Squeezed State Generation in Photonic Crystal Cavities

1. Squeezed State Generation in Photonic Crystal Cavities July 2 2008 Slide 1 (of 31)

2. 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)

3. Squeezed States Squeezed Coherent Slide 3 (of 31)

4. Generating Squeezing Hamiltonian for nondegenerate parametric down-conversion Equations of motion for a1 and a2 Slide 4 (of 31)

5. assume  real Define quadrature-phase amplitudes as The measure of correlation is given by Slide 5 (of 31)

6. As Slide 6 (of 31)

7. 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)

8. Feshbach Projection Define two operators Decompose Slide 8 (of 31)

9. Define a new basis as Slide 9 (of 31)

10. Rewrite eigenfunctions of full structure in terms of LQQ and LPP eigenfunctions and the field expansion takes the form Slide 10 (of 31)

11. 2 Cavity modes , Classical pump Single mode waveguide Nondegenerate parametric downconversion Problem tackled in Banaee thesis Slide 11 (of 31)

12. After making the RWA our Hamiltonian (w/o NLO) is… Where Slide 12 (of 31)

13. 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)

14. 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)

15. 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)

16. After FTing and setting Slide 16 (of 31)

17. Slide 17 (of 31)

18. Squeezed light out of the system (channel 1) The spectrum of squeezing for the X and Y quadratures is Where Slide 18 (of 31)

19. Aside: Homodyne detection for measuring squeezing Slide 19 (of 31)

20. Efficiency  fluctuations in autocorrelation are signal transmission local oscillator reflectance Related to squeezing by End Aside…. Slide 20 (of 31)

21. Then, X and Y quadratures Are given by Slide 21 (of 31)

22. Consider Slide 22 (of 31)

23. at threshold Slide 23 (of 31)

24. 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)

25. Slide 25 (of 31)

26. f1 = 209 THz(1.43m) 2f1 = 418 THz (721 nm) Slope Slide 26 (of 31)

27. Can estimate losses as Then, depending on the orientation of the crystal for an average power of 10 mW CW Slide 27 (of 31)

28. [111] [100] Little squeezing as below threshold Switching to pulsed laser with same average power gives ~ factor of 5 improvement Slide 28 (of 31)

29. Try a new system A = 0.2a B = 0.025a C = 0.2a Slide 29 (of 31)

30. Slide 30 (of 31)

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)

32. For this particular structure can get theoretical squeezing of ~ 70% at threshold Slide 32 (of 31)

33. 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)