Download Optical parametric down conversion

Parametric Down Conversion
Quantum optics seminar 77740
Winter 2008
Assaf 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
ω3
ω1
ω2
pump
pump
idler
Non linear medium
idler
signal
ω3
ω1
ω 2 = ω 3 − ω1
(optional)
PDC: the degenerate case
ω1 = ω2 = ω3 / 2
ω3
pump
Non linear medium
ω3
ω2 = ω3 / 2
pump
Signal, idler
Nonlinear medium
Non linear dependency of the polarization in the
electrical field:
r r
r
D = E + 4πP
c.g.s
r
t (1) r
t (2) r 2
t (3) r 3
P(t ) = χ E (t ) + χ E (t ) + χ E (t ) + ... =
r (1)
r (2)
r (3)
= P (t ) + P (t ) + P (t ) + ... =
r (1)
r NL
= P (t ) + P (t )
t
The susceptibility χ is a tensor.
r r
r (1)
r NL
r
r NL
(1)
D = E + 4π P (t ) + P (t ) = ε E + 4πP (t )
(
)
Substituting in Maxwell equation gives:
r
r
r
1 ∂ r
4π ∂ P
2
−∇ E + ∇(∇ ⋅ E ) + 2 2 E = −
c ∂t
c ∂t 2
2
2
r
r
∇ E >> ∇ (∇ ⋅ E )
r
∇ ⋅ D = 4πρ
r
r
1 dB
∇× E = −
c dt
r
∇⋅B = 0
r
r 1 dD 4π r
∇× H =
+
J
c dt
c
Assuming
( ρ = 0 , Slowly varying Amplitude)
and we get the wave equation in a medium:
2
r
r 1 ∂ r
4π ∂ P
2
−∇ E + 2 2 E = −
c ∂t
c ∂t 2
2
2
Representing the frequency component:
r
r
r
E (r , t ) = ∑ n En (r , t ) = ∑ n En (r ) ⋅ e− iωnt + c.c.
r NL
r NL
r NL
P (r , t ) = ∑ n P n (r , t ) = ∑ n P n (r ) ⋅ e− iωnt + c.c.
For every n we have:
r
ωn 2 (1) r
4πωn 2 NL
−∇ En (r ) − 2 ε En (r ) = −
Pn (r )
c
c
2
Properties of the second order susceptibility
Pi ( ω n + ω m ) =
∑ ∑ χ ( ) (ω
2
ijk
n
+ ω m , ω n , ω m ) E j (ω n ) E k (ω m )
j ,k n ,m
• Symmetries
t ( 2 ) reduce the number of numbers need to
describe χ .
t ( 2)
χ
•
is real for Lossless media.
• Kleinman symmetry: ( ωn , ωm ) is far away from
resonance
of the media.
ω
Pi ( ω n + ω m ) = ∑ ∑ χ (ijk2 ) E j (ω n ) E k (ω m )
j ,k n ,m
P ( ω3 ) = 4 d eff E ( ω1 ) E ( ω 2 )
For the degenerate case:
P ( 2 ω ) = 2 d eff E 2 ( ω)
Theory of classic Sum Frequency Generation
Inverse process of Harmonic Difference Generation
ω1
ω2
ω1
ω2
Non linear medium
ω 3 = ω1 + ω 2
We concentrate on the ω3 wave, defining:
r
i k z −ω t
E3 ( z, t ) = A3e ( 3 3 ) + c.c.
k3 =
n3ω3
,
c
n3 = ε(1) (ω3 )
P3 ( z , t ) = P3e − iω3t + c.c. = 4d eff E1E2e −iω3t + c.c. = 4d eff A1 A2ei ( k1+k2 ) e − iω3t
Substituting in
r
ωn 2 (1) r
4πωn 2 NL
−∇ En (r ) − 2 ε En (r ) = −
Pn (r )
c
c
2
16πd eff ω 3
d 2 A3
dA3
i ( k1 + k 2 − k3 ) z
2
+
ik
=
−
A
A
e
3
1 2
dz
d 2z
c2
gives
2
Slowly varying Amplitude approximation:
d 2 A3
dA3
<<
k
3
dz
d 2z
We find the dependency of A3 in A1 and A2
d eff ω 3
2
dA3
i∆kz
= 8πi
A
A
e
1 2
dz
k3c 2
(∆k = k1 + k2 − k3 )
By the same way we find also that
d eff ω1
dA1
* −i∆kz
= 8πi
A
A
e
3
2
2
dz
k1c
2
d eff ω 2
2
dA2
* −i∆kz
= 8πi
A
A
3 1 e
dz
k2c 2
Non depleting pump approximation:
A3 ( L) = 8πi
d eff ω 3
k3c
2
2
L
A1 A2 ∫ e
i∆kz
dz = 8πi
0
d eff ω 3
k3c 2
2
A3 << A1 , A2
⎛ e i∆kL − 1 ⎞
⎟⎟
A1 A2 ⎜⎜
⎝ i∆k ⎠
L =non linear medium length
nc
I 3 = 3 A3
2π
d eff ω 3
n3 c
2
2
64π
A
A2
=
1
2
2π
k3 c 4
2
2
4
d ω n
⎛ e i∆kL − 1 ⎞
2
2
⎛ ∆kL ⎞
⎜⎜
⎟⎟ = 32π eff 2 33 3 A1 A2 L2 sinc 2 ⎜
⎟
2
i
k
∆
k
c
⎝
⎠
⎝
⎠
3
2
2
4
2
I 3 = 512π 5
2
2 ⎛ ∆kL ⎞
sinc
L
⎜
⎟
2
n1 n 2 n3 λ3 c
⎝ 2 ⎠
d eff I 1 I 2
Implications:
•The intensity is quadratic in L if ∆k = 0 .
• I 3 ∝ I1 ⋅ I 2 .
•If ω 1 = ω 2 (Second Harmonic Generation), then
I 3 (ω3 = 2ω1 ) ∝ I12 (ω1 )
.
Method to achieve phase matching
k1 + k 2 − k 3 = 0
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)
Phase
Quasi
matching
Phase
• Quasi phase matching
matching
(LiNbO3, LiTaO3,
KTP - KTiOPO4)
No Phase
matching
From Ref. pic [1]•
Birefringence
k1 + k 2 − k 3 = 0
ω3 = ω1 + ω 2
n1ω1 + n2ω2 = n3ω3
n3 − n2 = (n1 − n2 )
assume ω3 ≥ ω2 ≥ ω1
ω1
ω3
For normal dispersion, n(ω ) is rising monotonically
Sellmeier BBO type 1
Sollution: using birefringence crystal
n(ω1 , j1 )ω1 + n(ω2 , j2 )ω2 = n(ω3 , j3 )ω3
j = e, o
n (λ )
no
ne (θ )
Example (for type 1 uniaxial)
ω3 = 2ω
j1 = j2 = o ,
( ω1 = ω2 = ω )
j3 = e
2 ⋅ n(ω , j1 = o)ω = n(ω3 , j3 = e)ω3 = n(ω3 , j3 = e)2 ⋅ ω
ne
λ[ µm]
ne θ (2ω , λ = 0.39 µm) = no (ω , λ = 0.78 µm)
Index ellipsoid for uniaxial crystals
ne
ẑ
•
•
r
k
x2
y2 z2
+ 2 + 2 =1
2
no
no
ne
θ
ne (θ )
no
•
ŷ
•
no
z is the optical axis (extraordinary).
x, y are ordinary axes.
Polarization D1 is on the x-y plane
and have no .
Polarization D2 have ne (θ ).
1
sin 2 (θ ) cos 2 (θ )
=
+
2
2
2
ne (θ )
ne
no
x̂
From Ref. pic [2]•
Properties of birefringence crystals
•
For uniaxial crystals:
Positive uniaxial
( ne > no )
Type 1 no 3ω3 = ne1ω1 + ne 2 ω2
Type 2 no 3ω3 = no1ω1 + ne 2 ω2
•
( ne < no )
ne 3ω3 = no1ω1 + no 2 ω2
ne 3ω3 = ne1ω1 + no 2 ω2
Propagating direction examples:
Collinear,
Ke
Ko
•
Negative uniaxial
Walk off: spatial, temporal.
non collinear
classical parametric down conversion is impossible
ω1 = ω3 = ω2 / 2
2
I 3 = 512π 5
2
2 ⎛ ∆kL ⎞
sinc
L
⎜
⎟
2
2
n1n2 n3λ3 c
⎝
⎠
d eff I1 I 2
If I1 ( z = 0) = I idler ( z = 0) = 0 there will be no PDC according to
the classical theory.
ω3
Non linear medium
ω3
ω2 = ω3 / 2
pump
0
pump
Signal, idler
ẑ
=> We have to use Quantum mechanics
Quantum Hamiltonian for Spontaneous
Parametric Down Conversion
Classical electrical energy in non linear media:
1
1
2
3
H=
B
r
t
d
r
+
d 3r
(
,
)
∫
∫
8π
8π
D ( r ,t )
1
1
2
3
=
B
r
t
d
r
+
d 3r
(
,
)
∫
∫
8π
8π
D ( r ,t )
∫ E (r , t ) ⋅ dD(r , t ) =
0
∫ E (r , t ) ⋅ d (ε
E + 4πP NL )
0
Pi = χ ij (ω1 ; ω1 ) E j (ω1 ) + χ ijk
(1)
(1)
( 2)
(ω1 ; ω1 − ω 2 , ω 2 ) E j (ω1 − ω 2 ) E k (ω 2 ) +
+ χ ijkl (ω1 ; ω1 − ω 2 − ω 3 , ω 2 , ω 3 ) E j (ω1 − ω 2 − ω 3 ) E k (ω 2 ) El (ω 3 ) + ....
( 3)
H=
1
1
2
3
B
(
r
,
t
)
d
r
+
d 3 r (ε (1) E 2 + X 1 (r ) + X 2 (r ) + ...)
∫
∫
8π
8π
1
(1)
′
d
ω
d
ω
χ
(ω , ω ′) Ei (r , ω ′) E j (r , ω )
ij
∫∫
2
1
( 2)
X 2 (r ) = 4π ∫∫∫ dωdω ′dω ′′χ ijk (ω ′′; ω − ω ′, ω ′) Ei (r , ω ′′) E j (r , ω ′ − ω ) E k (r , ω )
3
X 1 (r ) = 4π
Second quantization for the X 2 (r ) term:
X 2 (r ) = 4π
1
(2 )
dωdω ' dω"χ ijk (ω" ; ω − ω ' , ω ')Ei (r , ω")E j (r , ω '−ω )Ek (r , ω )
∫∫∫
3
(+ )
E j (ω ) = ∑ E jk aˆ jk e
Define:
(
i kr −ω jk t
)
k
E jk =
hω jk
j = 1,2
n jkV
V = quatization volume
â jk is the annihilation operator for j,k mode
(
H X 2 ∝ ∫ d 3 r χE (+ ) (ω ) E (− ) (ω − ω ' ) E (− ) (ω ' ' ) + χE (− ) (ω ) E (+ ) (ω − ω ' ) E (+ ) (ω ' ' )
)
In the degenerate case, when ω = ω ′ :
+
+
+
H X 2 ∝ hχ [aˆ0 (2ω )aˆ1 (ω )aˆ1 (ω ) + aˆ0 (2ω )aˆ1 (ω )aˆ1 (ω )]
Normal ordering, and assuming classical pump
a 0 = ν 0 e −2 ω t ,
+
+
nˆ1 (t ) = nˆ2 (t ) << ν 0
H X 2 ∝ hχ [aˆ1 (ω )aˆ1 (ω )ν 0 e
− i 2 ωt
+ c.c.]
2
χ = Coupling constant
Equations of motion
Slowly complex mode amplitudes
ˆ (t ) = aˆ (t )e iω1t
Α
1
1
ˆ (t ) = aˆ (t )e iω 2t
Α
2
2
ˆ (t ) 1
dΑ
1
ˆ (t ), Hˆ ] = −iχν Aˆ + (t )ei (ω1+ω2 −ω0 )t = −iχν Α
ˆ +
= [Α
χ2
1
0 2
0 2 (t )
dt
ih
ˆ (t ) 1
dΑ
2
ˆ (t ), Hˆ ] = −iχν Aˆ + (t )ei (ω1+ω2 −ω0 )t = −iχν Α
ˆ +
= [Α
χ2
2
0 1
0 1 (t )
dt
ih
ˆ (t )
d 2Α
2
2
1
ˆ
χ
ν
=
0 Α1 (t )
2
dt
ˆ (t )
d 2Α
2
2
2
ˆ
χ
ν
=
0 Α 2 (t )
2
dt
ˆ (t ) = Α
ˆ (0) cosh( χ ν t ) − ie iθ Α
ˆ + (0) sinh( χ ν t )
Α
1
1
0
2
0
ˆ (t ) = Α
ˆ (0) cosh( χ ν t ) − ie iθ Α
ˆ + (0) sinh( χ ν t )
Α
2
2
0
1
0
ν 0 =| ν 0 | e iθ
Photon statistics
r
The r’th moment of nˆ1 (t ) :
r
ˆ + r (t )Α
ˆ r (t ) | Ψ >= < 0 | Α
ˆ + r (t )Α
ˆ r (t ) | 0 > =
nˆ1 (t ) =< Ψ | Α
1
1
1, 2
1
1
1, 2
[
ˆ + (0) cosh( χ ν t ) + ie −iθ Α
ˆ (0) sinh( χ ν t )
=1, 2 < 0 | Α
1
0
2
0
(
)
] [Αˆ (0) cosh( χ ν
r
1
]
r
iθ ˆ +
t
)
−
ie
Α
(
0
)
sinh(
χ
ν
t
)
| 0 >1, 2 =
0
2
0
2
= − ie iθ sinh r (χ | ν 0 | t )Α 2 (0) | 0 >1, 2 =
+r
r
ˆ r (0 )Α
ˆ + r (0 ) | 0 > =
= sinh 2 r (χ | ν 0 | t )1, 2 < 0 | Α
2
2
1, 2
= sinh 2 r (χ | ν 0 | t )1, 2 < 0 | (nˆ 2 + 1)(nˆ 2 + 2 ).....(nˆ 2 + r ) | 0 >1, 2 =
= r!sinh 2 r (χ | ν 0 | t )
r
For nˆ 2 (t ) we get the same result:
r
nˆ2 (t ) = r!sinh 2 r (χ | ν 0 | t )
Result are similar to thermal photon statistics:
ρˆ =
exp( − Hˆ / K B T )
tr exp( − Hˆ / K B T )
[
(r )
nˆ k , s
=
(e
r!
β
−1
)
r
]
k = momentum
s = polarizati on
r
nˆ2 = ( nˆ 2 + 1)( nˆ 2 + 2)...( nˆ 2 + r )
Optical Parametric Oscilator
• Gain is asymptotically if
exponential if ∆k = 0 :
A1 (z ) ⇒
1
A1 (0 )e gz
2
g = 64π 2ω1 d eff A3 / ki c 4
2
2
2
• Tunable by changing the
phase matching condition, or
changing the cavity’s length.
• Better Finesse
ω p = ω3
ωs = ω1
ωi = ω2
From Ref. pic [1]•
Generating of squeezing state using OPO
Signal to noise
Source for single photons
H.W.P
P.B.S
Laser
ω
type 2
ω/2
colinear
crystal
ω/2
Q.W.P
|H >
ω
ω
POL
|V >
Homodyne
detection
P.B.S: Polarizing Beam Spliter
H.W.P: Half Wave Plate
Q.W.P: Quarter Wave Plate
POL: polarizer
Source for entanglement photons
Non collinear type 2 SPDC
1
iϕ
|
,
| V1 , H 2 > )
Ψ=
H
V
>
+
e
(
1
2
2
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