Czes ław Radzewicz Warsaw University Poland

1. Squeezing eigenmodesin parametric down-conversion Wojciech Wasilewski Czesław Radzewicz Warsaw University Poland Konrad Banaszek Nicolaus Copernicus University Toruń, Poland Alex Lvovsky University of CalgaryAlberta, Canada National Laboratory for Atomic, Molecular, and Optical Physics, Toruń, Poland

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

5. kp, wp k, w k’, w’ Three wave mixing wp =w+ w’

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

7. Quantization: etc. Input-output relations

8. The Bloch-Messiah theorem allows us to introduce eigenmodes for input and output fields: 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. A71, 055801(2005).

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 ynand fn • the squeezing parameters zn 

11. L Single pair generation regime kp, wp wp =w+w’ k, w Amplitude S  sin(Dk L/2)/Dk k’, w’ Dk = kp-k-k’

12. wp w’ w Single pair generation regime Amplitude S  Pump x sin(Dk L/2)/Dk

13. wp w’ w Single pair generation S(w,w’)=ei… w,w’|out =Σlj fj(w)gj(w’)

14. Gaussian approximation of S w2 D d w1+w2=wp Dk=0 w1

15. The wave function up to the two-photon term: “Classic” approach 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 which yields: The spectral amplitudes characterize pure squeezing modes

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

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

18. Spectral intensity of eigenmodes

19. Input and ouput modes

20. First mode vs. pump intensity

21. Homodyne detection

22. Noise budget

23. 1/LNL=1 2 4 ts 3 Detected squeezing vs. LO duration

24. Contribution of various modes tLO 15fs 30fs 50fs Mn n

25. Optimal LOs 4 5 3

26. SHG PDC – Optimizing homodyne detection

27. 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 Conclusions 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