Quantum Noise in Gravitational-wave Detectors Nergis Mavalvala @ LSU November 2004

Global network of detectors GEO VIRGO LIGO TAMA AIGO LIGO Detection confidence Source polarization Sky location LISA

Gravitational waves Transverse distortions of the space-time itself  ripples of space-time curvature Propagate at the speed of light Push on freely floating objects  stretch and squeeze the space transverse to direction of propagation  STRAIN Energy and momentum conservation require that the waves are quadrupolar  aspherical mass distribution

Astrophysics with GWs vs. E&M E&M (photons) GW Space as medium for field Spacetime itself ripples Accelerating charge Accelerating aspherical mass Absorbed, scattered, dispersed by matter Very small interaction; matter is transparent 10 MHz and up 10 kHz and down Light = not dark (but >95% of Universe is dark) Radiated by dark mass distributions  black holes, dark matter Very different information, mostly mutually exclusive Difficult to predict GW sources based on EM observations

Strength of GWs: e.g. Neutron Star Binary Gravitational wave amplitude (strain) For a binary neutron star pair R M M Quadrupole formalism is accurate to order of magnitude for most sources. Involves computing wave generation and radiation reaction from Einstein eqn. Weak internal gravity and stresses  nearly Newtonian source Kepler’s third law of planetary motion: period^2 = 4*pi^2*radius^3/(G*Msun) Distances  1 parsec = 3.26 l.y. = 3e18 cm r ~ 10^23 m ~ 10 Mpc (center of Virgo cluster) Distance of earth to center of galaxy ~ 30000 l.y. ~ 10 kpc h ~10-21 r

Effect of a GW on matter

GW detector at a glance L ~ 4 km For h ~ 10–21 Thermal noise  DL ~ 10-18 m Thermal noise  vibrations of mirrors due to finite temperature Seismic noise  ground motion due to natural and anthropogenic sources Shot noise  quantum fluctuations in the number of photons detected

Measurement and the real world How to measure the gravitational-wave? Measure the displacements of the mirrors of the interferometer by measuring the phase shifts of the light What makes it hard? GW amplitude is small External forces also push the mirrors around Laser light has fluctuations in its phase and amplitude

“Conventional” Interferometers Generation 1 (now!)

LIGO : Laser Interferometer Gravitational-wave Observatory 3 k m ( ± 1 s ) 4 km 2 km LA 4 km

Initial LIGO Sensitivity Goal

Science Runs and Sensitivity 2nd Science Run Feb - Apr 03 (59 days) S1 1st Science Run Sept 02 (17 days) S3 3rd Science Run Nov 03 – Jan 04 (70 days) LIGO Target Sensitivity Frequency (Hz) Strain (1/rtHz) S3 Duty Cycle H1 69% H2 63% L1 22% DL = strain x 4000 m 10-18 m rms

Signal-tuned Interferometers The Next Generation

Why a better detector? Astrophysics Factor 10 to 15 better amplitude sensitivity (Reach)3 = rate Factor 4 lower frequency bound NS Binaries Initial LIGO: ~20 Mpc Adv LIGO: ~350 Mpc BH Binaries Initial LIGO: 10 Mo, 100 Mpc Adv LIGO : 50 Mo, z=2 Stochastic background Initial LIGO: ~3e-6 Adv LIGO ~3e-9

A Quantum Limited Interferometer LIGO I Ad LIGO Seismic Suspension thermal Test mass thermal Quantum

Limiting Noise Sources: Optical Noise Shot Noise Uncertainty in number of photons detected a Higher circulating power Pbs a low optical losses Frequency dependence a light (GW signal) storage time in the interferometer Radiation Pressure Noise Photons impart momentum to cavity mirrors Fluctuations in number of photons a Lower power, Pbs Frequency dependence a response of mass to forces Shot noise: Laser light is Poisson distributed  sigma_N = sqrt(N) dE dt >= hbar  d(N hbar omega) >= hbar  dN dphi >= 1 Radiation Pressure noise: Pressure fluctuations are anti-correlated between cavities  Optimal input power depends on frequency

Initial LIGO

Signal-recycled Interferometer ℓ Cavity forms compound output coupler with complex reflectivity. Peak response tuned by changing position of SRM 800 kW 125 W Reflects GW photons back into interferometer to accrue more phase Signal Recycling signal

Advance LIGO Sensitivity: Improved and Tunable broadband detuned narrowband SQL  Heisenberg microscope analog If photon measures TM’s position too well, it’s own angular momentum will become uncertain. thermal noise

Sub-Quantum Interferometers Generation 2

Quantum Noise in Optical Measurements Measurement process Interaction of light with test mass Counting signal photons with a photodetector Noise in measurement process Poissonian statistics of force on test mass due to photons  radiation pressure noise (RPN) (amplitude fluctuations) Poissonian statistics of counting the photons  shot noise (SN) (phase fluctuations)

Free particle SQL uncorrelated 0.1 MW 1 MW 10 MW

Some quantum states of light Analogous to the phasor diagram Stick  dc term Ball  fluctuations Common states Coherent state Vacuum state Amplitude squeezed state Phase squeezed state McKenzie

Squeezed input vacuum state in Michelson Interferometer GW signal in the phase quadrature Not true for all interferometer configurations Detuned signal recycled interferometer  GW signal in both quadratures Orient squeezed state to reduce noise in phase quadrature X+ X- X+ X- X+ X-

Back Action Produces Squeezing f Squeezing produced by back-action force of fluctuating radiation pressure on mirrors b a Vacuum state enters anti-symmetric port Amplitude fluctuations of input state drive mirror position Mirror motion imposes those amplitude fluctuations onto phase of output field a1 a2 “In” mode at omega_0 +/- Omega  |in> = exp(+/- 2*j* beta) S(r, phi) |out> Heisenberg Picture: state does not evolve, only operators do. So |out> vacuum state is squeezed by factor sinh(r) = kappa/2 and angle phi = 0.5 arcot(kappa/2). Spectral densities assuming input vacuum state: S_b1 = exp(-2 r) ~ 1/kappa when kappa >> 1 S_b2 = exp(+2 r) ~ kappa S_{b1 b2} = 0

Conventional Interferometer with Arm Cavities Coupling coefficient k converts Da1 to Db2 k and squeeze angle f depends on I0, fcav, losses, f a b Amplitude  b1 = a1 Phase  b2 = -k a1 + a2 + h Radiation Pressure Shot Noise

If we could squeeze -k a1+a2 instead Optimal Squeeze Angle If we squeeze a2 shot noise is reduced at high frequencies BUT radiation pressure noise at low frequencies is increased If we could squeeze -k a1+a2 instead could reduce the noise at all frequencies “Squeeze angle” describes the quadrature being squeezed

Frequency-dependent Squeeze Angle

Squeezing – the ubiquitous fix? All interferometer configurations can benefit from squeezing Radiation pressure noise can be removed from readout in certain cases Shot noise limit only improved by more power (yikes!) or squeezing (eek!) Reduction in shot noise by squeezing can allow for reduction in circulating power (for the same sensitivity) – important for power-handling

Squeezed vacuum Requirements Generation methods Challenges Squeezing at low frequencies (within GW band) Frequency-dependent squeeze angle Increased levels of squeezing Generation methods Non-linear optical media (c(2) and c(3) non-linearites)  crystal-based squeezing (recent progress at ANU and MIT) Radiation pressure effects in interferometers  ponderomotive squeezing (in design & construction phase) Challenges Frequency-dependence  filter cavities Amplitude filters Squeeze angle rotation filters Low-loss optical systems

Sub-quantum-limited interferometer Narrowband Broadband Broadband Squeezed X+ X- Quantum correlations Input squeezing

Signal recycling mirror  quantum correlations Shot noise and radiation pressure (back action) noise are correlated (Buonanno and Chen, 2001) Optical field (which was carrying mirror displacement information) returns to the arm cavity Radiation pressure (back action) force depends on history of test mass (TM) motion Dynamical correlations Part of the light leaks out the SRM and contributes to the shot noise SN(t) RPN(t+t) BUT the (correlated) part reflected from the SRM returns to the TM and contributes to the RPN at a later time

Squeezing using nonlinear optical media

Vacuum seeded OPO ANU group  quant-ph/0405137

Squeezing using back-action effects

The principle A “tabletop” interferometer to generate squeezed light as an alternative nonlinear optical media Use radiation pressure as the squeezing mechanism Relies on intrinsic quantum physics of optical field-mechanical oscillator correlations Squeezing produced even when the sensitivity is far worse than the SQL Due to noise suppression a la optical springs

Noise budget

High circulating laser power High-finesse cavities Key ingredients High circulating laser power 10 kW High-finesse cavities 25000 Light, low-noise mechanical oscillator mirror 1 gm with 1 Hz resonant frequency Optical spring Detuned arm cavities

Optical Springs Modify test mass dynamics Suppress displacement noise (compared to free mass case) Why not use a mechanical spring? Thermal noise Connect low-frequency mechanical oscillator to (nearly) noiseless optical spring

Thermal Noise in Springs Why not use a mechanical spring? Displacements due to thermal noise introduced by the high frequency (mechanical) spring will wash out the effects of squeezing An optical spring with a high resonant frequency will not change the thermal force spectrum of the mechanical pendulum Use a low resonant frequency mechanical pendulum to minimize thermal noise Use an optical spring to produce a flat response out to higher frequencies

Optical spring Detuned cavity Positive detuning Detuning increases Cavity becomes longer Power in cavity decreases Radiation-pressure force decreases Mirror ‘restored’ to original position Cavity becomes shorter Power in cavity increases Mirror still ‘restored’ to original position

Noise budget – Equivalent displacement

In conclusion...

Next generation – quantum noise limited Squeezing being pursued on two fronts Nonlinear optical media Back-action induced correlations Other Quantum Non-Demolition techniques Evade measurement back-action by measuring of an observable that does not effect a later measurement Speed meters Optical bars Correlations between the SN and RPN quadratures

Gravitational-wave Interferometers Astrophysics Tests of General Relativity Quantum Measurement