Dispersive optomechanics:
A new approach to macroscopic quantum phenomena
Jack Harris, Jack Sankey, Ben Zwickl, Andrew Jayich, Brian Yang, Jeff Thompson
Departments of Physics and Applied Physics, Yale University, New Haven, CT
•Overview of
optomechanics
•High-Q MEMS integrated
with high-finesse cavities
•Laser cooling a mm-scale
object to 7 mK
•Strong “position-squared”
readout: towards quantum
jumps in a mechanical
oscillator
Steve Girvin (Yale)
Collaborators: Florian Marquardt (LMU)
Aash Clerk (McGill)
Quantum optomechanical systems, c. 1920
drawn by Niels Bohr
“Which path” / “Welcher Weg”
drawn by
J.M. Raimond
drawn by Niels Bohr
•Measure which path photon takes via
recoil (i.e., quantum radiation pressure)
•Does interference disappear?
Is this possible?
What other issues can be addressed?
Mechanical detectors coupled to EM resonators (circa 2008):
UCSB, Leiden
ENS
Yale
ENS
Cornell
NIST
LIGO
Vienna
kg
g
mg
mg
ng
pg
Mass
Caltech, Munich
MIT
ENS
JILA
IBM
Nanotubes, BECs,
Atoms, Ions…
Oregon
Mechanical detectors coupled to EM resonators:
•These devices span:
1017 in Mass
109 in Frequency
1011 in Length
•All probe the same physics, all are
described by the same Hamiltonian*
•Many within 102±1 of quantum
limited displacement detection
Goals:
Classical optical control of mechanical devices*
New type of quantum optics / mesoscopics
Quantum-limited force & displacement measurements
Generation of nonclassical light (squeezed, entnagled)
Quantization of macroscopic objects*
*Stay tuned
Unified description of optomechanical systems
Fixed
mirror
“spring”
x
Braginski, JETP (1967)
Pin
Optical mode
Optical
mode
“Movable”
mirror
Mechanical
mode
Mechanical
mode
Hˆ   ( xˆ )aˆ  aˆ  cbˆbˆ  dissipation & driving
Ηˆ (detuning)

Note: Radiation Force = 
xˆ
xˆ
For detuning  x:
 (  g (bˆ  bˆ))aˆ  aˆ  cbˆbˆ  dissipation & driving
Unified description of optomechanical systems
Fixed
mirror
“spring”
x
Pin
Optical mode
“Movable”
mirror
Mechanical
mode
1) Cavity detuning is proportional to x (so no quantum jumps)
2) A single element confines light, responds mechanically
Experimentally difficult!!
What are the possible quantum effects in this system?
•Radiation pressure shot noise (quantum back-action)
•Ground state of mechanical oscillator
•Quantum jumps of mechanical oscillator
What does it take to see these?
Quantum optomechanics:Why is radiation pressure shot noise hard to see?
Radiation pressure shot noise:
Thermal Langevin noise force:
S
(F )
BA
64 F 2 Pin

 c
Versus
(F )
SBA
F 2 PinQ
One possible figure of merit: R  ( F ) 
ST
T km
(F )
T
S
4kBT km

Q
Ratio of quantum to
thermal fluctuations
•Combine state-of-the-art mirrors with state-of-the-art MEMS
(Finesse = 1,000,000)
(Force sensitivity ~ 1 aN/Hz½ )
The main technical barrier to the quantum regime of optomechanics:
1 cm
Supermirror
Combine into
a single device
IBM
(Rugar et al)
Integrating good mirrors into good cantilevers…
High finesse mirrors:
High quality MEMS oscillators:
•Amorphous SiO2/Ta2O5 layers Strained, mechanically lossy
•Low mechanical loss
•Good figure / flatness Very stiff substrate
•Low spring constant
• At least 2 mm thick, 20 mm dia. Sets minimum k, m.
•Low mass
•Uncontaminated surface
•Substantial processing
(ion etching, ion milling, lithography, etc.)
Is there another approach?
Which doesn’t require optical and mechanical functionality in one device?
A new approach: dispersive optomechanical coupling
•Practical: Segregates optical andsupport
mechanical functionality
•Fundamental: Detuning, interaction Hamiltonian qualitatively different
“spring”
x
Fixed
mirror
dielectric slab
Fixed
mirror
cavity
detuning
Pin
Optical mode
slab position x
“Dispersive” coupling similar to off-resonant atom in a cavity
x
What does detuning actually look like?
c
  cos1  rc cos  4 x /   
L
rc
 / FSR
4
Cavity’s longitudinal resonances
4
3
3
rc = 0.000
2
1
2
1
0
0
2
4
.5
6
8
1
10
1.5
xk / 
12
2
x
What does detuning actually look like?
c
  cos1  rc cos  4 x /   
L
rc
 / FSR
4
4
3
2
1
rc = 0.300
rc = 0.000
3
2
1
0
0
2
4
.5
6
8
1
10
1.5
xk / 
12
2
x
What does detuning actually look like?
c
  cos1  rc cos  4 x /   
L
rc
 / FSR
4
4
3
2
1
rc = 0.600
rc = 0.300
rc = 0.000
3
2
1
0
0
2
4
.5
6
8
1
10
1.5
xk / 
12
2
x
What does detuning actually look like?
c
  cos1  rc cos  4 x /   
L
rc
 / FSR
4
4
3
2
1
rc = 0.880
rc = 0.600
rc = 0.300
rc = 0.000
3
2
1
0
0
2
4
.5
6
8
1
10
1.5
xk / 
12
2
x
• “Big” fixed mirrors provide high finesse
• Mechanical device doesn’t need to be very reflective
Nearly full optomechanical coupling even for
low-reflectivity cantilever!!
rc
 / FSR
4
4
3
2
1
rc = 0.995
rc = 0.880
rc = 0.600
rc = 0.300
rc = 0.000
3
2
1
0
0
2
4
.5
6
8
1
10
1.5
xk / 
12
2
The actual device: schematic
x
rc
Invar spacer
50 nm x 1 mm x 1 mm membrane
(Norcada, Inc.)
•10-6 Torr (ion pump)
•Vibration isolation
•T = 300 K
The actual device: membrane
1 mm
• 1 mm square:
cavity mode diameter
is 180 mm, so no
diffraction loss
• 50 nm thick SiN:
~/8 @ 1064 nm
rc = 0.35
Crucial questions:
•Optical properties (scatter / absorption)?
•Mechanical properties?
1 mm
•$20 each
Transmission (a.u.)
Laser detuning (GHz)
Optical characterization: cavity spectroscopy
2.24
1.12
0.00
532
1064
Membrane position (nm)
Fitting gives membrane reflectivity = 0.35
Consistent with 50 nm membrane with n = 2.2
Transmission resonances still look narrow…
Cavity finesse vs. membrane position
empty cavity finesse =175,000
Data
Fit (n = 2.15 + 0.00016 i )
Finesse
160,000
120,000
Low loss at nodes
of cavity mode
80,000
40,000
0
0
532
1064
Membrane position (nm)
1596
Calulcated finesse assuming better end mirrors…
Finesse
500,000
400,000
300,000
200,000
empty cavity finesse =
1,000,000
300,000
100,000
Theory:
finesse > 500,000
seems possible!
100,000
0
0
266
532
Membrane position (nm)
798
High loss - antinode
Amplitude (a.u.)
Mechanical ringdown to measure Q
T = 300 K
n0 = 134,000 Hz
Q = 1,120,000
1
(depends on strain
in membrane)
0
0
2
4
6
Time (s)
1
10
T = 300 mK
n0 = 119,000 Hz
Q = 11,300,000
Amplitude (a.u.)
0
0
8
40
80
Time (s)
120
160
• Q is 2 – 3 orders of
magnitude greater than
comparably sized devices
•7 aN/Hz½ force sensitivity
Within an order of
magnitude of worldrecord!
Seems possible to
couple state-of-the-art
cavities to state-of-theart MEMS…
using only commercial
devices!
What can we do with this device?
Operating in “linear detuning” regime,
dispersive coupling mimics other geometries:
But with easier fabrication
(high finesse, Q, etc.)
Ηˆ   ( xˆ )aˆ  aˆ  mbˆ bˆ
 (  g1 xˆ )aˆ  aˆ  mbˆbˆ
Transmission (a.u.)
For example, laser cooling…
Laser cooling the membrane (first results):
•Monitor membrane’s undriven motion
•Extract temperature from either Teff  k x
2
/ k B or Teff
Qtotal
T
Qmech
Amplitude of membrane motion
[m2/Hz]
•Two methods of extracting Teff agree (almost)
10-22
Pin=100 nW
Q ~ 105
Cavity finesse = 16,000
192 K
42 K
Increasing
laser power
10-24
28 K
10-26
1.5 K
10-28
Pin = 60 μW
Q ~ 100
0.3 K
0.5 K
10-30
116,000
117,000
118,000
119,000
Frequency [Hz]
• decrease in ω0 due to negative “optical spring”
• decrease in T due to “optical damping”
Laser cooling the membrane (improved setup):
(curves are not offset)
Sx(n) (m2/Hz)
10-26
Teff = 2.34 K ± 0.13 K
10-27
Teff = 253 mK ± 4.7 mK
10-28
Teff = 80 mK ± 1.8 mK
10-29
10-30
10-31
Teff = 13.3 mK ± 0.51 mK
Teff = 6.82 mK ± 0.61 mK
126
128
130
n (kHz)
•Theory predicts that same device should cool
from T = 300 mK to Teff = 1 mK << m
•Quantum ground state of a mm-scale object!
132
134
Laser frequency (fsr)
What else can we do with this device?
2
1
532
1064
Membrane displacement (nm)
Linear detuning:
Hˆ
  1  g1 xˆ  aˆ aˆ   bˆ bˆ
†
†

M
  g1 x0  bˆ  bˆ  aˆ aˆ  ...
†
†
 H , bˆ bˆ  0
†
Phonon number is not constant of
motion
Quadratic detuning:
Hˆ
  1  g 2 xˆ 2  aˆ aˆ   bˆ bˆ
†

†
M
†
†
  g 2 x0  bˆ bˆ  1  RW  aˆ aˆ  ...


†
H , bˆ bˆ  0
•Phonon number is constant of motion
•Cavity has a “shift-per-phonon”
•Quadratic coupling g2 diverges as rc ~ 1
Fundamentally different! Cavity only measures (displacement)2
Membrane quantum jumps?
In practice, one must measure cavity shift w/ SNR > 1
during membrane phonon lifetime
SNR 
Signal Power
(Averaging time) 
Noise PSD
Cavity shift per phonon is:
2
  ( x ) xZPF

L 2
(  )2
S
n
8 2c
2(1  rc )mm

 3c 3
Shot-noise-limited PSD is: S 

16 N 16 F 2 L2 Pin
Q
1
Thermal lifetime of n-phonon Fock state is:  n 
m n(n  1)  n (n  1)
Bath temperature
Membrane
temperature
(can be laser-cooled!)
Membrane quantum jumps: proposed experiment
Assumptions:
•Membrane is laser-cooled to ground state
•Cooling laser is then switched off
•Watch for 0  1 transition
Also include:
•Corrections to RWA & QND approx’s
Parameters
T = 300 mK
= 532 nm
Q = 12,000,000
x0 = 0.5 pm
m = 5 x 10-11 g
m = 2 x 100 kHz
Pin = 10 mW
F = 300,000
?!?!?
rc = 0.999
SNR = 1.0
Avoided crossings of higher-order cavity modes
TEM0,0 modes
3.36
Transmission (a.u.)
Laser detuning (GHz)
4.48
2.24
1.12
0.00
0
532
1064
1596
membrane displacement (nm)
Avoided crossings of higher-order cavity modes
Crossings made possible by
“dispersive” geometry.
“tweak” the input coupling…
Avoidedness controlled by…?
4.48
Laser detuning (MHz)
Laser detuning (GHz)
~32 kHz/nm2
3.36
2.24
1.12
0.00
0
532
1064
1596
membrane displacement (nm)
44.8 TEM
0,0
singlet
TEM2,0
triplet
0.00
~1.8 MHz/nm2
reff = 0.98
42.5
85.0
0.00
membrane displacement (nm)
Avoided crossings depend on membrane tilt:
66
44
22
0
30
0
15
Membrane displacement (nm)
Detuning (MHz)
Detuning (MHz)
88
44
22
0
0
17
Membrane displacement (nm)
Modelling the cavity eigenmodes:
Cavity eigenmodes given by Helmholtz equation:
   x, y , z  
2
2
n 2  x, y , z    x, y , z   0
c2
Solve using 4-mode degenerate perturbation theory:
4
i  x, y , z    ci , j (0)
j  x, y , z  
j 1
(0)
2
3
i  i(0)     (0)

n

1
d
x


j
i
j i
Unpertubed eigenmodes
Perturbation
Modelling the cavity eigenmodes:
Unpertubed eigenmodes (paraxial approximation):
i
(0)

H m ( 2 y / w) H n ( 2 z / w)
w  L2
m  n 1

e
 e

e

 y 2  z 2 / w2 i  m  n 1   ik y 2  z 2 / 2 R  ikx il / 2
e
m!n!
hermite-gauss transverse mode
guoy phase wavefront
curvature
w( x)  2( x 2  xR2 ) / kxR
Guoy phase: ( x )  tan 1 ( w2k / 2 R)
Wavefront radius of curvature: R ( x )  ( x 2  x R2 ) / x
Beam width:
Calculate overlap integrals over membrane volume:
-Analytically using approximate eigenmodes
-Numerically using exact eigenmodes
Main fitting parameters are:
-Coarse displacement of membrane from cavity waist
-Tilt of membrane relative to cavity axis
longitudinal
mode
Results of perturbation theory calculation:
Cavity detuning (MHz)
•Ignoring mode coupling gives sinusoidal detuning trivially
•Including coupling lifts degeneracies, makes crossings avoided:
132
tilt = 0.5 mrad
offset ~ 200 mm
Gaps set by offset from cavity waist
88
44
Triplet splitting set by tilt
0
200.5
200.6
Membrane displacement (mm)
Results of perturbation theory calculation:
Gaps at avoided crossings as a function of:
Offset
Tilt
264
176
Gap (MHz)
Gap (MHz)
132
88
44
0
88
0
0
200
400
Offset (mm)
-0.4
0
0.4
Tilt (mrad)
Some of the avoided crossings can be made “arbitrarily” sharp…
…equivalent to a very reflective membrane!
Comparing theory & experiment
•Red line is best fit
•Fitting parameter is the
position of membrane
Detuning (MHz)
44
Tilt = 0.0 mrad
22
0
•This is the first data; still limited
by drifts, vibrations, etc.
0
6
12
Membrane displacement (nm)
Tilt = 0.4 mrad
Detuning (MHz)
44
22
0
0
6
12
18
Membrane displacement (nm)
How strong can we make x2 coupling in the present setup?
•Detuning curvature = 20 MHz/nm2
110
•Equivalent to rc = 0.9995, as
required for QND
•Further improvements in rc possible
31
44
22
3.6MHz
MHz
Detuning (MHz)
88
22
transmission
0
24
0
6
12
18
Membrane displacement (nm)
•Data still limited by drift, vibrations,
but agreement with model is good
Caveat: x2 coupling is strong, but
membrane is no longer at a node,
so high finesse may be a problem
30
Open question: How is the system’s behavior modified
when gap in the cavity spectrum is smaller than m?
Dispersive coupling – a new type of optomechanics
•Commercial mirrors & MEMS: state-of-the-art optomechanics without microfab
•Laser cooling to 7 mK
•Couple directly to x2: Phonon QND, quantum jumps seem feasible
•Next: cryogenics
Jack
Sankey
(postdoc)
Ben
Zwickl
(grad)
Andrew
Jayich
(grad)
Brian
Yang
(grad)
Postdoc positions available
Jeff
Thompson
(now at Harvard)