## ABSTRACT

It appears from the description in Sec. 15.3 that a global state containing a single photon in each of the modes i = (q, s) and j = (q′, s′) has the form

|ψ〉 = |ψij〉 ⊗ ∏ k 6=i,j

|0k〉, (33.1)

where

|ψij〉 = |1i〉 ⊗ |1j〉 = aˆ†i (0)aˆ†j(0)|0i〉 ⊗ |0j〉. (33.2)

A Fock state containing precisely two photons also can be constructed by putting the two quanta into the same mode (say i), that is

|ψij〉 = |2i〉 ⊗ |0j〉 = 1√ 2

[ aˆ†i (0)

]2 |0i〉 ⊗ |0j〉. (33.3)

By linear superposition of Fock modes of the type given in Eq. (33.1) a general polychromatic two-photon state, |Φ2〉, can be formed:

|Φ2〉 = 1 2 L−3

∑ i,j

φij aˆ † i (0)aˆ

† j(0)|0〉. (33.4)

The factor 1/2 appearing in Eq. (33.4) originates in our wish to end up with a relation between the two-photon wave functions in direct and reciprocal space having the same structural form as in the one-photon case [Eq. (15.125)]. The quantities φij are amplitude weight factors characterizing the various |Φ2〉-states. The form of the superposition in Eq. (33.4) shows that only the sum φij + φji can play a physical role, and without loss of generality one therefore may set

φji = φij . (33.5)

The inner product of |Φ2〉 with itself is readily obtained utilizing that

〈0|aˆkaˆlaˆ†i aˆ†j|0〉 = δkiδlj + δkjδli. (33.6)

of

Thus,

〈Φ2|Φ2〉 =L −6

∑ i,j,k,l

= L−6

∑ i,j,k,l

φijφ ∗ kl (δkiδlj + δkjδli)

= L−6

∑ i,j

φij ( φ∗ij + φ

)

= L−6

∑ i,j

|φ∗ij |2. (33.7)

The two-photon state is normalized, i.e.,

〈Φ2|Φ2〉 = 1, (33.8)

provided the weight factors satisfy the condition

L−6

∑ i,j

|φij |2 → 1

|φs,s′(q,q′)|2 d 3q

(2π)3 d3q′

(2π)3

= 1. (33.9)

By acting on |Φ2〉 with the global number operator one obtains

Nˆ |Φ2〉 = 1 2 L−3

∑ i,k,l

φklaˆ † i aˆiaˆ

= 1

2 L−3

∑ i,k,l

( aˆ†kaˆi + δik

) aˆ†l |0〉

= 1

2 L−3

∑ i,k,l

φkl

[ aˆ†i aˆ

† l δik + aˆ

( aˆ†l aˆi + δil

)] |0〉

= 1

2 L−3

∑

∑ i,k

= 1

2 L−3

∑ i,j

(φij + φji) aˆ † i aˆ † j |0〉. (33.10)

In view of Eqs. (33.4) and (33.5), one finally has

Nˆ |Φ2〉 = 2|Φ2〉. (33.11)

The polychromatic state |Φ2〉 hence is an eigenstate for the global number operator with eigenvalue 2, a result which of course follows from the construction in Eq. (33.4).