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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
Tuesday, August 2, 2011
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
Tuesday, August 2, 2011
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
Tuesday, August 2, 2011
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?
Tuesday, August 2, 2011
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
Tuesday, August 2, 2011
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
Tuesday, August 2, 2011
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
Tuesday, August 2, 2011
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
Tuesday, August 2, 2011
Laser Interferometer Space Antenna• Arm length 5×109m detection band up to 1Hz (wavelength as low as 3×108m)
10
Tuesday, August 2, 2011
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
Tuesday, August 2, 2011
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
Tuesday, August 2, 2011
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
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
15
Tuesday, August 2, 2011
Initial LIGO Detector16
4 km
4 km
Arm Cavity signal light ~ h
LocalOscillator
~ freehorizontally
noise from light quantization
thermalfluctuations
Tuesday, August 2, 2011
“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
Tuesday, August 2, 2011
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
Tuesday, August 2, 2011
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
Tuesday, August 2, 2011
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
Tuesday, August 2, 2011
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
Tuesday, August 2, 2011
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
Tuesday, August 2, 2011
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
Tuesday, August 2, 2011
Origin of the Standard Quantum Limit• Quantum Mechanics: A & B cannot both be measured exactly if
24
A, B⎡⎣ ⎤⎦ ≠ 0
Tuesday, August 2, 2011
Origin of the Standard Quantum Limit• Quantum Mechanics: A & B cannot both be measured exactly if
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:
Tuesday, August 2, 2011
Origin of the Standard Quantum Limit• Quantum Mechanics: A & B cannot both be measured exactly if
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:
⇒ ΔxH (0) ⋅ ΔxH (t) ≥
t2M
Tuesday, August 2, 2011
Origin of the Standard Quantum Limit• Quantum Mechanics: A & B cannot both be measured exactly if
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:
⇒ ΔxH (0) ⋅ ΔxH (t) ≥
t2M
For duration τ this sets scale
ΔxQ ~ τ2M
Tuesday, August 2, 2011
Origin of the Standard Quantum Limit• Quantum Mechanics: A & B cannot both be measured exactly if
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:
⇒ ΔxH (0) ⋅ ΔxH (t) ≥
t2M
For duration τ this sets scale
ΔxQ ~ τ2M
Example: measure GW pulse through change in mirror positions between t = 0 and τ
xΔxQ
t = 0xΔxQ
t = τ
Tuesday, August 2, 2011
Origin of the Standard Quantum Limit• Quantum Mechanics: A & B cannot both be measured exactly if
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:
⇒ ΔxH (0) ⋅ ΔxH (t) ≥
t2M
For duration τ this sets scale
ΔxQ ~ τ2M
Measurement 1: measure x at t = 0 strongly, Δx(0) << ΔxQ ⇒ strongly perturb p
Example: measure GW pulse through change in mirror positions between t = 0 and τ
xΔxQ
t = 0xΔxQ
t = τ
Tuesday, August 2, 2011
Origin of the Standard Quantum Limit• Quantum Mechanics: A & B cannot both be measured exactly if
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:
⇒ ΔxH (0) ⋅ ΔxH (t) ≥
t2M
For duration τ this sets scale
ΔxQ ~ τ2M
Measurement 1: measure x at t = 0 strongly, Δx(0) << ΔxQ ⇒ strongly perturb p
Measurement 2: uncertainty of p enters further evolution of x ⇒ Δx(τ) >> ΔxQ
Example: measure GW pulse through change in mirror positions between t = 0 and τ
xΔxQ
t = 0xΔxQ
t = τ
Tuesday, August 2, 2011
Origin of the Standard Quantum Limit• Quantum Mechanics: A & B cannot both be measured exactly if
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:
⇒ ΔxH (0) ⋅ ΔxH (t) ≥
t2M
Trade-off: both have the same Δx(0)= Δx(τ)= ΔxQ ⇒ Standard Quantum Limit
For duration τ this sets scale
ΔxQ ~ τ2M
Measurement 1: measure x at t = 0 strongly, Δx(0) << ΔxQ ⇒ strongly perturb p
Measurement 2: uncertainty of p enters further evolution of x ⇒ Δx(τ) >> ΔxQ
Example: measure GW pulse through change in mirror positions between t = 0 and τ
xΔxQ
t = 0xΔxQ
t = τ
Tuesday, August 2, 2011
Origin of the Standard Quantum Limit• Quantum Mechanics: A & B cannot both be measured exactly if
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:
⇒ ΔxH (0) ⋅ ΔxH (t) ≥
t2M
Trade-off: both have the same Δx(0)= Δx(τ)= ΔxQ ⇒ Standard Quantum Limit
For duration τ this sets scale
ΔxQ ~ τ2M
Measurement 1: measure x at t = 0 strongly, Δx(0) << ΔxQ ⇒ strongly perturb p
Measurement 2: uncertainty of p enters further evolution of x ⇒ Δx(τ) >> ΔxQ
Example: measure GW pulse through change in mirror positions between t = 0 and τ
xΔxQ
t = 0xΔxQ
t = τ
“Heisenberg Microscope” treatmentno careful quantization of light
Tuesday, August 2, 2011
Origin of the Standard Quantum Limit• Quantum Mechanics: A & B cannot both be measured exactly if
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:
⇒ ΔxH (0) ⋅ ΔxH (t) ≥
t2M
Trade-off: both have the same Δx(0)= Δx(τ)= ΔxQ ⇒ Standard Quantum Limit
For duration τ this sets scale
ΔxQ ~ τ2M
Measurement 1: measure x at t = 0 strongly, Δx(0) << ΔxQ ⇒ strongly perturb p
Measurement 2: uncertainty of p enters further evolution of x ⇒ Δx(τ) >> ΔxQ
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
Example: measure GW pulse through change in mirror positions between t = 0 and τ
xΔxQ
t = 0xΔxQ
t = τ
“Heisenberg Microscope” treatmentno careful quantization of light
Tuesday, August 2, 2011
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
Tuesday, August 2, 2011
25
Standard Quantum Limit 10kg
Standard Quantum Limit 40kg
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
Tuesday, August 2, 2011
25
LIGO-III/Einstein Telescope must beat the SQL significantly in broad frequency band
Standard Quantum Limit 10kg
Standard Quantum Limit 40kg
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
Tuesday, August 2, 2011
• 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
Tuesday, August 2, 2011
• 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
carrierA cos ω0t
phasor diagram
Tuesday, August 2, 2011
• 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
phasequadrature
amplitudequadrature
E1
E2
carrierA cos ω0t
phasor diagram
Tuesday, August 2, 2011
• 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
phasequadrature
amplitudequadrature
E1
E2
carrierA cos ω0t
A co
s (ω 0t+
ϕ)
ϕ
anot
her c
arrie
r
phasor diagram
Tuesday, August 2, 2011
• 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
phasequadrature
amplitudequadrature
E1
E2
carrierA cos ω0t
E1
E2
A co
s (ω 0t+
ϕ)
ϕ
anot
her c
arrie
r
phasor diagram
Tuesday, August 2, 2011
• 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
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
Tuesday, August 2, 2011
• 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
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
Tuesday, August 2, 2011
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
Tuesday, August 2, 2011
• Heisenberg Uncertainty In the Frequency Domain
Quadratures, Homodyne Detection and Squeezing28
Sa1a1Sa2a2 − | Sa1a2 |2≥1
Minimum Uncertainty Gaussian States are:
vacuum statecoherent states
squeezed vacuuasqueezed states
Tuesday, August 2, 2011
• Heisenberg Uncertainty In the Frequency Domain
Quadratures, Homodyne Detection and Squeezing28
vacuum
E1
E2
Sa1a1Sa2a2 − | Sa1a2 |2≥1
Minimum Uncertainty Gaussian States are:
vacuum statecoherent states
squeezed vacuuasqueezed states
Tuesday, August 2, 2011
• Heisenberg Uncertainty In the Frequency Domain
Quadratures, Homodyne Detection and Squeezing28
coherent state
vacuum
E1
E2
Sa1a1Sa2a2 − | Sa1a2 |2≥1
Minimum Uncertainty Gaussian States are:
vacuum statecoherent states
squeezed vacuuasqueezed states
Tuesday, August 2, 2011
• Heisenberg Uncertainty In the Frequency Domain
Quadratures, Homodyne Detection and Squeezing28
coherent state
vacuum
E1
E2
Sa1a1Sa2a2 − | Sa1a2 |2≥1
Minimum Uncertainty Gaussian States are:
vacuum statecoherent states
squeezed vacuuasqueezed states
Tuesday, August 2, 2011
• Heisenberg Uncertainty In the Frequency Domain
Quadratures, Homodyne Detection and Squeezing28
coherent state
vacuum
E1
E2
Sa1a1Sa2a2 − | Sa1a2 |2≥1
Minimum Uncertainty Gaussian States are:
vacuum statecoherent states
squeezed vacuuasqueezed states
Tuesday, August 2, 2011
• Heisenberg Uncertainty In the Frequency Domain
Quadratures, Homodyne Detection and Squeezing28
coherent state
vacuum
E1
E2
Sa1a1Sa2a2 − | Sa1a2 |2≥1
Minimum Uncertainty Gaussian States are:
vacuum statecoherent states
squeezed vacuuasqueezed states
Tuesday, August 2, 2011
• Heisenberg Uncertainty In the Frequency Domain
Quadratures, Homodyne Detection and Squeezing28
coherent state
phasesqueezedvacuum
E1
E2
Sa1a1Sa2a2 − | Sa1a2 |2≥1
Minimum Uncertainty Gaussian States are:
vacuum statecoherent states
squeezed vacuuasqueezed states
Tuesday, August 2, 2011
• Heisenberg Uncertainty In the Frequency Domain
Quadratures, Homodyne Detection and Squeezing28
coherent state
phasesqueezed
amplitudesqueezed
vacuum
E1
E2
Sa1a1Sa2a2 − | Sa1a2 |2≥1
Minimum Uncertainty Gaussian States are:
vacuum statecoherent states
squeezed vacuuasqueezed states
Tuesday, August 2, 2011
• Heisenberg Uncertainty In the Frequency Domain
Quadratures, Homodyne Detection and Squeezing28
coherent state
phasesqueezed
amplitudesqueezed
amplitudesqueezed
vacuum
E1
E2
Sa1a1Sa2a2 − | Sa1a2 |2≥1
Minimum Uncertainty Gaussian States are:
vacuum statecoherent states
squeezed vacuuasqueezed states
Tuesday, August 2, 2011
• Heisenberg Uncertainty In the Frequency Domain
Quadratures, Homodyne Detection and Squeezing28
coherent state
phasesqueezed
amplitudesqueezed
amplitudesqueezed
vacuum
E1
E2
phasesqueezed
Sa1a1Sa2a2 − | Sa1a2 |2≥1
Minimum Uncertainty Gaussian States are:
vacuum statecoherent states
squeezed vacuuasqueezed states
Tuesday, August 2, 2011
Vacuum Fluctuation Point of View30
Carrier
InputTest Mass
End Test Mass
Arm CavityL ~ 4 km
Power RecyclingMirror
Carrier: Amplified by Resonance (Cavity & Recycling)
Tuesday, August 2, 2011
Vacuum Fluctuation Point of View30
Carrier
InputTest Mass
End Test Mass
Arm CavityL ~ 4 km
Power RecyclingMirror
Carrier: Amplified by Resonance (Cavity & Recycling)
Signal: anti-symmetric phase modulation, escape from “dark port”
anti-sym motion
anti-sym m
otion
Tuesday, August 2, 2011
Vacuum Fluctuation Point of View30
Carrier
InputTest Mass
End Test Mass
Arm CavityL ~ 4 km
Power RecyclingMirror
Carrier: Amplified by Resonance (Cavity & Recycling)
Signal: anti-symmetric phase modulation, escape from “dark port”
anti-sym motion
anti-sym m
otion
Vacuum Fluctuations of EM field enter oppositely from the detection port. Responsible for all quantum noises!
Tuesday, August 2, 2011
Vacuum Fluctuation Point of View30
Carrier
InputTest Mass
End Test Mass
Arm CavityL ~ 4 km
Power RecyclingMirror
Carrier: Amplified by Resonance (Cavity & Recycling)
Signal: anti-symmetric phase modulation, escape from “dark port”
anti-sym motion
anti-sym m
otion
Two Types of Noise:
Vacuum Fluctuations of EM field enter oppositely from the detection port. Responsible for all quantum noises!
Tuesday, August 2, 2011
Vacuum Fluctuation Point of View30
Carrier
InputTest Mass
End Test Mass
Arm CavityL ~ 4 km
Power RecyclingMirror
Carrier: Amplified by Resonance (Cavity & Recycling)
Signal: anti-symmetric phase modulation, escape from “dark port”
anti-sym motion
anti-sym m
otion
Two Types of Noise: shot noise
Shot Noisediscreteness of out-
going photons
Vacuum Fluctuations of EM field enter oppositely from the detection port. Responsible for all quantum noises!
Tuesday, August 2, 2011
Vacuum Fluctuation Point of View30
Carrier
InputTest Mass
End Test Mass
Arm CavityL ~ 4 km
Power RecyclingMirror
Carrier: Amplified by Resonance (Cavity & Recycling)
Signal: anti-symmetric phase modulation, escape from “dark port”
anti-sym motion
anti-sym m
otion
Two Types of Noise: shot noise
Shot Noisediscreteness of out-
going photons
Vacuum Fluctuations of EM field enter oppositely from the detection port. Responsible for all quantum noises!
and radiation-pressure noise
Radiation-Pressure Noiserandom radiation-pressure
force on mirrors
Tuesday, August 2, 2011
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
Tuesday, August 2, 2011
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
Tuesday, August 2, 2011
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
Tuesday, August 2, 2011
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
GW-induced
Back-Action-Induced
Tuesday, August 2, 2011
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
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.
Tuesday, August 2, 2011
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!!
Tuesday, August 2, 2011
Frequency Dependent Squeezing & Detection35
Standard Quantum Limit
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
• “Frequency dependent homodyne detection” plus input squeezing. [H.J. Kimble et al., 2001]
• Requires km-scale filters: losses destroy quantum correlations very easily!!
Tuesday, August 2, 2011
Frequency Dependent Squeezing & Detection35
Standard Quantum Limit
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% total loss
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!!
Tuesday, August 2, 2011
Frequency Dependent Squeezing & Detection35
Standard Quantum Limit
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% total loss
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!!
“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
Tuesday, August 2, 2011
Beating the SQL via “Back-Action Evasion”[Braginsky, Caves, Thorne, Vyatchanin 80s, ... , H.J. Kimble et al., 2001]
36
E1
E2
phasequadrature
amplitudequadrature
interferometer
E1
E2
mirr
or m
otio
n
injected vacuum
Tuesday, August 2, 2011
Beating the SQL via “Back-Action Evasion”[Braginsky, Caves, Thorne, Vyatchanin 80s, ... , H.J. Kimble et al., 2001]
36
E1
E2
phasequadrature
amplitudequadrature
interferometer
E1
E2
mirr
or m
otio
n
injected vacuum
GW-induced
Back-Action-Induced
Tuesday, August 2, 2011
Beating the SQL via “Back-Action Evasion”[Braginsky, Caves, Thorne, Vyatchanin 80s, ... , H.J. Kimble et al., 2001]
36
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.
Tuesday, August 2, 2011
Beating the SQL via “Back-Action Evasion”[Braginsky, Caves, Thorne, Vyatchanin 80s, ... , H.J. Kimble et al., 2001]
36
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.
Optical Losses can limit sensitivity!
Tuesday, August 2, 2011
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
Tuesday, August 2, 2011
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
Tuesday, August 2, 2011
Signal Recycling38
Carrierdi!erential motion
di!erential motion
Power RecyclingMirror
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
Tuesday, August 2, 2011
• 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
Tuesday, August 2, 2011
• 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
Tuesday, August 2, 2011
• 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
mechanical resonance
Tuesday, August 2, 2011
Optical Spring in “detuned cavities”40
X
FRAD(X)
restoringanti-restoring
FRAD does positive net work
FRAD does negative net work
anti-dampingdamping
Tuesday, August 2, 2011
Optical Spring in “detuned cavities”40
Positive Spring Constant + Delay ⇒ Negative Damping
X
FRAD(X)
restoringanti-restoring
FRAD does positive net work
FRAD does negative net work
anti-dampingdamping
Tuesday, August 2, 2011
Optical Spring in “detuned cavities”40
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)
restoringanti-restoring
FRAD does positive net work
FRAD does negative net work
anti-dampingdamping
Tuesday, August 2, 2011
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
Tuesday, August 2, 2011
• 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
Tuesday, August 2, 2011
Types of Dissipation in Bulk Material43
δT2 > δT1heatflow
δT1
thermoelastic dissipation
fast
slow
momentumflow
internal friction
Brownian Noise Thermoelastic Noise
Tuesday, August 2, 2011
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
Tuesday, August 2, 2011
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
Tuesday, August 2, 2011