the two-photon state generated by spontaneous parametric...

The Two-Photon State Generated by Spontaneous Parametric Down-Conversion

The Two-Photon State Generated bySpontaneous Parametric Down-Conversion

C. H. Monkenand

A. G. da Costa Moura

Universidade Federal de Minas GeraisBrazil


Funding

Conselho Nacional de DesenvolvimentoCientífico e Tecnológico

Instituto do Milênio de Informação Quântica

C A P E S


Interest: • Fundamental• Quantum state engineering


Spontaneous Parametric Down-Conversion

laser

down-convertedfields

nonlinearmedium

In first order, the interaction hamiltonian contains terms like

By energy conservation and phase matching,


optic axis

ordinaryextraordinary

The Refractive Index Surface



Type I Phase Matching

optic axis

Collinear

pump beam extraordinaryrefractive index surface(ellipsoid)

down-converted field ordinaryrefractive index surface (sphere)

dispersion



optic axis

noncollinear

pump beam extraordinaryrefractive index surface(ellipsoid)

down-converted field ordinaryrefractive index surface (sphere)




pump beam

crystal

down-convertedtwo-photon field



The Two-Photon State in the Paraxial Approximation

C. H. Monken, P. H. Souto Ribeiro, and S. Pádua, Phys. Rev. A 57, 3123 (1998)

C. K. Hong and L. Mandel, Phys. Rev. A 31, 2409 (1985)



The Two-Photon State in the Paraxial Approximation

1 photonplane wavemodes

two-photonplane wavespectrum


Paraxial Approximation in Uniaxial Media

Ordinary Polarization:


Paraxial Approximation in Uniaxial Media

Ordinary Polarization:

Extraordinary Polarization:




pump beamplane wavespectrum

BBO - Collinear type II SPDC

Lens f = 200 mm

U.V. mirror

B.S.P.

Counters

D1

D2

0 2 4 6 80

2

4

6

8

10

(a)

D1

vert

ical

position

(mm

) D1 counts/10000in 10 s

1.6 -- 2.01.2 -- 1.60.8 -- 1.20.4 -- 0.80.0 -- 0.4

0 2 4 6 80

2

4

6

8

10

(b)

Coincidencesin 10 s

200 -- 250150 -- 200100 -- 150

50 -- 1000 -- 50

Image transfer experiment

D1 horizontal position (mm)


D1

vert

ical

position

(mm

)

c

aperture

Laser Ar 351 nm





pump beamplane wavespectrum

D1 H-position (mm)

D1

V-p

osition

(mm

)

Calculated Measured coincidences

0

-2

2

-1

1

0-2 21-1

0

-2

2

-1

1

0-2 21-1

BS

LG

D2 fixed atthe origin

D1

Results I - No interferenceS. P. Walborn et al., Phys. Rev. A 69, 023811 (2004)

Results II - Interference Regime (Interferometer balanced)

BS

LG

D2 fixed atthe origin

D1

D1 H-position (mm)

D1

V-p

osition

(mm

)

Calculated Measured coincidences

0

-2

2

-1

1

0-2 21-1

0

-2

2

-1

1

0-2 21-1

S. P. Walborn et al., Phys. Rev. A 69, 023811 (2004)




pump laserwavenumber




collinear indexmismatch





walk-offvector*

* M. V. Fedorov et al., quant-ph/0612104v1 (2006)





walk-offvector

for 351nm –> 702nm in BBO



New Coordinates:

Monochromatic Approximation:



pump laserangularspectrum




displacementin x direction(walk-off)

For collinear 351nm 702nm in BBO





astigmatism*


*E. Abramochkin and V. Volostnikov, Opt. Commun. 83, 123 (1991)





astigmatism simplepropagation



collinear( )

noncollinear


Dependence on

Depends on

detectors in the far field scanned in opposite senses:

0 0 0 0

0



Dependence on (for ):

May “clip”

unless ( )

detectors in the far field scanned in the same sense:


-0.002 -0.001 0.001 0.002-0.002 -0.001 0.001 0.002-0.002 -0.001 0.001 0.002

Example: For collinear down-conversion in BBO

optic

axis

BBO - Collinear type II SPDC

Lens f = 200 mm

U.V. mirror

B.S.P.

Counters

D1

D2

0 2 4 6 80

2

4

6

8

10

(a)

D1

vert

ical

position

(mm

) D1 counts/10000in 10 s

1.6 -- 2.01.2 -- 1.60.8 -- 1.20.4 -- 0.80.0 -- 0.4

0 2 4 6 80

2

4

6

8

10

(b)

Coincidencesin 10 s

200 -- 250150 -- 200100 -- 150

50 -- 1000 -- 50

Image transfer experiment



D1

vert

ical

position

(mm

)

c

aperture

Laser Ar 351 nm


optic axis



astigmatism*


Factors that affect the transfer of angular spectrum

clipping walk-off

* The basic phenomenon behind the mode converter


Mode Converter

E. Abramochkin and V. Volostnikov, Opt. Commun. 83, 123 (1991)

M. W. Beijersbergen et al., Optics Commun. 96, 123 (1993)

M. Padgett, J. Arlt, N. Simpson, and L. Allen, Am. J. Phys. 64, 77 (1996)



All deviations from a constant in will affectthe transfer of angular spectrum.



Conclusions and Perspectives

The two-photon wave function carries the information containedin the pump beam angular spectrum: transverse profile, parity,orbital angular momentum, (Gouy phase?), etc.

The crystal anisotropy may affect this transfer of information.For a good transfer, very thin crystals or noncritical phasematching must be used.

The transfer of longitudinal properties of the pump beam is stillunder investigation.

the two-photon state generated by spontaneous parametric...

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