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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.

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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:

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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.

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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 !)

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

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

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Image Rotation Mirrors Top Mirror On Camera:

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

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How does the interferometer work? Input Beam

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How does the interferometer work? Input Beam

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How does the interferometer work? Input Beam

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How does the interferometer work? Input Beam

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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:

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

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Wigner Results: TEM-00 TheoryExperiment

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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.)

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

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Wigner Results: Two Beam Interference Glass plate

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Wigner Results: Two Beam Interference Theory Experiment

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

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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.

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