Measuring Two-Photon Orbital Angular Momentum Entanglement
G.F. Calvo, A. Picón and A. Bramon
Grup de F́ısica Teòrica, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain
(Dated: July 7, 2021)
We put forward an approach to estimate the amount of bipartite spatial entanglement of down-
converted photon states correlated in orbital angular momentum and the magnitude of the transverse
(radial) wave vectors. Both degrees of freedom are properly considered in our framework which only
requires azimuthal local linear optical transformations and mode selection analysis with two fiber
detectors. The coincidence distributions predicted by our approach give an excellent fit to the
distributions measured in a recent experiment aimed to show the very high-dimensional transverse
entanglement of twin-photons from a down-conversion source. Our estimate for the Schmidt number
is substantially lower but still confirms the presence of high-dimensional entanglement.
PACS numbers: 03.67.Mn, 42.50.Dv, 42.65.Lm
I. INTRODUCTION
The phenomenon of quantum entanglement whereby
distant systems can manifest perfectly random albeit per-
fectly correlated behavior is now recognized as the es-
sential ingredient to perform tasks which cannot be ac-
complished with classically correlated systems [1]. The
presence of entanglement has been traditionally revealed
in the violation of Bell-type inequalities [2]. However,
detecting such a violation does not provide in general a
measure of the amount of entanglement. This is par-
ticularly significant in systems correlated in multidimen-
sional degrees of freedom [3]. Several techniques have
been proposed to assess the presence of entanglement
for different quantum scenarios. These include state to-
mography [4, 5, 6], that yields a complete reconstruction
of a quantum state but requires many setting measure-
ments, entanglement witnesses [7], which detect some
entangled states with considerably less measurements,
and the experimental determination of concurrence [8, 9].
In Ref. [9], by using two copies of a down-converted
two-photon state entangled in polarization and trans-
verse momentum, the measurement of concurrence was
achieved with a single, local measurement on one of the
photons.
Although bipartite entanglement is well understood,
finding experimentally feasible procedures to quantify
it for systems correlated in multidimensional degrees of
freedom turns out to be quite challenging and relevant.
Indeed, access to higher dimensional Hilbert spaces in
which information can be encoded and manipulated has
recently attracted great interest, with proof-of-principle
demonstrations using quantum communication protocols
in three-level systems (qutrits), such as entanglement
concentration [10], quantum bit commitment [11] and
quantum coin-tossing [12]. Likewise, a complete char-
acterization of states hyperentangled in polarization, or-
bital angular momentum and frequency has been exper-
imentally implemented [13].
The aim of this paper is to address the problem of
how, by performing a certain set of local linear optical
operations affecting one of two multidimensional spatial
degrees of freedom in which two-photon states can be
entangled, it is possible to obtain an explicit measure of
the amount of bipartite transverse entanglement. Specif-
ically, according to the interplay between both spatial
degrees of freedom -orbital angular momentum (OAM)
and the magnitude of the radial wave vectors- fundamen-
tally different predictions are expected in the subsequent
joint photodetection process, via mode-selection analysis
with two fiber detectors preceded by azimuthal transfor-
mations (acting only on the OAM). The application of
our framework is compared with the results of a recent
experiment [14].
The paper is organized as follows: Section II presents
the general setting of the problem and includes the
Schmidt decomposition technique for describing bipar-
tite spatial entanglement. In Section III we reveal the
generic features that arise in the photodetection coinci-
dences according to the transverse structure of the two-
photon wave function. Section IV illustrates the results
found in Section III with an example of a realistic two-
photon wave function that yields a full analytical solution
to the problem. In Section V a closed-form expression
for the Schmidt number is obtained in terms of easily
accessible experimental parameters. We then proceed to
exploit this Schmidt number in two interesting examples
of optical transformations with azimuthal phase plates.
These enable us to estimate the amount of spatial entan-
glement of a two-photon source. Conclusions of the paper
are drawn in Section VI. Details of our calculations, to-
gether with some useful background material, have been
included in two Appendices.
II. MODAL SCHMIDT DECOMPOSITION FOR
TWO-PHOTON STATES
Two-photon pure quantum states are described in a
Hilbert space by a continuous bilinear superposition of
spatiotemporal multimode states. Sources of such non-
classical states of light are mostly realised in the process
of spontaneous parametric down-conversion [15], where
an intense quasimonochromatic laser pump illuminates
a crystal endowed with a quadratic nonlinearity produc-
ing pairs of photons (idler and signal). Conservation of
energy and momentum impose that the state be spec-
trally and spatially correlated. Here we explore entan-
glement involving spatial degrees of freedom that de-
pend on the transverse structure of these states. For
simplicity, we assume that the down-converted photons
are linearly polarized, monochromatic and frequency de-
generated. The two-photon state can then be written as
|ψ〉 =
dqidqsΦ(qi,qs)â
†(qi)â†(qs)|vac〉, where qi,s are
the transverse components of the idler and signal wave
vectors. Under conditions of paraxial and nearly collinear
propagation of the pump, idler and signal photons, the
two-photon amplitude Φ is given by [16] Φ(qi,qs) =
E(qi + qs)G(qi − qs). The function E represents the
transverse profile of the pump, whereas G originates from
the phase-matching condition in the longitudinal direc-
tion and depends on the specific orientation and cut of
the nonlinear crystal. Since the arguments of E and G
enforce correlations in different manifolds of the idler and
signal wave vector space, it is the global structure of Φ
the one dictating the entanglement degree of |ψ〉.
In order to extract the amount of entanglement con-
tained in Φ, one may resort to the Schmidt decomposi-
tion that provides the spatial information modes of the
two-photon pair. Suppose that the transverse spatial fre-
quency field of the pump beam has a Gaussian profile of
the form E(qi + qs) ∝ e−w
0|qi+qs|
2/4, where w0 is the
width (at the beam waist). The chosen pump profile
peaks when its argument vanishes. This imposes that
the idler and signal transverse wave vectors should be
mostly anticorrelated (qi ' −qs). Remarkably, it was
shown by Law and Eberly [17] that with such a Gaussian
profile E the normalised two-photon amplitude can be
expressed as
Φ(qi,qs) =
(−1)`
λ`n u`,n(qi)u−`,n(qs) , (1)
where u`,n(qi) = ei`φiv`,n(qi)/
2π and u−`,n(qs) =
e−i`φsv−`,n(qs)/
2π are the normalised polar Schmidt
mode functions for the idler and signal photons with
topological charge ` and radial index n corresponding
to eigenvalues λ`n (they satisfy 1 ≥ λ`n ≥ 0 and∑
`,n λ`n = 1) of the reduced density matrices for each
photon. Knowledge of λ`n yields a direct measure of the
degree of transverse entanglement given by the Schmidt
number [18] K = (
`,n λ
−1, which is the reciprocal
of the purity of the idler and signal density matrices,
it is invariant under free propagation and yields an av-
erage of the number of relevant spatial modes involved
in the decomposition. The larger the value of K, the
higher the transverse entanglement. For instance, prod-
uct states correspond to K = 1 (there is only one nonvan-
ishing eigenvalue equal to 1), whereas states with K > 1
are entangled. A distinguishing feature of decomposi-
tion (1) is that it represents Φ in terms of a perfectly
correlated discrete basis of paraxial eigenstates of the
FIG. 1: (color online). Two-photon coincidence detection
configuration. Down-converted idler and signal beams from a
nonlinear crystal traverse two optical systems (h(i,s)) which
include azimuthal phase plates and are coupled into single-
mode fibers (SMF) gated by a coincidence circuit (CC).
OAM operator along the direction of light propagation
(with corresponding eigenvalues `~) [19, 20], rather than
in a continuous plane-wave modal expansion. The precise
form of the radial idler and signal eigenmodes v`,n(qi)
and v−`,n(qs) depends on the specific phase-matching
function G. In particular, when G is approximated by
a constant there are striking consequences: Eq. (1) be-
comes a (non-normalisable) superposition in which all
OAM eigenstates have an equal weight and the radial de-
pendence becomes just a global factor. It is important to
emphasize that to attain decomposition (1) the appropri-
ate choice of the widths, wi and ws, for the idler and sig-
nal radial eigenmodes, has to be made. If the two-photon
amplitude is of the form Φ(qi,qs) = E(qi+qs)G(qi−qs),
then wi = ws ≡ wS , where wS is the so-called Schmidt
width. For widths different from wS an additional sum-
mation over the radial indexes occurs, and the perfect
correlation between the idler and signal radial modes is
absent (see Appendix A). Moreover, the fact that the
idler and signal mode functions are anticorrelated with
respect to their topological charge numbers is a conse-
quence of a more general process: the conservation of
OAM, which is transferred from the pump photon (car-
rying zero OAM for a Gaussian mode) to the down-
converted photon pair [21].
III. AZIMUTHAL TRANSFORMATIONS ON
THE TWO-PHOTON STATE
The usefulness of the Schmidt decomposition (1) be-
comes apparent when analyzing the propagation of two-
photon states through optical systems. Each of the in-
tervening modes evolves and transforms independently of
the others. It is also clear that the correlation proper-
ties displayed by the two-photon amplitude (1) are pre-
served in the position representation. Therefore, sup-
pose that the idler and signal photon beams, described
now by the paraxial two-photon wave function in the
transverse position representation [20, 22, 23] ψ(ri, rs) =
〈ri, rs|ψ〉 =
`,n(−1)
λ`nu`,n(ri)u−`,n(rs), are each
transmitted through different linear optical systems that
include diffractive (or refractive) azimuthal phase plates
(see Fig. 1). The role of the plates in each path is
to imprint an azimuth-dependent phase factor on the
incoming Schmidt modes. Their (separate) action on
the two-photon wave function can be represented by
the unitary and radially symmetric impulse response
functions h(i,s)(φ, φ′). These functions locally trans-
form the spiral harmonic mode content of each pho-
ton via ei`φ/
(i,s)
`′,` e
i`′φ/
2π, with h(i,s)`′,` =∫ 2π
dφ dφ′ei`φe−i`
′φ′h(i,s)(φ, φ′)/2π, so that the re-
sulting output two-photon wave function is
ψout(ri, rs) =
`i,`s,`,n
(−1)`
λ`n h
`s,−`
× ei`iφiv`,n(ri) ei`sφsv−`,n(rs)/2π , (2)
where the initial perfect correlation in OAM is lost, re-
maining only that corresponding to the radial modes.
Upon traversing their respective linear optical systems
the idler and signal photons, having OAM `i = `A, `s =
`B and radial indexes nA,B , are detected in coincidence.
By placing photodetectors at the output ports of suitable
arrays of deterministic mode sorter interferometers [24,
25], or computer-generated holograms [10], it is possible
to distinguish modes bearing different OAM. In practice,
the most straightforward procedure involves projecting
into single-mode fibers, where the propagated mode has a
fundamental Gaussian profile (`A = `B = nA = nB = 0).
The probability that the idler and signal photons will
be projected into modes u`A,nA and u`B ,nB is found to
be PnA,nB`A,`B = |
dridrsu∗`A,nA(ri)u
`B ,nB
(rs)ψout(ri, rs)|2,
with an additional incoherent (and weighted) sum over
the radial indexes nA,B when taking into account multi-
mode detection. This photodection probability is ex-
pressed in terms of the spatial overlap of the Schmidt and
the fiber modes at the planes where the phase plates are
located. Notice that their corresponding widths, wS and
wG, are not necessarily equal, and thus the orthogonal-
ity between these modes when their radial indexes differ
does not hold in general. To evaluate PnA,nB`A,`B , we de-
fine RA,B` ≡
rirsdridrsv
`A,nA
(ri)v
`B ,nB
×v(wS)`,n (ri)v
−`,n(rs), where the two different widths of
the radial modes have been specified for clarity. The co-
incidence probability can then be written as
PnA,nB`A,`B =
∣∣∣∣ ∞∑
(−1)`RA,B` h
`B ,−`
∣∣∣∣2 . (3)
All the radial dependence is contained in the functions
RA,B` that modulate the angular impulse response func-
tions h(i,s). We emphasize that although the radial part
of the Schmidt modes does not experience any signif-
icant transformation when the idler and signal beams
traverse their respective linear optical systems, its proper
inclusion in the detection process is essential. This is re-
flected in the structure of Eq. (3) with the presence of
`-dependent radial functions RA,B` , and is a consequence
of a non-constant phase-matching function G. Had G
been a constant then the input two-photon wave func-
tion (1) would have been represented by a common radial
function times a (non-normalisable) maximally entangled
superposition of spiral harmonic modes. In this limit-
ing case one derives a fundamentally different prediction
for the coincidences: P`A,`B ∝ |
`(−1)
`B ,−`|
where now all the radial dependence of the detected
modes appears only as a global function.
IV. AN EXACTLY SOLUBLE MODEL FOR THE
TWO-PHOTON AMPLITUDE
To illustrate the previous results, suppose that the
phase-matching function G is also Gaussian (see Ap-
pendix C, where a justification of this model is provided),
an approximation implying that photons are generated
near the phase-matching region of wave vectors where
the down-conversion process occurs most effectively. The
normalised two-photon amplitude reads
Φ(qi,qs) =
0|qi+qs|
2/4e−b
2|qi−qs|2/4 , (4)
where b < w0 plays the role of an effective width of
the phase-matching function, and depends on the non-
linear crystal thickness, although no specific relation is
assumed here. At variance with other models, where
G is often approximated by a constant (b → 0), the
two-photon amplitude (4) captures the relevant features
of the transverse wave-vector correlation between the
idler and signal photons, and provides an analytically
amenable model that yields explicit formulas for Eq. (3).
We show in Appendix A that for the two-photon ampli-
tude (4), the Schmidt mode functions belong to the well-
known Laguerre-Gaussian basis [17], or, more generally,
to a continuum family of spatial modes generated via
metaplectic mappings (rotations on the orbital Poincaré
sphere) from the Laguerre-Gaussian modes [20, 26, 27].
Furthermore, we also find that the Schmidt eigenvalues
are λ`n = (1 − ξ2)2ξ2|`|+4n, with ξ = (w0 − b)/(w0 + b).
This allows us to express the Schmidt number in the fol-
lowing closed form K = [(1 + ξ2)/(1− ξ2)]2. Large values
of K occur when b→ 0. The minimum K = 1 is attained
when b → w0 (the two-photon amplitude (4) becomes a
separable function in the idler and signal wave vectors).
The Schmidt width for any of the above families of eigen-
modes is always the same: wS =
2w0b. We stress that
the Schmidt width does not represent the actual cross-
section widths Wi,s of the idler and signal beams. These
widths, that can be measured experimentally, can also
be obtained by resorting to the partially reduced density
matrix of the idler and signal photons. For amplitude (4)
one finds Wi,s =
2w20 + b
2, which is consistent with the
widths employed in [28] in the thin crystal approximation
(b→ 0).
Let us focus on the most usual encountered situation
where the measured fiber modes are the fundamental
Gaussian modes (of width wG at the phase plates). Since
the involved eigenfunctions in the Schmidt decomposi-
tion (1) have cylindrical symmetry, we use the Laguerre-
Gaussian modes as the convenient computational basis to
derive the radial functions R` ≡ R
`A,B=nA,B=0
` (see Ap-
pendix B). Remarkably, it turns out that the functions
R` can be cast in terms of a sole parameter s
R`(s) =
Γ2(1 + |`|
Γ(1 + |`|)
; 1 + |`|; s2
s|`| , (5)
where F (a, b; c; d) denotes the hypergeometric function,
1 + ξ2 + (1− ξ2)(wS/wG)2
w20 − b2
w20 + b
2 + (2w0b/wG)2
. (6)
The parameter s corrects ξ by taking into account the
spatial overlapping of the Schmidt and the fiber modes.
Indeed, if and only if wS = wG, does s = ξ hold. When
b → 0 then s → 1, which corresponds to a constant
phase-matching function. The functions R`(s) increase
monotonically from R`(0) = 0 to R`(1) = 1 for ` 6= 0
(R0(s) = 1). The fact that all R`(s) depend on a single
parameter s will be exploited in the next Section to show
how one can estimate the Schmidt number in a particular
experimental scenario.
V. EXPERIMENTAL SCHMIDT NUMBER
Let us now examine the problem of measuring the
amount of transverse entanglement of two-photon sources
which, in our case, is characterised by the Schmidt num-
ber K. In principle, a complete quantum tomography
of the two-photon state could yield the desired amount
of entanglement [11], but generally this would require a
very large number of measurements, each for every possi-
ble pair of spatial modes, and this is technically demand-
ing. Here we propose an alternative approach. In our
simple model for the two-photon amplitude (4) we have
found that the radial part of the coincidence probabili-
ties (3) depends on a single parameter s. This parame-
ter s involves the characteristic widths appearing in the
two-photon amplitude (4), namely the pump beam width
w0 and the phase-matching width b, together with the
fiber mode width wG (at the phase plate locations). The
phase-matching width b could be measured by scanning
in the plane of detection, but in our case it is not neces-
sary. If, instead, one rotates the azimuthal phase plates
(maintaining the detectors fixed), then, the recording of
the coincidence distributions allows one to extract the
value of s as a fitting parameter for Ps ≡ P
nA=0,nB=0
`A=0,`B=0
via Eqs. (3) and (5). Therefore, if s is conceived as a
parameter to be directly measured (rather than b), the
amount of transverse entanglement can now be written
in the form
(1 + 2sµ2)2
(1− s)(1 + s+ 4sµ2)
, (7)
FIG. 2: (Left column) Normalised coincidence distributions
as a function of the relative orientation between the idler and
signal angular diaphragms. Solid, dashed and dotted curves
are calculated for s = 1, 0.7, and 0.4, respectively. (a) β = π/2
and (c) β = π, both with η = 0.5. (Right column) The
corresponding configurations of the angular diaphragms.
where µ ≡ w0/wG. This experimental Schmidt number
depends on quantities that are easily accessible: the fit-
ting parameter s and the widths w0 and wG (the presence
of the width ratio µ should be interpreted as a correcting
geometrical factor). It increases from K = 1, when s = 0,
to infinity as s→ 1.
By properly engineering the impulse response func-
tions h(i,s) it is possible to enhance the sensitivity of Ps
with s, thus improving the accuracy of the estimated K.
To this end, we consider two simple types of transpar-
ent azimuthal phase plates, and examine the dependence
of Eq. (3) when the idler and signal phase plates are
mutually rotated a relative angle α ≡ αi − αs. Their
dispersionless impulse response functions are of the form
h(i,s)(φ, φ′) = eiθi,s(φ)δ(φ− φ′). Owing to the initial per-
fect anticorrelation in OAM of the down-converted pho-
ton pairs, one expects that only when both photons are
subjected to complementary azimuthal transformations
the perfect anticorrelation is preserved and the coinci-
dences are maximal. As soon as the phase plates are ro-
tated in such a way that they are not longer oriented in
a complementary arrangement, the photon coincidences
decrease. Notice that the axes with respect of which the
angles αi and αs are taken do not coincide. For symmet-
ric phase plates these reference axes are inverted 180◦. In
what follows, we assume that each plate is characterised
by a noninteger parameter η = (n0 − 1)d/λ, where d is
the relative step height introduced by the plates, n0 their
refractive index, and λ the wavelength of the idler and
signal beams.
The first phase plate type we consider is an an-
gular diaphragm [see inset in Fig. 2(a)]. It con-
sists of a thin uniform dielectric circular slab with
a “cake-slice” indentation that subtends an angle β,
with a nonzero θi(φ) = −θs(φ) ≡ 2πη (mod 2π)
only if αi,s + β < φ < αi,s + 2π. Similar angu-
lar diaphragms have been employed in proof-of-principle
demonstrations of the uncertainty relation for angu-
lar position and OAM [29]. The probability (3) can
be cast as P(AD)s (α) =
R0(s)[(β − π)2 + π2 cot2(πη)]
FIG. 3: Normalised coincidence distributions as a function
of the relative orientation between the idler and signal spiral
phase plates. Solid, dashed and dotted curves are calculated
for s = 1, 0.7, and 0.4, respectively. (a) η = 0.5; and (b)
η = 4.5. (c) Configuration of the spiral phase plates.
`=1R`(s) cos(`α) sin
2(`β/2)/`2
. The main fea-
tures of the normalized coincidence P(AD)s (α)/P
s (0)
are: (i) it does not depend on the integer part of η
(it suffices to consider 0 < η < 1), the visibility be-
ing maximal when η = 1/2; (ii) the coincidence dis-
tributions are identical whether the aperture angle is
β or 2π − β and symmetrical around α = π; (iii) for
β 6= π and η fixed, the visibility diminishes as s decreases;
and (iv) the maximum visibility always occurs in the
limit s→ 1 (constant phase-matching function and thus
very high Schmidt number) where one has P(AD)s→1 (α) =
π[cot2(πη)− 1] + |2π − α− β|+ |α− β|
. Figures
2(a) and (c) depict the characteristic profiles of the (nor-
malised) P(AD)s when the aperture angles of the angular
diaphragms are β = π/2 and β = π [Figs. 2(b) and
(d) show, respectively, the configurations of the angu-
lar diaphragms that yield the coincidences (a) and (c)].
Comparing Figs. 2(a) and (c) one sees that the visibility
of the former exhibits a much stronger variation with s
than the latter. This suggests that the phase plate con-
figuration of Fig. 2(b) is preferred to that of Fig. 2(d)
for achieving a more accurate estimation of the Schmidt
number via Eq. (7). Finally, we should add that if η is an
integer number then the profiles of P(AD)s are constant
(independent of α).
The second type of azimuthal (refractive) component
is a spiral phase plate [see inset in Fig. 3(a)] [19, 30].
Its impulse response function includes the phase de-
pendence θi(φ) = −θs(φ) = ηφ, for αi,s < φ <
αi,s + 2π. In this case, the joint probability (3) can be
written as P(SPP )s (α) =
`R`(s)e
i`α(`+ η)−2
∣∣2. At
variance with P(AD)s , the coincidence distributions ex-
hibit the development of interference ripples as s de-
creases and η increases [see Figs. 3(a) and (b)]. When
s → 1 a parabolic profile is obtained P(SPP )s→1 (α) =
π2 cot2(πη) + (α− π)2
/ sin2(πη). In this limit the
coincidences become independent of the integer part of
η, and, as with angular diaphragms, the maximum coin-
cidence visibility is attained (when η = 1/2).
FIG. 4: Coincidence distribution dependence on the relative
orientation between idler and signal spiral phase plates (SPP
with η = 3.5). Circles are experimental values from Ref. [14],
solid and dashed curves correspond to P(SPP )s fitted with s =
0.66, and P(SPP )s→1 , respectively. The slight asymmetry in the
experimental values was probably caused by the presence of
a small anomaly in the center of the spiral phase plates [14].
(Inset) Schmidt number K as a function of the width ratio
µ = w0/wG for s = 0.66.
We have considered other types of azimuthal phase
plates and found the same parabolic dependence of the
coincidences as s→ 1. This is consistent with the above-
mentioned prediction that for very high transverse en-
tanglement the joint probability becomes independent of
the radial structure of the Schmidt modes. On the other
hand, the absence of a vanishing minimum in the co-
incidence distributions is a signature of a finite amount
of transverse entanglement. This effect gives rise to a
contribution in the photocounts which always exists re-
gardless of the detector efficiencies [see Fig. 2(a) and
Figs. 3(a),(b)]. It can however be diminished for cer-
tain phase plate configurations [see Fig. 2(c)] and/or by
employing a large wG (wG �
2w0b).
Spiral phase plates were recently used in an elegant ex-
periment aimed to show the high-dimensional spatial en-
tanglement of a two-photon state from a down-conversion
source [14]. The relevant parameters of the experiment
are: η = 3.5 for the phase plates, λ = 0.8 µm, pump
width w0 = 780 µm, and thickness of the nonlinear crys-
tal L = 1 mm [14]. According to the model presented
in Ref. [31], which corresponds to our P(SPP )s→1 , it was
concluded that K > 3700 ± 100. Figure 4 depicts the
predicted distribution for P(SPP )s (dashed line) together
with the measured coincidences reported in Ref. [14].
According to our approach, a probability distribution
P(SPP )s fitted with s = 0.66 (solid line in Fig. 4) shows
excellent agreement with the experimental results. Equa-
tion (7) then suggests that the corresponding Schmidt
number should be much smaller than the above K. How-
ever, we cannot perform a definite estimate for K since
the value of the fiber mode width wG at the location of
the spiral phase plates, needed to calculate µ = w0/wG
appearing in Eq. (7), was not provided in Ref. [14].
The inset in Fig. 4 plots K for a wide range of ratios
µ = w0/wG. For instance, using our fitting parameter
s = 0.66, together with w0 = 780 µm and assuming
wG & 250 µm, leads to µ . 3 and an estimate for K . 20,
at least two orders of magnitude smaller than the quoted
K in Ref. [14]. A very large value for K could only be ex-
pected if, in the experiment, the fiber mode width wG in
the plane containing the phase plates was much smaller
than the pump width w0.
VI. CONCLUSIONS
In view of the situation described above, it is rea-
sonable to conclude that additional experimental results
and further theoretical analyses -a recent example can
be found in Ref. [32]- are necessary and of interest in
order to clarify the problem of quantifying the amount
of transverse entanglement of photon pairs produced in
down-conversion sources. If future values for the Schmidt
number coming from accurate and solid measurements
confirm the estimate of Ref. [14] and cannot be repro-
duced within our approach, one should conclude that the
conventional Gaussian profile constitutes a poor approx-
imation for the phase-matching function G in Eq. (4).
However, the ability of our predicted coincidence distri-
butions to fit the experimental distributions of Ref. [14]
suggests the correctness of our analysis and that the value
for the Schmidt number quoted in Ref. [14] could be over-
estimated. In this case, our approach not only would pro-
vide a simple an accurate procedure to identify the rele-
vant spatial modes and extract the degree of transverse
entanglement; it would go in fact beyond by character-
izing the action of all local bipartite azimuthal optical
transformations on a broad family of two-photon states
using only two detectors. Our results could also be of in-
terest in other fields such as in identifying the intervening
spatial modes when preparing well-controlled superposi-
tions of photon states carrying OAM (to be coherently
transferred onto Bose-Einstein condensates [33, 34] or by
entangling photons with ensembles of cold atoms [35]), in
high-resolution ghost diffraction experiments with ther-
mal light [36], or in decoherence processes such as the en-
tanglement sudden death mediated by the simultaneous
action of several weak noise sources on bipartite photon
systems [37].
Acknowledgments
We thank S. Barnett, D. Diego, G. Molina-Terriza,
M.J. Padgett and S. Walborn for useful discussions
and gratefully acknowledge financial support from the
Spanish Ministry of Science and Technology through
Project FIS2005-01369, Juan de la Cierva Grant Pro-
gram, CONSOLIDER2006-00019 Program and CIRIT
Project SGR-00185.
APPENDIX A
In this Appendix we outline the derivation of the eigen-
values and eigenfunctions in the Schmidt decomposi-
tion (1) when the two-photon amplitude Φ is given by
Eq. (4). We start with some well-known facts.
For a general two-photon pure state, |ψ〉, it is possible
to express |ψ〉 as a bilinear sum of idler and signal basis
states |τi,s〉 belonging to the Hilbert space of the system
|ψ〉 =
τi,τs
Cτi,τs |τi〉 ⊗ |τs〉 , (A1)
where τi and τs label the set of quantum numbers for
the idler and signal photons, respectively, whereas the
coefficients Cτi,τs describe the probability amplitudes for
each tensor product of basis states. Our first aim is to
evaluate the coefficients Cτi,τs for the down-converted
state |ψ〉 =
dqidqsΦ(qi,qs)â†(qi)â†(qs)|vac〉. Let
uτi,s(qi,s) = 〈vac|â(qi,s)|τi,s〉 denote the basis wave func-
tions in the transverse momentum representation. Insert-
ing the closure relations
|τi,s〉〈τi,s| = 1̂, it can be
readily seen that the coefficients Cτi,τs are given by
Cτi,τs =
dqidqsΦ(qi,qs)u
(qi)u
(qs). (A2)
Since we are interested in elucidating the correlation
properties of the spatial degrees of freedom (OAM and
the magnitude of the transverse radial wave vectors), we
choose as the computational basis the complete set of
normalised Laguerre-Gaussian modes [20]
u`,n(q, φ) =
2π(|`|+ n)!
L|`|n
× exp
i`φ− i
(2n+ |`|)
≡ exp (i`φ) v`,n(q)/
2π, (A3)
where q and φ denote the radial and azimuthal variables
in momentum space, w is the mode width (at the beam
waist), and L|`|n (x) are the associated Laguerre polynomi-
als. The indices ` = 0,±1,±2, . . . and n = 0, 1, 2, . . . rep-
resent the winding (or topological charge) and the num-
ber of nonaxial radial nodes of the modes.
Combining Eqs. (4), (A2) and (A3), it follows that
ni,ns
`i,`s
dqi dqs e
−w20|qi+qs|
2/4e−b
2|qi−qs|2/4
× e−i`iφie−i`sφsv`i,ni(qi) v`s,ns(qs) . (A4)
By means of the well-known Anger-Jacobi identity
e−x cos(φi−φs) =
m=−∞(−1)
mIm(x)eim(φi−φs), where
Im(x) is the modified Bessel function of the first kind, the
two angular integrals yield the selection rule m = `i =
−`s ≡ `. This shows that the idler and signal photons are
perfectly anticorrelated with respect to their topological
charge, which is a manifestation of OAM entanglement.
Hence, Cni,ns`i,`s = C
ni,ns
`,−` δ``iδ−``s , where
ni,ns
`,−` = (−1)
`2w0b
qidqiqsdqse
−(w20+b
2)(q2i+q
× v`,ni(qi) v−`,ns(qs) I`
(w20 − b2)qiqs
. (A5)
The first of the radial integrals in (A5) can be performed
by resorting to the following formula∫ ∞
x|`|+1L|`|n (x
2)e−γx
I`(xy)dx
(γ − 1)ny|`|
2|`|+1γ|`|+n+1
4γ L|`|n
4(γ − 1)γ
valid for all real y, integers n ≥ 0 and `, and complex γ
(Re(γ) > 0).
The second radial integral can also be done analyti-
cally. However, a dramatic simplification occurs if the
widths wi and ws of the idler and signal radial modes,
which at this stage have not been specified, are properly
selected: wi = ws =
2w0b. In this case, one obtains
a second selection rule for the radial indices. Namely,
ni = ns ≡ n. That is, with such a choice of the widths,
which is unique, the idler and signal radial modes are
perfectly correlated. But this shows that it is precisely
for wi = ws =
2w0b when one derives the Schmidt de-
composition. Thus, the Schmidt width, wS , corresponds
to wS =
2w0b. The coefficients reduce then to
ni,ns
`i,`s
= (−1)`
(w0 + b)2
w0 − b
w0 + b
)|`|+2n
× δ``iδ−``sδnniδnns . (A6)
Let ξ = (w0−b)/(w0+b). It is now clear from Eq. (A6)
that the Schmidt eigenvalues are λ`n = (1− ξ2)2ξ2|`|+4n.
We therefore conclude that the Schmidt decomposition
of the two-photon amplitude (4) is
Φ(qi,qs) = (1− ξ2)
(−1)`ξ|`|+2nu`,n(qi)
× u−`,n(qs), (A7)
where u`,n(qi) = ei`φiv`,n(qi)/
2π and u−`,n(qs) =
e−i`φsv−`,n(qs)/
2π represent the idler and signal
Schmidt eigenmodes. These eigenmodes belong to the
Laguerre-Gaussian basis. It is worth mentioning that
although the Schmidt width is unique, other decom-
positions (actually infinitely many) are possible in dif-
ferent eigenmode bases. For instance, the Hermite-
Gaussian modes (in Cartesian variables and the same
2w0b) constitute another possible basis. In fact,
all eigenmode bases connected via the following unitary
(metaplectic) transformation Û(θ, ϕ) = exp(−iθ L̂ · uϕ),
where uϕ = (− sinϕ, cosϕ, 0) is a unit vector in the
equatorial plane of the so-called orbital Poincaré sphere
(parameterised by the spherical angles θ and ϕ) and L̂
is an angular momentum operator, form Schmidt bases
for (A7). The components of the angular momentum op-
erator: L̂x, L̂y, and L̂z are the three SU(2) generators.
Of these, only L̂z represents real spatial rotations along
the propagation of light, and it is thus the only compo-
nent associated with the orbital angular momentum of
light that can be measured in experiments [20].
APPENDIX B
In this second Appendix we explicitly show how to cal-
culate the radial functions R`(s) given in Eq. (5). For
the chosen two-photon amplitude (4) and the measure-
ment scenario, these radial functions depend on the spa-
tial overlap of the Schmidt and the fiber (fundamental
Gaussian) modes at the planes where the azimuthal phase
plates are located. We need to evaluate the integrals
rirsdridrsv
`A=0,nA=0
(ri)v
`B=0,nB=0
×v(wS)`,n (ri)v
−`,n(rs). We use the Schmidt eigenvalues
λ`n = (1 − ξ2)2ξ2|`|+4n found in Appendix A. The idler
and signal Schmidt modes v(wS)`,n (ri) and v
−`,n(rs) be-
long to the Laguerre-Gaussian mode basis. We stress
once again the fact that the corresponding fiber wG and
Schmidt wS widths are not necessarily equal. This im-
plies that the orthogonality relation for these modes,∫
rdr v(wG)`G,nG(r)v
`S ,nS
(r) = δ`G`SδnGnS , does not hold in
general when wS 6= wG.
We start by recalling a remarkable identity between
Laguerre polynomials L|`|n and the modified Bessel func-
tions I` [38]
n!L|`|n (x)L
n (y)
Γ(|`|+ n+ 1)
(xyz)−|`|/2
(x+ y)z
, (A8)
valid for all real x, y, integer ` and complex z (|z| < 1).
In detail, the integrals to calculate read
R` = (1− ξ2)ξ|`|
ridri
(|`|+ n)!
L|`|n
rsdrs
(|`|+ n)!
L|`|n
. (A9)
Inserting identity (A8) in (A9) and employing the follow-
ing result [38]∫ ∞
r e−γr
I`(2νr) dr =
Γ(1 + |`|
2Γ(1 + |`|)ν
M− 12 ,
valid for Re(γ) > 0 and all complex ν, with Ma,b repre-
senting the Whittaker functions, we obtain
Γ(1 + |`|
Γ(1 + |`|)
8(1− ξ2)3sw2S
ξ3w4G
dr exp
(2− s2)ξr2
(1− ξ2)sw2S
× M− 12 , |`|2
2sξr2
(1− ξ2)w2S
, (A10)
where we have introduced the parameter s = 2ξ/[1+ξ2 +
(1− ξ2)(wS/wG)2]. The remaining integral is carried out
through the use of formula
e−γ̄t
M− 12 ,
(t) =
Γ(1 + |`|
γ̄ − 1
) |`|
; 1 + |`|;
valid when γ̄ > 1/2. The function F (a, b; c; d) denotes
the hypergeometric function. Equation (A10) is finally
given by
1− ξ2
1 + (1−ξ
)2 Γ2(1 + |`|2 )Γ(1 + |`|)
; 1 + |`|; s2
s|`|. (A11)
This expression reduces to Eq. (5) when one leaves aside
all the `-independent factors (they do not play any sig-
nificant role in the normalised coincidence profiles). In
this way the two-photon detection probabilities (3) can
be cast in terms of the impulse response functions h(i)`A,`,
`B ,−` and the radial functions R`(s) that solely depend
on the parameter s (when `A = `B = 0).
APPENDIX C
In this appendix we would like to justify the model pro-
posed here, the double gaussian profile as phase matching
function.
1. Classical Energy for Non-linear Media
In this section we derive explicitly the energy of an
anisotropic and non-linear medium. This classical de-
scription enables us to find a reasonable quantized Hamil-
tonian which allow us to describe the PDC process. In
anisotropic and non-linear media we must be cautious in
the description of physical variables. For a deep analysis,
we begin by assuming that the polarization of the mate-
rial obeys the usual expansion from non-linear optics
Pi = εo[χ
ij Ej + χ
ijkEjEk + χ
ijklEjEkEl + ...] ,
(A12)
where χ(n) (with n > 1) represent the nonlinear suscep-
tibility tensors of order n+1 responsible for the coupling
of n + 1 fields. The energy of the electromagnetic field
can be written as
E · dD , (A13)
where the displacement vector field is defined as usual
D = �oE + P. In linear media, the energy density of
the electromagnetic field depends quadratically on the
electric field ∝ E2, whereas in non-linear (parametric)
media, polarization exhibits a dependence on higher or-
der electric field terms (A12). We can expand the energy
(A13) in two terms, one related to the linear contribution
of the electric field and the other one related to the non-
linear part. Taking the lowest order nonlinearity (A12),
the energy due to this contribution can be written as
[39, 40],
d3r χ̃
ijk Ei(r, t)Ej(r, t)Ek(r, t) (A14)
where the integration extends over the volume V of the
nonlinear medium. Notice that the second order suscep-
tibility is not the same as the susceptibility in equation
(A12). We will consider that the medium response to
the electric field is not instantaneous. Hence, the second
order polarization should be written as
i (r, t) = �o
dt1dt2 χ
ijk(t− t1, t− t2)
× Ej(r, t1)Ek(r, t2) .
(A15)
We can see, taking the Fourier transformation of the elec-
tric fields in equation (A15), that this specific form of the
susceptibility implies that different monochromatic waves
(with a well-defined frequency) of the electric field can
excite other frequencies. Hence, the energy of the elec-
tromagnetic field due to the bilinear susceptibility (A14)
can be expressed as [41],
∫ ∫ ∫
dωdω′dω′′ χ̃
ijk(ω, ω
′, ω′′)
× Ei(r, ω′′)Ej(r, ω − ω′)Ek(r, ω′) .
(A16)
This energy gives rise to the quantized Hamiltonian fre-
quently found in the literature [15] to describe the pro-
cess of spontaneous parametric down conversion, where
an incident beam interacts with a non-linear medium and
splits into two lower-frequency signal and idler photons.
2. Parametric Down-Conversion
If we have a crystal with nonzero χ(2), pumped with a
laser beam, there is a small, but non negligible, probabil-
ity that a pump photon will decay into an idler and signal
photon pair. We take the incident pump light beam to
be in the form of an intense quasi-monochromatic plane
wave propagating along the z-direction,
Ep(r, t) = �p unm(r⊥, z) e
iωpt , (A17)
where �p is the polarization of the pump beam and
unm(r⊥, z) is a slowly-variant-amplitude (for the mode
having indices n and m) along z which obeys the paraxial
equation in anisotropic media [42]. That is, the paraxial
equation for free propagation cannot be used and must be
modified. The pump beam is described classically, within
the undepleted approximation, which means that one as-
sumes that the input state is made up of a large number
of photons and only few of them interact with the non-
linear medium, producing pairs of correlated photons. In
other words, the pump beam, at the output of the crystal,
remains almost unaffected.
A form for the interaction Hamiltonian operator ĤI(t)
can be given motivated by the classical electromagnetic
field energy (A16), and reads (we are always in the inter-
action picture, though it can also be done in the Heisen-
berg picture [40]),
ĤI(t) =
ki,σi
ks,σs
ijk(ωp, ωi, ωs) (�
ki,σi
× (�∗ks,σs)
j(�p)
k unm(r⊥, z) â
ki,σi
ks,σs
e−i(qi+qs)·r⊥
× e−i[(kiz+ksz)z+(ωp−ωi−ωs)t] +H.c. ,
(A18)
where k(i,s) = q(i,s) + k(i,s)z ûz. In equation (A18) two
quantized electric fields are considered; they will be re-
sponsible for idler and signal photons, whereas the pump
beam is associated with the classical field [39]. As we
have mentioned before, we are carrying out the calcula-
tions under the interaction picture, where we know that
a state evolves as
|Φ(t)〉 = T
0 dτĤI (τ)
|Φo〉 ' |Φo〉 −
dτ : ĤI(τ) : |Φo〉
dτ ′ ĤI(τ)ĤI(τ
|Φo〉+ ... ,
(A19)
with T denoting temporal ordering. Since the interaction
is assumed to be weak enough, it suffices to expand the
exponential (A19) up to first order term. The action of
the Hamiltonian operator on the evolution of the initial
vacuum state yields the initial state plus a two-photon
state (contributions due to higher order processes involve
the generation of four, six,... photons, and are exceed-
ingly small). Notice that applying the Wick theorem in
the above expansion, temporal ordering becomes normal
ordering up to first order. Therefore,
|Φ(t)〉e = T
0 dτĤI (τ)
|Ω〉 ' |Ω〉 −
ki,σi
ks,σs
ijk(ωp, ωi, ωs) (�
ki,σi
ks,σs)
−i(kiz+ksz)z
r⊥ unm(r⊥, z) e
−i(qi+qs)·r⊥
−i(ωp−ωi−ωs)t/2 sin [(ωp − ωi − ωs)t/2]
(ωp − ωi − ωs)/2
|ki, σi; ks, σs〉 ,
(A20)
where we have performed the integration with respect to
time τ . Let L denote the length of the crystal along z,
whereas S corresponds to the transverse area (the vol-
ume of the crystal is V = LS). Equation (A20) is a gen-
eral expression for any input paraxial beam. Assuming
that the transverse area of the crystal S is much larger
than the characteristic width of the input beam, the inte-
gral with respect to the transverse position becomes the
transverse Fourier transform. Now taking into account
that the paraxial wave is propagating in an uniaxially
anisotropic crystal [42], this integral becomes
r⊥ unm(r⊥, z) e
−i(qi+qs)·r⊥ → eikpne(θ,ωp)ze
ne(θ,ωp)
2n2e(ωp)kp
q2y+q
n2e(θ,ωp)
n2o(ωp)
−iqxsθcθ
n2e(θ,ωp)
ñ2(ωp)
z eunm(qi + qs) , (A21)
where, for simplicity, we assume that the optical axis
is contained in the plane xz and the pump beam is lin-
early polarized along the x-direction (the pump beam has
extraordinary polarization). Here, θ refers to the angle
between the optical axis and the z-axis, and
q = qi + qs ; kp =
; cθ ≡ cos θ ; sθ ≡ sin θ ,
n2e(θ, ωp)
sin2 θ
n2e(ωp)
cos2 θ
n2o(ωp)
ñ2(ωp)
n2o(ωp)
n2e(ωp)
Knowledge of the pump spectrum at the input crystal
face enables, via equation (A21), to describe the evolu-
tion of the pump beam through the crystal. In order to
satisfy both the undepleted approximation and the fact
that the pump profile is assumed not to vary significantly
during the nonlinear propagation, the crystal must be
very thin. The z-dependence in equation (A21) will play
a crucial role to correctly describe the phase-matching
conditions below.
Defining
ne(θ, ωp)
n2e(ωp)
q2y + q
n2e(θ, ωp)
n2o(ωp)
−qx sin θ cos θ
n2e(θ, ωp)
ñ2(ωp)
we can perform the integral with respect to z in equation
(A20), resulting in the so-called phase-matching function:Z L
i[kpne(θ,ωp)−Θ−kiz−ksz ]z =
i∆(kp,ki,ks)L/2
sin [∆(kp,ki,ks)L/2]
∆(kp,ki,ks)/2
, (A22)
where the argument
∆(kp,ki,ks) ≡ kpne(θ, ωp)−Θ− kiz − ksz , (A23)
describes the wave vector mismatching. Therefore, equa-
tion (A20) can be cast (excluding the vacuum state) as
|Φ(t)〉(2ph)e = −
ki,σi
ks,σs
ijk(ωp, ωi, ωs) (�
ki,σi
)i(�∗ks,σs)
j(�p)
k ei∆(kp,ki,ks)L/2
sin [∆(kp,ki,ks)L/2]
∆(kp,ki,ks)/2
× ũnm(qi + qs) e−i(ωp−ωi−ωs)t/2
sin [(ωp − ωi − ωs)t/2]
(ωp − ωi − ωs)/2
|ki, σi; ks, σs〉 . (A24)
3. PDC in type II
In this section we are going to consider the paramet-
ric down conversion under type II phase-matching con-
ditions. This implies that the pump beam is extraor-
dinarily (linearly) polarized, whereas the idler and idler
photons are ordinarily (the polarization vector is perpen-
dicular to the optical axis) and extraordinarily polarized,
respectively. Our first aim will be to further simplify the
down-converted state (A24). If the interaction time is
sufficiently large, we can replace the temporal-sinc factor
in equation (A24) by a delta function,
sin [(ωp − ωi − ωs)t/2]
(ωp − ωi − ωs)/2
→ 2π δ(ωp − ωi − ωs) ,
This is a good approximation, and the oscillatory factor
e−i(ωp−ωi−ωs)t/2 can then be discarded by a regulariz-
ing procedure. In particular, one obtains the frequency
matching condition
ωp = ωi + ωs , (A25)
that is, energy conservation imposes that the annihila-
tion of the pump photon gives rise to the creation of the
idler and signal photons. In what follows, and effective
nonlinear susceptibility χeff will be used to represent
the relevant components of χ(2)ijk contracted with the cor-
responding components of the pump, idler and signal po-
larizations (we also absorb in χeff additional constants).
Considering the continuity of the wave vectors of signal
and idler photons (assuming that the volume of the crys-
tal is large enough), equation (A24) reduces to
|Φ〉(2ph)e = −
ks χeff (ωp, ωi, ωs) e
i∆(kp,ki,ks)L/2
sin [∆(kp,ki,ks)L/2]
∆(kp,ki,ks)L/2
eunm(qi + qs)
× δ(ωp − ωi − ωs) |ki, σo,ks, σe〉 .
Notice that the idler and signal photons have fixed po-
larization, ordinary (σo) and extraordinary (σe) respec-
tively.
For convenience, we change the integral in the z com-
ponent of the wave vector by the frequency. Therefore,
for the idler integral, the Jacobian due to the change of
variables is
qidkiz =
∂(qi, kiz)
∂(qi, ωi)
qidωi =
qidωi =˛̨̨̨
n2o(ωi)
ω2i no(ωi)
c2kiz
∂no(ωi)
qidωi ,
while for the signal integral, we have
d2qsdksz =
∣∣∣∣∂(qs, ksz)∂(qs, ωs)
∣∣∣∣ d2qsdωs =
∣∣∣∣∂ksz∂ωs
∣∣∣∣ d2qsdωs =∣∣∣∣∣∣∣
n3e(θ,ωs)
∂ne(θ,ωs)
− kszqsx sin 2θ ∂∂ωs
ñ2(ωs)
− q2sx
n′2e (θ,ωs)
n3e(ωs)
∂ne(ωs)
2 ksz
n2e(θ,ωs)
+ sin 2θ
ñ2(ωs)
∣∣∣∣∣∣∣ d2qsdωs . (A26)
Proof: For ordinary waves the dispersion relation in an
anisotropic medium is
ω2i n
o(ωi)
iz + q
Solving the quadratic equation for the variable kiz
kiz =
ω2i n
o(ωi)
− q2i .
Notice that we only choose the positive solution of kiz. Fi-
nally, taking the derivative with respect to ωi
n2o(ωi)
ω2i no(ωi)
c2kiz
∂no(ωi)
For extraordinary waves the dispersion relation in an
anisotropic uniaxial medium is more complex:
n2e(θ, ωs)
+ (qsx cosϕ+ qsy sinϕ)
ksz sin 2θ
ñ2(ωs)
(qsx cosϕ+ qsy sinϕ)
n′2e (θ, ωs)
(qsx sinϕ− qsy cosϕ)2
n2e(ωs)
,(A27)
where {θ, ϕ} are the polar and the azimuthal angles respec-
tively in spherical coordinates, with θ referring to the angle
subtended by the optical axis and the z-axis. For simplicity,
as we have mentioned before, we assume the optical axis to
be contained in the plane x-z (ϕ = 0). We have to take into
account the common definitions in anisotropic media
n2e(θ, ωs)
sin2 θ
n2e(ωs)
cos2 θ
n2o(ωs)
n′2e (θ, ωs)
cos2 θ
n2e(ωs)
sin2 θ
n2o(ωs)
ñ2(ωp)
n2o(ωp)
n2e(ωp)
In this case, performing the implicit derivative with respect
to ωs in the dispersion relation (A27),
n2e(θ, ωs)
2k2sz
n3e(θ, ωs)
∂ne(θ, ωs)
sin 2θ
ñ2(ωs)
+ kszqsx sin 2θ
ñ2(ωs)
n′2e (θ, ωs)
n3e(ωs)
∂ne(ωs)
. (A28)
where
ñ2(ωs)
n3o(ωs)
∂no(ωs)
n3e(ωs)
∂ne(ωs)
n′2e (θ, ωs)
cos2 θ
n3e(ωs)
∂ne(ωs)
sin2 θ
n3o(ωs)
∂no(ωs)
Solving the equation (A28) for ∂ksz
we find equation
(A26).
Substituting equations (A26) into the output state of
idler and signal photons (A26), we obtain a general ex-
pression for PDC in type II. So far, we have only assumed
the undepleted approximation and that the transverse
area of the crystal is much larger than the characteristic
width of the pump beam.
4. PDC in type II, collinear regime
We now focus on the collinear regime in PDC where
we only consider those idler and signal photons which
are produced nearly in the same direction as that of the
pump beam (in our framework along the z-axis). In this
particular case, the approximation q
(s,i)
(s,i)
� 1 is justified.
Therefore, equations (A26) reduce to
d2qidkiz =
no(ωi)
d2qidωi ,
for the idler, and
d2qsdksz =
ne(θ, ωs)
d2qsdωs ,
for the signal, where we have also made the reasonable
assumption that the variation of refractive indices with
respect to frequency is small. Thus, Jacobians due to
the change of variables are just refractive indices, which
depend on frequency.
In the collinear regime more simplifications can be
done. Let us explicitly write the argument of the phase
matching function (A23) as
∆(kp,ki,ks) = kpne(θ, ωp)− ksne(θ, ωs)− kino(ωi) +
sin 2θ
n2e(θ, ωp)
ñ2(ωp)
n2e(θ, ωs)
ñ2(ωs)
+ qix
sin 2θ
n2e(θ, ωp)
ñ2(ωp)
ne(θ, ωs)
2n2e(ωp)kp
(qiy + qsy)
+ (qix + qsx)
e(θ, ωp)
n2o(ωp)
n3e(θ, ωs)
2n2e(ωs)n
o(ωs)ks
ne(θ, ωs)
2n2e(ωs)ks
q2ix + q
2no(ωi)ki
(A29)
It is clearly an strongly anisotropic function. Notice that
the linear terms in the transverse wave vectors are re-
sponsible for the walk-off effect (the Poynting and wave
vectors of the signal and pump photons are not collinear,
i.e., their energy flows in a different direction from its cor-
responding phase front). We should mention a previous
study by Torres and coworkers [43] where the walk-off ef-
fect was taken into account to describe entangled photon
states generated in type I PDC.
The state for idler and signal photons within type II
PDC (A26) is
|Φ〉(2ph)e =
qidωi
qsdωs χeff (ωp, ωi, ωs)
× ei∆(kp,ki,ks)L/2 sinc [∆(kp,ki,ks)L/2] eunm(qi + qs)
× δ(ωp − ωi − ωs) |ki, σo,ks, σe〉
(A30)
where sinc(x) ≡ sin(x)/x. The symbol ∆Ω represents
the fact that the transverse idler and signal wave vectors
are constrained to be very small (this is the condition for
the collinear regime). This can be achieved experimen-
tally by using a suitable pin hole placed in the z axis.
Now, due to this restriction, only the maximum of the
sinc function contributes to the integral (the side lobes
being blocked). Furthermore, the action of the pin hole
truncating the sinc function can actually be modeled by
a simple exponential function. In addition, we can reg-
ularize the phase factor ei
|qi−qs|2 by just shifting our
coordinate system (z = 0) to the middle of the crystal,
see equation (A23). We will also focus, for simplicity, on
the frequency degenerated case, where the idler and sig-
nal photons have the same frequency (ωp = ωi/2 = ωs/2,
which can be realized by using appropriate narrow-band
filters in front of the photodetectors). Thus, the two-
photon state reads
|Φ〉(2ph)e = χeff (ωp)
d2qs e
−∆(kp,ki,ks)L/2
× ũnm(qi + qs) |ki, σo,ks, σe〉 .
(A31)
The effective susceptibility, due to the energy conserva-
tion and the frequency degeneration in idler and signal
photons, now depends only on the frequency of the pump
beam.
The argument of the phase-matching function G =
e−∆(kp,ki,ks)L/2 is still too complex. Since we wish to
maintain our framework at a sufficiently simple level, and
we will pursue a full analytical study, some further (and
drastic) approximations are required: we take all refrac-
tive indices (ordinary and extraordinary) to be identical
and independent of frequency. Hence, the argument of
the phase matching function reduces to
∆(kp,ki,ks) =
2kpneff
|qi − qs|2 , (A32)
which is, in contrast to equation (A29), an isotropic (with
cylindrical symmetry) function. The effective refractive
index neff can be interpreted as a suitable fitting pa-
rameter (in principle, it can be quite different from the
ordinary and extraordinary refractive indices).
In spite of the strong approximations performed above,
it is illustrative to determine their goodness by compar-
ing the sinc and the exponential phase-matching func-
tions when their arguments are given by equations (A29)
and (A32), respectively. Figure 5 depicts both profiles.
It can be seen that for sufficiently small transverse wave
vectors both profiles are coincident (by suitably choosing
neff ); the deviations arise, typically, when the pin hole is
already truncating both functions. This qualitatively jus-
tifies the validity of the approximations made above. Ac-
tually, this approximation was previously used (without
any justification) by Law and Eberly [17] to study bipar-
tite transverse entanglement from type II PDC. Further-
more, in other previous works, the sinc phase matching
function was even replaced by a constant [14, 31, 44, 45].
In summary, equation (A31) finally becomes (simpli-
fying the notation and absorbing the constant factors)
|ψ〉 =
qs Φ(qi,qs) |qi,qs〉 , (A33)
where Φ = EG is the two-photon amplitude, with
E(qi + qs) = ũnm(qi + qs) denoting the transverse
(spectrum) profile of the pump beam, and G(qi −
qs) = e
− L4kpneff |qi−qs|
being the phase-matching func-
tion. Since we will not focus in later sections on the po-
larization degree of freedom (recall that idler and signal
photons are assumed to be linearly polarized), we have
omitted its explicit dependence in (A33).
Equation (A33) will be exploited in (4) to carry out
our discussion about the OAM entanglement shared by
the downconverted idler and signal photons.
FIG. 5: Comparison between the sinc (solid curve) and the ex-
ponential (dotted curve) phase-matching functions when their
arguments are given by equations (A29) and (A32), respec-
tively. The shadowed region corresponds to the truncation
introduced by the pin hole.
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	INTRODUCTION
	MODAL SCHMIDT DECOMPOSITION FOR TWO-PHOTON STATES
	AZIMUTHAL TRANSFORMATIONS ON THE TWO-PHOTON STATE
	AN EXACTLY SOLUBLE MODEL FOR THE TWO-PHOTON AMPLITUDE
	Experimental Schmidt Number
	Conclusions
	Acknowledgments
	APPENDIX B
	APPENDIX C
	Classical Energy for Non-linear Media
	Parametric Down-Conversion
	PDC in type II
	PDC in type II, collinear regime
	References

  We put forward an approach to estimate the amount of bipartite spatial
entanglement of down-converted photon states correlated in orbital angular
momentum and the magnitude of the transverse (radial) wave vectors. Both
degrees of freedom are properly considered in our framework, which only
requires azimuthal local linear optical transformations and mode selection
analysis with two fiber detectors. The coincidence distributions predicted by
our approach give an excellent fit to the distributions measured in a recent
experiment aimed to show the very high-dimensional transverse entanglement of
twin photons from a down-conversion source. Our estimate for the Schmidt number
is substantially lower but still confirms the presence of high-dimensional
entanglement.

Measuring Two-Photon Orbital Angular Momentum Entanglement
G.F. Calvo, A. Picón and A. Bramon
Grup de F́ısica Teòrica, Universitat Autònoma de Barcelona, 08193 Bellaterra (Barcelona), Spain
(Dated: July 7, 2021)
We put forward an approach to estimate the amount of bipartite spatial entanglement of down-
converted photon states correlated in orbital angular momentum and the magnitude of the transverse
(radial) wave vectors. Both degrees of freedom are properly considered in our framework which only
requires azimuthal local linear optical transformations and mode selection analysis with two fiber
detectors. The coincidence distributions predicted by our approach give an excellent fit to the
distributions measured in a recent experiment aimed to show the very high-dimensional transverse
entanglement of twin-photons from a down-conversion source. Our estimate for the Schmidt number
is substantially lower but still confirms the presence of high-dimensional entanglement.
PACS numbers: 03.67.Mn, 42.50.Dv, 42.65.Lm
I. INTRODUCTION
The phenomenon of quantum entanglement whereby
distant systems can manifest perfectly random albeit per-
fectly correlated behavior is now recognized as the es-
sential ingredient to perform tasks which cannot be ac-
complished with classically correlated systems [1]. The
presence of entanglement has been traditionally revealed
in the violation of Bell-type inequalities [2]. However,
detecting such a violation does not provide in general a
measure of the amount of entanglement. This is par-
ticularly significant in systems correlated in multidimen-
sional degrees of freedom [3]. Several techniques have
been proposed to assess the presence of entanglement
for different quantum scenarios. These include state to-
mography [4, 5, 6], that yields a complete reconstruction
of a quantum state but requires many setting measure-
ments, entanglement witnesses [7], which detect some
entangled states with considerably less measurements,
and the experimental determination of concurrence [8, 9].
In Ref. [9], by using two copies of a down-converted
two-photon state entangled in polarization and trans-
verse momentum, the measurement of concurrence was
achieved with a single, local measurement on one of the
photons.
Although bipartite entanglement is well understood,
finding experimentally feasible procedures to quantify
it for systems correlated in multidimensional degrees of
freedom turns out to be quite challenging and relevant.
Indeed, access to higher dimensional Hilbert spaces in
which information can be encoded and manipulated has
recently attracted great interest, with proof-of-principle
demonstrations using quantum communication protocols
in three-level systems (qutrits), such as entanglement
concentration [10], quantum bit commitment [11] and
quantum coin-tossing [12]. Likewise, a complete char-
acterization of states hyperentangled in polarization, or-
bital angular momentum and frequency has been exper-
imentally implemented [13].
The aim of this paper is to address the problem of
how, by performing a certain set of local linear optical
operations affecting one of two multidimensional spatial
degrees of freedom in which two-photon states can be
entangled, it is possible to obtain an explicit measure of
the amount of bipartite transverse entanglement. Specif-
ically, according to the interplay between both spatial
degrees of freedom -orbital angular momentum (OAM)
and the magnitude of the radial wave vectors- fundamen-
tally different predictions are expected in the subsequent
joint photodetection process, via mode-selection analysis
with two fiber detectors preceded by azimuthal transfor-
mations (acting only on the OAM). The application of
our framework is compared with the results of a recent
experiment [14].
The paper is organized as follows: Section II presents
the general setting of the problem and includes the
Schmidt decomposition technique for describing bipar-
tite spatial entanglement. In Section III we reveal the
generic features that arise in the photodetection coinci-
dences according to the transverse structure of the two-
photon wave function. Section IV illustrates the results
found in Section III with an example of a realistic two-
photon wave function that yields a full analytical solution
to the problem. In Section V a closed-form expression
for the Schmidt number is obtained in terms of easily
accessible experimental parameters. We then proceed to
exploit this Schmidt number in two interesting examples
of optical transformations with azimuthal phase plates.
These enable us to estimate the amount of spatial entan-
glement of a two-photon source. Conclusions of the paper
are drawn in Section VI. Details of our calculations, to-
gether with some useful background material, have been
included in two Appendices.
II. MODAL SCHMIDT DECOMPOSITION FOR
TWO-PHOTON STATES
Two-photon pure quantum states are described in a
Hilbert space by a continuous bilinear superposition of
spatiotemporal multimode states. Sources of such non-
classical states of light are mostly realised in the process
of spontaneous parametric down-conversion [15], where
an intense quasimonochromatic laser pump illuminates
a crystal endowed with a quadratic nonlinearity produc-
ing pairs of photons (idler and signal). Conservation of
energy and momentum impose that the state be spec-
trally and spatially correlated. Here we explore entan-
glement involving spatial degrees of freedom that de-
pend on the transverse structure of these states. For
simplicity, we assume that the down-converted photons
are linearly polarized, monochromatic and frequency de-
generated. The two-photon state can then be written as
|ψ〉 =
dqidqsΦ(qi,qs)â
†(qi)â†(qs)|vac〉, where qi,s are
the transverse components of the idler and signal wave
vectors. Under conditions of paraxial and nearly collinear
propagation of the pump, idler and signal photons, the
two-photon amplitude Φ is given by [16] Φ(qi,qs) =
E(qi + qs)G(qi − qs). The function E represents the
transverse profile of the pump, whereas G originates from
the phase-matching condition in the longitudinal direc-
tion and depends on the specific orientation and cut of
the nonlinear crystal. Since the arguments of E and G
enforce correlations in different manifolds of the idler and
signal wave vector space, it is the global structure of Φ
the one dictating the entanglement degree of |ψ〉.
In order to extract the amount of entanglement con-
tained in Φ, one may resort to the Schmidt decomposi-
tion that provides the spatial information modes of the
two-photon pair. Suppose that the transverse spatial fre-
quency field of the pump beam has a Gaussian profile of
the form E(qi + qs) ∝ e−w
0|qi+qs|
2/4, where w0 is the
width (at the beam waist). The chosen pump profile
peaks when its argument vanishes. This imposes that
the idler and signal transverse wave vectors should be
mostly anticorrelated (qi ' −qs). Remarkably, it was
shown by Law and Eberly [17] that with such a Gaussian
profile E the normalised two-photon amplitude can be
expressed as
Φ(qi,qs) =
(−1)`
λ`n u`,n(qi)u−`,n(qs) , (1)
where u`,n(qi) = ei`φiv`,n(qi)/
2π and u−`,n(qs) =
e−i`φsv−`,n(qs)/
2π are the normalised polar Schmidt
mode functions for the idler and signal photons with
topological charge ` and radial index n corresponding
to eigenvalues λ`n (they satisfy 1 ≥ λ`n ≥ 0 and∑
`,n λ`n = 1) of the reduced density matrices for each
photon. Knowledge of λ`n yields a direct measure of the
degree of transverse entanglement given by the Schmidt
number [18] K = (
`,n λ
−1, which is the reciprocal
of the purity of the idler and signal density matrices,
it is invariant under free propagation and yields an av-
erage of the number of relevant spatial modes involved
in the decomposition. The larger the value of K, the
higher the transverse entanglement. For instance, prod-
uct states correspond to K = 1 (there is only one nonvan-
ishing eigenvalue equal to 1), whereas states with K > 1
are entangled. A distinguishing feature of decomposi-
tion (1) is that it represents Φ in terms of a perfectly
correlated discrete basis of paraxial eigenstates of the
FIG. 1: (color online). Two-photon coincidence detection
configuration. Down-converted idler and signal beams from a
nonlinear crystal traverse two optical systems (h(i,s)) which
include azimuthal phase plates and are coupled into single-
mode fibers (SMF) gated by a coincidence circuit (CC).
OAM operator along the direction of light propagation
(with corresponding eigenvalues `~) [19, 20], rather than
in a continuous plane-wave modal expansion. The precise
form of the radial idler and signal eigenmodes v`,n(qi)
and v−`,n(qs) depends on the specific phase-matching
function G. In particular, when G is approximated by
a constant there are striking consequences: Eq. (1) be-
comes a (non-normalisable) superposition in which all
OAM eigenstates have an equal weight and the radial de-
pendence becomes just a global factor. It is important to
emphasize that to attain decomposition (1) the appropri-
ate choice of the widths, wi and ws, for the idler and sig-
nal radial eigenmodes, has to be made. If the two-photon
amplitude is of the form Φ(qi,qs) = E(qi+qs)G(qi−qs),
then wi = ws ≡ wS , where wS is the so-called Schmidt
width. For widths different from wS an additional sum-
mation over the radial indexes occurs, and the perfect
correlation between the idler and signal radial modes is
absent (see Appendix A). Moreover, the fact that the
idler and signal mode functions are anticorrelated with
respect to their topological charge numbers is a conse-
quence of a more general process: the conservation of
OAM, which is transferred from the pump photon (car-
rying zero OAM for a Gaussian mode) to the down-
converted photon pair [21].
III. AZIMUTHAL TRANSFORMATIONS ON
THE TWO-PHOTON STATE
The usefulness of the Schmidt decomposition (1) be-
comes apparent when analyzing the propagation of two-
photon states through optical systems. Each of the in-
tervening modes evolves and transforms independently of
the others. It is also clear that the correlation proper-
ties displayed by the two-photon amplitude (1) are pre-
served in the position representation. Therefore, sup-
pose that the idler and signal photon beams, described
now by the paraxial two-photon wave function in the
transverse position representation [20, 22, 23] ψ(ri, rs) =
〈ri, rs|ψ〉 =
`,n(−1)
λ`nu`,n(ri)u−`,n(rs), are each
transmitted through different linear optical systems that
include diffractive (or refractive) azimuthal phase plates
(see Fig. 1). The role of the plates in each path is
to imprint an azimuth-dependent phase factor on the
incoming Schmidt modes. Their (separate) action on
the two-photon wave function can be represented by
the unitary and radially symmetric impulse response
functions h(i,s)(φ, φ′). These functions locally trans-
form the spiral harmonic mode content of each pho-
ton via ei`φ/
(i,s)
`′,` e
i`′φ/
2π, with h(i,s)`′,` =∫ 2π
dφ dφ′ei`φe−i`
′φ′h(i,s)(φ, φ′)/2π, so that the re-
sulting output two-photon wave function is
ψout(ri, rs) =
`i,`s,`,n
(−1)`
λ`n h
`s,−`
× ei`iφiv`,n(ri) ei`sφsv−`,n(rs)/2π , (2)
where the initial perfect correlation in OAM is lost, re-
maining only that corresponding to the radial modes.
Upon traversing their respective linear optical systems
the idler and signal photons, having OAM `i = `A, `s =
`B and radial indexes nA,B , are detected in coincidence.
By placing photodetectors at the output ports of suitable
arrays of deterministic mode sorter interferometers [24,
25], or computer-generated holograms [10], it is possible
to distinguish modes bearing different OAM. In practice,
the most straightforward procedure involves projecting
into single-mode fibers, where the propagated mode has a
fundamental Gaussian profile (`A = `B = nA = nB = 0).
The probability that the idler and signal photons will
be projected into modes u`A,nA and u`B ,nB is found to
be PnA,nB`A,`B = |
dridrsu∗`A,nA(ri)u
`B ,nB
(rs)ψout(ri, rs)|2,
with an additional incoherent (and weighted) sum over
the radial indexes nA,B when taking into account multi-
mode detection. This photodection probability is ex-
pressed in terms of the spatial overlap of the Schmidt and
the fiber modes at the planes where the phase plates are
located. Notice that their corresponding widths, wS and
wG, are not necessarily equal, and thus the orthogonal-
ity between these modes when their radial indexes differ
does not hold in general. To evaluate PnA,nB`A,`B , we de-
fine RA,B` ≡
rirsdridrsv
`A,nA
(ri)v
`B ,nB
×v(wS)`,n (ri)v
−`,n(rs), where the two different widths of
the radial modes have been specified for clarity. The co-
incidence probability can then be written as
PnA,nB`A,`B =
∣∣∣∣ ∞∑
(−1)`RA,B` h
`B ,−`
∣∣∣∣2 . (3)
All the radial dependence is contained in the functions
RA,B` that modulate the angular impulse response func-
tions h(i,s). We emphasize that although the radial part
of the Schmidt modes does not experience any signif-
icant transformation when the idler and signal beams
traverse their respective linear optical systems, its proper
inclusion in the detection process is essential. This is re-
flected in the structure of Eq. (3) with the presence of
`-dependent radial functions RA,B` , and is a consequence
of a non-constant phase-matching function G. Had G
been a constant then the input two-photon wave func-
tion (1) would have been represented by a common radial
function times a (non-normalisable) maximally entangled
superposition of spiral harmonic modes. In this limit-
ing case one derives a fundamentally different prediction
for the coincidences: P`A,`B ∝ |
`(−1)
`B ,−`|
where now all the radial dependence of the detected
modes appears only as a global function.
IV. AN EXACTLY SOLUBLE MODEL FOR THE
TWO-PHOTON AMPLITUDE
To illustrate the previous results, suppose that the
phase-matching function G is also Gaussian (see Ap-
pendix C, where a justification of this model is provided),
an approximation implying that photons are generated
near the phase-matching region of wave vectors where
the down-conversion process occurs most effectively. The
normalised two-photon amplitude reads
Φ(qi,qs) =
0|qi+qs|
2/4e−b
2|qi−qs|2/4 , (4)
where b < w0 plays the role of an effective width of
the phase-matching function, and depends on the non-
linear crystal thickness, although no specific relation is
assumed here. At variance with other models, where
G is often approximated by a constant (b → 0), the
two-photon amplitude (4) captures the relevant features
of the transverse wave-vector correlation between the
idler and signal photons, and provides an analytically
amenable model that yields explicit formulas for Eq. (3).
We show in Appendix A that for the two-photon ampli-
tude (4), the Schmidt mode functions belong to the well-
known Laguerre-Gaussian basis [17], or, more generally,
to a continuum family of spatial modes generated via
metaplectic mappings (rotations on the orbital Poincaré
sphere) from the Laguerre-Gaussian modes [20, 26, 27].
Furthermore, we also find that the Schmidt eigenvalues
are λ`n = (1 − ξ2)2ξ2|`|+4n, with ξ = (w0 − b)/(w0 + b).
This allows us to express the Schmidt number in the fol-
lowing closed form K = [(1 + ξ2)/(1− ξ2)]2. Large values
of K occur when b→ 0. The minimum K = 1 is attained
when b → w0 (the two-photon amplitude (4) becomes a
separable function in the idler and signal wave vectors).
The Schmidt width for any of the above families of eigen-
modes is always the same: wS =
2w0b. We stress that
the Schmidt width does not represent the actual cross-
section widths Wi,s of the idler and signal beams. These
widths, that can be measured experimentally, can also
be obtained by resorting to the partially reduced density
matrix of the idler and signal photons. For amplitude (4)
one finds Wi,s =
2w20 + b
2, which is consistent with the
widths employed in [28] in the thin crystal approximation
(b→ 0).
Let us focus on the most usual encountered situation
where the measured fiber modes are the fundamental
Gaussian modes (of width wG at the phase plates). Since
the involved eigenfunctions in the Schmidt decomposi-
tion (1) have cylindrical symmetry, we use the Laguerre-
Gaussian modes as the convenient computational basis to
derive the radial functions R` ≡ R
`A,B=nA,B=0
` (see Ap-
pendix B). Remarkably, it turns out that the functions
R` can be cast in terms of a sole parameter s
R`(s) =
Γ2(1 + |`|
Γ(1 + |`|)
; 1 + |`|; s2
s|`| , (5)
where F (a, b; c; d) denotes the hypergeometric function,
1 + ξ2 + (1− ξ2)(wS/wG)2
w20 − b2
w20 + b
2 + (2w0b/wG)2
. (6)
The parameter s corrects ξ by taking into account the
spatial overlapping of the Schmidt and the fiber modes.
Indeed, if and only if wS = wG, does s = ξ hold. When
b → 0 then s → 1, which corresponds to a constant
phase-matching function. The functions R`(s) increase
monotonically from R`(0) = 0 to R`(1) = 1 for ` 6= 0
(R0(s) = 1). The fact that all R`(s) depend on a single
parameter s will be exploited in the next Section to show
how one can estimate the Schmidt number in a particular
experimental scenario.
V. EXPERIMENTAL SCHMIDT NUMBER
Let us now examine the problem of measuring the
amount of transverse entanglement of two-photon sources
which, in our case, is characterised by the Schmidt num-
ber K. In principle, a complete quantum tomography
of the two-photon state could yield the desired amount
of entanglement [11], but generally this would require a
very large number of measurements, each for every possi-
ble pair of spatial modes, and this is technically demand-
ing. Here we propose an alternative approach. In our
simple model for the two-photon amplitude (4) we have
found that the radial part of the coincidence probabili-
ties (3) depends on a single parameter s. This parame-
ter s involves the characteristic widths appearing in the
two-photon amplitude (4), namely the pump beam width
w0 and the phase-matching width b, together with the
fiber mode width wG (at the phase plate locations). The
phase-matching width b could be measured by scanning
in the plane of detection, but in our case it is not neces-
sary. If, instead, one rotates the azimuthal phase plates
(maintaining the detectors fixed), then, the recording of
the coincidence distributions allows one to extract the
value of s as a fitting parameter for Ps ≡ P
nA=0,nB=0
`A=0,`B=0
via Eqs. (3) and (5). Therefore, if s is conceived as a
parameter to be directly measured (rather than b), the
amount of transverse entanglement can now be written
in the form
(1 + 2sµ2)2
(1− s)(1 + s+ 4sµ2)
, (7)
FIG. 2: (Left column) Normalised coincidence distributions
as a function of the relative orientation between the idler and
signal angular diaphragms. Solid, dashed and dotted curves
are calculated for s = 1, 0.7, and 0.4, respectively. (a) β = π/2
and (c) β = π, both with η = 0.5. (Right column) The
corresponding configurations of the angular diaphragms.
where µ ≡ w0/wG. This experimental Schmidt number
depends on quantities that are easily accessible: the fit-
ting parameter s and the widths w0 and wG (the presence
of the width ratio µ should be interpreted as a correcting
geometrical factor). It increases from K = 1, when s = 0,
to infinity as s→ 1.
By properly engineering the impulse response func-
tions h(i,s) it is possible to enhance the sensitivity of Ps
with s, thus improving the accuracy of the estimated K.
To this end, we consider two simple types of transpar-
ent azimuthal phase plates, and examine the dependence
of Eq. (3) when the idler and signal phase plates are
mutually rotated a relative angle α ≡ αi − αs. Their
dispersionless impulse response functions are of the form
h(i,s)(φ, φ′) = eiθi,s(φ)δ(φ− φ′). Owing to the initial per-
fect anticorrelation in OAM of the down-converted pho-
ton pairs, one expects that only when both photons are
subjected to complementary azimuthal transformations
the perfect anticorrelation is preserved and the coinci-
dences are maximal. As soon as the phase plates are ro-
tated in such a way that they are not longer oriented in
a complementary arrangement, the photon coincidences
decrease. Notice that the axes with respect of which the
angles αi and αs are taken do not coincide. For symmet-
ric phase plates these reference axes are inverted 180◦. In
what follows, we assume that each plate is characterised
by a noninteger parameter η = (n0 − 1)d/λ, where d is
the relative step height introduced by the plates, n0 their
refractive index, and λ the wavelength of the idler and
signal beams.
The first phase plate type we consider is an an-
gular diaphragm [see inset in Fig. 2(a)]. It con-
sists of a thin uniform dielectric circular slab with
a “cake-slice” indentation that subtends an angle β,
with a nonzero θi(φ) = −θs(φ) ≡ 2πη (mod 2π)
only if αi,s + β < φ < αi,s + 2π. Similar angu-
lar diaphragms have been employed in proof-of-principle
demonstrations of the uncertainty relation for angu-
lar position and OAM [29]. The probability (3) can
be cast as P(AD)s (α) =
R0(s)[(β − π)2 + π2 cot2(πη)]
FIG. 3: Normalised coincidence distributions as a function
of the relative orientation between the idler and signal spiral
phase plates. Solid, dashed and dotted curves are calculated
for s = 1, 0.7, and 0.4, respectively. (a) η = 0.5; and (b)
η = 4.5. (c) Configuration of the spiral phase plates.
`=1R`(s) cos(`α) sin
2(`β/2)/`2
. The main fea-
tures of the normalized coincidence P(AD)s (α)/P
s (0)
are: (i) it does not depend on the integer part of η
(it suffices to consider 0 < η < 1), the visibility be-
ing maximal when η = 1/2; (ii) the coincidence dis-
tributions are identical whether the aperture angle is
β or 2π − β and symmetrical around α = π; (iii) for
β 6= π and η fixed, the visibility diminishes as s decreases;
and (iv) the maximum visibility always occurs in the
limit s→ 1 (constant phase-matching function and thus
very high Schmidt number) where one has P(AD)s→1 (α) =
π[cot2(πη)− 1] + |2π − α− β|+ |α− β|
. Figures
2(a) and (c) depict the characteristic profiles of the (nor-
malised) P(AD)s when the aperture angles of the angular
diaphragms are β = π/2 and β = π [Figs. 2(b) and
(d) show, respectively, the configurations of the angu-
lar diaphragms that yield the coincidences (a) and (c)].
Comparing Figs. 2(a) and (c) one sees that the visibility
of the former exhibits a much stronger variation with s
than the latter. This suggests that the phase plate con-
figuration of Fig. 2(b) is preferred to that of Fig. 2(d)
for achieving a more accurate estimation of the Schmidt
number via Eq. (7). Finally, we should add that if η is an
integer number then the profiles of P(AD)s are constant
(independent of α).
The second type of azimuthal (refractive) component
is a spiral phase plate [see inset in Fig. 3(a)] [19, 30].
Its impulse response function includes the phase de-
pendence θi(φ) = −θs(φ) = ηφ, for αi,s < φ <
αi,s + 2π. In this case, the joint probability (3) can be
written as P(SPP )s (α) =
`R`(s)e
i`α(`+ η)−2
∣∣2. At
variance with P(AD)s , the coincidence distributions ex-
hibit the development of interference ripples as s de-
creases and η increases [see Figs. 3(a) and (b)]. When
s → 1 a parabolic profile is obtained P(SPP )s→1 (α) =
π2 cot2(πη) + (α− π)2
/ sin2(πη). In this limit the
coincidences become independent of the integer part of
η, and, as with angular diaphragms, the maximum coin-
cidence visibility is attained (when η = 1/2).
FIG. 4: Coincidence distribution dependence on the relative
orientation between idler and signal spiral phase plates (SPP
with η = 3.5). Circles are experimental values from Ref. [14],
solid and dashed curves correspond to P(SPP )s fitted with s =
0.66, and P(SPP )s→1 , respectively. The slight asymmetry in the
experimental values was probably caused by the presence of
a small anomaly in the center of the spiral phase plates [14].
(Inset) Schmidt number K as a function of the width ratio
µ = w0/wG for s = 0.66.
We have considered other types of azimuthal phase
plates and found the same parabolic dependence of the
coincidences as s→ 1. This is consistent with the above-
mentioned prediction that for very high transverse en-
tanglement the joint probability becomes independent of
the radial structure of the Schmidt modes. On the other
hand, the absence of a vanishing minimum in the co-
incidence distributions is a signature of a finite amount
of transverse entanglement. This effect gives rise to a
contribution in the photocounts which always exists re-
gardless of the detector efficiencies [see Fig. 2(a) and
Figs. 3(a),(b)]. It can however be diminished for cer-
tain phase plate configurations [see Fig. 2(c)] and/or by
employing a large wG (wG �
2w0b).
Spiral phase plates were recently used in an elegant ex-
periment aimed to show the high-dimensional spatial en-
tanglement of a two-photon state from a down-conversion
source [14]. The relevant parameters of the experiment
are: η = 3.5 for the phase plates, λ = 0.8 µm, pump
width w0 = 780 µm, and thickness of the nonlinear crys-
tal L = 1 mm [14]. According to the model presented
in Ref. [31], which corresponds to our P(SPP )s→1 , it was
concluded that K > 3700 ± 100. Figure 4 depicts the
predicted distribution for P(SPP )s (dashed line) together
with the measured coincidences reported in Ref. [14].
According to our approach, a probability distribution
P(SPP )s fitted with s = 0.66 (solid line in Fig. 4) shows
excellent agreement with the experimental results. Equa-
tion (7) then suggests that the corresponding Schmidt
number should be much smaller than the above K. How-
ever, we cannot perform a definite estimate for K since
the value of the fiber mode width wG at the location of
the spiral phase plates, needed to calculate µ = w0/wG
appearing in Eq. (7), was not provided in Ref. [14].
The inset in Fig. 4 plots K for a wide range of ratios
µ = w0/wG. For instance, using our fitting parameter
s = 0.66, together with w0 = 780 µm and assuming
wG & 250 µm, leads to µ . 3 and an estimate for K . 20,
at least two orders of magnitude smaller than the quoted
K in Ref. [14]. A very large value for K could only be ex-
pected if, in the experiment, the fiber mode width wG in
the plane containing the phase plates was much smaller
than the pump width w0.
VI. CONCLUSIONS
In view of the situation described above, it is rea-
sonable to conclude that additional experimental results
and further theoretical analyses -a recent example can
be found in Ref. [32]- are necessary and of interest in
order to clarify the problem of quantifying the amount
of transverse entanglement of photon pairs produced in
down-conversion sources. If future values for the Schmidt
number coming from accurate and solid measurements
confirm the estimate of Ref. [14] and cannot be repro-
duced within our approach, one should conclude that the
conventional Gaussian profile constitutes a poor approx-
imation for the phase-matching function G in Eq. (4).
However, the ability of our predicted coincidence distri-
butions to fit the experimental distributions of Ref. [14]
suggests the correctness of our analysis and that the value
for the Schmidt number quoted in Ref. [14] could be over-
estimated. In this case, our approach not only would pro-
vide a simple an accurate procedure to identify the rele-
vant spatial modes and extract the degree of transverse
entanglement; it would go in fact beyond by character-
izing the action of all local bipartite azimuthal optical
transformations on a broad family of two-photon states
using only two detectors. Our results could also be of in-
terest in other fields such as in identifying the intervening
spatial modes when preparing well-controlled superposi-
tions of photon states carrying OAM (to be coherently
transferred onto Bose-Einstein condensates [33, 34] or by
entangling photons with ensembles of cold atoms [35]), in
high-resolution ghost diffraction experiments with ther-
mal light [36], or in decoherence processes such as the en-
tanglement sudden death mediated by the simultaneous
action of several weak noise sources on bipartite photon
systems [37].
Acknowledgments
We thank S. Barnett, D. Diego, G. Molina-Terriza,
M.J. Padgett and S. Walborn for useful discussions
and gratefully acknowledge financial support from the
Spanish Ministry of Science and Technology through
Project FIS2005-01369, Juan de la Cierva Grant Pro-
gram, CONSOLIDER2006-00019 Program and CIRIT
Project SGR-00185.
APPENDIX A
In this Appendix we outline the derivation of the eigen-
values and eigenfunctions in the Schmidt decomposi-
tion (1) when the two-photon amplitude Φ is given by
Eq. (4). We start with some well-known facts.
For a general two-photon pure state, |ψ〉, it is possible
to express |ψ〉 as a bilinear sum of idler and signal basis
states |τi,s〉 belonging to the Hilbert space of the system
|ψ〉 =
τi,τs
Cτi,τs |τi〉 ⊗ |τs〉 , (A1)
where τi and τs label the set of quantum numbers for
the idler and signal photons, respectively, whereas the
coefficients Cτi,τs describe the probability amplitudes for
each tensor product of basis states. Our first aim is to
evaluate the coefficients Cτi,τs for the down-converted
state |ψ〉 =
dqidqsΦ(qi,qs)â†(qi)â†(qs)|vac〉. Let
uτi,s(qi,s) = 〈vac|â(qi,s)|τi,s〉 denote the basis wave func-
tions in the transverse momentum representation. Insert-
ing the closure relations
|τi,s〉〈τi,s| = 1̂, it can be
readily seen that the coefficients Cτi,τs are given by
Cτi,τs =
dqidqsΦ(qi,qs)u
(qi)u
(qs). (A2)
Since we are interested in elucidating the correlation
properties of the spatial degrees of freedom (OAM and
the magnitude of the transverse radial wave vectors), we
choose as the computational basis the complete set of
normalised Laguerre-Gaussian modes [20]
u`,n(q, φ) =
2π(|`|+ n)!
L|`|n
× exp
i`φ− i
(2n+ |`|)
≡ exp (i`φ) v`,n(q)/
2π, (A3)
where q and φ denote the radial and azimuthal variables
in momentum space, w is the mode width (at the beam
waist), and L|`|n (x) are the associated Laguerre polynomi-
als. The indices ` = 0,±1,±2, . . . and n = 0, 1, 2, . . . rep-
resent the winding (or topological charge) and the num-
ber of nonaxial radial nodes of the modes.
Combining Eqs. (4), (A2) and (A3), it follows that
ni,ns
`i,`s
dqi dqs e
−w20|qi+qs|
2/4e−b
2|qi−qs|2/4
× e−i`iφie−i`sφsv`i,ni(qi) v`s,ns(qs) . (A4)
By means of the well-known Anger-Jacobi identity
e−x cos(φi−φs) =
m=−∞(−1)
mIm(x)eim(φi−φs), where
Im(x) is the modified Bessel function of the first kind, the
two angular integrals yield the selection rule m = `i =
−`s ≡ `. This shows that the idler and signal photons are
perfectly anticorrelated with respect to their topological
charge, which is a manifestation of OAM entanglement.
Hence, Cni,ns`i,`s = C
ni,ns
`,−` δ``iδ−``s , where
ni,ns
`,−` = (−1)
`2w0b
qidqiqsdqse
−(w20+b
2)(q2i+q
× v`,ni(qi) v−`,ns(qs) I`
(w20 − b2)qiqs
. (A5)
The first of the radial integrals in (A5) can be performed
by resorting to the following formula∫ ∞
x|`|+1L|`|n (x
2)e−γx
I`(xy)dx
(γ − 1)ny|`|
2|`|+1γ|`|+n+1
4γ L|`|n
4(γ − 1)γ
valid for all real y, integers n ≥ 0 and `, and complex γ
(Re(γ) > 0).
The second radial integral can also be done analyti-
cally. However, a dramatic simplification occurs if the
widths wi and ws of the idler and signal radial modes,
which at this stage have not been specified, are properly
selected: wi = ws =
2w0b. In this case, one obtains
a second selection rule for the radial indices. Namely,
ni = ns ≡ n. That is, with such a choice of the widths,
which is unique, the idler and signal radial modes are
perfectly correlated. But this shows that it is precisely
for wi = ws =
2w0b when one derives the Schmidt de-
composition. Thus, the Schmidt width, wS , corresponds
to wS =
2w0b. The coefficients reduce then to
ni,ns
`i,`s
= (−1)`
(w0 + b)2
w0 − b
w0 + b
)|`|+2n
× δ``iδ−``sδnniδnns . (A6)
Let ξ = (w0−b)/(w0+b). It is now clear from Eq. (A6)
that the Schmidt eigenvalues are λ`n = (1− ξ2)2ξ2|`|+4n.
We therefore conclude that the Schmidt decomposition
of the two-photon amplitude (4) is
Φ(qi,qs) = (1− ξ2)
(−1)`ξ|`|+2nu`,n(qi)
× u−`,n(qs), (A7)
where u`,n(qi) = ei`φiv`,n(qi)/
2π and u−`,n(qs) =
e−i`φsv−`,n(qs)/
2π represent the idler and signal
Schmidt eigenmodes. These eigenmodes belong to the
Laguerre-Gaussian basis. It is worth mentioning that
although the Schmidt width is unique, other decom-
positions (actually infinitely many) are possible in dif-
ferent eigenmode bases. For instance, the Hermite-
Gaussian modes (in Cartesian variables and the same
2w0b) constitute another possible basis. In fact,
all eigenmode bases connected via the following unitary
(metaplectic) transformation Û(θ, ϕ) = exp(−iθ L̂ · uϕ),
where uϕ = (− sinϕ, cosϕ, 0) is a unit vector in the
equatorial plane of the so-called orbital Poincaré sphere
(parameterised by the spherical angles θ and ϕ) and L̂
is an angular momentum operator, form Schmidt bases
for (A7). The components of the angular momentum op-
erator: L̂x, L̂y, and L̂z are the three SU(2) generators.
Of these, only L̂z represents real spatial rotations along
the propagation of light, and it is thus the only compo-
nent associated with the orbital angular momentum of
light that can be measured in experiments [20].
APPENDIX B
In this second Appendix we explicitly show how to cal-
culate the radial functions R`(s) given in Eq. (5). For
the chosen two-photon amplitude (4) and the measure-
ment scenario, these radial functions depend on the spa-
tial overlap of the Schmidt and the fiber (fundamental
Gaussian) modes at the planes where the azimuthal phase
plates are located. We need to evaluate the integrals
rirsdridrsv
`A=0,nA=0
(ri)v
`B=0,nB=0
×v(wS)`,n (ri)v
−`,n(rs). We use the Schmidt eigenvalues
λ`n = (1 − ξ2)2ξ2|`|+4n found in Appendix A. The idler
and signal Schmidt modes v(wS)`,n (ri) and v
−`,n(rs) be-
long to the Laguerre-Gaussian mode basis. We stress
once again the fact that the corresponding fiber wG and
Schmidt wS widths are not necessarily equal. This im-
plies that the orthogonality relation for these modes,∫
rdr v(wG)`G,nG(r)v
`S ,nS
(r) = δ`G`SδnGnS , does not hold in
general when wS 6= wG.
We start by recalling a remarkable identity between
Laguerre polynomials L|`|n and the modified Bessel func-
tions I` [38]
n!L|`|n (x)L
n (y)
Γ(|`|+ n+ 1)
(xyz)−|`|/2
(x+ y)z
, (A8)
valid for all real x, y, integer ` and complex z (|z| < 1).
In detail, the integrals to calculate read
R` = (1− ξ2)ξ|`|
ridri
(|`|+ n)!
L|`|n
rsdrs
(|`|+ n)!
L|`|n
. (A9)
Inserting identity (A8) in (A9) and employing the follow-
ing result [38]∫ ∞
r e−γr
I`(2νr) dr =
Γ(1 + |`|
2Γ(1 + |`|)ν
M− 12 ,
valid for Re(γ) > 0 and all complex ν, with Ma,b repre-
senting the Whittaker functions, we obtain
Γ(1 + |`|
Γ(1 + |`|)
8(1− ξ2)3sw2S
ξ3w4G
dr exp
(2− s2)ξr2
(1− ξ2)sw2S
× M− 12 , |`|2
2sξr2
(1− ξ2)w2S
, (A10)
where we have introduced the parameter s = 2ξ/[1+ξ2 +
(1− ξ2)(wS/wG)2]. The remaining integral is carried out
through the use of formula
e−γ̄t
M− 12 ,
(t) =
Γ(1 + |`|
γ̄ − 1
) |`|
; 1 + |`|;
valid when γ̄ > 1/2. The function F (a, b; c; d) denotes
the hypergeometric function. Equation (A10) is finally
given by
1− ξ2
1 + (1−ξ
)2 Γ2(1 + |`|2 )Γ(1 + |`|)
; 1 + |`|; s2
s|`|. (A11)
This expression reduces to Eq. (5) when one leaves aside
all the `-independent factors (they do not play any sig-
nificant role in the normalised coincidence profiles). In
this way the two-photon detection probabilities (3) can
be cast in terms of the impulse response functions h(i)`A,`,
`B ,−` and the radial functions R`(s) that solely depend
on the parameter s (when `A = `B = 0).
APPENDIX C
In this appendix we would like to justify the model pro-
posed here, the double gaussian profile as phase matching
function.
1. Classical Energy for Non-linear Media
In this section we derive explicitly the energy of an
anisotropic and non-linear medium. This classical de-
scription enables us to find a reasonable quantized Hamil-
tonian which allow us to describe the PDC process. In
anisotropic and non-linear media we must be cautious in
the description of physical variables. For a deep analysis,
we begin by assuming that the polarization of the mate-
rial obeys the usual expansion from non-linear optics
Pi = εo[χ
ij Ej + χ
ijkEjEk + χ
ijklEjEkEl + ...] ,
(A12)
where χ(n) (with n > 1) represent the nonlinear suscep-
tibility tensors of order n+1 responsible for the coupling
of n + 1 fields. The energy of the electromagnetic field
can be written as
E · dD , (A13)
where the displacement vector field is defined as usual
D = �oE + P. In linear media, the energy density of
the electromagnetic field depends quadratically on the
electric field ∝ E2, whereas in non-linear (parametric)
media, polarization exhibits a dependence on higher or-
der electric field terms (A12). We can expand the energy
(A13) in two terms, one related to the linear contribution
of the electric field and the other one related to the non-
linear part. Taking the lowest order nonlinearity (A12),
the energy due to this contribution can be written as
[39, 40],
d3r χ̃
ijk Ei(r, t)Ej(r, t)Ek(r, t) (A14)
where the integration extends over the volume V of the
nonlinear medium. Notice that the second order suscep-
tibility is not the same as the susceptibility in equation
(A12). We will consider that the medium response to
the electric field is not instantaneous. Hence, the second
order polarization should be written as
i (r, t) = �o
dt1dt2 χ
ijk(t− t1, t− t2)
× Ej(r, t1)Ek(r, t2) .
(A15)
We can see, taking the Fourier transformation of the elec-
tric fields in equation (A15), that this specific form of the
susceptibility implies that different monochromatic waves
(with a well-defined frequency) of the electric field can
excite other frequencies. Hence, the energy of the elec-
tromagnetic field due to the bilinear susceptibility (A14)
can be expressed as [41],
∫ ∫ ∫
dωdω′dω′′ χ̃
ijk(ω, ω
′, ω′′)
× Ei(r, ω′′)Ej(r, ω − ω′)Ek(r, ω′) .
(A16)
This energy gives rise to the quantized Hamiltonian fre-
quently found in the literature [15] to describe the pro-
cess of spontaneous parametric down conversion, where
an incident beam interacts with a non-linear medium and
splits into two lower-frequency signal and idler photons.
2. Parametric Down-Conversion
If we have a crystal with nonzero χ(2), pumped with a
laser beam, there is a small, but non negligible, probabil-
ity that a pump photon will decay into an idler and signal
photon pair. We take the incident pump light beam to
be in the form of an intense quasi-monochromatic plane
wave propagating along the z-direction,
Ep(r, t) = �p unm(r⊥, z) e
iωpt , (A17)
where �p is the polarization of the pump beam and
unm(r⊥, z) is a slowly-variant-amplitude (for the mode
having indices n and m) along z which obeys the paraxial
equation in anisotropic media [42]. That is, the paraxial
equation for free propagation cannot be used and must be
modified. The pump beam is described classically, within
the undepleted approximation, which means that one as-
sumes that the input state is made up of a large number
of photons and only few of them interact with the non-
linear medium, producing pairs of correlated photons. In
other words, the pump beam, at the output of the crystal,
remains almost unaffected.
A form for the interaction Hamiltonian operator ĤI(t)
can be given motivated by the classical electromagnetic
field energy (A16), and reads (we are always in the inter-
action picture, though it can also be done in the Heisen-
berg picture [40]),
ĤI(t) =
ki,σi
ks,σs
ijk(ωp, ωi, ωs) (�
ki,σi
× (�∗ks,σs)
j(�p)
k unm(r⊥, z) â
ki,σi
ks,σs
e−i(qi+qs)·r⊥
× e−i[(kiz+ksz)z+(ωp−ωi−ωs)t] +H.c. ,
(A18)
where k(i,s) = q(i,s) + k(i,s)z ûz. In equation (A18) two
quantized electric fields are considered; they will be re-
sponsible for idler and signal photons, whereas the pump
beam is associated with the classical field [39]. As we
have mentioned before, we are carrying out the calcula-
tions under the interaction picture, where we know that
a state evolves as
|Φ(t)〉 = T
0 dτĤI (τ)
|Φo〉 ' |Φo〉 −
dτ : ĤI(τ) : |Φo〉
dτ ′ ĤI(τ)ĤI(τ
|Φo〉+ ... ,
(A19)
with T denoting temporal ordering. Since the interaction
is assumed to be weak enough, it suffices to expand the
exponential (A19) up to first order term. The action of
the Hamiltonian operator on the evolution of the initial
vacuum state yields the initial state plus a two-photon
state (contributions due to higher order processes involve
the generation of four, six,... photons, and are exceed-
ingly small). Notice that applying the Wick theorem in
the above expansion, temporal ordering becomes normal
ordering up to first order. Therefore,
|Φ(t)〉e = T
0 dτĤI (τ)
|Ω〉 ' |Ω〉 −
ki,σi
ks,σs
ijk(ωp, ωi, ωs) (�
ki,σi
ks,σs)
−i(kiz+ksz)z
r⊥ unm(r⊥, z) e
−i(qi+qs)·r⊥
−i(ωp−ωi−ωs)t/2 sin [(ωp − ωi − ωs)t/2]
(ωp − ωi − ωs)/2
|ki, σi; ks, σs〉 ,
(A20)
where we have performed the integration with respect to
time τ . Let L denote the length of the crystal along z,
whereas S corresponds to the transverse area (the vol-
ume of the crystal is V = LS). Equation (A20) is a gen-
eral expression for any input paraxial beam. Assuming
that the transverse area of the crystal S is much larger
than the characteristic width of the input beam, the inte-
gral with respect to the transverse position becomes the
transverse Fourier transform. Now taking into account
that the paraxial wave is propagating in an uniaxially
anisotropic crystal [42], this integral becomes
r⊥ unm(r⊥, z) e
−i(qi+qs)·r⊥ → eikpne(θ,ωp)ze
ne(θ,ωp)
2n2e(ωp)kp
q2y+q
n2e(θ,ωp)
n2o(ωp)
−iqxsθcθ
n2e(θ,ωp)
ñ2(ωp)
z eunm(qi + qs) , (A21)
where, for simplicity, we assume that the optical axis
is contained in the plane xz and the pump beam is lin-
early polarized along the x-direction (the pump beam has
extraordinary polarization). Here, θ refers to the angle
between the optical axis and the z-axis, and
q = qi + qs ; kp =
; cθ ≡ cos θ ; sθ ≡ sin θ ,
n2e(θ, ωp)
sin2 θ
n2e(ωp)
cos2 θ
n2o(ωp)
ñ2(ωp)
n2o(ωp)
n2e(ωp)
Knowledge of the pump spectrum at the input crystal
face enables, via equation (A21), to describe the evolu-
tion of the pump beam through the crystal. In order to
satisfy both the undepleted approximation and the fact
that the pump profile is assumed not to vary significantly
during the nonlinear propagation, the crystal must be
very thin. The z-dependence in equation (A21) will play
a crucial role to correctly describe the phase-matching
conditions below.
Defining
ne(θ, ωp)
n2e(ωp)
q2y + q
n2e(θ, ωp)
n2o(ωp)
−qx sin θ cos θ
n2e(θ, ωp)
ñ2(ωp)
we can perform the integral with respect to z in equation
(A20), resulting in the so-called phase-matching function:Z L
i[kpne(θ,ωp)−Θ−kiz−ksz ]z =
i∆(kp,ki,ks)L/2
sin [∆(kp,ki,ks)L/2]
∆(kp,ki,ks)/2
, (A22)
where the argument
∆(kp,ki,ks) ≡ kpne(θ, ωp)−Θ− kiz − ksz , (A23)
describes the wave vector mismatching. Therefore, equa-
tion (A20) can be cast (excluding the vacuum state) as
|Φ(t)〉(2ph)e = −
ki,σi
ks,σs
ijk(ωp, ωi, ωs) (�
ki,σi
)i(�∗ks,σs)
j(�p)
k ei∆(kp,ki,ks)L/2
sin [∆(kp,ki,ks)L/2]
∆(kp,ki,ks)/2
× ũnm(qi + qs) e−i(ωp−ωi−ωs)t/2
sin [(ωp − ωi − ωs)t/2]
(ωp − ωi − ωs)/2
|ki, σi; ks, σs〉 . (A24)
3. PDC in type II
In this section we are going to consider the paramet-
ric down conversion under type II phase-matching con-
ditions. This implies that the pump beam is extraor-
dinarily (linearly) polarized, whereas the idler and idler
photons are ordinarily (the polarization vector is perpen-
dicular to the optical axis) and extraordinarily polarized,
respectively. Our first aim will be to further simplify the
down-converted state (A24). If the interaction time is
sufficiently large, we can replace the temporal-sinc factor
in equation (A24) by a delta function,
sin [(ωp − ωi − ωs)t/2]
(ωp − ωi − ωs)/2
→ 2π δ(ωp − ωi − ωs) ,
This is a good approximation, and the oscillatory factor
e−i(ωp−ωi−ωs)t/2 can then be discarded by a regulariz-
ing procedure. In particular, one obtains the frequency
matching condition
ωp = ωi + ωs , (A25)
that is, energy conservation imposes that the annihila-
tion of the pump photon gives rise to the creation of the
idler and signal photons. In what follows, and effective
nonlinear susceptibility χeff will be used to represent
the relevant components of χ(2)ijk contracted with the cor-
responding components of the pump, idler and signal po-
larizations (we also absorb in χeff additional constants).
Considering the continuity of the wave vectors of signal
and idler photons (assuming that the volume of the crys-
tal is large enough), equation (A24) reduces to
|Φ〉(2ph)e = −
ks χeff (ωp, ωi, ωs) e
i∆(kp,ki,ks)L/2
sin [∆(kp,ki,ks)L/2]
∆(kp,ki,ks)L/2
eunm(qi + qs)
× δ(ωp − ωi − ωs) |ki, σo,ks, σe〉 .
Notice that the idler and signal photons have fixed po-
larization, ordinary (σo) and extraordinary (σe) respec-
tively.
For convenience, we change the integral in the z com-
ponent of the wave vector by the frequency. Therefore,
for the idler integral, the Jacobian due to the change of
variables is
qidkiz =
∂(qi, kiz)
∂(qi, ωi)
qidωi =
qidωi =˛̨̨̨
n2o(ωi)
ω2i no(ωi)
c2kiz
∂no(ωi)
qidωi ,
while for the signal integral, we have
d2qsdksz =
∣∣∣∣∂(qs, ksz)∂(qs, ωs)
∣∣∣∣ d2qsdωs =
∣∣∣∣∂ksz∂ωs
∣∣∣∣ d2qsdωs =∣∣∣∣∣∣∣
n3e(θ,ωs)
∂ne(θ,ωs)
− kszqsx sin 2θ ∂∂ωs
ñ2(ωs)
− q2sx
n′2e (θ,ωs)
n3e(ωs)
∂ne(ωs)
2 ksz
n2e(θ,ωs)
+ sin 2θ
ñ2(ωs)
∣∣∣∣∣∣∣ d2qsdωs . (A26)
Proof: For ordinary waves the dispersion relation in an
anisotropic medium is
ω2i n
o(ωi)
iz + q
Solving the quadratic equation for the variable kiz
kiz =
ω2i n
o(ωi)
− q2i .
Notice that we only choose the positive solution of kiz. Fi-
nally, taking the derivative with respect to ωi
n2o(ωi)
ω2i no(ωi)
c2kiz
∂no(ωi)
For extraordinary waves the dispersion relation in an
anisotropic uniaxial medium is more complex:
n2e(θ, ωs)
+ (qsx cosϕ+ qsy sinϕ)
ksz sin 2θ
ñ2(ωs)
(qsx cosϕ+ qsy sinϕ)
n′2e (θ, ωs)
(qsx sinϕ− qsy cosϕ)2
n2e(ωs)
,(A27)
where {θ, ϕ} are the polar and the azimuthal angles respec-
tively in spherical coordinates, with θ referring to the angle
subtended by the optical axis and the z-axis. For simplicity,
as we have mentioned before, we assume the optical axis to
be contained in the plane x-z (ϕ = 0). We have to take into
account the common definitions in anisotropic media
n2e(θ, ωs)
sin2 θ
n2e(ωs)
cos2 θ
n2o(ωs)
n′2e (θ, ωs)
cos2 θ
n2e(ωs)
sin2 θ
n2o(ωs)
ñ2(ωp)
n2o(ωp)
n2e(ωp)
In this case, performing the implicit derivative with respect
to ωs in the dispersion relation (A27),
n2e(θ, ωs)
2k2sz
n3e(θ, ωs)
∂ne(θ, ωs)
sin 2θ
ñ2(ωs)
+ kszqsx sin 2θ
ñ2(ωs)
n′2e (θ, ωs)
n3e(ωs)
∂ne(ωs)
. (A28)
where
ñ2(ωs)
n3o(ωs)
∂no(ωs)
n3e(ωs)
∂ne(ωs)
n′2e (θ, ωs)
cos2 θ
n3e(ωs)
∂ne(ωs)
sin2 θ
n3o(ωs)
∂no(ωs)
Solving the equation (A28) for ∂ksz
we find equation
(A26).
Substituting equations (A26) into the output state of
idler and signal photons (A26), we obtain a general ex-
pression for PDC in type II. So far, we have only assumed
the undepleted approximation and that the transverse
area of the crystal is much larger than the characteristic
width of the pump beam.
4. PDC in type II, collinear regime
We now focus on the collinear regime in PDC where
we only consider those idler and signal photons which
are produced nearly in the same direction as that of the
pump beam (in our framework along the z-axis). In this
particular case, the approximation q
(s,i)
(s,i)
� 1 is justified.
Therefore, equations (A26) reduce to
d2qidkiz =
no(ωi)
d2qidωi ,
for the idler, and
d2qsdksz =
ne(θ, ωs)
d2qsdωs ,
for the signal, where we have also made the reasonable
assumption that the variation of refractive indices with
respect to frequency is small. Thus, Jacobians due to
the change of variables are just refractive indices, which
depend on frequency.
In the collinear regime more simplifications can be
done. Let us explicitly write the argument of the phase
matching function (A23) as
∆(kp,ki,ks) = kpne(θ, ωp)− ksne(θ, ωs)− kino(ωi) +
sin 2θ
n2e(θ, ωp)
ñ2(ωp)
n2e(θ, ωs)
ñ2(ωs)
+ qix
sin 2θ
n2e(θ, ωp)
ñ2(ωp)
ne(θ, ωs)
2n2e(ωp)kp
(qiy + qsy)
+ (qix + qsx)
e(θ, ωp)
n2o(ωp)
n3e(θ, ωs)
2n2e(ωs)n
o(ωs)ks
ne(θ, ωs)
2n2e(ωs)ks
q2ix + q
2no(ωi)ki
(A29)
It is clearly an strongly anisotropic function. Notice that
the linear terms in the transverse wave vectors are re-
sponsible for the walk-off effect (the Poynting and wave
vectors of the signal and pump photons are not collinear,
i.e., their energy flows in a different direction from its cor-
responding phase front). We should mention a previous
study by Torres and coworkers [43] where the walk-off ef-
fect was taken into account to describe entangled photon
states generated in type I PDC.
The state for idler and signal photons within type II
PDC (A26) is
|Φ〉(2ph)e =
qidωi
qsdωs χeff (ωp, ωi, ωs)
× ei∆(kp,ki,ks)L/2 sinc [∆(kp,ki,ks)L/2] eunm(qi + qs)
× δ(ωp − ωi − ωs) |ki, σo,ks, σe〉
(A30)
where sinc(x) ≡ sin(x)/x. The symbol ∆Ω represents
the fact that the transverse idler and signal wave vectors
are constrained to be very small (this is the condition for
the collinear regime). This can be achieved experimen-
tally by using a suitable pin hole placed in the z axis.
Now, due to this restriction, only the maximum of the
sinc function contributes to the integral (the side lobes
being blocked). Furthermore, the action of the pin hole
truncating the sinc function can actually be modeled by
a simple exponential function. In addition, we can reg-
ularize the phase factor ei
|qi−qs|2 by just shifting our
coordinate system (z = 0) to the middle of the crystal,
see equation (A23). We will also focus, for simplicity, on
the frequency degenerated case, where the idler and sig-
nal photons have the same frequency (ωp = ωi/2 = ωs/2,
which can be realized by using appropriate narrow-band
filters in front of the photodetectors). Thus, the two-
photon state reads
|Φ〉(2ph)e = χeff (ωp)
d2qs e
−∆(kp,ki,ks)L/2
× ũnm(qi + qs) |ki, σo,ks, σe〉 .
(A31)
The effective susceptibility, due to the energy conserva-
tion and the frequency degeneration in idler and signal
photons, now depends only on the frequency of the pump
beam.
The argument of the phase-matching function G =
e−∆(kp,ki,ks)L/2 is still too complex. Since we wish to
maintain our framework at a sufficiently simple level, and
we will pursue a full analytical study, some further (and
drastic) approximations are required: we take all refrac-
tive indices (ordinary and extraordinary) to be identical
and independent of frequency. Hence, the argument of
the phase matching function reduces to
∆(kp,ki,ks) =
2kpneff
|qi − qs|2 , (A32)
which is, in contrast to equation (A29), an isotropic (with
cylindrical symmetry) function. The effective refractive
index neff can be interpreted as a suitable fitting pa-
rameter (in principle, it can be quite different from the
ordinary and extraordinary refractive indices).
In spite of the strong approximations performed above,
it is illustrative to determine their goodness by compar-
ing the sinc and the exponential phase-matching func-
tions when their arguments are given by equations (A29)
and (A32), respectively. Figure 5 depicts both profiles.
It can be seen that for sufficiently small transverse wave
vectors both profiles are coincident (by suitably choosing
neff ); the deviations arise, typically, when the pin hole is
already truncating both functions. This qualitatively jus-
tifies the validity of the approximations made above. Ac-
tually, this approximation was previously used (without
any justification) by Law and Eberly [17] to study bipar-
tite transverse entanglement from type II PDC. Further-
more, in other previous works, the sinc phase matching
function was even replaced by a constant [14, 31, 44, 45].
In summary, equation (A31) finally becomes (simpli-
fying the notation and absorbing the constant factors)
|ψ〉 =
qs Φ(qi,qs) |qi,qs〉 , (A33)
where Φ = EG is the two-photon amplitude, with
E(qi + qs) = ũnm(qi + qs) denoting the transverse
(spectrum) profile of the pump beam, and G(qi −
qs) = e
− L4kpneff |qi−qs|
being the phase-matching func-
tion. Since we will not focus in later sections on the po-
larization degree of freedom (recall that idler and signal
photons are assumed to be linearly polarized), we have
omitted its explicit dependence in (A33).
Equation (A33) will be exploited in (4) to carry out
our discussion about the OAM entanglement shared by
the downconverted idler and signal photons.
FIG. 5: Comparison between the sinc (solid curve) and the ex-
ponential (dotted curve) phase-matching functions when their
arguments are given by equations (A29) and (A32), respec-
tively. The shadowed region corresponds to the truncation
introduced by the pin hole.
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	INTRODUCTION
	MODAL SCHMIDT DECOMPOSITION FOR TWO-PHOTON STATES
	AZIMUTHAL TRANSFORMATIONS ON THE TWO-PHOTON STATE
	AN EXACTLY SOLUBLE MODEL FOR THE TWO-PHOTON AMPLITUDE
	Experimental Schmidt Number
	Conclusions
	Acknowledgments
	APPENDIX B
	APPENDIX C
	Classical Energy for Non-linear Media
	Parametric Down-Conversion
	PDC in type II
	PDC in type II, collinear regime
	References
