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/[email protected]
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
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