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Optomechanical Crystals in Cavity Opto- and Electromechanics

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Optomechanical Crystals in Cavity Opto- and Electromechanics Optomechanical Crystals in Cavity Opto- and Electromechanics Johannes Fink and Oskar Painter Institute for Quantum Information and Matter California Institute of Technology Announcements 2016: New Institute, brand new lab à Quantum integrated devices Deadline: Feb 7, 2016 (quantumids.com) - Circuit QED - Electro- and Optomechanics - Integrated microwave to optical link - Quantum communication & imaging Cavity Optomechanics Physics Hint = ~gomxaˆ †a = ~g0(b + b†)a†a xˆ = xzpf(b + b†) Aspelmeyer, Kippenberg, Marquardt Rev. Mod. Phys. 86 (2014) Why Mechanics? • Fundamental tests • Mechanics as a bus gravitation, decoherence connecting qubits, spins, photons, atoms, ... • Precision measurements displacements, masses, forces, • Mechanics as a toolbox accelerations storage, amplification, filtering, multiplexing, sensing, … • Mechanical circuits and arrays nonlinearities for QIP, collective dynamics Optomechanical Crystals Periodic Atomic Structure 0.5 nm àBandgaps for fo = 194 THz electron waves Periodically Placed Holes fm = 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 (g0) Coupling Strength • Phononic shield for high mechanical Q • Telecom wavelengths go = 1,100 kHz (experiment) • <n> << 1 at 10 mK J. Chan, et. al, Nature 478 (2011) Outline Lectures 1-3: Basics of OMCs Design & Engineering OMCs & Microwaves 1. Maxwell’s equations 1. OMC Band structures 1. Slot mode coupled PCs a) Basics a) Design b) Energy, mode 2. Linear and point defects b) Coupling volume and a) Basics c) Nonlinearities quantization b) 1D Nanobeam c) Symmetry and c) W1 Snowflake 2. Slot mode coupled ‘OMCs’ periodicity a) Design & coupling d) Band structures 3. OM Coupling b) Fabrication a) Boundary c) EIT 2. Acoustic wave equation b) Stress-Optical d) Ground state a) Basics c) Vacuum coupling e) TLS coupling b) Effective mass f) Wavelength c) Guided waves 4. Techniques conversion d) Band structures a) Fabrication b) Coupling 3. Outlook Optomechanical Crystals in Cavity Opto- and Electromechanics Basics of OMCs Maxwell’s Equations I Most general Linear and lossless Mixed dielectric medium Solutions are harmonic modes - No sources of light: With e.g. - Linear: - Isotropic Implications - No material dispersion - Lossless: is real and pos. - transversality - = 1 - “Master equation” à J. D. Joannopoulos, et. al, Princeton University Press (2008) Maxwell’s Equations II Procedure Eigenvalue Problem - For a given ε(r) - We can define operator Θ - Θ is linear and hermitian -> ω is real, modes are orthogonal 1D Example - Solve - Inner product - orthogonal modes to find mode profile Normalization - Then use e.g. - with - and to recover electric field profile (and make sure \ ) J. D. Joannopoulos, et. al, Princeton University Press (2008) Energy, Mode Volume, Quantization Variational Principle Quantization - Minimize EM energy functional -> minimize Uf to get lowest energy mode 2 2 ω0 /c subject to Physical Energy - For harmonic mode, time averaged ß max by definition Effective Mode Volume Relation to cavity / circuit QED - Depends on physics - Dipole moment - Electric field - Dipole coupling - Minimal possible in dielectric cavity ~ with ZPF (1D): with 2 - About ~ 0.01 μm3 (Si at 1550 nm) V usually normalized with |E(ratom)| J. D. Joannopoulos, et. al, Princeton University Press (2008) M. O. Scully, et. al, Cambridge University Press (1997) Symmetry and Bloch Waves Inversion Symmetry Continuous Translational Symmetry - Even odd - Operator - Solution (1D) - Homogeneous medium (3D): ε=1 -> plane waves -> disp. relation - Symmetry operator: Plane of glass - is also a valid mode with - Free space Band structure and α = 1 or -1 - Symmetry operations can be used to - Light line classify modes (without knowing the details of it) - Index guided J. D. Joannopoulos, et. al, Princeton University Press (2008) Periodicity and Bands Discrete Translational Symmetry Bloch Theorem (3D) - Bloch state vector - Reciprocal lattice vectors - Plane waves again Photonic Bands - Degenerate set - Operator - Reciprocal lattice vector - Transversality - Bloch - Periodicity - Brillouin zone à use MPB to get for a given J. D. Joannopoulos, et. al, Princeton University Press (2008) IBZ and Propagation Irreducible Brillouin Zone Polarization - e.g. Rotational symmetry: - 2D photonic crystals have symmetry - Allows only two different polarizations à Symmetries of the lattice are inherited by the bands TE: à Additional redundancy in the BZ TM: - In general: Bands have symmetries of point group (Rotation, Reflection, Inversion) Bloch wave propagation - IBZ of square Lattice - With time dependence à k is conserved à All scattering events are coherent - Group velocity J. D. Joannopoulos, et. al, Princeton University Press (2008) Photonic Band Gaps 1D Photonic Crystal Band structure - A multilayer film - Bloch state ε=(13,13) ε=(13,12) ε=(13,1) - BZ is 1D - PBG forms at where λ = 2 a - Consider only kz dielectric band air band - Layer width a/2 - Light line - Bandgap scales with Δε J. D. Joannopoulos, et. al, Princeton University Press (2008) Photonic Band Gaps 2D Photonic Crystal Band structure - A set of rods - 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 modes: TM: normal to plane E - Zero group velocity (standing waves) at X and M - Bloch state - Only TM has band gap à “symmetry BG” J. D. Joannopoulos, et. al, Princeton University Press (2008) Band Gaps & Slabs Triangular Lattice: Complete BG Triangular Lattice in a Slab - Compromise: weakly connected “rods” - 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: - Hexagonal BZ, BG for all polarizations - Out of plane radiation - TM –TE mixing - But no confinement to x-y plane J. D. Joannopoulos, et. al, Princeton University Press (2008) Acoustic Wave Equation Continuum mechanics Eigenvalue Problem λ ( p >> interatomic distances) - Material properties: - with operator elasticity tensor density displacement vector field Quantization again - Strain (relative deformation) - 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) 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 - Propagation - Polarization (SH), (SV) and (P) - Mirror symmetry: (-x+z), (+x-z), (+x+z) Material operator e.g. - Boundary: - Typical properties of SOI (Si) - Horizontal shear (SH) dispersion - Slab boundary couples SV and P modes - Form pair of solutions: λ = 1500 nm (-z) … flexural and (+z) … extensional T = 220 nm - Level repulsion causes low energy dispersion difference A. H. Safavi-Naeini and O. Painter, Springer (2014) 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) 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: 2 + 1 optical: 3 N - 3 - à phononic bandgap modes M. Eichenfield, et. Al. Optics Express 17 (2009) A. H. Safavi-Naeini and O. Painter, Optics Express 18 (2010) Optomechanical Crystals in Cavity Opto- and Electromechanics Design and Engineering 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) 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) 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) Point defects 1 Point defect in 1D Localization - Defect modes decay exponentially in crystal - Defect in multilayer film - 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) Point defects 2 1D Nanobeam cavity photonic phononic mech. and opt. cavity mode defect - 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 2 - Numerical optimization of geometry with fitness function, e.g. g0 /κ A. H. Safavi-Naeini and O. Painter, Optics Express 18 (2010) Linear defects Example: Waveguide in air - Introduce line defect in 2D crystal - One direction with discrete translational symmetry - ky in propagation
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