1. 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

2. Agenda Classical description Input-output relations Bloch-Messiah reduction Single-pair generation limit High-gain regime Optimizing homodyne detection

3. Fiber optical parametric amplifier Pump remains undepleted Pump does not fluctuate

4. Linear propagation High order effects Group velocity dispersion Group velocity Phase velocity

5. Three wave mixing k p,  p  p =  +  ’ k,  k ’,  ’

6. Classical optical parametric amplifier [See for example: M. Matuszewski, W. Wasilewski, M. Trippenbach, and Y. B. Band, Opt. Comm. 221, 337 (2003)]  (2) Linear propagation 3WM Interaction strength

7. Input-output relations Quantization: etc.

8. 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:

9. 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 

10. Squeezing modes  The operation of an OPA is completely characterized by: the mode functions  n  and  n the squeezing parameters  n

11. Single pair generation regime k p,  p  p =  +  ’ L k,  k ’,  ’ Amplitude S  sin(  k L/2)/  k  k = k p -k-k’

12. Single pair generation regime ’’  pp Amplitude S  Pump x sin(  k L/2)/  k

13. Single pair generation ’’  pp S( ,  ’ )=e i… ,  ’ |out  =Σ j f j (  )g j (  ’ )

14. Gaussian approximation of S 22 11    k=0 1+2=p1+2=p

15. “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)

16. Intense generation regime 1 mm waveguide in BBO 24 fs pump @ 400nm

17. Squeezing parameters RMS quadrature squeezing: e -2 

18. Spectral intensity of eigenmodes

19. Input and ouput modes

20. First mode vs. pump intensity

21. Homodyne detection

22. Noise budget

23. Detected squeezing vs. LO duration 1/L NL =1 2 3 4 ss

24. Contribution of various modes MnMn n 15fs  LO 30fs 50fs

25. Optimal LOs 3 4 5

26. Optimizing homodyne detection – SHG PDC

27. 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