Spatial Characterization Of Two-Photon States - GAP-Optique

2.2. Correlations between space and frequency
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 λ0 s = λ0 i = 810
nm at an angle ϕs,i = 10◦ .
Under the above conditions, figure 2.3 shows the purity of the spatial state
T r[ρ2 q] given by equation 2.12, as a function of the spatial filter width for different
frequency filter bandwidths. The half width at 1/e of the frequency filters
∆λn, is Bn = πc∆λn/(λ2 √
n ln 2). Figure 2.3 shows the presence of correlations
between space and frequency in all the situations for which the purity is smaller
than one.
According to figure 2.3, the spatial purity of the two-photon state gets closer
to one as ws (= wi) increase. Narrow filters in space (infinitely large ws,i) makes
the two-photon state separable in frequency and space. The separability and
the lack of correlations between frequency and space appear also in the“narrow
band” limit for the frequency. If ∆λs = ∆λi → 0 nm, then the purity is always
1 for any value of ws = wi.
When using typical interference filters, ∆λs = ∆λi ≈ 1 nm, the purity decreases
indicating space/frequency correlations. As the width of the frequency
filters increases, the purity converges quickly, the case of ∆λs = ∆λi = 10 nm
is almost equivalent to the case without frequency filters ∆λs, ∆λi → ∞.
In order to compare the theoretical results of this section with previous
experimental work, figure 2.4 shows the purity of the spatial state as a function
of the collection mode for three reported experiments. Figure 2.4 (a) shows the
purity for the data reported in reference [6]. The authors of that paper studied
a type-i spdc process in a 1 mm long liio3 crystal. They used a diode laser
with wavelength λp = 405 nm and bandwidth ∆λp = 0.4 nm as a pump beam.
And, by using monochromators with ∆λs,i = 0.2 nm, they detected pairs of
photons emitted at ϕs,i = 17◦ . Table 2.1 resumes these parameters.
The paper analyzes two pump waist values: wp = 30 µm and wp = 462 µm.
For those values, the authors demonstrated control of the frequency correlations
by changing the spatial properties of the pump beam. As the pump beam
influences the spatial shape of the generated photons, they implicitly reported
a correlation between space and frequency. This spatial to spectral mapping
exists at their collection mode ws = 133.48 µm for which the correlation indeed
appears according with our model.
Figure 2.4(b) shows T r[ρ2 q] for the case described in reference [46]. The
authors used a continuous wave pump beam with λp = 405 nm to generate
photons with λs,i = 810 nm in a 1.5 mm length bbo crystal. They detected the
photons after filtering with ∆λs,i = 10 nm interference filters. The pump beam
waist satisfied the condition wp >> L, and the generated photons propagated
at ϕs,i = 0◦ .
The experiment reported in reference [46] does not show evidence of spacefrequency
correlations. In agreement with this fact, the figure shows a lack
of correlation for the collinear case. However, under the conditions reported,
but in a non collinear configuration, it is not possible to neglect the space and
frequency correlations.
Finally figure 2.4 (c) plots T r[ˆρ 2 q] for the case described in reference [31].
In this paper, the authors illuminated a 2 mm long bbo crystal with a pump
beam with λp = 351.1 nm and wp ≈ 20 µm. Their crystal generated photons
at λs,i = 702 nm that propagated at 4o . They collected the photons after using
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