshallowerand contact. Figure 3, E and F, shows the regime of further broader with increasing Vsd.Thisdipshapelargely If we drive the system at higher RF powers enhanced RF driving. Now the nonlinearity is no resembles the broadening of Coulomb blockade (Fig. 3, C and D), we observe an asymmetric longer a perturbation of the spring constant, but peaks that occurs with increasing Vsd.Thus,we resonance peak, along with distinct hysteresis be- instead gives sharp peaks in the line shape and conclude from Fig. 2 that the single-electron tween upward and downward frequency sweeps. switching between several different metastable tuning oscillations are a mechanical effect that is Theoretically, this marks the onset of nonlinear modes [see further data in the SOM (20)]. At this adirectconsequenceofsingle-electrontunneling terms in the equation of motion, such as in the strong driving, we observe highly structured non- oscillations. well-studied Duffing oscillator (23, 24). k is mod- linear mechanical behavior that arises from the Besides softening, the charge fluctuations also ified by a large oscillation amplitude (x)thatis coupling of the resonator motion to the quantum provide a channel for dissipation of mechanical accounted for by replacing k with (k + ax2). The dot. energy. Figure 3A shows the resonance dip for time-averaged spring constant increases if a >0, In Fig. 3, we studied nonlinear coupling be- small RF power, with frequency traces in Fig. 3B. which is accompanied by a sharp edge at the high- tween the quantum dot and the mechanical res- In the Coulomb valleys, tunneling is suppressed frequency side of the peak; vice versa for a <0.In onator by applying a large RF driving force at a on April 26, 2010 Fig. 4. Spontaneous driving 05dI/dV (µS) 0 -10 dI/dV (µS) 5 Evidence for self-driven oscillations of the mechanical resonance

sd by single-electron tunneling. A B D I (A)Differentialconductance 2 (dI/dVsd)showingridgesof 10 5 sharp positive (deep red) and (nA) sd

negative (deep blue) spikes (ar- I B rows). ( )Similarridgesmea- ) sured on device two (trench Gate voltage-4.93 Vg nA www.sciencemag.org width = 430 nm) in the few- (mV) 5 (mV) 5 sd sd

– – V

hole regime (four-hole to V three-hole transition). Spikes in dI/dVsd appear as steplike ridges in current. (C)shows -10 Current ( the data from the upper half Self-oscillation of (B), but now as a three- -2 dimensional current plot. The threshold 0 0 ridges are entirely reproduc- Downloaded from ible. (D)(Inset)Coulombpeak -5.19 V (V) -5.17 -1.05 V (V) -0.95 Gate voltage g g E at V =0.5mVshowinglarge sd 284.5 switching-steps. (Main plot) From Steele et al (Kouwenhoven group, 2009) 14 Zoom-in view on data from C inset. (E)RFdrivenmechanical resonance measured for the 10 nA same Coulomb peak in (D) at adrivingpowerof–50 dB. Outside the switch region, the resonance has a narrow line shape and follows the soften- I f (MHz) ing dip from Figs. 2 and 3. At sd the first switch, the resonance position departs from the 283.5 expected position (indicated by dashed line). The mechan- 0.5 ical signal is strongly enhanced (nA)

in amplitude and displays a sd V ∆ I broad asymmetric line shape. V sd At the second switch, the res- g -0.75 onance returns to the frequency -4.935 -4.93 Vg (V) and narrow line shape expected at these powers.

1106 28 AUGUST 2009 VOL 325 SCIENCE www.sciencemag.org A coherent nanotube oscillator driven by electromechanical backaction

1. Measuring vibrations with a single-electron transistor

2. Creating a nanomechanical oscillator

3. Characterizing the oscillator

15 Excitation off

Bias on, excitation on

Gate voltage (mV)

Bias on, excitation off A coherent nanotube oscillator driven by electromechanical backaction

1. Measuring vibrations with a single-electron transistor

2. Creating a nanomechanical oscillator

3. Characterizing the oscillator

17 Measuring the output coherence

18 Statistics of the output coherence

Emission histograms Correlation function

19 How the oscillator works Electrical charge Electrical Bennet & Clerk (2007) Usmani, Blanter, Nazarov (2007) 20 How the oscillator works Electrical charge Electrical Bennet & Clerk (2007) Usmani, Blanter, Nazarov (2007) 21 Laser physics I: Injection locking Principle Implementation

Discovery: Huygens (1666) In lasers: Stover and Steier (1966)

22 Laser physics I: Injection locking

23 Laser physics II: Stabilising using feedback

24 Laser physics II: Stabilising using feedback

25 Why are nanomechanical lasers interesting?

• They connect the physics of backaction with the physics of lasers.

• They are amplifiers and transducers. Narrow linewidth means better sensitivity to perturbations.

• They are on-chip phonon generators for microscopy, information transfer, etc.

• They couple charge and motion on a mesoscopic scale.

26 Other nano-oscillators

NATURE COMMUNICATIONS | DOI: 10.1038/ncomms2827 Electrical ARTICLE Spin transfer torque Double quantum dot micromaser Josephson laser (Pribiag 2007) (Liu 2017) (Cassidy 2017) elf-sustained oscillations (SSOs) are distinct from the forced B I motion of a resonator because the oscillation period does bias Snot depend on the frequency of the driving force. Resonators in the SSO regime typically produce signals with a F J spectral line width, which is narrower than the intrinsic line L u VLF VHF width of the resonator. The reduced line width enhances the Φa 1 ! sensitivity of resonance-based force-gradient sensors . On the 1 !2 other hand, these oscillations are not desirable in experiments I

Mechanical that require observation of very small displacements, such as bias those induced by gravitational waves2 and quantum-mechanical Optomechanics 3–5 Optomechanics Vibrating SQUID (Grudininzero-point2010) fluctuations . Therefore,(Beardsley understanding 2010) the origin (Etaki 2013) 27 and behaviour of the SSO is essential. 20 129 kHz

Self-sustained mechanical oscillations are induced by strong 10 feedback between mechanical motion and the detector6–8. This ( µ V) can be engineered externally, if the amplifier is added as an 0

external element in the circuit9,10. An example of such an SQUID V 109 kHz –10 oscillator is an electronic amplifier, which applies a 9 –20 magnetomotive feedback force to a mechanical resonator . In B-field contrast, the feedback can also be due to intrinsic detector– 11,12 –3 –2 –1 0 1 2 3 resonator interaction . For example, an optomechanical I (µA) oscillator can be sustained by radiation pressure back-action11. bias In electromechanical systems, mesoscopic electronics are an Figure 1 | Device characteristics. (a) Colourized scanning electron important class of detectors. Devices such as the Atomic Point microscope image of the suspended 95 40 mm2 SQUID (red) at an angle  Contact13 and the Single Electron Transistor14,15 greatly enhance of 80 degrees to the substrate (blue). The Josephson junctions are located both the displacement resolution and the back-action. Despite in the loop. (b ) Schematic overview of the SQUID geometry and considerable theoretical interest16,17, direct observation of parameters. The resonator is represented by the deformed section of the sufficiently strong back-action and intrinsic SSO has not been loop marked in red. In reality, the displacement is out-of-plane and the reported in a mesoscopic electromechanical system. magnetic field in-plane; in the schematics, motion for convenience is

The displacement detector in this letter consists of a represented as in-plane. B, Ib, and ^a are the applied magnetic field, bias mechanical resonator, which contains a direct current (dc) current and magnetic flux, respectively. The two junctions have phase d1,2 of Superconducting QUantum Interference Device (SQUID) loop. the superconducting order parameter, u is the deflection of the resonator By virtue of its high flux-to-voltage responsivity, the SQUID coordinate from its equilibrium position, J is the circulating current around

enables accurate measurement of the vibrations of the resonator. the loop and FL is the Lorentz force on the resonator. The voltage across the In contrast to previous work where only part of the loop was loop is measured using a low-frequency (LF, o1 kHz) and a high-frequency 18 suspended , here we suspend the entire SQUID, forming a amplifier (HF, 41 kHz). The arrows point in the positive direction of the torsional resonator. We have previously studied the dynamic respective variables. (c) Measured SQUID voltage versus bias current. For back-action that the SQUID exerts on an embedded resonator each bias current in the main plot, a voltage-flux trace is measured over at when the resonator displacement causes only small flux least one flux quantum (inset). The red lines in the main figure show the variations19. Here we show that the torsional resonator is more minimum and maximum voltages (green dots) obtained from each trace. strongly coupled to the SQUID. The blue dots mark the bias at which the resonator measurements of Fig. 2 Specifically, we find that the behaviour on the upward and the are taken. (d) Finite-element simulation of the two lowest mechanical downward flank of the voltage-flux curve is strikingly different. resonance modes for a geometrically symmetrical SQUID. Red denotes high Although the upward flank shows the usual driven motion, the displacement amplitude, whereas blue denotes a small displacement. data for the downward flank display SSOs, as substantiated by the large amplitude, sharp profile, and, most importantly, the ring- shaped distribution of the waveform amplitude around zero. The and measuring the SQUID voltage at the same frequency. The mechanical vibrations cause large flux oscillations, on the order of lowest mechanical resonance modes are found at 109 and a flux quantum, ^0 h/2e 2 fWb. 129 kHz. The quality factor is extracted from a fit to the frequency ¼ ¼ response, yielding a quality factor of around 7,000 for both modes. Finite-element simulations show the nature of the Results eigenmodes to be a combination of flexural and torsional motion Voltage noise spectra. The torsional dc SQUID (Fig. 1a) is fab- (Fig. 1d). The lowest mode exhibits in-phase motion of the two ricated from an InAs-AlGaSb heterostructure, which is epitaxially arms, but is still detectable due to some arm asymmetry. The grown on a GaAs [111a] substrate (see Methods). Measurements piezo drive is used only to locate the eigenmodes and is turned off are performed at 20 mK in vacuum. To detect the out-of-plane for the measurements below. motion of the suspended SQUID loop, a parallel magnetic field is Voltage noise spectra of the two lowest modes are shown in applied (Fig. 1a,d). The field transduces the change in loop area Fig. 2a,b. The spectra are measured on the upward flank (blue into a change in magnetic flux. The SQUID transport char- data) and the downward flank (red data) of the voltage-flux curve. acteristics are shown in Fig. 1c (voltage-current) and its inset Compared with the spectra on the upward flank, the spectra on (voltage-flux). For the measurements below, the SQUID is the downward flank are much sharper, and the total power (the operated in a feedback loop that is set to the blue dots in Fig. 1c. area under the resonance peaks) increases strongly, indicating Note that the SQUID can be biased to either the upward or the oscillations with high amplitude. Figure 2c,d shows the power and downward flank of the voltage-flux curve (see Methods). the effective quality factor as a function of the bias current, The resonator displacement is detected by measuring the respectively. The effective quality factor Q is defined as the full- SQUID output voltage. The eigenmodes of the torsional SQUID width at half-maximum of the peaks divided by the peak are found by driving a piezo actuator with a sinusoidal voltage frequency9. The high amplitudes for the 109-kHz mode occur at

2 NATURE COMMUNICATIONS | 4:1803 | DOI: 10.1038/ncomms2827 | www.nature.com/naturecommunications & 2013 Macmillan Publishers Limited. All rights reserved. Summary

Maser Classical coherence signal Quadrature In-phase signal

Injection locking

Nanotube oscillator Emission Frequency

Feedback stabilisation Emission Frequency Ares et al. PRL 2016 117 170801 (2016) Wen et al. APL 113 153101 (2018) Schupp et al. arXiv:1810.05767 (2018) Wen et al. arXiv:1903.04474 (2019) 28