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.