1. Quantum mechanics on giant scales
MIT Corbitt, Ackley, Bodiya, Ottaway, Smith, Wipf
Caltech Bork, Heefner, Sigg, Whitcomb
AEI Chen, Ebhardt-Mueller, Rehbein
MIT, April 2008

2. Quantum noise in gravitational wave interferometers
Opening remark …
Nature 446 (April 2007)
Quantum noise in gravitational wave interferometers
Quantum states of light
Quantum behavior of giants

3. Basics of GW Detection Want very large L
Gravitational Waves “Ripples in space-time”
Stretch and squeeze the space transverse to direction of propagation
Want very large L
Example: Ring of test masses responding to wave propagating along z

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

5. GW detector at a glance Seismic noise
Ground motion (natural and anthropogenic)
Vibration isolation
20 kW
Thermal noise
Vibrations due to finite temperature
Low mechanical dissipation
Shot noise
Operate on dark fringe
High circulating power
10 W

6. What limits the sensitivity?
Seismic noise
Initial LIGO
Suspension thermal
Viscously damped pendulum
Shot noise
Photon counting statistics
SQL:
h(f) = sqrt(8*hbar/M)/Omega/L
Standard Quantum Limit

7. Ultimate limits ? Seismic gravity gradient
When ambient seismic waves pass near and under an interferometric gravitational-wave detector, they induce density perturbations in the Earth, which in turn produce fluctuating gravitational forces on the interferometer’s test masses.
Human gravity gradient
The beginning and end of weight transfer from one foot to the other during walking produces the strongest human-made gravity-gradient noise in interferometric gravitational-wave detectors (e.g. LIGO). The beginning and end of weight transfer entail sharp changes (time scale τ∼20 msec) in the horizontal jerk (first time derivative of acceleration) of a person’s center of mass.

8. LIGO future Enhanced LIGO Advanced LIGO
Make some upgrades with limited invasiveness to give ~2x improvement
Advanced LIGO
Extensive replacement of detector

9. Initial LIGO – S5 (now) Input laser power ~ 6 W Initial LIGO
Circulating power ~ 20 kW
Mirror mass
10 kg
Initial LIGO
SQL

10. Enhanced LIGO (Fall 2007) Input laser power ~ 30 W
Circulating power ~ 100 kW
Mirror mass
10 kg
Enhanced LIGO

11. Advanced LIGO (2011) Input laser power > 100 W
Circulating power > 0.5 MW
Mirror mass
40 kg

12. Quantum noise everywhere
Shot noise
Radiation pressure noise

13. Quantum Enhancement Squeezed state injection

14. Quantum Noise in an Interferometer
First proposed … C.M. Caves, PR D (1981)
Proof-of-principle demonstration … M. Xiao et al., PRL (1987)
More realistic configurations demonstrated …
Power Recycled Michelson K. McKenzie et al., PRL (2002)
Power and Signal Recycled Michelson H. Vahlbruch et al., PRL (2005)
Suspended prototype Goda et al. Nature Phys. (2008)
Laser X+ X- X+ X- X+ X- X+ X-

15. Squeezed Input Interferometer
Laser
GW Detector
SHG
Faraday isolator
The squeeze source drawn is an OPO squeezer, but it could be any other squeeze source, e.g. an optical parametric oscillator.
OPO
Homodyne Detector
Squeeze
Source
GW Signal

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

17. Squeezing measured… Goda et al., Opt. Lett. (2008)
Vahlbruch et al., PRL 97, (2007)

18. Squeezing injected @ the 40m prototype @ Caltech
Goda et al. (2007)

19. Radiation pressure and mirror oscillator coupling

20. Radiation pressure in Advanced LIGO
High power used to reduce shot noise
Radiation pressure of shot noise fluctuations exert fluctuating force on mirrors
Need to contend with
Additional force noise
Dynamical effects
Parametric instabilities
Optical spring
Control issues
Study radiation pressure effect on large masses to inform future GW detectors

21. Ponderomotive predominance
An experimental apparatus in which radiation pressure forces dominate over mechanical forces
Ultimate goals
Generation of squeezed states of light
Observation of quantum radiation pressure
Quantum state of the gram-scale mirror
Entanglement
En route
Optical cooling and trapping
Diamonds
Disclaimer
Any similarity to a gravitational wave interferometer is not merely coincidental. The name and appearance of lasers, mirrors, suspensions, sensors have been changed to protect the innocent.

22. Reaching the quantum limit in mechanical oscillators
To measure non-classical effects with large test masses (mirrors) typical of GW detectors
Thermal energy should be about 1 quantum (ħwosc = kBT)
The large inertia requires working at lower frequency (wosc  1/Mosc)
Desired temperature
Typical small object  wosc = 1 kHz and T = 50 nK  1010 below room temperature
Typical large object  wosc = 10 MHz and T = 0.5 mK
Need to cool with large cooling factor Need clever techniques for larger objects

23. Reaching the quantum limit in mechanical oscillators
One measure of quantumness is the thermal occupation number
Colder oscillator
Stiffer oscillator

24. TRAPPING Confine spatially
Cooling and Trapping
Two forces are useful for reducing the motion of
a particle
A restoring force that brings the particle back to equilibrium if it tries to move
Position-dependent force  SPRING
A damping force that reduces the amplitude of oscillatory motion
Velocity-dependent force  VISCOUS DAMPING
TRAPPING Confine spatially
COOLING Reduce velocity spread

25. Mechanical vs. optical forces
Mechanical forces  thermal noise
Stiffer spring (Wm ↑)  larger thermal noise
More damping (Qm ↓)  larger thermal noise
Optical forces do not affect thermal noise spectrum
True for any non-mechanical, ( non-dissipative or “cold” force), e.g. gravitation, electronic, magnetic
log f
Response reduced below
resonant frequency
Connect a high Q, low stiffness mechanical oscillator to a stiff optical spring

26. Optical springs and damping
Restoring
Damping
Anti-damping
Anti-restoring
Detune a resonant cavity to higher frequency (blueshift)
Opposite detuning than cold damping
Real component of optical force  restoring
But imaginary component (cavity time delay)  anti-damping
Unstable
Stabilize with feedback

27. Stable optical springs
Optical springs are always unstable if optical forces dominate over mechanical ones
Can be stabilized by feedback forces
Key idea: The optical damping depends on the response time of the cavity, but the optical spring does not So use two fields with a different response time
Fast response creates restoring force and small anti-damping
Slow response creates damping force and small anti-restoring force
Can do this with two cavities with different lengths or finesses
But two optical fields with different detunings in a single cavity is easier

28. Extreme optical stiffness Stable optical trap Optically cooled mirror
Experiments
Extreme optical stiffness Stable optical trap Optically cooled mirror

29. Experimental layout
10%
90%
5 W
1 m

30. Mechanical oscillator
We want to apply the stiffest possible optical spring to a weakly suspended oscillator
3” stainless steel ring is suspended as 1 Hz pendulum, with magnel/coil actuators for alignment and control
1 gram mirror is attached by two optical fibers, glued to side of mirror and to the ring
Depending on diameter of fibers and fabrication techniques, resonant frequency varies from 6 to 170 Hz, with Q's ranging from 3,000 to 20,000
Cladding removed from optical fibers with razor blade

31. Double suspension for mini mirror
Easy suspension
M ~ 1 g
f ~ 6.3 Hz
Q ~ 103 – 104
Poor thermal
noise, but
easy to
construct and
to control.
1cm
Coil/magnet pairs
for actuation(x5)‏

32. Vacuum chamber – MIT LASTI facility

33. Experimental Platform
Seismically isolated optical table
Vacuum chamber
10 W, frequency and
intensity stabilized laser
External vibration isolation

34. Suspensions
0.25 m long suspension 1 Hz fundamental resonance

35. Extreme optical stiffness
5 kHz K = 2 x 106 N/m Cavity optical mode  diamond rod
How stiff is it?
100 kg person  Fgrav ~ 1,000 N  x = F / k = 0.5 mm
Very stiff, but also very easy to break
Maximum force it can withstand is only ~ 100 μN or ~1% of the gravitational force on the 1 gm mirror
Replace the optical mode with a cylindrical beam of same radius (0.7mm) and length (0.92 m)  Young's modulus E = KL/A
Cavity mode 1.2 TPa
Compare to
Steel ~0.16 Tpa
Diamond ~1 TPa
Single walled carbon nanotube ~1 TPa (fuzzy)
Displacement / Force
Phase increases  unstable
Frequency (Hz)

36. Double optical spring  stable optical trap
Two optical beams  double optical spring
Carrier detuned to give restoring force
Subcarrier detuned to other side of resonance to give damping force with Pc/Psc = 20
Independently control spring constant and damping
Stable!
T. Corbitt et al., PRL (2007)

37. Optical cooling with double optical spring (all-optical trap for 1 gm mirror)
T. Corbitt et al., PRL (2007)
Increasing subcarrier detuning

38. Optical spring with cold damping
Experimental improvements
Reduce mechanical resonance frequency (from 172 Hz to 13 Hz)
Reduce frequency noise by shortening cavity (from 1m to 0.1 m)
Electronic feedback cooling instead of all optical
Mechanical Q = Cooling factor = 43000
Only cooling result to date where cooling factor exceeded the mechanical Q
T. Corbitt et al., PRL (2007)

39. What’s next…

40. Ponderomotive Interferometer
Two identical cavities
Optically recombined
Common-mode rejection cancels out laser noise

41. Interferometer configuration

43. Squeezing
7 dB
T. Corbitt et al., Phys. Rev A 73, (2006)

44. Light entanglement Logarithmic negativity Measure of CV entanglement
Degree to which density matrices of C and SC are not separable
Calculated using quantum Langevin approach
Cavity linewidth
Optical spring resonance
Entanglement (also squeezing)
C. Wipf et al. PRL (2008)

45. Even bigger mirror, even cooler
Working on reduction of excess noise in radiation pressure experiment at present
Meanwhile, current LIGO detectors are much more sensitive  operate at 10x above the standard quantum limit
LIGO just completed a two year science data taking run  opportunity!
But these interferometers don’t have strong radiation pressure effects  no optical spring or damping
Introduce a different kind of cold spring  restoring force from electronic feedback

46. Quantum measurement in Initial LIGO

47. Kilogram-scale mirrors (LIGO)‏
25cm diameter
10 cm thick
M ~ 10.8 kg ~ 1026 atoms
wosc = 2 p x 0.7 Hz

48. LIGO interferometers
Most sensitive degree of freedom is differential arm motion
Reduced mass is ¼ x 10.8 kg = 2.7 kg
Since optical force is insufficient, use electronic feedback to generate both spring and damping forces
Cold damping ↔ cavity cooling
Servo spring ↔ optical spring cooling

49. Servo spring Measurement performed at LIGO Hanford Observatory
Controller comprised a restoring force and a variable damping force
Measured response of servo spring for various damping gains
Chose 150 Hz as most sensitive measurement band
Deviations from perfect spring due to various filters for low frequency gain and high frequency cutoffs

50. Initial LIGO Cooling Lowest effective temperature = 1.4 mK
Cooling factor 2 x 108 below room temperature
Lowest occupation number = 234
Limited primarily by shot noise
Conservative estimate of center of mass motion
Uncertainty in limiting noise source leads to overestimate of COM motion
LIGO Scientific Collaboration, submitted to Nature (2008)

51. Summary

52. Classical radiation pressure effects
Stiffer than diamond
6.9 mK
Stable OS
Radiation pressure dynamics
Optical cooling
5 W
10%
90%
~1 m
Corbitt et al. (2007)

53. Quantum radiation pressure effects
Wipf et al. (2007)
Entanglement
Squeezing
Mirror-light entanglement
Squeezed vacuum generation

54. Initial LIGO Quantumness
1.4 mK
SQL

55. In conclusion
Radiation pressure effects observed and characterized in system with high optical power and 1 gram mirror
Extreme optical stiffness
Free and stable optical spring
Optical trapping technique that could lead to direct measurement of quantum behavior of a 1 gram object
Parametric instabilities (photon-phonon coupling)
Control system interaction
Testing extremely high power densities on small mirrors (20 kW in cavity  1 MW/cm2)
Nothing has blown up or melted (yet)
Establish path to observing radiation pressure induced squeezed light, entanglement and quantum states of truly macroscopic objects

56. In conclusion
Feedback cooling observed in Initial LIGO interferometers with occupation number N ~ 200
Next upgrade (Enhanced LIGO) should have N ~ 50
Advanced LIGO should operate at the SQL and lead to N ~1 (with an optical spring instead of a servo spring)

57. The End

58. Radiation-oscillator coupling  Amplitude-phase correlations
1. Light with amplitude fluctuations DA
incident on mirror
Movable mirror
Field transformation:
Incident laser light is in a coherent state (noise ball)
Coupling to mirror motion
Reflected light is in a squeezed state (noise ellipse)
2. Radiation pressure due to DA causes mirror to move by Dx
3. Phase of reflected light Df depends on mirror position and hence light amplitude, i.e DA  Dx  Df

59. Lessons learned so far…

60. Parametric instability
Acoustic drumhead modes (28 kHz, 45 kHz, 75 kHz, …) become unstable when detuned at high power
Viscous radiation pressure drives mode  parametric instability
Detuning in opposite direction reduces Q of the mode  cold damping
Mode stabilized through feedback to laser frequency
Parametric instability
(restoring force)
Optical damping
(anti-restoring force)
R = optical anti-damping/mechanical damping
R now greater than 100
T. Corbitt et al., Phys. Rev A 74, R (2006)

61. Control system Loop gain Between 100 and 1000 Hz GAIN < 1
System behaves freely
Important for observing squeezing
Optical spring requires major reshaping of electronic servo compensation
No optical spring
Measured data
With optical spring

62. Control system challenges
Digital control system (32 kS/sec)
Optical spring changes mechanical dynamics of mirrors  servo design
Parametric instabilities for modes at tens of kHz
Analog control forces applied directly on mirrors via magnet-coil pairs
Control system strong enough to maintain cavity resonance but not interfere with radiation pressure induced motion
5 Hz UGF (+ electronic viscous damping at Wos)
Large power buildup
Large dynamic range
Transients

63. Radiation-oscillator coupling  Amplitude-phase correlations
1. Light with amplitude fluctuations DA
incident on mirror
Movable mirror
Field transformation:
Incident laser light is in a coherent state (noise ball)
Coupling to mirror motion
Reflected light is in a squeezed state (noise ellipse)
2. Radiation pressure due to DA causes mirror to move by Dx
3. Phase of reflected light Df depends on mirror position and hence light amplitude, i.e DA  Dx  Df

64. Assumed experimental parameters
T. Corbitt et al., Phys. Rev A 73, (2006)