Strong quadratic and quartic coupling

1
Strong quadratic and quartic coupling in a
low-loss optomechanical system Jack Harris, Jack
Sankey, Ben Zwickl, Andrew Jayich, Brian
Yang Departments of Physics and Applied Physics,
Yale University, New Haven, CT, USA

• Can one observe energy quantization and quantum
  jumps in mechanical oscillators?
• The necessity of a strong x2 readout
• Progress towards this goal using avoided
  crossings

Steve Girvin (Yale) Florian Marquardt (LMU) Aash
Clerk (McGill) Andreas Nunnenkamp (Yale) Kjetil
Børkje (Yale)
Collaborators
2
Optomechanical systems photons coupling to
mechanics via radiation pressure
UCSB, Leiden
Yale
Caltech
ENS
NIST
Cornell
LIGO
Vienna
kg
mg
g
mg
ng
pg
Mass
Caltech, Laussane
MIT
ENS
JILA
IBM
Nanotubes, BECs, Atoms, Ions
Oregon
3
Top-down
Top-up

• Layered deposition
• ebeam / optical lithography

• Macrofab
• Self assembly?

Bottom-up
Bottom-down

• Self assembly
• Chemical synthesis

• Self assembly
• ebeam

4
Optomechanical Hamiltonian
Mechanical degree of freedom
This specifies the form of the optomechanical
coupling
Optical degree of freedom
Usually we are concerned with very small
dispalcements, so Taylor expand wcav(x)
Usually dominates
Radiation pressure
Dispersive coupling
Something else
Usually unimportant
photo-Duffing
Et cetera
5
Optomoechanical systems scientific goals
Fixed mirror
Observed classical effects Laser cooling Optical
bistability Nonlinear dynamics
spring
x
Pin

• Goals for the quantum regime
• Radiation pressure shot noise
• (quantum backaction)
• Squeezed light production
• Ground state cooling
• Quantum jumps of a MEMS

Movable mirror
6
linear versus quadratic optomechanical systems
Fixed mirror
spring
this specifies the form of the optomechanical
interaction
x
Pin
If the cavity detuning wcav x
Movable mirror
note that
Reflected light measures position, incompatible
with phonon eigenstate
If the cavity detuning wcav x2
Two major differences
Phonon number unchanged by readout
Cavity experiences a shift per phonon
proportional to the curvature of the detuning
Reflected light measures position squared,
equiv. to phonon number
7
Realizing quadratic optomechanical coupling

• Cavity detuning is quadratic in x when membrane
  is at a node

Fixed mirror
Fixed mirror
Pin
Optical mode
cavity detuning
slab position x
Detuning membranes overlap with optical
mode Dispersive coupling similar to
off-resonant atom in a cavity
8
The actual device
Transmission (a.u.)
TEM0,0 modes
2.24
spacer
Laser detuning (GHz)
Laser
FSR
1.12
1 mm x 1 mm x 50 nm SiNx membrane
0.00
532
1064
Membrane position (nm)

• Fits to membrane reflectivity 0.35
• Consistent with 50 nm thick membrane and n 2.2
• Quadratic detuning points allow x2 coupling

1 mm

• Membrane properties
• Resonant wm 2p 105 - 106 Hz
• Q factor 106 at 300K 107 at 300 mK
• Optical absorption

9
What does it take to see quantum jumps of the
membrane?
Measure single phonon cavity shift w/ SNR gt 1
during membrane phonon lifetime
Cavity shift per phonon
Shot noise
Thermal lifetime of n 0 Fock state
Technical goal Maximize , F, xzpf, Q
More subtle avoid unintentional position
measurement

• Membrane not exactly at sweet spot
  to lowest order
• Reflection versus transmission for a two-sided
  cavity but
  
• Finesse gradient in a one-sided cavity

Pref (x)
Ptran (x)
Pref (x)
r 1
10
Optical characterization cavity spectroscopy
2.24
Transmission (a.u.)
Laser detuning (GHz)
1.12
0.00
532
1064
Membrane position (nm)

• Fitting gives membrane reflectivity 0.35
• Consistent with 50 nm membrane with n 2.2
• Quadratic detuning points allow x2 measurement
• But 30 kHz/nm2 is too small by a factor of 100 !

Is phonon QND practical? Not yet.
11
Enhancing the quadratic coupling
4.48
TEM0,0 modes
3.36
Laser detuning (GHz)
2.24
Transmission (a.u.)
1.12
0.00
0
532
1064
1596
membrane displacement (nm)
12
Enhancing the quadratic coupling
Crossings made possible by dispersive
geometry. Curvature 1/gap Possible to
predict?
tweak the input coupling
4.48
32 kHz/nm2
TEM2,0 triplet
44.8
TEM0,0 singlet
3.36
Laser detuning (GHz)
Laser detuning (MHz)
2.24
0.00
1.8 MHz/nm2
1.12
42.5
85.0
0.00
membrane displacement (nm)
0.00
0
532
1064
1596
membrane displacement (nm)
13
Calculating these avoided crossings
Cavity eigenmodes are given by the Helmholtz
equation

• Can we solve this for a cavity with a membrane
  having arbitrary
• Position along the cavity axis?
• Tilt relative to the cavity axis?
• Thickness?

We know the solution without any membrane present

hermite-gauss transverse mode
guoy phase
wavefront curvature
plane wave
So try perturbation theory for the Helmholtz
equation
14
Calculating these avoided crossings
Similar to the Time-Independent Schroedinger
Equation where the membranes index of
refraction acts as a perturbing potential
New eigenmodes
Perturbation theory
New eigenvalues
Unpertubed eigenmodes
Perturbation
New eigenmodes and eigenfrequencies can be
calculated using only the unperturbed (empty
cavity) solutions!!
Physical picture
When membrane is at cavity waist and untilted,
symmetry is preserved light reflected from
membrane maintains its transverse
profile. Otherwise, membrane reflects light into
a different transverse profile. This couples
different transverse modes.
phase fronts
membrane
15
Theory reproduces gross structure
Comparing theory and experiment
theory
Overview, one more time
Detuning (GHz)
4.4
Membrane position (nm)
Membrane at waist, no tilt symmetry preserved
2.2
Detuning (GHz)
0.0
16
Checking the accuracy of the theory
Tilted membrane at waist
Tilted membrane 500 mm from waist
Gap size versus membrane position
Data
Theory
17
Maximizing detuning curvature
30 MHz/nm2 !
1,000 x improvement adequate for QND

• Theory requires more input to reproduce features
  exactly
• Qualitative agreement is very good
• Empirically, can be made very large

18
SiNx absorption has limited cavity finesse, cause
finesse gradient
empty cavity finesse 205,000
empty cavity finesse 18,000
16,000
160,000
14,000
120,000
Finesse
12,000
Finesse
80,000
10,000
40,000
8,000
6,000
0
0
532
1064
0
1064
x (nm)
x (nm)
Fit Imn 1.5 10-4
Fit Imn 2.3 10-4

• Membrane absorption keeps finesse low
• Finesse gradient displacement measurement

New result Stochiometric Si3N4 has at least 100
x lower absorption
52,000 50,000 48,000
Estimate Imn lt 1.5 10-6
Stochiometric membranes have same Q, 10 x higher
wm What about finesse gradient at crossings?
Finesse
0
1064
2128
thanks O. Painter, C. Regal
x (nm)
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Finesse gradient at crossings can be tuned to zero
Diagonal membrane tilt
Vertical membrane tilt

• Each mode has different finesse (mostly due to
  mirror inhomogeneities)

• At crossings, finesse values switch leading to
  strong finesse gradient

• is good for finesse gradient cooling (A.
  Clerk)
• is bad for phonon QND

• We can maximize or tune it to zero by
  varying the membrane tilt axis

20
Other functionality quartic optomechanical
coupling
Detuning (GHz)
zoom-in
Membrane position (nm)
Quartic detuning occurs as part of gross band
structure, not very sharp
21
Other functionality quartic optomechanical
coupling
Features in some crossings could be tuned for
sharper quartic coupling
TEM4,0, TEM3,1,TEM2,2,TEM1,3,TEM0,4
TEM2,0 TEM1,1,TEM0,2
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Dispersive coupling a new type of optomechanics

• Avoided crossings of cavity modes enable
  different optomechanical couplings
• Strong quadratic coupling may enable single
  phonon QND
• Strong quartic coupling may enable?
• Tunable finesse gradient weak for phonon QND,
  strong for cooling (A. Clerk)
• Stochiometric Si3N4 allows high finesse

Poster!
Ben Zwickl (grad)
Andrew Jayich (grad)
Jack Sankey (postdoc)
Brian Yang (grad)
Postdoc positions available
23
Mechanical ringdown to measure Q
T 300 K n0 134,000 Hz Q 1,120,000
1
(depends on strain in membrane)
Amplitude (a.u.)

• Q is 2 3 orders of magnitude greater than
  comparably sized devices
• 7 aN/Hz½ force sensitivity

0
0
2
4
6
8
10
Time (s)
1
Within an order of magnitude of world-record!
T 300 mK n0 119,000 Hz Q 11,300,000
Amplitude (a.u.)
Seems possible to couple state-of-the-art
cavities to state-of-the-art MEMS using only
commercial devices!
0
0
40
80
120
160
Time (s)
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Laser cooling the membrane (improved setup)
(curves are not offset)
10-26
Teff 2.34 K 0.13 K
10-27
Sx(n) (m2/Hz)
Teff 253 mK 4.7 mK
10-28
Teff 80 mK 1.8 mK
optimizing detuning
10-29
Teff 13.3 mK 0.51 mK
10-30
Teff 6.82 mK 0.61 mK
10-31
126
128
130
132
134
n (kHz)

• Theory predicts that same device should cool
  from T 300 mK to Teff 1 mK ltlt

• Quantum ground state of a mm-scale object! Stay
  tuned

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Results of perturbation theory calculation
Gaps at avoided crossings as a function of
264 176 88 0
Offset
Tilt
132 88 44 0
Gap (MHz)
Gap (MHz)
0
200
400
0
0.4
-0.4
Offset (mm)
Tilt (mrad)
Some of the avoided crossings can be made
arbitrarily sharp
equivalent to a very reflective membrane!
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Comparison between theory and experiment
Df (GHz)
Dx (nm)

• Red line is best fit from perturbation theory
  calculation
• Only fitting parameter is membrane position tilt
  thickness are known a priori

28
Avoided crossings depend on membrane tilt
88 66 44 22 0
44 22 0
Detuning (MHz)
Detuning (MHz)
0
17
0
15
30
Membrane displacement (nm)
Membrane displacement (nm)
29
The actual device schematic
x
rc
Invar spacer
Ultrastable laser (l 1,064 nm) Input optics
50 nm x 1 mm x 1 mm membrane

• Membrane properties
• Resonant wm 2p 105 - 106 Hz
• Q factor 106 at 300K 107 at 300 mK
• Optical absorption

1 mm
30
Realizing quadratic optomechanical coupling
rc
What does detuning actually look like?
x
4
rc 0.4
TEM0,0 Longitudinal mode 100,002
3
TEM0,0 Longitudinal mode 100,001
Cavity resonance frequency (w/FSR)
2
TEM0,0 Longitudinal mode 100,000
1
TEM0,0 Longitudinal mode 99,999
0
0
1
2
3
4
Membrane displacement from cavity waist (x/l)
31
Optical characterization cavity spectroscopy
2.24
Transmission (a.u.)
Laser detuning (GHz)
1.12
0.00
532
1064
Membrane position (nm)

• Fitting gives membrane reflectivity 0.35
• Consistent with 50 nm membrane with n 2.2
• Quadratic detuning points allow x2 measurement

Is phonon QND practical?
32
What does it take to see quantum jumps of the
membrane?

• Assumptions
• Membrane is laser-cooled to ground state
• Cooling laser is then switched off
• Watch for transition
• Also include
• Corrections to RWA QND approxs

Off by a factor of 100
26 kHz/nm2

• Parameters
• T 300 mK
• 1064 nm
• Q 12,000,000
• x0 0.5 pm
• m 5 x 10-11 g
• wm 2p x 100 kHz
• Pin 10 mW
• F 300,000
• 2 MHz/nm2
• SNR 1.0

2.24
Laser detuning (GHz)
1.12
0.00
?!?!?
532
1064
Membrane position (nm)
33
The actual device Amorphous SiN membrane in a
cavity
1 mm
1 mm

• Membrane properties
• 1 mm x 1 mm x 50 nm
• fundamental vibration wm 2p 105 - 106 Hz
• quality factor 106 at 300K 107 at 300 mK

• Cavity properties
• 7 cm long
• Finesse 40,000 w/ membrane at a node
• 38,000 w/ membrane away from node
• (200,000 should be possible)

Membrane absorbs slightly