1. Quantum Noise in Gravitational-wave Detectors
Nergis LSU November 2004

2. Global network of detectors
GEO
VIRGO
LIGO
TAMA
AIGO
LIGO
Detection confidence
Source polarization
Sky location
LISA

3. Gravitational waves
Transverse distortions of the space-time itself  ripples of space-time curvature
Propagate at the speed of light
Push on freely floating objects  stretch and squeeze the space transverse to direction of propagation  STRAIN
Energy and momentum conservation require that the waves are quadrupolar  aspherical mass distribution

4. Astrophysics with GWs vs. E&M
E&M (photons) GW
Space as medium for field
Spacetime itself ripples
Accelerating charge
Accelerating aspherical mass
Absorbed, scattered, dispersed by matter
Very small interaction; matter is transparent
10 MHz and up
10 kHz and down
Light = not dark (but >95% of Universe is dark)
Radiated by dark mass distributions  black holes, dark matter
Very different information, mostly mutually exclusive
Difficult to predict GW sources based on EM observations

5. Strength of GWs: e.g. Neutron Star Binary
Gravitational wave amplitude (strain)
For a binary neutron star pair R M M
Quadrupole formalism is accurate to order of magnitude for most sources.
Involves computing wave generation and radiation reaction from Einstein eqn.
Weak internal gravity and stresses  nearly Newtonian source
Kepler’s third law of planetary motion: period^2 = 4*pi^2*radius^3/(G*Msun)
Distances  1 parsec = 3.26 l.y. = 3e18 cm
r ~ 10^23 m ~ 10 Mpc (center of Virgo cluster)
Distance of earth to center of galaxy ~ l.y. ~ 10 kpc
h ~10-21 r

6. Effect of a GW on matter

7. GW detector at a glance L ~ 4 km For h ~ 10–21 Thermal noise 
DL ~ m
Thermal noise 
vibrations of mirrors due to finite temperature
Seismic noise 
ground motion due to natural and anthropogenic sources
Shot noise 
quantum fluctuations in the number of photons detected

8. Measurement and the real world
How to measure the gravitational-wave?
Measure the displacements of the mirrors of the interferometer by measuring the phase shifts of the light
What makes it hard?
GW amplitude is small
External forces also push the mirrors around
Laser light has fluctuations in its phase and amplitude

9. “Conventional” Interferometers Generation 1 (now!)

10. LIGO : Laser Interferometer Gravitational-wave Observatory3 k m ( 1 s )
4 km
2 km LA
4 km

11. Initial LIGO Sensitivity Goal

12. Science Runs and Sensitivity
2nd Science Run
Feb - Apr 03
(59 days) S1
1st Science Run
Sept 02
(17 days) S3
3rd Science Run
Nov 03 – Jan 04
(70 days)
LIGO Target Sensitivity
Frequency (Hz)
Strain (1/rtHz)
S3 Duty Cycle H1
69% H2
63% L1
22%
DL = strain x 4000 m 10-18 m rms

13. Signal-tuned Interferometers The Next Generation

14. Why a better detector? Astrophysics
Factor 10 to 15 better amplitude sensitivity
(Reach)3 = rate
Factor 4 lower frequency bound
NS Binaries
Initial LIGO: ~20 Mpc
Adv LIGO: ~350 Mpc
BH Binaries
Initial LIGO: 10 Mo, 100 Mpc
Adv LIGO : 50 Mo, z=2
Stochastic background
Initial LIGO: ~3e-6
Adv LIGO ~3e-9

15. A Quantum Limited Interferometer
LIGO I
Ad LIGO
Seismic
Suspension thermal
Test mass thermal
Quantum

16. Limiting Noise Sources: Optical Noise
Shot Noise
Uncertainty in number of photons detected a
Higher circulating power Pbs a low optical losses
Frequency dependence a light (GW signal) storage time in the interferometer
Radiation Pressure Noise
Photons impart momentum to cavity mirrors Fluctuations in number of photons a
Lower power, Pbs
Frequency dependence a response of mass to forces
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
 Optimal input power depends on frequency

17. Initial LIGO

18. Signal-recycled Interferometerℓ
Cavity forms compound output coupler with complex reflectivity. Peak response tuned by changing position of SRM
800 kW
125 W
Reflects GW photons back into interferometer to accrue more phase
Signal Recycling
signal

19. Advance LIGO Sensitivity: Improved and Tunable
broadband
detuned narrowband
SQL  Heisenberg microscope analog
If photon measures TM’s position too well, it’s own angular momentum will become uncertain.
thermal noise

20. Sub-Quantum Interferometers Generation 2

21. Quantum Noise in Optical Measurements
Measurement process
Interaction of light with test mass
Counting signal photons with a photodetector
Noise in measurement process
Poissonian statistics of force on test mass due to photons  radiation pressure noise (RPN) (amplitude fluctuations)
Poissonian statistics of counting the photons  shot noise (SN) (phase fluctuations)

22. Free particle SQL
uncorrelated
0.1 MW
1 MW
10 MW

23. Some quantum states of light
Analogous to the phasor diagram
Stick  dc term
Ball  fluctuations
Common states
Coherent state
Vacuum state
Amplitude squeezed state
Phase squeezed state
McKenzie

24. Squeezed input vacuum state in Michelson Interferometer
GW signal in the phase quadrature
Not true for all interferometer configurations
Detuned signal recycled interferometer  GW signal in both quadratures
Orient squeezed state to reduce noise in phase quadrature X+ X- X+ X- X+ X-

25. Back Action Produces Squeezingf
Squeezing produced by back-action force of fluctuating radiation pressure on mirrors
b a
Vacuum state enters anti-symmetric port
Amplitude fluctuations of input state drive mirror position
Mirror motion imposes those amplitude fluctuations onto phase of output field a1 a2
“In” mode at omega_0 +/- Omega  |in> = exp(+/- 2*j* beta) S(r, phi) |out>
Heisenberg Picture: state does not evolve, only operators do. So |out> vacuum state is squeezed by factor sinh(r) = kappa/2 and angle phi = 0.5 arcot(kappa/2).
Spectral densities assuming input vacuum state:
S_b1 = exp(-2 r) ~ 1/kappa when kappa >> 1
S_b2 = exp(+2 r) ~ kappa
S_{b1 b2} = 0

26. Conventional Interferometer with Arm Cavities
Coupling coefficient k converts Da1 to Db2
k and squeeze angle f depends on I0, fcav, losses, f a b
Amplitude  b1 = a1
Phase  b2 = -k a1 + a2 + h
Radiation Pressure
Shot Noise

27. If we could squeeze -k a1+a2 instead
Optimal Squeeze Angle
If we squeeze a2
shot noise is reduced at high frequencies BUT
radiation pressure noise at low frequencies is increased
If we could squeeze -k a1+a2 instead
could reduce the noise at all frequencies
“Squeeze angle” describes the quadrature being squeezed

28. Frequency-dependent Squeeze Angle

29. Squeezing – the ubiquitous fix?
All interferometer configurations can benefit from squeezing
Radiation pressure noise can be removed from readout in certain cases
Shot noise limit only improved by more power (yikes!) or squeezing (eek!)
Reduction in shot noise by squeezing can allow for reduction in circulating power (for the same sensitivity) – important for power-handling

30. Squeezed vacuum Requirements Generation methods Challenges
Squeezing at low frequencies (within GW band)
Frequency-dependent squeeze angle
Increased levels of squeezing
Generation methods
Non-linear optical media (c(2) and c(3) non-linearites)  crystal-based squeezing (recent progress at ANU and MIT)
Radiation pressure effects in interferometers  ponderomotive squeezing (in design & construction phase)
Challenges
Frequency-dependence  filter cavities
Amplitude filters
Squeeze angle rotation filters
Low-loss optical systems

31. Sub-quantum-limited interferometer
Narrowband
Broadband
Broadband Squeezed X+ X-
Quantum correlations
Input squeezing

32. Signal recycling mirror  quantum correlations
Shot noise and radiation pressure (back action) noise are correlated (Buonanno and Chen, 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
Part of the light leaks out the SRM and contributes to the shot noise
SN(t)
RPN(t+t)
BUT the (correlated) part reflected from the SRM returns to the TM and contributes to the RPN at a later time

33. Squeezing using nonlinear optical media

34. Vacuum seeded OPO
ANU group  quant-ph/

35. Squeezing using back-action effects

36. The principle
A “tabletop” interferometer to generate squeezed light as an alternative nonlinear optical media
Use radiation pressure as the squeezing mechanism
Relies on intrinsic quantum physics of optical field-mechanical oscillator correlations
Squeezing produced even when the sensitivity is far worse than the SQL
Due to noise suppression a la optical springs

37. Noise budget

38. High circulating laser power High-finesse cavities
Key ingredients
High circulating laser power
10 kW
High-finesse cavities
25000
Light, low-noise mechanical oscillator mirror
1 gm with 1 Hz resonant frequency
Optical spring
Detuned arm cavities

39. Optical Springs Modify test mass dynamics
Suppress displacement noise (compared to free mass case)
Why not use a mechanical spring?
Thermal noise
Connect low-frequency mechanical oscillator to (nearly) noiseless optical spring

40. Thermal Noise in Springs
Why not use a mechanical spring?
Displacements due to thermal noise introduced by the high frequency (mechanical) spring will wash out the effects of squeezing
An optical spring with a high resonant frequency will not change the thermal force spectrum of the mechanical pendulum
Use a low resonant frequency mechanical pendulum to minimize thermal noise
Use an optical spring to produce a flat response out to higher frequencies

41. Optical spring Detuned cavity
Positive detuning
Detuning increases
Cavity becomes longer
Power in cavity decreases
Radiation-pressure force decreases
Mirror ‘restored’ to original position
Cavity becomes shorter
Power in cavity increases
Mirror still ‘restored’ to original position

42. Noise budget – Equivalent displacement

43. In conclusion...

44. Next generation – quantum noise limited
Squeezing being pursued on two fronts
Nonlinear optical media
Back-action induced correlations
Other Quantum Non-Demolition techniques
Evade measurement back-action by measuring of an observable that does not effect a later measurement
Speed meters
Optical bars
Correlations between the SN and RPN quadratures

45. Gravitational-wave Interferometers
Astrophysics
Tests of General Relativity
Quantum Measurement