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 10th, 2006

OSA: Oct. 10th, 2006

Spontaneous Parametric Down-Conversion

NL Crystal

c

UV Pump

NIR, Type-II SPDC

signal

idler

c

kp

kski

Type-II

We will restrict ourselves to degenerate and nearly collinear SPDC taking,

2! s = 2! i = ! p

2ks ' 2ki ' kp


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?


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, is a fitting parameter, and

¾0 = ¡2R

kp0np

Ã (qs;qi) / exp½

¡12

·w2

0

2[1+ w20=¾0)2]

+ iµ

Lkp0

¡¾0

2[1+ (¾0=w20)2]

¶¸jqs + qi j

2¾

£ exp·¡

´L + iL4kp0

jqs ¡ qi j2¸


Entanglement

Entanglement criterion [1]:

More conveniently, we can use the Fedorov Ratio [2,3]:

[1] D’Angelo et al, PRL. 92, 233601 (2004).[2] Fedorov et al., PRA 69, 052117 (2004).[3] Chan and Eberly, quant-ph/0404093.

(qs qi)

qi

qs

xs

xi

(xs xi)

qi

qs

qi

qicond

¢ (qs + qi ) < min(¢ qs;¢ qi )

¢ (xs ¡ xi ) < min(¢ xs;¢ xi )

Rq ´ ¢ q¢ qcond

To quantify amount of entanglement, we use the Schmidt Number:


Propagation of Correlations

Near-field Far-fieldIntermediate-field

?


Propagation of Correlations (continued…)

x

xs

xi

xcond

q

qs

qi

qcond

z

z1

How does the Fedorov Ratio change on propagation?

What is going on here?


Experimental Set-up

BBO (2 mm)

PBS

Measure coincidence events as a function of xi and xs to map out wave function at various longitudinal positions.

jÃ(xs;xi )j2xi

xs

f=100 mm 1:1 imaging

¸ = 363:8 nmw0 = 850 ¹ m


Near-field Correlations

• Slits/knife edge placed in the imaged plane of the crystal.

• Strongly correlated photon positions.

• Rx = 9.61

• Theory predicts 31.6

xi (¹ m)

x s(¹

m)

Normalized Coincidence Rates


Far-field Correlations

• Slits/knife edge placed in the focal plane of the lens.

• Strongly anti-correlated photon positions.

• Rx = 28.52

• Theory predicts 76.3

xi (¹ m)

x s(¹

m)



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 1.92xi (¹ m)

x s(¹

m)



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.

jÃ(xs;xi )j2 = f (xs)g(xi )

Ã(xs;xi ) =p

f (xs)g(xi )eiÁ(xs ;x i )

jÃ(xs;xi ) + Ã(xs; ¡ xi )j2


Experiment to Detect Phase Entanglement

Same as before, but with a rotational shearing interferometer in idler arm.

Superimposes imagewith rotated version of itself jÃ(xs;xi ) + Ã(xs;¡ xi )j2


Theoretical Predictions

b

a

a

Fourth-order fringes yield information about entanglement independent of propagation distance.


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.


Acknowledgements

&

Supported by - the US Army Research Office through a MURI grant

Collaborators:J.H.

EberlyJ.P. Torres


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

entanglement propagation in photon pairs created by spdc

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