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

Transcription

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

2 Doubly-clamped pre-stressed silicon nitride string as Megahertz nanomechanical resonator fundamental flexural mode (in-plane) ~ 50 μm 200 nm x 100 nm 1 μm

3 OUTLINE 1. Dielectric transduction of high Q nanomechanical resonators: Actuation, detection, frequency tuning and mode coupling 2. Strongly coupled nanomechanical two-level system: Coherent control via radio frequency pulses 3. Temperature-dependent damping of SiN nanoresonators: Energy relaxation via TLS-mediated damping

4 OUTLINE 1. Dielectric transduction of high Q nanomechanical resonators: Actuation, detection, frequency tuning and mode coupling 2. Strongly coupled nanomechanical two-level system: Coherent control via radio frequency pulses 3. Temperature-dependent damping of SiN nanoresonators: Energy relaxation via TLS-mediated damping

5 Doubly-clamped pre-stressed silicon nitride beam as Megahertz nanomechanical resonator tensile force σa fundamental flexural mode bending rigidity EI 2 ( ) u x,t ρ A = 2 t + EI 4 ( ) u x,t x σa 4 2 ( ) u x,t x 2 tensile force σa 1 μm

6 High stress = High Q Tensile stress increases the stored energy for an a strongly unstressed stressed string: string: dominated by beam elongation (stress) U Q=2π ΔU mech diss = Re(E) Im(E) 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)

7 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 : σ= GPa E = 160 GPa σ= GPa E = 160 GPa amplitude [pm] detuning 10-5 mbar Q ~ 150,000 amplitude [mv] Q > 300, frequency 10-5 mbar Faust, Krenn, Manus, Kotthaus, Weig, Nature Comm. 3, 728 (2012)

8 Dielectric gradient field transduction An integrated platform for controlling high Q nanomechanical resonators Dielectric detection: Heterodyne detection w/ 3.5 GHz microwave cavity RF in RF out Dielectric actuation: Electrically induced gradient force out of plane modes: Ez Fz = py y 2 DC 2 DC RF V + V V Faust, Nature Comm. 3, 728 (2012) Dielectric mode coupling: Coupling spring induced by cross derivative of electric field kc 2 =α E zy Faust, Phys. Rev. Lett 109, (2012) 2 (in plane modes accordingly) Unterreithmeier, Nature 458, 1001 (2009). Dielectric frequency tuning: V DC -controlled effective spring constant (local field gradient at string position) out of plane modes: k+ keff ω 0 = m FDC w/ keff = z (in plane modes accordingly) elevated electrodes Rieger, Appl. Phys. Lett. 101, (2012)

9 Tuning in- and out-of-plane flexural modes Avoided crossing reminiscent of strong coupling in-plane mode in plane out of plane out-of-plane mode out of plane in plane Faust, Rieger, Seitner, Krenn, Manus, Kotthaus, Weig, Phys. Rev. Lett 109, (2012)

10 Two strongly coupled harmonic oscillators Theoretical description and analysis coupling strength k + 2k k Γ= m m 0 c 0 eff Γ/2π=7.77kHz damping γ 1 /2π= γ 2 /2π=83Hz eff frequency (MHz) Γ/2π Γ >> γ 1, strong coupling DC voltage (V) Faust, Rieger, Seitner, Krenn, Manus, Kotthaus, Weig, Phys. Rev. Lett 109, (2012)

11 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. Theramptime τ sets the final state: a diabatic / adiabatic transistion gets the system to point D / A I A 4. Measure oscillation energy at D and A (after delay δ) D signal power DC voltage

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

13 OUTLINE 1. Dielectric transduction of high Q nanomechanical resonators: Actuation, detection, frequency tuning and mode coupling 2. Strongly coupled nanomechanical two-level system: Coherent control via radio frequency pulses 3. Temperature-dependent damping of SiN nanoresonators: Energy relaxation via TLS-mediated damping

14 A classical nanomechanical two-level 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: out-of-plane upper in-plane in-plane lower upper out-of-plane lower This should allow to see also: Pöttinger & Lendi, Generalized Bloch equations for decaying systems, Phys. Rev. A 31, 1299 (1985) bath employ (generalized) Bloch equations to describe system dynamics apply radio frequency pulses to control the mechanical state

15 Rabi oscillations Demonstrating coherent control of a nanomechanical two level system Continous pumping leads to Rabi oscillations between the two states Oscillation frequency depends on pump amplitude and detuning T = 10 K Faust, Rieger, Seitner, Kotthaus, Weig, Nature Physics 9, 485 (2013)

16 Energy decay rate of lower and upper state and the average T 1 time Start from lower stateoruseπ 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)

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

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

19 Decoherence of nanomechanical modes A summary of the lessons learned T 1 = 4.31 ± 0.1 ms; T 2 = 4.35 ± 0.1 ms; T 2 * = 4.44 ± 0.1 ms T 1 = T 2 = T 2 * T 1 =T 2 negligible dephasing rate, here: 1 1 Γ ϕ = T T 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

20 OUTLINE 1. Dielectric transduction of high Q nanomechanical resonators: Actuation, detection, frequency tuning and mode coupling 2. Strongly coupled nanomechanical two-level system: Coherent control via radio frequency pulses 3. Temperature-dependent damping of SiN nanoresonators: Energy relaxation via TLS-mediated damping

21 Energy relaxation in high stress SiN strings Possible intrinsic pathways for energy relaxation Clamping losses: Resonant tunneling of resonator phonons into the support strongly suppressed by large mechanical impedance mismatch between resonator and clamps Q clamping 3,000,000 Wilson-Rae, Phys. Rev. B 77, (2008), Cole et al., Nature Comm. 2, 231 (2011), Rieger, Seitner, Kotthaus, Weig, Isacsson, submitted (2013) Thermoelastric damping: Damping arising from the coupling between the elastic strain field in the resonator caused by deformation and the temperature field Q clamping à Q clamping Lifshitz & Roukes, Phys. Rev. B 61, 5600 (2000) Damping from defects in the amorpous material: Glassy materials are known to exhibit configurational changes in the atomic structure commonly referred to as two-level defect fluctuators (TLS) Pohl et al., Rev. Mod. Phys. 74, 991 (2002); Vacher et al., Phys. Rev. B 72, (2005), Arcizet et al., Phys. Rev. A 80, (2009); Suh et al., Appl. Phys. Lett. 103, (2013).

22 Damping from bulk defects in glassy materials (AKA two-level systems at low temperature) TLS-type defect: Signatures of TLS: Temperature and frequency dependent damping Tunneling (only at very low temperatures) V Δ Thermally activated relaxation across barrier: defects with broad V and Δ distribution -> characteristic peak near 50 MHz Configurational change of the atomic structure which can be described by a double well potential Two level system at low temperatures Arrhenius peak: thermally activated relaxation over well-defined barrier, probably H defect -> characteristic peak near 200 MHz Pohl et al., Rev. Mod. Phys. 74, 991 (2002); Vacher et al., Phys. Rev. B 72, (2005); Arcizet et al., Phys. Rev. A 80, (2009); Suh et al., Appl. Phys. Lett. 103, (2013).

23 TLS signatures in micromechanical resonators Connecting micro- and nanomechanics to the results from glass physics Tobias Kippenberg lab: TLS in damping characteristics of a micromechanical SiO 2 resonator Keith Schwab lab: TLS in back-action evading measurement of micromechanical SiN membrane Arcizet et al., Phys. Rev. A 80, (2009); Suh et al., Appl. Phys. Lett. 103, (2013).

24 Temperature dependent damping (7 350 K) For the fundamental and third harmonic mode of the resonator fundamental mode: f 1 = 6.8 MHz third harmonic: f 3 = 20.2 MHz Damping Γ 1 (rad/s) Damping Γ 3 (rad/s) Temperature (K) Temperature (K) Faust, Rieger, Seitner, Kotthaus, Weig, arxiv:

25 TLS in SiN nanomechanical resonators (AKA two-level systems at low temperature) Thermally activated relaxation: defects with broad V and Δ distribution Arrhenius peak: well-defined barrier, probably H defect Inverse quality factor (10-6 ) Temperature (K) Inverse quality factor (10-6 ) Temperature (K) 2 1 γ 2 Δ Ωτ rel = Δ 2 ( Δ ) 2 2 ρ v T 2T1+Ω τ 0 Q d dv P,V sech w/ inverse hopping rate τ=τ 0 V exp sech T V 0 = 460E4 K, Λ C = 110E2K, log 10 τ 0 /s = 11.24E0.02 mode dependent background: n=1: (1.780E0.02) 10-6 ; n=3: (2.620E0.02) 10-6 Δ 2T Q, Va C ωτa exp = T T 2V 1 +ω τ exp T 1 a a 2 2 a a V a = 2354E10K, log 10 τ a /s = E0.02 mode dependent background: n=1: (3.080E0.02) 10-6 ; n=3: (4.370E0.02) 10-6

26 The role of TLS in energy relaxation Three-particle scattering relaxes momentum conservation Direct Relaxation relaxation via three-particle (two-particle scattering) inhibited involvingby a TLS conservation laws longitudinal bulk phonon mode Ω discrete resonator modes ω m TLS provide excess momentum: enable thermally excited bulk phonon to scatter off a resonator phonon. Large thermal population of bulk phonons: no re-emission of resonator phonons into the mode after TLS interaction (Γ ϕ 0) High-energy bulk phonon thermal relaxation rate: Γ = G vphl th ph th 2 m pathway for energy relaxation C L Γ Faust, Rieger, Seitner, Kotthaus, Weig, arxiv:

27 SUMMARY 1. Dielectric transduction of high Q nanomechanical resonators: efficient actuation, sensitive detection and controlled frequency tuning using electrically induced gradient fields Integrated control platform 2. Coherent control of a nanomechanical two-level system: Characterization of energy relaxation and dephasing of nanomechanical modes yields T 1 = T 2 = T 2 * Classical coherent manipulation 3. Damping of SiN nanoresonators: Coherence is limited by TLS-mediated damping; a pathway for energy relaxation is provided by three-particle scattering involving a TLS Increasing Q, T 1 and T 2 via material

28 MANY THANKS TO... the nanomechanics group in Munich and Konstanz Thomas Faust Philipp Paulitschke Johannes Rieger Maximilian Seitner Sebastian Stapfner Jörg P. Kotthaus Bert Lorenz Stephan Manus