nonclassical interferometry - geo600schnabel/vorl_ss06.pdf · nonclassical interferometry i....

Nonclassical InterferometrySlide presentation accompanying the

Lecture by Roman SchnabelSummer Semester 2006

Universität HannoverInstitut für Gravitationsphysik,

Max-Planck Institut für Gravitationsphysik (Albert Einstein Institut)

Callinstr. 38, D-30167 Hannover

[email protected]/personal/schnabel.html

Nonclassical InterferometryI. Introduction

1. Gravitational Wave Detection2. Classical Interferometry

II. Interferometer Quantum Noise: Basics and Tools3. Quantum Noise in Laser Interferometry4. Quantum Noise Spectral Densities5. Interferometer Readout: Homodyne versus Heterodyne Detection6. Quantum Noise of Sideband Modulation Fields7. Quadrature Input-Output Relations

III. Schemes for Nonclassical (Quantum-Non-Demolition) Interferometers8. Simple Michelson Interferometer with Squeezed Light Input9. “Variational Output” (Frequency dependent homodyne detection)

10. The Optical Spring Interferometer11. A Squeezed Light Upgraded GEO600 Detector? 12. The Speed Meter Idea and Sagnac Interferometer13. Optical Bars

3

I. Introduction 1. Gravitational Wave Detection

TopicsWhat are gravitational waves?Detectors on ground: GEO600, LIGO, VIRGO, TAMADetectors in space: LISAInterferometer quantum noise

Literature• P. Aufmuth and K. Danzmann, Gravitational wave detectors,

New Journal of Physics 7, 202 (2005), http://www.njp.org

• N. Robertson, Laser interferometric gravitational wave detectors, Classical and Quantum Gravity 17, R19 (2000).

4

Gravitational Waves

• GWs are the dynamical part of gravitation

• They carry hugh energies but hardly interact with anything

• They are ideal information carriers, almost no scattering or dissipation

• The whole universe is filled with GWs and has been transparent for them shortly after the big bang

5

Sources of Gravitational Waves

Supernovae

Binary systems

Big BangInflation

Accreting neutron stars Colliding supermassiveBlack Holes in Galaxies

Dark matter ?

NS

BH

6

The Gravitational Wave SpectrumSupernova inthe Milky Way

7

Laser

Photo-Diode

Beam splitter

Mirror

Mirror

⇒ Long arms

⇒ High laser power

Michelson Interferometer

Interferometric GW Detectors

VacuumnoiseVacuium noise needs

To be squeezed!

8

I. Introduction 2. Classical Interferometry

TopicsWaves and interferenceInterferometer topologiesThe beam splitterClassical description of the interferometerThe Michelson-Morley experiment

Literature• V. B. Braginski and F. Y. Khalili, Cambridge University Press (1995),

Quantum measurement. • P. R. Saulson, World Scientific (1994),

Fundamentals of Interferometric Gravitational Wave Detectors , 91 US$

9

∇2 v

E (v r ,t) −1c

∂2

∂t2

v E (v r ,t) = 0

I. 1. Classical Interferometry

Waves

Wave equation:

Solution for the electric field vector:

v E ω (r,t) = E0

ω α(v r ,t) e−iωt + α *(v r ,t) eiωt[ ] v p (v r ,t)

α(v r ,t)With dimensionless complex amplitude:

10


Waves

),( trvαDimensionless complex amplitude:

Monochromatic plane waves: ikzeztr −== 0)(),( ααα v

Gaussian beam, TEM00-mode:

( )222

0

2

0

0 ,2

exp),( yxizz

ikizz

tr +=⎟⎟⎠

⎞⎜⎜⎝

⎛+

−+

= ρραα v

ck 0, ω

=

11

0),(1),( 2

22 =

∂∂

−∇ trEtc

trE vvvv


WavesWave equation:

Equivalent solution for the electric field vector:

v E ω (r,t) = E0

ω X1(v r ,t) cosωt + X2(

v r ,t) sinωt[ ] v p (v r ,t)

X1(v r ,t) =α(v r ,t) + α*(v r ,t)

X2(v r ,t) = −i α(v r ,t) −α*(v r ,t)[ ]

α(v r ,t) =X1(

v r ,t) + iX2(v r ,t)2

Real and imaginary parts of the complex amplitude (real valued quadrature amplitudes):

12


Waves

Phasor diagram:

ℜ(α)

ℑ(α)

lα

mαml+α

φ

13


Waves

Phasor diagram:

lα

mα

ℜ(α)

ℑ(α)

Ω Sideband frequency modes

More dimensions to be added:Polarization modes (2),Spatial modes (inf.)

14

Homework 1Mach-Zehnder interferometer with 50% / 50% beam splitterHow depends output power from the differential phase in the two arms?

50/50

50/50inα

Iα

IIα1α

2αφΔ

?)(

?)(

2/12/1,

2/12/12/12/1

,0

2*22

1*11

2

1

2

1

=Δ==

=Δ==

−==

−=+=

⇒

⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−

=⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛−

=⎟⎟⎠

⎞⎜⎜⎝

⎛

Δ

Δ

Δ

φαα

φαα

αααα

αααααα

αα

ρττρ

αα

αρττρ

αα

φ

φ

φ

fn

fn

ee

e

inII

inIi

III

iIII

iII

I

inII

I

15


The Michelson-Morley experiment

16


The Michelson-Morley Experiment

17


The Michelson-Morley Experiment

18

II. Interferometer Quantum Noise: Basics and Tools 3. Quantum Noise in Laser Interferometry

TopicsCoherent statesQuantum model of the beam splitter Quantum model of the interferometerShot noise

Literature• C. M. Caves, Phys. Rev. D 23, 1693 (1981),

Quantum-mechanical noise in an interferometer.• C. M. Caves and B. L. Schumaker, Phys. Rev. A 31, 3068 (1985),

New formalism for two-photon quantum optics.• V. B. Braginski and F. Y. Khalili, Cambridge University Press (1995),

Quantum measurement.

19

I. 3. Quantum Noise in Laser Interferometry

The electric field vector at a fixed position for a certain polarization may be written in the following form:

Quantization of the Electromagnetic Field

ˆ E (t) = E0,k(ωk) ˆ X 1,k(t)cos(ωkt) + ˆ X 2,k(t)sin(ωkt)[ ]k∑

QuadraturesQuadrature amplitudes,(Amplitude quadrature amplitude,Phase quadrature amplitude)

or

ˆ E (t) = E0,k(ωk) ˆ a k(t)e−iωkt + ˆ a k†(t)eiωkt[ ]

k∑

20


The electric field vector at a fixed position for a certain polarization may be written in the following form:

ˆ E (t) = E0,k(ωk) ˆ a k(t)e−iωkt + ˆ a k†(t)eiωkt[ ]

k∑

Quantization of the Electromagnetic Field

ˆ a =ˆ X 1 + i ˆ X 2

2

ˆ a † =ˆ X 1 − i ˆ X 2

2

ˆ a , ˆ a †[ ]=1

ℜ( ˆ a ) ≡ ˆ X 1 /2 ˆ X 1 = ˆ X 1† = ˆ a + ˆ a †

ℑ( ˆ a ) ≡ ˆ X 2 /2 ˆ X 2 = ˆ X 2† = −i( ˆ a − ˆ a †)

ˆ X 1, ˆ X 2[ ]= 2i ⇒ Δ ˆ X 1Δ ˆ X 2 ≥1

annihilation op. in time domain:

creation operator in time domain:

21

Coherent state

)ˆ(aℜ

)ˆ(aℑ

α


α = ˆ a = α ˆ a α

Δ ˆ X 1 = Δ ˆ X 2 =1

Δ ˆ X 1

(Complex amplitude)

(Symmetric minimum uncertainty)

Phasor of the quantized field with Gaussian noise distribution(“ball on the stick”)

22

Coherent state

)ˆ(aℜ

)ˆ(aℑ

α

E

t


α = ˆ a = α ˆ a α

Δ ˆ X 1 = Δ ˆ X 2 =1

Δ ˆ X 1

(Complex amplitude)

(Symmetric minimum uncertainty)

23

Phase squeezed state


t

)ˆ(aℜ

)ˆ(aℑ

α

24

Amplitude squeezed state


t

)ˆ(aℜ

)ˆ(aℑ

α

25


State Representations of the Quantized Field

Fock States (Number States)

ˆ H = hωkk

∑ ˆ a k† ˆ a k +

12

⎛ ⎝ ⎜

⎞ ⎠ ⎟ = hωk

k∑

ˆ X 14

+ˆ X 24

⎛

⎝ ⎜

⎞

⎠ ⎟

ˆ a † ˆ a n = ˆ n n = n n

ˆ a n = n n −1

ˆ a † n = n +1 n +1 n =ˆ a †( )n

n!0

Hamiltonian of the electromagnetic field

26


Coherent States

ˆ a α =α α

α ˆ a † = α α*

α = e− α 2 / 2 αn

n!∑ n ⇒ P(n) = n α

2=

α2n

n!e− α 2

α = ˆ D (α) 0 = exp(αˆ a † −α* ˆ a ) 0

ˆ a † ˆ a = ˆ n = α ˆ n α =α*α = α 2 = Δ2 ˆ a † ˆ a ( )= ˆ a † ˆ a ( )2− ˆ a † ˆ a

2

Eigenvalue equation (complex)

Displacement operator

Poissonian distribution

Photon number expectation value Photon number variance

27

Photon Statistics – A Simple Experiment


1 2 3 t(ms)

Signal

Single Photon Detector

28

Photon Statistics – A Simple Experiment


Poissonian distribution, of expectation value k=4

k

enkknP k

n

=

= −

2

!),(

σ

29


ina

vaca

2c

1 c

1b

2b

Homework 3: Interferometer phasor picture

30

Quantum Model of the InterferometerMichelson interferometer with 50% / 50% beam splitter

ˆ c 1ˆ c 2

⎛

⎝ ⎜

⎞

⎠ ⎟ =

12

1 11 −1

⎛

⎝ ⎜

⎞

⎠ ⎟

ˆ b 1ˆ b 2

⎛

⎝ ⎜

⎞

⎠ ⎟ =

12

1 11 −1

⎛

⎝ ⎜

⎞

⎠ ⎟

eiφ / 2 00 e−iφ / 2

⎛

⎝ ⎜

⎞

⎠ ⎟

12

1 11 −1

⎛

⎝ ⎜

⎞

⎠ ⎟

ˆ a 1ˆ a 2

⎛

⎝ ⎜

⎞

⎠ ⎟

ˆ c 1ˆ c 2

⎛

⎝ ⎜

⎞

⎠ ⎟ =

12

eiφ / 2 e−iφ / 2

eiφ / 2 −e−iφ / 2

⎛

⎝ ⎜

⎞

⎠ ⎟

1 11 −1

⎛

⎝ ⎜

⎞

⎠ ⎟

ˆ a 1ˆ a 2

⎛

⎝ ⎜

⎞

⎠ ⎟ =

12

eiφ / 2 + e−iφ / 2 eiφ / 2 − e−iφ / 2

eiφ / 2 − e−iφ / 2 eiφ / 2 + e−iφ / 2

⎛

⎝ ⎜

⎞

⎠ ⎟

ˆ a 1ˆ a 2

⎛

⎝ ⎜

⎞

⎠ ⎟

ˆ c 1ˆ c 2

⎛

⎝ ⎜

⎞

⎠ ⎟ =

cos φ /2( ) sin φ /2( )sin φ /2( ) cos φ /2( )

⎛

⎝ ⎜

⎞

⎠ ⎟

ˆ a 1ˆ a 2

⎛

⎝ ⎜

⎞

⎠ ⎟

ˆ n 1 = ˆ c 1† ˆ c 1 = cos φ /2( )ˆ a 1

† + sin φ /2( )ˆ a 2†( )cos φ /2( )ˆ a 1 + sin φ /2( )ˆ a 2( )

= cos2 φ /2( )ˆ a 1† ˆ a 1 + sin2 φ /2( )ˆ a 2

† ˆ a 2 + cos φ /2( )ˆ a 1† sin φ /2( )ˆ a 2 + sin φ /2( )ˆ a 2

† cos φ /2( )ˆ a 1

ˆ n 1 = ˆ c 1† ˆ c 1 = α 2 cos2 Δφ

2⎛ ⎝ ⎜

⎞ ⎠ ⎟

31

n1 = c1†c1 = α 2 cos2 φ /2( ),

Δn12 = Δ c1

†c1( )2= α 2 cos2 φ /2( ),

n2 = c2†c2 = α2 sin2 φ /2( ),

Δn22 = Δ c2

†c2( )2= α 2 sin2 φ /2( ),

n12 = c1†c1 − c2

†c2 = α2 cos φ( ),

Δn122 = Δ c1

†c1 − c2†c2( )2

= α 2.


Shot Noise

Expectation value and variance at detector 2

Expectation value and variance for a differential detector 1-2


32


Shot Noise

SNR =1=dn2

Δn22

=sin φ /2( )cos φ /2( )α 2 dφ

α sin φ /2( )⇒ dφ =

1α cos φ /2( )

φ2

= 0 ⇒ dφ =1α

=1

ntotal

, Δφsn =1

ntotal

SNR =1=dn12

Δn122

=−sin φ( )α 2 dφ

α⇒ dφ =

1α sinφ

φ =π2

⇒ dφ =1α

=1

ntotal

, Δφsn =1

ntotal

Shot noise (coherent state noise) for a single detector at dark port

Shot noise for a differential detector at two half fringe ports

33

II. Interferometer Quantum Noise: Basics and Tools 4. Quantum Noise Spectral Densities

TopicsNoise spectral densities:

Shot noiseRadiation pressure noiseStandard Quantum Limit (SQL)

GEO 600

Literature• H. A. Haus, Electromagnetic Noise and Quantum Optical Measurement,

Springer, Berlin (2000)• C. Kittel, Elementary Statistical Physics, Wiley, New York (1958),• F. Reif, Fundamentals of Statistical Physics, McGraw-Hill, New York (1965),• H-A. Bachor and T. Ralph, Guide to Experiments in Quantum Optics,

Springer, Berlin (2003),• P. R. Saulson, World Scientific (1994),

Fundamentals of Interferometric Gravitational Wave Detectors.

34

I. 4. Quantum Noise Spectral Densities

Radiation Pressure Noise

SNR =1=dz

dzrp

=dz

dvmirrorτ=

dz mdPmirrorτ

, τ =

dz =dφ /2

ωc , dPmirror =

2hωc

12

dntotal

⇒ dφ =2hω2

mc2 τ dntotal , Δφrp =2hω2

mc2 τ Δntotal

Δφrp =2hω2

mc2 τ ntotal ∝ ntotal

Minimum phase signal in case of RPN and RPN standard deviation

time interval ofmeasurement

Coherent states

35


Shot Noise R-P Noise

τω

τω

τωφ

PLch

Pcz

P

sn

sn

sn

2

2

2

4

h

h

h

=Δ

=Δ

=Δ

Δφrp =4hω3τ 3P

m2c4

Δzrp =hωτ 3Pm2c2

Δhrp =4hωτ 3PL2m2c2

36


Standard-Quantum-Limit (SQL)

Δzmea ⋅ Δppert ≥h

2

Δzadd = Δppertτ /m ≥hτ

2mΔzmea

≡κ

Δzmea

Δzmea2 +

κ 2

Δzmea2

⎛

⎝ ⎜

⎞

⎠ ⎟

min

⇒ Δzmeaopt = κ =

hτ2m

= Δzaddopt

ΔzSQL = Δzmeaopt 2

+ Δzmeaopt 2

=hτm

Standard Quantum limit for a measurement of position(Heisenberg-Microscope approach [Braginski])

Perturbation on momentum due to positionmeasurement on mass m

Additional noise on positiondue to backaction after τ

Minimum of sum assuming no correlations

SQL

37


SQL

.22

,2

,42

2

mLz

Lh

mcz

mc

SQLSQL

SQLSQL

SQL

τ

τφω

τωφ

h

h

h

=Δ=Δ

=Δ=Δ

=Δ

Standard Quantum limit for a measurement of

Phase

Position

Strain induced by gravitational waves

38

The „single-sided spectral density“ integrated over all positive frequencies is equivalent to the full band variance of an infinitely long lasting measurement as considered before:

For a band limited measurement of duration τ :


Spectral Density

QuickTime™ and a YUV420 codec decompressor are needed to see this picture.

hdffSh Δ=∫∞

0

)(

( )fHzf

fhfSh Δ≡⎥⎦

⎤⎢⎣

⎡Δ

Δ=1,11,)( ττ

Gravitational waves will show a characteristic spectrum and the variance of the measured strain

Δ2h(t) will depend on frequency.

39


Spectral Density

( )( )

( ) ( ) hhhhhC

dththhthT

C

dtthT

h

hh

n

T

TTn

T

TT

22___________

2

2/

2/

2/

2/

)0(

)()(1lim)(

)(1lim

Δ≡−=−=⇒

−+−≡

≡

=

∫

∫

+

−∞→

+

−∞→

ττ

Time average:

Correlation function:

A proper definition of the spectral density requires a stationary random process that fulfills the ergodic hypothesis

40


Quantum Noise Spectral DensitiesSingle-sided linear noise spectral densities:

( )

( )2,

422,

2

,

22)(

22)(

42)(

fmLfS

fcmP

LfS

Pc

LfS

SQLh

RPNh

SNh

π

πω

ω

h

h

h

=

=

=

ττ

ττω

τω

12

12

14

2

2

22

4

2

mLh

cmP

Lh

Pc

Lh

SQL

RPN

SN

h

h

h

=Δ

⇒=Δ

=Δ

(White noise spectrum)

(Random walk spectrum)

(„Flicker“ noise spectrum)

41


Power-Recyclingmirror

Laser

Photo diode

600 m

600 m

1064 nm

Mirror masses:m=5.6 kg

Effective length:L=1200 m

10 kW

42

Quantum Noise of a Conventional MI

Shot noise


43

Quantum noise in phase quadrature


Standard quantum limit

(SQL)

Quantum noisewith increased laser

power (x100)

Shot noise

Radiation pressure noise


44

II. Interferometer Quantum Noise: Basics and Tools 5. Interferometer Readout:

Homodyne versus Heterodyne Detection

TopicsBalanced homodyne detection Homodyne detection in gravitational wave interferometersModulation/demodulation technique: heterodyne detectionBalanced sideband heterodyne detectionUnbalanced sideband heterodyne detectionQuantum noise in homodyne and heterodyne detection

Literature• A. Buonanno, Y. Chen, and N. Mavalvala, Phys. Rev. D 67, 122005 (2003),

Quantum noise in laser-interferometer gravitational-wave detectors with a heterodyne readout scheme.

45

(Squeezed) signal beamIntense local

oscillator

Phase shift θ

Electric current

V(ˆ i −) ≅ αLO2 ⋅ δ ˆ X 1 cosθ +δ ˆ X 2 sinθ

2= αLO

2 ⋅ δ ˆ X θ2 ≡ αLO

2 ⋅ Δ2 ˆ X θ

Balanced Homodyne Detection

Interf

ering

50/50

beam

splitte

r

II. 5. Homodyne versus heterodyne detection

ab

NaEa

46

BHDII. 5. Homodyne versus heterodyne detection

ˆ a Nˆ a E

⎛

⎝ ⎜

⎞

⎠ ⎟ =

12

1 11 −1

⎛

⎝ ⎜

⎞

⎠ ⎟

ˆ a eiθ

ˆ b

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟ =

12

ˆ a eiθ + ˆ b ˆ a eiθ − ˆ b

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

ˆ n N = ˆ a N† ˆ a N =

12

ˆ a †e−iθ + ˆ b †( ) ˆ a eiθ + ˆ b ( )=12

ˆ a † ˆ a † + ˆ a † ˆ b e−iθ + ˆ b † ˆ a eiθ + ˆ b † ˆ b ( )ˆ n E = ˆ a E

† ˆ a E =12

ˆ a †e−iθ − ˆ b †( ) ˆ a eiθ − ˆ b ( )=12

ˆ a † ˆ a † − ˆ a † ˆ b e−iθ − ˆ b † ˆ a eiθ + ˆ b † ˆ b ( )

ˆ n − = ˆ n N − ˆ n E ====

ˆ a † ˆ b e−iθ + ˆ b † ˆ a eiθ

α* + δˆ a †( ) β + δ ˆ b ( )e−iθ + β* + δ ˆ b †( ) b + α + δˆ a ( )eiθ

α*β + α*δ ˆ b + δˆ a †β + δˆ a †δ ˆ b ( )e−iθ + β*α + β*δˆ a + δ ˆ b †α + δ ˆ b †δˆ a ( )eiθ

2cosθ αβ + α ˆ X θb + β ˆ X θ

a + δˆ a †δ ˆ b e−iθ + δ ˆ b †δˆ a eiθ

ˆ X θ = ˆ X 1 cosθ + ˆ X 2 sinθ ≡ δˆ a e−iθ + δˆ a †eiθ

V ˆ n −( )≡ Δ2 ˆ n − = 2cosθ αβ + α ˆ X θb + β ˆ X θ

a + δˆ a †δ ˆ b e−iθ + δ ˆ b †δˆ a eiθ( )2− 2cosθ αβ( )2

Δ2 ˆ n − ≈ α 2Δ2 ˆ X θb ← α >> β , ˆ X θ

b,δˆ a

47


Quantum Noise / LinearizationCoherent state

E

t

ˆ X 1

ˆ X 2

aa ˆˆ δα +=

δ ˆ X 1 = δˆ a + δˆ a †

δ ˆ X 2 = −i δˆ a −δˆ a †( )

δˆ a ,δˆ a †[ ]=1

ˆ a † ˆ a = ˆ n = α 2

Δˆ n 2 = ˆ n 2 − ˆ n 2 = α 2

Quadrature noise operators

Commutator

Photon number expectation value

Photon number variance

δ ˆ X 12 δ ˆ X 2

2 ≥1 Heisenberg Uncertainty Relation

48


ina

vaca

2c

1 c

1b

2b

49

n1 = c1†c1 = α 2 cos2 φ /2( ),

Δn12 = Δ c1

†c1( )2= α 2 cos2 φ /2( ),

n2 = c2†c2 = α2 sin2 φ /2( ),

Δn22 = Δ c2

†c2( )2= α 2 sin2 φ /2( ),

n12 = c1†c1 − c2

†c2 = α2 cos φ( ),

Δn122 = Δ c1

†c1 − c2†c2( )2

= α 2.


Shot Noise


Expectation value and variance for a differential detector 1-2


50


GW interferometer with heterodyne readout

51


Conventional interferometer with balanced heterodyne and phase quadrature homodyne readout

52

II. Interferometer Quantum Noise: Basics and Tools 6. Quantum Noise of Sideband Modulation Fields

TopicsSideband modulations and quadrature fields„One-photon“ versus „Two-photon“ quantum opticsQuadrature noise operators in frequency domainCoherent and squeezed noisePhasor diagram in frequency domain

Literature• C. M. Caves, Phys. Rev. Lett. 31, 3068 (1985),

New Formalism for two-photon quantum optics.• H. J. Kimble, Y. Levin, A. B. Matsko, K. S. Thorne, and S. P. Vyatchanin,

Phys. Rev. D 65, 022002 (2001), Conversion of conventional gravitational-wave interferometers into quantum nondemolition interferometers by modifying their input and/or output optics.

53

II. 6. Quantum Noise of Sideband Fields

Quantum Optics in Frequency SpaceTime-dependent electric field operator:

ˆ E (t) = E0ˆ X 1(t)cos(ωt) + ˆ X 2(t)sin(ωt)[ ]

ˆ E (t) = E0ˆ X 1(t) +δ ˆ X 1(t)( )cos(ωt) +δ ˆ X 2(t)sin(ωt)[ ]

ˆ E (t) = E0ˆ X 1(t) + ˆ a 1(Ω)e−iΩt + ˆ a 1

†(Ω)eiΩt[ ]dΩ2π0

∞

∫⎧ ⎨ ⎩

⎫ ⎬ ⎭

cos(ωt) + ˆ a 2(Ω)e−iΩt + ˆ a 2†(Ω)eiΩt[ ]dΩ

2π0

∞

∫⎧ ⎨ ⎩

⎫ ⎬ ⎭

sin(ωt)⎡

⎣ ⎢

⎤

⎦ ⎥

ˆ a 1(Ω)e−iΩt + ˆ a 1†(Ω)eiΩt[ ]dΩ

2πf −Δf

f +Δf

∫ ≡ ˆ q 1 t, f ,Δf( )


2πf −Δf

f +Δf

∫ ≡ ˆ q 2 t, f ,Δf( )

⎫

⎬ ⎪ ⎪

⎭ ⎪ ⎪

ˆ q 1, ˆ q 2[ ]= ?

54


„Two-Photon“-Formalism

„One-Photon“-formalism approach:

„Two-Photon“-formalism approach:

ˆ E (t) = E0 ˆ a ωe−iωt + ˆ a ω† eiωt[ ] dω

2π0

∞

∫

ˆ E (t) = E0 e−iωt ˆ a +e−iΩt + ˆ a ωe+iΩt[ ] dΩ

2π0

∞

∫ + eiωt ˆ a +†e+iΩt + ˆ a −

†e−iΩt[ ] dΩ2π0

∞

∫⎧ ⎨ ⎩

⎫ ⎬ ⎭

ˆ a ± = ˆ a ω ±Ωω ± Ω

ω

Modulation frequencies

Modulation frequencies

55


„Two-Photon“-Formalismˆ a 1 = 1

2ˆ a + + ˆ a −

†( )= ω + Ω2ω

ˆ a ω +Ω + ω − Ω2ω

ˆ a ω−Ω†

ˆ a 2 = −i 12

ˆ a + − ˆ a −†( )

⎛

⎝ ⎜

⎞

⎠ ⎟ = −i ω + Ω

2ωˆ a ω +Ω −

ω − Ω2ω

ˆ a ω−Ω†

⎛

⎝ ⎜

⎞

⎠ ⎟

⇒ ˆ a + =12

ˆ a 1 + iˆ a 2( ) ˆ a − =12

ˆ a 1† + iˆ a 2

†( )

ˆ E (t) = E0 e−iωt ˆ a +e−iΩt + ˆ a −e

+iΩt[ ]dΩ2π0

∞

∫ + eiωt ˆ a +†e+iΩt + ˆ a −

†e−iΩt[ ]dΩ2π0

∞

∫⎧ ⎨ ⎩

⎫ ⎬ ⎭

ˆ E (t) = E0

e−iωt + eiωt( )2

ˆ a 1e−iΩt + ˆ a 1†e+iΩt[ ]dΩ

2π0

∞

∫ +i e−iωt − eiωt( )

2ˆ a 2e

−iΩt + ˆ a 2†e+iΩt[ ]dΩ

2π0

∞

∫⎧ ⎨ ⎪

⎩ ⎪

⎫ ⎬ ⎪

⎭ ⎪

ˆ E (t) = E0 2 cosωt ˆ a 1e−iΩt + ˆ a 1†e+iΩt[ ]dΩ

2π0

∞

∫ + sinωt ˆ a 2e−iΩt + ˆ a 2

†e+iΩt[ ]dΩ2π0

∞

∫⎧ ⎨ ⎩

⎫ ⎬ ⎭

56


„Two-Photon“-Formalism

ˆ a , ˆ a †[ ]=1

ˆ a 1 , ˆ a 1†[ ]= ˆ a 2, ˆ a 2

†[ ]=Ωω

ˆ a 1 , ˆ a 2[ ]= 0

ˆ a 1 , ˆ a 2†[ ]= ˆ a 1

†, ˆ a 2[ ]= i

ˆ q 1 , ˆ q 2[ ]∝ ˆ a 1 + ˆ a 1†( ), ˆ a 2 + ˆ a 2

†( )[ ]= ˆ a 1 , ˆ a 2[ ]+ ˆ a 1 , ˆ a 2†[ ]+ ˆ a 1

†, ˆ a 2[ ]+ ˆ a 1†, ˆ a 2

†[ ]= 2i

ˆ X 1 , ˆ X 2[ ]= 2i

(Usual discrete commutation relation of mode operators)

⇒

Compare with:

(Quadrature phase amplitudes are not hermitian)

57

δˆ a (ω) δˆ a ( ′ ω ) = 0

δˆ a = ˆ a − ˆ a

δˆ a (ω + Ω) δˆ a (ω − Ω) ≠ 0

Two-Photon DevicesII. 6. Quantum Noise of Sideband Fields

For independently excited modes:

For Two-Photon devices:

Optical Parametric AmplifierOptical Parametric Amplifier

58

Phasors in frequency spaceΩ

X1(Ω)

E 0

ω+Ω0

ω−Ω0

X2 (Ω)


59

II. Interferometer Quantum Noise: Basics and Tools 7. Quadrature Input-Output Relations

TopicsModular input-output-relations for optical devices:

- free propagation- beam splitters- cavities- interferometers

Literature• M. J. Collet and C. W. Gardiner, Phys. Rev. A 30, 1386 (1984),• Jan Harms, Diploma Thesis, Hannover University 2002,

Quantum Noise in the Laser-Interferometer Gravitational-Wave Detector GEO600

60

II. 7. Quadrature Input-Output Relations


Laser

Photo diode

v a ≡ˆ a 1 Ω( )ˆ a 2 Ω( )

⎛

⎝ ⎜

⎞

⎠ ⎟

v b ≡

ˆ b 1 Ω( )ˆ b 2 Ω( )

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

61

Quantum noise in phase quadrature



(SQL)

Quantum noisewith increased laser

power (x100)

Shot noise

Radiation pressure noise

II. 4. Quantum Noise Spectral Densities

62


LIGO Hanford LIGO Livingston

63



Laser

Photo diode

v a ≡ˆ a 1 Ω( )ˆ a 2 Ω( )

⎛

⎝ ⎜

⎞

⎠ ⎟

v b ≡

ˆ b 1 Ω( )ˆ b 2 Ω( )

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

64


65


Signal-recycling mirror


Laser

Photo diode

600 m

600 m

v a ≡ˆ a 1 Ω( )ˆ a 2 Ω( )

⎛

⎝ ⎜

⎞

⎠ ⎟

v b ≡

ˆ b 1 Ω( )ˆ b 2 Ω( )

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

66

GEO600 Quantum Noise (Design Values)II. 7. Quadrature Input-Output Relations

Phasequadrature

AmplitudeAmplitudequadraturequadrature

SQL

67

Q-I-O Relation (Notation)



2πf −Δf

f +Δf

∫ ≡ ˆ q 1 t, f ,Δf( )


2πf −Δf

f +Δf

∫ ≡ ˆ q 2 t, f ,Δf( )

ˆ a 1 =12

ˆ a + + ˆ a −†( )=

ω + Ω2ω

ˆ a ω +Ω +ω − Ω

2ωˆ a ω−Ω

†

ˆ a 2 = −i 12

ˆ a + − ˆ a −†( )

⎛

⎝ ⎜

⎞

⎠ ⎟ = −i ω + Ω

2ωˆ a ω +Ω −

ω − Ω2ω

ˆ a ω−Ω†

⎛

⎝ ⎜

⎞

⎠ ⎟

v a ≡ˆ a 1ˆ a 2

⎛

⎝ ⎜

⎞

⎠ ⎟ =

12

1 1−i i

⎛

⎝ ⎜

⎞

⎠ ⎟

ˆ a +ˆ a −

†

⎛

⎝ ⎜

⎞

⎠ ⎟ Quadrature amplitude vector:

68

Q-I-O Relation: Propagation/Radiation Pressure


v b ≡

ˆ b 1ˆ b 2

⎛

⎝ ⎜

⎞

⎠ ⎟ = eiΩl / c cos ω0l /c( ) −sin ω0l /c( )

sin ω0l /c( ) cos ω0l /c( )⎛

⎝ ⎜

⎞

⎠ ⎟

ˆ a 1ˆ a 2

⎛

⎝ ⎜

⎞

⎠ ⎟

=t P l ω0,Ω( )

ˆ a 1ˆ a 2

⎛

⎝ ⎜

⎞

⎠ ⎟

ˆ a 1ˆ a 2

⎛

⎝ ⎜

⎞

⎠ ⎟

ˆ b 1ˆ b 2

⎛

⎝ ⎜

⎞

⎠ ⎟

l

ˆ a 1ˆ a 2

⎛

⎝ ⎜

⎞

⎠ ⎟

ˆ b 1ˆ b 2

⎛

⎝ ⎜

⎞

⎠ ⎟

x

v b ≡

ˆ b 1ˆ b 2

⎛

⎝ ⎜

⎞

⎠ ⎟ =

t P 2x ω0,Ω( )

ˆ a 1ˆ a 2

⎛

⎝ ⎜

⎞

⎠ ⎟ = eiΩl / c cos ω02x /c( ) −sin ω02x /c( )

sin ω02x /c( ) cos ω02x /c( )⎛

⎝ ⎜

⎞

⎠ ⎟

ˆ a 1ˆ a 2

⎛

⎝ ⎜

⎞

⎠ ⎟

≈1 −ω02x /c

ω02x /c 1⎛

⎝ ⎜

⎞

⎠ ⎟

ˆ a 1ˆ a 2

⎛

⎝ ⎜

⎞

⎠ ⎟ ≡

t C x(P,m,Ω),ω0( )

ˆ a 1ˆ a 2

⎛

⎝ ⎜

⎞

⎠ ⎟

69

Q-I-O Relation: Beam Splitters


ˆ b 1ˆ c 1

⎛

⎝ ⎜

⎞

⎠ ⎟ =

ρ ττ −ρ

⎛

⎝ ⎜

⎞

⎠ ⎟

ˆ a 1ˆ d 1

⎛

⎝ ⎜

⎞

⎠ ⎟ ,

ˆ b 2ˆ c 2

⎛

⎝ ⎜

⎞

⎠ ⎟ =

ρ ττ −ρ

⎛

⎝ ⎜

⎞

⎠ ⎟

ˆ a 2ˆ d 2

⎛

⎝ ⎜

⎞

⎠ ⎟

⇒ˆ b 1ˆ b 2

⎛

⎝ ⎜

⎞

⎠ ⎟ = ρ

1 00 1

⎛

⎝ ⎜

⎞

⎠ ⎟

ˆ a 1ˆ a 2

⎛

⎝ ⎜

⎞

⎠ ⎟ + τ

1 00 1

⎛

⎝ ⎜

⎞

⎠ ⎟

ˆ d 1ˆ d 2

⎛

⎝ ⎜

⎞

⎠ ⎟ ,

ˆ c 1ˆ c 2

⎛

⎝ ⎜

⎞

⎠ ⎟ = τ

1 00 1

⎛

⎝ ⎜

⎞

⎠ ⎟

ˆ a 1ˆ a 2

⎛

⎝ ⎜

⎞

⎠ ⎟ − ρ

1 00 1

⎛

⎝ ⎜

⎞

⎠ ⎟

ˆ d 1ˆ d 2

⎛

⎝ ⎜

⎞

⎠ ⎟

ˆ a 1ˆ a 2

⎛

⎝ ⎜

⎞

⎠ ⎟

ˆ b 1ˆ b 2

⎛

⎝ ⎜

⎞

⎠ ⎟

+ −

ˆ c 1ˆ c 2

⎛

⎝ ⎜

⎞

⎠ ⎟

ˆ d 1ˆ d 2

⎛

⎝ ⎜

⎞

⎠ ⎟

⇒ˆ c 1ˆ c 2

⎛

⎝ ⎜

⎞

⎠ ⎟ =

1τ

1 00 1

⎛

⎝ ⎜

⎞

⎠ ⎟

ˆ a 1ˆ a 2

⎛

⎝ ⎜

⎞

⎠ ⎟ m

ρτ

1 00 1

⎛

⎝ ⎜

⎞

⎠ ⎟

ˆ b 1ˆ b 2

⎛

⎝ ⎜

⎞

⎠ ⎟ ,

ˆ d 1ˆ d 2

⎛

⎝ ⎜

⎞

⎠ ⎟ = m

ρτ

1 00 1

⎛

⎝ ⎜

⎞

⎠ ⎟

ˆ a 1ˆ a 2

⎛

⎝ ⎜

⎞

⎠ ⎟ +

1τ

1 00 1

⎛

⎝ ⎜

⎞

⎠ ⎟

ˆ b 1ˆ b 2

⎛

⎝ ⎜

⎞

⎠ ⎟

!

70

III. Schemes for Nonclassical Interferometers 8. Simple MI with Squeezed Light Input

TopicsQuadrature transfer function of a simple Michelson interferometer,

normalized to ist noise spectral densityA quantum noise MATLAB code Squeezed light input for a simple Michelson interferometerBeating the Standard Quantum Limit

Literature• H. J. Kimble, Y. Levin, A. B. Matsko, K. S. Thorne, and S. P. Vyatchanin,

Phys. Rev. D 65, 022002 (2001), Conversion of conventional gravitational-wave interferometers into quantum nondemolition interferometers by modifying their input and/or output optics

71

III. 8. Simple MI with Squeezed Light Input


Laser

Photo diode

1064 nm


Arm length:L=1200 m

200 kW

v a N ≡αLO

0⎛

⎝ ⎜

⎞

⎠ ⎟ +

v a '=αLO

0⎛

⎝ ⎜

⎞

⎠ ⎟ + 1/2v a

Ecv

Ncv

v a ≡ˆ a 1 Ω( )ˆ a 2 Ω( )

⎛

⎝ ⎜

⎞

⎠ ⎟

v b ≡

ˆ b 1 Ω( )ˆ b 2 Ω( )

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

v a E ≡αLO

0⎛

⎝ ⎜

⎞

⎠ ⎟ −

v a '

=αLO

0⎛

⎝ ⎜

⎞

⎠ ⎟ − 1/2v a

72

III. 8. Simple MI with Squeezed Light InputSimple Michelson

Interferometer without squeezed

field input

73



Laser

Photo diode

Squeezed vacuum state Faraday rotator

v a ≡ˆ a 1 Ω( )ˆ a 2 Ω( )

⎛

⎝ ⎜

⎞

⎠ ⎟

v b ≡

ˆ b 1 Ω( )ˆ b 2 Ω( )

⎛

⎝ ⎜ ⎜

⎞

⎠ ⎟ ⎟

74


Squeezed States

α+,α−,r,ϕ = ˆ S (r,ϕ) ˆ D (α+,α−) 0

ˆ S = exp r ˆ a + ˆ a −e−2iϕ − ˆ a +

† ˆ a −†e2iϕ( )[ ] , ˆ S † ˆ S =1 , ˆ S †(−r,ϕ) = ˆ S (r,ϕ)

cosh r( )=12

er + e−r( ) , sinh r( )=12

er −e−r( )

⇒ˆ S †(r,ϕ) ˆ a ± ˆ S (r,ϕ) = ˆ a ± cosh r( )− ˆ a ±

†e2iϕ sinh r( )ˆ S †(r,ϕ) ˆ a ±

† ˆ S (r,ϕ) = ˆ a ±† cosh r( )− ˆ a ±e

−2iϕ sinh r( )

ˆ a θ coh= α ˆ a θ α → ˆ a θ sqz

= α ˆ S † ˆ a θ ˆ S α

Squeezing operator (Degree and angle of squeezing)

Two-mode displacement operator

75

III. 8. Simple MI with Squeezed Light InputSimple Michelson

Interferometer with squeezed field

input (fixed orientation of

squeezing ellipse)

76


Total quantum noise, squeezed, r=1lower envelope


(SQL)

Shot noise

Radiation pressure

noise

Total quantum noise

Frequency Dependent Squeezed Light Input

77

III. Schemes for Nonclassical Interferometers 9. Variational Output

TopicsVariational output (frequency dependent homodyne detection)QND interferometry without injected squeezed states„Self-squeezing“: Ponderomotively squeezed lightRadiation pressure Kerr nonlinearityProposal for an experimental realization„Squeezed input – variational output“

Literature• H. J. Kimble, Y. Levin, A. B. Matsko, K. S. Thorne, and S. P. Vyatchanin,

Phys. Rev. D 65, 022002 (2001), Conversion of conventional gravitational-wave interferometers into quantum nondemolition interferometers by modifying their input and/or output optics

• R. Loudon, Phys. Rev. Lett., 47 , 815 (1981)Quantum Limit on the Michelson Interferometer used for Gravitational-Wave Detection

78

(Squeezed) signal beamIntense local

oscillator

Phase shift θ

Electric current

V(ˆ i −) ≅ αLO2 ⋅ δ ˆ X 1 cosθ +δ ˆ X 2 sinθ

2= αLO

2 ⋅ δ ˆ X θ2 ≡ αLO

2 ⋅ Δ2 ˆ X θ

Balanced Homodyne Detection

Interf

ering

50/50

beam

splitte

r


ab

NaEa

79

v a =ˆ X 1

a (Ω)ˆ X 2

a (Ω)

⎛

⎝ ⎜

⎞

⎠ ⎟

v b =

ˆ X 1b(Ω)

ˆ X 2b(Ω)

⎛

⎝ ⎜

⎞

⎠ ⎟

III. 9. Variational output


Laser

Photo diode

1064 nm


Arm length:L=1200 m

200 kW

Nav

Eav

Ecv

Ncv

80

Variational Homodyne Detection

SQL

116

π 316

π 516

π8

16π


81

Photodiode

Filter cavities

Frequency Dependent Homodyne Detection

Variational output interferometer. Two filter cavities are needed at the output. [Kimble et al., Phys. Rev. D 65, 022002 (2001)]


82

Faraday rotator

SHG

OPA

Photodiode

Filter cavities

“Squeezed Input“ / „Variational Output“

Squeezed input / variational output interferometer. Two filter cavities are needed at the output. [Kimble et al., Phys. Rev. D 65, 022002 (2001)]


83

Noise reduction by squeezed light, (-6 dB in variance)

“Squeezed Input“ / „Variational Output“III. 9. Variational output

Variationaloutput

84

III. Schemes for Nonclassical Interferometers 10. The Optical Spring Interferometer

Topics(Detuned) signal recycling: GEO600A quantum noise MATLAB code

for GEO600, a signal recycled optical spring interferometerBeating the SQL by the optical spring effect

Literature• A. Buonanno and Y. Chen, Class. Quantum Gravity 18, L95 (2001),

Optical noise correlations and beating the standard quantum limit in advanced gravitational-wave detectors.

• B. S. Sheard, M. B. Gray, C. M. Mow-Lowry, D. E. McClelland, and S. E. Whitcomb, Phys. Rev. A 69, R 051801 (2004),Observation and characterization of an optical spring.

85

III. 10. The optical spring interferometer

Signal-recyclingmirror


Laser

Photo diode

600 m

600 m

1064 nm


Effective length:L=1200 m

SR Power reflectivity: r2=0.99Detuning: Φ=0.0055 rad

10 kW

II. 9. The optical spring interferometer

East arm 600 m

North arm 600 m

600 m vacuum tube

87


88

Position Control

Magnetcoils

Electrostatic drive


89



Laser

Photo diode

Folded arm

Δx1

Δx2


90


Laser

v a

v b

v o

v i

v b = eiΩL / c 1 0

−k 1⎛

⎝ ⎜

⎞

⎠ ⎟ v a + 2k 1

hSQL

01

⎛

⎝ ⎜

⎞

⎠ ⎟

⎡

⎣ ⎢

⎤

⎦ ⎥

v b '≡

t R v a ' + v r

t P SR = eiΩx /c cos ω0x /c( ) −sin ω0x /c( )

sin ω0x /c( ) cos ω0x /c( )⎛

⎝ ⎜

⎞

⎠ ⎟

ˆ o 1ˆ o 2

⎛

⎝ ⎜

⎞

⎠ ⎟ = +

1τ

1 00 1

⎛

⎝ ⎜

⎞

⎠ ⎟

ˆ b 1'

ˆ b 2'

⎛

⎝ ⎜

⎞

⎠ ⎟ −

ρτ

1 00 1

⎛

⎝ ⎜

⎞

⎠ ⎟

ˆ a 1'

ˆ a 2'

⎛

⎝ ⎜

⎞

⎠ ⎟

ˆ i 1ˆ i 2

⎛

⎝ ⎜

⎞

⎠ ⎟ = −

ρτ

1 00 1

⎛

⎝ ⎜

⎞

⎠ ⎟

ˆ b 1'

ˆ b 2'

⎛

⎝ ⎜

⎞

⎠ ⎟ +

1τ

1 00 1

⎛

⎝ ⎜

⎞

⎠ ⎟

ˆ a 1'

ˆ a 2'

⎛

⎝ ⎜

⎞

⎠ ⎟

v a '

v b '

Interferometer


Propagation/Detuning

x

91


92

GEO600 Quantum Noise (Design Values)III. 10. The optical spring interferometer

Phasequadrature


(Without signal recycling)

SQL

93

III. Schemes for Nonclassical Interferometers 11. A Squeezed Light Upgraded GEO600 Detector?

TopicsOptical Spring Interferometer with

- squeezed input- variational output

Other noise sources than quantum in GEO600: Thermal, seismic noiseTable-top experimentsProposal for a squeezed light upgraded GEO600 detector

Literature• J. Harms, Y. Chen, S. Chelkowski, A. Franzen, H. Vahlbruch, K. Danzmann,

and R. Schnabel, Phys. Rev. D 68, 042001 (2003),Squeezed-input, optical-spring, signal-recycled gravitational-wave detectors.

• R. Schnabel, J. Harms, K. Strain, and K. Danzmann, Class. Quantum Grav. 21, S1045 (2003),Squeezed light for the interferometric detection of high-frequency gravitational waves.

94


Total quantum noise, squeezed


(SQL)

Shot noise

Radiation pressure

noise

Total quantum noise

Frequency Dependent Squeezed Light Input

95

Variational Homodyne Detection

SQL

116

π 316

π 516

π8

16π


96

Noise reduction by squeezed light, (-6 dB in variance)

“Squeezed Input“ / „Variational Output“III. 9. Variational output

Variationaloutput

97


Phasequadrature



SQL

98


Phasequadrature



SQL

[J. Harms et al., PRD (2003)]

-6 dB

99

Squeezed Light Input

Laser

Faraday Rotator

Photo diode

SHG

OPA

Filter cavities

Squeezed vacuuminput

III. 11. A Squeezed Light upgraded GEO600

100

Optical Spring SR Interferometers – Var.OutputIII. 11. A Squeezed Light upgraded GEO600

101

Optical Spring SR Interferometers – Var.OutputIII. 11. A Squeezed Light upgraded GEO600

Thermal noise Thermal noise ((internal internal substrate substrate + + coatingcoating))

102

Thermal noise

Thermal noise

SQL

Quantum and Thermal NoiseIII. 11. A Squeezed Light upgraded GEO600

Amplitude q., GEO 600 (expected)

Opto-mechanical resonance

Optical resonance

103

SQL

Squeezed Quantum Noise (-6 dB)

Thermal noise

Thermal noise


SQL

Amplitude q., GEO 600 (expected)

Opto-mechanical resonance

Optical resonance

squeezed vacuum

104

Phase squeezing

Squeezing at 45°

Phase squeezing

Amplitude squeezing

Phase squeezing

Squeezing at 45° Squeezing at -45°

Amplitude squeezing

Phase squeezing

Squeezing at 45° Frequency dependent Squeezing from a single filter cavity

Squeezed Quantum Noise (-6 dB)III. 11. A Squeezed Light upgraded GEO600

105

Amplitude quadratureϕ = 0°

ϕ = 0°

Amplitude quadratureϕ = 0°

ϕ = 0°ϕ = 45°

Detection of amplitude quadrature

ϕ = 0°

ϕ = 0°ϕ = 45°

ϕ = 90°

Variational Output?III. 11. A Squeezed Light upgraded GEO600

106

A Squeezed Light upgraded GEO600?


• Single filter cavity

• Squeezed light source

• Losses

• Proof of principle

107

Filter cavity

Laser

Faraday Rotator

Diode

SHG

OPA

Filter cavity locked to SR opposite sideband

SR mirror

1 W total power, 1 mW quantum noise limited > 100 Hz

Mode cleaner?


108


109


Pendulum

110

Squeezed Light Generation (OPA)Squeezed beam

Squeezed beam

Coherent Coherent beambeam


111

OPA layout

7%-MgO:LiNbO3Half-monolithic (hemilithic) crystal6.5mm x 2.5mm x 2.5 mm

Radius of curvature: 8 mm HR>99.99% at 1064 nm

Flat surfaceAR at 1064 nm and 532 nm

Output coupler: R=94%-96% at 1064 nm

Finesse ~ 150Waist ~ 30 μmFSR ~ 3.7 GHzγ ~ 25 MHz

A. Franzen

Squeezed Light Generation (OPA)III. 11. A Squeezed Light upgraded GEO600

112

Squeezed Light Generation (OPA)III. 11. A Squeezed Light upgraded GEO600

113

Losses on Squeezed Fields

Example 210 dB squeezing at => V1=0.1 (-10dB)source. V2=10 (+10dB)Assume efficiency:

η=0.83 => V1‘=0.25 (-6.0 dB)V2‘=8.5 (+ 9.3 dB)

Quadrature variances normalized to unity vacuum noise!

x [dB]=10log10V1

V1‘= ηV1 + (1-η)V2‘= ηV2 + (1-η)

Example 18 dB squeezing at => V1=0.16 (-8dB)source. V2=6.3 (+8dB)Assume efficiencies:Mode matching: 0.9Rotator: 0.9Detection: 0.9Isolator: 0.9Residual: 0.9

=> η= 0.6 => V1‘=0.5 (-3.0 dB)V2‘=4.2 (+6.2 dB)


114

LO

Homodyne detector

SHG

squeezed

OPA DetunedFiltercavity

SR

Power-recycling (PR)

squeezed

Squeezing Enhanced PR-SR Interferometer

Homodyne detector

SHG

squeezed

OPA10 MHz detuned filter cavity

SR

Power-recycling (PR)

squeezed

LO


115

Squeezing Enhanced PR-SR Interferometer

Shot noise

Broadband squeezing enhanced signals

Signal transfer of detuned signal recycling

Individual signals


116

A Squeezed Light upgraded GEO600?


2008 !

117

III. Schemes for Nonclassical Interferometers 12. Speed Meter Idea and Sagnac Interferometer

TopicsMeasurement backactionQuantum-Non-Demolition (QND) measurements and variablesSpeed meterQND-Sagnac-gravitational wave interferometer

Literature• V. B. Braginsky and F. Ya. Khalili, Rev. Mod. Phys. 68, 1 (1996),

Quantum nondemolition measurements: the route from toys to tools.• Y. Chen, Phys. Rev. D 67, 122004 (2003),

Sagnac interferometer as a speed-meter-type, quantum-nondemolition gravitational-wave detector.

• P. Purdue and Y. Chen, Phys. Rev. D 66, 122004 (2002), Practical speed meter designs for quantum nondemolition gravitational-wave interferometers.

118




2


2mΔzmea

≡κ

Δzmea

Δzmea2 +

κ 2

Δzmea2

⎛

⎝ ⎜

⎞

⎠ ⎟

min


hτ2m

= Δzaddopt


+ Δzmeaopt 2

=hτm





SQL

119




2


2mΔzmea

≡κ

Δzmea

Δzmea2 +

κ 2

Δzmea2

⎛

⎝ ⎜

⎞

⎠ ⎟

min


hτ2m

= Δzaddopt


+ Δzmeaopt 2

=hτm





SQL

120


Shot Noise

SNR =1=dn2

Δn22

=sin φ /2( )cos φ /2( )α 2 dφ

α sin φ /2( )⇒ dφ =

1α cos φ /2( )

φ2

= 0 ⇒ dφ =1α

=1

ntotal

, Δφsn =1

ntotal

SNR =1=dn12

Δn122

=−sin φ( )α 2 dφ

α⇒ dφ =

1α sinφ

φ =π2

⇒ dφ =1α

=1

ntotal

, Δφsn =1

ntotal

Shot noise (coherent state noise) for a single detector at dark port

Shot noise for a differential detector at two half fringe ports

121

III. 12. Speed Meter/Sagnac Interferometer

As in a Michelson interferometer bright and dark port are decoupled and can berecycled independently. Laser noise does not couple to the dark port.

122

III. 12. Speed Meter/Sagnac Interferometer

123

III. Schemes for Nonclassical Interferometers 13. Optical Bars

TopicsOptical bar topology of a gravitational wave interferometerWeakly coupled oscillatorsThree-mirror-cavity QND and speed meter property of optical bar / leverHow does optical bar relate to the signal-recycling topology?

Literature• V. B. Braginsky, M. L. Gorodetsky and F. Ya. Khalili, Phys. Lett. A 232, 340 (1997)

Optical bars in gravitational wave antennas.• P. Purdue, Phys. Rev. D 66, 122001 (2002),

Analysis of a quantum nondemolition speed-meter interferometer.• V. B. Braginsky and F. Ya. Khalili, Rev. Mod. Phys. 68, 1 (1996),

Quantum nondemolition measurements: the route from toys to tools.

124

GW Detection - The Beginning

Resonant bar antenna, Joe Weber, 1960 t=0

t=T/4

t=T/2

t=3T/4

t=T

125

1990

1995

M ~ 2000 kg, L ~ 3 mf ~ 900 Hz, Δf ~ 1 Hz

h ~ 4 × 10–19

Resonant Bar Detectors

1991

ALLEGROBaton Rouge, (USA)

EXPLORERGeneva, CERN, INFN

NAUTILUSFrascati, INFN (Italy)

1993NIOBEPerth, UWA (Australia)

1997AURIGALegnaro, INFN (Italy)

126

AURIGA Legnaro, INFN (Italy)

aluminum bar: length 3 m, diameter 60 cm, mass 2.3 tons,the mechanical quality factor is about 4x106 at 100 mK

Electro mechanical transducer

127

III. 13. Optical Bars

Coupled Oscillators

ϕ ≈ sinϕ

128


Coupled Oscillators

R

FR

DD

JD

mlmgl

lg 210120 +==== ωωω

~ torsional momentdue to spring

~ torsional momentdue to gravity

ϕ ≈ sinϕ ω1

ω0

129


Coupled Oscillators

R

FR

DD

JD

mlmgl

lg 210120 +==== ωωω

~ torsional momentdue to spring

~ torsional momentdue to gravity

ω1

ω0ω pend =

ω1 + ω0

2

ωbeat =ω1 −ω0

2

130


Coupled Cavities / Three-Mirror Cavities

α t

0⎛

⎝ ⎜

⎞

⎠ ⎟ =

t M 3

t L 2

t M 2

t L 1

t M 1

α i

α r

⎛

⎝ ⎜

⎞

⎠ ⎟

α t

0M3 M2 M1

L2 L1 α i

α r

α: complex amplitudes

131



t M 3

t L 2

t M 2

t L 1

t M 1 =

−iτ1τ 2τ 3

1 −ρ3

ρ3 −1⎛

⎝ ⎜

⎞

⎠ ⎟

e− ikL2 00 eikL2

⎛

⎝ ⎜

⎞

⎠ ⎟

×1 −ρ2

ρ2 −1⎛

⎝ ⎜

⎞

⎠ ⎟

e− ikL1 00 eikL1

⎛

⎝ ⎜

⎞

⎠ ⎟

1 −ρ1

ρ1 −1⎛

⎝ ⎜

⎞

⎠ ⎟

α t

0M3 M2 M1

L2 L1 α i

α r

132



τ 3M φ1,φ2( )=α t φ1,φ2( )

iα i

=τ1τ 2τ 3

ρ3e− iφ2 ρ2e

iφ1 − ρ1e−iφ1( )− eiφ2 eiφ1 − ρ1ρ2e

− iφ1( )

φ1 = kL1 =ωL1

c, φ2 = kL2 =

ωL2

c

α t

0M3 M2 M1

L2 L1 α i

α r

133


α2 (Power)

(Detuning Cavity 1)

Det

unin

g C

avity

2Coupled Cavities / Three-Mirror Cavities

A. Thüring

134



A. Thüring

Power (α2) in first cavity Power (α2) in second cavity

Power sloshing for anti-symmetric phase change

stable

unstable

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