Angular Two-Photon Interference and Angular Two-Qubit States
Anand Kumar Jha,1Jonathan Leach,2Barry Jack,2Sonja Franke-Arnold,2Stephen M. Barnett,3
Robert W. Boyd,1and Miles J. Padgett2
1Institute of Optics, University of Rochester, Rochester, New York 14627, USA
2Department of Physics and Astronomy, SUPA, University of Glasgow, Glasgow, United Kingdom
3Department of Physics, SUPA, University of Strathclyde, Glasgow, United Kingdom
(Received 7 August 2009; published 5 January 2010)
Using angular-position–orbital-angular-momentum entangled photons, we study angular two-photon
interference in a scheme in which entangled photons are made to pass through apertures in the form of
double angular slits, and using this scheme, we demonstrate an entangled two-qubit state that is based on
the angular-position correlations of entangled photons. The entanglement of the two-qubit state is
quantified in terms of concurrence. These results provide an additional means for preparing entangled
quantum states for use in quantum information protocols.
DOI: 10.1103/PhysRevLett.104.010501 PACS numbers: 03.67.Bg, 42.50.Ex, 42.50.Tx, 42.65.Lm
The signal and idler photons produced by parametric
down-conversion (PDC) are entangled in several different
degrees offreedom includingtime and energy, positionand
momentum, and angular position and orbital angular mo-
mentum (OAM). Entanglement of the two photons in a
given degree of freedom gives rise to two-photon coher-
ence in the corresponding domain, which manifests itself
as two-photon interference in that particular domain.
Several two-photon interference effects have been ob-
served in the temporal [1–5] and spatial [6–8] domains.
These effects have been used to test the foundations of
quantum mechanics [9–11] and are central to many appli-
cations as well [12–14].
The existence of a Fourier relationship between angular
position and OAM gives rise to angular interference—
interference in the OAM-mode distribution of a photon
field when it passes through an angular aperture [15–18].
Angular Fourier relationship in the context of angular-
position–OAM entanglement leads to two-photon interfer-
ence in the angular domain [19–21]. In this Letter, we
study angular two-photon interference in a scheme in
which entangled photons are made to pass through aper-
tures in the form of double angular slits and, using this
scheme, we demonstrate an entangled two-qubit state that
is based on the angular-position correlations of the down-
converted photons. Entangled two-qubit states are the nec-
essary ingredients for many quantum information proto-
cols [12–14], and they have previously been realized by
including polarization , time bin [4,5], frequency ,
position [7,8], transverse momentum [24,25], and OAM
[19–21]; however, to date, the angular-position correla-
tions had not been utilized. Therefore, the results presented
here not only demonstrate two-photon coherence effects in
the angular domain but also provide an additional means
for preparing entangled quantum states.
Let us consider the situation shown in Fig. 1(a). A
Gaussian pump beam produces signal and idler photons
by type-I degenerate PDC with noncollinear phase match-
ing. The state jctpi of the down-converted two-photon
field is given by [26,27]:
FIG. 1 (color online).
(see text for details). (b) An example of the phase pattern
impressed on the SLM. (c) Two-photon path diagrams showing
the four alternative pathways by which signal and idler photons
can pass through the angular slits and be detected in coincidence
at detectors Dsand Di.
(a) Schematic of the experimental setup
PRL 104, 010501 (2010)
8 JANUARY 2010
? 2010 The American Physical Society
Here s and i stand for signal and idler, respectively, and jli
represents an OAM eigenmode of order l, corresponding to
an azimuthal phase eil?. jclj2is the probability that the
signal and idler photons are generated in modes of order l
and ?l, respectively. The width of this mode probability
distribution is referred to as the spiral bandwidth of the
two-photon field . The signal and idler photons are
made to pass through double angular slits [as shown in
Fig. 1(a)] located in the image planes of the crystal. The
amplitude transmission functions of the individual angular
slits are given by
Ajað?jÞ ¼ 1
if ? ?=2 < ?j< ?=2
Ajbð?jÞ ¼ 1
if ? ? ?=2 < ?j< ? þ ?=2
where j ¼ s, i. There are, in principle, four alternative
pathways—represented by the two-photon path diagrams
pass through the apertures and get detected in coincidence
at detectors Dsand Di. In alternative 1 (4), the signal
photon passes through slit AsaðAsbÞ and the idler photon
through slit AiaðAibÞ. In alternative 2 (3), the signal photons
passes through slit AsaðAsbÞ and the idler photon through
slit AibðAiaÞ. We represent the states of the signal and idler
photons in alternatives 1, 2, 3, and 4 by jsaijiai, jsaijibi,
jsbijiai, and jsbijibi, respectively. Because of the strong
position correlations of the two photons in the image
planes of the crystal, only alternatives 1 and 4 have appre-
ciable probabilities. Therefore, the density matrix ? of the
two-qubit state thus prepared can be written in the angular-
position basis fjsaijiai;jsaijibi;jsbijiai;jsbijibig as:
? ¼ ?11jsaijiaihiajhsaj þ ?14jsaijiaihibjhsbj
þ ?41jsbijibihiajhsaj þ ?44jsbijibihibjhsbj;
where ?11and ?44are the probabilities that the signal and
idler photons are detected in alternatives 1 and 4, respec-
tively, with ?11þ ?44¼ 1. The off-diagonal term ?14is a
measure of coherence between alternatives 1 and 4, with
written as ?14¼
coherence and ? the argument of ?14.
We now write the density matrix ? in the OAM basis. By
taking the Fourier transforms of the amplitude transmis-
sion functions Asað?sÞ and Aiað?iÞ [15,16], corresponding
to each OAM mode in the summation of Eq. (1), we write
jsaijiai in the OAM basis as
where A is the normalization constant to ensure that
hiajhsajsaijiai ¼ 1. We evaluate jsaijiai by substituting
41; it is, in general, a complex number and can be
?ei?, where ? is the degree of
for Asað?sÞ and Aiað?iÞ from Eq. (2). In a similar manner,
we evaluate jsbijibi by substituting from Eq. (3). The
coincidence count rate Rsiof detectors Dsand Di, which
is the probability per ðunit timeÞ2that a photon is de-
tected at detector Dsin mode lsand another at detector
Diin mode li, is given by Rsi¼ihlijshlsj?jlsisjliii. Using
Eqs. (2)–(5), we find that
? f1 þ 2
?cos½ðlsþ liÞ? þ ??g:
The interference between the two alternatives manifests
itself in the periodic dependence on the angular separation
? and on the sum of the OAMs lsþ li. From Eq. (6),
ignoring the effects due to diffraction envelopes, the visi-
bility V of the coincidence fringes can be seen to be
The entanglement of a general two-qubit state can be
characterized in terms of Wootters’ concurrence [29,30],
which ranges from 0 to 1, with 1 corresponding to the
maximally entangled two-qubit state and 0 to a nonen-
tangled state. To calculate concurrence, we write the den-
sity matrix ? in the full 4 ? 4 form. The concurrence C is
then given by C ¼ maxf0;?1? ?2? ?3? ?4g. Here the
?is are the (positive) eigenvalues, in descending order, of
the operator R where R2¼
being the Pauli operator and ??the complex conjugate of
?. For the density matrix of Eq. (4), which has only two
nonzero diagonal elements, the concurrence C is
V ¼ 2?11?44
p?y? ?y???y? ?y
C ¼ 2j?14j ¼ 2
Comparing Eqs. (7) and (8), we see that the concurrence is
equal to the visibility of angular two-photon interference
C ¼ V:
In the setup of Fig. 1, the pump is a frequency-tripled,
mode-locked, Nd-YAG laser (Xcyte) with a pulse repeti-
tion frequency of 100 MHz at 355 nm. SLM denotes a
spatial light modulator from Hamamatsu, SMF a single
mode fiber, and F an interference filter with 10-nm band-
width, centered at710 nm. The 400 ?mdiameter Gaussian
pump beam was normally incident on a 3-mm-long crystal
of beta barium borate, phase matched for frequency degen-
erate type-I down-conversion with a semicone angle of the
down-converted beams of 3.5?. We note that for the given
pump beam and phase-matching parameters, the conserva-
tion of OAM is strictly obeyed in the down-conversion
process . The crystal plane was imaged, with a mag-
nification of about 5, onto the SLM planes, which were
PRL 104, 010501 (2010)
8 JANUARY 2010
then imaged onto the input facets of the SMFs with a
demagnification of about 380. The SLMs were used for
two purposes as illustrated in Fig. 1(b). One, they were
used for selecting out OAM modes ; and two, they
were used for simulating amplitude apertures  de-
scribed by Eqs. (2) and (3).
First of all, withoutanyapertures, the mode probabilities
jclj2were measured. Figure 2 shows the measured coinci-
dence counts plotted against l, with signal and idler pho-
tons being detected in modes of order l and ?l,
respectively. The mode probabilities jclj2were calculated
by normalizing the counts of Fig. 2.
Second, we verify the preparation of the two-qubit state
as represented by Eq. (4). Coincidence counts were mea-
sured with only one of the signal and one of the idler slits
(with ? ¼ ?=10 and ? ¼ ?=4) being displayed on the
SLMs and with both signal and idler photons being de-
tected in modes of order 0. Figure 3(a) shows the measured
coincidence detection probabilities of the signal and idler
photons in the four different alternatives. We find that the
probabilities ?22and ?33are negligibly small, showing that
the two-qubit state prepared in our experiment resembles
the state represented by Eq. (4) to a very good approxima-
tion. Therefore, as shown by Eq. (9), the entanglement of
the prepared two-qubit state can be characterized by mea-
suring the visibility of two-photon interference fringes in
the OAM basis.
Next, measurements were made in the OAM basis. Both
signal and idler slits, with ? ¼ ?=10 and ? ¼ ?=4, were
displayed on the SLMs. SLMiwas adjusted to successively
select out two different idler OAM modes: li¼ 2 and li¼
?2. For each selected idler mode li, coincidence counts
were measured as a function of the signal OAM mode ls.
Figure 3(b) shows the coincidence counts plotted against ls
for two different values of li. The solid dots are theoretical
fits based on Eq. (6), using the values of jclj2calculated
from Fig. 2. The visibility of the two-photon fringes is
92.8% (96.3%, after correcting for random coincidences),
within 2% experimental error. Thus, using Eq. (9), we find
that the concurrence of the prepared two-qubit state is
0.928 (0.963, after correcting for random coincidences).
However, as ?22and ?33are not precisely zero in our
experiment, it is desirable to quantify the error in the above
estimation of concurrence. Although a precise error calcu-
lation requires knowledge of all of the 16 different terms of
Letter, a realistic estimation can be obtained by modeling
the probabilities ?22and ?33as a small amount of noise in
the two-qubit density matrix ? in Eq. (4). The corrected
density matrix ?ðcÞ
?22jsaijibihibjhsaj þ ?33jsbijiaihiajhsbj and the visibility
VðcÞof angular two-photon fringes by VðcÞ¼ 2
with ?11þ ?22þ ?33þ ?44¼ 1. The concurrence CðcÞof
givenby ?ðcÞ¼ ? þ
OAM-mode order of signal and idler photons (l,-l)
Coincidence counts in 5 sec
FIG. 2 (color online).
abilities with the SLMs set for uniform reflectivity. Measured
coincidence counts are given as a function of l, the OAM-mode
order of the detected signal photon, with ?l being the OAM-
mode order of the idler photon. The solid dots are the expected
values based on the theoretical prediction of Ref.  [Eq. (10)];
the solid line through the dots is drawn as a visual guide. The
fitting parameters are an overall constant factor and the effective
beamwidth of down-converted modes as measured by the detec-
Measurements of the OAM-mode prob-
11 4433 22
Coincidence counts in 60 sec
FIG. 3 (color online).
probabilities ?11, ?22, ?33, and ?44. The measured probabilities
are shown by blue bars (black in the printed version); the
probabilities after correcting for random coincidences are shown
by light blue bars (gray in the printed version). (b) Measured
coincidence counts (light red, gray in the printed version) as
functions of lsfor two different values of li, with ? ¼ ?=10 and
? ¼ ?=4. The solid dots are theoretical fits obtained from
Eq. (6), the solid lines are visual guides, and the dashed lines
are measured random coincidences for the 10-ns coincidence
(a) Measured coincidence detection
PRL 104, 010501 (2010)
8 JANUARY 2010
the state takes the following form: CðcÞ¼ 2
, and with the above expression for visibility, it
can be written as
CðcÞ¼ VðcÞ? 2
which reduces to the formula given in Eq. (9) when
?22¼ ?33¼ 0.
Figs. 3(a) and 3(b), we find that the concurrence CðcÞof
the prepared two-qubit state is 0.875 (0.929, after cor-
recting for random coincidences), which differs from
the value obtained from Eq. (9) by about 6% (4% after
correcting for random coincidences).
Finally, to test the applicability of our method, we
perform a series of experiments with various values of
slit separation ?. Figure 4 shows the measured coincidence
counts as a function of lsfor four different values of ?:
?=6, ?=4, ?=2, and ?. The high visibility (between 85%
to 92%, without correcting for random coincidences) of
these plots shows that the angular-position correlations of
the signal and idler photons are almost uniform over the
entirerange of2?radians andthatthemethodisapplicable
over a wide range of ? values.
In conclusion, we have studied two-photon interference
in the angular domain and have reported experimental
demonstrations of an entangled two-qubit state that is
based on the angular-position correlations of the entangled
two-photon field. These results provide an additional
means for preparing entangled two-qubit states and con-
stitute a step towards better understanding angular-
position—OAM entanglement [26–28] and thus towards
finding novel ways of utilizing OAM basis for quantum
information science [19–21,32,34]. We believe that the
method presented in this Letter can be easily generalized
for preparing entangled two-qudit states, using apertures
with d angular slits.
Using thevalues in
We gratefully acknowledge financial support through a
MURI grant from the U.S. Army Research Office, through
DARPA/DSO, the Future and Emerging Technologies
(FET) programme within
Programme of the European Commission, HIDEAS
(No. FP7-ICT-221906), the UK EPSRC, RCUK, the
Royal Society, and the Wolfson Foundation. We would
like to thank Hamamatsu for their support of this work.
We thank M.N. O’Sullivan, L. Neves, and S. Agarwal for
useful discussions, and we also thank the referees of this
Letter for some important comments and suggestions.
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Coincidence counts in 60 sec
-10 -505 10-10-50510
FIG. 4 (color online).
tion of lsfor four different values of ?, with ? ¼ ?=10 and li¼
0. Each plot is an average of 24 different plots, taken with the
starting angles of both the signal and idler apertures rotated in
steps of 15?from 0?to 360?.
Measured coincidence counts as a func-
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