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SQUEEZING
Field quadratures
€
εx z,t( ) =ε0 sinkzsin ωt + φ( )=ε0 sinkz cosφ sinωt + sinφ cosωt( )=ε1 sinωt +ε2 cosωtε1 =ε0 sinkx cosφ; ε2 =ε0 sinkx sinφ
€
X1 t( ) =ε0V4ωε0 sinωt
X2 t( ) =ε0V4ωε0 cosωt
εx z,t( ) =4ωε0V
sinkx cosφX1 t( ) + sinφX2 t( )( )
Phasor representation
X2 q
uadr
atur
e
X1 quadrature
φ
€
ε0V4ωε0
Reminder: Light waves as harmonic oscillators
New coordinates
€
q t( ) =ε0V2ω 2ε0 sinωt
p t( ) =V
2µ0
B0 cosωt =ε0V2ε0 cosωt
p = ˙ q ˙ ̇ q = ˙ p = −ω 2q
Eem =12
p2 +ω 2q( )
€
q t( ) = mx t( )
p t( ) =1mpx t( )
€
X1 t( ) =ω2q t( ) X2 t( ) =
12ω
p t( )
Field Quadratures
€
X1 t( ) =ω2q t( ) X2 t( ) =
12ω
p t( )
€
ˆ q t( ) =
2ωˆ a + ˆ a †( )
ˆ p t( ) =1
2ωˆ a − ˆ a †( )
€
ˆ X 1 t( ) =12
ˆ a + ˆ a †( ) ˆ X 2 t( ) =12
ˆ a − ˆ a †( )
Uncertainty on field quadratures
€
En = n +12
⎛
⎝ ⎜
⎞
⎠ ⎟ ω
ΔxΔpx ≥2
⇒ ΔqΔp ≥ 2
ΔX1ΔX2 =ω2Δq 1
2ωΔp =
12ΔqΔp ≥ 1
4
€
q t( ) = mx t( )
p t( ) =1mpx t( )
Uncertainty in phasor representation
X2 q
uadr
atur
e
X1 quadrature
φ
Coherent states, shot noise and number-phase uncertainty
€
α = X1 + iX2 = α eiφ with α = X12 + X2
2( )ΔX1 = ΔX2 =
12
α2
= n
Δn = α +14
⎛
⎝ ⎜
⎞
⎠ ⎟
2
− α −14
⎛
⎝ ⎜
⎞
⎠ ⎟
2
= α = n
Δφ =uncertainty diameter
α=
12n
ΔnΔφ ≥ 12
X2 q
uadr
atur
e
X1 quadrature φ
X2 q
uadr
atur
e
X1 quadrature Δφ
LIGO Laser Interferometer Gravitational Wave Observatory
http://www.ligo.caltech.edu/
LIGO
http://www.ligo.caltech.edu/
LIGO
http://www.ligo.caltech.edu/
LIGO
http://www.ligo.caltech.edu/
LISA
http://lisa.nasa.gov
VIRGO
http://www.virgo.infn.it/
VIRGO
http://www.virgo.infn.it/
VIRGO
http://www.virgo.infn.it/
VIRGO
http://www.virgo.infn.it/
Squeezing the uncertainty
X2
X1
X2
X1
X2
X1
X2
X1
Δφ Δφ
Vacuum
X2
X1
1/2
1/2
Number states
X2
X1
Squeezing in classical harmonic oscillator
€
F x( ) = −kx 1−εsin2ω0t( ), ω0 =km
U x( ) =12
kx 2 1−εsin2ω0t( )
˙ ̇ x +ω02x = −ω0
2xεsin2ω0tx t( ) = B t( )cosω0t + C t( )sinω0t
˙ ̇ x t( ) = −ω02x t( ) −ω0
˙ B t( )sinω0t +ω0˙ C t( )cosω0t
−ω0˙ B t( )sinω0t +ω0
˙ C t( )cosω0t = −ω02εsin2ω0t B t( )cosω0t + C t( )sinω0t[ ]
− ˙ B t( )sinω0t + ˙ C t( )cosω0t = −ω0ε
2B t( )sinω0t + C t( )cosω0t[ ]
˙ B t( ) = +ω0ε
2B t( ); B t( ) = B0e
+εω0t
2
˙ C t( ) = −ω0ε
2C t( ); C t( ) = C0e
−εω 0t
2
Modulated force
Potential energy
Equation of motion
Trial solution and its 2nd derivative (B,C slowly varying, B’’=C’’=0)
Apply trigonometry to find this
Substitute into equation of motion
Equate sin and cos components
Displacement operator and creation of coherent states
€
ˆ D α( ) = eα ˆ a + −α∗ ˆ a
€
ˆ D −1 α( ) ˆ a ˆ D α( ) = ˆ a +α
€
= ˆ a +α( ) ˆ D −1 α( )α
€
ˆ D −1 α( ) ˆ a ˆ D α( ) ˆ D −1 α( )α = ˆ D −1 α( ) ˆ a α =α ˆ D −1 α( )α
€
ˆ a ˆ D −1 α( )α = 0
€
ˆ D −1 α( )α = 0
€
ˆ D α( ) 0 = α
1/2
1/2
Displacement of Vacuum
X2
X1
1/2
1/2
Squeezing operator
€
ˆ S z( ) = e12
z ˆ a 2−z* ˆ a +2( )
€
z = reiϕ
€
ˆ t = ˆ S z( ) ˆ a ̂ S + z( ) = ˆ a + z∗ ˆ a + +12!
z 2 ˆ a + 13!
z 2 z∗ ˆ a + +14!
z 4 ˆ a + ...
€
= ˆ a 1+12!
r2 +14!
r4 + ...⎛
⎝ ⎜
⎞
⎠ ⎟ + ˆ a +eiϕ r +
13!
r3 +15!
r5 + ...⎛
⎝ ⎜
⎞
⎠ ⎟
€
ˆ t = ˆ a cosh r + ˆ a +eiϕ sinh r
€
ˆ t ̂ S z( ) 0 = 0ˆ t ̂ S z( )α = tS z( )α
€
ˆ S z( )α = ˆ S z( ) ˆ D α( ) 0
Displacement of Squeezed Vacuum
X2
X1
Squeezing operator
€
ˆ t = ˆ a cosh r + ˆ a +eiϕ sinh r
€
ˆ t 1 =12
ˆ t + + ˆ t ( ) =12
ˆ a + + ˆ a ( ) cosh r + sinhr( ) = ˆ X 1er
ˆ t 2 =i2
ˆ t + − ˆ t ( ) =i2
ˆ a + − ˆ a ( ) cosh r − sinhr( ) = ˆ X 2e−r
€
α ˆ S + z( )ˆ t 1 ˆ S z( )α =α2
er
α ˆ S + z( )ˆ t 2 ˆ S z( )α =α2
e−r
Generation of squeezed light
€
S z( ) = e12za 2−z*a +2( )
χ(2)
ω
2ω
ω signal
pump idler
Detection of quadrature squeezed light
€
ε1 =12εLOe
iφLO +εS( )
ε2 =12εLOe
iφLO −εS( )εS =εS
X1 + iεSX 2
ε1 =12εLO cosφLO +εS
X1( ) + i εLO sinφLO +εSX 2( )[ ]
ε2 =12εLO cosφLO −εS
X1( ) + i εLO sinφLO −εSX 2( )[ ]
i1 − i2( )∝ε1ε1* −ε2ε2* ∝2εLO cosφLOεSX1 + sinφLOεS
X 2( )
Local Oscillator
Signal
- i1-i2
Detection of number squeezing
± i1 ± i2
input
Demonstrations of squeezed light I
Demonstrations of squeezed light II
Quantum noise in amplifiers
€
SNR =signal amplitude( )2
noise amplitude( )2 =I 2
ΔI 2
SNR =n 2
Δn( )2
G = eγL
Noise figure =SNRin
SNRout
= 2 − 1G
X2 q
uadr
atur
e
X1 quadrature
input
output
1/2
(G/2-1/4)1/2
Exercises
1. Use the definition of field quadratures X1(t) and X2(t) to verify what are their physical dimensions.
2. A ruby laser operating at 693nm emits pulses of energy 1mJ. Calculate the uncertainty in the phase of the laser light .
3. The proposed Laser Interferometer Space Antenna (LISA) experiment for gravity wave detection will use a standard Michelson interferometer (i.e. no power reciclying or cavity enhancement) with a laser operating at 1064 nm The length if the arms of the interferometer is 5×106 km and the power of the beams that form the interference pattern is ~10-11 W. Calculate the minimum strain that can be detected.
4. !! Demonstrate that an elastic force F=-kx(1-εsin2ω0t) modulated at twice the natural frequency (ε<<1) produces on an object of mass m oscillations that grow in time at one quadrature, while are damped at the other quadrature.