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Entanglement Propagation in Photon Pairs created by SPDC

Entanglement Propagation in Photon Pairs created by SPDC. Malcolm O’Sullivan-Hale , Kam Wai Chan, and Robert W. Boyd Institute of Optics, University of Rochester, Rochester, NY. Presented at OSA Annual Meeting, October 10 th , 2006. Spontaneous Parametric Down-Conversion. k. k. k. 2. 2.

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Entanglement Propagation in Photon Pairs created by SPDC

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  1. Entanglement Propagation in Photon Pairs created by SPDC Malcolm O’Sullivan-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 k k k 2 2 2 2 ' ' ! ! ! = = i i s s p p 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 2 ½ · µ ¶ ¸ ¾ L 1 R 2 w ¾ 0 2 0 ( ) j j à i ¡ + ¡ + q q q q / e x p i i ¡ s s ¾ 2 2 ; [ = ) ] [ ( = ) ] 2 2 = k 0 2 2 1 2 1 + + w ¾ ¾ w 0 0 k 0 0 0 p n 0 p p · ¸ L L i + ´ 2 j j £ ¡ ¡ q q e x p i s k 4 0 p 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

  5. qi Dqicond Entanglement Dqi qs ( ) ( ) ¢ ¢ ¢ ¢ i + < q q m n q q q R i i s s ; ´ d q ¢ c o n ( ) ( ) q ¢ ¢ ¢ i ¡ < x x m n x x i i s s ; 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] D’Angelo et al, PRL. 92, 233601 (2004). [2] Fedorov et al., PRA 69, 052117 (2004). [3] Chan and Eberly, quant-ph/0404093.

  6. Propagation of Correlations Intermediate-field ? Far-field Near-field

  7. Propagation of Correlations (continued…) qi Dqcond Dq qs xi Dxcond Dx xs How does the Fedorov Ratio change on propagation? z 1 z What is going on here?

  8. Experimental Set-up ¸ 3 6 3 8 n m = 2 : j ( ) j à x x i 8 5 0 s ; w ¹ m = 0 x x i s BBO (2 mm) f=100 mm 1:1 imaging PBS Measure coincidence events as a function of xi and xs to map out wave function at various longitudinal positions.

  9. Near-field Correlations ( ( ) ) x x ¹ ¹ m m i s • Slits/knife edge placed in the imaged plane of the crystal. • Strongly correlated photon positions. • Rx = 9.61 • Theory predicts 31.6 Normalized Coincidence Rates

  10. Far-field Correlations ( ( ) ) x x ¹ ¹ m m i s • Slits/knife edge placed in the focal plane of the lens. • Strongly anti-correlated photon positions. • Rx = 28.52 • Theory predicts 76.3 Normalized Coincidence Rates

  11. The Intermediate Zone? ( ( ) ) x x ¹ ¹ m m i s • 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

  12. Migration of Entanglement to Phase 2 j ( ) ( ) j p ( ) Ã Ã Á i ( ) + ( ¡ ) ( ) Ã f x x x x x x i i i s ; x x x g x e = s s i i 2 ; ; j ( ) j ( ) ( ) Ã f s s ; x x x g x = i i s s ; • At a particular propagation distance z0the 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.

  13. Experiment to Detect Phase Entanglement 2 j ( ) ( ) j à à + ¡ x x x x i i s s ; ; Same as before, but with a rotational shearing interferometer in idler arm. Superimposes image with rotated version of itself

  14. Theoretical Predictions b a a Fourth-order fringes yield information about entanglement independent of propagation distance.

  15. 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.

  16. Acknowledgements Collaborators: J.H. Eberly J.P. Torres & Supported by - the US Army Research Office through a MURIgrant

  17. THANK YOU! www.optics.rochester.edu/workgroups/boyd/nonlinear.html

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