1. Interferometric Measurement of Spatial Wigner Functions of Light Bryan Killett Brian J. Smith M. G. Raymer Funded by the NSF through the REU and ITR programs.

2. What is a spatial Wigner function? BUT - can we define a joint probability density function? In other words, something like: Pr(x,k x ). Given a normalized wave function (x), we know how to construct single-variable probability densities:

3. What is a spatial Wigner function? A straightforward definition of Pr(x,k x ) would have to… 2. Allow for a physical interpretation at every point. Therefore, Pr(x,k x ) needs to be non-negative everywhere. 1.Integrate to unity and satisfy the following relations: UNCERTAINTY PRINCIPLE! The spatial Wigner function corresponding to (x) is simply a function that fulfills the above condition.

4. The spatial Wigner function: This function has some other interesting properties: A single integral along the x axis gives the probability density for k x and vice-versa for the other axis. A double integral over all x and all k x results in unity. - It is sometimes negative! Thus, we call it a quasiprobability distribution. - It is uniquely related to (x) so a measurement of W(x,k x ) can be transformed to reveal (x). (not merely (x) 2 !)

5. How do we measure W(x,k x )? Top Mirror Output Beam Input Beam

6. The Top Mirror rotates the field. Clockwise (CW) Beam Counter-Clockwise (CCW) Beam x y y x x yx y Top Mirror

7. Image Rotation Mirrors Top Mirror On Camera:

8. How do we measure W(x,k x )? Only one output is measured. Input Beam

9. How does the interferometer work? Input Beam

10. How does the interferometer work? Input Beam

11. How does the interferometer work? Input Beam

12. How does the interferometer work? Input Beam

13. How does the interferometer work? The detector measures total power, which is the intensity of the output beam integrated over the face of the detector:

14. Wigner Results: TEM-00 (This mode is produced from a laser with standard spherical mirror resonators.)

15. Wigner Results: TEM-00 TheoryExperiment

16. Wigner Results: TEM-10 (This mode is produced from a laser with standard spherical resonators.) (This mode is produced from a laser with standard spherical mirror resonators.)

17. Wigner Results: TEM-10 Notice that W(x,k x ) is negative at some points. TheoryExperiment

18. Wigner Results: Two Beam Interference Glass plate

19. Wigner Results: Two Beam Interference Theory Experiment

20. Wigner Results: Simulated Decoherence Effects Experiment (No Decoherence) Experiment (With Decoherence)

21. Future work Examine single photon states like the Schrodinger Cat state. Analyze the effect of scattering and decoherence on the spatial Wigner function. Bell inequalities/entanglement measurements with two interferometers. See also a paper by our collaborators:Direct Measurement of the spatial Wigner function with area-integrated detection in Optics Letters Vol 28, #15, by E. Mukamel et. al.