Optomechanical Crystals in Cavity Opto- and Electromechanics · 2016. 11. 15. · Maxwell’s...

Johannes Fink and Oskar Painter

Institute for Quantum Information and Matter California Institute of Technology

Optomechanical Crystals in Cavity Opto- and Electromechanics

Announcements

2016: New Institute, brand new lab

à

-  Circuit QED -  Electro- and Optomechanics -  Integrated microwave to optical link -  Quantum communication & imaging

Quantum integrated devices (quantumids.com)

Deadline: Feb 7, 2016

Why Mechanics?

Cavity Optomechanics

Physics

x̂ = xzpf(b+ b

†)

Hint

= ~gom

x̂a†a = ~g0

(b+ b†)a†a

Aspelmeyer, Kippenberg, Marquardt Rev. Mod. Phys. 86 (2014)

•  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, …

Optomechanical Crystals

Coupling Strength

J. Chan, et. al, Nature 478 (2011)

go = 1,100 kHz (experiment)

0.5 nm Periodic Atomic Structure

500 nm

Periodically Placed Holes

àBandgaps for electron waves

fm = 5.7 GHz

fo = 194 THz

à Bandgaps for sound & light waves •  Independent routing of acoustic and optical waves

•  Strong co-localization of modes •  Large radiation pressure effect (g0) •  Phononic shield for high mechanical Q •  Telecom wavelengths •  <n> << 1 at 10 mK

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

Optomechanical Crystals in Cavity Opto- and Electromechanics

Basics of OMCs

Maxwell’s Equations I

J. D. Joannopoulos, et. al, Princeton University Press (2008)

Most general

Mixed dielectric medium

-  No sources of light: -  Linear: -  Isotropic -  No material dispersion -  Lossless: is real and pos. -  = 1 à

Linear and lossless

Solutions are harmonic modes With e.g.

Implications - transversality - “Master equation”

Maxwell’s Equations II

J. D. Joannopoulos, et. al, Princeton University Press (2008)

Procedure -  For a given ε(r)

-  Solve

to find mode profile

-  Then use e.g. to recover electric field profile (and make sure \ )

Eigenvalue Problem -  We can define operator Θ

-  Θ is linear and hermitian -> ω is real, modes are orthogonal

1D Example -  Inner product -  orthogonal modes

Normalization -  with

-  and

Energy, Mode Volume, Quantization

J. D. Joannopoulos, et. al, Princeton University Press (2008)

Variational Principle -  Minimize EM energy functional

-> minimize Uf to get lowest energy mode ω0

2/c2 subject to

Effective Mode Volume -  Depends on physics

-  Minimal possible in dielectric cavity ~ with

-  About ~ 0.01 μm3 (Si at 1550 nm)

Physical Energy -  For harmonic mode, time averaged

Relation to cavity / circuit QED

-  Dipole moment -  Electric field -  Dipole coupling

with ZPF (1D):

V usually normalized with |E(ratom)|2

M. O. Scully, et. al, Cambridge University Press (1997)

Quantization

ß max by definition

Symmetry and Bloch Waves

J. D. Joannopoulos, et. al, Princeton University Press (2008)

Inversion Symmetry -  Even odd

-  Symmetry operator:

-  is also a valid mode with and α = 1 or -1

-  Symmetry operations can be used to classify modes (without knowing the details of it)

Continuous Translational Symmetry -  Operator

-  Solution (1D)

-  Homogeneous medium (3D): ε=1 -> plane waves -> disp. relation

Plane of glass

-  Free space

-  Light line

-  Index guided

Band structure  

Periodicity and Bands

J. D. Joannopoulos, et. al, Princeton University Press (2008)

Discrete Translational Symmetry

-  Plane waves again

-  Degenerate set

-  Reciprocal lattice vector

-  Bloch

-  Brillouin zone

Bloch Theorem (3D)

-  Bloch state vector

-  Reciprocal lattice vectors

Photonic Bands

-  Operator

-  Transversality

-  Periodicity

à use MPB to get for a given

IBZ and Propagation

J. D. Joannopoulos, et. al, Princeton University Press (2008)

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

Band structure

-  PBG forms at where λ = 2 a

dielectric band air band

-  Bandgap scales with Δε

Photonic Band Gaps

J. D. Joannopoulos, et. al, Princeton University Press (2008)

1D Photonic Crystal -  A multilayer film

-  Bloch state

-  BZ is 1D

-  Consider only kz

-  Layer width a/2

-  Light line

ε=(13,13) ε=(13,12) ε=(13,1)  

Band structure -  k z = 0, r = 0.2 a, ε= (8.9, 1)

-  TM modes: -  Zero group velocity (standing waves) at X and M

-  Only TM has band gap à “symmetry BG”

Photonic Band Gaps

J. D. Joannopoulos, et. al, Princeton University Press (2008)

2D Photonic Crystal -  A set of rods

-  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

Band Gaps & Slabs

J. D. Joannopoulos, et. al, Princeton University Press (2008)

Triangular Lattice: Complete BG -  Compromise: weakly connected “rods”

-  Hexagonal BZ, BG for all polarizations

-  But no confinement to x-y plane

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

Eigenvalue Problem -  with operator

Acoustic Wave Equation

A. H. Safavi-Naeini and O. Painter, Springer (2014)

Continuum mechanics (λp >> interatomic distances)

-  Material properties:

elasticity tensor density displacement vector field

-  Strain (relative deformation)

-  Stress (Hooke’s law)

-  Newton’s law

-  Wave eqn.

Quantization again

-  Define ladder operators for each mode

-  Single phonon energy

-  With ZPF

-  And

Waves and Phonons 1 Guided waves

-  EM modes 2 transverse waves (different pol.) with:

-  Mechanical modes 2 transverse (shear) waves with 1 longitudinal (dilatational, pressure) wave with

A. H. Safavi-Naeini and O. Painter, Springer (2014)

Material

-  Typical properties of SOI (Si)

Phonons in a slab -  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

λ = 1500 nm T = 220 nm  

Waves and Phonons 2

A. H. Safavi-Naeini and O. Painter, Springer (2014)

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)

1D pad connector

-  Symmetries

-  à phononic bandgap

Phononic Band Structures 1D chain with basis

-  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 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

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

OMC Band Structures 1

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)

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

OMC Band Structures 3

A. H. Safavi-Naeini and O. Painter, Springer (2014)

Point defects 1

J. D. Joannopoulos, et. al, Princeton University Press (2008)

Point defect in 1D

-  Defect in multilayer film

-  Density of states

-  Defect allows localized mode -  νspecific “mirrors” for cavity -  Can “pull” or “push” a defect from any band

Localization

-  Defect modes decay exponentially in crystal -  Evanescent with complex k+iκ

-  Can approximate

-  Large k and small V at midgap

-  Strong confinement causes radiation loss

1D Nanobeam cavity -  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. g0

2/κ

Point defects 2

photonic phononic

defect mech. and opt. cavity mode

A. H. Safavi-Naeini and O. Painter, Optics Express 18 (2010)

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

Linear defects

J. D. Joannopoulos, et. al, Princeton University Press (2008)

Snowflake cavity

-  Change radius of snowflakes (quadratically)

-  Cavity modes:

Snowflake waveguide

-  Missing row of snowflakes

-  Band diagrams

Linear + Point defect in 2D

A. H. Safavi-Naeini and O. Painter, Optics Express 18 (2010)

Ey(r)  

Q(r)  

Optomechanical Coupling Small Perturbations

-  Get mode profiles Q(r) and e(r) -  Small modifications -  To first order

with

-  intuitively

A. H. Safavi-Naeini and O. Painter, Springer (2014)

Boundary perturbation

-  Deformation affects dielectric function -  High contrast step function across a boundary is

shifted -  Need to relate deformation to

Vacuum Coupling

-  Multiply with ZPF

-  Total coupling is the sum of both

overlap  

Photo elastic coupling

-  Strain affects the refractive index

-  With photo elastic tensor p -  Coupling:

‘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

-  Define a fitness function e.g. go2/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!

-  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)

Fabrication of OMCs

A. H. Safavi-Naeini and O. Painter, Springer (2014)

-  Fiber taper coupling

-  With adiabatic coupler

Coupling to OMCs

A. H. Safavi-Naeini and O. Painter, Springer (2014)

-  End fire

-  V-groove

50  μm  

J. D. Cohen et al., Opt. Express 21 (2013)

S. M. Meenehan et al., PRA 90 (2014)

Optomechanical Crystals in Cavity Opto- and Electromechanics

OMCs and Microwaves

Circuit QED + OMCs

Challenges

Optics •  Low loss •  Noise resilient à Communication

µw Circuits + Optomechanics: ‘Quantum Microwave Photonics‘

Why with microwaves? •  Less heating •  Circuit QED toolbox •  Fully engineered

GHz acoustics •  No active cooling •  Acoustic waveguides & circuits •  Phonon interference, entanglement

Microwaves •  Good qubits •  Very large g à Processing

AO transducer •  State synthesis and distribution •  Interface for circuits and atoms •  ‘Quantum Internet’

µw Circuits + Acoustic Cavities: ‘Microwave Phonon Circuits‘

•  Size mismatch à small gem •  Low bandwidth

•  Heating •  Quasiparticles

•  Losses & materials •  Complex fabrication

Microwave to Optical

Beat losses: Impedance matching:

Beam splitter like interaction Conversion efficiency

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)

SOI Sample @ RT

Tunable Photonic Crystal

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)

Experimental setup Mechanics 2D photonic crystal

!

o

/2⇡ ⇠ 200 THz

!

m

/2⇡ ⇠ 60 MHz

Q

o

⇠ 105

Q

m

⇠ 102

m

e↵

⇠ 7.8 pg

x

zpf

⇠ 4.1 fm

g

0,om

/2⇡ ⇠ 490 kHz

C

m

⇠ 1 fF

A. Pitanti, J. M. Fink, et al., Opt. Expr. 23 (2015)

Voltage tuning

Electromechanical Coupling

EM coupling

A. Pitanti, J. M. Fink, et al., Opt. Expr. 23 (2015)

Capacitive force And measured opt. tuneability α0=-2.5 pm/V2

EM coupling

Modulated capacitance

gem~ 50 MHz/nm

Stray: Cs~ 12 fF: à gext,em~ 15 Hz (Qs ~ 105, 8 GHz) à  Cem > 1 for n~ 105 à  Com > 1 for n ~ 102

Duffing response and phase locking

Nonlinear Mechanics

Nonlinear coupling Capacitive softening

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

Microwave Q

Setup

Coherent Response: EIT

à nd ~ 107 à Qm ~ 5x105 à G/π ~

400 kHz à Cmax ~ 104

Thermometry Calibration (C<<1)

Ti~ 220mK

Tf ~ 20 mK g0/2π ~ 41 Hz xzpf ~ 8 fm

Fluctuation dissipation theorem:  

Ground State Cooling

Strong coupling to microscopic TLS

Cavity QED physics

E0 ⇠ (~!/V )1/2g = E0d/~

D. Walls & G. Milburn, Quantum Optics (1994)

Facilitated by extreme electric field confinement à  ~120 V/m for single Photon

in the center of the gap

Vacuum Rabi splitting

ac Stark tuning

à  g/pi~1.8 MHz à  k/2pi~1.4 MHz

All-Microwave Wavelength Conversion

Beat losses: Impedance matching:

Beam splitter like interaction Conversion efficiency

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

à LW/2pi: 7 / 8 Hz à g0/2pi: 33 / 44 Hz

mode 1 @ 7.4 GHZ mode 2 @ 9.3 GHz  

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%

Dynamic Range: ~ 106

Bandwidth: ~ 1 kHz -10, -9 dBm à 0, 0 dBm

Data Theory  

More internal dynamics

Outlook: 01/2016 à

Acoustic mode •  Lower SQL input power •  needs large mutual L •  x2 detection •  which way experiment with phonons

Microwave to optical on Si (or Si3N4) •  On-chip coupling •  New Si fab process •  RT photon counting •  Teleportation, Quantum Illumination…

50  μm  

Intel i7: 109 transistors 103 contact pins •  On-chip demultiplexing •  Hardware protection (0-pi) •  Many body physics

Circuit QED + mechanics •  State synthesis and verification •  Nonlinearities •  Paramps •  Mech. tunability

Acknowledgements Alessandro Pitanti à Pisa Richard Norte à Delft Mahmoud Kalaee

Oskar Painter

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) http://ab-initio.mit.edu/book/

Cavity Optomechanics Nano- and Micromechanical Resonators Interacting with Light, Springer (2014) Chapter: Optomechanical Crystal Devices Amir H. Safavi-Naeini, Oskar Painter http://www.springer.com/fr/book/9783642553110