Beating Heisenberg’s Limit
Sean Leavey
Institute for Gravitational Research, University of Glasgow
Contents
1
How to Build a Gravitational Wave Interferometer
2
Beating the Standard Quantum Limit
3
Glasgow Sagnac-Speedmeter Experiment
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Gravitational Wave Interferometry
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How to Build a GW Interferometer
Build technical infrastructure
How long?
How much power?
How reflective to make the mirrors?
Topology?
Control it
Actuation
Control systems
Make it quieter than the thing you want to measure
Technical noise sources
Fundamental noise sources
Calibrate its signal in terms of h
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Choose an Arm Length
Arm length and mirror reflectivity determine the bandwidth of the
detector:
FSR
arcsin
π
FSR =
1 − r1 r2
√
2 r1 r2
c
2L
102
101
100
100
80
Phase [°]
fpole =
Signal [W / ° tuning]
103
60
40
20
0
100
101
102
103
Frequency [Hz]
104
105
106
For Advanced LIGO, fpole ≈ 10 kHz. NS/NS binary coalescences take
place around 40 Hz to 300 Hz.
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Choose a Power
Use as much power as possible!
More power provides more photons to interact with gravitational waves,
and can lower certain types of noise.
With higher powers, can run into stability and heat issues, but in
principle these are challenges that can be overcome.
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Choose Mirror Reflectivity
You might think we want as much mirror reflectivity as possible. This is
true in theory, but in practice the speed at which the control systems can
control the mirrors comes into play.
The cavity mirror reflectivities determine the cavity finesse:
F ≈
π
,
1 − r1 r2
which is related to how steep the resonance peaks are as cavity mirrors
move...
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Choose Mirror Reflectivity
100
10-1
-2
10150000
100000
50000
0
50000
Frequency Offset [Hz]
100000
150000
100
10-1
10-2
10-3
10-4
-5
10150000
100000
50000
0
50000
Frequency Offset [Hz]
100000
150000
p
Amplitude [ W ]
p
Amplitude [ W ]
Higher finesse leads to narrower resonances...
...and it’s harder to catch the mirror as it swings through.
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Aside: The Mizuno Limit
It turns out that the sensitivity of a Michelson interferometer is
determined in the ideal case solely by laser wavelength λ, bandwidth
∆fBW and cavity power .
Mizuno Limit
r
h̃ &
2~λ ∆fBW
πc
Jun Mizuno’s thesis, Section 3.1.3, Eq. 3 (p56)
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Actuation
Mirror control is required to get the
interferometer into a locked state and keep
it there.
This is usually achieved via coil and magnet
actuation on the suspensions holding the
mirrors.
Actuator noise can be significant. For
instance, stray magnetic fields can couple
into the actuators and represent a
displacement noise source.
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Actuation
For low noise actuation directly on the test mass, electrostatic drives
(ESDs) are used. These are low range but low noise actuators.
Credit: Wittel et. al. (arXiv)
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Operating Point
A GW interferometer needs to be at its operating point to be optimally
sensitive, with each mirror’s position controlled to within as little as
10−12 m.
Operating Point
Amplitude
When each cavity within the interferometer is on resonance.
1.0
0.5
0.0
0.5
1.0
0
1
2
3
4
5
Waves
6
7
8
9
10
i.e. each cavity fits an integer number of half-wavelengths.
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Sensing and Control
All mirrors are subject to
noise from seismic activity on
the Earth.
We usually only care about
cavity mirror motion, since
their motion is enhanced by
the cavity.
Suspending optics from
pendulums can provide a
great deal of isolation, but
this is most effective at
higher frequencies.
How can we sense cavity
mirror motion?
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Sensing and Control
We want the laser (carrier) to be resonant in each cavity.
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Sensing and Control
Adding a resonant electro-optic modulator (EOM) lets us superimpose
modulation sidebands on the main laser light (carrier).
Amplitude w.r.t. carrier [arb. units]
1.0
0.8
E = E0 e i(ωt+β sin(ωm t))
β iωm t β −iωm t
iωt
≈ E0 e
1+ e
− e
2
2
0.6
0.4
0.2
0.0
20
15
10
5
0
5
Frequency [MHz]
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10
15
20
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Sensing and Control
The sidebands are not resonant in the cavity and so are mainly reflected
by the first mirror.
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Sensing and Control
Phase changes caused by the movement of the cavity mirrors beat with
the reflected sidebands to produce signal sidebands.
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Sensing and Control
We add a photodiode to view the light reflected by the cavity.
This photodiode is demodulated at the same frequency as the EOM is
modulated.
This is called Pound-Drever-Hall control.
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Sensing and Control
Error Signal [arb. units]
Pound-Drever-Hall Signal Example
1.0
0.5
0.0
0.5
1.0
150
160
170
180
End Mirror Tuning [°]
190
200
210
Notice that the signal is bipolar. Left of resonance, the cavity gets
shorter and the signal is positive. Right of resonance, the cavity is longer
and the signal is negative.
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Sensing and Control
Error Signal [arb. units]
Pound-Drever-Hall Signal Example
1.0
0.5
0.0
0.5
1.0
150
160
170
180
End Mirror Tuning [°]
190
200
210
The slope of the signal at the zero crossing determines the optical gain
of the interferometer to that particular degree of freedom.
Cavity finesse, a function of the mirror reflectivies, determines the slope
of the error signal1 .
1 For
a simple Fabry-Perot cavity.
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Sensing and Control
For more complicated interferometers, there are many degrees of freedom.
LY − LX
2
LY + LX
CARM =
2
MICH = lY − lX
lY + lX
PRCL = lP +
2
lY + lX
SRCL = lS +
2
ETMY
DARM =
+ filter cavities,
mode cleaners, etc...
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Arm Cavity Y
ITMY
EOM PRM
Arm Cavity X
BS
SRM
ITMX
ETMX
PD
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Sensing and Control
Each degree of freedom is controlled with a control loop. The error signal
from each photodiode is fed back to actuators on mirrors.
Advanced LIGO
Interferometric Control Scheme
J. Kissel, for the ISC Group
G1200632-v3
DARM
TMSY
α MICH
ERM
ETMY
00
Hz
β SCRL
~2
~100 Hz
~40 kHz
CARM
ITMY
Input
Mode
Cleaner
MC3
CP
MCPY
~1 Hz
ϕm
FI
MC2
Laser
ERM
ITMX
BS
TMSX
PR3
MC1
PR2
PRM
?
CP
Power
Recycling
Cavity
SR2
High Bandwidth Cavity
Length Control Signal
MICH
~20 Hz
High Bandwidth Cavity
Angular Control Signal
Test Mass Quad Sus (QUAD)
Signal
Recycling
Cavity
PRCL
~20 Hz
Beam Splitter / Fold Mirror Triple Sus (BSFM)
HAM Large Triple Sus (HLTS)
Transmission Monitor Double Sus (TMTS)
SR3
?
SRCL
~20 Hz
~10 Hz
SR3
~10 Hz
SRM
DC
PD
FI
HAM Small Triple Sus (HSTS)
CommHard,
DiffHard,
CommSoft,
DiffSoft,
ETM
X
Output Mode Cleaner Double Sus (OMCS)
Faraday Single Sus (OFIS)
HAM Auxiliary Single Sus (HAUX)
Output
Mode
Cleaner
HAM Tip-Tilt Single Sus (HTTS)
Credit: Jeff Kissel (LIGO-G1200632)
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From mirror motion to h
Once you’ve got a locked interferometer, you need to calibrate its output
(volts) in terms of strain (h). This is a whole other topic in itself!
In GEO-HF, for instance, the
calibration is done by
sequentially calibrating the
ESD to the CARM degree of
freedom, then the input
mode cleaners to the laser
stabilisation. The laser’s
PZT then provides the
absolute strain calibration.
MFn
PR lock
37.16
MHz
~
AGC
MCn
PDPR
EOM1
MC1
EOM2
EOM3
MC1 lock
25.25
MHz
~
13
MHz
MC2 lock
MC2
BS
EOM4
~
MPR
PDBSs
MCe
PDO
Credit: Leong et al. (DOI: 10.1088/0264-9381/29/6/065001)
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ESD Calibration Factor (Arbitrary Units)
Aside: GEO calibration over time
0.17
0.16
0.15
01.02.07
06.02.05
01.07.09
28.07.09 28.01.10
21.06.11
16.08.11
04.02.10
18.02.05
30.06.09
0.14
30.07.09
0.13
0.12
23.12.05
11.05.06
30.07.05
MC1 length calibration
free−swinging MI calibration
photon pressure calibration
0.11
0.1
0
1
2
3
4
5
Time [Years since Dec 18, 2004]
6
7
Credit: Leong et al. (DOI: 10.1088/0264-9381/29/6/065001)
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Sources of Noise
Test masses need to be quieter than the thing you want to measure. You
also need to measure the strain to the required precision without
introducing too much additional noise.
That means the laser, mirrors, actuators, control systems and
photodetectors in practice set the sensitivity of the interferometer.
These fall into two categories:
Technical noise sources
Fundamental noise sources
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Technical Noise Sources
Technical noises represent sources that exist due to the equipment,
materials or techniques used, or the temperature at which the
interferometer operates.
Thermal noise is a major barrier to sensitivity, both in the test masses
and suspensions. This is one of the reasons why LIGO Voyager plans to
go cryogenic.
Seismic noise couples ground motion into the test masses at low
frequencies where suspensions can perform little damping.
Electronic noise represents noise present in components used to obtain
signals from and control the interferometer, such as photodiode dark
current, Johnson noise, etc.
Others include laser noise, oscillator noise, gravity gradient noise, etc...
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Fundamental Noise: Shot Noise
14
Voltage [arb. units]
12
10
1
hS (f ) =
L
8
6
4
r
~cλ
2πP
2
0
0.0
0.2
0.4
Time [arb. units]
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0.6
0.8
1.0
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Radiation Pressure Noise
1.0
∆x [arb. units]
0.5
0.0
hRP (f ) =
0.5
1.0
0.0
0.2
0.4
Time [arb. units]
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0.6
0.8
1
mf 2 L
r
~P
2π 3 cλ
1.0
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The Standard Quantum Limit
Radiation Pressure Noise
r
1
~P
hRP (f ) =
2
mf L 2π 3 cλ
Shot Noise
1
hS (f ) =
L
r
~cλ
2πP
Together, radiation pressure
and shot noise combine to
place a hard limit on
sensitivity at all frequencies
for a standard interferometer
setup - the SQL. An
interferometer only touches
the SQL at one frequency
but the limit applies to all
frequencies.
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Advanced LIGO
Quantum noise
Seismic noise
Suspension thermal noise
Coating Brownian noise
Substrate Brownian noise
Total noise
−22
Strain [1/√Hz]
10
−23
10
−24
10
1
10
2
10
Frequency [Hz]
3
10
arXiv: 1103.2728
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Contents
1
How to Build a Gravitational Wave Interferometer
2
Beating the Standard Quantum Limit
3
Glasgow Sagnac-Speedmeter Experiment
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Beating the SQL
State of the art detectors are limited by quantum noise nearly over their
entire detection range. How will we improve this performance?
Manipulating the shape of
light: squeezing
Local readout
of an optical
bar
Optomechanically coupled
optical springs
Credit: Stefan Hild, Hollberg workshop, Glasgow 2014
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Beating the SQL
So far we used Michelson interferometers to derive strain, by
continuously measuring the displacement of the mirrors.
Displacement measurements are subject to Heisenberg’s Uncertainty
Principle. Thus:
[x̂ (t) , x̂ (t + δt)] 6= 0
and
[x̂ (t) , p̂ (t)] 6= 0
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Position vs Speed Measurement
However, in the 1930s, John von Neumann showed
that some observables can be measured in pairs
continuously without encountering the Heisenberg
uncertainty.
One such pair is momentum, which manifests itself
as the speed at which a test mass moves:
[p̂ (t) , p̂ (t + δt)] = 0
Sean Leavey | Beating Heisenberg’s Limit
Figure: John von
Neumann. By LANL
[Public domain], via
Wikimedia Commons
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Sagnac Speed-Meter Configuration
It turns out that a zero-area Sagnac interferometer is automatically a
speed meter.
φCW ∝ xN (t) + xE (t + δt)
φCCW ∝ xE (t) + xN (t + δt)
∆φ = [xN (t) − xN (t + δt)]
− [xE (t) − xE (t + δt)]
∆φ ≈δt (ẋE (t) − ẋN (t))
Differential phase is proportional to
test mass speed.
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Aside: Getting h from speed
There is nothing special required to get strain information from the
speed-meter interferometer.
We simply integrate over the speed to get the position, then use existing
calibration techniques to get h.
Voltage [arb. units]
0.0
Voltage / ω [arb. units]
1.0
0.0
0.5
0.5
1.0
1.0
0.5
0.5
1.0
0.0
0.2
0.4
Time [arb. units]
0.6
0.8
1.0
We don’t measure the ‘DC’ position of each mirror, thus we avoid the
Uncertainty Principle.
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Cancellation of Back-Action Noise
Two counter-propagating beams hit each
cavity mirror at a particular time. The two
beams impart radiation pressure with a
certain phase. To obtain cancellation, the
phase difference between these beams has
to be 180◦ .
Can see that:
cavity power imbalance
reflectivity imbalance
losses
will degrade phase difference and thus
cancellation of radiation pressure noise.
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Sensitivity Improvement
10-16
Michelson
Sagnac
Displacement [m/sqrt(Hz)]
10-17
10-18
10-19
10-20 2
10
103
Frequency [Hz]
104
105
This is a lossless case. Losses degrade sensitivity and will be important to
quantify (more later).
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Einstein Telescope
The Speed-Meter configuration has the potential to offer great
improvements in sensitivity in future detectors.
Figure: N. V. Voronchev, S. P. Tarabrin, S. L. Danilishin, arXiv:1503.01062
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Contents
1
How to Build a Gravitational Wave Interferometer
2
Beating the Standard Quantum Limit
3
Glasgow Sagnac-Speedmeter Experiment
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Aims
Create an ultra-low noise speed-meter testbed
which is dominated by radiation pressure noise
Demonstrate reduced radiation pressure noise
over an equivalent Michelson
Gain experience and understanding of
speed-meters for future detector design
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A
C1.4M ERC grant for
“high risk, high gain”
research
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Glasgow Speed-Meter Design
In vacuum, seismically isolated
Triangular cavities
1 g and 100 g cavity mirrors
Cavity round-trip length 2.8 m
In-vacuum, suspended balanced
homodyne detector at audio
frequencies
Electrostatic drives for direct
actuation on test masses
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Optical Layout
0.5
M9
0.0
M11
M14
input beam
M3b
M12 M13
M5
LaserStab
M6
M2b
M1b
M4
M8
M7
M3a
M10
M1a
M16
M15
PhD3
M2a
PhD4
−0.5
−0.5
0.0
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0.5
1.0
1.5
2.0
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Vacuum System
Two 1 m diameter connected tanks. Each tank contains rubber stacks
and multiple 30 kg plates for seismic isolation.
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Status: Vacuum System
Bridge structure joins the topmost
stacks together so their residual
seismic motion is common. This
makes longitudinal seismic motion
common to all mirrors.
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Status: Vacuum System
Simulations show that the seismic motion should be damped sufficiently
within the 100 Hz to 1 kHz measurement region.
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Status: Cleanliness
Residual gas analysis shows lots of long-chained hydrocarbons. Probably
oil from manufacturing metal structures. Hydrocarbons can settle on
mirrors and burn into their surface, leading to loss. Cleaning in progress.
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New Suspension Territory
Advanced LIGO: 42 kg
Sean Leavey | Beating Heisenberg’s Limit
AEI 10 m Prototype:
100 g
Glasgow Speed-Meter:
1g
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Status: Auxiliary Suspensions Completed
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Status: Control at Low Frequencies
One caveat to the speed-meter is that its response vanishes at low
frequencies. Below a certain frequency, the interferometer’s cavity
mirrors are uncontrollable due to technical noise in the control system.
Sagnac Speedmeter Response
10 11
BHD SSM
10 10
W / m / sqrt(Hz)
10 9
10 8
10 7
10 6
10 5
10 -1
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10 0
10 1
10 2
Frequency (Hz)
10 3
10 4
10 5
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Status: Control at Low Frequencies
This doesn’t happen in a Michelson, where the response is flat (since it’s
a displacement measurement).
PDH Response
W / m / sqrt(Hz)
10 7
10 6
M9 PDH Cavity A
10 5
10 -1
10 0
10 1
10 2
10 3
10 4
10 5
Frequency (Hz)
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Status: Control at Low Frequencies
Work is ongoing to find a way to pick off some light between the arm
cavities to undertake a position measurement at low frequencies in
addition to the velocity measurements at higher frequencies.
Carrier
RF sidebands
Signal sidebands
G1
Arm
B
DARM
PDH
-
H1
-
PDH
n1
H2
SSM
n2
BHD
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Arm
A
G2
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Status: Quantifying the Effect of Losses
Losses in the Sagnac Speed Meter degrade sensitivity. They reintroduce
back-action noise and so we see an extra 1f effect on top of the 1f effect
already present.
10-16
Sagnac 15 ppm
Sagnac 30 ppm
Sagnac 100 ppm
Sagnac lossless
Michelson lossless
Displacement [m/sqrt(Hz)]
10-17
10-18
10-19
10-20 2
10
103
Frequency [Hz]
104
105
Credit: Danilishin et. al (arXiv: 1412.0931)
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Status: Electrostatic Drive
We plan to build the first real-world demonstration
of the fringe field ESD design outlined previously.
Simulations suggest we need approximately 800 V
actuation range.
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Still to come in 2015
Finish in-vacuum installation of clean (!) infrastructure
Design and build remaining suspensions
Finalise input optics (mode cleaner, etc.)
Test ESD concept (parallel experiment, possibly combining BHD
test)
Installation of computerised control and data acquisition system (+
wiring)
Create sensing and control scheme
Set final test mass requirements, order optics
(2016) Start to assemble final experiment and take some data!
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Thanks for listening!
http://speed-meter.eu/
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