NON-LINEAR OPTICS AND QUANTUM OPTICSscienze-como.uninsubria.it/andreoni/allegati/lectures/lezione 1.pdfBIBLIOGRAPHY • Y.R. Shen The principles of nonlinear optics John Wiley & Sons

NON-LINEAR OPTICS AND QUANTUM OPTICS

Non classical states• squeezed states (degenerate parametric down conversion

and second harmonics generation)• entangled states (non-degenerate parametric down conversion )• conditional states

Non-linear beam splitter (sum frequency generation)

BIBLIOGRAPHY

• Y.R. Shen The principles of nonlinear optics John Wiley & Sons (New York, 1984)• M. Schubert, B. Wilhelmi Nonlinear optics and quantum electronics John Wiley & Sons (New York, 1986)• V.G. Dmitriev, G.G. Gurzadyan, D.N. Nikogosyan Handbook of nonlinear optical crystal Springer-Verlag (Berlin Heidelberg, 1990)• P.N. Butcher, D. Cotter The elements of nonlinear optics Cambridge University Press (Cambridge, 1990) • B.E.A. Saleh, M.C. Klein Fundamentals of photonics John Wiley & Sons (New York, 1991)• A.C. Newell, J.V. Moloney Nonlinear optics Addison-Wesley (Redwood City, 1992)• D.L. Mills Nonlinear optics Springer-Verlag (Berlin Heidelberg, 1991)• Handbook of Photonics – Editor-in-Chief M.C. Gupta - CRC Press (Boca Raton New York, 1997)• R.W. Boyd Nonlinear optics Academic Press (San Diego, 1992)•G.S. He, S.H. Liu Physics of nonlinear optics World Scientific (Singapore, 1999)

Classical

Quantum• L. Mandel and E. Wolf Optical coherence and quantum optics Cambridge University Press (Cambridge, 1995)• R. Loudon The quantum theory of light (third edition) Oxford University Press (Oxford, 2000)• U. Leonhardt Measuring the quantum state of light Cambridge University Press (Cambridge, 1997)

MAXWELL EQUATIONS IN DIELECTRIC MEDIA

2 22

02 2 20

1c t t

µ∂ ∂∇ − =

∂ ∂E PE

0 0

;

0 ; 0 ;

t t

ε µ

∂ ∂∇ × = ∇ × = −

∂ ∂∇ ⋅ = ∇ ⋅ =

= + =

D BH E

D BD E P B H

for homogeneous and isotropic media we can derive a wave equation

where c0 is the propagation velocity in vacuum

if the medium is weakly nonlinear, we can write:

2 (3) 30 0 NL2 4 ...dε χ χ ε χ= + + + = +P E E E E P

and thus:2 2

2 NL02 2 2

1c t t

µ∂ ∂∇ − =

∂ ∂E PE where c is the propagation velocity in the medium

Second-order nonlinear optics 2NL 2d=P E

Third-order nonlinear optics (3) 3NL 4χ=P E

Second-order nonlinear opticsCOUPLED-WAVE THEORY OF THREE-WAVE MIXING

( ) ( )2 2

2 2NL0 NL2 2 2

1 with 2t d tc t t

µ∂ ∂∇ − = =

∂ ∂E PE P E

( ) ( ) ( ) ( )* *

1,2,3 1, 2, 3

1 1, with ; 2 2

q q qi t i t i tq q q q q q q

q q

t E e E e E e E Eω ω ω ω ω−− −

= =± ± ±

⎡ ⎤= + = = − =⎣ ⎦∑ ∑E r r r r

If we suppose that the three waves interacting in the medium, have distinct frequencies ω1 , ω2 and ω3 , and one frequency is the sum or the difference of the other two,

(frequency matching condition ω3 = ω1 + ω2) we get three equations:

( ) ( )

( ) ( )

( ) ( )

3 21

3 12

1 23

222 *1

1 0 3 2 3 22

222 *2

2 0 3 1 3 12

222 3

3 0 1 2 1 22

2

2

2

i ti t

i ti t

i ti t

E e d E E ec

E e d E E ec

E e d E E ec

ω ωω

ω ωω

ω ωω

ω µ ω ω

ω µ ω ω

ω µ ω ω

−

−

+

⎧⎛ ⎞∇ + = − −⎪⎜ ⎟

⎝ ⎠⎪⎪⎛ ⎞⎪ ∇ + = − −⎨⎜ ⎟

⎝ ⎠⎪⎪⎛ ⎞⎪ ∇ + = − +⎜ ⎟⎪⎝ ⎠⎩

( ) ( ) ( ) ( )NL

, 1, 2, 3

1,2

q ri tq r

q r

t d E E e ω ω+

=± ± ±

= ∑P r r r ( ) ( )2 2NL

2, 1, 2, 3

12

q ri tq r q r

q r

d E E et

ω ωω ω +

=± ± ±

∂= − +

∂ ∑P

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

2 2 2 *1 1 0 1 3 2

2 2 2 *2 2 0 2 3 1

2 2 23 3 0 3 1 2

2

2

2

k E d E E

k E d E E

k E d E E

µ ω

µ ω

µ ω

⎧ ∇ + = −⎪⎪ ∇ + = −⎨⎪

∇ + = −⎪⎩

r r r

r r r

r r r

Nondegenerate three-wave mixing

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

2 2 2 *1 1 0 1 3 1

2 2 23 3 0 3 1 1

2k E d E E

k E d E E

µ ω

µ ω

⎧ ∇ + = −⎪⎨

∇ + = −⎪⎩

r r r

r r r

Degenerate three-wave mixing ω1 = ω2

COLLINEAR THREE-WAVE MIXING: PLANE-WAVE SOLUTION

( ) ( ) ( )2 qik zq q q q qE E z a z eη ω −= =r

( ) ( ) ( ) ( ) ( )*

1,2,3

1, 22

q q q qi k z t i k z tq q q q

q

t a z e a z eω ωη ω − − −

=

⎡ ⎤= +⎣ ⎦∑E r

0 00q

q qn nε µηη = =

( )( )

( )2

2

2q

q q qq

E zI z a zω

η= = ( ) ( ) ( ) 2q

q qq

I zz aφ

ω= = z photon flux density [ph/(s·m2)]

We suppose that the envelope aq(z) is slowly varying with z and use the slowly varying envelope approximation (SVEA)

( ) ( )2

2 22

qik z qq q

d ak a z e

dz−⎡ ⎤∇ + =⎣ ⎦

22 qq q q

dai k k a

dz− − 2

q qk a+ 2q qik z ik zqq

dae i k e

dz− −

⎡ ⎤≈ −⎢ ⎥

⎢ ⎥⎣ ⎦SVEA

* 13 2

* 23 1

31 2

i k z

i k z

i k z

da iga a edzda iga a edzda iga a edz

− ∆

− ∆

∆

⎧ = −⎪⎪⎪ = −⎨⎪⎪ = −⎪⎩

3 2 1k k k k∆ = − −

31 2 30

1 2 3

2g dn n nω ω ω η= coupling coefficient

detuning

where

PARAMETRIC APPROXIMATION

• undepleted pump a3(z) = a3(0) :

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

* 2 22 21 3 22

1 1 3 32 23

* 2 22 22 3 12

2 2 3 32 23

0 00 cosh sinh

2 2

0 00 cosh sinh

2 2

ki z

ki z

ka az za z e a k i kk

ka az za z e a k i kk

γγ γ

γ

γγ γ

γ

∆−

∆−

⎧ ⎧ ⎫∆ −⎪ ⎪⎡ ⎤ ⎡ ⎤⎪ = − ∆ + − ∆⎨ ⎬⎢ ⎥ ⎢ ⎥⎪ ⎣ ⎦ ⎣ ⎦− ∆⎪ ⎪⎩ ⎭⎪⎨

⎧ ⎫⎪ ∆ −⎪ ⎪⎡ ⎤ ⎡ ⎤= − ∆ + − ∆⎪ ⎨ ⎬⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎪ − ∆⎪ ⎪⎩ ⎭⎩

( )

( )

* 13 2

* 23 1

0

0

i k z

i k z

da iga a edzda iga a edz

− ∆

− ∆

⎧ = −⎪⎪⎨⎪ = −⎪⎩

( )3 30 2ga γ=

* 1 32

* 2 31

2

2

i k z

i k z

da i a edzda i a edz

γ

γ

− ∆

− ∆

⎧ = −⎪⎪⎨⎪ = −⎪⎩

• undepleted reference field a2(z) = a2(0) :

( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

* 2 22 21 2 32

1 1 2 22 22

2 22 23 2 123 3 2 22 2

2

0 00 cos sin

2 2

0 00 cos sin

2 2

ki z

ki z

ka az za z e a k i kk

ka az za z e a k i kk

γγ γ

γ

γγ γ

γ

∆−

∆

⎧ ⎧ ⎫∆ −⎪ ⎪⎡ ⎤ ⎡ ⎤⎪ = + ∆ + + ∆⎨ ⎬⎢ ⎥ ⎢ ⎥⎪ ⎣ ⎦ ⎣ ⎦+ ∆⎪ ⎪⎩ ⎭⎪⎨

⎧ ⎫⎪ ∆ +⎪ ⎪⎡ ⎤ ⎡ ⎤= + ∆ − + ∆⎪ ⎨ ⎬⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦⎪ + ∆⎪ ⎪⎩ ⎭⎩

( )

( )

* 13 2

31 2

0

0

i k z

i k z

da iga a edzda iga a edz

− ∆

∆

⎧ = −⎪⎪⎨⎪ = −⎪⎩

( )2 20 2ga γ=

*1 2

3

2 21

2

2

i k z

i k z

da i a edzda i a edz

γ

γ

− ∆

∆

⎧= −⎪⎪

⎨⎪ = −⎪⎩

UNDEPLETED REFERENCE FIELD up-conversion: a1(0) ≠ 0 ; a2(z) = a2(0) ; a3(0) = 0

( ) ( )

( ) ( )

2 22 2 21 1 2 22 2

2

2 22 1 23 22 2

2

0 cos sin2 2

0sin

2

ki z

ki z

z k za z a k i k ek

a za z i k ek

γ γγ

γγ

γ

∆−

∆

⎧ ⎧ ⎫∆⎪ ⎪⎡ ⎤ ⎡ ⎤⎪ = + ∆ + + ∆⎨ ⎬⎢ ⎥ ⎢ ⎥⎪ ⎣ ⎦ ⎣ ⎦+ ∆⎪ ⎪⎪ ⎩ ⎭⎨⎪ ⎡ ⎤= − + ∆⎪ ⎢ ⎥⎣ ⎦+ ∆⎪⎩

( ) ( )

( ) ( )

2 222 22

1 1 22 22 22 2

222 22

3 1 22 22

0 cos2

0 sin2

z kz kk k

zz kk

γφ φ γ

γ γ

γφ φ γ

γ

⎧ ⎧ ⎫∆⎪ ⎪⎡ ⎤= + ∆ +⎪ ⎨ ⎬⎢ ⎥⎣ ⎦+ ∆ + ∆⎪ ⎪ ⎪⎩ ⎭⎨⎪ ⎡ ⎤= + ∆⎪ ⎢ ⎥⎣ ⎦+ ∆⎩

Photon-flux densities

( ) ( )

( ) ( ) ( )

2 21 1 22 2

2

3 1 2

0 arctan tan 2 2

0 0 2 2

k z kz k zk

kz z

γγ

π

⎧ ⎧ ⎫∆ ∆⎪ ⎪⎡ ⎤⎪Λ = Λ + + ∆ −⎨ ⎬⎢ ⎥⎪ ⎣ ⎦+ ∆⎪ ⎪⎨ ⎩ ⎭⎪ ∆⎪Λ = Λ + Λ − +⎩

Phases

fl up-conversion in phase matching :

( ) ( )

( ) ( )

21 1 2

23 1 2

0 cos2

0 sin2

zz

zz

φ φ γ

φ φ γ

⎧ ⎡ ⎤=⎪ ⎢ ⎥⎪ ⎣ ⎦⎨

⎡ ⎤⎪ = ⎢ ⎥⎪ ⎣ ⎦⎩

( ) ( )

( ) ( ) ( )

1 1

3 1 2

0

0 02

z

z πΛ = Λ⎧

⎪⎨

Λ = Λ + Λ −⎪⎩

( )( )

2 23

1

sin0 2

zI zI

γ= =KThe efficiency of up-conversion is:

2

2

fl up-conversion with phase mismatch ∆k :

( )( )

22 22 2

232 2

21 22

sin2

0 42

zkI z zI zk

γγ

γ

⎡ ⎤+ ∆⎢ ⎥⎣ ⎦= =⎡ ⎤+ ∆⎢ ⎥⎣ ⎦

K

the effect of the the phase mismatch is the reduction of the conversion efficiency:

For weak coupling γ2 z 1

( )( )

22

232 2

1

sin2

0 42

zkI z zI zk

γ

⎡ ⎤∆⎢ ⎥⎣ ⎦= =⎡ ⎤∆⎢ ⎥⎣ ⎦

K

2sinc2zk⎡ ⎤∆⎢ ⎥⎣ ⎦

2zk∆ππ 2π2π

UNDEPLETED REFERENCE FIELD down-conversion: a1(0) = 0 ; a2(z) = a2(0) ; a3(0) ≠ 0

( ) ( )

( ) ( )

* 2 22 3 2

1 22 22

2 22 2 23 3 2 22 2

2

0sin

2

0 cos sin2 2

ki z

ki z

a za z i k ek

z k za z a k i k ek

γγ

γ

γ γγ

∆−

∆

⎧ ⎡ ⎤= − + ∆⎪ ⎢ ⎥⎣ ⎦+ ∆⎪⎪⎨ ⎧ ⎫⎪ ∆⎪ ⎪⎡ ⎤ ⎡ ⎤= + ∆ − + ∆⎨ ⎬⎪ ⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦+ ∆⎪ ⎪⎪ ⎩ ⎭⎩

( ) ( )

( ) ( )

222 22

1 3 22 22

2 222 22

3 3 22 22 22 2

0 sin2

0 cos2

zz kk

z kz kk k

γφ φ γ

γ

γφ φ γ

γ γ

⎧ ⎡ ⎤= + ∆⎪ ⎢ ⎥⎣ ⎦+ ∆⎪⎨

⎧ ⎫∆⎪ ⎪ ⎪⎡ ⎤= + ∆ +⎨ ⎬⎪ ⎢ ⎥⎣ ⎦+ ∆ + ∆⎪ ⎪⎩ ⎭⎩

Photon-flux densities

( ) ( ) ( )

( ) ( )

1 3 2

2 23 3 22 2

2

0 0 2 2

0 arctan tan 2 2

kz z

k z kz k zk

π

γγ

∆⎧Λ = Λ − Λ − −⎪⎪

⎧ ⎫⎨ ∆ ∆⎪ ⎪⎡ ⎤⎪Λ = Λ − + ∆ +⎨ ⎬⎢ ⎥⎪ ⎣ ⎦+ ∆⎪ ⎪⎩ ⎭⎩

Phases

UNDEPLETED PUMP a1(0) ≠ 0 (signal) ; a2(0) = 0 (idler) ; a3(z) = a3(0) (pump)

( ) ( )

( ) ( )

2 22 2 21 1 3 32 2

3

* 2 23 1 2

2 32 23

0 cosh sinh2 2

0sinh

2

ki z

ki z

z k za z a k i k ek

a za z i k ek

γ γγ

γγ

γ

∆−

∆−

⎧ ⎧ ⎫∆⎪ ⎪⎡ ⎤ ⎡ ⎤⎪ = − ∆ + − ∆⎨ ⎬⎢ ⎥ ⎢ ⎥⎪ ⎣ ⎦ ⎣ ⎦− ∆⎪ ⎪⎪ ⎩ ⎭⎨⎪ ⎡ ⎤= − − ∆⎪ ⎢ ⎥⎣ ⎦− ∆⎪⎩

( ) ( )

( ) ( )

222 23

1 1 32 23

222 23

2 1 32 23

0 1 sinh2

0 sinh2

zz kk

zz kk

γφ φ γ

γ

γφ φ γ

γ

⎧ ⎧ ⎫⎪ ⎪⎡ ⎤= + − ∆⎪ ⎨ ⎬⎢ ⎥⎣ ⎦− ∆⎪ ⎪ ⎪⎩ ⎭⎨⎪ ⎡ ⎤= − ∆⎪ ⎢ ⎥⎣ ⎦− ∆⎩

Photon-flux densities

( ) ( )

( ) ( ) ( )

2 21 1 32 2

3

2 3 1

0 arctan tanh 2 2

0 0 2 2

k z kz k zk

kz z

γγ

π

⎧ ⎧ ⎫∆ ∆⎪ ⎪⎡ ⎤⎪Λ = Λ + − ∆ −⎨ ⎬⎢ ⎥⎪ ⎣ ⎦− ∆⎪ ⎪⎨ ⎩ ⎭⎪ ∆⎪Λ = Λ − Λ − −⎩

Phases

UNDEPLETED PUMP

( ) ( )

( ) ( )

21 1 3

22 1 3

0 cosh2

0 sinh2

zz

zz

φ φ γ

φ φ γ

⎧ ⎡ ⎤=⎪ ⎢ ⎥⎪ ⎣ ⎦⎨

⎡ ⎤⎪ = ⎢ ⎥⎪ ⎣ ⎦⎩

( ) ( )

( ) ( ) ( )

1 1

2 3 1

0

0 02

z

z πΛ = Λ⎧

⎪⎨

Λ = Λ − Λ −⎪⎩

fl parametric amplification in phase matching |γ3| >|∆k|, ∆k = 0:

( )( )

2 31

1

cosh0 2

zI zI

γ= =KThe efficiency of parametric amplification is:

fl parametric amplification out of phase matching |γ3|á|∆k|, ∆k ≠ 0 :

( ) ( )2

332

1 1 0 14

kz zez eγ

γφ φ∆

−⎧ ⎫⎪ ⎪≈ +⎨ ⎬⎪ ⎪⎩ ⎭

( ) ( )( )

2 23

3 32 21 10

1

00 4

k kz z z

kI z I e e e

I

γγ γ

∆ ∆− −

∆ =

−Γ = = = ΓThe signal amplification is:

UNDEPLETED PUMP

|γ3| < |∆k|

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

222 3

22 231 1 3 1 122 22

3 3

222 23

2 1 3 2 3 1223

tan20 1 sin ; 0 arctan

2 2

0 sin ; 0 0 2 2 2

zk kz kz k z z

k k

z kz k z zk

γγφ φ γ

γ γ

γ πφ φ γγ

⎧ ⎧ ⎫⎡ ⎤∆ ∆ −⎪ ⎪ ⎪⎧ ⎫ ⎢ ⎥ ∆⎪ ⎪ ⎪ ⎪⎡ ⎤ ⎣ ⎦= + ∆ − Λ = Λ + −⎨ ⎬ ⎨ ⎬⎢ ⎥⎣ ⎦∆ −⎪ ⎪ ∆ −⎪ ⎪⎩ ⎭⎨ ⎪ ⎪⎩ ⎭

∆⎡ ⎤= ∆ − Λ = Λ − Λ − −⎢ ⎥⎣ ⎦∆ −

⎪⎪⎪

⎪⎪⎪⎪⎩

fl parametric generation of superfluorescence |γ3|Ü |∆k| :

( ) ( )2

231 1 20 1 sin

2zz k

kγ

φ φ⎧ ⎫⎪ ⎪⎡ ⎤≈ + ∆⎨ ⎬⎢ ⎥∆ ⎣ ⎦⎪ ⎪⎩ ⎭

The signal amplification is:

( ) ( )( )

2 22 2 2

23 31 13 02 22

1

sin sin0 2 2sin0 2 4

2 2

k

z zk kzI z I zI k z zk k

γ γγ ∆ =

⎡ ⎤ ⎡ ⎤∆ ∆⎢ ⎥ ⎢ ⎥− ⎡ ⎤ ⎣ ⎦ ⎣ ⎦Γ = ≈ = = Γ⎢ ⎥∆ ⎣ ⎦ ⎡ ⎤ ⎡ ⎤∆ ∆⎢ ⎥ ⎢ ⎥⎣ ⎦ ⎣ ⎦

PHASE MATCHING

The efficiency of the parametric processes is maximum in condition of phase matching

fl nonlinear materials in which the phase mismatch can be modified.

PHASE MATCHING k3 = k1 + k2FREQUENCY MATCHING ω3 = ω1 + ω2

( ) ( ) ( )( ) ( ) ( )

3 3 3 3 1 1 1 1 2 2 2 2

3 3 3 3 1 1 1 1 2 2 2 2

cos cos cos

sin sin sin

n n n

n n n

ω ω θ ω ω θ ω ω θ

ω ω θ ω ω θ ω ω θ

= +⎧⎪⎨

= +⎪⎩

k1

k2

k3 noncollinear

1 3θ θ−

2 3θ θ−

OPTICALLY ANISOTROPIC CRYSTALSas the nonlinear media

UNIAXIAL and BIAXIALcrystals

k2k1collinear

k3

( ) ( ) ( )3 3 3 1 1 1 2 2 2 n n nω ω ω ω ω ω= +

kZ

Y

X

Characterized by the presence of a special direction called optical-axis (Z-axis).

The plane containing the Z-axis and the wave vector k is called the

principal plane.

Zk k

E E

90°

The light beam whose polarization is normal to the principal plane is called ordinary beam (o-beam)

The refractive index no of the o-beam does not depend on the propagation direction

Zk

k90°

EE

The light beam whose polarization is parallel to the principal plane is called extraordinary beam (e-beam)

The refractive index nextr of the e-beam depends on the propagation direction being a function of the angle θ between the Z axis and the vector k :

1 22 2

extr 2 2o e

cos sinnn n

θ θ−

⎛ ⎞= +⎜ ⎟

⎝ ⎠

UNIAXIAL CRYSTALS

Note that in general no= no(ω) ; ne = ne(ω)and they are given by dispersion relations such as

Sellmeier relations

The refractive indices of the ordinary (no) and extraordinary (ne) beams

in the plane normal to the Z-axis are called the principal values.

ne> no positive crystalno> ne negative crystal

α

PHASE-MATCHING CONDITIONSuniaxial crystals

PM I : ω1Æo , ω2Æ o , ω3Æ e

( ) ( ) ( )( ) ( ) ( )

( ) ( )( ) ( )

( ) ( ) ( )

3 1 2

3 3 3 3 1 1 1 1 2 2 2 2

3 3 3 3 1 1 1 1 2 2 2 2

1 1 o 1

2 2 o 2

1 22 2

3 3 2 2o 3 e 3

, cos cos cos

, sin sin sin

cos sin,

n n n

n n n

n n

n n

nn n

ω ω ωω ω α θ ω ω θ ω ω θ

ω ω α θ ω ω θ ω ω θ

ω ω

ω ω

α αω αω ω

−

= +⎧⎪ = +⎪⎪ = +⎪⎪ =⎨⎪ =⎪⎪ ⎛ ⎞⎪ = +⎜ ⎟⎪ ⎝ ⎠⎩

( ) ( ) ( )( ) ( ) ( )

( ) ( )

( ) ( )( )

PM II : ω1Æo , ω2Æ e , ω3Æ e

( )( )

( ) ( ) ( )

3 1 2

3 3 3 3 1 1 1 1 2 2 2 2 2

3 3 3 3 1 1 1 1 2 2 2 2 2

1 1 o 1

1 22 22 2

2 2 2 2 2o 3 e 3

1 22 2

3 3 2 2o 3 e 3

, cos cos , , cos

, sin sin , , sin

cos sin, ,

cos sin,

n n n

n n n

n n

nn n

nn n

ω ω ωω ω α θ ω ω θ ω ω θ α θ

ω ω α θ ω ω θ ω ω θ α θ

ω ω

α θ α θω θ α

ω ω

α αω αω ω

−

−

= +⎧⎪ = +⎪⎪ = +⎪⎪ =⎪⎨

⎛ ⎞− −⎪ = +⎜ ⎟⎝ ⎠

⎛ ⎞= +⎜ ⎟

⎝ ⎠⎩

⎪⎪⎪⎪⎪

15 22sin cos sin 3effd d dα α ϕ= −⇒ PM I

Optimization for PM I: crystal cut at 90ϕ =

222 cos cos3effd d α ϕ=⇒ PM II

Optimization for PM II: crystal cut at 0ϕ =

Many possible phase-matched interactions depending on theangles between the fields, on the wavelength and on the tuning angle

k1

k2

k3

1 3θ θ−

2 3θ θ−

32.5 37.5 40 42.5 45 47.5 50

-15

-10

-5

5

0.35 0.45 0.5 0.55 0.6 0.65 0.7

-60

-40

-20

20

40

60

0.35 0.45 0.5 0.55 0.6 0.65 0.7

-60

-40

-20

20

40

60

0.35 0.45 0.5 0.55 0.6 0.65 0.7

-60

-40

-20

20

40

60

(deg)

(deg)α

2 ( m)λ µ

(deg)

( )2 3θ θ−( )1 3θ θ−Internal phase-matching angles in BBO I for λ3 = 0.349 µm

External phase-matching angles

2 ( m)λ µ 2 ( m)λ µ

(deg)(deg) α = 34°, θcut = 34° α = 34°, θcut = 22.8°

α = 34°

NON COLLINEAR TYPE I INTERACTION SCHEME

k2

k3

k1

E1E2

E3

Zoptical axis

y

x

z

Y

X

α

1ϑ2ϑ

3ϑ

0

Phase mismatch

22 31

22 31

cos sinsin cos

d d dd d d

α αα α

+

−

= += −

( ) ( ) ( )3

1 2 3 0, ,

1 1 2 2 3 3

2,

g dn n n

ω ω ω ηω ω ω α+ − + −= 2 2g g g+ −= +

( ) ( )2 2ˆ ˆ ˆˆ ˆ g g g g g g g+ − + − + −= + + = +w y z y z

3 1 2∆ = − −k k k k

( ) ( ) ( ) ( )3

1 2 3 022 31

1 1 2 2 3 3

2cos sin,effg d d

n n nω ω ω ηα α

ω ω ω α= +

( ) ( ) ( )3 3 3eff y zg a g a g a+ −= +r r r

( ) ( ) ( ){ }

( ) ( ) ( ){ }

( ) ( ) ( ){ }

0 11 1 1 1

1

0 22 2 2 2

2

0 33 3 3 3

3

ˆ 2, exp . .2

ˆ 2, exp . .2

ˆ 2, exp . .2

t a i t c cn

t a i t c cn

t a i t c cn

η ω ω

η ω ω

η ω ω

⎧= − ⋅ − +⎡ ⎤⎪ ⎣ ⎦

⎪⎪⎪ = − ⋅ − +⎡ ⎤⎨ ⎣ ⎦⎪⎪⎪ = − ⋅ − +⎡ ⎤⎣ ⎦⎪⎩

xE r r k r

xE r r k r

wE r r k r

Fields

Parameters

( ) ( ) ( ) ( )( ) ( ) ( ) ( )

( ) ( ) ( ) ( )

*1 1 3 2

*2 2 3 1

2

3 3 1 2

ˆ exp ˆ exp

ˆ exp

eff

eff

eff

a ig a a i

a ig a a i

ga i a a ig

⎧⎪ ⋅ = − ⋅⎪⎪ ⋅ = − ⋅⎨⎪⎪ ⋅ = − ⋅⎪⎩

k r r r k r

k r r r k r

k r r r k r

∇ ∆

∇ ∆

∇ ∆

MAXWELL EQUATIONS For non-collinear type I interaction out of phase matching

undepleted pump a3(r) = a3(0):

( ) ( ) ( )

( ) ( )

* 2 31 1 2

1

2 2

ˆ ˆ ˆ ˆ ˆ2 0 cosh sinh 0 sinh expˆ ˆ2 2 2 2

ˆ ˆ 0 cosh sinh

2

i k iAa a Q Q a Q iQ Q

i ka a Q QQ

⎧ ⎫⎧ ⎫⎡ ⎤ ⎡ ⎤ ⎡ ⎤⎛ ⎞ ⎛ ⎞ ⎛ ⎞∆ ∆ ∆ ∆ ⋅ ∆ ∆⎡ ⎤⎪ ⎪ ⎪ ⎪ ⎛ ⎞= ⋅ + ⋅ + ⋅ × − ⋅⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎨ ⎨ ⎬ ⎬⎜ ⎟ ⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎢ ⎥∆ ⋅ ⎝ ⎠⎣ ⎦⎢ ⎥ ⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠ ⎝ ⎠⎪ ⎪⎪ ⎪⎣ ⎦ ⎣ ⎦ ⎣ ⎦⎩ ⎭⎩ ⎭

⎡ ⎤⎛ ⎞∆ ∆ ∆= ⋅ +⎢ ⎥⎜ ⎟

⎢ ⎥⎝ ⎠⎣ ⎦

k k k k k kr r r r rk k

kr r ( )* 1 31

2

ˆ ˆ ˆ20 sinh expˆ ˆ2 2 2iAa Q iQ

⎧⎪⎪⎪⎨

⎧ ⎫⎧ ⎫⎡ ⎤ ⎡ ⎤⎪ ⎛ ⎞ ⎛ ⎞∆ ⋅ ∆ ∆⎡ ⎤⎪ ⎪ ⎪ ⎪ ⎛ ⎞⋅ + ⋅ × − ⋅⎢ ⎥ ⎢ ⎥⎨ ⎨ ⎬ ⎬⎜ ⎟ ⎜ ⎟⎪ ⎜ ⎟⎢ ⎥∆ ⋅ ⎝ ⎠⎣ ⎦⎢ ⎥ ⎢ ⎥⎝ ⎠ ⎝ ⎠⎪ ⎪⎪ ⎪⎣ ⎦ ⎣ ⎦⎪ ⎩ ⎭⎩ ⎭⎩

k k k k kr r rk k

( )( )2

3 2

1 2

4ˆ ˆ ˆ ˆ

AQ k= −∆

∆ ⋅ ∆ ⋅k k k k( )3 3 0effA g a=ˆ k∆ =∆ ∆k k

in phase matching 1 2ˆ ˆ ˆ ˆ ˆ ˆ0 and k∆ = ∆ ⋅ = ∆ ⋅ ≡ ∆ ⋅k k k k k k

( ) ( ) ( )

( ) ( ) ( )

3

3

/ 2*3 31 1 2

/ 2*3 32 2 1

ˆ ˆ 0 cosh 0 sinhˆ ˆ ˆ ˆ

ˆ ˆ 0 cosh 0 sinhˆ ˆ ˆ ˆ

iPM

iPM

A Aa a a e

A Aa a a e

λ π

λ π

+

+

⎧ ⎧ ⎫⎡ ⎤ ⎡ ⎤⎪ ⎪= ∆ ⋅ + ∆ ⋅⎪ ⎨ ⎬⎢ ⎥ ⎢ ⎥∆ ⋅ ∆ ⋅⎪ ⎪⎣ ⎦ ⎣ ⎦⎪ ⎩ ⎭⎨

⎧ ⎫⎡ ⎤ ⎡ ⎤⎪ ⎪⎪ = ∆ ⋅ + ∆ ⋅⎨ ⎬⎢ ⎥ ⎢ ⎥⎪ ∆ ⋅ ∆ ⋅⎪ ⎪⎣ ⎦ ⎣ ⎦⎩ ⎭⎩

r k r k rk k k k

r k r k rk k k k

QUANTUM DESCRIPTION OF THE PROCESSES

Three-wave mixing 1 2 3ω ω ω< <

( )( ){ }

† † †1 2 3 1 2 3

† † †1 2 3 1 2 3

ˆ ˆ ˆ ˆ ˆ ˆ

ˆ ˆ ˆ ˆ ˆ ˆexp

H a a a a a a

U i a a a a a a

κ

τ

= +

= − +

quantum Hamiltonian

Evolution operator

The Heisenberg equations of motion derived by the quantum Hamiltonian correspond to the classical Maxwell equations

†11 2 3

†22 1 3

33 1 2

ˆ 1 ˆ ˆ ˆ,

ˆ 1 ˆ ˆ ˆ,

ˆ 1 ˆ ˆ ˆ,

da a H i a adt ida a H i a adt i

da a H i a adt i

κ

κ

κ

⎡ ⎤= = −⎣ ⎦

⎡ ⎤= = −⎣ ⎦

⎡ ⎤= = −⎣ ⎦

Note that the coupling coefficient depends on all the parametersof the interaction, possibly including the phase mismatch

By mapping time evolution into spatial evolution, we obtain that quantum equations are formally equivalent to classic equations, for operators instead of field-amplitudes

SPONTANEOUS DOWN CONVERSION

If we now consider the Hamiltonian for undepleted pump field, that can be analytically solved, we get

†12

†21

ˆ ˆ

ˆ ˆ

da i adtda i adt

κγ

κγ

⎧ = −⎪⎪⎨⎪ = −⎪⎩

( )* † †1 2 1 2ˆ ˆ ˆ ˆH a a a aκ γ γ= +

( ) ( ){ }* † †1 2 1 2ˆ ˆ ˆ ˆexpU S i i a a a aτγ τ γ γ→ − = − +

( ) [ ] ( ) [ ] ( ) ( )( ) [ ] ( ) [ ] ( ) ( )

* *1 1 2 1 2

* *2 2 1 2 1

ˆ ˆ ˆ ˆ ˆ0 cosh 0 sinh 0 0

ˆ ˆ ˆ ˆ ˆ= 0 cosh 0 sinh 0 0

i i

i i

a a t a e t a e a

a a t a e t a e a

φ φ

φ φ

κγ κγ µ ν

κγ κγ µ ν

⎧ = + = +⎪⎨

+ = +⎪⎩

2 2 1µ ν− =

which is the two-mode squeezing transformation originating the “twin-beam”

2twb

0

1 n

n

n nψ ξ ξ∞

=

= − ∑ ie φ νξµ

=

3 ˆˆ ˆA

tκγ → ∆ ⋅∆ ⋅

k rk k

where we can identify

Experimental system to generate spontaneous down conversion:TWA = travelling-wave optical parametric amplifier

Laser Nd:YLF

L BBO349 nm

Nd:YLF laser mode-locked, amplifiedλ F= 1047 nm, λSH = 523 nm, λTH= 349 nmPulse time duration 4.7 ps @ 349 nm, Energy per pulse 360 µJ, rep-rate 500 Hz

β-BaB2O4 (BBO)Cut for type I (ooe) interactionθcut = 22.8°Dimensions 10×10×3 mm3

Laser

Crystal

28.7° 29.5° 32.1° 32.8° 33.5° 34.0°

-0.2 -0.1 0.1 0.2

-0.2

-0.1

0.1

0.2

-0.2 -0.1 0.1 0.2

-0.2

-0.1

0.1

0.2

-0.2 -0.1 0.1 0.2

-0.2

-0.1

0.1

0.2

-0.2 -0.1 0.1 0.2

-0.2

-0.1

0.1

0.2

-0.2 -0.1 0.1 0.2

-0.2

-0.1

0.1

0.2

-0.2 -0.1 0.1 0.2

-0.2

-0.1

0.1

0.2

-0.2 -0.1 0.1 0.2

-0.2

-0.1

0.1

0.2

34.7°

-0.2 -0.1 0.1 0.2

-0.2

-0.1

0.1

0.2

-0.2 -0.1 0.1 0.2

-0.2

-0.1

0.1

0.2

-0.2 -0.1 0.1 0.2

-0.2

-0.1

0.1

0.2

-0.2 -0.1 0.1 0.2

-0.2

-0.1

0.1

0.2

-0.2 -0.1 0.1 0.2

-0.2

-0.1

0.1

0.2

-0.2 -0.1 0.1 0.2

-0.2

-0.1

0.1

0.2

35.3° 35.9° 36.4° 37.0° 38.0° 39.2°

Experiment Simulation

STATISTICAL PROPERTIES

The state of a quantum system is fully described by the statistical operator ρρ can be represented on different bases, such as

- on the number states (Fock states)

- on the coherent states

- P representation

, 0n mn n m mρ ρ

∞

=

= ∑2 2

2

1 d dρ α α ρ β β α βπ

= ∫2( )P dρ α α α α=∫

where P(α) is real and normalized 2( ) 1P dα α =∫but it is not positive in all cases, so that it cannot be interpreted as a probability distribution in classical sense.

Moreover P(α) sometimes does not exist.

{ }† * * * 2( ) ( )a aN tr e e e P dη η ηα η αχ η ρ α α− −≡ = ∫

* * 22

1( ) ( )NP e dη α ηαα χ η ηπ

−= ∫if the P-representation exists

* * 22

1( ) ( )W e dη α ηαα χ η ηπ

−= ∫If the P-representation exists, we have

22 ' 22( ) ( ) 'W e P dα αα α απ

− −= ∫

Alternatively we can use the simmetric characteristic function

and define the Wigner function{ }† *

( ) a atr eη ηχ η ρ −≡

The P-representation exists if and only if the Fourier transform of the normally-ordered characteristic function, , exists( )Nχ η

PARAMETRIC DOWN CONVERSION

Statistical properties of one of the fields produced/amplified by the TWA

1) Initial state for fields 1 and 2 is a pure coherent state 10 20,α α

( ) ( )†1 1 2ˆ ˆ ˆ( ) 0 0a t a aµ ν= +10 20 10 20, ,ρ α α α α=

2 2 * *1 1( ) expNχ η η ν ηα η α⎡ ⎤= − + −⎣ ⎦ being the mean value of1α 1a

21

2 21( ) expP

α αα

πν ν

⎡ ⎤−= −⎢ ⎥

⎢ ⎥⎣ ⎦ ( )2

122

2 ( )2( ) exp1 21 2

tW

α αα

νπ ν

⎡ ⎤−= −⎢ ⎥

++ ⎢ ⎥⎣ ⎦

Photon number distribution

( ) ( )( )

( ) ( )22 2

1 112 2 22

| | | |exp1 11

n

nnp n Lνα α

ν ν νν+

⎡ ⎤ ⎛ ⎞⎜ ⎟= − −⎢ ⎥⎜ ⎟+ +⎢ ⎥ +⎣ ⎦ ⎝ ⎠

2) Initial state for fields 1 and 2 is the vacuum state 10 200, 0α α= =

( )2

22

22( ) exp1 21 2

Wα

ανπ ν

⎡ ⎤= −⎢ ⎥

++ ⎢ ⎥⎣ ⎦

2

2 21( ) expP

αα

πν ν

⎡ ⎤= −⎢ ⎥

⎢ ⎥⎣ ⎦

Photon number distribution

( ) ( )( )

2

121

n

np nν

ν+=

+

The Wigner function can be reconstructed by optical tomographythat makes use of the data from homodyne detection

complete information about the quantum stateall the elements of the density matrix

The photon number distribution can be obtained from theWigner function, but it can also be measured separately, withoutmaking use of homodyne detection

partial information about the quantum stateonly the diagonal elements of the density matrix

There are many different features of classical and quantum statesthat can be used for characterizing them:

- with respect to the photon number distribution:Poissonian, sub-Poissonian and super-Poissonian states

direct measurement of the Fano factor ( )2 nF

nσ

=

- with respect to the Wigner function: Gaussian or non gaussian-states

tomographic reconstruction of the Wigner function