# Characterising and Tailoring Spatial Correlations in Multi-Mode Parametric
Downconversion

Vatshal Srivastav<EMAIL_ADDRESS>Institute of Photonics and Quantum Sciences
(IPAQS), Heriot-Watt University, Edinburgh, UK  Natalia Herrera Valencia
Institute of Photonics and Quantum Sciences (IPAQS), Heriot-Watt University,
Edinburgh, UK  Saroch Leedumrongwatthanakun  Institute of Photonics and
Quantum Sciences (IPAQS), Heriot-Watt University, Edinburgh, UK  Will
McCutcheon  Institute of Photonics and Quantum Sciences (IPAQS), Heriot-Watt
University, Edinburgh, UK  Mehul Malik<EMAIL_ADDRESS>Institute of
Photonics and Quantum Sciences (IPAQS), Heriot-Watt University, Edinburgh, UK

###### Abstract

Photons entangled in their position-momentum degrees of freedom (DoFs) serve
as an elegant manifestation of the Einstein-Podolsky-Rosen paradox, while also
enhancing quantum technologies for communication, imaging, and computation.
The multi-mode nature of photons generated in parametric downconversion has
inspired a new generation of experiments on high-dimensional entanglement,
ranging from complete quantum state teleportation to exotic multi-partite
entanglement. However, precise characterisation of the underlying position-
momentum state is notoriously difficult due to limitations in detector
technology, resulting in a slow and inaccurate reconstruction riddled with
noise. Furthermore, theoretical models for the generated two-photon state
often forgo the importance of the measurement system, resulting in a
discrepancy between theory and experiment. Here we formalise a description of
the two-photon wavefunction in the spatial domain, referred to as the
collected joint-transverse-momentum-amplitude (JTMA), which incorporates both
the generation and measurement system involved. We go on to propose and
demonstrate a practical and efficient method to accurately reconstruct the
collected JTMA using a simple phase-step scan known as the
$2D\pi$-measurement. Finally, we discuss how precise knowledge of the
collected JTMA enables us to generate tailored high-dimensional entangled
states that maximise discrete-variable entanglement measures such as
entanglement-of-formation or entanglement dimensionality, and optimise
critical experimental parameters such as photon heralding efficiency. By
accurately and efficiently characterising photonic position-momentum
entanglement, our results unlock its full potential for discrete-variable
quantum information science and lay the groundwork for future quantum
technologies based on multi-mode entanglement.

##  I Introduction

Figure 1: (a) Spontaneous parametric down conversion (SPDC): A nonlinear
process where a coherent pump photon at wavelength $\lambda_{P}$ interacts
with a nonlinear crystal ($\chi^{(2)}$) and generates two daughter photons
(signal and idler) entangled in their position-momentum degrees-of-freedom.
(b) Joint-transverse-momentum-amplitude (JTMA): The density plot of a 2D slice
of the JTMA $[F(q_{sx}=0,q_{sy},q_{ix}=0,q_{iy})]$. The JTMA represents the
transverse-momentum correlations between signal and idler photons and is
characterized by two parameters, $\sigma_{P}$ and $\sigma_{S}$ (see Eq. (2)).
The pump bandwidth parameter $\sigma_{P}$ is inversely propositional to the
beam waist of the pump at the crystal plane ($w_{P}$) and dictates the
strength of the correlations, while the generation bandwidth parameter
$\sigma_{S}$ is determined by the length of the crystal $L_{z}$ and the
wavelength of the pump $\lambda_{P}$, and determines the modal generation
bandwidth of the JTMA.

The Einstein, Podolsky, and Rosen (EPR) paradox lies at the heart of quantum
mechanics Einstein  _et al._ (1935). Using the paradigmatic example of two
quantum particles sharing perfect correlations (or anti-correlations) between
their complementary properties of position and momentum, EPR postulated an
inconsistency between local realism and the completeness of quantum mechanics
Reid  _et al._ (2009). A physical realisation of the original EPR experiment
proved challenging, and much of the subsequent theoretical and experimental
work focused on a discrete version of the EPR paradox postulated by Bohm and
formalised by Bell’s inequality Bohm and Aharonov (1957); Bell (1964); Brunner
_et al._ (2014). While discrete-variable experiments such as ones based on
polarisation Clauser  _et al._ (1969); Aspect  _et al._ (1982) have laid the
foundation for the quantum technologies of today, the exploration of
continuous quantum properties in the vein of the original EPR
gedankenexperiment has recently flourished Shin  _et al._ (2019); Wasak and
Chwedeńczuk (2018); Keller  _et al._ (2014); Ndagano  _et al._ (2020); Edgar
_et al._ (2012); Moreau  _et al._ (2012), thanks to a series of experimental
advances and several practical motivations.

Pairs of photons produced in nonlinear spontaneous parametric downconversion
(SPDC) provide a natural platform for tests of EPR entanglement. Photons
generated in SPDC are correlated/anti-correlated in their position and
momentum owing to the conservation of energy and momentum that governs this
process Boyd (2008); Hong and Mandel (1985); Malygin  _et al._ (1985);
Schneeloch and Howell (2016); Brambilla  _et al._ (2010); Walborn  _et al._
(2010). While this source was adapted for the earliest violations of Bell’s
inequality based on discrete-variable polarisation entanglement, the ability
to harness its inherent position-momentum correlations has led to a recent
explosion of interest in high-dimensional entanglement of photonic spatial
modes Erhard  _et al._ (2020a); Friis  _et al._ (2019); Bavaresco  _et al._
(2018)—ranging from demonstrations of high-dimensional Bell-like inequalities
Dada  _et al._ (2011), composite quantum state teleportation Wang  _et al._
(2015), to exotic forms of multi-photon entanglement Malik  _et al._ (2016);
Erhard  _et al._ (2018); Hiesmayr  _et al._ (2016); Erhard  _et al._ (2020b).
High-dimensional (qudit) entanglement also provides significant advantages
over qubit-based systems in the form of increased information capacity
Valencia  _et al._ (2020); Cao  _et al._ (2020); Cozzolino  _et al._ (2019);
Erhard  _et al._ (2017); Steinlechner  _et al._ (2017); Mafu  _et al._ (2013)
and robustness to noise Ecker  _et al._ (2019); Zhu  _et al._ (2021); Huber
and Pawłowski (2013); Vertesi  _et al._ (2010), making it a very promising
platform for next-generation quantum technologies such as device-independent
quantum cryptography Acín  _et al._ (7 06). Thus, the ability to efficiently
and accurately characterise the underlying two-photon state entangled in its
continuous position-momentum degrees-of-freedom is of paramount importance.

Modelling the generation of entangled photons in a continuum of modes allows
for the identification of the effective number of entangled modes that are
present in the system Miatto  _et al._ (2012a); Schneeloch and Howell
(2016)—the so-called generation bandwidth. In addition, precise knowledge of
the continuous position-momentum correlations is crucial for accurately
tailoring spatial mode bases that maximise metrics relevant to discrete-
variable quantum information processing, such as the entanglement-of-formation
($E_{\textrm{oF}}$), entanglement dimensionality, and the state fidelity.
Experimental reconstruction of a position-momentum entangled state presents
some unique challenges—detector technology limits one to scanning through the
position/momentum space of interest with a single-mode detector, which
inherently introduces loss and involves very long measurements times Howell
_et al._ (2004); Schneeloch  _et al._ (2019). Recent work has pushed the
capabilities of arrayed single-photon detectors to reconstruct such states
faster Edgar  _et al._ (2012); Moreau  _et al._ (2012); Defienne  _et al._
(2018); Ndagano  _et al._ (2020), however these techniques still suffer from
resolution limits, loss, and an associated large noise background.

In this work, we formulate a theoretical model for a two-photon position-
momentum entangled state—the collected joint-transverse-momentum-amplitude
(JTMA)—that incorporates the generation as well as the measurement system used
in an experiment. We propose and demonstrate a practical and fast method to
fully characterise the collected JTMA using a simple $\pi$-phase step scan,
akin to a classical knife-edge measurement of a laser beam. Our method, known
as the $2D\pi$-measurement, allows us to measure the state parameters
independent of our knowledge of the optical system and crystal properties.
While here we implement our measurement technique with programmable phase-only
spatial light modulators (SLMs), its simplicity enables it to be performed
with low-cost components such as a microscope glass slide. We demonstrate the
versatility of our measurement scheme by implementing it on two experiments in
the continuous-wave near-infrared and pulsed telecom wavelength regimes.
Finally, we discuss how accurate knowledge of the collected JTMA enables us to
generate tailored discrete-variable high-dimensional entangled states that
maximise a desired property such as entanglement-of-formation or entanglement
dimensionality, or optimise experimental measures such as photon heralding
efficiency. Our methods have significant potential implications for
entanglement-based quantum technologies as well as fundamental tests of
quantum mechanics, and can be translated to other continuous degrees of
freedom such as time-frequency in a straightforward manner.

##  II Theory

###  II.1 Collected Bi-photon JTMA

As shown in Fig. 1a, the process of spontaneous parametric downconversion
(SPDC) results in the generation of a two-photon state whose correlations in
momentum space can be well approximated by a function that we call the joint-
transverse-momentum-amplitude (JTMA). In Appendix A, we derive the full JTMA
of the two-photon wavefunction produced in Type-II SPDC by a periodically
poled nonlinear crystal designed to achieve phase-matching at degenerate
frequencies in the collinear configuration. At degenerate frequencies and
imposing a Gaussian transverse pump profile across the crystal, we show how it
approximates the well-known form Schneeloch and Howell (2016); Miatto  _et
al._ (2012a):

$F(\textbf{q}_{s},\textbf{q}_{i})=\mathcal{N}_{1}\underbrace{\exp\bigg{(}\frac{-|\textbf{q}_{s}+\textbf{q}_{i}|^{2}}{2\sigma_{P}^{2}}\bigg{)}}_{\textrm{Pump
profile}}\times\underbrace{\text{
sinc}\bigg{(}\frac{|\textbf{q}_{s}-\textbf{q}_{i}|^{2}}{\sigma_{S}^{2}}\bigg{)}}_{\textrm{Phase-
matching condition}},$ (1)

where $\textbf{q}_{s}(\textbf{q}_{i})$ is the transverse-momentum wave-vector
for the signal (idler) photon and $\mathcal{N}_{1}$ is a normalization
constant. The first term in Eq. (1) represents the transverse wave-vector
components of the pump, while the second term represents the phase-matching
condition imposed on the down-conversion process by the nonlinear crystal. The
pump width parameter $\sigma_{P}$ depends on the $1/e^{2}$ beam radius of the
pump’s intensity profile at the crystal plane ($w_{p}$), while the generation
width parameter $\sigma_{S}$ is determined by the crystal length $L_{z}$ and
pump wavevector inside the crystal $\textit{k}_{p}$. These parameters are
defined as Miatto  _et al._ (2012b):

$\sigma_{P}=\frac{\sqrt{2}}{w_{p}},\qquad\sigma_{S}=\sqrt{\frac{4\textit{k}_{p}}{L_{z}}}.$
(2)

The pump wavevector $\textit{k}_{p}=n_{P}\tfrac{2\pi}{\lambda_{P}}$, where
$n_{P}$ is the refractive index of the nonlinear crystal at pump wavelength
$\lambda_{P}$. The interplay between the parameters $\sigma_{P}$ and
$\sigma_{S}$ determines the momentum correlations between the signal and idler
photon. In the case where $\sigma_{P}<\sigma_{S}$, the parameter $\sigma_{P}$
dictates the strength of the momentum correlations, whilst $\sigma_{S}$
represents the generation width of the JTMA function (see Fig. 1b). For
instance, a very broad (plane wave) pump beam has $\sigma_{P}\ll\sigma_{S}$,
and the approximation $\sigma_{P}\rightarrow 0$ can be made. Here, the pump
contribution to the JTMA approximates to
$\delta(\textbf{q}_{s}+\textbf{q}_{i})$, which results in perfect anti-
correlations between the signal and idler transverse momenta. In contrast, for
a non-zero $\sigma_{P}$, the degree of correlations decreases as the
$\sigma_{P}$ increases. For very thin crystals ($L_{z}\rightarrow 0$), the
generation width of momentum correlation tends to infinity
($\sigma_{S}\rightarrow\infty$). However, as shown in Baghdasaryan  _et al._
(2021), this approximation breaks for small beam waist $w_{p}$, revealing the
importance of a finite value of $\sigma_{S}$ and $\sigma_{P}$ in reality.

Figure 2: Collected JTMA: (a) The density plot shows a 2D slice of the
collected JTMA $[G(q_{sy}=0,q_{sx},q_{iy}=0,q_{ix})]$ from Eq. (II.1), where
the Gaussian distributions on the axes $q_{sx}$ and $q_{ix}$ represent the
collection modes $\mathcal{C}(\textbf{q}_{n}|\sigma_{C})$ that suppress the
Sinc sidelobes of the generated JTMA $F(\textbf{q}_{s},\textbf{q}_{i})$ shown
in Fig. 1b. (b) Collected JTMA under the collection-limited (CL)
approximation, which is valid if $\sigma_{S}\gtrapprox\sqrt{2}\sigma_{C}$,
resulting in the double-Gaussian form shown in Eq. (8).

Thus far, we have only discussed the generated two-photon state, which depends
solely on the pump and crystal parameters $\sigma_{P}$ and $\sigma_{S}$.
However, the measured momentum correlations also depend on the configuration
of the detection system. Photonic spatial modes are routinely measured in the
laboratory via a combination of a holographic spatial light modulator (SLM)
and single-mode fibre (SMF) that together act as a spatial-mode filter Qassim
_et al._ (2014); Bouchard  _et al._ (2018). To model the effects of such a
spatial-mode filter on the JTMA, we first consider the effect of phase-only
SLMs placed in the Fourier plane of the nonlinear crystal. By implementing a
diffractive hologram, an SLM can be used to apply an arbitrary amplitude and
phase function $\Phi^{(x)}(x)$ on an incident light field, which is then
projected onto the Gaussian single-mode of the collection fibre. Collection
modes have been considered in the context of spatial-mode entanglement Miatto
_et al._ (2012a). For convenience, we choose to work in momentum space at the
crystal plane so the collection mode takes the form
$\mathcal{C}(\textbf{q}|\sigma_{C})=(\sqrt{\pi}\sigma_{C})^{-\mbox{$\textstyle\frac{1}{2}$}}\exp\\{-\tfrac{|\textbf{q}|^{2}}{\sigma_{C}^{2}}\\}$,
characterised by a collection bandwidth $\sigma_{C}$ in transverse angular
momentum units. However, since the collection mode commutes with the SLM
holograms, here we incorporate it into the state itself. We can then consider
the collected bi-photon JTMA, which allows us to model all the properties of
the state that we can access and manipulate with SLM holograms. Since the
collection mode:

$\displaystyle G(\textbf{q}_{s},\textbf{q}_{i})$ $\displaystyle=$
$\displaystyle\mathcal{C}(\textbf{q}_{s}|\sigma_{C})\times\mathcal{C}(\textbf{q}_{i}|\sigma_{C})\times
F(\textbf{q}_{s},\textbf{q}_{i})$ $\displaystyle=$
$\displaystyle\tfrac{\mathcal{N}_{1}}{\sqrt{\pi}\sigma_{C}}\underbrace{\exp\bigg{(}-\frac{|\textbf{q}_{s}|^{2}}{\sigma_{C}^{2}}\bigg{)}\exp\bigg{(}-\frac{|\textbf{q}_{i}|^{2}}{\sigma_{C}^{2}}\bigg{)}}_{\text{Collection
widths}}$ $\displaystyle\times$
$\displaystyle\underbrace{\exp\bigg{(}\frac{-|\textbf{q}_{s}+\textbf{q}_{i}|^{2}}{2\sigma_{P}^{2}}\bigg{)}}_{\text{Correlations
strength}}\times\underbrace{\text{sinc}\bigg{(}\frac{|\textbf{q}_{s}-\textbf{q}_{i}|^{2}}{\sigma_{S}^{2}}\bigg{)}}_{\text{Generation
width}}.$

The choice of $\sigma_{C}$ (relative to $\sigma_{S}$ and $\sigma_{P}$) limits
the entanglement dimension and collection efficiency that can be achieved. In
practice, $\sigma_{C}$ can be carefully set through a choice of the optical
system parameters (discussed in detail in Section. 3). As can be seen in Fig.
2a, the effect of including Gaussian collection modes with a specific
$\sigma_{C}$ suppresses the Sinc sidelobes of the generated JTMA (Fig. 1b). We
can then write the collected two-photon coincidence (joint) probability of
detecting the signal and idler photons when displaying the hologram functions
$\Phi_{s}$ and $\Phi_{i}$ respectively, as

$\textbf{Pr}(\Phi_{s},\Phi_{i})=\biggl{|}\int d^{2}\textbf{q}_{s}\int
d^{2}\textbf{q}_{i}\Phi_{s}(\textbf{q}_{s})\Phi_{i}(\textbf{q}_{i})G(\textbf{q}_{s},\textbf{q}_{i})\biggr{|}^{2}.$
(4)

###  II.2 $2D\pi$-Measurement

To characterize the collected JTMA of a two-photon state, one requires to know
the parameters $\sigma_{P}$, $\sigma_{S}$, and $\sigma_{C}$. While these
parameters can be calculated from the optical system properties, one would
practically need to be able to measure them independently and verify that an
experimentally generated state is indeed close to what theory predicts.
Additionally, this capability is of particular relevance when the optical
system is complex, unknown, or inaccessible.

Here, we introduce a simple measurement scheme that we call the
$2D\pi$-measurement, which allows us to estimate the parameters $\sigma_{P}$
and $\sigma_{C}$, and obtain an accurate two-photon JTMA. This measurement is
related to the classical knife-edge measurement routinely used to measure the
transverse profile of a laser beam Arnaud  _et al._ (1971). The
$2D\pi$-measurement can be thought of as a two-photon phase-only knife-edge,
where a $\pi$-phase step is scanned across both the signal and the idler
photons, resulting in a 2D function containing information about the two-
photon JTMA. The $\pi$-phase step is easily implemented via phase-only spatial
light modulators (SLMs) placed in each path. The choice of using a phase edge
over amplitude prevents high photon loss during the measurement, making the
$2D\pi$-measurement an efficient alternative to knife-edge or post-selection
slit-based experiments Strekalov  _et al._ (1995); Pittman  _et al._ (1995);
Howell  _et al._ (2004); Jha  _et al._ (2010); Just  _et al._ (2013); Paul
_et al._ (2014); Chen  _et al._ (2019). Here, we assume the apparent
rotational symmetry of the JTMA in the joint $(q_{x},q_{y})$ planes, and scan
the $\pi$-phase discontinuous profiles for both signal and idler SLMs along
the $x$-axis.

In particular, as illustrated in Fig. 3b, the SLMs display

$\displaystyle\Phi_{n}(q_{nx};a_{n})=\begin{cases}1&q_{nx}<a_{n}\\\
-1&q_{nx}>a_{n},\end{cases}$ (5)

where $n\in\\{s,i\\}$ (signal/idler). By applying the hologram functions
$\Phi_{s}$ and $\Phi_{i}$, the two-photon coincidence probability in Eq. (4)
can be expanded to

$\displaystyle\begin{split}\text{Pr}(a_{s},a_{i})&=\biggl{|}\int
dq_{sy}dq_{iy}\bigg{[}\int^{a_{s}}_{-\infty}dq_{sx}\int^{a_{i}}_{-\infty}dq_{ix}G(\textbf{q}_{s},\textbf{q}_{i})\\\
&+\int_{a_{s}}^{\infty}dq_{sx}\int_{a_{i}}^{\infty}dq_{ix}G(\textbf{q}_{s},\textbf{q}_{i})-\int^{a_{s}}_{-\infty}dq_{sx}\int_{a_{i}}^{\infty}\\\
&dq_{ix}G(\textbf{q}_{s},\textbf{q}_{i})-\int_{a_{s}}^{\infty}dq_{sx}\int^{a_{i}}_{-\infty}dq_{ix}G(\textbf{q}_{s},\textbf{q}_{i})\bigg{]}\biggr{|}^{2}\end{split}$
(6)

Due to the Sinc dependence in Eq. (II.1), the integrals in Eq. (6) do not have
any known closed analytic forms. Under certain approximations of $\sigma_{P}$
and $\sigma_{S}$, one can further simplify the expression in Eq. (6) to ease
its numerical evaluation and data fitting. For instance, the approximation of
a nearly plane-wave pump beam ($\sigma_{P}\rightarrow 0$) simplifies Eq. (6)
to an integral that can be solved straightforwardly. However, the finite
apertures in any optical system lead to a non-zero uncertainty in the momentum
of the pump, thus resulting in an invalid approximation in practice.

Figure 3: Experimental setup: (a) A laser is used to pump a nonlinear ppKTP
crystal to generate a pair of photons entangled in their transverse position-
momentum via Type-II spontaneous parametric downconversion (SPDC). Our
experiment is performed on two parallel setups with a continuous-wave laser
diode pump at 405nm and a pulsed Ti:Sapph pump at 775nm. The pump photons are
filtered by a dichroic mirror (DM) and the downconverted photons are separated
with a polarising-beam-splitter (PBS). The signal and idler photons are
incident on two phase-only spatial light modulators (SLM) that are used for
performing the $2D\pi$-measurement in the transverse position DoF. The
filtered photons are collected by a combination of telescopes (L4 and L5) and
aspheric lenses (A) coupling to single-mode fibres (SMF), followed by
detection through either single-photon-avalanche photodiodes (for
$\lambda_{s1},\lambda_{i1}=810$nm) or superconducting nanowire detectors (for
$\lambda_{s2},\lambda_{i2}=1550$nm). A coincidence counting logic (CC) is used
for recording time-coincident photon detection events within a coincidence
window of 0.2 ns. (b) Examples of diffractive computer generated holograms
implemented on the signal and idler SLMs for performing the
$2D\pi$-measurement in the $x$-direction. (c) 2D profile of the experimental
setup showing the relevant parameters $w_{P}$ and $w_{C}$. The pump beam
radius ($w_{P}$) at the crystal plane determines the pump width parameter
$\sigma_{P}$ (Eq. (2)). The back-propagated collection mode radius ($w_{C}$)
at the SLM determines the collection width parameter ($\sigma_{C}$) at the
crystal, which depends on the single-mode fibres and optical collection system
lenses used.

Here, we discuss a more practical approximation that accounts for the effect
of collection optics and is thus justified from an experimental point of view.
We assert that when the generation bandwidth parameter $\sigma_{S}$ is
sufficiently large compared to the collection bandwidth parameter $\sigma_{C}$
(whilst $\sigma_{P}$ remains relatively small), we can replace the Sinc
argument in Eq. (II.1) with a Gaussian function. This relies on a comparison
of the Gaussian envelopes determined by $\sigma_{C}$ and the Sinc envelope
determined by $\sigma_{S}$. In particular, transforming to the sum and
difference coordinates,
$\textbf{q}_{\pm}=\tfrac{1}{\sqrt{2}}(\textbf{q}_{s}\pm\textbf{q}_{i})$ we
have the collected JTMA

$\displaystyle\begin{split}&G(\textbf{q}_{+},\textbf{q}_{-})=\\\
&\tfrac{\mathcal{N}_{1}}{\sqrt{\pi}\sigma_{C}}\exp\biggl{\\{}\frac{-|\textbf{q}_{+}|^{2}}{\tilde{\sigma}_{P}^{2}}\biggr{\\}}\exp\biggl{\\{}\frac{-|\textbf{q}_{-}|^{2}}{\sigma_{C}^{2}}\biggr{\\}}\textrm{sinc}\biggl{(}\frac{-2|\textbf{q}_{-}|^{2}}{\sigma_{S}^{2}}\biggr{)},\end{split}$
(7)

where
$\tilde{\sigma}_{P}=\bigl{(}\frac{1}{\sigma_{C}^{2}}+\frac{1}{\sigma_{P}^{2}}\bigr{)}^{-\frac{1}{2}}$
and for $\sigma_{P}$ relatively smaller than $\sigma_{C}$,
$\tilde{\sigma}_{P}\approx\sigma_{P}$. We then approximate the product of the
Gaussian envelope ($\sigma_{C}$) and the Sinc argument ($\sigma_{S}$) to a
Gaussian, which is valid if $\sigma_{S}\gtrapprox\sqrt{2}\sigma_{C}$ (see
Appendix B). Under this collection-limited (CL) approximation, the collected
bi-photon JTMA ($G^{CL}$) reads

$\displaystyle\begin{split}G(\textbf{q}_{+},\textbf{q}_{-})&\approx
G^{CL}(\textbf{q}_{+},\textbf{q}_{-})\\\
&=\tfrac{\mathcal{N}_{1}}{\sqrt{\pi}\sigma_{C}}\exp\biggl{\\{}-\frac{|\textbf{q}_{s}+\textbf{q}_{i}|^{2}}{2\tilde{\sigma}_{P}^{2}}\biggr{\\}}\exp\biggl{\\{}-\frac{|\textbf{q}_{s}-\textbf{q}_{i}|^{2}}{2\sigma_{C}^{2}}\biggr{\\}}.\end{split}$
(8)

Notice that instead of $\sigma_{S}$, $\sigma_{C}$ now determines the width of
the JTMA (see Fig. 2b). Even though we have gotten rid of the Sinc dependence
in the collection limited approximation, there is no analytic expression for
the $2D\pi$-measurement. Still, we can derive some contributions analytically,
from which we can recover $\sigma_{P}$ and $\sigma_{C}$. One of the relevant
contributions is $\text{Pr}(a,-a)$, which is given as

$\displaystyle\text{Pr}(a,-a)=\bigg{|}N-N^{\prime}\exp\bigg{(}-\frac{2a^{2}}{\sigma_{C}^{2}}\bigg{)}\biggr{|}^{2},$
(9)

where $N$ and $N^{\prime}$ has absorbed all the constants that are independent
of $a$. The expression of $\text{Pr}(a,-a)$ provides the information about the
collection width parameter $\sigma_{C}$. Another crucial contribution that
retrieves $\sigma_{P}$ is $\text{Pr}(a,a)$, which is given as

$\displaystyle\begin{split}\text{Pr}(a,a)&=\mathcal{A}\biggl{[}\sqrt{2}\sigma_{P}\text{
}e^{-2a^{2}\big{(}\frac{1}{\tilde{\sigma}_{P}^{2}}+\frac{1}{\sigma_{C}^{2}}\big{)}}-2\sqrt{\pi}\text{
}|a|\text{ }e^{\frac{-2a^{2}}{\sigma_{C}^{2}}}\\\ &\times\text{
erfc}\bigg{(}\frac{\sqrt{2}|a|}{\tilde{\sigma}_{P}}\bigg{)}-\frac{\pi\sigma_{C}}{2\sqrt{2}}\bigg{\\{}1-2\text{erf}\bigg{(}\frac{\sqrt{2}|a|}{\sigma_{C}}\bigg{)}\bigg{\\}}\biggr{]}^{2},\end{split}$
(10)

where erfc is the complementary error function. By fitting the expressions Eq.
(9) and Eq. (10) to the experimental data corresponding to $\text{Pr}(a,-a)$
and $\text{Pr}(a,a)$, we can obtain $\sigma_{C}$ and $\sigma_{P}$ that
describe the collected JTMA, which subsequently characterizes the spatial
correlations produced in spontaneous parametric downconversion. For a detailed
calculation of $\text{Pr}(a,-a)$ and $\text{Pr}(a,a)$, please refer to
Appendix C.

##  III Experiment and Results

Figure 4: Experimental results: $2D\pi$-measurement
[$\mathcal{C}_{o}$Pr$(a_{s},a_{i})$] for (a) $\lambda_{s1},\lambda_{i1}=810$nm
and (e) $\lambda_{s2},\lambda_{i2}=1550$nm, where $\mathcal{C}_{o}$ is
proportional to two-photon coincidence count rates. We obtain $\sigma_{C}$ and
$\sigma_{P}$ by fitting $\text{Pr}(a,-a)$ [(b) and (f)] and $\text{Pr}(a,a)$
[(c) and (g)]. Figures (d) and (h) show the experimentally determined
collected JTMA at both wavelengths, where (d) for
$\lambda_{s1},\lambda_{i1}=810$nm, $\sigma_{C}=103.2\pm 1.8$rad/mm and
$\sigma_{P}=7.45\pm 1.51$rad/mm, and (h) for
$\lambda_{s2},\lambda_{i2}=1550$nm, $\sigma_{C}=72.5\pm 2.3$rad/mm and
$\sigma_{P}=3.85\pm 0.31$rad/mm.

To demonstrate the versatility of the $2D\pi$-measurement, we perform an
experimental implementation at two different pump wavelengths (CW 405nm and
femtosecond-pulsed 775nm). The general setup is the same for both wavelengths
(see Fig. 3a). A laser is shaped by a telescope (L1 and L2) to pump a 5mm long
periodically poled nonlinear ppKTP crystal that generates a pair of down-
converted photons (for
$405\text{nm}\rightarrow\lambda_{s1},\lambda_{i1}=810$nm, for
$775\text{nm}\rightarrow\lambda_{s2},\lambda_{i2}=1550$nm) entangled in their
transverse position-momentum DoF via Type-II SPDC. After removing the pump
with a dichroic-mirror (DM), the generated photons are separated with a
polarising-beam-splitter (PBS) and made incident onto two phase-only spatial
light modulators (SLMs and SLMi) placed in the Fourier plane of the crystal
via lens L3 ($f=250$mm). The spatial field at the SLM plane is directly
related to the transverse momentum space at the crystal plane via
$\textbf{q}=\tfrac{2\pi x}{f\lambda}$, where $f$ is the focal length of L3 and
$\lambda$ is the signal/idler wavelength. We perform the $2D\pi$-measurement
using diffractive holograms displayed on the SLMs (Fig. 3b), together with the
collection of single-mode fibres (SMFs), allow for arbitrary spatial-mode
projective measurements to be performed on the incoming photons.

The optical system parameters (lenses L1–L5 and A) are judiciously chosen.
First, in order to obtain a highly correlated JTMA, the telescope system of
lenses L1 and L2 is chosen to maximise the pump radius $w_{P}$ at the crystal
plane, thus minimising the pump width parameter $\sigma_{P}$ for the strength
of the momentum correlation, while ensuring that the pump beam is not
truncated by the crystal aperture. Next, consider the back-propagated beam
from the SMF to the ppKTP crystal (shown in red in Fig. 3c). The aspheric lens
A and the optical system of lenses L3–L5 are chosen such that the collection
width parameter $\sigma_{C}$ meets the condition
$\sigma_{S}\gtrapprox\sqrt{2}\sigma_{C}$, allowing us to work under the
collection-limited JTMA approximation (see previous section). The telescope
system L4 and L5 has also been referred to in our previous work as an
“intensity-flattening telescope” (IFT) as it effectively broadens the back-
propagated Gaussian envelope of the collection mode such that higher order
modes associated with the edges of the JTMA are measured efficiently, while
the lower order modes are suppressed Bouchard  _et al._ (2018).

The photons are detected by single-photon-avalanche photodiodes for
$\lambda_{s1},\lambda_{i1}=810$nm and superconducting nanowire detectors
(SNSPD) for $\lambda_{s2},\lambda_{i2}=1550$nm, which are connected to a
coincidence counting logic (CC) with a coincidence window of 0.2ns. We
characterize the collected JTMA at the Fourier plane of the crystal located at
the SLM planes. The plots in Figs. 4a and 4e show the data obtained for the
$2D\pi$-measurement performed at both wavelengths
($\lambda_{s1},\lambda_{i1}=810$nm and $\lambda_{s2},\lambda_{i2}=1550$nm),
while the reconstructed JTMAs are shown in Figs. 4d and 4h. We obtain
$\sigma_{C}$ and $\sigma_{P}$ by fitting the closed-form expression of
$\text{Pr}(a,-a)$ and $\text{Pr}(a,a)$ (Eqs. (9) and (10)) to the experimental
data. It is worth noting that the feature corresponding to $\sigma_{P}$ is
also present in the visibility of $\text{Pr}(a,-a)$, which is shown in the
fitting curves in Figs. 4b and 4f (refer to Eq. (45)). Those features,
therefore, provide a sensitive measure of the correlation strength
$\sigma_{P}$ even when the resolution of the scan is coarse, unlike slit-based
measurements where the trade-off between the slit size and photon flux is the
issue.

[t]  : Predicted and measured parameters describing the generated and collected joint-transverse-momentum-amplitude (JTMA) obtained from the $2D\pi$-measurement. $\lambda$ |  ${w_{p}}$ |  $\sigma_{P}^{\textrm{pre}}$ |  $\sigma_{P}^{\textrm{meas}}$ |  $\sigma_{S}^{\textrm{pre}}$ |  $\sigma_{C}^{\textrm{pre}}$ |  $\sigma_{C}^{\textrm{meas}}$
---|---|---|---|---|---|---
(nm) |  ($\mu$m) |  (rad/mm) |  (rad/mm) |  (rad/mm) |  (rad/mm) |  (rad/mm)
810 |  $188\pm 5.3$ |  $7.52\pm 0.21$ |  $7.45\pm 1.51$ |  $151.1\pm 3.1$ |  $106.5\pm 12.6$ |  $103.2\pm 1.8$
1550 |  $450\pm 5.3$ |  $3.14\pm 0.04$ |  $3.85\pm 0.48$ |  $106.7\pm 2.1$ |  $76.7\pm 8.01$ |  $72.5\pm 2.3$
The predicted pump width and generation width parameters
($\sigma_{P}^{\textrm{pre}}$ and $\sigma_{S}^{\textrm{pre}}$) are obtained
from the pump waist $w_{p}$ and crystal length according to Eq. (2). The
predicted collection width parameter ($\sigma_{P}^{\textrm{pre}}$) is
calculated from measurements of the optical collection system (please see main
text for details). The measured parameters ($\sigma_{P}^{\textrm{meas}}$ and
$\sigma_{C}^{\textrm{meas}}$) are obtained from the $2D\pi$-measurement
performed on the two-photon state. Both predicted and measured values of
$\sigma_{C}$ meet the condition $\sigma_{S}\gtrapprox\sqrt{2}\sigma_{C}$ that
allows us to operate under the collection-limited (CL) approximation (please
see Eq. (8)).

The measured values of the pump and collection width parameters
($\sigma_{P}^{\textrm{meas}}$, $\sigma_{C}^{\textrm{meas}}$) obtained from the
$2D\pi$-measurement are reported in Table III and agree with their predicted
values ($\sigma_{P}^{\textrm{pre}}$, $\sigma_{C}^{\textrm{pre}}$), which are
calculated from our knowledge of the optical system parameters. The predicted
value of the pump width parameter $\sigma_{P}^{\textrm{pre}}$ for both
wavelengths is calculated from the $1/e^{2}$ pump radius at the crystal plane
(Eq. (2)), and the predicted collection width parameter
$\sigma_{C}^{\textrm{pre}}$ is calculated by back-propagating the width of the
fundamental Gaussian mode of the SMFs to the crystal plane (Fig. 3c). The
error propagation is analysed by taking into account $\pm$0.5mm uncertainties
of the measured distances between lenses and focal lengths.

##  IV Tailoring High-Dimensional Entanglement

Figure 5: Optimising $T_{ab}$ for 810nm two-photon state by tailoring the
pixel mask in 31 dimensions: Figures (a-c) [(d-f)] show the collected JTMA,
pixel mask holograms, and resulting state coefficients $T_{ab}$, before
[after] optimisation. The square regions in the JTMA plots in (a,d) indicate
the values of $q_{sx}$ and $q_{ix}$ used to generate the pixel masks holograms
in (b,e). The green square regions represent the pixels that are correlated
($T_{aa}$ elements), while the red ones are responsible for cross-talk between
them ($T_{ab}$ elements $\forall a\neq b$). Note that while the 2D JTMA
($x$-dimension) is shown here, calculating all the elements of $T_{ab}$
require integration over the entire 4D expression for the JTMA (Eq. 12), i.e.,
$q_{sx}$, $q_{sy}$, $q_{ix}$ and $q_{iy}$. The green dashed circles represent
the maximum size of the detectable area, which is further constrained by the
finite collection bandwidth parameter $\sigma_{C}$. In (a) and (b), the pixels
are set to the same size with a fixed spacing between them, and the resulting
state (c) shows significant cross-talk between discrete spatial modes. The
dimensionality of entanglement ($d_{ent}$) is calculated to be 26 in the
Hilbert space of $d=31$ pixels Bavaresco  _et al._ (2018). In (d) and (e), we
show how one can tailor the pixel mask from knowledge of the JTMA such that
the coincidence probability $\text{Pr}(\Phi_{s},\Phi_{i})$ is equal for all
pixels, while minimizing the cross-talk between them at the expense of total
counts. This ensures that the state shown in (f) is optimized to be in the
Schmidt basis while also being close to maximally entangled. Consequently, the
dimensionality of entanglement is increased to its maximal value of
$d_{ent}=31$.

Once accurate knowledge of the continuous position-momentum two-photon state
characterised by the JTMA has been obtained, one may want to discretise such a
state for use in quantum information applications based on discrete variables
Dada  _et al._ (2011); Mafu  _et al._ (2013); Bavaresco  _et al._ (2017);
Herrera Valencia  _et al._ (2020). For instance, to harness discrete variable
high-dimensional entanglement, one needs to choose an appropriate modal basis
in which to work. The design of such discrete modal bases is often informed by
the Schmidt decomposition of the entire bi-photon wavefunction Straupe  _et
al._ (2011); Walborn and Pimentel (2012). However, at the expense of lower
count rates one can design modal bases to optimise for more general figures of
merit such as entanglement-of-formation ($E_{\textrm{oF}}$), heralding
efficiency, and measurement fidelity, while taking into account the types of
devices used (for example, phase-only SLMs).

We begin with a theoretical treatment of how our continuous two-photon state
is discretised via specific projective modal measurements. In practice, we
display holograms $\\{\Phi_{s}(\textbf{q}_{s})\\}^{a}$ and
$\\{\Phi_{i}(\textbf{q}_{i})\\}^{b}$ (where $|\Phi_{s}(\textbf{q}_{s})|\leq
1$) on the SLMs to generate a subnormalised post-selected state
$\left|\right.\\!{\psi^{(PS)}}\\!\left.\right\rangle$ in the _standard_
discrete modal basis
$\\{\left|\right.\\!{\hat{e}_{n}}\\!\left.\right\rangle\\}_{a}$:

$\left|\right.\\!{\psi^{(PS)}}\\!\left.\right\rangle=\sum_{ab}T_{ab}\left|\right.\\!{\hat{e}_{s}^{a}}\\!\left.\right\rangle\left|\right.\\!{\hat{e}_{i}^{b}}\\!\left.\right\rangle,$
(11)

where the complex elements $T_{ab}$ are given by

$T_{ab}=\int d^{2}\textbf{q}_{s}\int
d^{2}\textbf{q}_{i}\Phi_{s}^{a}(\textbf{q}_{s})\Phi_{i}^{b}(\textbf{q}_{i})G(\textbf{q}_{s},\textbf{q}_{i}).$
(12)

Here, $G(\textbf{q}_{s},\textbf{q}_{i})$ is the collected JTMA (collection-
limited in our case) whose form can be obtained via the 2D$\pi$ measurement
described in the preceding sections. The holograms
$\Phi_{s}^{a}(\textbf{q}_{s})$ and $\Phi_{i}^{b}(\textbf{q}_{i})$ can be
constructed in a manner to ensures that the associated discrete modal bases
are orthonormal (see Appendix D).

Now we can determine the probabilities associated with measuring generalised
projectors (arbitrary coherent superpositions) both in the two-photon and
single-photon case. We can measure an arbitrary normalised vector
$\left|\right.\\!{\vec{v}_{s}}\\!\left.\right\rangle=\sum_{a}v^{a}_{s}\left|\right.\\!{\hat{e}^{a}_{s}}\\!\left.\right\rangle$
by constructing hologram $\Phi_{s}^{\vec{v}}(\textbf{q}_{s})$ given by

$\Phi^{\vec{v}_{s}}_{s}(\textbf{q}_{s})=A^{\vec{v}_{s}}\sum_{a}v_{s}^{a}\Phi^{a}_{s}(\textbf{q}_{s}),$
(13)

with $A^{\vec{v}}$ chosen such that
$\max_{\textbf{q}}|\Phi^{\vec{v}}_{s}(\textbf{q})|\leq 1$. The preceding
condition ensures that a hologram does not increase energy and can only add
loss. The two-photon coincidence probability for measuring our state in modes
$\left|\right.\\!{\vec{v}_{s}}\\!\left.\right\rangle$ and
$\left|\right.\\!{\vec{v}_{i}}\\!\left.\right\rangle$ is given by

$\textbf{Pr}(\vec{v}_{s},\vec{v}_{i})=(A^{\vec{v}_{s}}A^{\vec{v}_{i}})^{2}|\left\langle\right.\\!{\vec{v}_{s}}\\!\left.\right|\left\langle\right.\\!{\vec{v}_{i}}\\!\left.\right|\left|\right.\\!{\psi^{(PS)}}\\!\left.\right\rangle|^{2}.$
(14)

Similarly, the probability of measuring a signal photon (inclusive of a
possible idler photon) in mode
$\left|\right.\\!{\vec{v}_{s}}\\!\left.\right\rangle$ depends on the
collection mode of only signal $(\mathcal{C}(\textbf{q}_{s}|\sigma_{C}))$ and
is expressed as

$\begin{split}\textbf{Pr}(\vec{v}_{s})&=\int d^{2}\textbf{q}_{i}\left|\int
d^{2}\textbf{q}_{s}\Phi^{\vec{v}_{s}}_{s}(\textbf{q}_{s})\mathcal{C}(\textbf{q}_{s}|\sigma_{C})F(\textbf{q}_{s},\textbf{q}_{i})\right|^{2},\\\
\end{split}$ (15)

where the product of generated JTMA $F(\textbf{q}_{s},\textbf{q}_{i})$ and the
signal’s collection mode $\mathcal{C}(\textbf{q}_{s}|\sigma_{C})$ can be
further simplified as discussed in Appendix B (see Eq. (39)).

With knowledge of the collected JTMA ($G(\textbf{q}_{s},\textbf{q}_{i})$), a
variety of figures of merit might be considered when designing such modal
bases (corresponding to holograms $\Phi_{s}^{a}(\textbf{q}_{s})$ and
$\Phi_{i}^{b}(\textbf{q}_{i})$) and choosing which measurements to make in a
given basis. Here, we take the example of disjoint discrete spatial modes
(“pixel basis” Herrera Valencia  _et al._ (2020)) defined by macro-pixels in
the SLM plane to exemplify some of these merits. In the pixel basis, the size
of each pixel, the spacing between them, and the size of the complete pixel
mask are important parameters to take into account in the basis design. One
may vary these parameters to optimise for the following properties:

Schmidt basis: A standard discrete basis for the post-selected two-photon
state (Eq. (11)) can be designed such that it corresponds to the Schmidt basis
where the coincidence cross-talk between modes is minimized, and thus
suppressing the off-diagonal elements of $T_{ab}$ ($T_{ab}\rightarrow 0$
$\forall a\neq b$). In the case of the pixel basis, this corresponds to
choosing the spacing between pixels to be at least equal to the pump width
parameter (appropriately propagated to position coordinates at the SLM plane),
which determines the JTMA correlation strength
$\sigma_{P}^{(x)}=\frac{f\lambda}{2\pi}\sigma_{P}$ (see Fig. 5).

Entanglement dimensionality: When constructing a standard discrete basis,
there is a limit on the entanglement dimensionality—the maximal number of
correlated modes that can be considered whilst remaining in the Schmidt basis.
Information about this can be deduced from the JTMA: the accessible number of
generated entangled modes, related to the reciprocal of the marginal state
purity (often known as the Schmidt number Law and Eberly (2004)), can be
estimated through $\sigma_{P}$ and $\sigma_{S}$ Schneeloch and Howell (2016).
However, as we have shown here, $\sigma_{S}$ is often constrained by the
collection width parameter $\sigma_{C}$, knowledge of which can be used to
estimate the perhaps more relevant number of collected entangled modes. For
the pixel basis, this involves an optimisation of the number of correlated
macro-pixels one can fit within the collected area, while having appreciable
count rates.

Maximal entanglement: Optimising the standard discrete basis such that the
coincidence probability for each mode is equal (while remaining in the Schmidt
basis) imposes $T_{ab}$ is proportional to the identity matrix and thus a
maximally entangled state. This maximises entropic quantifiers of entanglement
such as entanglement-of-formation ($E_{\textrm{oF}}$) and can be achieved for
instance, by optimally varying pixel size as a function of radial distance
from the optic axis (see Fig. 5 for a detailed example).

Basis-dependent efficiency: One can find bases where all the holograms can be
efficiently realised by maximising $A^{\vec{v}}$ for all elements of a basis.
For instance, with disjoint pixels, all bases mutually unbiased to the
standard basis obtain $A^{\vec{v}}=\sqrt{d}$ for all elements. This ensures
that all measurements maximise photon flux, thus drastically reducing
measurement times Herrera Valencia  _et al._ (2020).

Heralding efficiency: The heralding efficiency, or the probability that the
detection of a photon in one mode (signal) indicates a photon in the other
(the heralded photon or idler), is normally studied in a symmetric
configuration, i.e. the same collection parameters apply to both photons Grice
_et al._ (2011); Dixon  _et al._ (2014). The inherent multi-mode nature of the
JTMA opens up an alternate way to tune heralding efficiencies in an asymmetric
manner, i.e. with different collection parameters and resultant heralding
efficiencies for each photon Ljunggren and Tengner (2005). We can define a
one-sided heralding efficiency in this case, where measuring the signal photon
in mode $\left|\right.\\!{\vec{v}_{s}}\\!\left.\right\rangle$ heralds the
presence of an idler photon in mode
$\left|\right.\\!{\vec{v}_{i}}\\!\left.\right\rangle$ with an efficiency

$\eta^{s\rightarrow
i}=\frac{\textbf{Pr}(\vec{v}_{s},\vec{v}_{i})}{\textbf{Pr}(\vec{v}_{s})}.$
(16)

Therefore, designing holograms $\Phi_{i}^{\vec{v}}(\textbf{q}_{i})$ to
maximise Eq. (16), and choosing bases such that $A^{\vec{v}_{i}}$ is large can
lead to high one-sided heralding efficiency (see Appendix D). For instance,
increasing the size of the idler pixel than that of the signal optimises the
heralding efficiency, resulting in a larger overlap of heralded photons on the
idler side.

We have recently implemented some of the above techniques experimentally in
order to rapidly certify high-fidelity entangled states with entanglement
dimensionalities up to $d=55$, entanglement-of-formation up to
$E_{\textrm{oF}}=4$ ebits Herrera Valencia  _et al._ (2020), and to violate
high-dimensional steering inequalities in dimension up to $d=15$ Designolle
_et al._ (2021). These experiments were performed in the pixel basis, where
through knowledge of the JTMA obtained via the $2D\pi$-measurement, a post-
selected state closest to a maximally entangled state was realised. The SLM
holograms were designed to minimise the cross-talk between modes, while
simultaneously equalising photon count rates for all discrete modes across
$\sigma_{C}$ (in a manner similar to the procedure discussed in Fig. 5).
Furthermore, high dimensional entanglement witnesses exploiting only bases
mutually unbiased to the standard pixel basis ensured high basis-dependent
efficiencies and minimised measurement times.

Figure 6: LG radial-mode entanglement: (a) The knowledge of collected JTMA
($G(\textbf{q}_{s},\textbf{q}_{i})$) determines the maximum size of the mode
waist (blue dotted circle in (b)) to witness entanglement in 7 dimensions in
$p=0,\dots,10$ radial LG modes Valencia  _et al._ (2021).

While the above examples have used the disjoint and discrete pixel basis,
knowledge of the JTMA can be used to optimise other spatial mode bases such as
the Laguerre-Gaussian (LG) basis, which plays a significant role in classical
and quantum optics Krenn  _et al._ (2017). In the LG basis, correlations in
both the azimuthal and radial components depend on the relationship between
the pump and the down-converted signal/idler mode waists Miatto  _et al._
(2011); Salakhutdinov  _et al._ (2012), indicating the effective number of
detectable Schmidt modes that are entangled in the full transverse field
Miatto  _et al._ (2012b); Straupe  _et al._ (2011). The ability to determine
the collected JTMA allows us to experimentally adjust $\sigma_{P}$ and
$\sigma_{C}$ such that the correlations are maximised. While the degree of
correlations relates to $\sigma_{P}$, information of $\sigma_{C}$ sets a limit
on the size of modes we can optimally measure (see Fig. 6), which is of
particular importance when dealing with modes that have radial dependence.
Taking these considerations into account, we were able to recently certify
entanglement dimensionalities of up to 26 in a 43-dimensional radial and
azimuthal LG space Valencia  _et al._ (2021), demonstrating the potential of
the JTMA in harnessing the full capabilities of high-dimensional entanglement.

##  V Conclusion and Discussion

In our work, we have studied the spatial wavefunction of position-momentum
entangled bi-photon states generated in collinear Type-II SPDC. We define the
collected joint-transverse-momentum amplitude (JTMA), a function that
characterizes bi-photon state in the momentum degree-of-freedom while
incorporating the effects of the measurement system. We propose a method to
efficiently and accurately characterize the collected JTMA using phase-only
modulated holograms, and experimentally demonstrate it on two identical
entanglement sources at different wavelengths. From knowledge of the collected
JTMA, we discuss how one can tailor discrete-variable high-dimensional
entangled states via projective measurements in several spatial mode bases.
Our techniques can be used to generate high-fidelity, high-dimensional
entangled states of light, which can be further optimised for properties such
as maximal entanglement and single photon heralding efficiencies. The utility
of our characterisation methods is evident in some recent works, where we have
used knowledge of the JTMA to tailor diverse kinds of high-dimensional
entangled states of light with record quality and dimensionalities in device-
dependent Herrera Valencia  _et al._ (2020); Valencia  _et al._ (2021) as well
as one-sided device-independent platforms Designolle  _et al._ (2021).

Information about the JTMA could be used for implementing and optimising
arbitrary spatial-mode projective measurements, which are required for
violating Bell-like inequalities proposed for high-dimensional systems Vertesi
_et al._ (2010); Collins  _et al._ (2002); Salavrakos  _et al._ (2017).
Additionally, knowledge of the JTMA can be used for tuning one-sided photon
heralding efficiencies Iskhakov  _et al._ (2016); Ramelow  _et al._ (2013);
Castelletto  _et al._ (2005), which play a significant role in device-
independent tests of quantum mechanics and the related field of device-
independent QKD. The ability to control the correlations/anti-correlations of
an entangled pair of photons using the measured JTMA parameters could enable
the engineering of quantum states with tailored spatial and spectral
properties Brambilla  _et al._ (2010), which could be used to boost the
performance of quantum-enhanced imaging and metrology Malik and Boyd (2014);
Basset  _et al._ (2019). In addition, our methods for characterizing the JTMA
can be translated to other degrees-of-freedom such as time-frequency Jha  _et
al._ (2008); Kues  _et al._ (2017); Martin  _et al._ (2017), and could enable
the characterization of the full spatio-temporal bi-photon wavefunction, which
would have a wide-ranging impact on entangled-based quantum technologies.

###### Acknowledgements.

This work was made possible by financial support from the QuantERA ERA-NET Co-
fund (FWF Project I3773-N36), the UK Engineering and Physical Sciences
Research Council (EPSRC) (EP/P024114/1), and the European Research Council
(Starting Grant PIQUaNT).

## References

  * Einstein  _et al._ (1935) A. Einstein, B. Podolsky, and N. Rosen,  _Can quantum-mechanical description of physical reality be considered complete?_Phys. Rev. 47, 777 (1935).
  * Reid  _et al._ (2009) M D Reid, P D Drummond, W P Bowen, E G Cavalcanti, P K Lam, H A Bachor, U L Andersen, and G Leuchs,  _Colloquium: The Einstein-Podolsky-Rosen paradox: From concepts to applications,_Reviews of Modern Physics 81, 1727 (2009).
  * Bohm and Aharonov (1957) D. Bohm and Y. Aharonov,  _Discussion of experimental proof for the paradox of einstein, rosen, and podolsky,_Phys. Rev. 108, 1070 (1957).
  * Bell (1964) John S. Bell,  _On the Einstein Podolsky Rosen paradox,_Physics Physique Fizika 1, 195 (1964).
  * Brunner  _et al._ (2014) Nicolas Brunner, Daniel Cavalcanti, Stefano Pironio, Valerio Scarani, and Stephanie Wehner,  _Bell nonlocality,_Reviews of Modern Physics 86, 419 (2014).
  * Clauser  _et al._ (1969) John Clauser, Michael Horne, Abner Shimony, and Richard Holt,  _Proposed Experiment to Test Local Hidden-Variable Theories,_Phys. Rev. Lett. 23, 880 (1969).
  * Aspect  _et al._ (1982) Alain Aspect, Philippe Grangier, and Gérard Roger,  _Experimental realization of einstein-podolsky-rosen-bohm gedankenexperiment: A new violation of bell’s inequalities,_Phys. Rev. Lett. 49, 91 (1982).
  * Shin  _et al._ (2019) D.K. Shin, B.M. Henson, S.S. Hodgman, Tomasz Wasak, Jan Chwedeńczuk, and Truscott A.G.,  _Bell correlations between spatially separated pairs of atoms,_Nat Commun 10, 4447 (2019), arXiv:1811.05681.
  * Wasak and Chwedeńczuk (2018) Tomasz Wasak and Jan Chwedeńczuk,  _Bell inequality, einstein-podolsky-rosen steering, and quantum metrology with spinor bose-einstein condensates,_Phys. Rev. Lett. 120, 140406 (2018), arXiv:1710.10879.
  * Keller  _et al._ (2014) Michael Keller, Mateusz Kotyrba, Florian Leupold, Mandip Singh, Maximilian Ebner, and Anton Zeilinger,  _Bose-einstein condensate of metastable helium for quantum correlation experiments,_Phys. Rev. A 90, 063607 (2014), arXiv:1406.1322.
  * Ndagano  _et al._ (2020) Bienvenu Ndagano, Hugo Defienne, Ashley Lyons, Ilya Starshynov, Federica Villa, Simone Tisa, and Daniele Faccio,  _Imaging and certifying high-dimensional entanglement with a single-photon avalanche diode camera,_npj Quantum Inf 6, 94 (2020), arXiv:2001.03997.
  * Edgar  _et al._ (2012) M P Edgar, D S Tasca, F Izdebski, R E Warburton, J Leach, M Agnew, G S Buller, R.W. Boyd, and Miles J Padgett,  _Imaging high-dimensional spatial entanglement with a camera,_Nature Communications 3, 984 (2012).
  * Moreau  _et al._ (2012) Paul-Antoine Moreau, Joé Mougin-Sisini, Fabrice Devaux, and Eric Lantz,  _Realization of the purely spatial einstein-podolsky-rosen paradox in full-field images of spontaneous parametric down-conversion,_Phys. Rev. A 86, 010101 (2012), arXiv:1204.0990.
  * Boyd (2008) Robert W. Boyd,  _Nonlinear Optics_ (Academic Press, 2008).
  * Hong and Mandel (1985) C. K. Hong and L. Mandel,  _Theory of parametric frequency down conversion of light,_Phys. Rev. A 31, 2409 (1985).
  * Malygin  _et al._ (1985) A. Malygin, A. Penin, and Alexander Sergienko,  _Spatiotemporal grouping of photons in spontaneous parametric scattering of light,_ Soviet Physics Doklady (1985).
  * Schneeloch and Howell (2016) James Schneeloch and John C Howell,  _Introduction to the transverse spatial correlations in spontaneous parametric down-conversion through the biphoton birth zone,_Journal of Optics 18, 53501 (2016), arXiv:1502.06996.
  * Brambilla  _et al._ (2010) E. Brambilla, L. Caspani, L. A. Lugiato, and A. Gatti,  _Spatiotemporal structure of biphoton entanglement in type-II parametric down-conversion,_Phys. Rev. A 82 (2010), 10.1103/PhysRevA.82.013835, arXiv:1205.1637.
  * Walborn  _et al._ (2010) S.P. Walborn, C.H. Monken, S. Pádua, and P.H. Souto Ribeiro,  _Spatial correlations in parametric down-conversion,_Physics Reports 495, 87 (2010), arXiv:1010.1236.
  * Erhard  _et al._ (2020a) Manuel Erhard, Mario Krenn, and Anton Zeilinger,  _Advances in High Dimensional Quantum Entanglement,_Nature Reviews Physics 2, 365 (2020a).
  * Friis  _et al._ (2019) Nicolai Friis, Giuseppe Vitagliano, Mehul Malik, and Marcus Huber,  _Entanglement certification from theory to experiment,_Nature Reviews Physics 1, 72 (2019).
  * Bavaresco  _et al._ (2018) Jessica Bavaresco, Natalia Herrera Valencia, Claude Klockl, Matej Pivoluska, Paul Erker, Nicolai Friis, Mehul Malik, and Marcus Huber,  _Measurements in two bases are sufficient for certifying high-dimensional entanglement,_Nature Physics 14, 1032 (2018).
  * Dada  _et al._ (2011) Adetunmise C Dada, Jonathan Leach, Gerald S Buller, Miles J Padgett, and Erika Andersson,  _Experimental high-dimensional two-photon entanglement and violations of generalized Bell inequalities,_Nature Physics 7, 677 (2011), arXiv:1104.5087.
  * Wang  _et al._ (2015) Xi-Lin Wang, Xin-Dong Cai, Zu-En Su, Ming-Cheng Chen, Dian Wu, Li Li, Nai-Le Liu, Chao-Yang Lu, and Jian-Wei Pan,  _Quantum teleportation of multiple degrees of freedom of a single photon,_Nature 518, 516 (2015).
  * Malik  _et al._ (2016) Mehul Malik, Manuel Erhard, Marcus Huber, Mario Krenn, Robert Fickler, and Anton Zeilinger,  _Multi-photon entanglement in high dimensions,_Nature Photonics 10, 248–252 (2016), arXiv:1509.02561.
  * Erhard  _et al._ (2018) Manuel Erhard, Mehul Malik, Mario Krenn, and Anton Zeilinger,  _Experimental Greenberger-Horne-Zeilinger entanglement beyond qubits,_Nat Photon 12, 759 (2018), arXiv:1708.03881.
  * Hiesmayr  _et al._ (2016) B. C. Hiesmayr, M. J. A. de Dood, and W. Löffler,  _Observation of four-photon orbital angular momentum entanglement,_Phys. Rev. Lett. 116, 073601 (2016), arXiv:1808.00005.
  * Erhard  _et al._ (2020b) Manuel Erhard, Mehul Malik, Mario Krenn, and Anton Zeilinger,  _Experimental creation of multi-photon high-dimensional layered quantum states,_npj Quantum Inf 88 (2020b), arXiv:2001.06253.
  * Valencia  _et al._ (2020) Natalia Herrera Valencia, Suraj Goel, Will McCutcheon, Hugo Defienne, and Mehul Malik,  _Unscrambling entanglement through a complex medium,_Nature Physics 16, 1112 (2020), arXiv:1910.04490.
  * Cao  _et al._ (2020) Huan Cao, She-Cheng Gao, Chao Zhang, Jian Wang, De-Yong He, Bi-Heng Liu, Zheng-Wei Zhou, Yu-Jie Chen, Zhao-Hui Li, Si-Yuan Yu, Jacquiline Romero, Yun-Feng Huang, Chuan-Feng Li, and Guang-Can Guo,  _Distribution of high-dimensional orbital angular momentum entanglement over a 1km few-mode fiber,_Optica 7, 232 (2020), arXiv:1811.12195.
  * Cozzolino  _et al._ (2019) Daniele Cozzolino, Beatrice Da Lio, Davide Bacco, and Leif Katsuo Oxenløwe,  _High-dimensional quantum communication: Benefits, progress, and future challenges,_Advanced Quantum Technologies 2, 1900038 (2019), arXiv:1910.07220.
  * Erhard  _et al._ (2017) Manuel Erhard, Mehul Malik, and Anton Zeilinger,  _A quantum router for high-dimensional entanglement,_Quantum Science and Technology 2, 014001 (2017), arXiv:1605.05947.
  * Steinlechner  _et al._ (2017) Fabian Steinlechner, Sebastian Ecker, Matthias Fink, Bo Liu, Jessica Bavaresco, Marcus Huber, Thomas Scheidl, and Rupert Ursin,  _Distribution of high-dimensional entanglement via an intra-city free-space link,_Nature Communications 8, 15971 (2017).
  * Mafu  _et al._ (2013) Mhlambululi Mafu, Angela Dudley, Sandeep Goyal, Daniel Giovannini, Melanie McLaren, Miles J. Padgett, Thomas Konrad, Francesco Petruccione, Norbert Lütkenhaus, and Andrew Forbes,  _Higher-dimensional orbital-angular-momentum-based quantum key distribution with mutually unbiased bases,_Phys. Rev. A 88, 032305 (2013), arXiv:1402.5810.
  * Ecker  _et al._ (2019) Sebastian Ecker, Frédéric Bouchard, Lukas Bulla, Florian Brandt, Oskar Kohout, Fabian Steinlechner, Robert Fickler, Mehul Malik, Yelena Guryanova, Rupert Ursin, and Marcus Huber,  _Overcoming Noise in Entanglement Distribution,_Phys. Rev. X 9, 041042 (2019), arXiv:1904.01552.
  * Zhu  _et al._ (2021) F. Zhu, M. Tyler, N. H. Valencia, M. Malik, and J. Leach,  _Is high-dimensional photonic entanglement robust to noise?_AVS Quantum Science 3, 011401 (2021), arXiv:1908.08943.
  * Huber and Pawłowski (2013) Marcus Huber and Marcin Pawłowski,  _Weak randomness in device-independent quantum key distribution and the advantage of using high-dimensional entanglement,_Phys. Rev. A 88, 032309 (2013), arXiv:1301.2455.
  * Vertesi  _et al._ (2010) Tamás Vertesi, Stefano Pironio, and Nicolas Brunner,  _Closing the Detection Loophole in Bell Experiments Using Qudits,_Phys. Rev. Lett. 104, 60401 (2010), arXiv:0909.3171.
  * Acín  _et al._ (7 06) Antonio Acín, Nicolas Brunner, Nicolas Gisin, Serge Massar, Stefano Pironio, and Valerio Scarani,  _Device-Independent Security of Quantum Cryptography against Collective Attacks,_Physical Review Letters 98, 230501 (2007-06).
  * Miatto  _et al._ (2012a) F M Miatto, H Di Lorenzo Pires, S M Barnett, and M P van Exter,  _Spatial Schmidt modes generated in parametric down-conversion,_Eur. Phys. J. D 66, 263 (2012a).
  * Howell  _et al._ (2004) John C Howell, Ryan S Bennink, Sean J Bentley, and R W Boyd,  _Realization of the Einstein-Podolsky-Rosen Paradox Using Momentum- and Position-Entangled Photons from Spontaneous Parametric Down Conversion,_Phys. Rev. Lett. 92, 210403 (2004).
  * Schneeloch  _et al._ (2019) J Schneeloch, C C Tison, M L Fanto, Paul M Alsing, and Gregory A Howland,  _Quantifying entanglement in a 68-billion-dimensional quantum state space,_Nat. Commun. , 10:2785 (2019), arXiv:1804.04515.
  * Defienne  _et al._ (2018) Hugo Defienne, Matthew Reichert, and Jason W Fleischer,  _General Model of Photon-Pair Detection with an Image Sensor,_Physical Review Letters 120, 203604 (2018), arXiv:1801.09019.
  * Miatto  _et al._ (2012b) F M Miatto, D Giovannini, J Romero, S Franke-Arnold, S M Barnett, and M J Padgett,  _Bounds and optimisation of orbital angular momentum bandwidths within parametric down-conversion systems,_Eur. Phys. J. D 66, 178 (2012b), arXiv:1112.3910.
  * Baghdasaryan  _et al._ (2021) Baghdasar Baghdasaryan, Fabian Steinlechner, and Stephan Fritzsche,  _Justifying the thin-crystal approximation in spontaneous parametric down-conversion for collinear phase matching,_Phys. Rev. A 103, 063508 (2021).
  * Qassim  _et al._ (2014) Hammam Qassim, Filippo M Miatto, Juan P Torres, Miles J Padgett, Ebrahim Karimi, and Robert W Boyd,  _Limitations to the determination of a Laguerre-Gauss spectrum via projective, phase-flattening measurement,_Journal of the Optical Society of America B 31, A20 (2014), arXiv:1401.3512.
  * Bouchard  _et al._ (2018) Frédéric Bouchard, Natalia Herrera Valencia, Florian Brandt, Robert Fickler, Marcus Huber, and Mehul Malik,  _Measuring azimuthal and radial modes of photons,_Optics Express 26, 31925 (2018), arXiv:1808.03533.
  * Arnaud  _et al._ (1971) J. A. Arnaud, W. M. Hubbard, G. D. Mandeville, B. de la Clavière, E. A. Franke, and J. M. Franke,  _Technique for fast measurement of gaussian laser beam parameters,_Appl. Opt. 10, 2775 (1971).
  * Strekalov  _et al._ (1995) D Strekalov, Alexander Sergienko, and D Klyshko,  _Observation of two-photon “ghost” interference and diffraction,_Phys. Rev. Lett. 74, 3600 (1995).
  * Pittman  _et al._ (1995) T B Pittman, Y H Shih, D V Strekalov, and A V Sergienko,  _Optical imaging by means of two-photon quantum entanglement,_Phys. Rev. A 52, R3429 (1995).
  * Jha  _et al._ (2010) Anand Kumar Jha, Jonathan Leach, Barry Jack, Sonja Franke-Arnold, Stephen M. Barnett, Robert W. Boyd, and Miles J. Padgett,  _Angular two-photon interference and angular two-qubit states,_Phys. Rev. Lett. 104, 010501 (2010).
  * Just  _et al._ (2013) J Just, A. Cavanna, M.V. Chekhova, and G. Leuchs, New J. Phys 15, 083015 (2013).
  * Paul  _et al._ (2014) E. C. Paul, M. Hor-Meyll, P. H.Souto Ribeiro, and S. P. Walborn,  _Measuring spatial correlations of photon pairs by automated raster scanning with spatial light modulators,_Scientific Reports 4 (2014).
  * Chen  _et al._ (2019) Lixiang Chen, Tianlong Ma, Xiaodong Qiu, Dongkai Zhang, Wuhong Zhang, and Robert W. Boyd,  _Realization of the einstein-podolsky-rosen paradox using radial position and radial momentum variables,_Phys. Rev. Lett. 123, 060403 (2019).
  * Bavaresco  _et al._ (2017) Jessica Bavaresco, Marco Túlio Quintino, Leonardo Guerini, Thiago O Maciel, Daniel Cavalcanti, and Marcelo Terra Cunha,  _Most incompatible measurements for robust steering tests,_Phys. Rev. A 96, 22110 (2017), arXiv:1704.02994.
  * Herrera Valencia  _et al._ (2020) Natalia Herrera Valencia, Vatshal Srivastav, Matej Pivoluska, Marcus Huber, Nicolai Friis, Will McCutcheon, and Mehul Malik,  _High-Dimensional Pixel Entanglement: Efficient Generation and Certification,_Quantum 4, 376 (2020), arXiv:2004.04994.
  * Straupe  _et al._ (2011) S S Straupe, D P Ivanov, A A Kalinkin, I B Bobrov, and S P Kulik,  _Angular Schmidt modes in spontaneous parametric down-conversion,_Phys. Rev. A 83, 60302 (2011), arXiv:1103.4077.
  * Walborn and Pimentel (2012) S. P. Walborn and A. H. Pimentel,  _Generalized Hermite-Gauss decomposition of the two-photon state produced by spontaneous parametric down conversion,_Journal of Physics B: Atomic, Molecular and Optical Physics 45, 165502 (2012), arXiv:arXiv:1208.4336.
  * Law and Eberly (2004) C K Law and J H Eberly,  _Analysis and interpretation of high transverse entanglement in optical parametric down conversion,_Phys. Rev. Lett. 92, 127903 (2004).
  * Grice  _et al._ (2011) W. P. Grice, R. S. Bennink, D. S. Goodman, and A. T. Ryan,  _Spatial entanglement and optimal single-mode coupling,_Phys. Rev. A 83, 023810 (2011).
  * Dixon  _et al._ (2014) P. Ben Dixon, Danna Rosenberg, Veronika Stelmakh, Matthew E. Grein, Ryan S. Bennink, Eric A. Dauler, Andrew J. Kerman, Richard J. Molnar, and Franco N. C. Wong,  _Heralding efficiency and correlated-mode coupling of near-ir fiber-coupled photon pairs,_Phys. Rev. A 90, 043804 (2014), arXiv:1407.8487.
  * Ljunggren and Tengner (2005) Daniel Ljunggren and Maria Tengner,  _Optimal focusing for maximal collection of entangled narrow-band photon pairs into single-mode fibers,_Phys. Rev. A 72, 062301 (2005), arXiv:quant-ph/0507046.
  * Designolle  _et al._ (2021) Sébastien Designolle, Vatshal Srivastav, Roope Uola, Natalia Herrera Valencia, Will McCutcheon, Mehul Malik, and Nicolas Brunner,  _Genuine high-dimensional quantum steering,_Phys. Rev. Lett. 126, 200404 (2021), arXiv:2007.02718.
  * Valencia  _et al._ (2021) Natalia Herrera Valencia, Vatshal Srivastav, Saroch Leedumrongwatthanakun, Will McCutcheon, and Mehul Malik,  _Entangled ripples and twists of light: radial and azimuthal laguerre–gaussian mode entanglement,_Journal of Optics 23, 104001 (2021), arXiv:2104.04506.
  * Krenn  _et al._ (2017) Mario Krenn, Mehul Malik, Manuel Erhard, and Anton Zeillinger,  _Orbital angular momentum of photons and the entanglement of Laguerre-Gaussian modes,_Phil.Trans. R Soc. A 375, 20150442 (2017).
  * Miatto  _et al._ (2011) Filippo M Miatto, Alison M Yao, and Stephen M Barnett,  _Full characterization of the quantum spiral bandwidth of entangled biphotons,_Phys. Rev. A 83, 33816 (2011), arXiv:1011.5970.
  * Salakhutdinov  _et al._ (2012) V D Salakhutdinov, E R Eliel, and W Löffler,  _Full-Field Quantum Correlations of Spatially Entangled Photons,_Phys. Rev. Lett. 108, 173604 (2012), arXiv:1204.5758.
  * Collins  _et al._ (2002) Daniel Collins, Nicolas Gisin, Noah Linden, Serge Massar, and Sandu Popescu,  _Bell inequalities for arbitrarily high-dimensional systems,_Phys. Rev. Lett. 88, 040404 (2002), arXiv:quant-ph/0106024.
  * Salavrakos  _et al._ (2017) Alexia Salavrakos, Remigiusz Augusiak, Jordi Tura, Peter Wittek, Antonio Acín, and Stefano Pironio,  _Bell inequalities tailored to maximally entangled states,_Phys. Rev. Lett. 119, 040402 (2017), arXiv:1607.04578.
  * Iskhakov  _et al._ (2016) Timur Sh. Iskhakov, Vladyslav C Usenko, Ulrik L Andersen, Radim Filip, Maria V Chekhova, and Gerd Leuchs,  _Heralded Source of Bright Mesoscopic Sub-Poissonian Light,_Opt. Lett 41, 2149 (2016), arXiv:1511.08460.
  * Ramelow  _et al._ (2013) Sven Ramelow, Alexandra Mech, Marissa Giustina, Simon Groblacher, Witlef Wieczorek, Jörn Beyer, Adriana Lita, Brice Calkins, Thomas Gerrits, Sae Woo Nam, Anton Zeilinger, and Rupert Ursin,  _Highly efficient heralding of entangled single photons,_Optics Express 21, 6707 (2013), arXiv:1211.5059.
  * Castelletto  _et al._ (2005) S Castelletto, I P Degiovanni, and V Schettini,  _Spatial and spectral mode selection of heralded single photons from pulsed parametric down-donversion,_Optics Express 13, 6709 (2005).
  * Malik and Boyd (2014) Mehul Malik and Robert W Boyd,  _Quantum Imaging Technologies,_Riv Nuovo Cimento 37, 273 (2014).
  * Basset  _et al._ (2019) Marta Gilaberte Basset, Frank Setzpfandt, Fabian Steinlechner, Erik Beckert, Thomas Pertsch, and Markus Gräfe,  _Perspectives for Applications of Quantum Imaging,_Laser & Photonics Reviews 59, 1900097 (2019).
  * Jha  _et al._ (2008) Anand Kumar Jha, Mehul Malik, and Robert W. Boyd,  _Exploring energy-time entanglement using geometric phase,_Phys. Rev. Lett. 101, 180405 (2008), arXiv:1711.05079.
  * Kues  _et al._ (2017) Michael Kues, Christian Reimer, Piotr Roztocki, Luis Romero Cortes, Stefania Sciara, Benjamin Wetzel, Yanbing Zhang, Alfonso Cino, Sai T Chu, Brent E Little, David J Moss, Lucia Caspani, Jose Azana, and Roberto Morandotti,  _On-chip generation of high-dimensional entangled quantum states and their coherent control,_Nature 546, 622 (2017).
  * Martin  _et al._ (2017) Anthony Martin, Thiago Guerreiro, Alexey Tiranov, Sébastien Designolle, Florian Fröwis, Nicolas Brunner, Marcus Huber, and Nicolas Gisin,  _Quantifying photonic high-dimensional entanglement,_Phys. Rev. Lett. 118, 110501 (2017), arXiv:1701.03269.
  * Yang  _et al._ (2008) Zhenshan Yang, Marco Liscidini, and J. E. Sipe,  _Spontaneous parametric down-conversion in waveguides: A backward Heisenberg picture approach,_Phys. Rev. A 77, 033808 (2008).

##  Appendix A Joint-transverse-momentum-amplitude (JTMA) of Type-II SPDC with
Periodic Poling

We consider the case of Type-II SPDC (whereby the pump, signal and idler are
polarised on the $e$, $e$ and $o$ axes respectively), for periodically poled
crystals designed to achieve phase-matching at degenerate signal/idler
frequencies in the colinear configuration. To account for all these
contributions, we derive an expression for the entire bi-photon state, and
apply approximations to arrive at the JTMA stated in the main text (Eq. (1)).
Following the conventional asymptotic fields approach, the first-order
nonlinear Hamiltonian in the backward-Heisenberg picture, $H^{NL}(t)$ (see
reference Yang  _et al._ (2008) for a detailed account) takes the form,

$\begin{split}\int H^{NL}(t)dt\propto\int dt\int d^{3}\textbf{k}_{i}\int
d^{3}\textbf{k}_{s}\int d^{3}\textbf{k}_{p}\int
d^{3}\textbf{x}\exp{(\textit{i}(\textbf{k}_{s}+\textbf{k}_{i}-\textbf{k}_{p})\cdot\textbf{x}})\chi^{(2)}(\textbf{x})\\\
\times\exp{(\textit{i}(\omega(\textbf{k}_{s})+\omega(\textbf{k}_{i})-\omega(\textbf{k}_{p}))t)}\text{
}\alpha_{p}(\textbf{k}_{p})\hat{a}_{s}^{\dagger}(\textbf{k}_{s})\hat{a}_{i}^{\dagger}(\textbf{k}_{i})+h.c.\end{split}$
(17)

The temporal integral can be extended over all time, leading to the energy-
matching term, $\int dt\exp{(\textit{i}\Delta\omega
t)}=\delta(\Delta\omega)=\delta(\omega(\textbf{k}_{s})+\omega(\textbf{k}_{i})-\omega(\textbf{k}_{p}))$.
The $d^{3}\textbf{x}$ integral over a crystal of dimensions
$L_{x}^{\perp}\times L_{y}^{\perp}\times L_{z}$, situated about the origin can
be evaluated as,

$\begin{split}\int_{-L_{x}^{\perp}/2}^{L_{x}^{\perp}/2}&dx_{x}\int_{-L_{y}^{\perp}/2}^{L_{y}^{\perp}/2}dx_{y}\int_{-L_{z}/2}^{L_{z}/2}dx_{z}\exp{(\textit{i}(\textbf{k}_{s}+\textbf{k}_{i}-\textbf{k}_{p})\cdot\textbf{x}})\chi^{(2)}(\textbf{x})\\\
&=\operatorname{sinc}\left(\frac{L_{x}^{\perp}}{2}(q_{sx}+q_{ix}-q_{px})\right)\operatorname{sinc}\left(\frac{L_{y}^{\perp}}{2}(q_{sy}+q_{iy}-q_{py})\right)\operatorname{sinc}\left(\frac{L_{z}}{2}\Delta
k_{z}\right)\\\
&\approx\delta(\textbf{q}_{s}+\textbf{q}_{i}-\textbf{q}_{p})\operatorname{sinc}\left(\frac{L_{z}}{2}\Delta
k_{z}\right),\end{split}$ (18)

where we introduce notation $\textbf{q}_{n}$ for the transverse components of
$\textbf{k}_{n}$ (for $n=s,i,p$), and the longitudinal wavevector mismatch,
$\Delta k_{z}$ (to be defined), contains a contribution from the periodic
poling structure with the period $\Lambda$. The final approximation will hold
for crystals with transverse extents ($L_{x}^{\perp}$, $L_{y}^{\perp}$) that
are sufficiently large.

Using the approximations above, the integral in Eq. (17) can be written as

$\begin{split}\int H^{NL}(t)dt\propto\int d^{3}\textbf{k}_{i}\int
d^{3}\textbf{k}_{s}\int
d^{3}\textbf{k}_{p}\delta(\Delta\omega)\delta(\textbf{q}_{s}+\textbf{q}_{i}-\textbf{q}_{p})\operatorname{sinc}\left(\frac{L_{z}}{2}\Delta
k_{z}\right)\alpha_{p}(\textbf{k}_{p})\hat{a}_{s}^{\dagger}(\textbf{k}_{s})\hat{a}_{i}^{\dagger}(\textbf{k}_{i})+h.c.\end{split}$
(19)

The frequency of each field is dependent only on the modulus of the wave-
vector (and its polarisation, assuming close to colinear propagation).
Furthermore, we need to consider fields only with positive $z$ components,
which invites a change of integration variables from Cartesian components of k
to transverse wave-vector, q, and the modulus of the wave-vector,
$|\textbf{k}|$. Note that such a coordinate change results in a lack of clear
correspondence of the mode operators to their strictly orthonormal
counterparts, however this mathematical convenience remains suitable for
obtaining a description of the biphoton state. Hence, we can express the
$z$-components of k’s in terms of their modulus and transverse components,

$k_{z}=\sqrt{\lvert\textbf{k}\rvert^{2}-\lvert\textbf{q}\rvert^{2}}=|\textbf{k}|\sqrt{1-\frac{|\textbf{q}|^{2}}{|\textbf{k}|^{2}}}\simeq|\textbf{k}|\bigg{(}1-\frac{1}{2}\frac{|\textbf{q}|^{2}}{|\textbf{k}|^{2}}\\\
-\frac{1}{8}\frac{|\textbf{q}|^{4}}{|\textbf{k}|^{4}}+O(\frac{|\textbf{q}|^{6}}{|\textbf{k}|^{6}})\bigg{)},$
(20)

where we make use of a close-to-colinear approximation,
$|\textbf{q}|\ll|\textbf{k}|$. Differentiating w.r.t. $|\textbf{k}|$, we have

$dk_{z}=(1-\frac{|\textbf{q}|^{2}}{|\textbf{k}|^{2}})^{-1/2}d|\textbf{k}|\simeq(1+\frac{1}{2}\frac{|\textbf{q}|^{2}}{|\textbf{k}|^{2}}+O(\frac{|\textbf{q}|^{4}}{|\textbf{k}|^{4}}))d|\textbf{k}|,$
(21)

allowing the transformation,

$d^{3}\textbf{k}=d^{2}\textbf{q}\,dk_{z}\approx
d^{2}\textbf{q}\,d|\textbf{k}|\,(1+\frac{1}{2}\frac{|\textbf{q}|^{2}}{|\textbf{k}|^{2}}).$
(22)

The phase-matching contribution arising from the longitudinal wave-vector
mismatch can now be expressed as

$\begin{split}\Delta k_{z}&=k_{sz}+k_{iz}-k_{pz}+\tfrac{2\pi}{\Lambda}\\\
&=\sqrt{|\textbf{k}_{s}|^{2}-|\textbf{q}_{s}|^{2}}+\sqrt{|\textbf{k}_{i}|^{2}-|\textbf{q}_{i}|^{2}}-\sqrt{|\textbf{k}_{p}|^{2}-|\textbf{q}_{p}|^{2}}+\tfrac{2\pi}{\Lambda}\\\
&=|\textbf{k}_{s}|\sqrt{1-\frac{|\textbf{q}_{s}|^{2}}{|\textbf{k}_{s}|^{2}}}+|\textbf{k}_{i}|\sqrt{1-\frac{|\textbf{q}_{i}|^{2}}{|\textbf{k}_{i}|^{2}}}-|\textbf{k}_{p}|\sqrt{1-\frac{|\textbf{q}_{p}|^{2}}{|\textbf{k}_{p}|^{2}}}+\tfrac{2\pi}{\Lambda}\\\
&\approx|\textbf{k}_{s}|+|\textbf{k}_{i}|-|\textbf{k}_{p}|-\frac{1}{2}\bigg{[}\frac{|\textbf{q}_{s}|^{2}}{|\textbf{k}_{s}|}+\frac{|\textbf{q}_{i}|^{2}}{|\textbf{k}_{i}|}-\frac{|\textbf{q}_{p}|^{2}}{|\textbf{k}_{p}|}\bigg{]}+\tfrac{2\pi}{\Lambda}\end{split}$
(23)

To proceed, we make use of the dispersion relations for the various fields,
for which we make first-order expansions (no group velocity dispersion) about
the degenerate, energy-matched frequencies
$\omega_{0}:=\omega_{s0}=\omega_{i0}=\omega_{p0}/2$, so that for the pump
field,

$\begin{split}|\textbf{k}_{p}|=k^{(e)}_{p}(\omega_{p})\simeq
k^{(e)}_{p}(\omega_{p0})+(\omega_{p}-\omega_{p0})\frac{1}{v_{gp}}=k_{p0}+\frac{\tilde{\omega}_{p}}{v_{gp}}=k_{p0}+\tilde{k}_{p}\end{split}$
(24)

where $k^{(e)}_{p}(\omega_{p})$ is the dispersion relation for polarisation on
the extraordinary axis expanded about $\omega_{p0}$ (the central pump
frequency), $\tilde{\omega}_{p}=(\omega_{p}-\omega_{p0})$ is the frequency
offset, and $v_{gp}$ and $k_{p0}$ are the group velocity and wave-vector of
the pump field at $\omega_{p0}$, polarised accordingly. Similarly, for signal
and idler fields,

$\begin{split}|\textbf{k}_{s}|=k^{(e)}_{s}(\omega_{s})\simeq
k^{(e)}_{s}(\omega_{0})+(\omega_{p}-\omega_{0})\frac{1}{v_{gs}}=k_{s0}+\frac{\tilde{\omega}_{s}}{v_{gs}}=k_{s0}+\tilde{k}_{s}\\\
|\textbf{k}_{i}|=k^{(o)}_{i}(\omega_{i})\simeq
k^{(o)}_{i}(\omega_{0})+(\omega_{i}-\omega_{0})\frac{1}{v_{gi}}=k_{i0}+\frac{\tilde{\omega}_{i}}{v_{gi}}=k_{i0}+\tilde{k}_{i}\end{split}{}$
(25)

This allows us to consider the wave-vector mismatch for colinear generation
($\textbf{q}\rightarrow 0$) at degeneracy ($\tilde{\omega}\rightarrow 0$),

$\begin{split}\Delta k_{z0}:=k_{s0}+k_{i0}-k_{p0}.\end{split}$ (26)

Thus, rewriting $\Delta k_{z}$ at degeneracy as

$\Delta k_{z}=\Delta k_{z0}+\tfrac{2\pi}{\Lambda}$ (27)

For sources optimised in this regime, periodic poling of the crystal is used
to cancel this contribution, thereby achieving phase-matching ($\Delta
k_{z}=0$). Thus, we set the poling period, $\tfrac{2\pi}{\Lambda}=-\Delta
k_{z0}$.

Noting that we may write the energy-matching term as
$\delta(\Delta\omega)=v^{-1}_{gp}\delta(\tfrac{v_{gs}}{v_{gp}}\tilde{k}_{s}+\tfrac{v_{gi}}{v_{gp}}\tilde{k}_{i}-\tilde{k}_{p})$,
then imposing that the pump field may be expressed as a separable function of
q and $\tilde{k}$ so that
$\alpha_{p}(\textbf{k})\approx\alpha_{p}(\textbf{q})\tilde{\alpha}_{p}(\tilde{k})$,
and using the approximation made in Eq. (22), we can perform the integrals
$d^{3}\textbf{k}_{p}$ in Eq. (19) to arrive at the form,

$\begin{split}\int H^{NL}(t)dt&\propto\int
d^{2}\textbf{q}_{i}d\tilde{k}_{i}\int
d^{2}\textbf{q}_{s}d\tilde{k}_{s}v^{-1}_{gp}\times\bigg{(}1+\frac{1}{2}\frac{|\textbf{q}_{s}|^{2}}{(k_{s0}+\tilde{k}_{s})^{2}}\bigg{)}\bigg{(}1+\frac{1}{2}\frac{|\textbf{q}_{i}|^{2}}{(k_{i0}+\tilde{k}_{i})^{2}}\bigg{)}\bigg{(}1+\frac{1}{2}\frac{|\textbf{q}_{s}+\textbf{q}_{i}|^{2}}{(k_{p0}+\tfrac{v_{gs}}{v_{gp}}\tilde{k}_{s}+\tfrac{v_{gi}}{v_{gp}}\tilde{k}_{i})^{2}}\bigg{)}\\\
&\quad\times\text{sinc}\biggl{(}\frac{L_{z}}{2}\bigg{(}\tilde{k}_{s}+\tilde{k}_{i}-\bigg{(}\frac{v_{gs}}{v_{gp}}\tilde{k}_{s}+\frac{v_{gi}}{v_{gp}}\tilde{k}_{i}\bigg{)}-\frac{1}{2}\biggl{[}\frac{|\textbf{q}_{s}|^{2}}{k_{s0}+\tilde{k}_{s}}+\frac{|\textbf{q}_{i}|^{2}}{k_{i0}+\tilde{k}_{i}}-\frac{|\textbf{q}_{s}+\textbf{q}_{i}|^{2}}{k_{p0}+\tfrac{v_{gs}}{v_{gp}}\tilde{k}_{s}+\tfrac{v_{gi}}{v_{gp}}\tilde{k}_{i}}\biggr{]}\biggr{)}\biggr{)}\\\
&\quad\times\alpha_{p}(\textbf{q}_{s}+\textbf{q}_{i})\tilde{\alpha}_{p}(\tfrac{v_{gs}}{v_{gp}}\tilde{k}_{s}+\tfrac{v_{gi}}{v_{gp}}\tilde{k}_{i})\,\hat{a}_{s}^{\dagger}(\textbf{k}_{s})\hat{a}_{i}^{\dagger}(\textbf{k}_{i})+h.c.\end{split}$
(28)

Even for relatively broad-band fields we have $\tilde{k}\ll k_{0}$ for pump,
signal and idler, so we can approximate $\tilde{k}+k_{0}\approx k_{0}$ in the
Sinc quotients, and for close to co-linear generation
$\frac{|\textbf{q}_{i}|^{2}}{(k_{i0}+\tilde{k}_{i})^{2}}\ll 1$, so we can
neglect polynomial terms in front of the Sinc of the order
$\mathcal{O}(|\textbf{q}|^{2}/k_{0}^{2})$ and above, to obtain,

$\begin{split}\int H^{NL}(t)dt&\propto\int
d^{2}\textbf{q}_{i}d\tilde{k}_{i}\int
d^{2}\textbf{q}_{s}d\tilde{k}_{s}\textrm{
}\text{sinc}\biggl{(}\frac{L_{z}}{2}\bigg{(}\tilde{k}_{s}+\tilde{k}_{i}-\bigg{(}\frac{v_{gs}}{v_{gp}}\tilde{k}_{s}+\frac{v_{gi}}{v_{gp}}\tilde{k}_{i}\bigg{)}-\frac{1}{2}\biggl{[}\frac{|\textbf{q}_{s}|^{2}}{k_{s0}}+\frac{|\textbf{q}_{i}|^{2}}{k_{i0}}-\frac{|\textbf{q}_{s}+\textbf{q}_{i}|^{2}}{k_{p0}}\biggr{]}\biggr{)}\biggr{)}\\\
&\quad\times\alpha_{p}(\textbf{q}_{s}+\textbf{q}_{i})\tilde{\alpha}_{p}(\tfrac{v_{gs}}{v_{gp}}\tilde{k}_{s}+\tfrac{v_{gi}}{v_{gp}}\tilde{k}_{i})\,\hat{a}_{s}^{\dagger}(\textbf{k}_{s})\hat{a}_{i}^{\dagger}(\textbf{k}_{i})+h.c.\end{split}$
(29)

At degeneracy, where $\tilde{k}_{s}=\tilde{k}_{i}=0$, we have,

$\begin{split}\int H^{NL}(t)dt&\propto\int
d^{2}\textbf{q}_{i}d\tilde{k}_{i}\int
d^{2}\textbf{q}_{s}d\tilde{k}_{s}\textrm{
}\text{sinc}\biggl{(}\frac{L_{z}}{4}\biggl{[}\frac{|\textbf{q}_{s}+\textbf{q}_{i}|^{2}}{k_{p0}}-\frac{|\textbf{q}_{s}|^{2}}{k_{s0}}-\frac{|\textbf{q}_{i}|^{2}}{k_{i0}}\biggr{]}\biggr{)}\times\alpha_{p}(\textbf{q}_{s}+\textbf{q}_{i})\,\hat{a}_{s}^{\dagger}(\textbf{k}_{s})\hat{a}_{i}^{\dagger}(\textbf{k}_{i})+h.c.\end{split}$
(30)

To simplify further, we define a scaled transverse momenta,

$\begin{split}\tilde{\textbf{q}}_{s}&:=\left(\frac{k_{p0}}{k_{s0}}-1\right)^{\tfrac{1}{2}}\textbf{q}_{s}:=c_{s}\textbf{q}_{s}\,\end{split}$
(31)

and analogously for the idler. We write
$\varepsilon:=(1-\frac{1}{c_{s}c_{i}})$ and express the product of the Sinc
function and pump profile $\alpha_{p}$ as:

$F(\textbf{q}_{s},\textbf{q}_{i})\propto\text{sinc}\biggl{(}\frac{L_{z}}{4k_{p0}}\biggl{[}(2+\varepsilon)|\tilde{\textbf{q}}_{-}|^{2}+\varepsilon|\tilde{\textbf{q}}_{+}|^{2}\biggr{]}\biggr{)}\times\alpha_{p}({\textbf{q}}_{s}+{\textbf{q}}_{i}),$
(32)

where $|\tilde{\textbf{q}}_{\pm}|:=\tfrac{\left|c_{s}\textbf{q}_{s}\pm
c_{i}\textbf{q}_{i}\right|}{\sqrt{2}}$. We define the above expression as the
joint-transverse-momentum-amplitude (JTMA).

For $\lambda_{s}=1550$nm, we have $c_{s}=0.9677$, $c_{i}=1.0139$ and for
$\lambda_{s}=810$nm, $c_{s}=0.9979$, $c_{i}=1.047$. We approximate
$\varepsilon\approx 0$ because $\varepsilon=-0.0193$ at 1550nm and
$\varepsilon=0.043$ at 810nm. Under this approximation, the JTMA can be
simplified to:

$\begin{split}F(\textbf{q}_{s},\textbf{q}_{i})\propto\text{sinc}\biggl{(}\frac{L_{z}}{2k_{p0}}|\tilde{\textbf{q}}_{-}|^{2}\biggr{)}\times\alpha_{p}({\textbf{q}}_{s}+{\textbf{q}}_{i}).\end{split}$
(33)

We see that colinear Type-II phase-matching with periodic poling deviates
somewhat from the idealised JTMA in the main text (Eq. 1) owing to the non-
unit $c_{s}$, $c_{i}$, and the non-vanishing $\varepsilon$. However, this
discrepancy becomes increasingly benign for the collection-limited systems in
consideration in which the Sinc contribution becomes dominated instead by the
collection optics.

Rewriting the normalised JTMA for $c_{s}=c_{i}\approx 1$ (and remaining at
degeneracy $\lambda_{s}=\lambda_{i}$), we impose a Gaussian transverse pump
profile across the crystal, to obtain

$F(\textbf{q}_{s},\textbf{q}_{i})=\mathcal{N}_{1}\textrm{sinc}\biggl{(}\frac{|\textbf{q}_{s}-\textbf{q}_{i}|^{2}}{\sigma_{S}^{2}}\biggr{)}\times\textrm{exp}\biggl{(}-\frac{|\textbf{q}_{s}+\textbf{q}_{i}|^{2}}{2\sigma_{P}^{2}}\biggr{)},$
(34)

where $\sigma_{S}=\sqrt{4k_{p0}/L_{z}}$ and $\mathcal{N}_{1}$ is a
normalisation constant. Here the transverse pump momentum profile
$\alpha_{P}(\textbf{q}_{p})$ is a Gaussian, where $w_{P}$ is the pump radius
in position space, so that $\sigma_{P}=\frac{\sqrt{2}}{w_{P}}$.

Transforming to the Fourier plane with 2$f$-lens configuration, we have
($\textbf{q}\rightarrow\frac{2\pi\textbf{x}}{f\lambda}$),

$\begin{split}F\left(\textbf{x}_{s},\textbf{x}_{i}\right)\propto\text{sinc}\biggl{(}\frac{1}{\sigma_{S}^{2}}\left|\frac{2\pi}{f\lambda_{s}}{\textbf{x}}_{s}-\frac{2\pi}{f\lambda_{i}}{\textbf{x}}_{i}\right|^{2}\biggr{)}\alpha_{p}\biggl{(}-\frac{1}{2\sigma_{P}^{2}}\left|\frac{2\pi}{f\lambda_{s}}{\textbf{x}}_{s}+\frac{2\pi}{f\lambda_{i}}{\textbf{x}}_{i}\right|^{2}\biggr{)}.\end{split}$
(35)

##  Appendix B Collected JTMA to Collection Limited JTMA

Figure 7: Comparison between product of Sinc and Gaussian term
($f(|\textbf{q}_{-}|)$) and the collection Gaussian envelope
($g(|\textbf{q}_{-}|)$): For $\sigma_{S}\gtrapprox\sqrt{2}\sigma_{C}$, the
product of the Sinc term and Gaussian envelope can be approximated by a
Gaussian with width $\sigma_{C}$

We introduce the effect of the collection mode in
$F(\textbf{q}_{s},\textbf{q}_{i})$ by considering collected JTMA
$G(\textbf{q}_{+},\textbf{q}_{-})$ given in Eq.
(LABEL:eq:collectedJTMASumDiff). To approximate the collected JTMA to
collection limited JTMA, we investigate the approximation of the product of
Sinc and the collection Gaussian envelope:

$\displaystyle\begin{split}f(|\textbf{q}_{-}|)=\mathcal{N}\exp\biggl{\\{}\frac{-|\textbf{q}_{-}|^{2}}{\sigma_{C}^{2}}\biggr{\\}}\textrm{sinc}\biggl{(}\frac{2|\textbf{q}_{-}|^{2}}{\sigma_{S}^{2}}\biggr{)}\,,\end{split}$
(36)

with $\mathcal{N}$ a normalisation factor, by the Gaussian function,

$\displaystyle\begin{split}g(|\textbf{q}_{-}|)=\mathcal{N}^{\prime}\exp\biggl{\\{}\frac{-|\textbf{q}_{-}|^{2}}{\sigma_{C}^{2}}\biggr{\\}}\,.\end{split}$
(37)

These terms have inner product,

$\displaystyle\begin{split}\int
d^{2}\textbf{q}_{-}\mathcal{N}\exp\biggl{\\{}\frac{-|\textbf{q}_{-}|^{2}}{\sigma_{C}^{2}}\biggr{\\}}\textrm{sinc}\biggl{(}\frac{2|\textbf{q}_{-}|^{2}}{\sigma_{S}^{2}}\biggr{)}\mathcal{N}^{\prime}\exp\biggl{\\{}\frac{-|\textbf{q}_{-}|^{2}}{\sigma_{C}^{2}}\biggr{\\}}\geq
0.99\end{split}$ (38)

for $\sigma_{S}>1.4161\sigma_{C}\gtrapprox\sqrt{2}\sigma_{C}$. Slightly looser
approximation sees an inner product of 0.95 achieved at
$\sigma_{S}>\tfrac{3}{2\sqrt{2}}\sigma_{C}$.

Furthermore, in the case of calculating the singles count rates in Eq. (15),
in which only the herald photon imparts a collection envelope we have a mildly
stronger condition for the approximation to hold,

$\displaystyle\begin{split}\mathcal{N}\exp\biggl{\\{}\frac{-|\textbf{q}_{-}|^{2}}{2\sigma_{C}^{2}}\biggr{\\}}\textrm{sinc}\biggl{(}\frac{2|\textbf{q}_{-}|^{2}}{\sigma_{S}^{2}}\biggr{)}\approx\mathcal{N}^{\prime}\exp\biggl{\\{}\frac{-|\textbf{q}_{-}|^{2}}{2\sigma_{C}^{2}}\biggr{\\}}\,,\end{split}$
(39)

leading to an additional factor of $\sqrt{2}$ arising in the equivalent
conditions above, ie. $\sigma_{S}\gtrapprox 2\sigma_{C}$ implies greater than
0.99 inner product.

##  Appendix C Calculations of $\text{Pr}(a,-a)$ and $\text{Pr}(a,a)$

Figure 8: Illustration of regions in collected JTMA to approximate
$\text{Pr}(a,-a)$ and $\text{Pr}(a,a)$: (a) For $\text{Pr}(a,-a)$, the shaded
regions $G^{CL}_{++}$ and $G^{CL}_{--}$ are equal due to the symmetry about
$q_{+x}=0$ (green, dashed). (b) Similarly, for $\text{Pr}(a,a)$, the regions
$G^{CL}_{-+}$ and $G^{CL}_{+-}$ are equal due to symmetry across $q_{-x}=0$.
In (c) $G^{CL}_{+-}=\tilde{R}_{1}+\tilde{R}_{2}+\tilde{R}_{3}$, which is
calculated from various integrals (See Eq. (46) and (47))

For convenience, the expression of two photon coincidence probability in Eq.
(6) can be written in the reduced form

$\displaystyle\text{Pr}(a_{s},a_{i})=\bigl{|}G_{++}+G_{--}-G_{-+}-G_{+-}\bigr{|}^{2}.$
(40)

For the signal photon, the subscript $+$ represents the integration performed
in the interval $[a_{s},\infty)$ and $-$ corresponds to the integration in the
interval $(-\infty,a_{s}]$ (same for idler).

To calculate $\text{Pr}(a,-a)$, we expand the integral for $\text{Pr}(a,-a)$
in terms of integral by regions $G^{CL}_{++}$, $G^{CL}_{--}$, $G^{CL}_{-+}$,
and $G^{CL}_{+-}$ (see Fig. 8a). From symmetry, we have $G^{CL}_{--}$ =
$G^{CL}_{++}$, and with Eq. (40) we write $\text{Pr}(a,-a)$ as,

$\text{Pr}(a,-a)=|N-4G^{CL}_{--}|^{2},$ (41)

where
$N=G^{CL}_{--}+G^{CL}_{++}+G^{CL}_{-+}+G^{CL}_{+-}=\pi\sqrt{2\pi}\mathcal{N}_{1}\frac{\sigma_{C}^{3}\sigma_{P}^{2}}{\sigma_{C}^{2}+\sigma_{P}^{2}}$.
Assuming $\sigma_{P}\ll\sigma_{C}$ in the bow-tie region formed by
$G^{CL}_{--}$ and $G^{CL}_{++}$, the factor
$\exp\bigl{\\{}-\frac{(q_{sx}-q_{ix})^{2}}{2\sigma_{C}^{2}}\bigr{\\}}$ is
approximated by $\exp\big{(}-\frac{2a^{2}}{\sigma_{C}^{2}}\big{)}$, which
reduces the expression in Eq. (41) to

$\displaystyle\text{Pr}(a,-a)$
$\displaystyle\approx\bigg{|}N-4\int_{-\infty}^{a}\int_{-\infty}^{-a}\mathcal{G}^{DG}(q_{sx},q_{ix};a)\text{d}q_{sx}\text{d}q_{ix}\bigg{|}^{2}$
(42)

where

$\mathcal{G}^{DG}(q_{sx},q_{ix};a)\propto\exp\bigg{(}-\frac{(q_{sx}+q_{ix})^{2}}{2\tilde{\sigma}_{P}^{2}}\bigg{)}\exp\bigg{(}-\frac{2a^{2}}{\sigma_{C}^{2}}\bigg{)},$
(43)

for which there exist a simple analytic solution,

$\begin{split}\Pr(a,-a)&\approx\left|N-N^{\prime}\exp\bigg{(}-\frac{2a^{2}}{\sigma_{C}^{2}}\bigg{)}\right|^{2}\\\
N^{\prime}&:=\sqrt{2\pi}\mathcal{N}_{1}\frac{\pi\sigma_{C}^{3}\sigma_{P}^{3}}{\left(\sigma_{C}^{2}+\sigma_{P}^{2}\right)^{3/2}}\end{split}$
(44)

given in Eq. (9). One can write the expression for the visibility $V$ in terms
of $N$ and $N^{\prime}$ as

$V=\frac{|N|^{2}-|N-N^{\prime}|^{2}}{|N|^{2}+|N-N^{\prime}|^{2}}\approx\frac{\sigma_{C}^{2}-|\sigma_{C}-\sigma_{P}|^{2}}{\sigma_{C}^{2}+|\sigma_{C}-\sigma_{P}|^{2}}\quad\textrm{
for }\sigma_{P}\ll\sigma_{C}.$ (45)

For a fixed value of $\sigma_{C}$, obtained from the fit of $\Pr(a,-a)$, one
can also get the pump width parameter $\sigma_{P}$ from the Eq. (45), given
the visibility of the experimental data.

Therefore, for $\text{Pr}(a,a)$, we follow the same steps as $\text{Pr}(a,-a)$
of writing the complete integral as the sum of integrals over several regions
(see Fig. 8b). Here, the integrals $G^{CL}_{-+}$ and $G^{CL}_{+-}$ are equal
($G^{CL}_{+-}=G^{CL}_{-+}\equiv\tilde{R}$). We aim to partition the space into
areas which have closed-form integral solutions and a bow-tie region which can
be well approximated under the previous reasoning. We expand
$\tilde{R}=\tilde{E}+F-(\tilde{F}_{1}+\tilde{F}_{2})$, where the contributions
are given as,

$\begin{split}\tilde{E}&=\iint_{\mathcal{L}}G^{CL}(q_{+x},q_{-x})\text{d}q_{+x}\text{d}q_{-x},\qquad
F=\frac{1}{2}\int_{a}^{3a}\int_{-a}^{a}\mathcal{G}^{DG}(q_{sx},q_{ix};a)\text{d}q_{sx}\text{d}q_{ix},\\\
\tilde{F}_{1}&=\frac{1}{2}\int^{a}_{-\infty}\int^{-a}_{-\infty}\mathcal{G}^{DG}(q_{sx},q_{ix};a)\text{d}q_{sx}\text{d}q_{ix},\qquad\tilde{F}_{2}=\frac{1}{2}\int^{\infty}_{3a}\int^{\infty}_{a}\mathcal{G}^{DG}(q_{sx},q_{ix};a)\text{d}q_{sx}\text{d}q_{ix},\end{split}$
(46)

where $G^{CL}(q_{+x},q_{-x})$ is integrated over the region
$\mathcal{L}\in(q_{-x}\leq\sqrt{2}x)$. The rationale behind this expansion can
be seen when we further express each terms in the sum of $\tilde{R}$ into the
regions shown in Fig. 8c,

$\begin{split}\tilde{E}&=\tilde{R}_{2}+\tilde{R}_{3}+\tilde{R}_{4}+\tilde{R}_{6},\qquad
F=\frac{1}{2}(\tilde{R}_{1}+\tilde{R}_{2})=\tilde{R}_{1},\\\
\tilde{F}_{1}&=\frac{1}{2}(\tilde{R}_{5}+\tilde{R}_{4})=\tilde{R}_{4},\qquad\tilde{F}_{2}=\frac{1}{2}(\tilde{R}_{7}+\tilde{R}_{6})=\tilde{R}_{6},\\\
\tilde{R}&=\tilde{E}+F-(\tilde{F}_{1}+\tilde{F}_{2})=\tilde{R}_{1}+\tilde{R}_{2}+\tilde{R}_{3}.\end{split}$
(47)

We solve for $N-4\tilde{R}$ with the help of integral tables given for error
functions and the fact that $\text{Pr}(a,a)$ is an even function, and we get,

$N-4\tilde{R}=\mathcal{A}\biggl{[}\sqrt{2}\sigma_{P}\text{
}e^{-2a^{2}\big{(}\frac{1}{\tilde{\sigma}_{P}^{2}}+\frac{1}{\sigma_{C}^{2}}\big{)}}-2\sqrt{\pi}\text{
}|a|\text{ }e^{\frac{-2a^{2}}{\sigma_{C}^{2}}}\times\text{
erfc}\bigg{(}\frac{\sqrt{2}|a|}{\tilde{\sigma}_{P}}\bigg{)}-\frac{\pi\sigma_{C}}{2\sqrt{2}}\bigg{\\{}1-2\text{erf}\bigg{(}\frac{\sqrt{2}|a|}{\sigma_{C}}\bigg{)}\bigg{\\}}\biggr{]}.$
(48)

To fit $\text{Pr}(a,-a)$ and $\text{Pr}(a,a)$, one requires the estimated
location of origin from the experiment data, which we can obtain from

$\textrm{Pr}(a,-\infty)\propto\lvert\textrm{erf}\bigg{(}\frac{\sqrt{2}(a-a_{s})}{\sigma_{C}}\bigg{)}\rvert^{2},\textrm{Pr}(-\infty,a)\propto\lvert\textrm{erf}\bigg{(}\frac{\sqrt{2}(a-a_{i})}{\sigma_{C}}\bigg{)}\rvert^{2},$
(49)

where $\text{Pr}(a,-\infty)$ is minimum for $a=a_{s}$ ($\text{Pr}(-\infty,a)$
is minimum for $a=a_{i}$). Hence, the location of origin coincides with the
location of the respective minimas ($q_{sx}=a_{s},q_{ix}=a_{i}$).

##  Appendix D Mode Basis Design and Heralding Efficiency Details

Figure 9: Heralding of the idler photon state by projection onto a signal
photon. The signal photon is projected onto the measurement mode,
$M_{s}(\textbf{q}_{s})=\Phi_{s}(\textbf{q}_{s})\mathcal{C}(\textbf{q}_{s}|\sigma_{C})$
(turquoise, left axis) determined by the collection mode, $\mathcal{C}_{s}$
(red, left axis) and a pixel hologram $\Phi_{s}$, of diameter $d_{1}$. The
resultant heralded idler photon state, $\psi^{\Phi_{s}}_{i}(\textbf{q}_{i})$,
(blue, bottom), is determined by the projection of $M_{s}(\textbf{q}_{s})$
onto the JTMA (contours), see Eq. (56), which owing to the relative size of
$\sigma_{C}$, can be approximated by the correlation term (density plot) (see
Appendix B). The finite width of the JTMA leads to contributions arising from
regions $F_{0}$ and $F_{2}$ that result in a heralded photon with a width
larger than $d_{1}$. The idler measurement mode,
$M_{i}(\textbf{q}_{i})=\Phi_{i}(\textbf{q}_{i})\mathcal{C}(\textbf{q}_{i}|\sigma_{C})$
(blue region, lower plot axes), is choosen to have an increased pixel diameter
$d_{2}>d_{1}$, leading to an increased coincidence probability given by the
inner product, $\langle M_{i}^{\Phi_{s}},\psi^{\Phi_{s}}_{i}\rangle$ (Eq.
(57)), whilst the (inclusive) singles probability of the signal detection is
$\langle\psi^{\Phi_{s}}_{i},\psi^{\Phi_{s}}_{i}\rangle$ (Eq. (56)) independent
of $d_{2}$. Hence, an increased $d_{2}$ leads to an increase in the one-sided
heralding efficiency (Eq. (16)).

To produce a postselected state experimentally, we project holograms
$\\{\Phi_{s}(\textbf{q}_{s})\\}_{a}$ and $\\{\Phi_{i}(\textbf{q}_{i})\\}_{b}$
on the SLMs with an additional constraint $|\Phi_{s}(\textbf{q}_{s})|\leq 1$.
This results in a state characterised by the complex elements, $T_{ab}$, given
by,

$T_{ab}=\int d^{2}\textbf{q}_{s}\int
d^{2}\textbf{q}_{i}\Phi_{s}^{a}(\textbf{q}_{s})\Phi_{i}^{b}(\textbf{q}_{i})G(\textbf{q}_{s},\textbf{q}_{i}).$
(50)

describing the subnormalised postselected state,
$\left|\right.\\!{\psi^{(PS)}}\\!\left.\right\rangle$, which is defined on the
corresponding normalised discrete mode basis
$\\{\left|\right.\\!{\hat{e}^{a}_{s}}\\!\left.\right\rangle\\}_{a}$,

$\begin{split}\left|\right.\\!{\psi^{(PS)}}\\!\left.\right\rangle&:=\sum_{ab}T_{ab}\left|\right.\\!{\hat{e}_{s}^{a}}\\!\left.\right\rangle\left|\right.\\!{\hat{e}_{i}^{b}}\\!\left.\right\rangle,\end{split}$
(51)

where, to ensure
$\\{\left|\right.\\!{\hat{e}^{a}_{s}}\\!\left.\right\rangle\\}_{a}$ are
normalised, we write

$|\langle{\hat{e}_{s}^{a}}|{\hat{e}_{s}^{a}}\rangle|^{2}=1=N^{a}_{s}\bigg{|}\int
d^{2}\textbf{q}_{s}\Phi_{s}^{a}(\textbf{q}_{s})\mathcal{C}(\textbf{q}_{s}|\sigma_{C})\bigg{|}^{2}=N^{a}_{s}\bigg{|}\int
d^{2}\textbf{q}_{s}M^{a}_{s}(\textbf{q}_{s})\bigg{|}^{2}$ (52)

with $N_{s}^{a}$ as normalisation factors (similar for idler). We construct
the holograms so that
$\left|\right.\\!{\hat{e}_{n}^{a}}\\!\left.\right\rangle$ ($n=s,i$) are
orthogonal and form the discrete mode basis, which can easily be achieved by
for instance making $\\{\Phi_{s}(\textbf{q}_{s})\\}_{a}$ disjoint (pixels).
The resultant state can be understood as the full biphoton state filtered
through the collection mode apertures defined by $\hat{L}_{s}$ and
$\hat{L}_{i}$,

$\displaystyle\begin{split}\hat{L}_{s}&:=\sum_{a}\frac{1}{\sqrt{N^{a}_{s}}}|{\hat{e}^{a}_{s}}\rangle\langle{\hat{e}^{a}_{s}}|\\\
\left|\right.\\!{\psi^{(PS)}}\\!\left.\right\rangle&:=\hat{L}_{s}\hat{L}_{i}\left|\right.\\!{\psi^{(bi)}}\\!\left.\right\rangle.\\\
\end{split}$ (53)

When measuring arbitrary vectors in this subspace, for instance
$\left|\right.\\!{\vec{v}^{s}}\\!\left.\right\rangle=\sum_{a}v^{s}_{a}\left|\right.\\!{\hat{e}^{a}}\\!\left.\right\rangle$,
with $|\langle\vec{v}^{s}|\vec{v}^{s}\rangle|=1$, one constructs the holograms
as

$\begin{split}\Phi^{\vec{v}}_{s}(\textbf{q}_{s})=A^{\vec{v}}\sum_{a}v_{a}\Phi^{a}_{s}(\textbf{q}_{s})\end{split}$
(54)

with $A^{\vec{v}}$ chosen so that
$\max_{\textbf{q}}|\Phi^{\vec{v}}_{s}(\textbf{q})|\leq 1$. These holograms
result in the measurement statistics,

$\begin{split}\Pr(\vec{v}^{s},\vec{v}^{i})&=\left(A^{\vec{v}^{s}}A^{\vec{v}^{i}}\right)^{2}\left|\left\langle\right.\\!{\vec{v}^{s}}\\!\left.\right|\left\langle\right.\\!{\vec{v}^{i}}\\!\left.\right|\left|\right.\\!{\psi^{(PS)}}\\!\left.\right\rangle\right|^{2}\\\
\end{split}$ (55)

The choice of $A^{\vec{v}}$ results in an effective change in the
postselection probability. Similarly, probability of obtaining a (inclusive)
single signal photon when measuring these states is

$\begin{split}\psi^{\Phi_{s}}_{i}(\textbf{q}_{i})&:=\int
d^{2}\textbf{q}_{s}\Phi^{\vec{v}^{s}}_{s}(\textbf{q}_{s})\mathcal{C}(\textbf{q}_{s}|\sigma_{C})F(\textbf{q}_{s},\textbf{q}_{i})\\\
\Pr(\vec{v}^{s})&=\int
d^{2}\textbf{q}_{i}\left|\psi^{\Phi_{s}}_{i}(\textbf{q}_{i})\right|^{2}\\\
&=\left(A^{\vec{v}^{s}}\right)^{2}\int
d^{2}\textbf{q}_{i}\left|\left\langle\right.\\!{\vec{v}^{s}}\\!\left.\right|\left\langle\right.\\!{\textbf{q}_{i}}\\!\left.\right|\hat{L}_{s}\left|\right.\\!{\psi^{bi}}\\!\left.\right\rangle\right|^{2}\\\
\end{split}$ (56)

with $\psi^{\Phi_{s}}_{i}(\textbf{q}_{i})$ describing the pure heralded idler
photon state after heralding with $\Phi^{\vec{v}^{s}}_{s}$ on the signal. The
coincidence probability can be written similarly in terms of this heralded
idler photon state,

$\begin{split}\Pr(\vec{v}^{s},\vec{v}^{i})&=\left|\int
d^{2}\textbf{q}_{i}M^{\Phi_{s}}_{i}(\textbf{q}_{i})\psi^{\Phi_{s}}_{i}(\textbf{q}_{i})\right|^{2}\\\
\end{split}$ (57)

The one-sided heralding efficiency is given by

$\begin{split}\eta^{s\rightarrow
i}=\frac{\Pr(\vec{v}^{s},\vec{v}^{i})}{\Pr(\vec{v}^{s})}\end{split}$ (58)

which can be optimised by ensuring the idler mode basis has large overlap with
the heralded photons from the signal, as well as choosing measurements for
which $A^{\vec{v}_{i}}$ may be large. In Fig. 9 we depict the increased
heralding efficiency associated to increasing the size of one party’s pixel
relative to the other.