1. Quantum Noise in Advanced Gravitational Wave Interferometers
NSF Proposal Review
Nergis Mavalvala PAC 13, LLO December 2002

2. Quantum Noise Measurement process Noise in measurement process
Interaction of light with test masses
Counting signal photons with a PD
Noise in measurement process
Poissonian statistics of force on test mass due to photons  radiation pressure noise (RPN)
Poissonian statistics of counting the photons  shot noise (SN)

3. Strain sensitivity limit due to quantum noise
Shot Noise
Uncertainty in number of photons detected a
(Tunable) interferometer response  Tifo depends on light storage time of GW signal in the interferometer
Radiation Pressure Noise
Photons impart momentum to cavity mirrors Fluctuations in the number of photons a
Shot noise:
Laser light is Poisson distributed  sigma_N = sqrt(N)
dE dt >= hbar  d(N hbar omega) >= hbar  dN dphi >= 1
Radiation Pressure noise:
Pressure fluctuations are anti-correlated between cavities

4. Classical Optics - Quantum Optics
Amplitude
Phase
Power
Phasor representation
Length  amplitude
Angle  phase
Quantum optics
Annihilation operator
Creation operator
Photon number
and not Hermitian  NOT observables

5. Defining Observables Define a Hermitian operator pair
is the amplitude quadrature operator
is the phase quadrature operator
Linearization
DC term + fluctuations
Fluctuations in amplitude and phase quadratures
McKenzie

6. Heisenberg and QND Heisenberg Uncertainty Principle
Measure position of a particle very precisely
Its momentum very uncertain
Measurement of its position at a later time uncertain since
Quantum non-demolition (QND)
Evade measurement back-action by measuring of an observable that does not effect a later measurement
Good QND variables (observables)
Momentum of a free particle since [p, H] = 0
Quadrature components of EM field (explained in next few viewgraphs)

7. ‘Ball on stick’ representation
Analogous to the phasor diagram
Stick  dc term
Ball  fluctuations
Common states
Coherent state
Vacuum state
Amplitude squeezed state
Phase squeezed state
McKenzie

8. Vacuum State in a Michelson
Michelson on dark fringe  laser contributes only correlated noise
GW signal from anti-correlated effect  phase quadrature
Vacuum noise couples in at the beamsplitter
Phase quadrature fluctuations: Shot Noise
Amplitude quadrature fluctuations: Radiation Pressure Noise
So all the light we use in the interferometer has quantum noise on it. This top diagram represents the vacuum state fluctuations. This is a minimum uncertainty state, with equal variances in the phase and amplitude quadratures. This noise is everywhere that there isn’t already a laser beam.
But why do we care for the vacuum state in the Michelson? We know the laser beam has quantum noise on it, but it is correlated noise i.e. common to both arms, so it will cancel back at the beam-splitter.
The reason is because at the unused port of the beam-splitter the vacuum mode is entering. It interferes with the input beam, and passes on it’s noise statistics, causing anti-correlated noise on each arm’s electric field. When the beams recombine on the beam-splitter noise couples to the the output, which limits the sensitivity of the interferometer.
The vacuum mode fluctuations couple into two types of noise – the amplitude quadrature noise of the vacuum couples into phase quadrature noise on the electric fields in the arms, producing anti-correlated phase shifts, which we call. SHOT NOISE.
The phase quadrature noise of the vacuum state couples to amplitude quadrature noise on the electric fields in the arms, producing anti-correlated amplitude fluctuations, which gives rise to. RADIATION PRESSURE NOISE.
SHOT NOISE scales with 1/sqrt(POWER), at the beam-splitter so it can be reduced by increasing the power of the laser or power recycling factor, where-as RADIATION PRESSURE NOISE scales with sqrt(POWER), so it increases with power at the beam-splitter.
As I mentioned earlier, the first generation detectors are be limited by SHOT NOISE but NOT by RADIATION PRESSURE NOISE, and so is the experiment we did. So I will discuss reducing SHOT noise, and not RAD PRESSURE NOISE. So how do we reduce it?
McKenzie

9. Squeezed State in a Michelson
Inject squeezed state into the dark port of Michelson to replace vacuum
Amplitude squeezed state oriented to reduce noise in the signal (phase) quadrature
Squeezing! The HUP says that the product of the variance in the two quadratures must be greater or equal to 1, so if we are limited only by shot noise, and we don’t care about radiation pressure noise, we can use a squeezed state with reduced noise in one quadrature that we care about, at the expense of the other quadrature which we don’t.
So we want to reduce shot noise, which means we need to inject a amplitude squeezed state. This is what one looks like 3dB. The noise in the amplitude quadrature is ½ whist the noise in the phase quadrature twice as much as it was.
So we inject this into the unused port of the beam-splitter. And since it is there , then vacuum state no longer enters. What does squeezing do in a MIchelson?
McKenzie

10. Squeezed State in a Michelson
GW signal in the phase quadrature
Orient squeezed state to reduce noise in phase quadrature1
Since a local oscillator ‘selects’ the phase quadrature, where there is a GW signal, we need only worry about this quadrature. This diagram shows the Michelson arm electric fields with the vacuum noise and with squeezed noise at one point in time where the is a GW signal. On the diagram on the left, the vacuum noise has enter the dark port and interfered with the input beam putting noise on each of the electric fields in the arms, which adds up back at the beam-splitter, so that the GW signal has vacuum noise on it. The noise in the plane of the signal is the noise on the signal.
You can see that noise is reduced if we inject squeezing in the dark port as in the figure on the right. The squeezing is injected and interferes with the input beam. The noise on the electric fields in each arm is less and when recombined it turns out that the noise in the plane of the signal is much less.
So that is the theory behind the experiment, reduced shot noise on the GW signal.
McKenzie

11. Standard Quantum Limit
“Traditional” treatment (Caves, PRD 1980)
Shot noise and radiation pressure noise uncorrelated
Vacuum fluctuations entering output port of the beam splitter superpose N1/2 fluctuations on the laser light
Optimal Pbs for a given Tifo
Standard quantum limit in GW detectors
Limit to TM position (strain) sensitivity for that optimal power for a given Tifo and frequency
Minimize total quantum noise (quadrature sum of SN and RPN) for a given frequency and power)

12. Signal recycling mirror  quantum correlations
Shot noise and radiation pressure (back action) noise are correlated (Buonanno and Chen, PRD 2001)
Optical field (which was carrying mirror displacement information) returns to the arm cavity
Radiation pressure (back action) force depends on history of test mass (TM) motion
Dynamical correlations t
SN(t)
RPN(t+t)
Part of the light leaks out the SRM and contributes to the shot noise BUT the (correlated) part reflected from the SRM returns to the TM and contributes to the RPN at a later time

13. New quantum limits Quantum correlations 
SQL no longer meaningful
Optomechanical resonance (“optical spring”)
Noise cancellations possible
Quantization of TM position not important (Pace, et. al, 1993 and Braginsky, et. al, 2001)
GW detector measures position changes due to classical forces acting on TM
No information on quantized TM position extracted

14. Quantum Manipulation: AdLIGO as an example
“Control” the quantum noise
Many knobs to turn: Optimize ifo sensitivity with
Choice of homodyne (DC) vs. heterodyne (RF) readout
RSE detuning  reject noise one of the SB frequencies
Non-traditional modulation functions
Non-classical light?

15. Quantum manipulation: Avenues for AdLIGO+
Squeezed light
Increased squeeze efficiency
Non-linear susceptibilities
High pump powers
Internal losses
Low (GW) frequencies
QND readouts
Manipulation of sign of SN-RPN correlation terms
Manipulation of signal vs. noise quadratures (KLMTV, 2000)
Squeezed vacuum into output port
ANU, 2002

16. Recent interest in quantum effects and squeezing
AdLIGO is quantum-noise-limited at almost all frequencies
Quantum correlations allow for manipulation of quantum noise
Squeezing is more readily achieved except
More is better (> 6 dB)
Not in GW band (100 Hz)

17. This proposal at a glance
Squeezed
Source
Interferometer
Nergis Mavalvala (PI)
Stan Whitcomb (co-PI)
Post-doc (TBD)
Keisuke Goda (Grad)
Thomas Corbitt (Grad)
10 dB
AdLIGO
100 Hz
AdLIGO+
$ 400k/yr

18. Squeezed source 10 dB 100 Hz Internal losses Injection losses
c(2) and c(3) non-linearities
Pump power
Photodetection efficiency
100 Hz
Reduce seed noise
Correlate multiple squeezed beams

19. Squeezed source Advance LIGO AdLIGO + Signal tuned
Quantum noise limited
SEM  SN and RPN correlated  QND
Squeezed input
Evaluation in the presence of other noise limits and technical noise
AdLIGO +
Speedmeters
White light interferometers
All reflective interferometers
Ponderomotive squeezing

20. Experimental Program
Set up a Quantum Measurement (QM) lab within LIGO Laboratory at MIT
Goals
Develop techniques for efficient generation of squeezed light
Explore QND techniques for below QNL readouts of the GW signal (AdLIGO and beyond)
Trajectory
Build OPA squeezer (generation n+)
Build table-top (or suspended optics) experiments to test open questions in below QNL interferometric readouts

21. Programmatics: People
People involved (drafted, conscripted)
PIs
Nergis Mavalvala (PI)
Stan Whitcomb (co-PI)
1 to 1.5 post-docs
David Ottaway and/or TBD (?)
Yanbei Chen (MIT Fellowship?)
Graduate students (2+)
Keisuke Goda and Thomas Corbitt
Collaborators, advisors, sages
Lam, McClelland (ANU)
Fritschel, Shoemaker, Weiss, Zucker (MIT)
Visitors
McKenzie (ANU). Sept. to Dec. 2002

22. Programmatics: $$
MIT seed funds supporting presently on-going experimental work
NSF request (average over 3 years)
350 k$/year
Salaries – 100 k$/yr
1 post-doc
(2+1) graduate students
1 month summer salary for NM
Equipment – 100 k$/yr
Lab equipment
Non-linear crystal development
Travel – 20k$/yr

23. Programmatics: Facilities
Space
Optics labs (NW17-069, NW17-034)
Physical environment
Possibly share seismically and acoustically quiet environment of LASTI for QND tests involving suspended optics
Research equipment
Share higher-power, shot-noise-limited, frequency-stabilized laser
In-house low loss optics, low noise photodetection capabilities
Suspended optics interferometer

24. Programmatics: When? Squeezing
Summer, 2002
Gain experience with OPA squeezer (with Lam, Buchler at ANU)
Fall, 2002 – Summer, 2003
Building OPA squeezer with “borrowed” crystals (with Goda)
Pathfinder program to learn about remedies for deficiencies in present non-linear crystals
Formation of “10 dB consortium” for next-generation crystal development (with Lam)
2003 and 2004
Acquire new crystals
Testing with high-power, low-noise pump
Beyond
Attack open questions in the field subject to personnel, interests, funding and recent developments (and LIGO I status)

25. Programmatics: When? Interferometry
2002
AdLIGO quantum noise and optimal readout scheme (with Buonanno and Chen)
Prospects for squeezing (with Corbitt)
Experimental test of simple speedmeter (Whitcomb, Chen, DeVine at ANU)
2003
Prospects for squeezing in AdLIGO+ (with Corbitt)
Experimental tests
Beyond
Attack open questions in the field subject to personnel, interests, funding and recent developments (and LIGO I status)

26. If we succeed…