1. Australia-Italy Workshop October 2005
Recent Developments toward Sub-Quantum-Noise-Limited Gravitational-wave Interferometry
Australia-Italy Workshop October 2005

2. Advanced LIGO

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

4. 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

5. Initial LIGO

6. Sub-Quantum Interferometers

7. 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)

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

9. In the presence of correlations
Output = Shot + Radiation pressure + Signal
Heisenberg uncertainty principle in spectral domain
Follows that

10. How to correlate quadratures?
Make noise in each quadrature not independent of each other
(Nonlinear) coupling process needed
Squeezed states of light and vacuum

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

12. 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

13. Heisenberg Uncertainty Principle
No measurement can be completely deterministic in two non-commuting observables.
E.g. Position and momentum of a particle
Similarly for the electromagnetic field

14. Quantum Non-demolition
Evade measurement back-action by measuring of an observable that does not effect a later measurement
Good QND variables (observables)
Momentum of a particle
Quadrature fields
Atomic spin states
Back-action is also key concept in quantum controls
Time evolution of position operator depends on momentum, but not vice versa.

15. 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

16. 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-

17. Back Action Produces Squeezing
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 b1 b2 f
Squeezing produced by back-action force of fluctuating radiation pressure on mirrors 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

18. Frequency-dependent coupling constant
Couples radiation pressure to mirror motion
Newton’s law
Cavity pole a b
Amplitude  b1 = a1
Phase  b2 = -k a1 + a2 + h
Radiation Pressure
Shot Noise

19. “Squeeze angle” describes the quadrature being squeezed
Optimal Squeeze Angle
“Squeeze angle” describes the quadrature being squeezed
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

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

21. Frequency-dependent Squeeze Angle
Narrowband
Broadband
Conventional

22. 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

23. 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
Radiation pressure effects in interferometers  ponderomotive squeezing
Challenges
Frequency-dependence  filter cavities
Amplitude filters
Squeeze angle rotation filters
Low-loss optical systems

24. Squeezing using nonlinear optical media

25. Non-linear crystals Optical Parametric Amplification (OPA)
Three (or four) wave mixing
Pump (532nm)
Seed (1064nm)

26. Optical Parametric Amplification
Annihilation operator
Amplitude and phase quadrature operators
Amplification
Violates commutation relation
Need to include vacuum state

27. Optical parametric amplification
Need phase sensitive amplification (parametric amplification)
Amplify amplitude quadrature by
Deamplify phase quadrature by
Product of the gains is unity
Commutation relations satisfied

28. Optical Parametric Oscillator

29. Squeezing at audio frequencies Next-generation crystals in use
What’s new since the 90’s?
Squeezing at audio frequencies
Next-generation crystals in use
Filter cavities for frequency-dependent squeezing
Noise couplings to establish fundamental limits
Detailed calculations of noise budget
Photo-thermal noise not a problem
Pump noise coupling being considered

30. Typical Experimental Setup

31. Squeezed Vacuum

32. Low frequency squeezing at ANU
ANU group  quant-ph/
ANU group  quant-ph/

33. What’s next Most important goal for GW interferometry
Stably locked squeezed vacuum state (no coherent amplitude) injection into a GW interferometer prototype
Issues to work out
Coupling into interferometer dark port through output mode cleaner etc
Error signals for optimum quadrature
Previous test always with squeezed light?
Wu, Kimble et al. (1987)
McKenzie et al. (2002) ...

34. Optical Layout

35. Sensitivity Results
And this is our results. This is the power detected as a function of frequency. This first curve is the signal (which is a mock GW signal induced by modulating the arm 5.46MHz) for the a simple Michelson. This is the noise floor, which is shot noise and the signal peaks its head above the noise floor.
The next curve is the simple michelson with squeezing injected into the dark port. You can see the noise floor is lower by about 2 dB. And the signal looks a fair bit larger.
The next curve is without squeezing, but with power recycling. The power recycling mirror reflectivity is 90% and gave us a power recycling factor of 4. The signal is increased by this amount.
The final plot is taken with power recycling and squeezing. The reduction in noise floor is about 2.3 dB. This is the money plot and shows that not only is squeezing in a gravitational wave detector is viable and does enhance a the sensitivity by reducing shot noise, it also interacts favorably with power recycling. That is, the shot noise reduction is increased for the power recycling case over the simple Michelson case.
Why is the improvement greater for PR than for simple Michelson? For two reasons, the first is to do with the control scheme of the differential mode of the Michelson. We use offset locking, which means the Michelson is slightly offset from a dark fringe, allowing a small amount of input power to the photo-detector, which is also the local oscillator. Since it lets a small amount of the input beam through to the detector, by symmetry from the other direction a small amount of the squeezing is lost through the Michelson and towards to the laser. Since the power in the Micheson is higher with power recycling the offset is less, so the squeezing that is transmitted towards the laser is reduced. The second reason is that the power recycled michelson is a cavity, with the power mirror as the front mirror and whole Michelson is effectively the other mirror. Since the squeezing and signal is outside the bandwidth of the cavity the power recycled Michelson effectively like a high reflectivity mirror.
So the result are good, but could be better, if the circulator, which we used to inject the squeezing had less loss.
K.McKenzie et al. Phys. Rev. Lett., (2002)

36. Squeezing using back-action effects

37. The principle
A “tabletop” interferometer to generate squeezed light as an alternative to 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

38. The Ponderomotive Interferometer

39. High circulating laser power High-finesse cavities
Key ingredients
High circulating laser power
10 kW
High-finesse cavities
15000
Light, low-noise mechanical oscillator mirror
1 gm with 1 Hz resonant frequency
Optical spring
Detuned arm cavities

40. Optical Springs Modify test mass dynamics
Suppress displacement noise (compared to free mass case)
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
Connect low-frequency mechanical oscillator to (nearly) noiseless optical spring
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. Detuned cavity for optical spring
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. Assumed experimental parameters

43. Noise budget

44. Noise budget – Equivalent displacement

45. What do we already know? Detailed simulation of noise couplings
Uses first fully quantum mechanical simulation code for a GW interferometer (Corbitt)
Used in AdLIGO simulations (Fritschel and Popescu)
“Exported” to Hannover and Glasgow (Schnabel and Strain)
Location and infrastructure
LASTI laser, vacuum envelop and seismic isolation
Cavity geometrical parameters
Mini-mirror suspensions

46. High finesse cavity tests
What’s next
Design completion
Suspension
Control system
High finesse cavity tests
Fixed mini-mirror – optical tests
Suspended mini-mirror – includes mirror dynamics and radiation-pressure coupling
Complete interferometer

47. Why is this interesting/important?
First ever demonstration of ponderomotive squeezing
Probes quantum mechanics of optical field-mechanical oscillator coupling at 1 g mass scales
Test of low noise optical spring
Suppression of thermal noise
Simulations and techniques useful for AdLIGO and other GW interferometers
Quantum optical simulation package
Michelson detuning
Role of feedback control in these quantum systems

48. Other QND ideas

49. Speed meters
Principle  weakly coupled oscillators
Energy sloshes between the oscillators
p phase shift after one slosh cycle
Driving one oscillator excites the other

50. Implementation of a speed meter
homodyne detection
sloshing cavity
Position signal from arm cavity enters “sloshing” cavity
Exits “sloshing” cavity with p phase shift
Re-enters arm cavity and cancels position signal
Remaining signal  relative velocity of TMs
Purdue and Chen, Phys. Rev. D 66, (2002)

51. QND techniques needed to do better
Conclusions
Advanced LIGO is expected to reach the quantum noise limit in most of the band
QND techniques needed to do better
Squeezed states of the EM field appears to be a promising approach
Factors of 2 to 5 improvements foreseeable in the next decade
Not fundamental but technical
Need to push on this to be ready for third generation instruments