Workshop on Nano-Opto-Electro-Mechanical Systems Approaching the Quantum Regime September 2010

Transcription

1 Workshop on Nano-Opto-Electro-Mechanical Systems Approaching the Quantum Regime 6-10 September 2010 A Phonon-Tunneling Approach to Support-Induced Dissipation of Nanomechanical Resonators Ignacio WILSON-RAE Technische Universitaet Munchen Physik Department I T34 James Franck Strasse, Garching GERMANY

2 A phonon-tunneling approach to support-induced dissipation of nanomechanical resonators Ignacio Wilson-Rae Technische Universität München Workshop on Nano-Opto-Electro-Mechanical Systems Approaching the Quantum Regime, Trieste, September 2010 T-34 Prof. W. Zwerger

3 Outline 1 Motivation 2 Theory Master formula for Q 3 Free-free resonators (Aspelmeyer group, Vienna) 4 Stressed membranes (Parpia-Craighead coll., Cornell) 5 Conclusions

4 Motivation: high-q nanomechanics Technological applications: signal-processing (on-chip UHF narrow-band filters) ultra-sensitive sensors at the molecular scale high resolution scanning-probe force microscopy Quantum limited control of a single macroscopic mechanical degree of freedom Initialize oscillator with long phonon lifetime in its quantum ground state. Quantum signatures: phonon jumps, non-classical states, Schrödinger cats... D. Rugar et al, Nature 2004

5 Motivation: high-q nanomechanics Technological applications: signal-processing (on-chip UHF narrow-band filters) ultra-sensitive sensors at the molecular scale high resolution scanning-probe force microscopy Quantum limited control of a single macroscopic mechanical degree of freedom Quantum optics analogue : nanomechanical J-C model, TL atom superconducting qubit, quantum dot (QD) or NV center. D. Rugar et al, Nature 2004 O Connell et al., Nature 2010

6 Motivation: high-q nanomechanics Room temperature cavity-assisted sideband-cooling to the ground-state: k B T κq < 1 with κ < ω m or n opt ( kb T ω m Q ) 2/3 fq Hz. Importance of fq Predicting the mechanical dissipation remains a challenge. Relevant mechanisms at low T and high-vacuum: two-level fluctuators anharmonic effects / phonon-phonon interactions (?) radiation of phonons (elastic-waves) into the supports

7 Motivation: high-q nanomechanics Importance of fq Predicting the mechanical dissipation remains a challenge. Relevant mechanisms at low T and high-vacuum: two-level fluctuators anharmonic effects / phonon-phonon interactions (?) radiation of phonons (elastic-waves) into the supports Which is the fundamental limit set by the design? Temperature and scale independent Other mechanisms add incoherently 1/Q tot = i 1/Q i

8 Quantum dissipation of mechanical motion Caldeira-Leggett model Generalized equation of motion ˆX R (t)+ t 0 dt γ(t t ) ˆXR (t )+ω 2 R ˆX R (t) =ˆξ(t) determined by environmental force spectral density I(ω). I(ω) =ω dt γ(t) eiωt, 2ω R 1 Q = I(ω R) ω R, γ(t) is the symmetric dissipation-kernel [γ(t) δ(t) in Ohmic case or in weak coupling limit] and ˆξ(t) the environmental noise. Markov approximation may not suffice for quantum nanomechanics e.g. structured spin-boson model for detector-resonator system, strongly non-linear resonators.

9 Quantum dissipation of mechanical motion Caldeira-Leggett model Generalized equation of motion ˆX R (t)+ t 0 dt γ(t t ) ˆXR (t )+ω 2 R ˆX R (t) =ˆξ(t) determined by environmental force spectral density I(ω). Dissipation induced by unavoidable coupling to vibrations of the substrate elastic wave radiation clamping loss. (M.C. Cross and R. Lifshitz, PRB 2001; D.M. Photiadis and J.A. Judge APL 2004) Objectives: General understanding of limit imposed on Q by the design of a small mechanical resonator. Microscopic derivation of quantum dissipative dynamics of a macroscopic mechanical degree of freedom. IWR, Phys. Rev. B 77, (2008).

10 Theory Abrupt junctions with supports Cantilever Bridge Support d y z L d L Supports Ideal limit: phonon tunneling between the beam and the supports is the only source of dissipation upper bound for Q- values. Derivation from elasticity theory (λ a) of phononic modes of the beam coupled to its substrate scattering modes. d/l provides a natural small parameter.

11 Theory Abrupt junctions with supports Cantilever Bridge Support d y z L d L Supports Derivation from elasticity theory (λ a) of phononic modes of the beam coupled to its substrate scattering modes. d/l provides a natural small parameter. Closed system open system representations related by canonical transformation reduction of I(ω) to overlaps between scattering modes and resonator mode. General expressions for I(ω) in terms of: (i) support DOS or (ii) transmission at beam-support junction.

12 Spectral densities and Q-values for beams General Relations (3D) Monolithic structure (σ = 1 3 ) Compression I n (ω) Q n Q n (L, w, t) ω Q n ρ s ct 3 L δµ b c c 3ũc(α)k n δ L 2 πwt 1 n + δ 2 Typical value for 150MHz Torsion ω 3 Q n ω 2 n ρ s c 5 t L 2δµ b r 2 c 5 t ũt(α, γ z )k 3 n 4.1 δ w 2 L 4 π 3 t 6 1 (n + δ 2 ) Vertical bending ω Q n ρ s ct 3 L 4δC n µ b c v 3ũv(α)k n L 5 δc n π 4 wt 4 «4 3π k n L Horizontal bending ω Q n ρ s ct 3 L 4δC n µ b c hũh(α)k 3 n L 5 δc n π 4 tw 4 «4 3π k n L 2D slab supports 1/f noise: I n,β =v = ω n Q n, I n,v = ω2 n. Q n ω Bridge with single support interference effects can modify the scalings of Q and frequency dependence of I(ω).

13 Q-solver for generic geometry Spectral density: I(ω) π 2ρ s ρ R ω R ω q S d S (σ q ū R σ R ū q ) 2 δ[ω ω(q)] Master formula for dissipation: 1 Q π 2ρ s ρ R ω 3 R q S d S ( ) σ (0) q ū R σ R ū (0) 2 q valid for isolated resonance and kd 1, resonator mode satisfies: non-singular S 0 = free BC singular S 0 = clamped BC and support modes the converse. δ[ω R ω(q)]

14 Q-solver for generic geometry Master formula for dissipation: 1 Q π 2ρ s ρ R ω 3 R q S d S ( ) σ (0) q ū R σ R ū (0) 2 q valid for isolated resonance and kd 1, resonator mode satisfies: non-singular S 0 = free BC singular S 0 = clamped BC and support modes the converse. (i) Pedestal geometries: contact area substrate, e.g. microspheres, microdisks, microtoroids, micropillars. δ[ω R ω(q)] (ii) Planar geometries: contact area substrate, e.g. beams, in-plane modes of plates, flexural modes of plates.

15 Q-solver for generic geometry Master formula for dissipation: 1 Q π ( ) 2ρ s ρ R ωr 3 d S σ (0) q ū R σ R ū (0) 2 q δ[ω R ω(q)] q S (i) Pedestal geometries: contact area substrate, e.g. microspheres, microdisks, microtoroids, micropillars. (ii) Planar geometries: contact area substrate, e.g. beams, in-plane modes of plates, flexural modes of plates. Fermi golden rule: free BC H int = S d S σ > ū < clamped BC H int = S d S σ < ū > More general than small contact area valid for generic high-q resonance provided ω R /ω R 1. Reduction to decoupled resonator mode + free wave propagation in the substrate.

16 Q-solver for generic geometry Master formula for dissipation: 1 Q π 2ρ s ρ R ω 3 R q S d S ( ) σ (0) q ū R σ R ū (0) 2 q δ[ω R ω(q)] (i) Pedestal geometries: contact area substrate, e.g. microspheres, microdisks, microtoroids, micropillars. (ii) Planar geometries: contact area substrate, e.g. beams, in-plane modes of plates, flexural modes of plates. Flexural modes of symmetric plate geometry inscribed in circle. Q-solver: use FEM for resonator mode and cylindrical modes for support substrate 1/2-space (with Garrett Cole).

17 Free-free DBR-micromirror structures High-reflectivity DBR-micromirrors as used in optomechanical Fabry-Perot setups (Q c c 10 3 ). Auxiliary beams act as noise filters provided their resonances are avoided for generic positioning Q f f /Q c c (w/w s ) 2.

18 Free-free DBR-micromirror structures High-reflectivity DBR-micromirrors as used in optomechanical Fabry-Perot setups (Q c c 10 3 ). Approximately preserves ω 0, S eff and V eff Free-free design provides a tool to study impact of geometry on Q.

19 Free-free DBR-micromirror structures Single fit parameter: internal dissipation offset Q exp /Q exp 260% ( 80%) vs. f /f 20% ( 10%) (f 2MHz). At nodal positioning Q th vs. Q c c Antisymmetric mode:

20 Stressed membranes Dispersive optomechanics Stress reduces internal dissipation (e.g. two-level systems) What is the effect of stress on clamping losses? Assume high stress regime t 2 /D 2 σ/e R 1 Drums: 1 = 4π2 ζ nm ρ R t Q nm ρ s D γ η 3 γũn,γ

21 Stressed membranes Dissipation for different harmonics of a given resonator.

22 Stressed membranes Square (D = 252.3µ t = 12.5nm): E s = 148 GPa, ρ s = 3.75gcm 3,1/Q int = Drum (D = 14.5µ t = 110nm): E s = 323 GPa, 1/Q int =

23 Stressed membranes Dissipation 1Q (a) Frequency MHz Quality Factor Q (b) Frequency MHz Dissipation 1Q (c) Frequency MHz Quality Factor Q (d) Frequency MHz

24 Stressed membranes Dissipation 1Q10 5 Dissipation 1Q (a) Frequency MHz (c) Frequency MHz Quality Factor Q Frequency MHz Frequency MHz Special classes of high-q harmonics (n, 1) n>0 for drum and (n, n) n>1 for square (circles) nodal lines intersect periphery at evenly spaced points [fq = Hz for (6, 6)]. Quality Factor Q (b) (d)

25 Destructive interference effects Special classes of high-q harmonics (n, 1) n>0 for drum and (n, n) n>1 for square (circles) nodal lines intersect periphery at evenly spaced points [fq = Hz for (6, 6)].

26 Destructive interference effects c R /c γ σρ s /E s ρ R 1 resonant λ in substrate D.

27 Destructive interference effects Analytical approximations for drum: ρ s c 3 t Q 01 2π 2 σ R cr 2 ω 01ũ0(ν s ) ( ) ρ 3 R Es D = ρ s σ t ( Q n1 n 2n c ) 2n s, Q 01 c R νs =1/3 Q nm Q n1 ( ζn1 ζ nm ) 2n+1. c R /c γ σρ s /E s ρ R 1 resonant λ in substrate D.

28 Destructive interference effects Analytical approximations for drum: ρ s c 3 t Q 01 2π 2 σ R cr 2 ω 01ũ0(ν s ) ( ) ρ 3 R Es D = ρ s σ t ( Q n1 n 2n c ) 2n s, Q 01 c R νs =1/3 Q nm Q n1 ( ζn1 ζ nm ) 2n+1. For square: damping rate constant as n, m are increased with fixed ratio.

29 Destructive interference effects Analytical approximations for drum: ρ s c 3 t Q 01 2π 2 σ R cr 2 ω 01ũ0(ν s ) ( ) ρ 3 R Es D = ρ s σ t ( Q n1 n 2n c ) 2n s, Q 01 c R νs =1/3 Q nm Q n1 ( ζn1 ζ nm ) 2n+1. Azimuthal harmonics exhibit exponential suppression of phonon radiation asymptotically mute, e.g. fq clamp Hz for n 6 and thickness t < 250 nm.

30 Circle vs. square Quality Factor Q Nodal Lines Q-value of the nth harmonic in the series (n, 1) n 0 [(n, n) n 1 ] for dimensions D (c) n =(ζ n1 /ζ 01 )D (c) 0 [D (s) n = nd (s) 0 ], D (c) 0 = 2ζ 01 D (s) 0 /π with D(s) 0 = 50 µm = f n = 7.99 MHz.

31 Collaborators Free-free geometries: Garrett Cole, Katharina Werbach, Michael R. Vanner, and Markus Aspelmeyer (IQOQI and U. Vienna). arxiv: Stressed membranes: Rob Barton, Darren Southworth, Scott Verbridge, Rob Ilic, Harold Craighead, and Jeevak Parpia (Cornell).

32 Conclusions Clamping-loss limited Q-values have a strong geometric character. Generic Q-solver for guiding design optimization quantitative understanding of support engineering. Mapping out of mechanical mode shape provides neat experimental test of the theory. arxiv: Striking non-monotonic behavior of membrane harmonics successfully explained. Destructive interference effects can lead to effectively clamping-loss free harmonics (requires 2D geometry). Stress increases the clamping loss. Typical optomechanical setups are limited by clamping loss.