Diploma thesis

3.2 Pure heralded single photons
3.2.1 Quantum networks and heralding
The fidelity of pure heralded quantum states is of critical importance to quantum
networks. As a source of single photons, a lot of attention has been paid to the
process of spontaneous parametric downconversion. Signal and idler exhibit strict
photon number correlations, which can be utilised to herald the existence of the
signal photon by detecting the corresponding idler photon, resulting in a heralded
single photon source.
One easy way to separate the two-photon-state, after it left the waveguide, is
by the use of type-II downconversion and a polarizing beam splitter (PBS) (Figure
3.23). Detecting the idler photon to herald the presence of the signal photon is,
Figure 3.23: Heralding single photons with type-II PDC
in this case, equivalent to discarding all information about the idler photon. This
measurement process is modelled as a partial trace over the Hilbert space belonging
to the idler wave packet.
T ri |ψs,i〉 〈ψs,i| = ρs
(3.38)
After this detection the signal photon, in general, will not be in a pure single photon
state, but in a mixed single photon state ρs. If and only if the two-photon-state
|ψs,i〉 can be written as a product state |ψs〉 ⊗ |ψi〉, the heralded single photon will
be in a pure state.
T ri (|ψs〉 〈ψs| ⊗ |ψi〉 〈ψi|) = |ψs〉 〈ψs| ⊗ T ri |ψi〉 〈ψi| = |ψs〉 〈ψs| (3.39)
Two-photon-states that can be decomposed into a product state are called decorrelated
two-photon-states. Under what conditions is it possible to generate these
signal and idler photons into a product state? From the two-photon-state deduced
in Equation 3.23 we can immediately see that signal and idler states have to be
separable:
� ∞ � ∞
|ψs,i〉 = A
0 0 � ∞
!
= A
0
dωs dωif(ωs, ωi)â † sâ †
i |vac〉
dωs fs(ωs)â † s |0〉 ⊗
� ∞
0
dωi fi(ωi)â †
i |0〉 . (3.40)
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