Hyperentanglement in Quantum

Interferometry

Mete Atat�ure,1 Giovanni Di Giuseppe,2 Matthew D. Shaw,2 Alexander V. Sergienko,1;2

Bahaa E. A. Saleh,2 and Malvin C. Teich1;2

Quantum Imaging Laboratory,1Department of Physics and 2Department of Electrical and Computer Engineering,

Boston University, 8 Saint Mary's Street, Boston, MA 02215

Abstract

The role of hyperentanglement in quantum interference is explored us-

ing spontaneous parametric down-conversion and various aperture con�gura-

tions. We have developed a general theory that includes hyperentanglement

in wavevector and polarization, which is consistent with a broad range of ex-

periments. In particular, the theory explains the observed modi�cation of the

usual triangular interference dip, including its asymmetrization and inversion.

This study lends insight to a general class of quantum states that is useful for

quantum information processing.

I. INTRODUCTION

In the nonlinear-optical process of spontaneous parametric down-conversion (SPDC) [1],

in which a laser beam illuminates a nonlinear-optical crystal, pairs of photons are generated

in a state that can be entangled [2] concurrently in frequency, momentum, and polarization.

A signi�cant number of experimental e�orts designed to verify the entangled nature of such

states were carried out using a single kind of entanglement, such as entanglement in energy

[3], momentum [4], or polarization [5]. In general, the entangled-photon quantum state is

not factorizable into single-parameter functions that carry their own entanglement indepen-

dently. Consequently any attempt to access one of these features is a�ected by the presence

of the others. The mainstream approach to investigating quantum interference to date has

been to eliminate the dependence of the quantum state on all forms of entanglement other

than the one under consideration. For example, when investigating polarization entangle-

ment spectral and spatial �ltering are typically imposed in an attempt to restrict attention

to polarization alone.

A more general approach to this problem is to consider, and make use of, the hyperentan-

gled quantum state from the outset. As a result, the observed quantum-interference pattern

in one feature, such as polarization, can be modi�ed at will by controlling the dependence of

1

the state on the other parameters, such as frequency and transverse wavevector. This strong

interdependence has its origin in the nonfactorizability of the quantum state into product

functions of the separate forms of entanglement.

In this paper we theoretically and experimentally study how the polarization quantum-

interference pattern, presented as a function of relative temporal delay between the photons

of an entangled pair, is modi�ed by controlling the optical system through di�erent kinds

of spatial apertures. Although the experimental work presented here focuses on continuous-

wave-pumped type-II SPDC, the theoretical formalism is general, so that the approach is

also suitable for pulse-pumped [6,7], as well as Type-I SPDC.

Our study leads to a deeper physical understanding of hyperentangled photon states

and, concomitantly, provides a route for engineering these states for speci�c applications,

including quantum information processing.

II. HYPERENTANGLED-STATE FORMALISM

With this motivation we present a complete multidimensional analysis of the entangled-

photon state generated via type-II SPDC. To admit a broad range of possible experimental

schemes we consider, in turn, three distinct stages in any experimental apparatus: the

generation, propagation, and detection of the quantum state [8].

A. Generation

By virtue of the relatively weak interaction in the nonlinear crystal, we consider the

two-photon state generated within the con�nes of �rst-order time-dependent perturbation

theory:

j(2)i � i

�h

tZt0

dt0 Hint(t0) j0i : (1)

Here Hint(t0) is the interaction Hamiltonian, (t0; t) is the duration of the interaction, and j0i

is the initial vacuum state. The interaction Hamiltonian governing this phenomenon is [9]

Hint(t0) � �(2)

ZV

dr E(+)p

(t0; r) E(�)o

(t0; r) E(�)e

(t0; r) + H:c: ; (2)

where �(2) is the second-order susceptibility and V is the volume of the nonlinear medium in

which the interaction takes place. The operator E(�)j

(t0; r) represents the positive- (negative-) frequency portion of the jth electric-�eld operator, with the subscript j representing the

pump (p), ordinary (o), and extraordinary (e) waves at time t0 and position r, and H.c. standsfor Hermitian conjugate. Because of the high intensity of the pump �eld it can be represented

by a classical c-number, rather than as an operator, with an arbitrary spatiotemporal pro�le

given by

Ep(r; t) =

Zdk

p

~Ep(k

p)eikp�re�i!p(kp)t ; (3)

2

where ~Ep(k

p) is the complex-amplitude pro�le of the �eld as a function of the wavevector

kp.

We decompose the three-dimensional wavevector kpinto a two-dimensional transverse

wavevector qpand frequency !

p, so that Eq. (3) takes the form

Ep(r; t) =

Zdq

pd!

p

~Ep(q

p;!

p)ei�pzeiqp�xe�i!pt ; (4)

where x spans the transverse plane perpendicular to the propagation direction z. In a

similar way the ordinary and extraordinary �elds can be expressed in terms of the quantum-

mechanical creation operators ay(q; !) for the (q; !) modes as

E(�)j

(r; t) =

Zdq

jd!

je�i�jze�iqj�xei!jt ay

j(q

j; !

j) ; (5)

where the subscript j = o; e. The longitudinal component of k, denoted �, can be written

in terms of the (q; !) pair as [8,10]

� =

vuut"n(!; �)!c

#2� jqj2 ; (6)

where c is the speed of light in vacuum, � is the angle between k and the optical axis of the

nonlinear crystal (see Fig. 1), and n(!; �) is the index of refraction in the nonlinear medium.

Note that the refractive index n(!; �) is the extraordinary refractive index, ne(!; �), when

calculating � for extraordinary waves, and it is the ordinary refractive index, no(!), for

ordinary waves.

Substituting Eqs. (4) and (5) into Eqs. (1) and (2) yields the wavefunction at the output

of the nonlinear crystal of thickness L:

j(2)i �Zdq

odq

ed!

od!

e�(q

o;q

e;!

o; !

e)ay

o

(qo; !

o)ay

e

(qe; !

e)j0i ; (7)

with

�(qo;q

e;!

o; !

e) = ~E

p(q

o+ q

e;!

o+ !

e)L sinc

�L�

2

�e�i

L�

2 : (8)

Here � = �p� �

o� �

ewhere �

j(j = p; o; e) is related to the indices (q

j; !

j) via relations

similar to Eq. (6). The nonseparability of the function �(qo;q

e;!

o; !

e) in Eqs. (7) and (8),

recalling (6), is the hallmark of concurrent hyperentanglement.

B. Propagation

Propagation between the planes of generation and detection is characterized by the

transfer function of the optical system. The biphoton probability amplitude [9] at the

space-time coordinates (xA; t

A) and (x

B; t

B), where detection takes place, is de�ned by,

A(xA;x

B; t

A; t

B) = h0jE(+)

A(x

A; t

A)E

(+)

B(x

B; t

B)j(2)i : (9)

3

The explicit forms of the quantum �elds present at the detection locations are represented

by [8]

E(+)A

(xA; t

A) =

Zdq d! e�i!tA [H

Ae(x

A;q;!)a

e(q; !) +H

Ao(x

A;q;!)a

o(q; !)] ;

E(+)B

(xB; t

B) =

Zdq d! e�i!tB [H

Be(x

B;q;!)a

e(q; !) +H

Bo(x

B;q;!)a

o(q; !)] ; (10)

where the transfer function Hij(i = A;B and j = e; o) describes the propagation of a (q; !)

mode from the nonlinear-crystal output plane to the detection plane. Substituting Eqs. (7)

and (10) into Eq. (9) yields a general form for the biphoton probability amplitude:

A(xA;x

B; t

A; t

B) =

Rdq

odq

ed!

od!

e�(q

o;q

e;!

o; !

e)

�hH

Ae(x

A;q

e;!

e)H

Bo(x

B;q

o;!

o) e�i(!etA+!otB)

+HAo(x

A;q

o;!

o)H

Be(x

B;q

e;!

e) e�i(!otA+!etB)

i: (11)

By choosing explicit forms of the functions HAe, H

Ao, H

Be, and H

Bo, the overall biphoton

probability amplitude can be altered as desired.

C. Detection

The formulation of the detection process requires some knowledge of the apparatus to be

used. Slow detectors, for example, impart temporal integration while �nite area detectors

impart spatial integration. One extreme case is realized when the temporal response of a

point detector is spread negligibly with respect to the characteristic time scale of SPDC,

namely the inverse of down-conversion bandwidth. In this limit the coincidence rate reduces

to

R = jA(xA;x

B; t

A; t

B)j2 : (12)

On the other hand, quantum-interference experiments typically make use of slow bucket

detectors. Under these conditions, the coincidence count rate R is readily expressed in

terms of the biphoton probability amplitude as

R =Zdx

Adx

Bdt

Adt

BjA(x

A;x

B; t

A; t

B)j2 : (13)

III. HYPERENTANGLED-STATE MANIPULATION

In this section we apply the mathematical description presented above to speci�c

quantum-interferometry con�gurations. Since the evolution of the state is ultimately de-

scribed by the general transfer function Hij, an explicit form of this function is needed for

each con�guration considered. At �rst glance it seems that, for particular experimental

arrangements, Hijcan be separated into di�raction-dependent and -independent terms. In

fact, most of the con�gurations used in quantum interferometry experiments to date can be

4

mathematically described by separable transfer functions. Free space, apertures, and lenses,

for example, can be considered as di�raction-dependent elements while beam splitters, tem-

poral delays, and waveplates can be considered as di�raction-independent elements.

Almost all quantum-interference experiments performed to date have a common feature,

namely that the transfer function Hijin Eq. (11), with i = A;B and j = o; e, can be

separated into di�raction-dependent and -independent terms as

Hij(x

i;q;!) = T

ijH(x

i;q;!) ; (14)

where the di�raction-dependent terms are grouped in H and the di�raction-independent

terms are grouped in Tij(see Fig. 2). For collinear SPDC con�gurations, for example, in

the presence of a relative optical-path delay � between the ordinary and the extraordinary

polarized photons, as illustrated in Fig. 3(a), Tijis simply

Tij= (e

i� e

j) e�i!��ej ; (15)

where the symbol �ejis the Kronecker delta so that �

ee= 1 and �

eo= 0. The unit vector e

i

describes the orientation of the polarization analyzers in the experimental apparatus, while

ejis the unit vector that describes the polarization of the down-converted photons.

Using the expression for Hijgiven in Eq. (14) in the general biphoton probability am-

plitude given in Eq. (11), we construct a compact expression for all systems that can be

separated into di�raction-dependent and -independent elements as:

A(xA;x

B; t

A; t

B) =

Rdq

odq

ed!

od!

e�(q

o;q

e;!

o; !

e)

�hTAeH(x

A;q

e;!

e)T

BoH(x

B;q

o;!

o) e�i(!etA+!otB)

+ TAoH(x

A;q

o;!

o)T

BeH(x

B;q

e;!

e) e�i(!otA+!etB)

i: (16)

Given the general form of the biphoton probability amplitude for a separable system, we

now proceed to investigate several speci�c experimental arrangements.

For simplicity and clarity the pump �eld ~Ep(q

p;!

p) in Eq. (4) is chosen to be a monochro-

matic plane wave described by

~Ep(q

p;!

p) = �(q

p)�(!

p) ; (17)

which does not e�ect the hyperentangled nature of the state. In this limit the nonfactoriz-

ability of the state is due solely to the phase-matching, which allows for a crisper observation

of the e�ect. For quantum-state engineering using hyperentanglement, it is clear that the

spatiotemporal properties of the pump can also be useful (see Appendix). For the experi-

mental work presented in this paper the angle separation between eiand e

jis 45�, so that T

ij

can be simpli�ed by using (ei� e

j) = � 1p

2[5]. Substituting this into Eq. (14), the biphoton

probability amplitude becomes

A(xA;x

B; t

A; t

B) =

Rdq

odq

ed!

od!

e�(q

o+ q

e) �(!

o+ !

e� !0

p)L sinc

�L�

2

�e�i

L�

2 e�i!e�

�hH(x

A;q

e;!

e)H(x

B;q

o;!

o) e�i(!etA+!otB)

�H(xA;q

o;!

o)H(x

B;q

e;!

e) e�i(!otA+!etB)

i; (18)

5

which simpli�es to

A(xA;x

B; t

A; t

B) =

Rdq d! L sinc

�L�

2

�e�i

L�

2 e�i!� e�i!0p(tA+tB)

�hH(x

A;q;!)H(x

B;�q;!0

p� !) e�i!(tA�tB)

�H(xA;�q;!0

p� !)H(x

B;q;!) ei!(tA�tB)

i: (19)

Using this form for the biphoton probability amplitude in Eq. (13), we can now investigate

the behavior of the quantum-interference pattern for systems described by di�erent transfer

functions H.

The di�raction-dependent elements in most of these experimental arrangements are il-

lustrated in Fig. 3(b). To describe this system mathematically via the function H, we need

to derive the impulse response function, also known as the point-spread function for optical

systems. A typical aperture diameter of b = 1 cm at a distance d = 1 m from the crystal

output plane yields b4=4�d3 < 10�2 using � = 0:5 �m, which indicates the validity of the

Fresnel approximation. We therefore proceed with the calculation of the impulse response

function in this approximation. Without loss of generality we now present a two-dimensional

(one longitudinal and one transverse) analysis of the impulse response function (extension

to three dimensions is straightforward).

Referring to Fig. 3(b), the overall propagation through this system can be broken into

free-space propagation from the nonlinear-crystal output surface (x; 0) to the plane of the

aperture (x0; d1), free-space propagation from the aperture plane to the thin lens (x00; d1+d2),and �nally free-space propagation from the lens to the plane of detection (x

i; d1 + d2 + f),

with i = A;B. Free-space propagation of a monochromatic spherical wave with frequency

! from (x; 0) to (x0; d1) over a distance r is

ei!

cr = ei

!

c

pd2

1+(x�x0)2 � ei

!

cd1e

i !

2cd1

(x�x0)2

: (20)

The spectral �lter is mathematically represented by a function F(!) and the aperture by

the function p(x). In the (x0; d1) plane, at the location of the aperture, the impulse response

function of the optical system between planes x and x0 takes the form

h(x0; x;!) = F(!) p(x0)ei!c d1ei !

2cd1(x�x0)2

: (21)

Also, the impulse response function for the single-lens system from the plane (x0; d1) to theplane (x

i; d1 + d2 + f), as shown in Fig. 3(b), is

h(xi; x0;!) = ei

!

c(d2+f)e�i

!x2

i

2cf[ d2f�1]e�i

!xix0

cf : (22)

Combining this with Eq. (21) provides

h(xi; x;!) = F(!) ei!c (d1+d2+f)e�i

!x2

i

2cf[ d2f�1]e

i !x2

2cd1

Zdx0p(x0)e

i!x02

2cd1 e�i!

cx

0

hx

d1+xi

f

i; (23)

which is the impulse response function of the entire optical system from the crystal output

plane to the detector input plane. We use this impulse response function to determine the

transfer function of the system in terms of transverse wavevectors via

6

H(xi;q;!) =

Zdxh(x

i;x;!) eiq�x ; (24)

so that the transfer function explicitly takes the form

H(xi;q;!) =

"ei

!

c(d1+d2+f)e�i

!jxi j2

2cf[ d2f�1]e�i

cd1

2!jqj2 ~P

!

cfxi� q

!#F(!) ; (25)

where the aperture function ~P�!

cf

xi� q

�is de�ned by

~P

!

cfxi� q

!=Zdx0p(x0)e�i

!x0�x

i

cf eiq�x0

: (26)

Using Eq. (25) we can now describe the propagation of the modes of the system from the

crystal to the detection planes. Since no birefringence is assumed for any material in the

system considered to this point, this transfer function is identical for both polarization modes

(o,e).

Recalling that the analysis is carried out in the Fresnel approximation and that the

SPDC �elds are assumed to be quasi-monochromatic, we can derive, in the absence of

spectral �ltering, an analytical form for the coincidence-count rate de�ned in Eq. (13):

R(� ) = R0 [1� V (� )] ; (27)

where R0 is the coincidence rate when there is no quantum interference, and

V (� ) = �

�2�

LD� 1

�sinc

"!0pL2M2

4cd1

�

LD�

�2�

LD� 1

�#~PA

�!0

pLM

4cd1

2�

LDe2

!~PB

!0pLM

4cd1

2�

LDe2

!:

(28)

Here D = 1=uo� 1=u

ewith u

jdenoting the group velocity for the j-polarized photon

(j = o; e), M = @ lnne(!0

p

=2; �OA)=@�e [11], and �(x) = 1 � jxj for �1 � x � 1, and zero

otherwise. A derivation of Eqs. (27) and (28), along with the de�nitions of all quantities

in this expression, is presented in the Appendix. The function ~Pi(with i = A;B) is the

normalized Fourier transform of the squared magnitude of the aperture function pi(x); it is

given by

~Pi(q) =

R Rdy p

i(y)p�

i(y) e�iy�qR R

dy pi(y)p�

i(y)

: (29)

The pro�le of the function ~Piwithin Eq. (28) plays a key role in the results presented in this

paper. The common experimental practice is to use extremely small apertures and thereby

reach the one-dimensional plane wave limit. In the context of Eq. (28), this results in ~Pi

functions that are broad in comparison with � so that � determines the shape of the quantum

interference pattern. The sinc function in Eq. (28) is approximately equal to unity for all

practical situations and therefore plays an insigni�cant role. This function represents the

di�erence between a one-dimensional model which results in a perfect triangular interference

7

dip, R(� ) = R0

h1 � �

�2�LD

� 1�i, and a three-dimensional model in the presence of a very

small on-axis aperture.

Note that for symmetric apertures, jpi(x)j = jp

i(�x)j, the functions ~P

iare also symmetric

by virtue of Eq. (29). However, within Eq. (28) the centers of symmetry for these functions,

which are located at � = 0, are shifted with respect to the center of symmetry of the

function �, which is symmetric around � = LD=2. Consequently, as observed previously

[7], the behavior of Eq. (28), which contains products of these functions, is asymmetrical.

When the apertures are wide, the pi(x) are broad functions which result in narrow ~P

i, so

that the interference pattern is strongly in uenced by the particular shape of the functions~Pi. If, in addition, the apertures are spatially shifted in the transverse plane, the ~P

ibecome

oscillatory functions that result in sinusoidal modulation of the interference pattern. This

can also result in inversion of the dip into a peak for certain ranges of the delay � , as will

be discussed subsequently. In short, it is clear from Eq. (28) that V (� ) can be altered

dramatically by selecting the aperture pro�le.

A. Quantum Interference with Circular Apertures

Of practical interest is the e�ect of the aperture shape, via the function ~P (q), on polar-

ization quantum interferometry. Therefore, we now consider the experimental arrangement

illustrated in Fig. 3(a) in the presence of a circular aperture with diameter b.

The pump was a single-mode cw argon-ion laser with a wavelength of 351.1 nm and

a power of 200 mW. The pump light was delivered to a �-BaB2O4 (BBO) crystal with a

thickness of 1.5 mm. The crystal was aligned to produce collinear and degenerate Type-

II spontaneous parametric down-conversion. Residual pump light was removed from the

signal and idler beams with a fused-silica dispersion prism. The collinear beams were then

sent through a delay line comprised of a crystalline quartz element (fast axis orthogonal

to the fast axis of the BBO crystal) whose thickness could be varied to alter the delay

between the photons of a down-converted pair. The photon pairs were then sent to a non-

polarizing beam splitter. Each arm of the polarization intensity interferometer following

this beam splitter contained a Glan-Thompson polarization analyzer at 45�, a convex lens

to focus the incoming beam, and an actively quenched Peltier-cooled single-photon-counting

avalanche photodiode detector [denoted Diwith i = A;B in Fig. 3(a)]. No spectral �ltering

was used in the selection of the signal and idler photons for detection. The counts from

the detectors were conveyed to a coincidence counting circuit with a 3-ns coincidence-time

window. Corrections for accidental coincidences were not necessary.

The mathematical representation of this aperture is given in terms of the Bessel function

J1,

~P(q) = 2J1 (b jqj)b jqj : (30)

The observed normalized coincidence rates (quantum-interference patterns) from a 1.5-

mmBBO crystal (symbols), along with the expected theoretical curves (solid), are displayed

in Fig. 4 as a function of relative optical delay for various values of aperture diameter b.

Clearly the observed interference pattern is more asymmetric for larger values of b. As the

8

aperture becomes wider the phase-matching condition between the pump and the generated

down-conversion allows more (q; !) modes to be admitted. Therefore the (q; !) modes

that have less overlap with the other modes introduce distinguishability. This inherent

distinguishability, in turn, a�ects the quantum-interference pattern in the form of loss of

visibility and asymmetry in the shape of the pattern.

The theoretical plots of the visibility of the quantum-interference pattern at the full-

compensation delay � = LD=2, as a function of the crystal length, is plotted in Fig. 5 for

various aperture diameters. Full visibility is expected only in the limit of extremely thin

crystals, or with the use of extremely small aperture diameters corresponding to the one-

dimensional limit of the formalism presented here. As the crystal thickness increases, the

visibility is seen to depend more dramatically on the aperture diameter. If the pump �eld is

pulsed, then there are additional limitations on the visibility that emerge as a result of the

broad spectral bandwidth of the pump �eld [6,7]. The experimentally observed visibility for

various aperture diameters, using the 1.5-mm thick BBO crystal employed in our experiments

(symbols), agrees well with the theory.

B. Quantum Interference with Slit Apertures

For the majority of quantum-interference experiments involving relative optical-path

delay, circular apertures are the norm. In this section we consider the use a vertical slit

aperture to investigate the transverse symmetry of the generated photon pairs. Since the

experimental arrangement of Fig. 3(a) remains identical aside from the aperture, Eq. (19)

still holds and the aperture function p(x) takes the explicit form

~P(q) = sin(b e1 � q)b e1 � q

sin(a e2 � q)a e2 � q : (31)

The data shown by squares in Fig. 6 is the observed normalized coincidence rate for a

1.5-mm BBO in the presence of a vertical slit aperture with a = 7 mm and b = 1 mm.

The quantum-interference pattern is highly asymmetric and has low visibility, and indeed is

similar to that obtained using a wide circular aperture (see Fig. 4). The solid curve is the

theoretical quantum-interference pattern expected for the vertical slit aperture used.

In order to investigate the transverse symmetry, the complementary experiment has also

been performed using a horizontal slit aperture. For the horizontal slit, the parameters a

and b in Eq. (31) are interchanged so that a = 1 mm and b = 7 mm. The data shown

by triangles in Fig. 6 is the observed normalized coincidence rate for a 1.5-mm BBO in

the presence of this aperture. The most dramatic e�ect observed is the symmetrization of

the quantum-interference pattern and the recovery of the high visibility, despite the wide

aperture along the horizontal axis. A practical bene�t of such a slit aperture is that the

count rate is increased dramatically, which is achieved by limiting the range of transverse

wavevectors along the optical axis of the crystal to induce indistinguishability and allowing

a wider range along the orthogonal axis to increase the collection e�ciency of the SPDC

photon pairs. This �nding is of signi�cant value, since a high count rate is required for many

applications of entangled photon pairs and, indeed, many researchers have suggested more

complex means of generating high- ux photon pairs [12].

9

Noting that the vertical axis is where the optical axis of the crystal lies, we deduce

from these results that the dominating portion of distinguishability is, as expected, along

the optical axis. The axis orthogonal to that (the horizontal axis in this case) provides a

negligible contribution to distinguishability, so that almost full visibility can be achieved

despite the wide aperture along the horizontal axis.

C. Quantum-Interference with Increased Acceptance Angle

One of the biggest obstacles for accessing a wider range of transverse wavevectors is the

dispersive elements in the system. One or more dispersion prisms, for example, are used to

separate the intense pump �eld from the down-converted photons. These elements behave as

spectral and spatial �lters in combination with the limited aperture sizes considered earlier

[13]. Consequently, the combination of the spatial and spectral �ltering serves to reduce the

range of the hyperentanglement accessed by the experiments.

To increase the limited acceptance angle of the detection system in order to utilize the

multi-parameter features of the entangled-photon pairs, we carry out experiments using

the alternate setup shown in Fig. 7. In particular, a dichroic mirror is used instead of a

prism. Moreover, by reducing the separation of the crystal and the aperture plane, the

e�ective acceptance angle is increased. This allows us to access a greater range of transverse

wavevectors in our experimental scheme, which facilitates the observation of the e�ects

discussed previously.

Using this experimental arrangement, we repeated the circular-aperture experiments,

the results of which were presented in Fig. 4. Figure 8 displays the observed quantum-

interference patterns (normalized coincidence rates) from a 1.5-mm BBO crystal (symbols)

along with the expected theoretical curves (solid) as a function of relative optical delay for

various values of the aperture diameter b. The e�ective aperture diameter for the interference

pattern with the lowest visibility (circles) is now determined not by an aperture placed 1 m

away from the nonlinear crystal, as in Fig. 3(a), but rather by the dimensions of the optical

system, as will be discussed later.

With these results in the absence of dispersive elements (prisms), and with wider ac-

ceptance angles, we verify once more that the observed asymmetric pattern of the quantum

interference is consistent with the multi-parameter structure of SPDC. It has been asserted

that the asymmetry in the quantum-interference pattern could be caused by the presence of

dispersive prisms, which were referred to as asymmetric linear optical elements in Ref. [14].

The formalism presented here fully explains the experimental observations presented in this

work, as well as in previous work that focused on pulse-pumped SPDC [7] without the use

of any �tting parameters.

D. Shifted-Aperture E�ects

In the analysis presented so far, the optical elements in the system are placed concentri-

cally about the longitudinal (z) axis. In this condition, the sole aperture before the beam

splitter in the con�guration, shown in Fig. 3(a), yields the same transfer function as two

identical apertures placed in each arm after the beam splitter, shown in Fig. 7. In this

10

section we show that the observed quantum-interference pattern is also sensitive to a rela-

tive shift of the apertures in the transverse plane. To account for this, we must include an

additional factor in Eq. (28):

cos

�!pLM

4cd1

2�

LDe2 � (sA � s

B)

�; (32)

where si(with i = A;B) is the displacement of each aperture from the longitudinal (z)

axis. This extra factor provides yet another element of control on the quantum-interference

pattern for a given aperture form.

1. Quantum Interference with Shifted-Slit Apertures

First, we consider the example of two slit apertures, one in each arm of the interferometer,

which can be shifted with respect to the longitudinal (z) axis. The aperture function in

Eq. (31) then takes the form

~Pi(q) =

sin(b e1 � q)b e1 � q

sin(a e2 � q)a e2 � q eisi�q (33)

with i = A;B. A spatially shifted aperture introduces a phase factor into the aperture

function which, in turn, results in a sinusoidal modulation of the quantum-interference

pattern.

The two sets of data shown in Fig. 9 represent the observed normalized coincidence

rates for a 1.5-mm BBO crystal in the presence of identical apertures placed in each arm as

shown in Fig. 7. The square points correspond to the use of 1�7 mm horizontal slits. The

circular points correspond to the same apertures rotated 90 degrees to form vertical slits.

Since this con�guration, as shown in Fig. 7, accesses a wider range of acceptance angles, the

dimensions of the other optical elements become relevant as e�ective apertures in the system.

Although the apertures themselves are aligned symmetrically, an e�ective vertical shift of

jsA� s

Bj = 1 mm is induced by the relative displacement of the two polarization analyzers.

The solid curves in Fig. 9 are the theoretical plots for the two aperture orientations.

2. Quantum Interference with Shifted-Ring Apertures

For the same experimental con�guration as shown in Fig. 7, but now using an annular

aperture rather than a slit aperture in one of the arms, and a 7 mm circular aperture in

the other, yields the results shown in Fig. 10. The annular aperture function with internal

diameter a = 2 mm and external diameter b = 4 mm then becomes

~Pi(q) =

2

b� a

"J1 (b jqj)jqj � J1 (a jqj)

jqj

#eisi�q (34)

with i = A;B. The symbols are the experimental results for various values of the relative

shift jsA� s

Bj as denoted in the legend. Note that, for certain values of the relative optical-

path delay (� ), the interference pattern displays inversion in the form of a peak rather than

the familiar dip expected in this type of experiment.

11

IV. CONCLUSION

In summary, we observe that the hyperentangled nature of the two-photon state gener-

ated by SPDC allows transverse features, represented by their wavevectors, to play a role

in polarization-based quantum interference experiments. The interference patterns gener-

ated in these experiments are, as a result, governed by the pro�les of the apertures in the

optical system which admit wavevectors in speci�ed directions. The quantitative agreement

between the experimental results using a variety of aperture pro�les, and the theoretical

results from the formalism presented in this paper, con�rm this interplay. In contrast to

the usual single-direction polarization entangled state, the wide-angle polarization entangled

state o�ers a richness that can be exploited in a variety of applications involving quantum

information processing.

APPENDIX

The purpose of this Appendix is to derive Eq. (28) using Eqs. (13), (19), and (25).

To obtain an analytical solution within the Fresnel approximation we assume quasi-

monochromatic �elds and perform an expansion in terms of a small frequency spread around

the central frequency (!0p=2) associated with degenerate down-conversion, and small trans-

verse components jqj with respect to the total wavevector kjfor collinear down-conversion.

In short, we use the fact that j�j � !0p

=2, with � = ! � !0p

=2, and jqj2 � jkj2. In these

limits we obtain

�o(!;q)� K

o+! � !0

p

=2

uo

� jqj22K

o

(A1)

�e(!;q)� K

e� ! � !0

p=2

ue

� jqj22K

e

+Me2 � q ; (A2)

where the explicit forms for Kj, u

jj = o; e, and Me2 are [11]:

Kj= jk

jj(!;q)j!0

p

2;q=0

1

uj

=@�

j(!;q)

@!

�����!0p

2;q=0

(A3)

Me2 =jk

ejrqjke

j�e

�����!0p

2;q=0

M =@ lnn

e(!; �

e)

@�e

�����!0p

2;�e=�OA

: (A4)

Using the results in Eqs. (A1), (A2) and (A3) we can now provide an approximate form for

�, which is the argument of the sinc function in Eq. (19), as

� � �D� + 2cjqj2!0p

+Me2 � q (A5)

where D = 1uo

� 1ue

.

Using this approximate form for � in the integral representation of sinc(x)

sinc

�L�

2

�e�i

L�

2 =Z 0

�Ldz e�iz� ; (A6)

12

with the assumption that L� d1, we obtain Eq. (27) with

R0 =

Zd�

Z 0

�Ldz e�iD�z

Z 0

�Ldz0 eiD�z

0 J0(z; z0) ; (A7)

V (� ) =1

R0

Zd� e�2i��

Z 0

�Ldz e�iD�z

Z 0

�Ldz0 e�iD�z

0 JV(z; z0) ; (A8)

where the functions

J0(z; z0) =

!0p

2cd1

!2exp

"�i !

0p

8cd1M2(z2 � z0 2)

#

� ~PA

"!0p

4cd1M(z � z0)e2

#~PB

"� !0

p

4cd1M(z � z0)e2

#; (A9)

JV(z; z0) =

!0p

2cd1

!2exp

"�i !

0p

8cd1M2(z2 � z0 2)

#

� ~PA

"!0p

4cd1M(z + z0)e2

#~PB

"� !0

p

4cd1M(z + z0)e2

#; (A10)

are derived by carrying out the integrations over the variables x and q.

Performing the remaining integrations leads us to Eq. (28). This equation allows robust

and rapid numerical simulations of quantum-interferometric measurement to be obtained.

For the simulations provided in this work, with pump wavelength of 351 nm, the calculated

values of M and D are 0.0711 and 248 fsec/mm, respectively.

In the case of type-I SPDC, both photons of a generated pair have ordinary polarization.

Consequently, the vector Me2 does not appear in the expansions of the wavevectors, unless

the pump �eld itself has transverse wavevector components. Since the jqj2 term in the

expansion of �, as given in Eq. (A5), is smaller than Me2 � q, similar e�ects in type-I

quantum-interferometric measurements are expected to be smaller.

Acknowledgments.| This work was supported by the National Science Foundation. The

authors would like to thank A. F. Abouraddy and M. C. Booth for valuable suggestions.

13

REFERENCES

[1] S. E. Harris, M. K. Oshman, and R. L. Byer, Phys. Rev. Lett. 18, 732 (1967); D. Magde

and H. Mahr, Phys. Rev. Lett. 18, 905 (1967).

[2] E. Schr�odinger, Naturwissenschaften 23, 807 (1935); 23, 823 (1935); 23, 844 (1935)

[Translation: J. D. Trimmer, Proc. Am. Phil. Soc. 124, 323 (1980); reprinted in Quan-

tum Theory and Measurement, edited by J. A. Wheeler and W. H. Zurek (Princeton

University Press, Princeton, 1983)].

[3] C. K. Hong, Z. Y. Ou, and L. Mandel, Phys. Rev. Lett. 59, 2044 (1987); P. G. Kwiat,

A. M. Steinberg, and R. Y. Chiao, Phys. Rev. A 47, R2472 (1993).

[4] J. G. Rarity and P. R. Tapster, Phys. Rev. Lett. 64, 2495 (1990).

[5] Z. Y. Ou and L. Mandel, Phys. Rev. Lett. 61, 50 (1988); Y. H. Shih and C. O. Alley,

Phys. Rev. Lett. 61, 2921 (1988); Y. H. Shih and A. V. Sergienko, Phys. Lett. A 191,

201 (1994); P. G. Kwiat, K. Mattle, H. Weinfurter, A. Zeilinger, A. V. Sergienko, and

Y. H. Shih, Phys. Rev. Lett. 75, 4337 (1995).

[6] G. Di Giuseppe, L. Haiberger, F. De Martini, and A. V. Sergienko, Phys. Rev. A 56,

R21 (1997); T. E. Keller and M. H. Rubin, Phys. Rev. A 56, 1534 (1997); W. P. Grice,

R. Erdmann, I. A. Walmsley, and D. Branning, Phys. Rev. A 57, R2289 (1998);

J. Pe�rina, Jr., A. V. Sergienko, B. M. Jost, B. E. A. Saleh, and M. C. Teich, Phys.

Rev. A 59, 2359 (1999).

[7] M. Atat�ure, A. V. Sergienko, B. M. Jost, B. E. A. Saleh, and M. C. Teich, Phys. Rev.

Lett. 83, 1323 (1999); M. Atat�ure, G. Di Giuseppe, M. D. Shaw, A. V. Sergienko,

B. E. A. Saleh, and M. C. Teich, submitted.

[8] B. E. A. Saleh, A. F. Abouraddy, A. V. Sergienko, and M. C. Teich, Phys. Rev. A 62,

043816 (2000).

[9] D. N. Klyshko, Photons and Nonlinear Optics (Gordon and Breach Science Publishers,

New York, 1988).

[10] T. B. Pittman, D. V. Strekalov, D. N. Klyshko, M. H. Rubin, A. V. Sergienko, and

Y. H. Shih, Phys. Rev. A 53, 2804 (1996).

[11] B. R. Mollow, Phys. Rev. A 8, 2684 (1973); M. H. Rubin, Phys. Rev. A 54, 5349 (1996).

[12] P. G. Kwiat, E. Waks, A. G. White, I. Appelbaum, and P. H. Eberhard, Phys. Rev.

A. 60, R773 (1999); S. Tanzilli, H. De Riedmatten, W. Tittel, H. Zbinden, P. Baldi,

M. De Micheli, D. B. Ostrowsky, and N. Gisin, Electron. Lett. 37, 26 (2001); C. Kurt-

siefer, M. Oberparleiter, and H. Weinfurter, LANL eprint quant-ph/0101074.

[13] For a review of spectral tuning through angular dispersive elements, see, for example,

J.-C. Diels and W. Rudolph, Ultrashort Laser Pulse Phenomena (Academic Press, San

Diego, 1996) Chap. 2.

[14] Y. H. Kim, V. Berardi, M. V. Chekhova, and Y. H. Shih, Phys. Rev. A 64, 011801(R)

(2001).

14

FIGURES

Figure 1

k

OA

e3

e1

e2

θ

κq

M

θOA

FIG. 1. Decomposition of a three-dimensional wavevector (k) into longitudinal (�) and trans-

verse (q) components. The angle between the optical axis of the nonlinear crystal (OA) and the

wavevector k is �. The angle between the optical axis and the longitudinal axis (e3) is denoted

�OA. The spatial walk-o� of the extraordinary polarization component of a �eld travelling through

the nonlinear crystal is characterized by the quantity M .

BBO Hij (xi,q;ω) xi

BBO xiH(xi,q;ω)

TTTTij

(a)

(b)

FIG. 2. (a) Illustration of the generalized setup for observation of quantum interference using

SPDC. BBO represents a beta-barium borate nonlinear optical crystal, Hij(x

i;q;!) is the transfer

function of the system, and the detection plane is represented by xi. (b) For most experimental

con�gurations the transfer function can be factorized into di�raction-dependent [H(xi;q;!)] and

di�raction-independent (Tij) components.

15

(a)(b)

RelativeDelay τ

702 nm

351 nm702 nm

BBO

351 nmAr+ Laser

DABeam

Splitter

DB

PolarizationAnalyzers

CoincidenceCircuit

SpectralFilter

Aperture

BBO p(x)

d2d1 f

LensFFFF (ω)D

Prism

FIG. 3. (a) Schematic of the experimental setup for observation of quantum interference using

cw-pumped type-II collinear SPDC (see text for details). (b) Detail of the path from the crystal

output plane to the detector input plane. F(!) represents an (optional) �lter transmission function,

p(x) represents an aperture function, and f is the focal length of the lens.

-150 0 150 300 4500.0

0.2

0.4

0.6

0.8

1.0

PRISM

NO

RM

AL

IZE

D C

OIN

CID

EN

CE

-CO

UN

T R

AT

E

RELATIVE OPTICAL-PATH DELAY (fsec)

APERTURE DIAMETER 10.0 mm 8.5 mm 6.0 mm 5.0 mm 3.0 mm

1-D MODEL

FIG. 4. Normalized coincidence-count rate R(�)=R0, as a function of the relative optical-path

delay � , for di�erent diameters of a circular aperture. The symbols are the experimental results

and the solid curves are the theoretical plots for each aperture diameter as denoted in the legend.

No �tting parameters are used. The dashed curve represents the one-dimensional (1-D) plane-wave

theory.

16

0 2 4 6 8 100.0

0.2

0.4

0.6

0.8

1.0 1-D

3

1

2

4 5

b = 7.5

0.5

0.25 mm

CO

INC

IDE

NC

E V

ISIB

ILIT

Y

CRYSTAL THICKNESS (mm)

FIG. 5. Solid curves represents theoretical coincidence visibility of the quantum-interference

pattern as a function of crystal thickness, for various circular-aperture diameters b. The dashed line

represents the one-dimensional (1-D) plane-wave limit of the multi-parameter formalism. Visibility

is calculated for a relative optical-path delay � = LD=2. The squares represent experimental data

collected using a 1.5-mm-thick BBO crystal, and aperture diameters as indicated on the plot.

-150 0 150 300 4500.0

0.2

0.4

0.6

0.8

1.0

PRISM

APERTURE ORIENTATION

VERTICAL HORIZONTAL

NO

RM

AL

IZE

D C

OIN

CID

EN

CE

-CO

UN

T R

AT

E

RELATIVE OPTICAL-PATH DELAY (fsec)

FIG. 6. Normalized coincidence-count rate as a function of the relative optical-path delay for

a 1 � 7-mm horizontal slit (triangles). Experimental results for a vertical slit are indicated by

squares. Solid curves are the theoretical plots for the two orientations.

17

RelativeDelay τ

702 nm

351 nm702 nm

BBO

351 nmAr+ Laser

DABeam

Splitter

DB

PolarizationAnalyzers

CoincidenceCircuit

SpectralFilter

Aperture B

DichroicMirror

Aperture A

FIG. 7. Schematic of alternate experimental setup for observation of quantum interference us-

ing cw-pumped type-II collinear SPDC. The con�guration illustrated here makes use of a dichroic

mirror in place of the prism used in Fig. 3(a), thereby admitting greater acceptance of the trans-

verse-wave components. The dichroic mirror re ects the pump wavelength (351 nm) while trans-

mitting a broad wavelength range that includes the bandwidth of the SPDC. The single aperture

shown in Fig. 3(a) is replaced by identical apertures placed in the two arms of the interferometer.

-150 0 150 300 4500.0

0.2

0.4

0.6

0.8

1.0

APERTURE DIAMETER 7 mm 4 mm 2 mm

DICHROIC MIRROR

NO

RM

AL

IZE

D C

OIN

CID

EN

CE

-CO

UN

T R

AT

E

RELATIVE OPTICAL-PATH DELAY (fsec)

FIG. 8. Normalized coincidence-count rate as a function of the relative optical-path delay, for

di�erent diameters of an aperture that is circular in the con�guration of Fig. 7. The symbols are

the experimental results and the solid curves are the theoretical plots for each aperture diameter

as denoted in the legend. No �tting parameters are used. The behavior of the interference pattern

is similar to that observed in Fig. 4; the dependence on the diameter of the aperture is slightly

stronger in this case.

18

-150 0 150 300 4500.0

0.2

0.4

0.6

0.8

1.0

APERTURE ORIENTATION

VERTICAL HORIZONTAL

DICHROIC MIRROR

NO

RM

AL

IZE

D C

OIN

CID

EN

CE

-CO

UN

T R

AT

E

RELATIVE OPTICAL-PATH DELAY (fsec)

FIG. 9. Normalized coincidence-count rate as a function of the relative optical-path delay for

identical 1 � 7-mm horizontal slits placed in each arm in the con�guration of Fig. 7 (squares).

Experimental results are also shown for two identical 1� 7-mm vertical slits, but shifted by 1 mm

with respect to each other (circles). Solid curves are the theoretical plots for the two orientations.

-150 0 150 300 4500.0

0.4

0.8

1.2

1.6DICHROIC MIRROR

NO

RM

AL

IZE

D C

OIN

CID

EN

CE

-CO

UN

T R

AT

E

RELATIVE OPTICAL-PATH DELAY (fsec)

RELATIVE SHIFT 2.5 mm 1.0 mm 0.0 mm

FIG. 10. Normalized coincidence-count rate as a function of the relative optical-path delay, for

an annular aperture (internal and external diameters of 2 and 4 mm, respectively) in one of the

arms in the con�guration of Fig. 7. A 7-mm circular aperture is placed in the other arm. The

symbols are the experimental results for the relative shifts as denoted in the legend. The solid

curves are the theoretical plots without any �tting parameters.

19