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June 2011Mathematical Problems in Industry

Modeling photon generation

C. J. McKinstrie

Bell Laboratories, Alcatel-Lucent

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2MPI, June 2011

Outline of the talk

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Optical pulse propagation in a fiber

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Four-wave mixing (FWM)

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Selected applications of FWM

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Scientific goals of the project

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Coupled-mode equations (CMEs)

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Input-output equations (IOEs)

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Schmidt and adjoint

decompositions

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Summary

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3MPI, June 2011

Scalar nonlinear Schrodinger equation

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Light-wave propagation in a fiber is governed by the generalized nonlinear Schrodinger equation (NSE)

dz

A(t) = -

αA(t) + iβ(idt

)A(t) + iγ|A(t)|2A(t).

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NSE governs wave propagation in a variety of weakly-nonlinear media.

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Includes convection, dispersion, (gain) loss, nonlinear phase modulation (PM) and four-wave mixing (FWM).

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Excludes time-dependent fiber responses, which cause stimulated Brillouin

and Raman scattering (SBS and SRS), and wave steepening.

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Excludes polarization-dependent fiber responses, which cause polarization rotation and enable vector FWM.

[G. Agrawal, Nonlinear Fiber Optics (Elsevier, 2006); R. Boyd, Nonlinear Optics (Elsevier, 2008); L. Mollenauer, Solitons in Optical Fibers (Elsevier, 2006).]

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4MPI, June 2011

Parametric devices are enabled by four-wave mixing

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In four-wave mixing (FWM), weak sidebands (s and i) are driven by strong

pumps (p and q).

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Modulation instability (MI): 2πp

→ πs

+ πi

(πj

is a photon with frequency ωj

).

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Phase conjugation (PC): πp

+ πq

→ πs

+ πi

.

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Bragg scattering (BS), or frequency conversion (FC): πs

+ πq

→ πp

+ πi

.

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MI and PC amplify signals, but add excess noise, whereas BS frequency converts signals without adding noise.

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By varying the pump and signal frequencies, one can control whether MI, PC and BS occur separately or simultaneously.

[C. McKinstrie, J. Sel. Top. Quantum Electron. 8, 538 and 956 (2002).]

ωs q ip

BS

ωs i qp

PC

s p iω

MI

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5MPI, June 2011

Degenerate four-wave mixing

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In degenerate FWM, also called modulation interaction (MI), a strong pump (p) drives a weak signal and idler (s, i). The frequency-matching (FM) condition is 2ωp

=

ωs

+ ωi

.

dz

As

= i(βs

+ 2γ|Ap

|2)As

+ iγAp2Ai

*,

dz

Ap

≈

i(βp

+ γ|Ap

|2)Ap

,

dz

Ai

= i(βi

+ 2γ|Ap

|2)As

+ iγAp2As

*.

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Remove pump phase factor: Aj

(z) = Bj

(z)exp[i(βp

+ γP)z], where P = |Ap

|2.

dz

Bs

= i(βs

-

βp

+ γP)Bs

+ iγBp2Bi

*,

dz

Bi

= i(βi

-

βp

+ γP)Bi

+ iγBp2Bs

*.

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Conjugate the i-equation and look for eigenvalues

(MI wavenumbers) k.

k = (δs

–

δi

)/2 ±

[(δs

+ δi

)2/4 -

(γP)2]1/2, where δj

= βj

-

βp

+ γP.

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Define the (wavenumber) mismatch δ

= (δs

+ δi

)/2 = (βs

– 2βp

+ βi

)/2 + γP.

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If |δ| > γP, then k is real; the MI is stable, (s and i) sidebands do not grow.

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If |δ| < γP, then k is imaginary; the MI is unstable, sidebands grow.

[C. McKinstrie, J. Sel. Top. Quantum Electron. 8, 538 & 956 (2002).]

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6MPI, June 2011

When is the MI unstable?

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Expand the wavenumbers

about the pump frequency.

βj

(ωj

) = β0

(ωp

) + β1

(ωp

)(ωj

–

ωp

) + β2

(ωp

)(ωj

–

ωp

)2/2; ωs,i

= ωp

± ω,

2δ

= [β0

(ωp

) + β1

(ωp

)ω

+ β2

(ωp

)ω2/2] -

2β0

(ωp

)

+

[β0

(ωp

) -

β1

(ωp

)ω

+ β2

(ωp

)ω2/2] + 2γP = β2

(ωp

)ω2

+ 2γP.

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If β2

(ωp

) > 0 (normal dispersion), then |δ| > γP; MI is stable.

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If -4γP < β2

(ωp

)ω2

< 0 (anomalous dispersion), then |δ| > γP; MI is unstable.

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The maximal spatial growth rate γP

is attained when ω

= (2γP/|β2

|)1/2.

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In the presence of higher-order dispersion, extra gain bands can exist.

-75

-55

-35

-15

5

1560 1570 1580 1590 1600

λ0(dB)

(nm)

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7MPI, June 2011

Input-output equations for MI

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Let Bs

= Cs

exp[i(δs

-

δi

)z/2] and Bi

= Ci

exp[i(δi

–

δs

)z/2]. Then the MI equations can be written in the symmetric form

dz

Cs

= iδCs

+ iγBp2Ci

*, dz

Ci

* = -iδCi

* -

iγ(Bp

*)2Cs

,

where the (common) mismatch δ

= (δs

+ δi

)/2.

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The solutions of the MI equations can be written in the input-output form

Cs

(z) = μ(z)Cs

(0) + ν(z)Ci

(0), Ci

*(z) = ν*(z)Cs

(0) + μ*(z)Ci

*(0),

where the transfer (Green) functions

μ(z) = cos(kz) + iδsin(kz)/k, ν(z) = iγBp2sin(kz)/k

and the MI wavenumber

k = [δ2

- (γP)2]1/2.

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Notice that |μ(z)|2

- |ν(z)|2

= 1, from which it follows that

|Cs

(z)|2

- |Ci

(z)|2

= [|μ(z)|2

- |ν(z)|2][|Cs

(0)|2

- |Ci

(0)|2] = |Cs

(0)|2

- |Ci

(0)|2.

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Sideband photons are created in pairs (linear theory)!

[C. McKinstrie, Opt. Express 12, 5037 (2004).]

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8MPI, June 2011

Conservation equations for MI

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With pump-depletion included, the nonlinear MI equations are

dz

As

= i(βs

+ 2γ|Ap

|2)As

+ iγAp2Ai

*,

dz

Ap

= i(βp

+ γ|Ap

|2)Ap

+ i2As

Ai

Ap

*,

dz

Ai

= i(βi

+ 2γ|Ap

|2)As

+ iγAp2As

*.

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The signal equation implies that

dz

|As

|2

= iγAp2Ai

*As

* -

iγ(Ap

*)2Ai

As

.

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By combining this and similar equations, one obtains the Manley-Rowe-Weiss (MRW) equations

dz

(|As

|2

+ |Ap

|2

+ |Ai

|2) = 0,

dz

(|As

|2

- |Ai

|2) = 0.

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Dim(|A|2) = E/T and the photon energies ≈

hω0

. Dim(|A|2/hω0

) = 1/T (photon flux).

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Photons are created and destroyed in pairs (2 pump or 2 sideband

photons):

2πp

↔ πs

+ πi

, where πj

is a photon with frequency ωj

.

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MRW and FM imply energy conservation: dz

(|As

|2ωs

+ |Ap

|2ωp

+ |Ai

|2ωi

) = 0.

[J. Manley, Proc. IRE 44, 904 (1956), M. Weiss, Proc. IRE 45, 1012 (1957).]

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9MPI, June 2011

Radiation generation in photonic-crystal fiber

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Singly-resonant OPO: PCF (l = 1.3 m, γ

= 110/Km-W), pulsed pump (τ

= 8 ps, λ

≈

710 nm, P > 15 W), dichroic

mirrors. Frequency shifts from 20 –

170 THz.•

Performance was limited by pump-sideband walk-off.

[Y. Xu, Opt. Lett. 33, 1351 (2008); S. Murdoch (2009).]

(S: 770–1150 nm)

(aS: 510-690 nm)

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10MPI, June 2011

Broad-bandwidth amplification for communication

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Parametric amplifiers have broader gain bandwidths than their competitors.

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The current record bandwidth is 150 nm (signal plus idler).

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Perpendicular pumps provide signal-polarization-independent gain.

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Standard system with 128 channels at 10 Gb/s

requires 51 nm bandwidth.

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Latest system (AL 1830) with 88 channels at 100 Gb/s

requires 35 nm.

15

20

25

30

35

1560 1580 1600

Parametric AmpRaman Amp, one pumpErbium Fiber Amp (shifted 30 nm)

Wavelength (nm)

Gai

n (d

B)

[R. Jopson

(2004); J. Chavez Boggio, Photon. Technol. Lett. 21, 612 (2009).]

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11MPI, June 2011

HOM interference for quantum information science

λ/2 plate angle, θ

(rads)

4-Fo

ld C

oinc

iden

ce R

ate

Purity = 1Purity = 0

λ/2@θ

D

BA

C

λ/2@θ

D

BA

C

Purity = 86 ±

2%

[O. Cohen, PRL 102, 123603 (2009).]

•Hong-Ou-Mandel interference: Good wave-packets →

high-visibility fringes.

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12MPI, June 2011

General scientific goals

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High-gain FWM amplifies input signals and generates idlers, or generates signals and idlers from noise.

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Low-gain FWM generates photon pairs (1 signal and 1 idler photon).

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FWM also frequency converts input signals without gain.

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Tutorial discussion pertained to monochromatic, or continuous-wave (CW), pump, signal and idler waves. In many applications, the waves are pulses.

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What are the generated photon pulses (wave-packets) like and how do they depend on the system (fiber and pump) parameters? How do we tailor them for applications in quantum information science?

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What input pulses optimize the operations of FWM processes (amplification, frequency conversion, pulse reshaping)?

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13MPI, June 2011

Coupled-mode equations

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Signal and idler evolution is governed by the coupled-mode equation (CME)

dX/dz

= iAX

+ iBX*,

where A is hermitian

and B is symmetric (QM).

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S and I each have 1 F-component: X = [As

,Ai

]t.

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S and I have 1 F-

and 2 P-components: X = [Asx

,Asy

,Aix

,Aiy

]t.

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S and I have n F-components: X is a 2n x 1 vector.

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S and I have n F-

and 2 P-components: X is a 4n x 1 vector.

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Question: Under what conditions can the CME be solved analytically?

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If A and B are simultaneously diagonalizable, then dxj

/dz

= iαj

xj

+ iβj

xj+,

where xj

is an e-amplitude, and αj

and βj

are e-values (1-mode squeezing).

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In most applications, A and B are not simultaneously diagonalizable!

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14MPI, June 2011

Singular value (Schmidt) decomposition

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Every complex matrix M = UDV+, where U and V are unitary and D is diagonal.

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The columns of U are e-vectors of MM+

and the columns of V are e-vectors of M+M. These e-vectors are called Schmidt modes.

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The entries of D (Schmidt coefficients σ) are the square roots of the (common) non-negative e-values of MM+

and M+M.

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M = UDV+

is equivalent to M = Σj

Uj

σj

Vj+.

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Consider Y = MX, where X and Y are input and output vectors: M resolves input modes ―

dilates mode amplitudes ―

projects output modes.

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SVD common in numerical mathematics (e.g. least squares minimization, linear equations). Recently became common in theoretical quantum

optics.

[G. Stewart, SIAM Rev. 35, 551 (1993).]

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15MPI, June 2011

Input-output equations

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Consider the IOE X(z) = M(z)X(0) + N(z)X*(0); M and N are transfer matrices.

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Abbreviate as Y = MX + NX*; X and Y are input and output vectors.

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Individual SVDs

M = Uμ

Dμ

Vμ+

and N = Uν

Dν

Vν+

are not useful by themselves.

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QM requires that MM+

- NN+

= I, i.e. Uμ

Dμ2Uμ

+

- Uν

Dν2Uν

+

= I. Hence, Um

= Uν

and μj

2

–

νj2

= 1.

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The inverse transformation is X = M+Y –

NtY*. QM requires that M+M –

NtN* = I, i.e. Vμ

Dμ2Vμ

+

- Vν

*Dν2Vν

t

= I. Hence, Vμ

= Vν

*.

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M and N have simultaneous SVDs: Y = UDμ

V+X + UDν

VtX*.

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Define yj

= Uj+Y and xj

= Vj+X. Then yj

= μj

xj

+ νj

xj+.

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Each input-mode operator is related to a single output-mode operator by a 1-mode squeezing transformation (simple)!

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Comments: The Schmidt coefficients and modes depend on z.

The theorem does not tell us how to determine M and N.

[S. Braunstein, Phys. Rev. A 71, 055801 (1995); C. McKinstrie, Opt. Commun. 282, 583 (2009).]

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16MPI, June 2011

The adjoint

method works well

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dX/dz

= iAX

+ iBX* and its conjugate can be rewritten as dY/dz

= iLY, where Y = [X,X*]t

and L = [A, B; B*, A*] is not self-adjoint.

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Determine the e-vectors (LEj

= λj

Ej

) and adjoint

e-vectors (L+Fj

= λj

*Fj

).

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The e-vectors are bi-orthogonal: Fj+Ek

= δjk

or F+E = I.

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Adjoint

decomposition: L = Σj

Ej

λj

Fj+

= EDλ

F+.

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exp(iLz) = Σj

Ej

exp(iλj

z)Fj+

and exp(iLz2

)exp(iLz1

) = exp[iL(z1

+z2

)].

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Comment: The e-vectors Ej

and Fj

do not depend on z and the propagation coefficients exp(iλj

z) depend simply on z. Transfer matrices concatenate.

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Question: What is the physical significance of the e-modes, which involve xj

and xj+, so are not superposition modes?

[C. McKinstrie, Opt. Commun. 282, 583 (2009).]

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17MPI, June 2011

Follow the breadcrumbs!

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Consider the CME dY/dz

= iLY. L is almost hermitian:

L = [A, B; -B*, -A*] = S1

H where S1

= [I,0; 0, -I] and H = [A, B; B*, A*]. Properties of e-values and e-vectors?

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Consider the IOE Y(z) = T(z)Y(0): T = [M, N; N*, M*] is symplectic:

TS2

Tt

= S2

, where S2

= [0, I; -I, 0]. (MM+

- NN+

= I, MNt

– NMt

= 0.)

•

The CME and IOE can be reformulated in terms of real quantities:

L →

real non-symmetric and T →

real symplectic?

[R. Littlejohn, Rep. Phys. 138, 193 (1986); R. Gilmore, Lie Groups (Dover, 2006); C. McKinstrie, J. Sel. Top. Quantum Electron. (2011).]

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18MPI, June 2011

Summary

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CMEs

govern a variety of parametric processes in classical communications and quantum information science.

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It is important to determine the natural input and output modes of these processes, and how they depend on the system parameters.

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The mathematics of CMEs

(IOEs) involve Schmidt and adjoint

decompositions.

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The relation between these decompositions requires further study!