Transcript Finding the Carrier Envelope Offset Phase via Optical Rectification

Squeezed State Generation in Photonic
Crystal Cavities
July 2 2008
Slide 1 (of 31)
Outline
• What are squeezed states?
• Field Quantization Formalism of Viviescas
for open optical cavities
• Using NLO to generate squeezed states
• Measuring squeezed states
Slide 2 (of 31)
Squeezed States
Squeezed
Coherent
Slide 3 (of 31)
Generating Squeezing
Hamiltonian for nondegenerate parametric down-conversion
Equations of motion for a1 and a2
Slide 4 (of 31)
assume  real
Define quadrature-phase amplitudes as
The measure of correlation is given by
Slide 5 (of 31)
As
Slide 6 (of 31)
Field Quantization in Open
Cavities
Vector potential satisfies the wave equation
We can write the exact eigenmodes of entire system as
Slide 7 (of 31)
Feshbach Projection
Define two operators
Decompose
Slide 8 (of 31)
Define a new basis as
Slide 9 (of 31)
Rewrite eigenfunctions of full structure in terms of LQQ and LPP
eigenfunctions
and the field expansion takes the form
Slide 10 (of 31)
Problem tackled in Banaee
thesis
2 Cavity modes
,
Classical pump
Single mode waveguide
Nondegenerate parametric
downconversion
Slide 11 (of 31)
After making the RWA our Hamiltonian (w/o NLO) is…
Where
Slide 12 (of 31)
To add in the NLO consider a classical pump field
Then the NL term in the Hamiltonian becomes
Coupling term g is described by
Slide 13 (of 31)
Applying Heisenberg equations of motion for the cavity and
the reservoir
We can integrate the reservoir operators from some initial
time
by expressing the same operators in terms of their values at
and taking the FT
Slide 14 (of 31)
where
by substituting this into the above equations, and moving to a
frame rotating at
assume weak coupling, and make
Markov approximation
Slide 15 (of 31)
After FTing and setting
Slide 16 (of 31)
Slide 17 (of 31)
Squeezed light out of the system (channel 1)
The spectrum of squeezing for the X and Y quadratures is
Where
Slide 18 (of 31)
Aside: Homodyne detection for
measuring squeezing
Slide 19 (of 31)
Efficiency 
fluctuations in autocorrelation are
signal transmission
local oscillator
reflectance
Related to squeezing by
End Aside….
Slide 20 (of 31)
Then, X and Y quadratures
Are given by
Slide 21 (of 31)
Consider
Slide 22 (of 31)
at threshold
Slide 23 (of 31)
Numerical Estimation of
squeezing
a = 420 nm
Simulated for
Al0.3Ga0.7As
band gap ~ 689 nm
pump beam ~720 nm
 = 0.15a
h = 200 nm
r = 0.29a
Slide 24 (of 31)
Slide 25 (of 31)
f1 = 209 THz(1.43m)
2f1 = 418 THz (721 nm)
Slide 26 (of 31)
Slope
Can estimate losses as
Then, depending on the orientation of the crystal for an
average power of 10 mW CW
Slide 27 (of 31)
[111]
[100]
Little squeezing as below threshold
Switching to pulsed laser with same average power gives
~ factor of 5 improvement
Slide 28 (of 31)
Try a new system
A = 0.2a
B = 0.025a
C = 0.2a
Slide 29 (of 31)
Slide 30 (of 31)
f1 = 208 THz(1.44m)
2f1 = 416 THz (722 nm)
Can estimate losses as
Then, depending on the orientation of the crystal for an
average power of 10 mW CW
Slide 31 (of 31)
For this particular structure can get
theoretical squeezing of ~ 70% at threshold
Slide 32 (of 31)
Summary
• Briefly reviewed a new formalism for
calculating cavity resonances coupled to a
bath
• Showed how the formalism was used
along with NLO to estimate a squeezed
spectrum
• Still to come… A more thorough
comparison of this technique with the
Green’s function formalism
Slide 33 (of 31)