Silicon nitride strings as nanomechanical resonators

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1 Silicon nitride strings as nanomechanical resonators Eva Weig Summer School: Quantum Optomechanics & Nanomechanics, École de Physique des Houches,

2 Silicon nitride strings as nanomechanical resonators: Exploring high Q, mode coupling, coherent control and more Eva Weig Summer School: Quantum Optomechanics & Nanomechanics, École de Physique des Houches,

3 Doubly-clamped pre-stressed, amorphous silicon nitride string as Megahertz nanomechanical resonator SiN SiO 2 Si fundamental flexural mode (in-plane) 200 nm x 100 nm Quantum Optomechanics & Nanomechanics, Les Houches 1 mm

, Nature 430, 329 (2004) [2] Moser et al.

single e - spin / zeptonewton sensitivity, nano MRI? [3] Chaste et al.

, Nature Nano 7, 602 (2012) for fundamental experiments quantum ground state: [6] Teufel et al.

4 Why study nanomechanical resonators? Because of a broad range of fascinating applications in practical applications force sensing: [1] Rugar et al., Nature 430, 329 (2004) [2] Moser et al., Nature Nano 8, 493 (2013) mass sensing / mass spectrometry: yoctogram sensitivity artificial nose? single e - spin / zeptonewton sensitivity, nano MRI? [3] Chaste et al., Nature Nano 7, 301 (2012) [4] Hanay et al., Nature Nano 7, 602 (2012) for fundamental experiments quantum ground state: [6] Teufel et al., Nature 475, 359 (2011) [7] Chan et al., Nature 478, 89 (2011) non-classical physics: the first quantum machine [5] O Connell et al., Nature 464, 679 (2010) entanglement? quantum gravity? [8] Safavi-Naeini et al., PRL 108, (2012) [9] Palomaki et al., Science 342, 710 (2013) [10] Marshall et al., PRL 91, (2003) [11] Pikovski et al., Nature Phys. 8, 393 (2012) 4

5 Doubly-clamped pre-stressed, amorphous silicon nitride string Taking a peek at Euler Bernoulli beam theory Assumptions: 1: homogeneous, isotropic and elastic material 2: transverse (flexural) vibrations 3: long and thin beam (L >> W, H): only bending rigidity and elongation; neglect shear distortion and rotary inertia 3 : this is equivalent to the condition of uniaxial stress (only s xx 0; s E ) xx xx 3 : or to: a neutral (stress-free) plane in the center of the beam (Euler-Bernoulli hypothesis) z x 5 y 1 mm

6 Doubly-clamped pre-stressed, amorphous silicon nitride string Taking a peek at Euler Bernoulli beam theory tensile force sa SiN SiO 2 Si bending rigidity EI z 6 x y tensile force sa 1 mm

7 Taking a peek at Euler Bernoulli theory of a long and thin vibrating string under tensile stress u restoring forces: F x, t EI bend 3 3 u x,t x elong F x,t sa u x,t x use Newton s law: 2 u x,t A dx F 2 tot x dx, t Ftot x, t t Euler-Bernoulli equation: u x,t u x,t u x,t A EI sa t x x bending elongation see e.g. Landau & Lifshitz Theory of Elasticity, Timoshenko et al. Vibration Problems in Engineering, 7

EI sa 0 2 4 2 t x x u x u position time t u u 2 EI u x sa u x A u

8 Taking a peek at Euler Bernoulli theory of a long and thin vibrating string under tensile stress u u x,t u x,t u x,t A EI sa t x x u x u position time t u u 2 EI u x sa u x A u x u t u t 0 mode shapes, eigenfrequencies, effective mass dynamics 8

9 The nanomechanical resonator as harmonic oscillator Effective mass, damping, amplitude and phase spectrum equation of motion eff m u t m u t ku t F t eff ansatz u t u e 0 F t f e 0 i t i t solution f 1 f 1 u e m eff ( 0 ) i m eff ( 0 ) i m eff L 1 2 meff A u 2 x dx u x 0 = 0.5 m for a string u t with tan f 1 0 m eff ( 0 ) e i t z 9

The nanomechanical resonator as harmonic oscillator Effective mass, damping, amplitude and phase spectrum u f 1 f 1 e 0 0 0 2 2 m 2 2 2

10 The nanomechanical resonator as harmonic oscillator Effective mass, damping, amplitude and phase spectrum u f 1 f 1 e m eff ( 0 ) i m eff ( 0 ) i m eff L 1 2 meff A u 2 x dx u x 0 = 0.5 m for a string u t with tan f 1 0 m eff ( 0 ) e i t z 10

11 Beyond the harmonic approximation: Bistability develops for large amplitude vibration in the Duffing regime 2 3 f i ut ut 0 ut u t e m Duffing t u ) A (db) u ) P drive (dbm) f (Hz) 11

12 Eigenmodes of nanomechanical resonators For a doubly clamped string with built-in tensile stress image: Cornell Univ. L = 11.6 mm L = 17.5 mm L = 35 mm f n 2 2 n EI sl n s L A n 2L high stress SiN on Si: E = 160 GPa s = 830 MPa = 2,800 kg/m 3 for n =

High stress = High Q Tensile stress increases the stored energy for a strongly stressed string: dominated by beam elongation (stress) U Q = 2 U mech diss dominated

13 High stress = High Q Tensile stress increases the stored energy for a strongly stressed string: dominated by beam elongation (stress) U Q = 2 U mech diss dominated by flexural rigidity / bending (Young s modulus) see also: Unterreithmeier et al., Phys. Rev. Lett. 105, (2010) Yu et al., Phys. Rev. Lett. 108, (2012) 13

14 amplitude [pm] Ultra-high Q SiN resonators at 300 K Tensile stress of SiN film deposited on Si/SiO2 vs. fused silica wafer high stress SiN on Si: high stress SiN on SiO 2 : s = GPa E = 160 GPa s = GPa E = 160 GPa detuning 10-5 mbar Q ~ 150,000 amplitude [mv] Q > 300,000 Qf ~ Hz frequency 10-5 mbar 14 Verbridge et al., J.Appl. Phys. 99, (2006) Faust, Krenn, Manus, Kotthaus, Weig, Nature Comm. 3, 728 (2012)

15 OUTLINE 1. Cavity optomechanics with SiN strings: Large Q-factors & dielectric control 2. Dielectric control of SiN strings: Dynamics of strongly coupled nanomechanical modes 3. Parametric pumping of strongly coupled modes: Mechanical (2) and (3) processes 15

16 OUTLINE 1. Cavity optomechanics with SiN strings: Large Q-factors & dielectric control 2. Dielectric control of SiN strings: Dynamics of strongly coupled nanomechanical modes 3. Parametric pumping of strongly coupled modes: Mechanical (2) and (3) processes 16

17 Cavity optomechanics An (incomplete) overview of a rapidly growing field,, km 17

18 Cavity optomechanics An (incomplete) overview of a rapidly growing field Karrai (Munich) Bouwmeester (UCSB) Aspelmeyer (Wien) Heidmann (Paris) Mavalvala (MIT) LIGO Vahala (Caltech) Kippenberg (MPQ) Bowen (Queensland) Lipson (Cornell) Favero (Paris) Stamper-Kurn Lehnert (JILA Boulder) Teufel (NIST Boulder) Schwab (Caltech) Sillanpää (Aalto) Tang (Yale) Painter (Caltech) Painter (Caltech) Harris (Yale) Weig/Kotthaus/Kippenberg

nanomechanical strings (Weig group, LMU,

(Kippenberg group, MPQ, now EPFL) Armani et

19 Microtoriod & nanostring Combining high finesse and high Q high Q SiN nanomechanical strings (Weig group, LMU, now Konstanz) high Q silica microtoriod (Kippenberg group, MPQ, now EPFL) Armani et al., Nature 421, 925 (2003). 50 µm Anetsberger, Arcizet, Unterreithmeier, Riviere, Schliesser, Weig, Kotthaus, Kippenberg, Nature Physics 5, 909 (2009).

G/(2) (MHz/nm) Dispersive optomechanical

the toriod s evanescent field distance x 0

20 G/(2) (MHz/nm) Dispersive optomechanical coupling Approaching the nanoresonator into the toriod s evanescent field distance x 0 microcavity negative frequency shift microcavity linewidth constant cavity frequency pull parameter: G x 0 d dx opt vacuum OM coupling g G x G 0 ZPF no measurable cavity linewidth broadening x purely dispersive coupling 0 2m mech 20

21 Intracavity power Sub-femtometer displacement sensitivity of a nanomechanical resonator at room temperature cavity resonance laser l = 1548 nm m m Optical frequency Displacement sensitivity: S xx = 570 am/hz 1/2 Anetsberger, Weig, Kotthaus, Kippenberg et al., Nature Physics 5, 909 (2009). Anetsberger, Weig, Kotthaus, Kippenberg et al., Phys. Rev. A 82, (R) (2010). see also Teufel et al., Nature Nano 4, 820 (2009). for similar work on quantum limited displacement sensing at 5 fm/hz 1/2 with a superconducting circuit 21

laser m Optical frequency m Optical frequency Stokes processes dominate, i.e. emission of phonons into resonator Optomechanical pumping Anti-Stokes processes dominate, i.

22 Intracavity power Intracavity power Dynamical backaction on the nanomechanical resonator induced by the cavity photons radiation pressure Blue detuned laser cavity resonance laser Resolved sideband regime: Red detuned laser m Optical frequency m Optical frequency Stokes processes dominate, i.e. emission of phonons into resonator Optomechanical pumping Anti-Stokes processes dominate, i.e. absorption of resonator phonons Optomechanical cooling 22

frequency oscillation amplitude threshold effective zero damping coherent

23 Intracavity power Dynamical backaction-induced coherent self-oscillation in the resolved sideband regime ( m > cavity ) blue detuned laser laser m m Optical frequency oscillation amplitude threshold effective zero damping coherent self-oscillation Anetsberger, Weig, Kotthaus, Kippenberg et al., Nature Physics 5, 909 (2009). 23

24 OUTLINE 1. Cavity optomechanics with SiN strings: Large Q-factors & dielectric control 2. Dielectric control of SiN strings: Dynamics of strongly coupled nanomechanical modes 3. Parametric pumping of strongly coupled modes: Mechanical (2) and (3) processes 24

Dielectric gradient field transduction An integrated platform to

Heterodyne detection w/ 3.5 GHz microwave cavity Faust, Nature Comm.

cross derivative of electric field z Fy y k c z 25 ΔC m (t) in out C

Lett 109, 037205 (2012) V Dielectric actuation: Electrically induced

Unterreithmeier, Nature 458, 1001 (2009) see also: Schmid, APL 89,

Dielectric frequency tuning: V DC -controlled effective spring

25 Dielectric gradient field transduction An integrated platform to control high Q nanomechanical resonators Dielectric detection: Heterodyne detection w/ 3.5 GHz microwave cavity Faust, Nature Comm. 3, 728 (2012) Dielectric mode coupling: Coupling spring induced by cross derivative of electric field z Fy y k c z 25 ΔC m (t) in out C L C m (0) E 2 2 z y Faust, Phys. Rev. Lett 109, (2012) V Dielectric actuation: Electrically induced gradient force e.g. out of plane mode: Ez Fz py y 2 DC 2 DC RF V V V Unterreithmeier, Nature 458, 1001 (2009) see also: Schmid, APL 89, (2006). Dielectric frequency tuning: V DC -controlled effective spring constant (local field gradient at string position) e.g. out of plane mode: FD C k eff z k k 0 m V eff 2 DC 0 z y elevated electrodes V DC Rieger, Appl. Phys. Lett. 101, (2012)

writefield piezo drive (optional) single layer capacitor

26 Dielectric gradient field transduction Combining dielectric actuation and heterodyne detection chip writefield piezo drive (optional) single layer capacitor C by 3.5 GHz mw resonator (l/4 microstrip) feed lines (L-coupled) 26

27 Tuning in- and out-of-plane flexural mode of a resonator in elevated electrode layout Faust, Rieger, Seitner, Krenn, Manus, Kotthaus, Weig, Phys. Rev. Lett 109, (2012) 27

28 Tuning in- and out-of-plane flexural modes Avoided crossing reminiscent of strong coupling in-plane mode g/2 = 7.77 khz out-of-plane mode 1 /2= 2 /2=83 Hz strong coupling: g >> 1,2 Faust, Rieger, Seitner, Krenn, Manus, Kotthaus, Weig, Phys. Rev. Lett 109, (2012) 28

DC voltage ramp across coupling region 3.

29 Time-resolved dynamics of coupled modes Measurement sequence 1. State initialization at point I by constant drive 2. DC voltage ramp across coupling region 3. Final state depends on ramp time t: a diabatic / adiabatic transistion gets the system to point D / A 4. Measure oscillation energy at D and A (after delay d) I A D signal power DC voltage 29

Padiabatic 1e e t 2 2 e t D see also: L. Novotny, Am. J. Phys.

30 A classical analogue of Landau-Zener physics Establishing time-domain control of nanoresonator state Landau-Zener dynamics with additional decay: I A Pdiabatic e 2 2 Padiabatic 1e e t 2 2 e t D see also: L. Novotny, Am. J. Phys. 78, 1199 (2010) decay time 1/=1.9ms Faust, Rieger, Seitner, Krenn, Manus, Kotthaus, Weig, Phys. Rev. Lett 109, (2012) 30

31 Landau-Zener-Stückelberg interference? Double passages through the avoided crossing 31

32 Landau-Zener-Stückelberg interference at room temperature? Does coherence last till 300 K? 32

A classical nanomechanical two-mode system and its two hybrid modes as basis states of a Bloch sphere On resonance, the +45 and -45 mechanical hybrid modes form a two-mode system: upper

33 A classical nanomechanical two-mode system and its two hybrid modes as basis states of a Bloch sphere On resonance, the +45 and -45 mechanical hybrid modes form a two-mode system: upper lower see also: Pöttinger & Lendi, Generalized Bloch equations for decaying systems, Phys. Rev. A 31, 1299 (1985) Frimmer & Novotny, The classical Bloch equations, Am. J. Phys. 82, 947 (2014) 33

dm t M t M dt T R M y t upper lower Control the mechanical state via radio frequency pulses see also: Pöttinger & Lendi, Generalized Bloch

34 A classical nanomechanical two-mode system and its two hybrid modes as basis states of a Bloch sphere On resonance, the +45 and -45 mechanical hybrid modes form a two-mode system: Bloch equations upper lower dmx t Mx t dt T 2 2 z z 0 1 M y t dmy t My t Mx t RMz t dt T dm t M t M dt T R M y t upper lower Control the mechanical state via radio frequency pulses see also: Pöttinger & Lendi, Generalized Bloch equations for decaying systems, Phys. Rev. A 31, 1299 (1985) Frimmer & Novotny, The classical Bloch equations, Am. J. Phys. 82, 947 (2014) bath 34

pulses T = 10 K Faust, Rieger, Seitner, Kotthaus, Weig, Nature Physics 9, 485 (2013)

35 Rabi oscillations Demonstrating coherent control of a nanomechanical two level system Continous pumping leads to Rabi oscillations between the two states Calibrate and /2 pulses T = 10 K Faust, Rieger, Seitner, Kotthaus, Weig, Nature Physics 9, 485 (2013) see also: Okamoto et al., Nature Physics 9, 480 (2013) Phonons 2015, Nottingham

T 1,u and T 1,l are different for upper and lower state Rate average T 1 =1/(1/T 1,u

36 Energy decay rate of lower and upper state and the average T 1 time Start from lower state or use pulse to transform all energy to the upper state Energy relaxation times T 1,u and T 1,l are different for upper and lower state Rate average T 1 =1/(1/T 1,u +1/T 1,l ) T = 10 K Faust, Rieger, Seitner, Kotthaus, Weig, Nature Physics 9, 485 (2013) 36

rotation in the Bloch sphere Second /2 pulse rotates back to the z-axis T

37 Ramsey fringes Determining the T 1 and T 2 * times First /2 pulse creates a superpostition state Small detuning of pump pulses leads to slow rotation in the Bloch sphere Second /2 pulse rotates back to the z-axis T = 10 K Faust, Rieger, Seitner, Kotthaus, Weig, Nature Physics 9, 485 (2013) 37

reversing all external influences Final 3/2 pulse rotates back to the z axis

38 Hahn echo experiment Determining the phase coherence time T 2 First /2 pulse initiates a superposition state A pulse after t/2 flips the state by 180, reversing all external influences Final 3/2 pulse rotates back to the z axis T = 10 K Faust, Rieger, Seitner, Kotthaus, Weig, Nature Physics 9, 485 (2013) 38

39 Dissipation and decoherence of nanomechanical modes A summary of the lessons learned T 1 = ms; T 2 = ms; T 2 * = ms T 1 = T 2 = T 2 * T 1 =T 2 negligible elastic phase relaxation, here: 1 1 No factor of 2 T T missing here! 2 1 uncommon for a solid state system coherence time solely limited by the energy loss rates of the two states T 2 =T 2 * no measurable inhomogeneous broadening all phonons reside in the same collective mechanical mode stable external fields 39

500 120 defect TLS V two-level system Pohl et al., Rev. Mod. Phys. 74, 991 (2002) see also for evidence of TLS in nanomechanical systems: Arcizet et al., Phys. Rev. A 80, 021803 (2009) Suh et al.

40 Γ 3 (rad/s) Damping Γ 1 (rad/s) Damping Γ 3 1 (rad/s) Damping Γ 1 (rad/s) What limits the dissipation of SiN nanostring resonators? Intrinsic damping from material defects, so called two-level systems (TLS) Signatures of TLS in the temperature and frequency dependent damping: defect TLS V two-level system Pohl et al., Rev. Mod. Phys. 74, 991 (2002) see also for evidence of TLS in nanomechanical systems: Arcizet et al., Phys. Rev. A 80, (2009) Suh et al., Appl. Phys. Lett. 103, (2013) H(?) Temperature (K) 500 f 1 = 6.8 MHz Faust, Rieger, Seitner, Kotthaus, Weig, Phys. Rev. B 89, R (2014)

41 A pathway to higher Q Single crystal materials under strong tensile stress s = 200 MPa for InGaP nanoresonator see Cole et al., Appl. Phys. Lett. 104, (2014) Epitaxial resonator material substantially reduces number of defects Quantum Optomechanics & Nanomechanics, Les Houches

42 OUTLINE 1. Cavity optomechanics with SiN strings: Large Q-factors & dielectric control 2. Dielectric control of SiN strings: Dynamics of strongly coupled nanomechanical modes 3. Parametric pumping of strongly coupled modes: Mechanical (2) and (3) processes 42

43 Parametric oscillation Self-oscillation resulting from parametric frequency modulation Mathieu equation:. 2 Y t Y t ( d(t)) Y t 0 with 0 d(t) d cos( Pt) d cos(20t) d P 20 / 2 0 P Frequency modulation at twice the eigenfrequency: Instability and spontaneous self-oscillation for d 0 Q 43

44 Parametric oscillation Self-oscillation resulting from parametric frequency modulation Mathieu equation:. 2 Y t Y t ( d(t)) Y t 0 with 0 d(t) d cos( t) d cos(2 V t) P 0 DC self-oscillation Frequency modulation at twice the eigenfrequency: Instability and spontaneous self-oscillation for d 0 Q 44

Parametric oscillation An ubiquitous concept in physics in

future applications Josephson parametric oscillators: e.g.

(1892) [2] Bell Labs (~1958) Optical parametric oscillators

oscillators: parasitic decoupling, mechanical computing?

45 Parametric oscillation An ubiquitous concept in physics in commercial products Voltage-controlled oscillators (VCO): for future applications Josephson parametric oscillators: e.g. based on varactor diode [1] Fitzgerald, Phys. Soc. Lond. (1892) [2] Bell Labs (~1958) Optical parametric oscillators (OPO): MIRA [5] Yurke et al., PRA 39, 2519 (1989) [6] Wilson et al., PRL 105, (2010) Micromechanical parametric oscillators: parasitic decoupling, mechanical computing? [3] Giordane & Miller, PRL 14, 973 (1965) [7] Turner et al., Nature 396, 149 (1998) [4] [8] Mahboob & Yamaguchi, Nat. Nano 3, 275 ( 08) 45

46 Parametric actuation of two coupled modes using a single modulation frequency f P 2 f C = 2(f OOP + f IP )/2 46

47 Parametric actuation of two coupled modes using a single modulation frequency f P 2 f C = 2(f OOP + f IP )/2 47

48 Parametric actuation of two coupled modes using a single modulation frequency f P 2 f C = 2(f OOP + f IP )/2 48

49 SUMMARY 1. Cavity optomechanics with SiN strings: Hybrid optomechanical system with high Q resonator and cavity Sensitive displacement detection and dynamical backaction 2. Dielectric control of SiN strings: Versatile toolbox for strongly coupled resonator modes Landau-Zener dynamics and Stückelberg interference Full Bloch sphere control of classical two-level system 3. Parametric pumping of strongly coupled modes: 49

50 Some more research topics origins of high Q gradient fields strong coupling synchronization nonlinear dynamics Nb cavities hybrid NOMS integrated NOMS CNT NOMS fully suspended NEMS PhD and postdoc openings charge shuttling GaAs nanowires nanowire arrays cellular force sensing 50 contact eva.weig@uni-konstanz.de

51 PhD and/or Postdoc opening: Cavity nano-optomechanics with atomic-scale resonators fiber-based micro-fabry-pérot cavity microstructured SiN chip suspended carbon nanotube S. Stapfner et al., Cavity-enhanced optical detection of carbon nanotube Brownian motion, Appl. Phys. Lett. 102, (2013) More info: or 51

PhD and/or Postdoc opening: Cavity nano-optomechanics with atomic-scale resonators fiber-based micro-fabry-pérot cavity microstructured SiN chip suspended carbon nanotube

52 PhD and/or Postdoc opening: Cavity nano-optomechanics with atomic-scale resonators fiber-based micro-fabry-pérot cavity microstructured SiN chip suspended carbon nanotube / atomically thin membrane Collaborators: I. Favero, A. Hüttel, G. A. Steele, J. Reichel, D. Hunger More info: eva.weig@uni-konstanz.de or 52

53 MANY THANKS TO the nanomechanics group in Munich and Konstanz Thomas Faust Johannes Rieger Stephan Manus Jörg P. Kotthaus Johannes Kölbl Louis Kukk Stephan Schneider Maximilian Seitner Maximilian Bückle Felix Rochau Katrin Gajo Juliane Doster COLLABORATORS: H. Ribeiro (McGill) M. Hartmann (Heriot-Watt) A. Ridolfo (TUM) A. Mehdi (TUM) A. Isacsson (Chalmers) 53

54 Registration & more information: 54

55 SUMMARY 1. Cavity optomechanics with SiN strings: Hybrid optomechanical system with high Q resonator and cavity Sensitive displacement detection and dynamical backaction 2. Dielectric control of SiN strings: Versatile toolbox for strongly coupled resonator modes Landau-Zener dynamics and Stückelberg interference Full Bloch sphere control of classical two-level system 3. Parametric pumping of strongly coupled modes: Degenerate and non-degenerate parametric oscillation ( (2) ) Degenerate four-wave mixing ( (3) ) 55

Coherent control and TLS-mediated damping of SiN nanoresonators. Eva Weig

Coherent control and TLS-mediated damping of SiN nanoresonators Eva Weig Doubly-clamped pre-stressed silicon nitride string as Megahertz nanomechanical resonator fundamental flexural mode (in-plane) ~