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

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

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

Global network of detectors GEO VIRGO LIGO TAMA AIGO LIGO LISA

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

Quantum noise everywhere Shot noise Radiation pressure noise

Radiation pressure – mirror oscillator coupling

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

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.

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

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

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

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

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

Phase II cavity 10% 90% 5 W

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)

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)

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

Quantum states of a giant 1 gm mirror  1022 atoms

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

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

Cooler mirror T. Corbitt et al. quant-ph/0705.1018 (2007)

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 + 0.05^2 + 0.02^2) = 0.07 T decreased by factor of 40000 nosc increased by factor of 200 Damping alone cannot change nosc

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

What’s next…

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

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)

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

The End NSF LIGO Lab Thorne, Girvin, Harris

Lessons learned so far…

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, 021802R (2006)

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

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