and the coincidence of two transmitted photons
for different combinations of g is measured.
In our experiment, polarization-entangled photon pairs (uncorrected average visibility 97.99 T
0.03%) at 810 nm were created using a type II
nonlinear crystal in a Sagnac-type configuration
(26, 27). The SLM in the transfer setup is programmed such that the reflected photons acquire
l multiples of 2p azimuthal phase (l quanta of
OAM), which leads to complex patterns when
OAM is large (Fig. 1, B to D). Therefore, we
used a high-resolution SLM (1920 × 1080, full
HD; Holoeye Photonics AG, Berlin) with small
pixel size (8 mm). Nonetheless, for values of l ≥
300 we observed a clear reduction of mode transformation efficiency, which is the main limiting
factor (22). This is only a technical limitation that
can be overcome by higher-resolution SLMs or
novel techniques for creating photons with higher
l values (28). After the transfer setups, the modes
are enlarged to fit the masks (laser-cut black cardboard) and transmitted photons are focused to
bucket detectors.
As a demonstration of the flexibility of our
setup, we created two-dimensional spatial mode
entanglement with highly asymmetric OAM
states, in which one photon is transferred to l =
T10 and the second photon to l = T100 (Fig. 3A).
Because of its intrinsic conservation of angular
momentum, the SPDC process could not create
this asymmetric state directly. We then transferred
both photons to l = T100, showing the ability to
create OAM modes with very high difference
in quantum number (Fig. 3B). The highest value
of OAM per single photon where strong correlations were still measurable was l = T300 for
both photons (Fig. 3C). The decrease in mode
transformation efficiency of the SLM, however,
strongly affects the coincidence rate (about 1 coincidence count per minute in the maximum) and
therefore the statistical significance of our results.
To demonstrate successful transfer, we constructed an entanglement witness [similar to (29)],
which verifies entanglement if the sum of two
visibilities in two mutually unbiased bases is
above the classical bound of (21/2 + 1)/2 ≈ 1.21
(22, 29). The data for the visibilities were taken
in addition to the fringe measurements (apart from
l = T300) with longer integration. For the asymmetric OAM state l = T10/T100, we achieved a
witness value of 1.48 T 0.01. When both photons
were transferred to l = T100, the witness value
was 1.55 T 0.01. Both values were calculated
without any correction of the data and violate
the classical limit by ~30 standard deviations,
demonstrating the successful entanglement transfer. Because of the significantly smaller creation and detection efficiencies (hence a lower
pair detection rate) for l = T300, we corrected
for accidental coincidence counts (22), yielding
a value of 1.6 T 0.3 for our entanglement witness. With a statistical significance of more than
80%, we thus violate the bound for separable
states with photons that each carry l = T300
quanta of OAM.
Fig. 3. Measured coincidence
counts as a function of the angle
of one mask and different angles
of the other mask. The measured
coincidence counts (points) show
a sinusoidal dependence (fitted
lines) and depend only on the dif-
ference between the angles of
the masks, which is a clear signa-
ture of nonclassical correlations.
(A) The first photon is transferred
to l = T10ℏ and the second to
l = T100ℏ, showing the ability
to create asymmetric OAM entan-
gled states. (B) Both photons are
transferred to l = T100ℏ. (C)
Both photons carry l = T300ℏ
and nonclassical correlations can
still be measured. Here, the count
rate decreased significantly (about
1 coincidence count per minute)
primarily because of limited con-
version efficiency. The integration
times in (A), (B), and (C) were 2
min, 9 min, and 64 min, respec-
tively, for each data point. Error
bars in all plots (if large enough
to be seen) are estimated from
Poissonian count statistics.
Fig. 4. Measurements
of remote angular sensi-
tivity enhancement. (A)
Normalized coincidence
count rates where one pho-
ton is projected on diag-
onal polarization, and the
second photon is either
kept polarization-encoded
while the polarizer is ro-
tated (green triangles) or
transferred to l = T10ℏ
(blue diamonds), to l =
T100ℏ (red squares), or
to l = T300ℏ (black cir-
cles) while the appropri-
ate mask is rotated. The
errors are estimated as-
suming Poissonian count
statistics. (B) From the
steepest part of the fringes
(0°), it is possible to cal-
culate the corresponding
angular sensitivity limited
by statistical fluctuations
for different numbers of
detected pairs. The dashed
lines are the theoretically
expected sensitivities (as-
suming 100% visibility and Poissonian fluctuation) and the points are the measured values. To illustrate
the enhancement for 100 detected pairs, we measured the angular position of the randomly rotated mask
by correcting the change in the coincidence counts with a rotation of the remote polarizer. The right panel
shows histograms of 20 different random angles that were measured for each arrangement. For l =
T300ℏ, the limit of our high-precision rotation stage (T0.016°) was determined with the polarizer in a
low-precision mount (T1°). To reach the same precision without OAM-induced angular resolution en-
hancement, about 3.3 million detected pairs would have been necessary.
