Some Ideas About a Vacuum Squeezer

A.GiazottoINFN-Pisa

Some considerations on 2 very important quantum noises: Shot noise and Radiation pressure

1)Shot Noise: Uncertainty prin. ΔφΔN1

The phase of a coherent light beam fluctuates as

The phase produced by GW signal is

The measurability condition is i.e.

i.e shot noise decreases by increasing W 1/2F

2

1

4

cFL

LFhsig

~

~

~~ sigFL

c

FL

W

hh

2

1

4

~

FW

W LASER 2W FW

Wt

h

N

1

2)Radiation Pressure Noise

The photon number fluctuations create a fluctuating momentum

on the mirrors of the FP cavities :

The spectral force on the mirrors is:

For the measurability condition this force should be smaller than

Riemann force:

The measurability cond. for Shot noise and Radiation Pressure noise is:

Whc

F

h

W

c

hF

t

P

~

h

Wt

c

hFN

c

hFP

WhLcM

FhWh

c

FLhMFRiem

22 2

2

1

~~.

2

21

2162

2

222~

2

2

L

cFL

WF

hWFh

cLMh

The term WF2 produces dramatic effects on the sensitivity; minimizing h with respect to W we obtain the Standard Quantum limit:

216

2

F

cMW

MLh

1

LimitQuantumStandar

~

This limit can only be reduced either by increasing L and mirror masses M or by using a Squeezed Vacuum

Not to scale

Radiation Pressure

Shot Noise ĥ

BS pow.103 W

104 W

105 W

106 W

107 W

Hz 10 100 1000

Noise Budget

1 10 100 1000 1000010-2710-2610-2510-2410-2310-2210-2110-2010-1910-1810-1710-1610-1510-1410-1310-12

Virgo 28-3-2001http://www.virgo.infn.it/michele.punturo@pg.infn.it

Radiation Pressure Quantum Limit Wire Creep Absorption Asymmetry Acoustic Noise Magnetic Noise Distorsion by laser heating Coating phase reflectivity

h(f)

[1/s

qrt(H

z)]

Frequency [Hz]

Total Seismic Noise Newtonian (Cella-Cuoco) Thermal Noise (total) Thermal Noise (Pendulum) Thermal Noise (Mirror) Mirror thermoelastic noise Shot Noise

MLΩ

1h~

SQL

skskskk aah

,,, 2

1Η

In Quantum Electrodynamics it is shown that Electric field Ex and magnetic field Hy of a z-propagating em wave do not commute and satisfy the following commutation relation:

)(),(),,( 213

1021 rr

zitrBtrE yx

We may expand the vector potential and the Hamiltonian as a sum of creation and anihilation operators:

..),(,

)(,, cheatrA

sk

tkrisksk

Where s is the polarization From E&H commutation relations we obtain

jiqkjqsk aa ,,,, ,

In interferometric detectors, GW produce sidebands at frequency ω0±Ω, where 2π.10<Ω<2π.10000 is the acoustical

band; this process involves the emission of two correlated photons of frequency ω0±Ω: tie 0

ε cosΩt

200

00titi

titc

ti eec

ee)()()cos(

ω0-Ω ω0 ω0+Ω

Consequently it is appropriate to deal with two correlated photon processes. The positive frequency of the fluctuating electric field is:

AcC

deaeaCeE tititi

in0

0

2

20

)()(

Where a±=aω±Ω satisfy the following commutation relations:

)(,, '''

2aaaa

the prime means that a is evaluated at ω±Ω’.

The interaction of EM field Vacuum Fluctuation with a real systems:The quadrature formalism

Where

ttattaC

chd

eaeaeCEEE tititiininin

0200

0 20

sin)(cos)(

..)(

/

)()(

0222

0200

dti

dti

eata

eata

)(Re)(

)(Re)(

//

2

2

2

0

i

aaa

aaa

)(

)(

/

We can then describe the evolution of the emf vacuum fluctuation propagating in a optical system by means of the QUADRATURE STATE:

)(/

)(

2

0

a

a Intensity Fluct.

Phase Fluct.

If we include the carrier amplitude α, the total field is defined:

Where α=(2W/hν0)1/2 is the amplitude of

the carrier in a coherent state and W is the laser power entering the cavity.See Fig.1. We may rewrite Ein after

reflection on a mirror moving by δX (keeping first order terms in a0, π/2 only)

ttattaAc

E ininin 020002

sin)(cos))(( ,/,

From eq.1 it is evident the role of a0,in and a π/2,in in creating

radiation pressure and phase fluctuations respectively and also shows that reflection on a mirror moving by δX gives a phase shift contributing only to π/2 quadrature..

Eq. 1

t0 Fig.1

210 /a212 // a

Xat

aa

c

Xt

atE in

inininin

411

21 20

020

0,/

,,/, cos)(cos)()(

Fabry-Perot Cavity

L

(T1,R1,B) (T2,R2,B)

F+Q=

2

0

π/b

b

2

0

2

0

π/π/a

a

2

0

π/c

c

2

0

2

0

π/π/ G

G

P

P

2

0

2

0

π/π/ H

H

E

E

LossesVacuum Fluctuations

Classical Field δL1

δL2

Time Decomposition of amplitudes in small+large (P+G) components

c

δLc

LtδLL

tωπ/Gc

Ltπ/PR

c

δLc

LtδLL

tωGc

LtPR

tωtπ/aπ/αTtωtaαTtωπ/Gπ/PtωGP

12222

0sin22

2112222

0cos02

01

0sin2210cos0010sin220cos00

2

0

2

0

π/π/ Q

Q

F

F

Decomposition of amplitudes in small+large (P+G) components

c

L

cc

Lt

c

L

cc

LtG

c

LtPR

c

L

cc

Lt

c

L

cc

LtG

c

LtPR

ttaTttaTtGPtGP

c

Lt

c

LtG

c

LtPR

c

Lt

c

LtG

c

LtPR

ttaTttaTtGPtGP

Lc

LtL

c

LRR

TG

c

LGR

c

LGRTG

c

LGR

c

LGRTG

c

Lt

c

LtGR

c

Lt

c

LtGR

tTtTtGtG

c

Lc

LtLL

tGc

LtPR

c

Lc

LtLL

tGc

LtPR

ttaTttaTtGPtGP

20cos0

20sin0cos

20sin0

20cos0sin2/

22/1

20cos0

20sin0sin

20sin0

20cos0cos0

201

0sin2/10cos01sincos

20sin0cos

20cos0sin2/

22/1

20sin0sin

20cos0cos0

201

0sin2/10cos01sincos

1222

2011

12

0cos2/12

0sin012/12/

20sin2/1

20cos01010

20sin0cos

20cos0sin2/1

20sin0sin

20cos0cos01

0sin2/10cos010sin2/0cos0

12222

0sin2/2

2/112222

0cos02

01

0sin2/2/10cos0010sin2/2/0cos00

02/2/000

02/2/000

Large Components

L

(T1,R1,B) (T2,R2,B)

F+Q

2

0

π/b

b

2

0

2

0

π/π/a

a

2

0

π/c

c

2

0

2

0

π/π/ G

G

P

P

2

0

2

0

π/π/ H

H

E

E

Contributions to small components

Small Components

c

LRR

c

LReR

Tc

LRR

c

LeL

c

LReR

w

wB

a

aT

P

P

Gc

LRR

c

Lc

LtL

aTc

LReRP

c

L

cGR

c

L

cGR

c

LePR

c

LePRtaTtP

c

L

cGRc

L

cGRc

LePRc

LePRtaTtP

c

L

cGR

c

L

cGR

c

L

c

LtPR

c

L

c

LtPRtaTtP

c

L

cGR

c

L

cGR

c

L

c

LtPR

c

L

c

LtPRtaTtP

c

Lt

c

LtcGR

c

Lt

c

Lt

c

LtPR

c

Lt

c

Lt

cGR

c

Lt

c

Lt

c

LtPR

ttaTttaTttPttP

c

Li

π/

c

Li

c

Li

π/π/

π/

c

Li

c

Li

c

Li

c

Li

c

Li

2011

21

12

2122

21

2

2

2122

21

20sin

20cos001

20cos

20sin012/12/

2

0cos2

0sin001

2

0sin2/1

2

0cos01010

20sin

20cos001

20cos

220sin

2012/12/

20cos

20sin001

20sin

22/1

20cos

201010

20cos0cos

20sin0sin02/1

20sin0cos

20cos0sin

22/1

20cos0sin

20sin0cos001

20sin0sin

20cos0cos

201

0sin2/10cos010sin2/0cos0

0

2

1

2

0010

0

2

1

2

0

2

01

2

0

01010

2

1

02/1

2

2/1

2

02/1

22

02/12/1

02/1

Fout and Qout fields

The out Fields

Gc

LRR

π/Q

Q

π/

c

LRc

Li

eR

c

LR

T

c

LRR

Lc

Li

eL

cπ/w

w

T

B

π/a

a

c

LRc

Li

eR

c

LRc

Li

eT

π/a

aR

π/F

F

π/c

LR

c

LRc

Li

eR

c

LRc

i

eRT

c

LRR

Lc

Li

eL

cπ/w

wB

π/a

aT

c

LRc

Li

eR

c

LRc

i

e

π/E

E

c

LRR

π/T

c

LR

c

Lc

LtL

c

LRR

π/T

c

LRc

Li

eR

c

LRR

c

Lc

Li

eL

π/w

wB

π/a

aT

c

LRc

i

eπ/E

E

Gc

LR

c

Lc

LtL

Pc

LRc

i

eE

c

L

c

Lc

LtL

Gc

L

c

Lc

LtL

Gc

Lc

i

ePc

Lc

i

ePE

c

L

c

Lc

LtL

Gc

L

c

Lc

LtL

Gc

Lc

i

ePc

Lc

i

ePE

c

Lt

c

Lt

c

Lc

LtL

Gc

Lt

c

Lt

c

LtP

c

Lt

c

Ltc

Lc

LtL

Gc

Lt

c

Lt

c

LtPtEtE

201

2

0

2

0

20

2

11

2

202

12011

12

022

0

1

2

2

0

20

2

11

20

2

212

01

2

0

2

02

201

20

2

11

20

2

112

011

12022

022

01

20

2

11

20

2

2

0

2011

2

01

2

20

1202

2011

2

01

20

2

11

2

201

1202

2

022

012

0

2

2

0

2

20

120

20

2

20sin

12022/

20cos

12020

20cos

2

2/2

0sin

2

02/

20cos

12022/

20sin

12020

20sin

2

2/2

0cos

2

00

20cos0cos

20sin0sin

12

022/2

0sin0cos2

0cos0sin2

2/

20cos0sin

20sin0cos

12

02

02

0sin0sin2

0cos0cos2

00sin

2/0cos

0

L

(T1,R1,B) (T2,R2,B)

F+Q

2

0

π/b

b

2

0

2

0

π/π/a

a

2

0

π/c

c

2

0

2

0

π/π/ G

G

P

P

2

0

2

0

π/π/ H

H

E

E

Matrix Inversion

Rationalization of Fout and Qout fields

2

021

20cos121

20111

20

2

0

2

02

202

12

0cos121

2011

120

221

2

020

2

122

01

20

2

421

20cos

2

121

20

2

11

2

0

21

20cos121

2011

2011

1

421

20cos

2

121

20

2

11

2

20sin

2

1

2

20cos

2

11

20cos

2

112

0sin

2

1

20sin

2

12

0cos

2

11

20

2

11

1

1

π/Rc

LR

c

LRRR

c

LR

π/Q

Q

π/c

LR

Rc

LR

c

LRR

c

Lc

Li

eL

T

π/w

w

c

LRc

Li

eBTπ/a

aR

c

LRc

Li

e

c

Li

eRc

Lc

Li

eR

c

LRc

Li

eR

π/F

F

Rc

LR

c

LRR

c

LRR

c

Li

eRc

Lc

Li

eR

c

LRc

Li

eR

c

Lc

Li

eRc

Lc

Li

eR

c

Lc

Li

eRc

Lc

Li

eR

c

Lc

Li

eRc

Lc

Li

eR

c

LRc

Li

eR

cbad

ac

bd

dc

ba

Ponderomotive action

21

20cos121

0

220

20111

1202

2

0

12

2

0

421

20cos

21212

12

0cos121

20

211.2

011

2020

2

212

04

2/2/0020

22

04

20

2)(

412

0

22

02

02

2

021

20cos121

2011

20120

2

021

20cos121

2011

12

0

21

20cos121

2

02

20

20111

1202

2

0

12

2

04

21

20cos

2121

20

211

12

0

2/2/002

0)(

20sin2/2/0cos002

0)(

RcLR

π/cLRc

LRRRcLc

LieL

π/w

w

TB

π/a

a

cLi

eRcLc

LieRRc

LR

cLRc

LieRc

LRR

π/Mc

Th

GtPGtPMc

h

McicW

LcLi

eL

π/c

LR

π/c

LR

RcLR

cLRR

π/Tπ/GG

π/RcLR

cLRR

Tπ/G

G

RcLR

π/cLRc

LRRRcLc

LieL

π/w

w

TB

π/a

a

cLi

eRcLc

LieR

cLRc

LieR

Tπ/P

P

GtPGtPh

tic

W

tGtPtGtPh

tic

W

Mirror displacement due to Rad. Press. Fluctuations

L

(T1,R1,B) (T2,R2,B)

F+Q

2

0

π/b

b

2

0

2

0

π/π/a

a

2

0

π/c

c

2

0

2

0

π/π/ G

G

P

P

2

0

2

0

π/π/ H

H

E

E

Intracavity Power Fluctuations

Phase shifts due to:

2

02

011

12

0

2

0

0

220

21

20cos121

2011

421

20cos

2

121

20

2

11120

221

0

220

21

20cos121

2011

421

20cos

2

121

20

2

11

21

20cos121

0

220

20111

1202

2

0

1

2

2

0

421

20cos

2

12121

20cos121

20

2

11.2

011

2020

22

020

418

2

020

2

122

01

20

2

421

20cos

2

121

20

2

11

2

0

π/c

LRR

Rc

LR

π/Q

Q

π/c

LR

Rc

LR

c

LRR

c

Li

eRc

Lc

Li

eR

c

LRc

Li

eR

c

Lc

Li

eL

T

π/c

LR

Rc

LR

c

LRR

c

Li

eRc

Lc

Li

eR

c

LRc

Li

eR

Rc

LR

π/c

LR

c

LRRR

c

Lc

Li

eL

π/w

w

T

B

π/a

a

c

Li

eRc

Lc

Li

eRRc

LR

c

LRc

Li

eRc

LRR

π/Mc

h

T

π/w

w

c

LRc

Li

eBTπ/a

aR

c

LRc

Li

e

c

Li

eRc

Lc

Li

eR

c

LRc

Li

eR

π/F

F

Ext.+int. vacuums

Pondero-motive action

External Forces

Classical Field

Ponderomotive action at Resonance

2

0

2

0

0

22

11

11120

2

0

2

11

2002112

02

22001

222002

11

11

20

22

020

8

2

02

11

2

12

2

02

11

1

2

2

0

2

0

2

0

0

2

11

12

11

21

120

2

0

2

11

2002112

02

22001

222002

1111

.41

20

22

020

8

2

02

122

01

2

2

11

1

2

0

2

2

2

2

12

0cos

π/π/Q

Q

π/

c

Li

eR

R

c

Lc

Li

eL

π/R

π/π/Rc

Lc

Li

eL

π/π/wwT

Bπ/π/aa

c

Li

eR

R

Mc

h

π/w

w

c

Li

eR

c

Li

eBT

π/a

a

c

Li

eR

Rc

Li

e

π/F

F

π/π/Q

Q

π/R

c

Li

eR

T

c

Lc

Li

eL

π/R

π/π/Rc

Lc

Li

eL

π/π/wwT

Bπ/π/aa

c

Li

eRR

T

Mc

h

π/w

wc

Li

eBTπ/a

aRc

Li

e

c

Li

eRπ/F

F

c

L

Resonance condition

The Squeezed Out Vacuum State

Squeezing factor

Squeezed Out Vacuum State

1

012

0

20

4022

02

02

0

2

0

1220

222

2032

1

012

0

204

1

0002

022

2032

2

022

0

2

0

11

11

2

211

1

0

2

11

1112

0

20

2

1

00

11112

022

11

211

20

2208

2

02

11

2

12

2

02

11

1

2

2

0

2

0

2

0

0

0

2

11

11120

2

1

00

1

2022

11

211

20

22

2002

0

8

2

02

11

2

12

2

02

11

1

2

2

0

02

c

LcLi

eLhW

FwK

π/w

wFB

Kaπ/a

a

π/F

F

cL

Mc

WFK

Lc

Li

eLh

WFwFBa

Mc

WF

π/w

wFB

π/a

a

π/F

F

RFc

Li

eR

cLi

eR

R

c

LcLi

eLh

W

wRR

Ba

cLi

eR

R

Mc

W

π/w

w

cLi

eR

cLi

eBT

π/a

a

cLi

eR

RcLi

e

π/F

F

π/π/Q

Q

cLi

eR

R

c

LcLi

eL

wT

Ba

cLi

eR

R

Mc

h

π/w

w

cLi

eR

cLi

eBT

π/a

a

cLi

eR

RcLi

e

π/F

F

π/

FP Ponderomotive Cavity Output State

Squeezing Factor

Summary

02/2

02~02

0

2

0

122

02

22

2032

0

320~

12

wKπ/w

wFB

hLKaπ/a

a

π/F

F

cL

Mc

WFK

hW

cFhLLc

Li

eL

FP Ponderomotive Cavity Optical Matrix

1

0~

02/2

022

01

01

2

0 hLwKπ/w

wFB

π/a

a

Kπ/F

F

Why is K the squeezing factor?

Squeezed State

02

0

2

0KVV

V

outVoutV

//

Covariance Matrix 221det

21

1

KTrCC

KK

KC

The eigenvalues λ and the eigevectors V, in the large K approximation, are:

11

11

2

2

,

,

KVK

KVK

tg Φ=K The squeezing factor is then . For having V- parallel to Vπ/2out we must rotate the ellipse by an angle Φ.

K

1

V+

V-

Φ

Vπ/2out

V0 out

The relevant parameters for measuring squeezing are: 1) the eigenvalue is giving the size of the ellipse minor axis i.e. the inverse of the squeezing factor. 2) the eigenvectors V which gives the tangent of the rotation angle Φ needed to bring the ellipse minor axis parallel to π/2 quadrature

What happens if K1 K2

022

0

012

0UKU

Ut

VKV

Vr

//

If the squeezed quadrature state is the overlap of two states having squeezing factors K1 and K2 respectively:

The covariance matrix C is:

22

221

22222211

22

221

2122

12

22

121

KtKrTrCtrKKC

KtKrKtKr

KtKrC

det

The eigenvalues λ and the eigevectors V, in the large K approximation, are:

22

21

222

221222

2111

2/

0

1

22

2122

222

1

tKrK

tKrKtgtrKK

tKrKVtKrK

Eq.s 7

V+

V-

Φ

Vπ/2out

V0 out

Topology of the vacuum reflected by a resonant PC

022/

02

,0,2/

,0w

Kw

wFB

inKa

ina

ina

21221

2312

22t11

222

2t121

2/21

2/22114/2221

1

2/

0

1

22

21

4

222

FBKFB

FBK

tKK

tKKtg

FB

FBK

KtrK

tKrKV

trK

Our squeezed state is:

V+

V-

Φ

Vπ/2out

V0 out

Losses are then responsible for squeezing degrade; infact the squeezing factor in presence of losses becomes and in the limit becomes

FBK

FBK

22

2

1

1

122 FBK FB

1

Rotation in Quadrature Space

A Detuned FP cavity may produce rotations in Quadrature Space.

The rotation operator is:

K

hLπ/a

KaK

K

hLπ/a

hLπ/aK

Ka

hπ/aKK

AB

tg

K

KK

K

K

K

AKB

haKBA

KahLπ/aABa

KahLπ/aBAa

KahLπ/a

a

BA

AB

BA

π/

~2

01

~2

~2

10

~2

?11

1

11

1

21

1

1

~?1

220

~20

0~

20

0~

2

022 2

By operating it Out of Resonance (Ld =LRES+ΔL) on the Squeezed state we would like to obtain the optimally rotated state

0

~2

0Kahπ/a

a

haK

π/

~?1

2

V+

V-

Φ

Vπ/2out

V0 out

cdLi

edRcdLc

dLi

edR

dRcdL

RcdLi

ecdL

RcdLi

edR

422

0cos

2

21

20

22

0

2

1

22422

0cos

2

21

20cos21

24

12

0sin21

2

20sin21

22

0cos21

24

1

BA

AB

BA

cdLi

edRcdLc

dLi

edR

cdL

dRcdLi

ecdLi

edRcdL

dRcdLi

e

cdL

dRcdLi

ecdL

dRcdLi

ecdLi

edR

Detuning Evaluation

The condition B=-AK, in the approximation 2ΩLRES/c<<1, gives:

When the detuning is , the I.C. classical amplitude becomes half max.:

11~20

1~11

211

111120sin1

221

2221

4

221

4221

24

221

42421

26262

221

424210

21

62

10222

12

221

4

021

24

10021

2

11

111122

11221

2111

20sin

112

sin1

22

sin11

~20

~2

2cos11

2sin1

RcL

RK

R

RR

cLLimK

RKRe

RKRKeReeReeRKRReeKR

c

L

ReeRc

L

K

ReeR

c

LR

K

Re

hπ/aKahπ/ac

LReeRa

c

LRe

c

Li

c

Li

c

Li

c

Li

c

Li

c

Li

c

Li

c

Li

c

Li

c

Li

c

Li

c

Li

c

Li

c

Li

c

Li

c

Li

2

012

2~

2

021

20cos121

12

0

201

11

12

01 π/Rπ/R

c

LR

c

Lc

L

R

Tπ/G

G

11~2

0R

cL

0 1-R1cL2

0

G0

Frequency behavior and finesse of a Detuned cavity

It is remarkable that a detuned cavity produces rotations in the quadrature space having the same K frequency behavior i.e. 1/Ω2. Had the ratio B/A not the same 1/Ω2 functional character of K, a frequency independent optimal squeezing could not be obtained. Infact, by expanding B/A in series of 1/Ω2 we obtain:

1

22

2032

2114...2

31

12

2121

c

L

Mc

WFK

dRRcdL

cdL

dRdR

AB

d

M

W

c

dFL

cdL

dRMc

dRWF

dRdR

dF0

2

824

2

212032

1

11

1

From this equation it is possible to evaluate the detuned cavity finesse Fd.

Radiation Pressure and Shot noise in a Michelson Interferometer

Effects of the vacuum fluctuations entering the ports of the Beam Splitter of a FP Michelson interferometer

FP1 L1,R1,δl1,w1

W1,D1

BS FP2

W2, D2

L2,R2 ,δl2,w2

U

Laser W,ω0

2/

0

2/

0

bb

2/

0

2/v0v

W1,2 Power entering FP cavities L1,2 FP cavity length D1,2 FP-BS Length R1,2 FP entrance mir. Transmitt.

δl1,2 FP mirrors displacement b, v Injected Vacuum fluct. w1,, w2 Internal Vacuum fluct. α, β Classical Fields

With reference to the previous diagram we obtain for the output field U:

1

0~

2222

0,22/22/,2

0,222

222

2)0v0(2/v2/

0v0/202

21

1

0~

1112

0,12/12/,1

0,111

212

1)0v0(2/v2/

0v0/102

21

hLi

ewKw

wFB

ie

Kbb

bcDie

hLi

ewKw

wFB

ie

Kbb

bcDieU

)1/(1

/

JR

JF

cJ

FJ

LJ

22

2032

Mc

JFJW

JK

cJLieJR

cJLie

hJW

J /21

/

0

322

2,1J

FP cavities lossesFP Ponderomotive GW signal

When the two FP cavities are equal i.e: D=D1=D2 +nλ/2 L=L1=L2 +nλ/2 K=K1=K2 Γ= Γ1= Γ2

K=K1=K2 we obtain:

This Eq. clearly shows that the only contribution to the noise is given by the vacuum quadrature operators v0,π/2 entering the BS

from the dark fringe port; in this symmetric configuration there is no noise contribution from b0,π/2 .This is a fundamental result

showing that we may SQL by injecting in the BS black port a squeezed vacuum.

1

0~2

02/2/

022

2

0v2/v0v/02

21

hLiewKw

wFB

ieK

cDieU

Sum of two incoherent vacuums

If we inject a K squeezed state with carrier βeiφ coherent with laser carrier α0 we obtain (for the sake of simplicity we omit vacuums fields due to losses):

Initial StateVacuum and and Classical fieldentering BS Laser port.

00

2/b0b

Squeezed Vacuum and and Classical field entering BS Black port.

sin

cos

/2/v0v

K

K

hLKKbKb

KbcLFDie

hLKKbKb

KbcLFDie

cLFDie

cLFDie

cLFDie

cLFDie

U

~222)0v0(/2/v2/

0v0/22202

21

~111)0v0(/2/v2/

0v0/11102

21

/22202/11102sin

/22202cos0

/11102cos0

Let us inject this signal in a detuned cavity:

K

hLL

K

KKKK

K

KKb

KK

hLLK

KKKKKbK

hLLKKKbKKK

Kin

K

KU

~2211)21(

0v)21(

02/v

~2211

1210v)21(0

12/v

~22110v21021

12/v

,0v

11

11

In conclusion, if we inject in the BS black port a K squeezed state, for having a squeezing K the following inequalicies should hold:

KKKKK

KKK

1

1

21

21

)(

sin

cos~

22110v210211

2/v

0v

hLLKKKbKKK

KU

A Vacuum Squeezer conceptual Diagram

Ponderomotive 1

K1

ITF-Squeezer relative phase

alignment

Φ LASER

K2

Pon

der

omot

ive

2

(10-3, 1)

(10-3, 1)

Detuned

Recycling Mirror

Detuning

Phase ModulatorWD

2W

Polarization

Rotators

PC

ωM

L

L+Δ

Squeezed Vacuum output

Vπ/2out

V0out

All Locking amplifiers are low-pass filtered

10 Hz

Ponderomotive Cavities Equalization

ψ

PC

Detuning

(T2=60,R2=40)

Variable BS (t2=1-r2, r2)

Variable Attenuator (η)

Polarizer

Squeezed Vacuum

LASER

ITF-Squeezer relative phase

alignment

Arm 2 L +Δ

Arm 1 L

Squeezed vac. carrier ampl.

ωM

Φ

K2

Ponderomotive 1

K1

ωM

ωM

Detuned

Ponderomotive 2

ωM

PC

ωM

(T2=1- σ2,σ2=10-6)

PolarizationRotators

1

0~

2222

0,22/22/,2

0,222

222

2)0v0(2/v2/

0v0/202

21

1

0~

1112

0,12/12/,1

0,111

212

1)0v0(2/v2/

0v0/102

21

hLi

ewKw

wFB

ie

Kbb

bcDie

hLi

ewKw

wFB

ie

Kbb

bcDieU