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)
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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
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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)
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°
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34.7°
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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
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