Optomechanical Crystals in Cavity Opto- and Electromechanics

Transcription

1 Optomechanical Crystals in Cavity Opto- and Electromechanics Johannes Fink and Oskar Painter Institute for Quantum Information and Matter California Institute of Technology

2 Announcements 2016: New Institute, brand new lab à Quantum integrated devices (quantumids.com) Deadline: Feb 7, Circuit QED - Electro- and Optomechanics - Integrated microwave to optical link - Quantum communication & imaging

3 Cavity Optomechanics Physics H int = ~g omˆxa a ˆx = x zpf (b + b ) Aspelmeyer, Kippenberg, Marquardt Rev. Mod. Phys. 86 (2014) Why Mechanics? Fundamental tests gravitation, decoherence Precision measurements displacements, masses, forces, accelerations Mechanical circuits and arrays nonlinearities for QIP, collective dynamics Mechanics as a bus connecting qubits, spins, photons, atoms,... Mechanics as a toolbox storage, amplification, filtering, multiplexing, sensing,

4 Optomechanical Crystals Periodic Atomic Structure 0.5 nm à Bandgaps for electron waves f o = 194 THz Periodically Placed Holes f m = 5.7 GHz à Bandgaps for sound & 500 nm light waves Independent routing of acoustic and optical waves Strong co-localization of modes Large radiation pressure effect (g 0 ) Phononic shield for high mechanical Q g o = 1,100 khz (experiment) Telecom wavelengths <n> << 1 at 10 mk Coupling Strength J. Chan, et. al, Nature 478 (2011)

5 Outline Lectures 1-3: Basics of OMCs 1. Maxwell s equations a) Basics b) Energy, mode volume and quantization c) Symmetry and periodicity d) Band structures 2. Acoustic wave equation a) Basics b) Effective mass c) Guided waves d) Band structures Design & Engineering 1. OMC Band structures 2. Linear and point defects a) Basics b) 1D Nanobeam c) W1 Snowflake 3. OM Coupling a) Boundary b) Stress-Optical c) Vacuum coupling 4. Techniques a) Fabrication b) Coupling OMCs & Microwaves 1. Slot mode coupled PCs a) Design b) Coupling c) Nonlinearities 2. Slot mode coupled OMCs a) Design & coupling b) Fabrication c) EIT d) Ground state e) TLS coupling f) Wavelength conversion 3. Outlook

6 Optomechanical Crystals in Cavity Opto- and Electromechanics Basics of OMCs

7 Maxwell s Equations I Most general Linear and lossless Mixed dielectric medium Solutions are harmonic modes - No sources of light: - Linear: - Isotropic - No material dispersion - Lossless: is real and pos. - = 1 With e.g. Implications - transversality - Master equation à J. D. Joannopoulos, et. al, Princeton University Press (2008)

8 Maxwell s Equations II Procedure - For a given ε(r) Eigenvalue Problem - We can define operator Θ - Θ is linear and hermitian -> ω is real, modes are orthogonal - Solve 1D Example - Inner product - orthogonal modes to find mode profile - Then use e.g. Normalization - with to recover electric field profile (and make sure \ ) - and J. D. Joannopoulos, et. al, Princeton University Press (2008)

9 Energy, Mode Volume, Quantization Variational Principle - Minimize EM energy functional Quantization -> minimize U f to get lowest energy mode ω 02 /c 2 subject to Physical Energy - For harmonic mode, time averaged ß max by definition Effective Mode Volume - Depends on physics - Minimal possible in dielectric cavity ~ with - About ~ 0.01 μm 3 (Si at 1550 nm) J. D. Joannopoulos, et. al, Princeton University Press (2008) Relation to cavity / circuit QED - Dipole moment - Electric field - Dipole coupling with ZPF (1D): V usually normalized with E(r atom ) 2 M. O. Scully, et. al, Cambridge University Press (1997)

10 Symmetry and Bloch Waves Inversion Symmetry - Even odd Continuous Translational Symmetry - Operator - Solution (1D) - Homogeneous medium (3D): ε=1 -> plane waves -> disp. relation - Symmetry operator: Plane of glass - is also a valid mode with and α = 1 or -1 - Free space Band structure - Symmetry operations can be used to classify modes (without knowing the details of it) - Light line - Index guided J. D. Joannopoulos, et. al, Princeton University Press (2008)

11 Periodicity and Bands Discrete Translational Symmetry Bloch Theorem (3D) - Bloch state vector - Reciprocal lattice vectors - Plane waves again Photonic Bands - Degenerate set - Reciprocal lattice vector - Bloch - Operator - Transversality - Periodicity - Brillouin zone à use MPB to get for a given J. D. Joannopoulos, et. al, Princeton University Press (2008)

12 IBZ and Propagation Irreducible Brillouin Zone - e.g. Rotational symmetry: à Symmetries of the lattice are inherited by the bands à Additional redundancy in the BZ - In general: Bands have symmetries of point group (Rotation, Reflection, Inversion) - IBZ of square Lattice Polarization - 2D photonic crystals have symmetry - Allows only two different polarizations TE: TM: Bloch wave propagation - With time dependence à k is conserved à All scattering events are coherent - Group velocity J. D. Joannopoulos, et. al, Princeton University Press (2008)

13 Photonic Band Gaps 1D Photonic Crystal - A multilayer film Band structure - Bloch state - BZ is 1D - Consider only kz ε=(13,13) ε=(13,12) ε=(13,1) - PBG forms at where λ = 2 a dielectric band air band - Layer width a/2 - Light line - Bandgap scales with Δε J. D. Joannopoulos, et. al, Princeton University Press (2008)

14 Photonic Band Gaps 2D Photonic Crystal - A set of rods Band structure - k z = 0, r = 0.2 a, ε= (8.9, 1) - Band gap in x-y plane - Can prevent light to propagate in any direction in this plane - Modes in x-y plane are TE: H normal to plane or TM: E normal to plane - Bloch state - TM modes: - Zero group velocity (standing waves) at X and M - Only TM has band gap à symmetry BG J. D. Joannopoulos, et. al, Princeton University Press (2008)

15 Band Gaps & Slabs Triangular Lattice: Complete BG - Compromise: weakly connected rods - Hexagonal BZ, BG for all polarizations Triangular Lattice in a Slab - Index guiding in z direction - Forms quasi photonic band gap (only for guided modes below light cone) - Band modes decay as exp( i (k + i κ) z) - Avoid leakage: - Out of plane radiation - TM TE mixing - But no confinement to x-y plane J. D. Joannopoulos, et. al, Princeton University Press (2008)

16 Acoustic Wave Equation Continuum mechanics (λ p >> interatomic distances) Eigenvalue Problem - Material properties: elasticity tensor density displacement vector field - Strain (relative deformation) - with operator Quantization again - Define ladder operators for each mode - Stress (Hooke s law) - Single phonon energy - Newton s law - With ZPF - Wave eqn. - And A. H. Safavi-Naeini and O. Painter, Springer (2014)

17 Waves and Phonons 1 Guided waves Phonons in a slab - EM modes 2 transverse waves (different pol.) with: - Mechanical modes 2 transverse (shear) waves with 1 longitudinal (dilatational, pressure) wave with Material - Typical properties of SOI (Si) λ = 1500 nm T = 220 nm - Propagation - Polarization (SH), (SV) and (P) - Mirror symmetry: (- x + z ), (+ x - z ), (+ x + z ) operator e.g. - Boundary: - Horizontal shear (SH) dispersion - Slab boundary couples SV and P modes - Form pair of solutions: (- z ) flexural and (+ z ) extensional - Level repulsion causes low energy dispersion difference A. H. Safavi-Naeini and O. Painter, Springer (2014)

18 Waves and Phonons 2 Phonons in a beam - Additional boundary condition - Boundaries also couples SH modes - 2 flexural modes and one extensional (+ x + z ) - One additional torsional mode (- x - z ) A. H. Safavi-Naeini and O. Painter, Springer (2014)

19 Phononic Band Structures 1D chain with basis 1D pad connector - Symmetries - Dispersion relation - Acoustic is linear at small k - Band gap scales with Δm (and K) - For N > 2 masses: acoustic: optical: 3 N - 3 modes - à phononic bandgap M. Eichenfield, et. Al. Optics Express 17 (2009) A. H. Safavi-Naeini and O. Painter, Optics Express 18 (2010)

20 Optomechanical Crystals in Cavity Opto- and Electromechanics Design and Engineering

21 OMC Band Structures 1 Quasi - 1D Nanobeam crystal Lattice: photonic bands: phononic bands: - Symmetry points: Γ(k=0), M (k=π/a) - Optics: Fundametal TE modes in black - Mechanics: Extensional modes shown in black A. H. Safavi-Naeini and O. Painter, Optics Express 18 (2010)

22 OMC Band Structures 2 Quasi 2D Cross crystal Lattice: photonic bands: phononic bands: (even, vertical sym.) à Bad choice for OMC à Great choice for phononic shield A. H. Safavi-Naeini and O. Painter, Springer (2014)

23 OMC Band Structures 3 Quasi 2D Snowflake crystal Lattice: photonic bands: phononic bands: (even, vertical sym.) à Higher symmetry à Independent tuning a-2r (phononics) and w (photonics) à Great choice for OMC A. H. Safavi-Naeini and O. Painter, Springer (2014)

24 Point defects 1 Point defect in 1D - Defect in multilayer film Localization - Defect modes decay exponentially in crystal - Evanescent with complex k+iκ - Can approximate - Density of states - Large k and small V at midgap - Defect allows localized mode - νspecific mirrors for cavity - Can pull or push a defect from any band - Strong confinement causes radiation loss J. D. Joannopoulos, et. al, Princeton University Press (2008)

25 Point defects 2 1D Nanobeam cavity photonic phononic defect mech. and opt. cavity mode - Push optical defect for X point ß further from light cone - Pull mechanical defect from Γ point ß constructive overlap with optical mode - Choose a quadratic scaling of the defect ß minimize wave package in real and reciprocal space - Numerical optimization of geometry with fitness function, e.g. g 02 /κ A. H. Safavi-Naeini and O. Painter, Optics Express 18 (2010)

26 Linear defects Example: Waveguide in air - Introduce line defect in 2D crystal - One direction with discrete translational symmetry - k y in propagation direction is conserved - Projected band structure for dielectric rods: for a (k y,ω 0 ), choose any k x (continuous regions) - Guided band inside the BG - Coupling to and guiding of traveling photons and phonons J. D. Joannopoulos, et. al, Princeton University Press (2008)

27 Linear + Point defect in 2D Snowflake waveguide - Missing row of snowflakes Snowflake cavity - Change radius of snowflakes (quadratically) - Band diagrams - Cavity modes: E y (r) Q(r) A. H. Safavi-Naeini and O. Painter, Optics Express 18 (2010)

28 Optomechanical Coupling Small Perturbations - Get mode profiles Q(r) and e(r) - Small modifications Photo elastic coupling - Strain affects the refractive index - To first order with - With photo elastic tensor p - Coupling: - intuitively Vacuum Coupling - Multiply with ZPF overlap Boundary perturbation - Deformation affects dielectric function - High contrast step function across a boundary is shifted - Need to relate deformation to - Total coupling is the sum of both A. H. Safavi-Naeini and O. Painter, Springer (2014)

29 Recipe for designing an OMC - Conceive a suitable design lattice / unit cell - Get the material parameters - Simulate photonic band structure in MPB - Simulate phononic band structure in Comsol - Optimize the design by hand for good band gaps - Simulate the band structures of different defect perturbations (tuning) - Now simulate the full cavity in Comsol (use all available symmetries) - Extract frequencies, Q opt, g om and check overlap of modes - Simulate Q mech using a perfectly matched layer - Maybe add a phononic shield to improve Q mech if possible - Define a fitness function e.g. g o2 /κ and do numerical optimization of the design i.e. vary defect size, depth, perturbation - Test if design is robust, i.e. remove symmetries in simulation, introduce fabrication defects - Try to fabricate and test it!

30 Fabrication of OMCs - SOI substrate - ZEP resist kev EBPG - Optimized C4F8 / SF6 plasma etch - 49% HF release - Repeated piranha cleaning + H termination (1:20 HF in water) A. H. Safavi-Naeini and O. Painter, Springer (2014)

31 Coupling to OMCs - Fiber taper coupling - End fire - With adiabatic coupler - V-groove S. M. Meenehan et al., PRA 90 (2014) 50 μm A. H. Safavi-Naeini and O. Painter, Springer (2014) J. D. Cohen et al., Opt. Express 21 (2013)

32 Optomechanical Crystals in Cavity Opto- and Electromechanics OMCs and Microwaves

33 Circuit QED + OMCs µw Circuits + Optomechanics: Quantum Microwave Photonics µw Circuits + Acoustic Cavities: Microwave Phonon Circuits Microwaves Good qubits Very large g à Processing Optics Low loss Noise resilient à Communication GHz acoustics No active cooling Acoustic waveguides & circuits Phonon interference, entanglement AO transducer State synthesis and distribution Interface for circuits and atoms Quantum Internet Why with microwaves? Less heating Circuit QED toolbox Fully engineered Losses & materials Complex fabrication Challenges Size mismatch à small g em Low bandwidth Heating Quasiparticles

34 Microwave to Optical Beam splitter like interaction Conversion efficiency Beat losses: Impedance matching: Safavi-Naeini, A. H. et al., NJP 13, (2011) Wang, Y. and Clerk, A., PRL 108, (2012) Vitali, D. et al., PRL 109, (2012)

35 A. Di Falco, APL 92 (2008) A. Safavi-Naeini, et al, APL 97 (2010) Winger, M. et al., Opt. Expr. 19, (2011) Sun, X. et al., APL 101, (2012) Tunable Photonic Crystal A. Pitanti, J. M. Fink, et al., Opt. Expr. 23 (2015) SOI RT! o /2 200 THz! m /2 60 MHz Q o 10 5 Q m 10 2 m e 7.8 pg x zpf 4.1 fm g 0,om /2 490 khz C m 1fF 2D photonic crystal Mechanics Experimental setup

36 Electromechanical Coupling EM coupling Voltage tuning Capacitive force And measured opt. tuneability α 0 =-2.5 pm/v 2 Modulated capacitance EM coupling Stray: C s ~ 12 ff: g em ~ 50 MHz/nm à g ext,em ~ 15 Hz (Q s ~ 10 5, 8 GHz) à C em > 1 for n~ 10 5 à C om > 1 for n ~ 10 2 A. Pitanti, J. M. Fink, et al., Opt. Expr. 23 (2015)

37 Nonlinear Mechanics Nonlinear coupling Capacitive softening Duffing response and phase locking à Directions: thermal squeezing, amplification, efficient AO modulation (go to vacuum) à Linear range sufficient for state conversion however not sideband resolved (get an inductor) à But want better g 0, lower C s, and a low loss substrate A. Pitanti, J. M. Fink, et al., Opt. Expr. 23 (2015)

38 Slot Mode Coupled OMCs Can we do even better? - Remove substrate! - Increase mechanical frequency (sideband resolution) à Slot mode coupled OMCs on stressed silicon nitride M. Davanco et al., Opt. Express 20 (2012) K. E. Grutter et al., arxiv (2015)

39 Silicon Nitride Chip Design Schematic Circuit High Z Inductors: à Can get as low as Cs~2 ff à new circuit element: a linear superinductor à Great for coupling any small dipole moment object! à Localizes charge on capacitor

40 Silicon Nitride Chip Design Expected coupling: à Expect Hz à Total Impedance is about 4 Rq à Cs due to additional wiring, cross overs, loading, ground, Acoustic bandgap:

41 Si3N4 Through Chip Membrane Devices Etch through Si wafer leaving 300 nm thick transsi 3 N 4 membrane 32 LC circuits On 4x4 membranes Transmission Lines

42 On-membrane circuit Double cavity device Top coil

43 On-membrane circuit Double cavity device Nanobeam center

44 Fabrication Key fab steps Gap view

45 Setup and basic Characteristics Setup Microwave Q

46 Coherent Response: EIT à n d ~ 10 7 à Q m ~ 5x10 5 à G/π ~ 400 khz à C max ~ 10 4

47 Thermometry Calibration (C<<1) T i ~ 220mK Fluctuation dissipation theorem: g 0 /2π ~ 41 Hz x zpf ~ 8 fm T f ~ 20 mk

48 Ground State Cooling

49 Strong coupling to microscopic TLS Cavity QED physics Vacuum Rabi splitting g = E 0 d/~ E 0 (~!/V ) 1/2 à g/pi~1.8 MHz à k/2pi~1.4 MHz ac Stark tuning Facilitated by extreme electric field confinement à ~120 V/m for single Photon in the center of the gap D. Walls & G. Milburn, Quantum Optics (1994)

50 All-Microwave Wavelength Conversion Beam splitter like interaction Conversion efficiency Beat losses: Impedance matching: Safavi-Naeini, A. H. et al., NJP 13, (2011) Wang, Y. and Clerk, A., PRL 108, (2012) Hill, J. T. et al., Nat. Commun. 3, (2012)

51 Cooling Run & EIT mode 7.4 GHZ mode 9.3 GHz à LW/2pi: 7 / 8 Hz à g0/2pi: 33 / 44 Hz

52 Wavelength Conversion Theory: Efficiency calibration: Conversion: P signal = - 60 dbm ~ 26 photons R. W. Andrews, Nat. Phys. 10 (2014)

53 Wavelength Conversion Conversion efficiency vs. C1, C2: ~ 60% Data Theory Bandwidth: ~ 1 khz -10, -9 dbm à 0, 0 dbm Dynamic Range: ~ 10 6

54 Outlook: 01/2016 à Acoustic mode Lower SQL input power needs large mutual L x 2 detection which way experiment with phonons Circuit QED + mechanics State synthesis and verification Nonlinearities Paramps Mech. tunability Microwave to optical on Si (or Si3N4) On-chip coupling New Si fab process RT photon counting Teleportation, Quantum Illumination More internal dynamics Intel i7: 10 9 transistors 10 3 contact pins 50 μm On-chip demultiplexing Hardware protection (0-pi) Many body physics

55 Acknowledgements Alessandro Pitanti à Pisa Richard Norte à Delft Mahmoud Kalaee Oskar Painter

56 References Molding the flow of light John D. Joannopoulos, Steven G. Johnson, Joshua N. Winn, and Robert D. Meade. Princeton University Press, second edition (2008) Cavity Optomechanics Nano- and Micromechanical Resonators Interacting with Light, Springer (2014) Chapter: Optomechanical Crystal Devices Amir H. Safavi-Naeini, Oskar Painter