Interferometric Measurement of Spatial Wigner Functions of Light Bryan Killett Brian J. Smith M. G....
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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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Transcript of Interferometric Measurement of Spatial Wigner Functions of Light Bryan Killett Brian J. Smith M. G....
Interferometric Measurement of Spatial Wigner Functions
of LightBryan Killett
Brian J. SmithM. G. Raymer
Funded by the NSF through the REU
and ITR programs.
What is a spatial Wigner function?
2)()Pr( xx
2
)()Pr(
dxexk xikx
x
BUT - can we define a “joint probability density” function?
In other words, something like: Pr(x,kx).
x
x
Pk where
Given a normalized wave function (x), we know how to construct single-variable probability densities:
What is a spatial Wigner function?
A straightforward definition of Pr(x,kx) would have to…
2. Allow for a physical interpretation at every point. Therefore,
Pr(x,kx) needs to be non-negative everywhere.
1. Integrate to unity and satisfy the following relations:
UNCERTAINTY PRINCIPLE!
dxkxk xx ),Pr()Pr(
xx dkkxx ),Pr()Pr(
The spatial Wigner function corresponding to (x) is simply a function that fulfills the above condition.
The spatial Wigner function:
')'()'(),( '2* dxexxxxkxW xikx
x
This function has some other interesting properties:
A single integral along the x axis gives the probability density for kx and vice-versa for the other axis.
A double integral over all x and all kx 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,kx) can be transformed to reveal (x). (not merely (x)2 !)
How do we measure W(x,kx)?
Top Mirror
Output Beam
Input Beam
The “Top Mirror” rotates the field.
Clockwise(CW) Beam
Counter-Clockwise (CCW) Beam
x
y
y
xx
yx
y
Top Mirror Top Mirror
Image Rotation Mirrors
Top MirrorOn Camera:
How do we measure W(x,kx)?
)()(
Setup Classical
xEx
Only one output is measured.
Input Beam
How does the interferometer work?
Input Beam
)'(0 xE
)'()'( 0 xExE
How does the interferometer work?
)'()'( 0 xxExE
Input Beam
)'(0 xE
)'()'( 0 xExE
How does the interferometer work?
'xikxe
Input Beam
)'(0 xE
)'()'( 0 xxExE
How does the interferometer work?
'0'
021 )'()'()'( xikxik
outputxx exxEexxExE
)'()'()'( xExExE CCWCWoutput
Input Beam
)'(0 xE
)'()'( 0 xxExE 'xikxe
How does the interferometer work?
'0'
021 )'()'()'( xikxik
outputxx exxEexxExE
),(CC 21 xkxWI
'0'
021 )'()'()'( xikxik
outputxx exxEexxExE
The detector measures total power, which is the intensity ofthe output beam integrated over the face of the detector:
')'(2dxxEI output
')'()'( '2*0021 dxexxExxECC xikx
Wigner Results: TEM-00
(This mode is producedfrom a laser with standard
spherical mirror resonators.)
Wigner Results: TEM-00
Theory Experiment
Wigner Results: TEM-10
(This mode is producedfrom a laser with standard
spherical resonators.)(This mode is produced
from a laser with standardspherical mirror
resonators.)
Wigner Results: TEM-10
Notice that W(x,kx)is negative at some
points.
Theory Experiment
Wigner Results: Two Beam Interference
Glass plate
Wigner Results: Two Beam Interference
Theory Experiment
Wigner Results: Simulated Decoherence Effects
Experiment(No Decoherence)
Experiment(With Decoherence)
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 spatialWigner function with area-integrated detection” in Optics Letters Vol 28, #15, by E. Mukamel et. al.
