Diploma thesis

3.2.2 Frequency entanglement of two-photon-states
One possibility to quantify the spectral amount of entanglement is to perform a
Schmidt decomposition of the two-photon state. It is a transformation into a unique
set of orthonormal functions:
|ψs,i〉 = �
n
� λn |ψ n s 〉 ⊗ |ψ n i 〉 . (3.52)
The λn are the Schmidt coefficients and ψn s , ψn i are called Schmidt functions. With
the help of the λn it is possible to quantify the amount of entanglement. If only
one nonzero λn exists, we are coping with an uncorrelated two-photon-state. A high
number of Schmidt coefficients is equal to an entanglement in the frequency domain.
There are several possibilities to quantify the amount of entanglement. For example
the cooperativity parameter K, it equals 1 for an uncorrelated two-photon-state
and rises with successive contribution of Schmidt modes.
K =
1
� ∞
n=0 λ2 n
(3.53)
The entropy of entanglement is an alternative figure of merit and starts from 0
for an uncorrelated two-photon-state, also rising with increasing correlations.
S = −
∞�
λn log2(λn) (3.54)
n=0
The Schmidt decomposition, for a pure two-photon state, is performed by a numerical
singular value decomposition of the corresponding JSA. To illustrate this we
performed Schmidt decompositions of a correlated and an uncorrelated two-photonstate.
The Schmidt numbers in Figure 3.27 belong to the JSA plotted in Figure 3.26.
This two-photon-state exhibits a vast amount of Schmidt numbers, slowly decaying
for higher Schmidt modes ψ n s , ψ n i
, and hence of high cooperativity K = 33.93 and
entropy of entanglement S = 5.32.
In the case of a mostly uncorrelated JSA as plotted in Figure 3.30, the first Schmidt
number approaches 1 and the two-photon state is almost perfectly decorrelated
(K=1.20 and S=0.68) (Figure 3.31). Most signal and idler photons are emitted in
the first pair of Schmidt modes (Figures 3.32 and 3.33).
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