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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

exploring thecorrelationsofentangledphotonsinvariables

including polarization [22], time bin [4,5], frequency [23],

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]:

jctpi ¼

X

1

l¼?1

cljlisj?lii:

(1)

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

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0031-9007=10=104(1)=010501(4)010501-1

? 2010 The American Physical Society

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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 [28]. 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

or else

0; (2)

Ajbð?jÞ ¼ 1

if ? ? ?=2 < ?j< ? þ ?=2

or else

0;

(3)

where j ¼ s, i. There are, in principle, four alternative

pathways—represented by the two-photon path diagrams

[3]ofFig.1(c)—bywhichthedown-convertedphotonscan

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

?14¼ ??

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

jsaijiai¼AX

?X

where A is the normalization constant to ensure that

hiajhsajsaijiai ¼ 1. We evaluate jsaijiai by substituting

(4)

41; it is, in general, a complex number and can be

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

?11?44

p

?ei?, where ? is the degree of

l

cl

X

1

2?

l0

1

2?

Z?

Z?

??d?sAsað?sÞe?iðl0?lÞ?sjl0is

l00

??d?iAiað?iÞe?iðl00þlÞ?ijl00ii;

(5)

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

????????

?11?44

Rsi¼A2?4

16?4

X

l

clsinc

?

ðls? lÞ?

2

?

sinc

?

ðliþ lÞ?

2

?????????

2

? f1 þ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

p

?cos½ðlsþ liÞ? þ ??g:

(6)

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

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

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

with

?

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

?:

(7)

ffiffiffiffi?

p?y? ?y???y? ?y

ffiffiffiffi?

p,

?y¼

0

i

?i

0

?

C ¼ 2j?14j ¼ 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

?11?44

p

?:

(8)

Comparing Eqs. (7) and (8), we see that the concurrence is

equal to the visibility of angular two-photon interference

fringes.

C ¼ V:

(9)

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 [31]. The crystal plane was imaged, with a mag-

nification of about 5, onto the SLM planes, which were

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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 [32]; and two, they

were used for simulating amplitude apertures [33] 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

thetwo-qubitdensitymatrixandisbeyondthescopeofthis

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

is then

?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

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

?11?44

p

?,

2500

5000

7500

(-6, 6)

OAM-mode order of signal and idler photons (l,-l)

Coincidence counts in 5 sec

0

(-3, 3)

(0, 0)

(3, -3)

(6, -6)

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. [28] [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-

tion system.

Measurements of the OAM-mode prob-

(a)

Probability

0.5

0.4

0.3

0.2

0.1

0

ρρρρ

11 4433 22

0.486

0.016

0.028

0.460

-10-50

ls

5 10

100

200

100

200

(b)

Coincidence counts in 60 sec

li=−2

li=+2

0

0

0.495

0.470

0.025

0.019

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

detection window.

(a) Measured coincidence detection

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the state takes the following form: CðcÞ¼ 2

2?22?33

, and with the above expression for visibility, it

can be written as

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

?11?44

p

? ?

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

p

CðcÞ¼ VðcÞ? 2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffi

experimental

?22?33

p

;

(10)

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.

theSeventhFramework

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Coincidence counts in 60 sec

100

200

300

0

100

200

0

ls

ls

= /2

β π

= /4

β π

β=π

β π

=

/6

100

200

0

100

200

0

-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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