1. Quantum mechanics on giant scales
Quantum nature of light
Quantum states of mirrors
Nergis Stanford University, November 2008

2. Outline
Challenges to reaching a quantum state for a macroscopic object
Thermally driven fluctuations
Optical trapping of mirrors
Strong interactions of light with mirrors
Cold mirrors
Single external degree of freedom
March toward the quantum world
Observing quantum effects in macroscopic objects

3. A quantum mechanical oscillator
Quantum mechanics 1
Particle in a harmonic potential well with simple and familiar Hamiltonian
This mechanical oscillator seems such a tangible quantum device
But there is no experimental system as yet that requires a quantum description for a macroscopic mechanical oscillator

4. Macroscopic oscillators in the quantum regime
Useful for making very sensitive position or force measurements
Gravitational wave detectors
Atomic force microscopes
Explore the quantum-classical boundary on all size scales
Ground state cooling
Direct observation of quantum effects
Superpositions
Entanglement
Decoherence
Tools for quantum information systems

5. 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

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

7. 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

8. The optical spring effect and optical trapping of mirrors

9. 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

10. How to make an optical spring?
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

11. Optical springs and damping
Restoring
Damping
Anti-damping
Anti-restoring
Detune a resonant cavity to higher frequency (blueshift)
Real component of optical force  restoring
But imaginary component (cavity time delay)  anti-damping
Unstable
Stabilize with feedback
Cavity cooling
Blue shift (higher f) optical spring
Red shift (lower f)  cavity cooling

12. 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

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

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

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

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

18. Extreme optical stiffness
5 kHz K = 2 x 106 N/m Cavity optical mode  diamond rod
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
Displacement / Force
Phase increases  unstable
Frequency (Hz)

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

20. 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)

21. Supercold mirrors Toward observing mirror quantum states

22. 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)

23. 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)

24. Quantum measurement in gravitational wave detectors

25. 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 

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

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

28. 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

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

30. 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

31. 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

32. Cavity cooling
200x
1012x
UCB

33. In the (near?) future: Observable quantum effects

34. 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

35. Quantum states of light
Coherent state (laser light)
Squeezed state
Two complementary observables
Make on noise better for one quantity, BUT it gets worse for the other
DX1 and DX2 associated with amplitude and phase uncertainty X1 X2

36. 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) X1 X2
Laser X1 X2
Arbitrarily below shot noise
Shot noise limited  (number of photons)1/2 X1 X2 X1 X2
Squeezed vacuum
Vacuum fluctuations

37. 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

38. 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

39. 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

40. A radiation pressure dominated interferometer
Key ingredients
Two identical cavities with 1 gram mirrors at the ends
High circulating laser power
Common-mode rejection cancels out laser noise
Optical spring effect to suppress external force (thermal) noise

41. 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)

42. 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)

43. 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

44. Closing remarks

45. 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)

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

47. Initial LIGO Quantumness
SQL

48. In conclusion
MIT experiments in the extreme radiation pressure dominated regime have yielded several important classical results
Extreme optical stiffness  few MegaNewton/m
Stiff and stable optical spring  optical trapping of mirrors
Optical cooling of 1 gram mirror  few milliKelvin
Established path toward quantum regime where we expect to observe radiation pressure induced squeezed light, entanglement and quantum states of very macroscopic objects

49. In conclusion
LIGO detectors operate close to the standard quantum limit
An excellent testbed for observing quantum behavior in macroscopic objects
Feedback cooling in Initial LIGO interferometers achieved occupation number N ~ 200
Present upgrade (Enhanced LIGO, 2009) should have N ~ 50
A major upgrade (Advanced LIGO, 2015) should operate at the Standard Quantum Limit and lead to N ~1
Of course, they will also detect gravitational waves

50. And now for the most important part…

51. Cast of characters MIT Collaborators Timothy Bodiya Thomas Corbitt
Sheila Dwyer
Keisuke Goda
Nicolas Smith
Christopher Wipf
Eugeniy Mikhailov
Edith Innerhofer
David Ottaway
Sarah Ackley
Jason Pelc
MIT LIGO Lab
Collaborators
Yanbei Chen
Caltech MQM group
Stan Whitcomb
Daniel Sigg
Rolf Bork
Alex Ivanov
Jay Heefner
Caltech 40m Lab
Kirk McKenzie
David McClelland
Ping Koy Lam
Helge Müller-Ebhardt
Henning Rehbein

52. Thanks to… Our colleagues at Funding from LIGO Laboratory
The LIGO Scientific Collaboration
Funding from
Sloan Foundation
MIT
National Science Foundation

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

54. Gravitational, optical, electronic, magnetic...
Cooling
Γ = damping rate
Ω = resonant frequency
(Teff – T0) Γ0 + (Teff – T1)Γ1 = 0
Teff = (T1Γ1 + T0 Γ0) / (Γ0 + Γ1)‏
Teff ≈ T0 Γ0/ Γ1 Γ0 Ω0
Environment T0
Mass
Teff
Cooling factor limited
by Ω0 / Γ0
Γ1 (>>Γ0 )
Gravitational, optical, electronic, magnetic...
T1(<< T0)‏

55. Gravitational, optical, electronic, magnetic...
Dilution
Γ = damping rate
Ω = resonant frequency
Cooling factor limited
by Ω1 / Γ0
Maximize Ω1
Minimize Ω0 , Γ0 Γ0 Ω0
Environment T0
Mass
Teff Γ1
Ω1(>>Ω0)
Use second spring to stiffen and cool system
Gravitational, optical, electronic, magnetic...
T1(<< T0)‏

56. A note about calibration
Sensor noise
What we want to measure +
Mirror +
Force noise
Mirror Position
Controller
What we measure
For each frequency band, we assume the worst case scenario for force or sensor noise in order to estimate the real mirror motion

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

58. Quantum noise in gravitational wave interferometers
Opening remark …
Nature 446 (April 2007)
Quantum noise in gravitational wave interferometers
Quantum behavior of macroscopic objects (“giants”)
Quantum states of light

59. Ground state cooling
At room temperature
With optical trapping

60. A mirror phonon can be absorbed by the photon, increasing the photon energy (damping);
The photon can emit the phonon, decreasing the photon energy (potential acoustic instability).
Anti Stokes process— absorption of phonons
Stokes process — emission of phonons

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

62. The Standard Quantum Limit
SQL
Limit to the precision with which light can be used to measure the position of a particle
Balance of photon shot noise (uncertainty in the number of photons) and radiation pressure due to the fluctuating photon number (back action)

63. Advanced LIGO Quantum noise everywhere

64. Origin of the Quantum Noise Vacuum fluctuations

65. 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) X1 X2
Laser X1 X2
Arbitrarily below shot noise
Shot noise limited  (number of photons)1/2 X1 X2 X1 X2
Squeezed vacuum
Vacuum fluctuations

66. Quantum Enhancement Squeezed state injection

67. How to squeeze?
My favorite way
A tight hug

68. Squeezing injection in Advanced LIGO
Laser
Prototype GW detector
SHG
Faraday isolator
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

69. Quantum enhancement 2.9 dB or 1.4x
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)

70. Squeezing injection in Advanced LIGO
Laser
GW Detector
SHG
Faraday isolator
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

71. Advanced LIGO with squeeze injection
Radiation pressure
Shot noise