1. Quantum States of Light and Giants
MIT Corbitt, Ackley, Bodiya, Ottaway, Smith, Wipf
Caltech Bork, Heefner, Sigg, Whitcomb
AEI Chen, Ebhardt-Mueller, Rehbein
ITAMP, May 2007

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
LISA

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

6. Quantum noise everywhere
Shot noise
Radiation pressure noise

7. Radiation pressure – mirror oscillator coupling

8. 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

9. Ponderomotive predominance
An experimental apparatus in which radiation pressure forces dominate over mechanical forces
Ultimate goals
Generation of squeezed states of light
Quantum ground state of the gram-scale mirror
Mirror-light entanglement
En route
Optical cooling and trapping
Diamonds
Parametric instabilities
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.

10. 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

11. 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
log f
Response reduced below
resonant frequency
Connect a high Q, low stiffness mechanical oscillator to a stiff optical spring

12. 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

13. 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

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

15. Phase II cavity
10%
90%
5 W

17. 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
Independently control spring constant and damping
T. Corbitt et al., PRL (2007)

18. Diamonds? 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
Frequency (Hz)

19. Optical cooling Increasing subcarrier detuning
T. Corbitt et al., PRL (2007)
Increasing subcarrier detuning

20. Quantum states of a giant
1 gm mirror  1022 atoms

21. Reaching quantum ground state
Number of quanta in mode
Decoherence time
Number of oscillations before mode decays
Optical spring
 nosc increases
Optical damping
 nosc conserved

22. Experimental improvements
Lower frequency mechanical resonance  13 Hz
Shorter cavity (0.1 m)  less frequency noise
Some acoustic features
Feedback damping

23. Cooler mirror
T. Corbitt et al. quant-ph/ (2007)

24. Cool mirror Without optical trap With optical trap
based on Thomas's FIT of the coldest OS mode:
Teff = 0.8 K
Omega_eff = 2*pi*1800 Hz
Q_eff = 6
==> Nbar_eff = 1/(1.38e-23*0.8/(1.05e-34*2*pi*1800*6)) = 6.5e-7
compared with no OS case:
T = 295 K
Omega = 2*pi*172 Hz
Q = 3200
==> Nbar_m = 1/(1.38e-23*295/(1.05e-34*2*pi*172*3200)) = 8.9e-8
Nbar_eff/Nbar_m = 7.2
Error on Nbar = sqrt(delta_T^2 + delta_Omega^2 + delta_Q^2)
where deltas are fractional errors (5% for temp., 5% for frequency of
modes, 2% for Q).
==> delta_Nbar = sqrt(0.05^ ^ ^2) = 0.07
T decreased by factor of 40000
nosc increased by factor of 200 Damping alone cannot change nosc

25. Approaching a quantum state
Ponderomotive experiment with two cavities
Without optical trapping
With optical trap at 1 kHz
Hbar/kB = 1.05e-34/1.38e-23 = 7.6e-12

26. What’s next…

28. Approaching quantum regime
Single cavity measurement was limited by frequency noise of the laser
At present technical noise in two cavity system is factor of ~100 away from design noise floor
Improving rapidly
What might we measure in this setup?
Quantum radiation pressure
Squeezing
Entanglement
Ultimate limit comes from the quantum noise of the optical fields
Quantum radiation pressure
Assuming 6.3 Hz for our resonator and a Q of 10^4, for viscous damping we get R=0.25. For structural, we do about 10x better (at a frequency
100x higher, ~600 Hz), for 2.5.
To explain the disparity (to show how we improve things), going from Jack's 1e-8, we have a factor of
(172/6.3)^2 ~ 10^3 from the lower frequency resonator
(10 / 1e-3) ~ 10^4 from using 10 W instead of 1 mW
~ 3 from higher Q
~ 10 from structural damping

29. Light entanglement Logarithmic negativity Measure of CV entanglement
To what degree are the density matrices of C and SC not separable
Calculated using quantum Langevin approach
Cavity linewidth
Entanglement (also squeezing)
Optical spring resonance
C. Wipf et al. (2007)

30. 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

31. The End
NSF
LIGO Lab
Thorne, Girvin, Harris

32. Lessons learned so far…

33. 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)

34. 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

35. 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