Some Ideas About a Vacuum Squeezer

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• Some Ideas About a Vacuum Squeezer
• A.Giazotto
• INFN-Pisa

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Some considerations on 2 very important
quantum noises Shot noise and
Radiation pressure 1)Shot Noise
Uncertainty prin. ?f?N?1 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
FW W
LASER 2W FW
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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
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The term WF2 produces dramatic effects on the
sensitivity minimizing h with respect to W we
obtain the Standard Quantum limit

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

We may expand the vector potential and the
Hamiltonian as a sum of creation and anihilation
operators
Where s is the polarization
From EH commutation
relations we obtain
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In interferometric detectors, GW produce
sidebands at frequency ?0O, where
2p.10ltOlt2p.10000 is the acoustical band this
process involves the emission of two correlated
photons of frequency ?0O

e cosOt
?0-O ?0 ?0O
Consequently it is appropriate to deal with two
correlated photon processes. The positive
frequency of the fluctuating electric field is
Where aa?O satisfy the following commutation
relations
the prime means that a is evaluated at ?O.
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The interaction of EM field Vacuum Fluctuation
with a real systemsThe quadrature
formalism
Where
 
We can then describe the evolution of the emf
vacuum fluctuation propagating in a optical
system by means of the QUADRATURE STATE
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If we include the carrier amplitude a, the total
field is defined 

Where a(2W/h?0)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 dX (keeping first order terms in a0, p/2
only)  
Eq. 1
From eq.1 it is evident the role of a0,in and a
p/2,in in creating radiation pressure and phase
fluctuations respectively and also shows that
reflection on a mirror moving by dX gives a phase
shift contributing only to p/2 quadrature..
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Fabry-Perot Cavity
Vacuum Fluctuations
Classical Field
Losses
dL2
dL1
FQ
L
(T2,R2,B)
(T1,R1,B)
Time Decomposition of amplitudes in smalllarge
(PG) components
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Rationalization of Fout and Qout
fields

Matrix
Inversion
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Ponderomotive action



Intracavity Power Fluctuations
Mirror displacement due to Rad. Press.
Fluctuations
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Phase shifts due
to
Ext.int. vacuums Pondero-motive
action External Forces Classical Field
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Summary
Squeezing Factor
FP Ponderomotive Cavity Output State
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What happens if K1? K2

If the squeezed quadrature state is the overlap
of two states having squeezing factors K1 and K2
respectively
The covariance matrix C is
Eq.s 7
The eigenvalues ? and the eigevectors V, in the
large K approximation, are
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Rotation in
Quadrature Space A Detuned FP cavity may produce
rotations in Quadrature Space. The rotation
operator is
By operating it Out of Resonance (Ld LRES?L)
on the Squeezed state we
would like to obtain the optimally rotated
state

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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/O2.
Had the ratio B/A not the same 1/O2 functional
character of K, a frequency independent optimal
squeezing could not be obtained. Infact, by
expanding B/A in series of 1/O2 we
obtain
From this equation it is possible to evaluate the
detuned cavity finesse Fd.
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With reference to the previous diagram we obtain
for the output field U

FP cavities losses
FP Ponderomotive
GW signal
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