Czesław Radzewicz Warsaw University Poland Konrad Banaszek Nicolaus Copernicus University Toruń, Poland Alex Lvovsky University of Calgary Alberta, Canada Squeezing eigenmodes in parametric down- conversion National Laboratory for Atomic, Molecular, and Optical Physics, Toruń, Poland Wojciech Wasilewski
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Czesław Radzewicz Warsaw University Poland Konrad Banaszek Nicolaus Copernicus University Toruń, Poland Alex Lvovsky University of Calgary Alberta, Canada.
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Czesław RadzewiczWarsaw University
Poland
Konrad BanaszekNicolaus Copernicus University Toruń, Poland
Alex LvovskyUniversity of Calgary
Alberta, Canada
Squeezing eigenmodesin parametric down-conversion
Squeezing eigenmodesin parametric down-conversion
National Laboratory for Atomic, Molecular, and Optical Physics, Toruń, Poland
[See for example: M. Matuszewski, W. Wasilewski, M. Trippenbach, and Y. B. Band,Opt. Comm. 221, 337 (2003)]
c(2)
Linear propagation
3WMInteraction
strength
Input-output relationsInput-output relations
Quantization: etc.
DecompositionDecomposition
As the commutation relations for the output field operators must be preserved, the two integral kernels can be decomposed using the Bloch-Messiah theorem:
S. L. Braunstein,Phys. Rev. A 71, 055801 (2005).
The Bloch-Messiah theorem allows us to introduce eigenmodes for input and output fields:
Squeezing modesSqueezing modes
The characteristic eigenmodes evolve according to:
• describe modes that are described by pure squeezed states
• tell us what modes need to be seeded to retain purity
a(0) a(z)
.... ....
G1G2G3G4U V
bin bout
.... ....
a(0) a(z)
Squeezing modesSqueezing modes
a(0) a(z)
.... ....
G1G2G3G4U V
bin bout
.... ....
a(0) a(z)
The operation of an OPA is completely characterized by:• the mode functions nand n• the squeezing parameters n
Single pair generation regimeSingle pair generation regime
kp, p
p = + ’
L
k,
k’, ’Amplitude S sin(k L/2)/k
k = kp-k-k’
Single pair generation regimeSingle pair generation regime
’
pAmplitude S Pump x sin(k L/2)/k
Single pair generationSingle pair generation
’
p
S(,’)=ei… ,’|out
=Σ j fj()gj(’)
Gaussian approximation of SGaussian approximation of S
2
1
k=0
1+2=p
“Classic” approach“Classic” approach
Schmidt decomposition for a symmetric two-photon wave function:
C. K. Law, I. A. Walmsley, and J. H. Eberly,Phys. Rev. Lett. 84, 5304 (2000)
We can now define eigenmodes which yields:
The spectral amplitudes characterize pure squeezing modes
The wave function up to the two-photon term:
W. P. Grice and I. A. Walmsley, Phys. Rev. A 56, 1627 (1997);T. E. Keller and M. H. Rubin, Phys. Rev A 56, 1534 (1997)