Parametric Down Conversion

Quantum optics seminar 77740Winter 2008Assaf Shaham

context

• Introduction• Theory of classic Sum Frequency

Generation• Quantum Hamiltonian for PDC• Applications

Non linear optics • Creation of new frequencies in the system.• Usually, the new frequencies have small amplitude

relative to the input frequencies.

Harmonic Difference Generation

1ω132 ωωω −=

3ωNon linear medium

2ω

3ω pump

idler

pump

idler1ω

signal

(optional)

PDC: the degenerate case

2/321 ωωω == Non linear medium 3ω pump

3ω2/32 ωω =pump Signal, idler

Nonlinear medium

Non linear dependency of the polarization in the electrical field:

4D E P= + πr r r

(1) (2) 2 (3) 3

(1) (2) (3)

(1)

( ) ( ) ( ) ( ) ...

( ) ( ) ( ) ...

( ) ( )NL

P t E t E t E t

P t P t P t

P t P t

= χ + χ + χ + =

= + + + =

= +

r r r rt t t

r r r

r r

( )(1) (1)4 ( ) ( ) 4 ( )NL NLD E P t P t E P t= + π + = ε + πr r r r r r

The susceptibility is a tensor.χt

c.g.s

Substituting in Maxwell equation gives:

Assuming ( , Slowly varying Amplitude) and we get the wave equation in a medium:

Representing the frequency component:

For every n we have:

2 22

2 2 2

1 4( ) PE E Ec t c t

∂ π ∂−∇ +∇ ∇⋅ + = −

∂ ∂

rr r r

2 ( )E E∇ >> ∇ ∇ ⋅r r

2 22

2 2 2

1 4 PE Ec t c t

∂ π ∂−∇ + = −

∂ ∂

rr r

( , ) ( , ) ( ) . .

( , ) ( , ) ( ) . .

n

n

i tn nn n

i tNL NL NLn nn n

E r t E r t E r e c c

P r t P r t P r e c c

− ω

− ω

= = ⋅ +

= = ⋅ +

∑ ∑∑ ∑

r r r

r r r

2 22 (1)

2

4( ) ( ) ( )NLn nn n nE r E r P r

c cω πω

−∇ − ε = −r r

Jcdt

Ddc

H

BdtBd

cE

D

rr

r

r

rr

r

π

πρ

41

0

1

4

+=×∇

=⋅∇

−=×∇

=⋅∇

0=ρ

Properties of the second order susceptibility( ) ( ) ( )2

, ,

, , ( ) ( )i n m ijk n m n m j n k mj k n m

P E Eω + ω = χ ω + ω ω ω ω ω∑ ∑

• Symmetries reduce the number of numbers need to describe .

• is real for Lossless media.• Kleinman symmetry: is far away from

of the media.

For the degenerate case:

( )2χt

( )2χt

( ),n mω ωresonanceω

( ) ( )

( )

2

, ,

3 1 2

( ) ( )

4 ( ) ( )

i n m ijk j n k mj k n m

eff

P E E

P d E E

ω + ω = χ ω ω

ω = ω ω

∑∑

( ) 22 2 ( )P d Eω = ωeff

Theory of classic Sum Frequency GenerationInverse process of Harmonic Difference Generation

2ω1ω

3 1 2ω = ω + ω

Non linear medium1ω

2ω

3ω

( )13 33 3 3, ( )nk n

cω

= = ε ω( )3 33 3( , ) . .i k z tE z t A e c c−ω= +r

We concentrate on the wave, defining:

tikkieff

tieff

ti eeAAdcceEEdccePtzP 32133 )(212133 4..4..),( ωωω −+−− =+=+=

2 22 (1)

2

4( ) ( ) ( )NLn nn n nE r E r P r

c cω πω

−∇ − ε = −r r

Substituting in gives

zkkkieff eAAcd

dzdA

ikzdAd )(

212

233

323

2321

162 −+−=+

ωπ

Slowly varying Amplitude approximation:dzdA

kzdAd 3

323

2

<<

We find the dependency of in and

By the same way we find also that

Non depleting pump approximation:

3A 1A 2A

kzieff eAAck

di

dzdA ∆= 212

3

233 8

ωπ

kzieff eAAck

di

dzdA ∆−= *

2321

211 8

ωπ

kzieff eAAck

di

dzdA ∆−= *

1322

222 8

ωπ

⎟⎟⎠

⎞⎜⎜⎝

⎛∆−

==∆

∆∫ kieAA

ckd

idzeAAck

diLA

kLieff

Lkzieff 188)( 212

3

23

0212

3

23

3

ωπ

ωπ

L =non linear medium length

⎟⎠⎞

⎜⎝⎛ ∆=⎟⎟

⎠

⎞⎜⎜⎝

⎛∆−

==∆

2sinc32164

22222

22

1323

34

32

22

2142

3

43

2232

33

3kLLAA

ck

ndki

eAAck

dcnA

cnI eff

kLieff ω

πω

πππ

213 , AAA <<

( )321 kkkk −+=∆

⎟⎠⎞

⎜⎝⎛ ∆=

2sinc512 22

23321

212

53

kLLcnnn

IIdI eff

λπ

23 3 1 1 1( 2 ) ( )I Iω = ω ∝ ω

Implications:

•The intensity is quadratic in L if .• .•If (Second Harmonic Generation), then

.

0=∆k

3 1 2I I I∝ ⋅1 2ω = ω

Method to achieve phase matching 0321 =−+ kkk

We need to control the refraction index n for every

frequency:• Angle tuning - using birefringence properties of the

nonlinear crystal. (KDP - KH2PO4, BBO - Beta-BaB2O4)

• Temperature tuning(lithium niobate LiNbO3)

• Quasi phase matching(LiNbO3, LiTaO3,

KTP - KTiOPO4)

Phasematching

QuasiPhasematching

No Phasematching

•From Ref. pic [1]

Birefringence 0321 =−+ kkk

332211 ωωω nnn =+

( )3

12123 ωωnnnn −=−

213 ωωω +=

123 ωωω ≥≥assume

For normal dispersion, is rising monotonically )(ωn

Sollution: using birefringence crystal

Sellmeier BBO type 1

)(λn

][ mµλ

)78.0,()39.0,2( mnmn oe µλωµλωθ ===

( )

ωωωωωω

ωωωωω

⋅=====⋅===

===

2),(),(),(2,

2

333331

321

213

ejnejnojnejojj

Example (for type 1 uniaxial)

oejjnjnjn

,),(),(),( 333222111

==+ ωωωωωω

on

en

)(θen

Index ellipsoid for uniaxial crystals

• z is the optical axis (extraordinary).• x, y are ordinary axes.

• Polarization D1 is on the x-y plane and have .

• Polarization D2 have .

en

on

on

kr

12

2

2

2

2

2

=++eoo n

zny

nx

x

y

z

( )θen

θ

on( )θen

( ) ( )2

2

2

2

2

cossin)(

1

oee nnnθθ

θ+=

•From Ref. pic [2]

Properties of birefringence crystals

• For uniaxial crystals:Positive uniaxial Negative uniaxial

Type 1Type 2

• Propagating direction examples:Collinear, non collinear

• Walk off: spatial, temporal.

( )e on n> ( )e on n<

3 3 1 1 2 2o e en n nω = ω + ω 3 3 1 1 2 2e o on n nω = ω + ω

3 3 1 1 2 2o o en n nω = ω + ω 3 3 1 1 2 2e e on n nω = ω + ω

Ko

Ke

classical parametric down conversion is impossible

2/231 ωωω ==

⎟⎠⎞

⎜⎝⎛ ∆=

2sinc512 22

23321

212

53

kLLcnnn

IIdI eff

λπ

If there will be no PDC according to the classical theory.

=> We have to use Quantum mechanics

0)0()0(1 ==== zIzI idler

Non linear medium 3ω

2/32 ωω =3ω

pump Signal, idler

pump

z0

Quantum Hamiltonian for Spontaneous Parametric Down Conversion

Classical electrical energy in non linear media:

∫ ∫∫

∫ ∫∫

+⋅+=

=⋅+=

),(

0

)1(332

),(

0

332

)4(),(81),(

81

),(),(81),(

81

trDNL

trD

PEdtrErdrdtrB

trdDtrErdrdtrBH

πεππ

ππ

....)()()(),,;(

)()(),;()();(

32321323211)3(

2212211)2(

111)1(

+−−−−+

+−−+=

ωωωωωωωωωωωχ

ωωωωωωωχωωωχ

lkjijkl

kjijkjiji

EEE

EEEP

∫∫∫

∫∫

∫∫

−′′′′′−′′′′′=

′′′=

++++=

),(),(),(),;(314)(

),(),(),(214)(

...))()((81),(

81

)2(2

)1(1

212)1(332

ωωωωωωωωχωωωπ

ωωωωχωωπ

εππ

rErErEdddrX

rErEddrX

rXrXErdrdtrBH

kjiijk

jiij

Second quantization for the term:

Define:

is the annihilation operator for j,k mode

In the degenerate case, when :

Normal ordering, and assuming classical pump

)(2 rX

jka

ωω ′=

volumenquatizatioV =

)](ˆ)(ˆ)2(ˆ)(ˆ)(ˆ)2(ˆ[ 1101102ωωωωωωχ aaaaaaH X

+++ +∝ h

2021

200 )(ˆ)(ˆ, νν ω <<== − tntnea t

.].)(ˆ)(ˆ[ 20112

cceaaH tiX +∝ −++ ωνωωχh

( ) ( )( ) ( ) ( ) ( )ωωωωωωωωχωωωπ ,',",',';""'314 2

2 rErErEdddrX kjiijk −−= ∫∫∫( ) ( )

2,1

ˆ)(

==

∑=−+

jVn

E

eaEE

jk

jkjk

k

tkrijkjkj

jk

ω

ω ω

h

( ) ( ) ( ) ( ) ( ) ( )( )∫ ++−−−+ −+−∝ )''()'()()''()'()(32

ωωωωχωωωωχ EEEEEErdH X

Coupling constant=χ

Equations of motionSlowly complex mode amplitudes

ti

ti

etat

etat2

1

)(ˆ)(ˆ)(ˆ)(ˆ

22

11

ω

ω

=Α

=Α

)(ˆ)(ˆ]ˆ),(ˆ[1)(ˆ

)(ˆ)(ˆ]ˆ),(ˆ[1)(ˆ

10)(

1022

20)(

2011

021

2

021

2

tietAiHtidt

td

tietAiHtidt

td

ti

ti

+−++

+−++

Α−=−=Α=Α

Α−=−=Α=Α

χνχν

χνχν

ωωωχ

ωωωχ

h

h

θ

θ

θ

νν

νχνχ

νχνχ

νχ

νχ

i

i

i

e

tiett

tiett

tdt

td

tdt

td

||

)sinh()0(ˆ)cosh()0(ˆ)(ˆ)sinh()0(ˆ)cosh()0(ˆ)(ˆ

)(ˆ)(ˆ

)(ˆ)(ˆ

00

01022

02011

22

02

22

2

12

02

21

2

=

Α−Α=Α

Α−Α=Α

Α=Α

Α=Α

+

+

Photon statisticsThe r’th moment of :

For we get the same result:

Result are similar to thermal photon statistics:

( ) ( ) ( ) ( )

[ ] [ ]( ) ( )

( ) ( ) ( )( ) ( )( ) ( )( )tr

rnnnt

t

tie

tiettiet

tttttn

r

r

rrr

rrri

riri

rrrrr

||sinh!

0|ˆ.....2ˆ1ˆ|0||sinh

0|0ˆ0ˆ|0||sinh

0|)0(||sinh

0|)sinh()0(ˆ)cosh()0(ˆ)sinh()0(ˆ)cosh()0(ˆ|0

0|ˆˆ|0|ˆˆ|)(ˆ

02

2,12222,102

2,1222,102

2

2,120

2,1020102012,1

2,1112,1111

νχ

νχ

νχ

νχ

νχνχνχνχ

θ

θθ

=

=>+++<=

=>ΑΑ<=

=>Α−=

=>Α−ΑΑ+Α<=

=>ΑΑ<>=ΨΑΑΨ=<

+

+

+−+

++

)(ˆ1 tn r

)(ˆ2 tn r ( )trtn rr ||sinh!)(ˆ 02

2 νχ=

[ ]( )

( )rr

sk

B

B

ern

TKHtrTKH

1!ˆ

)/ˆexp()/ˆexp(ˆ

,−

=

−−

=

β

ρ

onpolarizatismomentumk

==

)ˆ)...(2ˆ)(1ˆ(ˆ 2222 rnnnn r +++=

Optical Parametric Oscilator• Gain is asymptotically if

exponential if :

• Tunable by changing the phase matching condition, orchanging the cavity’s length.

• Better Finesse

3ωω =p 1ωω =s

2ωω =i

( ) ( ) gzeAzA 021

11 ⇒42

322

12 /64 ckAdg ieffωπ=

0=∆k

•From Ref. pic [1]

Generating of squeezing state using OPO

Signal to noise

Source for single photons

Laser ωcrystalcolineartype 2

P.B.S

Homodyne

detection

H.W.P Q.W.P

ω

2/ω

ω

2/ω >H|

>V|POL

P.B.S: Polarizing Beam SpliterH.W.P: Half Wave PlateQ.W.P: Quarter Wave PlatePOL: polarizer

Source for entanglement photons

Non collinear type 2 SPDC

( )1 2 1 21 | , | ,2

iH V e V HϕΨ = > + >•From Ref. pic [3]

references

• R.W. Boyd, Nonlinear optics, 2nd edition, Academic Press, New York, 2003.• L. Mandel, E Wolf, Optical coherence and Quantum optics, Cambridge university press, New

York, 1995.• Wu L.A, Kimble H.j., Hall J.L., Wu Huifa, Generation of squeezed states by parameteric down

conversion, Phys rev Lett 57, 2520 – 2523, 1986.• Breitenbach G, Schiller S, Mlynek J, Measurement of the quantum states of squeezed light,

Nature, 387, 471-475, 1997. • Rubin M.H., Klyshko D.N., Shin Y.H., Sergienko A.V., Theory of two photon

entanglement in type-II optical parametric down-conversion , Phys. Rev. A 50, 5122-5133 ,1994.• Kwiat P. G., Mattle K., Weinfurter H., Zeilinger A., Sergienko A. V., Shih Y.,

New High-Intensity source of Polarization-entangled photon pairs, Phys. Rev. Lett. 75, 4337-4341, 1995.

• Halevi A., Thesis Work, The Hebrew University of Jerusalem, 2008.

Pictures from:• [1] RP Photonics: http://www.rp-photonics.com/img/qpm.png• [2] http://www.cooper.edu/engineering/projects/gateway/ee/solidmat/modlec3/node8.html• [3] Geiger Mark S. , Keller S. James, Creation and Characterization of entangled photons Via

second Harmonic Generation of Infrared Diodes, Kenyon College, Gambier OH. 2006:http://biology.kenyon.edu/HHMI/posters_2005/GeigerM.jpg