1. Nergis Mavalvala Aspen February 2004
Quantum Noise, Quantum Correlations and (the demise of) the Standard Quantum Limit in Gravitational-Wave Interferometers
Nergis Mavalvala Aspen February 2004

2. Origins (and demise?) of the SQL
Yanbei’s talk (next)
Michelson interferometer
Vacuum fluctuations enter anti-symmetric port
Conventional interferometer
Laser fields and noise fields stored in arm cavities  frequency-dependent phase shifts
Signal-tuned interferometer
Detuned cavities  optical spring effects
SN and RPN correlations
Speed meters, optical bars, feedback
Back-action evading measurements

3. Squeezing
Squeezed vacuum input is the one elixir we know of that is common to all these schemes
Requirements
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 (Roman’s talk Tuesday am)
Radiation pressure effects in interferometers  ponderomotive squeezing (Thomas’s talk Tuesday am)
Challenges
Frequency-dependence  filter cavities
Amplitude filters (Corbitt et al.)
Squeeze angle rotation filters (KLMTV, B&C, Harms et al.)
Low-loss optical systems

4. Astrophysical considerations
How can astrophysics of GW sources help us decide between all these different options for QND interferometers?
Forget it, no worries!
Technical considerations will be more important
Wait it out!
Astrophysics (upper limits and/or detections) from Initial and Adv. LIGO will guide the way
Act now!
Use present-day astrophysics to help set priorities

5. Quantum Noise in Optical Measurements
Measurement process
Interaction of light with test mass
Counting signal photons with a PD
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)

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

7. In the presence of correlations
Heisenberg uncertainty principle in spectral domain
Obvious that

8. 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

9. Conventional Interferometer
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

10. The coupling coefficient
A rather important entity since all QND effects in GW interferometers arise because of this coupling of radiation pressure to mirror motion
Newton’s law
Cavity pole

11. Ponderomotive Squeezingb1 b2 f
Squeezing produced by back-action force of fluctuating radiation pressure on mirrors
b a 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

12. 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

13. Variable quadrature readout
Squeeze angle (ponderomotive or input) depends on frequency
Easy to see  RPN dominates at low freqs SN dominates at high freqs.
Optimal to detect different quadratures at different frequencies
But how to get ?
Optical cavities as filters

14. Cavities as filters
Cavity applies phase shift a( f )
Control a ( f ) via cavity detuning
Upper and lower GW sidebands f0  fgw  two cavities
GW signal with filtering required at ~100 Hz  filter cavities ~ arm cavities

15. Filter cavities (KLMTV)
Detect optimal quadrature at each frequency
 Variational readout
homodyne
detection
Rotate squeeze
angle in freq.
dependent way
 FD squeezing
Need low loss cavities
Kimble, Levin, Matsko, Thorne, and Vyatchanin, Phys. Rev. D 65, (2001)

16. What about heterodyne readout?
Kentaro’s talk
Homodyne and balanced heterodyne give just one quadrature
AND additional noise due to quantum fluctuations at 2 fm
Unbalanced heterodyning is a variable-quadrature readout
Heisenberg Uncertainty Principle still applies  simultaneous measurement of both quadratures cannot beat SQL in a broadband way fm fm
Heterodyning is naturally a variable quadrature readout
Buonanno, Chen, and Mavalvala, Phys. Rev. D 67, 1220XX (2003)

17. A Quantum Limited Interferometer
LIGO I
LIGO II
Seismic
Suspension thermal
Test mass thermal
Quantum

18. Wednesday and Friday talks
How will we get there?
Wednesday and Friday talks
Seismic noise
Active isolation system
Mirrors suspended as fourth (!!) stage of quadruple pendulums
Thermal noise
Suspension  fused quartz; ribbons
Test mass  higher mechanical Q material, e.g. sapphire
Optical noise
Input laser power  increase to ~200 W
Tailor interferometer response  detuning of signal recycling cavity

19. 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
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

20. Sub-quantum-limited interferometer
Thomas’s talk (this session) X+ X-
Quantum correlations
Input squeezing

21. Squeezing – the ubiquitous fix?
All configurations can benefit from squeezing
Radiation pressure noise can be removed from readout
Frequency-dependent homodyne detection (a.k.a. “variational” or “variable quadrature” readout)
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)

22. Squeezed light production
Methods
Non-linear crystals
Ponderomotive
Get fixed squeeze angle
Want frequency-dependent squeeze angle

23. What about losses?

24. Astrophysics of Sources
Very few techniques achieve broadband improvements  global optimization for all sources is not possible
Buonanno and Chen, gr-qc/

25. The End

26. New quantum limits Quantum correlations  Quantum non-demolition
SQL no longer meaningful
Optomechanical resonance (“optical spring”)
Noise cancellations possible
Quantum non-demolition
Evade measurement back-action by measuring of an observable that does not effect a later measurement
Correlations between the SN and RPN quadratures
“Quantum non-demolition for free?”

27. Vacuum State in a Michelson
Michelson on dark fringe  laser contributes only correlated noise
GW signal from anti-symmetric mode  phase quadrature
Vacuum noise couples in at the anti-symmetric input of 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

28. 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-

29. Input vacuum state gets squeezed in an interferometer
Ponderomotive 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 X+ X- X- X+

30. Initial LIGO

31. Standard Quantum Limit in GW interferometers
Enforced only by the light’s quantum noise, not by the TMs
Decompose into RPN and SN
The SQL 

32. The SQL (contd…)
But   
No correlations 

33. 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

34. 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

35. QND in AdLIGO and beyond
QND interferometers
Manipulation of SN-RPN correlation terms
Manipulation of signal vs. noise quadratures (KLMTV, 2000)
Squeezed vacuum into output port
Squeezed light
Increased squeeze efficiency
Non-linear susceptibilities
High pump powers
Internal losses
Squeeze at low (GW) frequencies
ANU, 2002

36. If we succeed with squeezing…

37. Production of squeezed light
Non-linear crystals
Optical Parametric Amplification (OPA)
Three wave mixing
Pump (532nm)
Seed (1064nm)

38. OPA Process Phase dependent Lines of force Compresses along phase axis
Stretches along amplitude axis

39. Squeezing Experimental Schematic

40. Proposed squeezing experiment

41. Squeezed State in a Michelson
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
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

42. 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

43. Squeezed source wish list
10 dB
Internal losses
Injection losses
c(2) and c(3) non-linearities
Pump power
Photodetection efficiency
100 Hz
Reduce noise in parametric oscillator
Correlate multiple squeezed beams