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Gravitational-Wave Detectorsfundamental principles

Yanbei ChenCalifornia Institute of Technology

Tuesday, August 2, 2011

Overview

• Gravitational waves: their interaction with light and matter.

- two different coordinate systems

• Qualitative understanding of a ground-based interferometer’s sensitivity

- order-of-magnitude formula

• Calculation of interferometer’s quantum-limited sensitivity

- quantization of light

- shot noise

- radiation-pressure noise

• Thermal Fluctuations and their influence to sensitivity

- suspension thermal noise

- internal thermal noise

2


Gravitational Wave Spacetime in the TT Gauge• Plane wave solution to Linearized Einstein’s Equation, in the Transverse Traceless (TT)

gauge

3

hTTij (t,x) = h+ (t −N ⋅x)eij+ + h× (t −N ⋅x)eij

×gij = ηij + hij ,

N : wave propagation direction

e+ij , e×ij : polarization tensors

If forms an orthonormal basis, then we can set (e1,e2 ,N)

e+ij = (e1)i (e1)i − (e2 )i (e2 )ie×ij = (e1)i (e2 )i + (e1)i (e2 )i

This is called the TT gauge because hij is transverse, and trace-free.

Example:

e1 = eθ , e2 = eϕ , N = er ,

Question: How are Light and Masses Affected by GW (Metric Perturbation)?

x

y

z N

e1 = eθ

e2 = eϕ


Response of Laser Interferometers: TT Gauge• Propagation of Light in the TT gauge

4

gab∇a∇bΦ = 0⇔ ∂µ ( −ggµν∂νΦ) = 0the scalar

wave equation

Φ = Aexp(ikµxµ + iδφ) = Aexp −iωt + ik ⋅x + iδφ( )

δφ(t0 + L,x0 + kL) =ω2c

ki k jhij (t0 + ξ,x0 + kξ)dξ0

L

∫

flat-space solution plus additional phase due to GW

ku = (ω ,k) =ω (1, k)

4-wavevector ang freq 3-wavevector propagation direction

kµ∂µδφ =hTTµνk

µkν

2δϕ slowly

varying

additional phase accumulates along rays as wave propagates

kµ

z

t

ray o

f flat

spac

e

propag

ation

dire

ction

δϕ ac

cum

ulate

s

no lo

nger

a ra

y!

Light propagation is modified by GW


Response of Free Masses• Equation of motion

• For test masses moving at low coordinate speeds (same order as h), up to leading order, we have

• In TT gauge: mass equation of motion is not influenced by GW. • It is then simple to describe an idealized, free-mass GW detector

5

Duα

dτ= Fα ⇒ duα

dτ+ Γα

βγuβuγ = Fα

dv j

dt= F j − Γ j

00 +O(h2 ) = F j +O(h2 )

Fα: non-gravitational force

z

t

δϕ

A B

Free Masses A and B, at constant coordinate locations

carrying ideal clockscan determine δϕ by measuring additional

phase shift of light.

But how can we measure phase?


Interferometry• A, B, and C freely falling• A sends light to B and C• B and C reflect light with no phase shift• A compares phase between light from B and light

from C.

• This gives

• This compares the phase of A against itself, therefore eliminates phase noise

6

A BC

t

δϕ1

δϕ2

δϕ3

δϕ4

(δϕ1 + δϕ2) - (δϕ3+δϕ4)

Interferometer gets signal from GW-induced phase modulationnot from test-mass motion


Short-Armed Interferometer• For small separation (much less than reduced wavelength): only local GW matters

• Example, plane wave along z direction

7

δφ(t0 + L,x0 + kL) = ω0

2cki k jhij (...)dξ

0

L

∫ ⇒ δφ(t,x) = ω0

cL2k i k jhij (t,x)⎡

⎣⎢⎤⎦⎥

phase modulation according to “stretching/squeezing of spacetime”

+ polarization × polarization

hxx = −hyy hxy = hyx

stretching

squeezingsqueezing stretching

optimal placement of Michelson arms: in the x-y plane, adapted to polarization


Ground-Based Laser Interferometers• LIGO/VIRGO/GEO600/TAMA/LCGT/AIGO ... all have arm less than wavelength (arm

length up to 4km, frequency up to a few kHz, or wavelength down to ~10km)

8


δφ(t0 + L,x0 + kL) =ω0

cLH pe

pij k

i k j

2e− iΩ(t−N⋅x) e− iΩL (1−N⋅k ) −1

−iΩL(1−N ⋅ k)

... plane wave with propagation direction N

Response of Laser Interferometers: TT Gauge• For larger separation (~ reduced wavelength): oscillatory nature matters

9

hp = Hpe− iΩ(t−N⋅x) , p = +,×

same as beforethis favors

k orthogonal to N(transverse wave)proportional to L

additional phase factorthis favors k along N(phase coherence)and prefers small L

~1/L for large L

GW along z

z

x y

+ polarized

k

θ

0 10 20 30 40 50 600

1

2

3

4

L

∆Φ

GW phase shift for a fixed Ω and varying L

π/2

π/8π/4


Laser Interferometer Space Antenna• Arm length 5×109m detection band up to 1Hz (wavelength as low as 3×108m)

10


Local Lorentz Frame• Another gauge is easier sometimes for describing small-sized detectors• Fermi Normal Coordinate System for one particular test mass

• In this frame, test mass feels tidal force (proportional to mass and distance from center)

• Effect on propagation of light suppressed by • Michelson Interferometer gets signal from mass motion, not phase modulation of light.

11

ds2 = −(1+ R0l0mxl xm )dt 2 − 4

3R0ljmx

l xmdx jdt + δ ij −13Riljmx

l xm⎛⎝⎜

⎞⎠⎟ dx

idx j

x j = 1

2hjkTT xk + F j

R0l0m = − 1

2hTTlm ~

h2

R0ljm ~ Rjlkm ~h2

(L / )2

+ polarization

stretching

squeezing

× polarization

squeezing stretching

hxy = hyxhxx = −hyy


Bar Detector In the Local Lorentz Frame12

Joe Weber

ρ ∂2u(t, x)∂t 2

+ ργ ∂u(t, x)∂t

+Y ∂2u(t, x)∂x2

= ρ2xh(t)

L/2-L/2

z

x

∂u(t, x)∂x x=±L /2

= 0

piezoelectricsensor

Voltage is V = PΔ ∂u∂x x=0

Δ

maximum sensitivity at resonant frequencies


Gravitational-Wave Detectors

Yanbei ChenCalifornia Institute of Technology


Summary from last time• Two gauges for describing GW’s influence on matter and light

• TT gauge- GW adds additional phase to light- GW does not move mirrors

• Local Lorentz Frame- GW moves mirrors- GW does not directly add more phase shift

14


Overview

• Gravitational waves: their interaction with light and matter.

- two different coordinate systems

• Qualitative understanding of a ground-based interferometer’s sensitivity

- order-of-magnitude formula

• Calculation of interferometer’s quantum-limited sensitivity

- quantization of light

- shot noise

- radiation-pressure noise

• Thermal Fluctuations and their influence to sensitivity

- suspension thermal noise

- internal thermal noise

15


Initial LIGO Detector16

4 km

4 km

Arm Cavity signal light ~ h

LocalOscillator

~ freehorizontally

noise from light quantization

thermalfluctuations


“Noise Curve”17

noise from light quantization

noise from thermal

fluctuations

squa

re r

oot

of n

oise

sp

ectr

al d

ensi

ty

Spectral Density: Noise Power Per Frequency Band


Spectral Density• Definition 1:

• Definition 2:

• Definition 3:

18

x(Ω)x*(Ω ') = 12S(Ω)δ (Ω−Ω ') Noise Power

Per Frequency Band

C(t) ≡ x(t)x(0) , S(Ω) = C(t)eiΩt dt−∞

+∞

∫ Fourier Transform of Two-TimeCorrelation Function

S(Ω) = limT→+∞

2T

x(t)eiΩt dt−T /2

T /2

∫2

For signal with duration τ and frequency near 1/τ, error is

δ x ~ Sx (1 /τ ) /τ


Michelson Interferometer: Sensitivity Estimate

• Resolvable phase: 1/(Number of Photons)1/2

• Photon Number: Power×Duration/(Energy of Photon)

19

input light

X-arm Y-arm

phasor diagram

α = 2πλLh

α

δh = λ

2πLω0

I0τ⇒ Sh =

λ2πL

ω0

I0= 7.5 ⋅10−21 4km

L⎛⎝⎜

⎞⎠⎟5WI0

⎛⎝⎜

⎞⎠⎟

1/2

Hz−1/2assuming λ=1μm

rms error “shot noise” spectral density(noise power/frequency band)

initial LIGO 2×10-23

factor of 300-400 away!

Laser LightI0, ω0

beamsplitter

test-mass mirror

test-mass mirror

Lh/2

Lh/2

X arm

Y arm

+ polarizad plane GW along z axis


Resonant Enhancement of Sensitivity20

Inserting Fabry-Perot cavities in order to increase power and increase sensitivity

cavity perfectend

mirror

power transmissivityT

length L

Laser LightI0, ω0

ωres determined by: round-trip path integer wavelengths

2 1 0 1 20

1

2

3

4

5

Ω0Ωres

I circ

Icircmax = 2

TI0

γ = Tc4L

peakcirculating

power

bandwidth

actual circulating

powerIcirc =

Imaxcirc

1+ (ω0 −ω res )2 / γ 2

Time constant:

τ = 1 / γ = 4L /Tc

“Mean Number of Bounces”

2 /T


GW

S h

Resonant Enhancement of Sensitivity21

Laser LightI0, ω0 bea

msp

litter

test-mass mirror

test-mass mirror

Lh/2

Lh/2

arm cavity

armcavity

power recyclingmirror

LIGO I:

Power-recycling gain ~ 50 -- 60[noise ~ 1/(Mich. input power)1/2]

# of bounces in arm cavity ~40

Total factor of improvement ~300

Improvement is only below the bandwidth of the cavity.

Above bandwidth:

within light storage time, GW already changessign. Resonant enhancement deteriorates!

ΩGW > γ ⇔ τGW < τ storageno cavity

with

cav

ity

# ofbouncesCavity Gain = 2 /T

1+ (Ω / γ )2

Shshot = λ

2πLω0

I0

1+ (Ω / γ )2

2 /T


Radiation Pressure Noise22

Laser LightI0, ω0

beamsplitter

test-mass mirror

test-mass mirror

Lh/2

Lh/2

X arm

Y arm

+ polarizad plane GW along z axis

⇒ SF = 4ω0I0

c2

⇒ Sx =1

mΩ24ω0I0

c2

⇒ Shrad pres = 1

mΩ2L4ω0I0

c2 δP = δN ⋅ 2ω0

c= N 2ω0

c= I0τω0

2ω0

crms momentum of mirror

given by photon # fluctuation

δF = δP /τ = 4ω0I0

c2τ

without cavity ...

with cavity gain

Sh

rad pres = 1mΩ2L

4ω0I0

c22 /T

1+ (Ω / γ )2

GW

S h

without cavity ~Ω -2

with cavity


Standard Quantum Limit23

Shshot = λ

2πLω0

I0= 1Lc2

I0ω0

1+ (Ω / γ )2

2 /T

If we place the two types of noise together

Sh

rad pres = 1mΩ2L

4I0ω0

c22 /T

1+ (Ω / γ )2= 2mΩ2L

I0ω0

c22 /T

1+ (Ω / γ )2

Their dependences on power and cavity gain are opposite

GW

S h 1 × power

10 × power

1/10 × power

GW

S h

Total Noise Never Surpasses

the Standard Quantum Limit


Origin of the Standard Quantum Limit• Quantum Mechanics: A & B cannot both be measured exactly if

24

A, B⎡⎣ ⎤⎦ ≠ 0



24

A, B⎡⎣ ⎤⎦ ≠ 0

xH (t) = xH (0)+ pH (0)

Mt ⇒ xH (0), xH (t)[ ] = xH (0), xH (0)+ pH (0)

Mt⎡

⎣⎢⎤⎦⎥= itM

≠ 0

• Position x(t) of a free mass, which we attempt to measure:



24

A, B⎡⎣ ⎤⎦ ≠ 0

xH (t) = xH (0)+ pH (0)

Mt ⇒ xH (0), xH (t)[ ] = xH (0), xH (0)+ pH (0)

Mt⎡

⎣⎢⎤⎦⎥= itM

≠ 0


⇒ ΔxH (0) ⋅ ΔxH (t) ≥

t2M



24

A, B⎡⎣ ⎤⎦ ≠ 0

xH (t) = xH (0)+ pH (0)

Mt ⇒ xH (0), xH (t)[ ] = xH (0), xH (0)+ pH (0)

Mt⎡

⎣⎢⎤⎦⎥= itM

≠ 0


⇒ ΔxH (0) ⋅ ΔxH (t) ≥

t2M

For duration τ this sets scale

ΔxQ ~ τ2M



24

A, B⎡⎣ ⎤⎦ ≠ 0

xH (t) = xH (0)+ pH (0)

Mt ⇒ xH (0), xH (t)[ ] = xH (0), xH (0)+ pH (0)

Mt⎡

⎣⎢⎤⎦⎥= itM

≠ 0


⇒ ΔxH (0) ⋅ ΔxH (t) ≥

t2M


ΔxQ ~ τ2M

Example: measure GW pulse through change in mirror positions between t = 0 and τ

xΔxQ

t = 0xΔxQ

t = τ



24

A, B⎡⎣ ⎤⎦ ≠ 0

xH (t) = xH (0)+ pH (0)

Mt ⇒ xH (0), xH (t)[ ] = xH (0), xH (0)+ pH (0)

Mt⎡

⎣⎢⎤⎦⎥= itM

≠ 0


⇒ ΔxH (0) ⋅ ΔxH (t) ≥

t2M


ΔxQ ~ τ2M

Measurement 1: measure x at t = 0 strongly, Δx(0) << ΔxQ ⇒ strongly perturb p


xΔxQ

t = 0xΔxQ

t = τ



24

A, B⎡⎣ ⎤⎦ ≠ 0

xH (t) = xH (0)+ pH (0)

Mt ⇒ xH (0), xH (t)[ ] = xH (0), xH (0)+ pH (0)

Mt⎡

⎣⎢⎤⎦⎥= itM

≠ 0


⇒ ΔxH (0) ⋅ ΔxH (t) ≥

t2M


ΔxQ ~ τ2M


Measurement 2: uncertainty of p enters further evolution of x ⇒ Δx(τ) >> ΔxQ


xΔxQ

t = 0xΔxQ

t = τ



24

A, B⎡⎣ ⎤⎦ ≠ 0

xH (t) = xH (0)+ pH (0)

Mt ⇒ xH (0), xH (t)[ ] = xH (0), xH (0)+ pH (0)

Mt⎡

⎣⎢⎤⎦⎥= itM

≠ 0


⇒ ΔxH (0) ⋅ ΔxH (t) ≥

t2M

Trade-off: both have the same Δx(0)= Δx(τ)= ΔxQ ⇒ Standard Quantum Limit


ΔxQ ~ τ2M




xΔxQ

t = 0xΔxQ

t = τ



24

A, B⎡⎣ ⎤⎦ ≠ 0

xH (t) = xH (0)+ pH (0)

Mt ⇒ xH (0), xH (t)[ ] = xH (0), xH (0)+ pH (0)

Mt⎡

⎣⎢⎤⎦⎥= itM

≠ 0


⇒ ΔxH (0) ⋅ ΔxH (t) ≥

t2M



ΔxQ ~ τ2M




xΔxQ

t = 0xΔxQ

t = τ

“Heisenberg Microscope” treatmentno careful quantization of light



24

A, B⎡⎣ ⎤⎦ ≠ 0

xH (t) = xH (0)+ pH (0)

Mt ⇒ xH (0), xH (t)[ ] = xH (0), xH (0)+ pH (0)

Mt⎡

⎣⎢⎤⎦⎥= itM

≠ 0


⇒ ΔxH (0) ⋅ ΔxH (t) ≥

t2M



ΔxQ ~ τ2M



More Carefully: wavefunction broadening at t = τ caused by radiation-pressure force at t = 0,effect can be removed if shot noise t = τ is correlated with it


xΔxQ

t = 0xΔxQ

t = τ

“Heisenberg Microscope” treatmentno careful quantization of light


25

initial LIGO achieved in

2007

Advanced LIGO (current plan)• Construction of parts: 2008 --• Installation

•begins: late 2010•finishes: mid 2012

•2015: earliest possible operationEinstein Telescope

Generations of LIGO vs. the SQL[Abramovici et al., 1992]

10 x

10 x


25

Standard Quantum Limit 10kg



2007





10 x

10 x


25

LIGO-III/Einstein Telescope must beat the SQL significantly in broad frequency band




2007





10 x

10 x


• Optical field close to ω0 can ben written in the quadrature representation

• Act as modulations when superimposed with single-frequency carrier at ω0

Quadratures, Homodyne Detection and Squeezing26

E(t) = E1(t)cosω0t + E2 (t)sinω0t E1,2 (t) : slowly varying

E1

E2

phasor diagram






E1

E2

carrierA cos ω0t

phasor diagram






phasequadrature

amplitudequadrature

E1

E2

carrierA cos ω0t

phasor diagram






phasequadrature

amplitudequadrature

E1

E2

carrierA cos ω0t

A co

s (ω 0t+

ϕ)

ϕ

anot

her c

arrie

r

phasor diagram






phasequadrature

amplitudequadrature

E1

E2

carrierA cos ω0t

E1

E2

A co

s (ω 0t+

ϕ)

ϕ

anot

her c

arrie

r

phasor diagram






phasequadrature

amplitudequadrature

E1

E2

carrierA cos ω0t

E1

E2E1 cos (ϕ) + E2 sin (ϕ)

- E1 s

in (ϕ

) + E

2 cos

(ϕ)

A co

s (ω 0t+

ϕ)

ϕ

anot

her c

arrie

r

phasor diagram






phasequadrature

amplitudequadrature

E1

E2

carrierA cos ω0t

E1

E2E1 cos (ϕ) + E2 sin (ϕ)

- E1 s

in (ϕ

) + E

2 cos

(ϕ)

A co

s (ω 0t+

ϕ)

ϕ

anot

her c

arrie

r

Homodyne Detection

phasor diagram


Quantum “Two-Photon” Formalism

• One mode, approximated as within a cylinder of cross sectional area A

• ... normalization is chosen such that

• Therefore

• Quadratures (two photon formalism)

27

EH (t) =

dω2π

2πωAc

aωe− iωt + h.c.⎡⎣ ⎤⎦

0

+∞

∫

H = E2 (x)

4πAdx∫ = dω

2πω aωa

†ω + a†ωaω⎡⎣ ⎤⎦

0

+∞

∫

aω ,a†ω '⎡⎣ ⎤⎦ = 2πδ (ω −ω ')

EH (t) =

4πω0

Aca1(t)cosω0t + a2 (t)sinω0t[ ]

a1(Ω) =aω0+Ω

+ a†ω0−Ω

2, a2 (Ω) =

aω0+Ω− a†

ω0−Ω

2i

Hermitian Operators

[a1(t),a1(t ')] = [a2 (t),a2 (t ')] = 0, [a1(t),a2 (t)] = iδ (t − t ')analogous to conjugate coordinates and momenta


• Heisenberg Uncertainty In the Frequency Domain


Sa1a1Sa2a2 − | Sa1a2 |2≥1

Minimum Uncertainty Gaussian States are:

vacuum statecoherent states

squeezed vacuuasqueezed states




vacuum

E1

E2

Sa1a1Sa2a2 − | Sa1a2 |2≥1







coherent state

vacuum

E1

E2

Sa1a1Sa2a2 − | Sa1a2 |2≥1







coherent state

phasesqueezedvacuum

E1

E2

Sa1a1Sa2a2 − | Sa1a2 |2≥1







coherent state

phasesqueezed

amplitudesqueezed

vacuum

E1

E2

Sa1a1Sa2a2 − | Sa1a2 |2≥1







coherent state

phasesqueezed

amplitudesqueezed

amplitudesqueezed

vacuum

E1

E2

Sa1a1Sa2a2 − | Sa1a2 |2≥1







coherent state

phasesqueezed

amplitudesqueezed

amplitudesqueezed

vacuum

E1

E2

phasesqueezed

Sa1a1Sa2a2 − | Sa1a2 |2≥1





Time-Domain Picture29

vacuum

amplitudesqueezed

phasesqueezed


Vacuum Fluctuation Point of View30

Carrier

InputTest Mass

End Test Mass

Arm CavityL ~ 4 km

Power RecyclingMirror

Carrier: Amplified by Resonance (Cavity & Recycling)



Carrier

InputTest Mass

End Test Mass

Arm CavityL ~ 4 km



Signal: anti-symmetric phase modulation, escape from “dark port”

anti-sym motion

anti-sym m

otion



Carrier

InputTest Mass

End Test Mass

Arm CavityL ~ 4 km




anti-sym motion

anti-sym m

otion

Vacuum Fluctuations of EM field enter oppositely from the detection port. Responsible for all quantum noises!



Carrier

InputTest Mass

End Test Mass

Arm CavityL ~ 4 km




anti-sym motion

anti-sym m

otion

Two Types of Noise:




Carrier

InputTest Mass

End Test Mass

Arm CavityL ~ 4 km




anti-sym motion

anti-sym m

otion

Two Types of Noise: shot noise

Shot Noisediscreteness of out-

going photons




Carrier

InputTest Mass

End Test Mass

Arm CavityL ~ 4 km




anti-sym motion

anti-sym m

otion

Two Types of Noise: shot noise

Shot Noisediscreteness of out-

going photons


and radiation-pressure noise

Radiation-Pressure Noiserandom radiation-pressure

force on mirrors


Interaction between quadrature field and test mass

• Equations of motion for a simple Michelson

• Adding cavity...

31

b1 = e2iΩLa1

b2 = e2iΩLa2 + e

iΩL 2I0ω0

c2x

x = Lh + 2 2mΩ2c

ω0I0eiΩLa1

... motion of mirror phase modulates light

... amplitude modulation drives mirror

a

b

a’

b’ Michelson

simply need to add equations

aj = Rbj + T ′aj′bj = Tbj − R ′aj


Input-Output Relation of Fabry Perot Michelson• Input Output Relation

• Features- Gain in transfer function due to cavity is

- Amplitude of light in cavity decays as , noting that

- Assuming vacuum input state- SQL limited when measuring output phase quadrature (b2)- Can surpass SQL if measure an appropriate quadrature- If input state can be squeezed vacuum, then not SQL limited, either,

32

b1 = e2iβa1

b2 = e2iβ (a2 − Ka1)+ e

iβ K hhSQL

β = arctanΩ

γ, K = 2γΘ3

Ω2 (Ω2 + γ 2 ), Θ3 = 8ω0Icirc

mLc, hSQL = 8

mΩ2L2

2 /TΩ2 + γ 2

e−γ t

e2iβ = −Ω + iγΩ + iγ

= −1+ 2iγΩ + iγ


Beating the SQL via “Back-Action Evasion”[Braginsky, Caves, Thorne, Vyatchanin 80s, ... , H.J. Kimble et al., 2001]

33

E1

E2

phasequadrature

amplitudequadrature

interferometer

E1

E2

mirr

or m

otio

n

injected vacuum



33

E1

E2

phasequadrature

amplitudequadrature

interferometer

E1

E2

mirr

or m

otio

n

injected vacuum

GW-induced

Back-Action-Induced



33

E1

E2

phasequadrature

amplitudequadrature

interferometer

E1

E2

mirr

or m

otio

n

injected vacuum

GW-induced

Back-Action-Induced

• Careful combination of output quadratures give:

- only Sensing Noise & GW-induced motion: Back Action Evasion

• The back-action-evading output quadrature is frequency dependent.


Frequency Dependent Homodyne Detection34

Kimble et al., 2001


Frequency Dependent Squeezing & Detection35

Standard Quantum Limit

1 10 100 1000 1041025

1024

1023

1022

1021

1020

f Hz

S hf1

Hz

• “Frequency dependent homodyne detection” plus input squeezing. [H.J. Kimble et al., 2001]

• Requires km-scale filters: losses destroy quantum correlations very easily!!




1 10 100 1000 1041025

1024

1023

1022

1021

1020

f Hz

S hf1

Hz

lossless, with filters

1 10 100 1000 1041025

1024

1023

1022

1021

1020

f Hz

S hf1

Hz

10 dB input squeezing






1 10 100 1000 1041025

1024

1023

1022

1021

1020

f Hz

S hf1

Hz


1 10 100 1000 1041025

1024

1023

1022

1021

1020

f Hz

S hf1

Hz


1% total loss

1 10 100 1000 1041025

1024

1023

1022

1021

1020

f Hz

S hf1

Hz






1 10 100 1000 1041025

1024

1023

1022

1021

1020

f Hz

S hf1

Hz


1 10 100 1000 1041025

1024

1023

1022

1021

1020

f Hz

S hf1

Hz


1% total loss

1 10 100 1000 1041025

1024

1023

1022

1021

1020

f Hz

S hf1

Hz



“Loss Limit”

1 10 100 1000 1041025

1024

1023

1022

1021

1020

f Hz

S hf1

Hz 10dB input squeezing and

1% loss

Loss Limit = 1/5 SQL

loss ⋅squeeze factor( )1/4



36

E1

E2

phasequadrature

amplitudequadrature

interferometer

E1

E2

mirr

or m

otio

n

injected vacuum



36

E1

E2

phasequadrature

amplitudequadrature

interferometer

E1

E2

mirr

or m

otio

n

injected vacuum

GW-induced

Back-Action-Induced



36

E1

E2

phasequadrature

amplitudequadrature

interferometer

E1

E2

mirr

or m

otio

n

injected vacuum

GW-induced

Back-Action-Induced




Optical Losses can limit sensitivity!


Further Developments37

“Loss Limit” ~ 1/5 SQL

Position Meter

Speed Meter

1 10 100 1000 1041025

1024

1023

1022

1021

1020

f Hz

S hf1

Hz


Further Developments37

“Loss Limit” ~ 1/5 SQL

Position Meter

Speed Meter

1 10 100 1000 1041025

1024

1023

1022

1021

1020

f Hz

S hf1

Hz

1 10 100 1000 1041025

1024

1023

1022

1021

1020

f Hz

S hf1

Hz Strawman Einstein Telescope curve

• low frequency: increase mirror mass by 2 ~ 4

• high frequency: separate interferometer


Signal Recycling38

Carrierdi!erential motion

di!erential motion


Signal RecyclingMirror

signal fed back with phase and gain, changes optical resonance structure

Meers, 1988

``moves’’ photons into a ``useful band’’, sensitive to GWs at non-zero

frequency

1Sh ( f )

df∫ ∝ I


• High frequencies: dominated by optical resonance

• Low frequencies:

• “new” resonant frequency depends on (power)1/2

• mirror motion amplified around resonance, sensitivity surpass SQL.

Advanced LIGO’s Quantum Noise39







optical resonance







optical resonance

mechanical resonance


Optical Spring in “detuned cavities”40

X

FRAD(X)

restoringanti-restoring



X

FRAD(X)


FRAD does positive net work

FRAD does negative net work

anti-dampingdamping



Positive Spring Constant + Delay ⇒ Negative Damping

X

FRAD(X)




anti-dampingdamping



Positive Spring Constant + Delay ⇒ Negative Damping

Effects of “Strong” Linear Quantum

Measurement

1. Strong Back Action Noise.(Standard Quantum Limit)

2. Strong change of dynamics. (Optical Spring)

X

FRAD(X)




anti-dampingdamping


Summary• Ground-based GW detectors are close to being quantum limited

• 3rd generation detectors will surpass the SQL, at which Heisenberg Uncertainty of mirrors makes it difficult to improve sensitivity simply by increasing power

• in order to surpass the SQL one can - introduce correlations between radiation-pressure and shot noise- modify mirror dynamics by light

41


• Why are there thermal noises?- Each eigenmode of mechanical motion has kT thermal energy- [optical spring resonance does not have kT thermal energy, because it is opto-

mechanical]- although we detect GW off mechanical resonance• However, if there are many mechanical modes, where do we start?

Origin of Thermal Noise42

suspensionwire

substrate

coat

ing

Fluctuation-Dissipation Theorem:

Sx ( f ) =kBTπ 2 f 2

Re[Z( f )]

Z( f ) ≡ v( f )F( f )

impedancereal part of impedance gives

dissipation

fluctuation

this arises because both fluctuation and dissipationoriginate from x’s coupling to heat bath


Types of Dissipation in Bulk Material43

δT2 > δT1heatflow

δT1

thermoelastic dissipation

fast

slow

momentumflow

internal friction

Brownian Noise Thermoelastic Noise


Suspension Thermal Noise• dissipations in the suspension system

limits the Q factor of the pendulum

44

suspensionwire

substrate

coat

ing

assuming structural damping uniformthroughout the wire


Internal Thermal Noise• Substrate- Brownian: loss due to internal friction- Thermoelastic: loss due to internal heat flow, induced by non-uniform compression

• Coating- Brownian (this dominates in the current mirrors, due to high internal friction in coating)- Thermoelastic (next largest, due to layered structure of coating)

45

substrate

coat

ing

force with pressure profileequal to power profile

of optical mode

various dissipations will correspond to various types of noise

although coating is much thinner than substratedissipation in the coating is much larger

than that in the substrate


coating thermal noise

Advanced LIGO Noise Budget46


suspension thermal noise


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