Entanglement Propagation in Photon Pairs created by SPDC

1
Entanglement Propagation in Photon Pairs created
by SPDC

• Malcolm OSullivan-Hale, Kam Wai Chan, and Robert
  W. Boyd
• Institute of Optics, University of Rochester,
  Rochester, NY

Presented at OSA Annual Meeting, October 10th,
2006
2
Spontaneous Parametric Down-Conversion
Type-II
We will restrict ourselves to degenerate and
nearly collinear SPDC taking,
c
ks
ki
kp
3
Motivation

• Entangled position and momentum spaces in SPDC
  involve high dimensional Hilbert spaces.
• Questions
• Effects of free-space propagation on the spatial
  correlations between photons?
• Implications about the entanglement present in
  the system?

4
Theory of SPDC
Given a Gaussian pump field (waist w0, curvature
R), the two-photon wave function can be written
in momentum-space as,
where qs,i are transverse wave vectors, L is the
crystal length, h is a fitting parameter, and
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Entanglement
Entanglement criterion 1
D(qs qi)
Dqi
Dqs
To quantify amount of entanglement, we use the
Schmidt Number
More conveniently, we can use the Fedorov Ratio
2,3
D(xs- xi)
Dxi
Dxs
1 DAngelo et al, PRL. 92, 233601 (2004). 2
Fedorov et al., PRA 69, 052117 (2004). 3 Chan
and Eberly, quant-ph/0404093.
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Propagation of Correlations
Near-field
Far-field
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Propagation of Correlations (continued)
How does the Fedorov Ratio change on propagation?
z
1
z
What is going on here?
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Experimental Set-up
BBO (2 mm)
f100 mm
11 imaging
PBS
Measure coincidence events as a function of xi
and xs to map out wave function at various
longitudinal positions.
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Near-field Correlations
Normalized Coincidence Rates

• Slits/knife edge placed in the imaged plane of
  the crystal.
• Strongly correlated photon positions.
• Rx 9.61
• Theory predicts 31.6

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Far-field Correlations
Normalized Coincidence Rates

• Slits/knife edge placed in the focal plane of the
  lens.
• Strongly anti-correlated photon positions.
• Rx 28.52
• Theory predicts 76.3

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The Intermediate Zone?

• Slits moved behind image plane (12 cm behind
  image plane or 14.5 cm after crystal).
• Little residual spatial correlations.
• Rx 1.19
• Theory predicts 1.92

Normalized Coincidence Rates
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Migration of Entanglement to Phase

• At a particular propagation distance z0 the
  amplitude-squared wave function will become
  separable, although the wave function itself is
  entangled, i.e.
• The entanglement has migrated to phase. We need
  interferometric methods to get information about
  the entanglement.

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Experiment to Detect Phase Entanglement
Same as before, but with a rotational shearing
interferometer in idler arm.
Superimposes image with rotated version of itself
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Theoretical Predictions
b
a
a
Fourth-order fringes yield information about
entanglement independent of propagation distance.
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Conclusions

• We have experimentally investigated the
  propagation of spatial correlations in SPDC.
• The apparent disappearance of entanglement can be
  explained by the migration of entanglement from
  intensity to the phase of the wave function.
• Rotational shearing interferometers should enable
  us to recover the entanglement information in the
  intermediate zones.

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Acknowledgements
Collaborators J.H. Eberly J.P. Torres

Supported by - the US Army Research Office
through a MURI grant
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• THANK YOU!

www.optics.rochester.edu/workgroups/boyd/nonlinear
.html