1. Quantum mechanics on giant scales
Gravitational wave detectors
Quantum nature of light
Quantum states of mirrors
Nergis UMass, March 2010

2. Outline Quantum limit in gravitational wave detectors
Origins of the quantum limit
EM vacuum fluctuations
Interactions of light with mirrors
Getting past the quantum limit
Experiments
Quantum optics
Quantum optomechanics
Necessary building blocks in the classical regime
Progress toward the quantum regime

3. Gravitational waves (GWs)
Prediction of Einstein’s General Relativity (1916)
Indirect detection led to Nobel prize in 1993
Ripples of the space-time fabric
GWs stretch and squeeze the space transverse to direction of propagation
Emitted by accelerating massive objects
Cosmic explosions
Compact stars orbiting each other
Stars gobbling up stars
“Mountains” on stellar crusts 

4. GW detector at a glance Mirrors hang as pendulums Quasi-free particles
Respond to passing GW
Filter external force noise
4 km
20 kW
Optical cavities
Mirrors facing each other
Builds up light power
Lots of laser power P
Signal  P
Noise 
10 W

5. LIGO: Laser Interferometer Gravitational-wave Observatory3 k m ( 1 s )
MIT
4 km
NSF
Caltech LA
4 km

6. Quantum noise in Initial LIGO
Shot noise
Photon counting statistics
Radiation pressure noise
Fluctuating photon number exerts a fluctuating force

7. Advanced LIGO Quantum noise limited
Radiation pressure noise
Stronger measurement  larger backaction
Shot noise
More laser power  stronger measurement

8. Origin of the Quantum Noise Vacuum fluctuations

9. Quantum states of light
Heisenberg Uncertainty Principle
Coherent state (laser light)
Squeezed state
Two complementary observables
Make on noise better for one quantity, BUT it gets worse for the other
X1 and X2 associated with amplitude and phase X1 X2

10. Quantum Noise in an Interferometer
Caves, Phys. Rev. D (1981)
Slusher et al., Phys. Rev. Lett. (1985)
Xiao et al., Phys. Rev. Lett. (1987)
McKenzie et al., Phys. Rev. Lett. (2002)
Vahlbruch et al., Phys. Rev. Lett. (2005)
Goda et al., Nature Physics (2008)
Radiation pressure noise
Quantum fluctuations exert fluctuating force  mirror displacement X1 X2
Laser X1 X2
Shot noise limited  (number of photons)1/2
Arbitrarily below shot noise X1 X2 X1 X2
Vacuum fluctuations
Squeezed vacuum

11. Quantum Enhancement Squeezed state injection

12. Squeezing injection in Advanced LIGO
Laser
GW Detector
SHG
Faraday isolator
Squeezing
source
The squeeze source drawn is an OPO squeezer, but it could be any other squeeze source, e.g. ponderomotive squeezer.
OPO
Homodyne Detector
Squeeze
Source
GW Signal

13. Advanced LIGO with squeeze injection
Radiation pressure
Shot noise

14. How to squeeze photon states?
Need to simultaneously amplify one quadrature and de-ampilify the other
Create correlations between the quadratures
Simple idea  nonlinear optical material where refractive index depends on intensity of light illumination

15. Squeezed state generation
Vacuum (shot)
Noise power (dBm/rtHz)
Squeezed
Dark
Time (s)
Frequency (Hz)
Goda et al., Opt. Lett. (2008)
Vahlbruch et al., New J. Phys. 9, 371 (2007)

16. Squeezing injection in a prototype interferometer
K. Goda, O. Miyakawa, E. E. Mikhailov, S. Saraf, R. Adhikari, K.McKenzie, R. Ward, S. Vass, A. J. Weinstein, and N. Mavalvala, Nature Physics 4, 472 (2008)
2.9 dB or 1.4x

17. Radiation pressure
The other side of the quantum optical coin

18. Radiation pressure rules!
Experiments in which radiation pressure forces dominate over mechanical forces
Opportunity to study quantum effects in macroscopic systems
Observation of quantum radiation pressure
Generation of squeezed states of light
Quantum ground state of the gram-scale mirror
Entanglement of mirror and light quantum states
Classical light-oscillator coupling effects en route (dynamical backaction)
Optical cooling and trapping
Light is stiffer than diamond

19. Reaching the quantum limit in mechanical oscillators
The goal is to measure non-classical effects with large objects like the (kilo)gram-scale mirrors
The main challenge  thermally driven mechanical fluctuations
Need to freeze out thermal fluctuations Zero-point fluctuations remain
One measure of quantumness is the thermal occupation number
Want N  1
Colder oscillator
Stiffer oscillator

20. Reaching the quantum limit in macroscopic mechanical oscillators
Large inertia requires working at lower frequency (Wosc  1/√Mosc)
To reach
For small m-oscillator Wosc = 10 MHz and T = 0.5 mK
For larger objects Wosc = 1 kHz and T = 50 nK
1010 below room temperature !

21. 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
Fluctuation-dissipation theorem
Connect a high Q, low stiffness mechanical oscillator to a stiff optical spring  DILUTION
Dilution – a fraction of the energy of the oscillator is stored in the optical field instead of in the elastic flexing of the wire, or in the acoustic modes
The optical spring shifts the oscillator's resonant frequency while leaving its mechanical losses unchanged. The mechanical quality factor $Q_M$, as limited by those losses, is increased by the factor $\Omega_{\rmeff} / \Omega_M$, where $\Omega_M$ is the natural frequency of the free mechanical oscillator.
We refer to this as ``optical dilution'', analogous to the phenomenon of ``damping dilution'' that accounts for the fact that the $Q$ of the pendulum mode can be much higher than the mechanical $Q$ of the material of which it is made~\cite{saulsonPRD1990,dilution}.
This mitigation of intrinsic thermal noise is possible because a fraction of the energy is stored in the (noiseless) gravitational field. In the case of the pendulum, the dilution factor depends on the amount of elastic energy stored in the flexing wire compared to the energy stored in the gravitational field -- approximated by the ratio of the gravitational spring constant to the mechanical spring constant.
Optical dilution accounts for the fact that thermal noise in our mechanical oscillator is reduced due to energy stored in the optical field (the optical spring force acts similar to the gravitational force).
True for any non-mechanical force ( non-dissipative or “cold” force ), e.g. gravitation, electronic, magnetic

22. The optical spring effect and optical trapping of mirrors

23. Cavity length or laser wavelength
Optical cavities
Light storage device
Two mirrors facing each other
Interference  standing wave
Intracavity power
Cavity length or laser wavelength

24. How to make an optical spring? Radiation pressure force
Detune a resonant cavity to higher frequency (blueshift)
Change in cavity mirror position changes intracavity power
Change in radiation-pressure exerts a restoring force on mirror
Time delay in cavity response introduces a viscous anti-damping force x P

25. Optical springs and damping
Restoring
Damping
Anti-damping
Anti-restoring
Radiation pressure of light in an optical cavity  force on mirror
Detune a resonant cavity to higher frequency (blueshift)
Real component of optical force  restoring
But imaginary component (cavity time delay)  anti-damping
Unstable
Can stabilize with feedback
Cavity cooling
Optical spring
Blue shift (flaser > fcavity) optical spring
Red shift (flaser < fcavity)  cavity cooling

26. Classical Experiments
Extreme optical stiffness Stable optical trap Optically cooled mirror

27. Experimental cavity setup
10%
90%
5 W
Optical fibers
1 gram mirror
Coil/magnet pairs
for actuation (x5)‏

28. Multicolor optical cavity
Two colors resonant at different (adjacent) orders
Each can have arbitrary detuning
Intracavity power
Cavity length or laser wavelength

29. 10 W, frequency and
intensity stabilized laser
External vibration isolation

30. Extreme optical stiffness
5 kHz K = 2 x 106 N/m Cavity optical mode  diamond rod
Very stiff, but also very easy to break
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
Displacement / Force
Phase increases  unstable
Frequency (Hz)

31. 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., Phys. Rev. Lett 98, (2007)

32. Optical cooling with double optical spring (all-optical trap for 1 gm mirror)
Increasing subcarrier detuning
T. Corbitt, Y. Chen, E. Innerhofer, H. Müller-Ebhardt, D. Ottaway, H. Rehbein, D. Sigg, S. Whitcomb, C. Wipf and N. Mavalvala, Phys. Rev. Lett 98, (2007)

33. Active feedback cooling
Measure mirror displacement
Filter displacement signal
Feed it back to mirror as a force
Controller
PDH
Laser
Continuous measurement  measurement-induced decoherence
EOM
PBS
QWP
Continuous measurement  measurement-induced decoherence

34. Optical spring with active feedback cooling
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
Cooling factor = 43000
Teff = 6.9 mK N = 105
Mechanical Q = 20000
Cooling factor larger than mechanical Q because Gamma = Omega_eff/Q. The OS increases Omega but doesn’t affect Gamma (OS is non-mechanical), so Q must increase to keep Gamma constant.
T. Corbitt, C. Wipf, T. Bodiya, D. Ottaway, D. Sigg, N. Smith, S. Whitcomb, and N. Mavalvala, Phys. Rev. Lett 99, (2007)

35. Quantum measurement in gravitational wave detectors

36. Even bigger, even cooler
Initial LIGO detectors much more sensitive  operate at 10x above the standard quantum limit
But these interferometers don’t have strong radiation pressure effects (yet)  no optical spring or damping
Introduce a different kind of cold spring  use electronic feedback to generate both restoring and damping forces
Cold damping ↔ cavity cooling
Servo spring ↔ optical spring cooling
SQL

37. Active feedback cooling + spring
Measure mirror displacement
Filter displacement signal
Feed it back to mirror as a force
Controller
PDH
Laser
EOM
PBS
QWP

38. Cooling the kilogram-scale mirrors of Initial LIGO
Teff = 1.4 mK N = 234 T0/Teff = 2 x 108
Mr ~ 2.7 kg ~ 1026 atoms
Wosc = 2 p x 0.7 Hz
LIGO Scientific Collaboration

39. Other cool oscillators

40. Some other cool oscillators
Toroidal microcavity  g
NEMS  g
AFM cantilevers  10-8 g
Micromirrors  10-7 g
SiN3 membrane  10-8 g
NEMs capacitively coupled to SET (Schwab group, Maryland (now Cornell)
Kippenberg group (Munich)
Harris group (Yale)
Bouwmeester group (UCSB)
Aspelmeyer group (Vienna)
LIGO-MIT group
LIGO
LIGO  103 g
Minimirror  1 g

41. Next steps … Marching on toward the quantum limit

42. Radiation pressure rules!
Experiments in which radiation pressure forces dominate over mechanical forces
Opportunity to study quantum effects in macroscopic systems
Observation of quantum radiation pressure
Quantum ground state of the gram-scale mirror
Generation of squeezed states of light
Entanglement of light and mirror quantum states
Classical light-oscillator coupling effects en route (dynamical backaction)
Optical cooling and trapping
Light is stiffer than diamond

43. Radiation pressure: Another way to squeeze light
Create correlations between light quadratures using a movable mirror
Amplitude fluctuations of light impart fluctuating momentum to the mirror
Mirror displacement is imprinted on the phase of the light reflected from it

44. Radiation pressure: Another way to squeeze light
Create correlations between light quadratures using a movable mirror
Amplitude fluctuations of light impart fluctuating momentum to the mirror
Mirror displacement is imprinted on the phase of the light reflected from it

45. Classical noise, be vanquished
Squeezed
Vacuum fluctuations
Two identical cavities with 1 gram mirrors at the ends
Common-mode rejection cancels out laser noise

46. Squeezing Squeezing 7 dB or 2.25x
T. Corbitt, Y. Chen, F. Khalili, D.Ottaway, S.Vyatchanin, S. Whitcomb, and N. Mavalvala, Phys. Rev A 73, (2006)

47. Present status
Blue curve = noise with 50 mW of input power and detuning = 1
Red line = noise level required to observe sqz and quant. rp with 5 W of input power

48. Thermal noise, be vanquished!
All glass suspension
Bonded with vacseal
Glass fibers drawn in-house
Large “ears” to isolate mirror from fiber bending point
Many iterations on assembly and handling
18 hours

49. Present status 4x
Scattered light?

50. Closing remarks

51. Classical radiation pressure effects
Stiffer than diamond
6.9 mK
Stable OS
Radiation pressure dynamics
Optical cooling
10%
90%
5 W
~0.1 to 1 m
Corbitt et al. (2007)

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

53. LIGO Quantumness
N = 234
SQL
N = 1

54. And now for the most important part…

55. Cast of characters MIT Collaborators Thomas Corbitt Christopher Wipf
Timothy Bodiya
Sheila Dwyer
Nicolas Smith
Edith Innerhofer
MIT LIGO Lab
Collaborators
Yanbei Chen and group
Stan Whitcomb
Daniel Sigg
Rolf Bork
Alex Ivanov
Jay Heefner
LIGO Scientific Collaboration

56. The End Gravitational wave detectors Quantum nature of light
Quantum states of mirrors

57. Trapping and cooling Dynamic backaction cooling
Stable optical trap with two colors
Stiff!
Stable!
T. Corbitt et al., Phys. Rev. Lett 98, (2007)

58. Entanglement
Two systems are entangled when their individual states cannot be recovered separately
Correlate two optical fields by coupling to mechanical oscillator
Quantum state of each light field not separable (determine by measuring density matrix)
Quantify the degree of non-separability using logarithmic negativity
Entanglement
C. Wipf, T. Corbitt, Y. Chen, and N. Mavalvala, New J. Phys./ (2008)

59. Cavity cooling
200x
1012x
Cornell,CU
UCB