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Squeezing eigenmodes in parametric downconversion Konrad Banaszek Nicolaus Copernicus University Toruń, Poland Wojciech Wasilewski Czesław Radzewicz Warsaw University Poland Alex Lvovsky University of Calgary Alberta, Canada National Laboratory for Atomic, Molecular, and Optical Physics, Toruń, Poland Agenda • • • • • • Classical description Input-output relations Bloch-Messiah reduction Single-pair generation limit High-gain regime Optimizing homodyne detection Fiber optical parametric amplifier tp (2) c • Pump remains undepleted • Pump does not fluctuate Linear propagation High order effects Group velocity dispersion Group velocity Phase velocity Three wave mixing kp, wp k, w wp =w+ w’ k’, w’ Classical optical parametric amplifier (2) c Linear propagation Interaction strength 3WM [See for example: M. Matuszewski, W. Wasilewski, M. Trippenbach, and Y. B. Band, Opt. Comm. 221, 337 (2003)] Input-output relations Quantization: etc. Decomposition 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 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) .... .... bin a(0) .... U bout G1 G2 G3 G4 a(z) V .... Squeezing modes The operation of an OPA is completely characterized by: • the mode functions yn and fn • the squeezing parameters zn a(0) a(z) .... .... bin a(0) .... U bout G1 G2 G3 G4 a(z) V .... Single pair generation regime kp, wp wp = w + w’ k, w k’, w’ L Amplitude S sin(Dk L/2)/Dk Dk = kp-k-k’ Single pair generation regime w’ wp Amplitude S Pump x sin(Dk L/2)/Dk w Single pair generation w’ wp S(w,w’)=ei… w,w’|out =Σ lj fj(w)gj(w’) w Gaussian approximation of S w2 D d w1+w2=wp Dk=0 w1 “Classic” approach 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) 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 The spectral amplitudes which yields: characterize pure squeezing modes Intense generation regime • 1 mm waveguide in BBO • 24 fs pump @ 400nm Squeezing parameters RMS quadrature squeezing: e-2z Spectral intensity of eigenmodes Input and ouput modes |y0| =|f0| 2 arg y0 arg f0 2 First mode vs. pump intensity LNL=1/15mm |y0| 2 arg y0 LNL=100mm Homodyne detection fLO Noise budget Detected squeezing vs. LO duration 1/LNL=1 2 4 ts 3 Contribution of various modes tLO 15fs 30fs 50fs Mn n Optimal LOs 4 53 Optimizing homodyne detection SHG PDC – Conclusions • The Bloch-Messiah theorem allows us to introduce eigenmodes for input and output fields • For low pump powers, usually a large number of modes becomes squeezed with similar squeezing parameters • Any superposition of these modes (with right phases!) will exhibit squeezing • The shape of the modes changes with the increasing pump intensity! • In the strong squeezing regime, carefully tailored local oscillator pulses are needed. • Experiments with multiple beams (e.g. generation of twin beams): fields must match mode-wise. • Similar treatment applies also to Raman scattering in atomic vapor WW, A. I. Lvovsky, K. Banaszek, C. Radzewicz, quant-ph/0512215 A. I. Lvovsky, WW, K. Banaszek, quant-ph/0601170 WW, M.G. Raymer, quant-ph/0512157