Spatial Characterization Of Two-Photon States - GAP-Optique

2. Correlations and entanglement
Signal
purity Filters
1
0
0.1 Collection mode 1mm
0nm
1nm
10nm
→∞
Figure 2.6: The correlations between photons can not be suppressed by filtering
in only one degree of freedom. The purity of the signal state is only maximum
when using ultra-narrow spatial and frequency filters. In this case, the 1 nm filters
are broad enough to allow the correlations, and the 10 nm filters are equivalent to
infinitely broad ones. The parameters used to generate this figure are listed in table
2.2.
The differences between the matrix C and matrix B in the last section, are
due to the order of the primed and unprimed variables in the arguments of the
mode functions in equations 2.14 and 2.21.
The next part of this section uses equation 2.20, to calculate the values of
the purity of the signal photon for different spdc configurations.
2.3.3 Numerical calculations
Figure 2.6 shows the space-frequency purity of the signal photon as a function
of the spatial filter width for different values of the frequency filter width.
Consider a degenerate type-i spdc configuration, where a pump beam with
wavelength λ 0 p = 405 nm, and beam waist wp = 400 µm illuminates a lithium
iodate (liio3) crystal with length L = 1 mm, and negligible Poynting vector
walk-off (ρ0 = 0). The crystal emits signal and idler photons with a wavelength
λ 0 s = λ 0 i = 810 nm at an angle ϕs,i = 10 ◦ , and the collection modes for signal
and idler are assumed to be equal (ws = wi). All these parameters are listed
in the first column of table 2.2.
As was discussed in the first part of this section, it is possible to achieve
maximal separability between the photons by using infinitely narrow filters
in both space and frequency. In the region of small values of ws (= wi) a
considerable correlation between signal and idler exists even in the case of
infinitely narrow frequency filters. Different values for the signal photon purity
can be achieved by changing the filter width, as shown by figure 2.6.
Additionally, the figure shows how the purity T r[ˆρ 2 signal ] is confined between
the values obtained for ∆λs = ∆λi → 0 nm and ∆λs, ∆λi → ∞. This limits
can be tailored by modifying the other parameters of the spdc configuration.
Figure 2.7 shows T r[ˆρ 2 signal ] as a function of the pump beam waist wp for
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