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

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

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

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

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

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 !

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

The optical spring effect and optical trapping of mirrors

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

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

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

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

Classical Experiments Extreme optical stiffness Stable optical trap Optically cooled mirror

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

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

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

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)

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

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, 150802 (2007)

Supercold mirrors Toward observing mirror quantum states

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, 150802 (2007)

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, 160801 (2007)

Quantum measurement in gravitational wave detectors

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 

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

LIGO: Laser Interferometer Gravitational-wave Observatory 3 k m ( ± 1 s ) MIT 4 km NSF Caltech LA 4 km http://ligo.mit.edu

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

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

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

Some other cool oscillators Toroidal microcavity  10-11 g NEMS  10-12 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

Cavity cooling 200x 1012x UCB

In the (near?) future: Observable quantum effects

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

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

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

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

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

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

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

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, 023801 (2006)

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./283659 (2008)

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

Closing remarks

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)

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

Initial LIGO Quantumness SQL

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

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

And now for the most important part…

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

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

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

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)‏

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)‏

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

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

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

Ground state cooling At room temperature With optical trapping

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

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

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)

Advanced LIGO Quantum noise everywhere

Origin of the Quantum Noise Vacuum fluctuations

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

Quantum Enhancement Squeezed state injection

How to squeeze? My favorite way A tight hug

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

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)

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

Advanced LIGO with squeeze injection Radiation pressure Shot noise