In this appendix we derive explicit expressions for the scattered two-photon density matrix in Eq. (6.9), recapitulated,
ρ(out)qr,q′r′ = X
klk′l′
AklA∗
k′l′uqkurlu∗q′k′u∗r′l′. (6.13)
The random unitary matrix ensemble will be used for averaging over different real- izations of disorder. Any unitary integral of the form
ua1b1...uanbnu∗c1d1...u
∗
cndn
is nonzero if and only ifc∈P(a) andd∈P(b), whereP(x) is the collection of all different permutations ofx, [147, 148]. Logically, there are four essentially different
nonzero second-order integrals possible, being [147, 148] α1 ≡ u11u11u∗11u∗11= 2M M2(M+ 1), (6.14) α2 ≡ u11u12u∗11u∗12= M M2(M+ 1), (6.15) α3 ≡ u11u22u∗11u∗22= M2 M2(M+ 1)(M −1), (6.16) α4 ≡ u11u22u∗12u∗21= − M M2(M+ 1)(M −1), (6.17)
where M is the dimension of the random unitary matrix. When applying this knowledge to Eq. (6.13) it is found that the density matrix elements can take four different values ρ(out)qr,q′r′ = X klk′l′ AklA∗ k′l′uqkuqlu∗qk′uql∗′ , q=q′ =r=r′ X klk′l′ AklA∗ k′l′uqkurlu∗qk′u∗rl′ ,(q=q′)6= (r=r′) X klk′l′ AklA∗ k′l′uqkurlu∗rk′u∗ql′ ,(q=r′)6= (r=q′) 0 ,elsewhere. (6.18)
Now it is a matter of bookkeeping to perform the summation and using the integrals in Eqs. (6.14)-(6.17). The nonzero contributions in the summations of Eq. (6.18) are listed in Tab. 6.2. By applying this list to Eq. (6.18) we obtain
ρ(out)qr,q′r′ = α1D+α2(T − D) +α2(S − D) , q=q′=r=r′ α2D+α3(T − D) +α4(S − D) ,(q=q′)6= (r=r′) α2D+α4(T − D) +α3(S − D) ,(q=r′)6= (r=q′) 0 ,elsewhere , (6.19)
where we introduced three functionals of the inputAkl being
T ≡ M X k=1 M X l=1 AklA∗ kl (6.20) D ≡ M X k=1 AkkA∗ kk, (6.21) S ≡ M X k=1 M X l=1 AklA∗lk. (6.22)
uqkuqlu∗
qk′u∗ql′ uqkurlu∗qk′u∗rl′ uqkurlu∗rk′u∗ql′
k=l=k′=l′ α1 α2 α2
(k=k′)6= (l=l′) α2 α3 α4
(k=l′)6= (l=k′) α2 α4 α3
Table 6.2: Listing of nonzero outcomes for the three unitary integrals in Eq. (6.18) for different combinations of the summation indicesk,l,k′, andl′(note thatq6=r). The coefficientsα
i are defined in Eqs. (6.14)-(6.17).
The symbolsT, D ∈[0,T], and S ∈[−(T − D),+(T − D)] stand fortotal power, diagonal power, andsymmetry parameter, respectively.
By substituting Eqs. (6.14)-(6.17) for the unitary integralsαi in Eq. (6.19) we find ρ(out)qr,q′r′ = 1 M T +S M+ 1 , q=q′=r=r′ 1 M(M−1) M T − S M+ 1 ,(q=q′)6= (r=r′) 1 M(M−1) M S − T M+ 1 ,(q=r′)6= (r=q′) 0 ,elsewhere (6.23)
We now make two remarks of mathematical interest. The first remark is thatT and S are functionals ofAkl that are conserved under transformation via any unitary scattering matrix. This can easily be demonstrated by combining the appropriate elements ofρ(out)qr,qr to obtain the averageT andS in the output state being
T(out) = X qr ρ(out)qr,qr=M ρ (out) 11,11+M(M −1)ρ (out) 12,12=T, (6.24) S(out) = X qr ρ(out)qr,rq=M ρ (out) 11,11+M(M −1)ρ (out) 12,21=S. (6.25)
The second remark is that the unitary integrals in Eqs. (6.14)-(6.17) are uniquely determined by stating thatT is a conserved quantity for anyM and any Akl. In other words, by combining Eqs. (6.24) and (6.19) one retrieves Eqs. (6.14)-(6.17). Actually, the unitary integrals are also uniquely determined by stating thatS is a conserved quantity for anyM and anyAkl. So by combining Eqs. (6.25) and (6.19) one retrieves Eqs. (6.14)-(6.17) as well.
Finally, we restrict ourselves to T = 1, which corresponds to normalized two- photon quantum states in Eq. (6.1). Furthermore, we concentrate on the on- diagonal elements of the scattered density matrix only. The relative strength of
these elements is given by the ratio F ≡ ρ (out) 11,11 ρ(out)12,12 = 1 +S 1 + 1− S M−1 , (6.26)
which we will call the pairing factor, whereS ∈[−1,1] andF ∈[0,2]. The pairing factor is the two-photon detection probability observed with two detectors looking into the same spatial channel divided by the two-photon detection probability ob- served with two detectors looking into different spatial channels. Explicit values for these probabilities are easily retrieved by using
Trρ(out)qr,q′r′ =M ρ (out)
11,11+M(M−1)ρ (out)
12,12= 1, (6.27)
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