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) •
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 direction is conserved
- Projected band structure for dielectric rods:
for a (ky,ω0), choose any kx (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) Linear + Point defect in 2D
Snowflake waveguide Snowflake cavity
- Missing row of snowflakes - Change radius of snowflakes (quadratically)
- Band diagrams
- Cavity modes:
Ey(r)
Q(r)
A. H. Safavi-Naeini and O. Painter, Optics Express 18 (2010) Optomechanical Coupling
Small Perturbations Photo elastic coupling
- Get mode profiles Q(r) and e(r) - Strain affects the refractive index - Small modifications
- To first order - With photo elastic tensor p with - Coupling:
- intuitively overlap Boundary perturbation
Vacuum Coupling - Deformation affects dielectric function - High contrast step function across a boundary is - Multiply with ZPF shifted - Need to relate deformation to
- Total coupling is the sum of both
A. H. Safavi-Naeini and O. Painter, Springer (2014) ‘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, Qopt, gom and check overlap of modes
- Simulate Qmech using a perfectly matched layer
- Maybe add a phononic shield to improve Qmech if possible 2 - Define a fitness function e.g. go /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! Fabrication of OMCs
- SOI substrate - ZEP resist
- 100 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) Coupling to OMCs
- Fiber taper coupling - End fire
- With adiabatic coupler
S. M. Meenehan et al., PRA 90 (2014) - V-groove
50 μm
A. H. Safavi-Naeini and O. Painter, Springer (2014) J. D. Cohen et al., Opt. Express 21 (2013) Optomechanical Crystals in Cavity Opto- and Electromechanics
OMCs and Microwaves Circuit QED + OMCs
µw Circuits + Optomechanics: µw Circuits + Acoustic Cavities: ‘Quantum Microwave Photonics‘ ‘Microwave Phonon Circuits‘
Microwaves Optics GHz acoustics • Good qubits • Low loss • No active cooling • Very large g • Noise resilient • Acoustic waveguides & circuits à Processing à Communication • Phonon interference, entanglement
AO transducer Why with microwaves? • State synthesis and distribution • Less heating • Interface for circuits and atoms • Circuit QED toolbox • ‘Quantum Internet’ • Fully engineered
Challenges
• Losses & materials • Size mismatch à small gem • Heating • Complex fabrication • Low bandwidth • Quasiparticles Microwave to Optical
Beam splitter like interaction
Conversion efficiency
Beat losses: Impedance matching:
Safavi-Naeini, A. H. et al., NJP 13, 013017 (2011) Wang, Y. and Clerk, A., PRL 108, (2012) Vitali, D. et al., PRL 109, (2012) Tunable Photonic Crystal A. Pitanti, J. M. Fink, et al., Opt. Expr. 23 (2015)
! /2⇡ 200 THz SOI Sample @ RT o ⇠ ! /2⇡ 60 MHz m ⇠ Q 105 o ⇠ Q 102 m ⇠ m 7.8pg e↵ ⇠ x 4.1fm zpf ⇠ g /2⇡ 490 kHz 0,om ⇠ C 1fF m ⇠ 2D photonic crystal Mechanics Experimental setup
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) Electromechanical Coupling
EM coupling Voltage tuning Capacitive force
And measured opt. tuneability 2 α0=-2.5 pm/V
Modulated capacitance
EM coupling
Stray: Cs~ 12 fF:
à gext,em~ 15 Hz 5 (Qs ~ 10 , 8 GHz) 5 à Cem > 1 for n~ 10 2 gem~ 50 MHz/nm à Com > 1 for n ~ 10
A. Pitanti, J. M. Fink, et al., Opt. Expr. 23 (2015) 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 g0, lower Cs, and a low loss substrate
A. Pitanti, J. M. Fink, et al., Opt. Expr. 23 (2015) 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 1508.05919 (2015) 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 Silicon Nitride Chip Design
Expected coupling:
à Expect 40-50 Hz à Total Impedance is about 4 Rq à Cs due to additional wiring, cross overs, loading, ground, …
Acoustic bandgap: Si3N4 Through Chip Membrane Devices
Etch through Si wafer leaving 300 nm thick
transSi3N4 membrane
32 LC circuits On 4x4 membranes
Transmission Lines On-membrane circuit
Double cavity device Top coil On-membrane circuit
Double cavity device Nanobeam center Fabrication
Key fab steps Gap view Setup and basic Characteristics
Setup
Microwave Q Coherent Response: EIT
7 à nd ~ 10 5 à Qm ~ 5x10 à G/π ~ 400 kHz 4 à Cmax ~ 10 Thermometry Calibration (C<<1)
Ti~ 220mK
Fluctuation dissipation theorem:
g0/2π ~ 41 Hz Tf ~ 20 mK xzpf ~ 8 fm Ground State Cooling Strong coupling to microscopic TLS
Cavity QED physics Vacuum Rabi splitting
1/2 g = E0d/~ E0 (~!/V ) ⇠ 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) All-Microwave Wavelength Conversion
Beam splitter like interaction
Conversion efficiency
Beat losses: Impedance matching:
Safavi-Naeini, A. H. et al., NJP 13, 013017 (2011) Wang, Y. and Clerk, A., PRL 108, (2012) Hill, J. T. et al., Nat. Commun. 3, (2012) Cooling Run & EIT mode 1 @ 7.4 GHZ mode 2 @ 9.3 GHz
à LW/2pi: 7 / 8 Hz à g0/2pi: 33 / 44 Hz Wavelength Conversion
Theory: Efficiency calibration:
R. W. Andrews, Nat. Phys. 10 (2014) Conversion:
Psignal = - 60 dBm ~ 26 photons Wavelength Conversion
Conversion efficiency vs. C1, C2: ~ 60% Bandwidth: ~ 1 kHz -10, -9 dBm à 0, 0 dBm Data Theory
Dynamic Range: ~ 106 Outlook: 01/2016 à
Acoustic mode Circuit QED + mechanics • Lower SQL input power • State synthesis and verification • Nonlinearities • Paramps • needs large mutual L • Mech. tunability • x2 detection • which way experiment with phonons
Microwave to optical on Si (or Si3N4) More internal dynamics • On-chip coupling • New Si fab process Intel i7: • RT photon counting 109 transistors • Teleportation, Quantum Illumination… 103 contact pins
50 μm
• On-chip demultiplexing • Hardware protection (0-pi) • Many body physics Acknowledgements
Alessandro Pitanti à Pisa Richard Norte à Delft Mahmoud Kalaee
Oskar Painter References
Molding the flow of light Cavity Optomechanics Nano- and Micromechanical Resonators John D. Joannopoulos, Steven G. Johnson, Interacting with Light, Springer (2014) Joshua N. Winn, and Robert D. Meade. Princeton University Press, second edition (2008) Chapter: Optomechanical Crystal Devices Amir H. Safavi-Naeini, Oskar Painter
http://ab-initio.mit.edu/book/
http://www.springer.com/fr/book/ 9783642553110