# High-dimensional maximally entangled photon pairs in parametric down-
conversion

Richard Bernecker<EMAIL_ADDRESS>Theoretisch-Physikalisches
Institut, Friedrich-Schiller-Universität Jena, Max-Wien-Platz 1, D-07743 Jena,
Germany  Helmholtz-Institut Jena, Fröbelstieg 3, D-07743 Jena, Germany
Baghdasar Baghdasaryan  Institut für Angewandte Physik, Friedrich-Schiller-
Universität Jena, Albert-Einstein-Str. 6, D-07745 Jena, Germany  Stephan
Fritzsche  Theoretisch-Physikalisches Institut, Friedrich-Schiller-Universität
Jena, Max-Wien-Platz 1, D-07743 Jena, Germany  Helmholtz-Institut Jena,
Fröbelstieg 3, D-07743 Jena, Germany

###### Abstract

Photon pairs generated from spontaneous parametric down-conversion are a well-
established method to realize entangled bipartite photonic systems. Laguerre-
Gaussian modes, which carry orbital angular momentum (OAM), are commonly
exploited to engineer high-dimensional entangled quantum states
experimentally. For Hilbert spaces with dimension $d>2$, maximally entangled
states (MES) help to improve the capacity and security of quantum
communication protocols, among several other promising features. However, the
direct generation of MES in well-defined high-dimensional subspaces of the
infinite OAM basis has remained a challenge. Here, we formalize how the
spatial distribution of the pump beam and the nonlinear profile of the crystal
can be utilized to generate MES without additional spatial filtering of OAM
modes within a subspace. We illustrate our approach with maximally entangled
qutrits ($d=3$) and ququints ($d=5$).

††preprint: APS/123-QED

##  I Introduction

The generation and control of entangled photons have evolved into an
indispensable tool for the extensive research and development of quantum
technologies in branches like quantum cryptography, quantum computation, and
quantum imaging [1, 2, 3, 4]. Great interest has focused on the entanglement
between high-dimensional quantum systems with dimensionality $d>2$, so-called
qudits [5]. In comparison to the frequently explored qubits ($d=2$), qudits
provide higher information capacities and improved security for quantum
communication protocols [6, 7, 8] or a notable increase of efficiency for
quantum-computational tasks [9, 10].

An important class of bipartite states are maximally entangled states (MESs)
like the famous Bell states for $d=2$. MESs are an ideal target for
applications in high-dimensional systems because of the complete non-
separability of the two distinct subsystems and perfect quantum correlations
within the total system [11].

High-dimensional quantum states can be realized with regard to different
degrees of freedom (DOFs) of light, such as frequency [12], photon number
[13], and temporal modes [14]. Much attention has been devoted to orbital
angular momentum (OAM) states of light [15, 16, 17] due to facile scalability
in dimensions [18, 19] and their application for long-distance quantum
communication in free space or through mode fibers [20, 21].

Spontaneous parametric down-conversion (SPDC) is widely employed to generate
entangled bipartite systems [22]. In SPDC, a photon pair or biphoton is
emitted after the interaction of a pump photon with a crystal of second-order
nonlinearity. The generated photon pairs intrinsically possess high-
dimensional entanglement in OAM, which has been demonstrated experimentally
[23, 24]. In particular, the spatial entanglement between the photon pairs is
distributed over many OAM modes, also known as spiral bandwidth [25, 26]. The
amplitudes of the OAM modes within the spiral bandwidth must be uniformly
distributed to ensure maximal entanglement.

However, the step from high-dimensional entangled states to MESs is difficult
to realize in experiments [27]. Until the present, post-selection procedures
like ”Procrustean filtering” have been applied to equalize the OAM amplitudes
of the biphoton state [28, 29, 30], but limit the number of usable photon
pairs and often require additional steps in the set-up. An open question
concerns the efficient generation of a MES from SPDC within a well-defined
subspace of the infinite OAM space.

Spatially engineered pump beams have been a popular way to shape the OAM
amplitudes of the biphoton state [31, 32]. For instance, an optimal weighted
superposition of Laguerre-Gaussian modes [33, 34, 35, 36] can equalize the
amplitude of desired OAM modes. However, this approach does not automatically
lead to the generation of MESs, due to the presence of additionally created
OAM modes through the engineered pump beam.

Another well-known technique to manipulate the biphoton state concerns the use
of tailored nonlinear crystals. Periodically poled crystals [37] with a sinc-
like phase-matching function (PMF) are favored to optimize the photon-pair
source brightness. Recently, innovative domain-engineering techniques [38, 39,
40, 41] have introduced alternative PMFs like in Gaussian shape [42, 43],
which minimize the spectral and spatial correlations between the photon pairs
without post-filtering [44, 45]. So far, customized PMF have not been
investigated for generating maximally entangled photon pairs.

In this theoretical work, we aim to combine the pump beam and crystal
engineering techniques in order to maximize the entanglement between photon
pairs. We formulate conditions for the spatial profile of the pump and the PMF
of the crystal that optimize the generation of desired high-dimensional MESs.
Our method of simultaneous pump and crystal tuning eliminates the need for
spatial post-selection in the OAM subspace.

##  II Theoretical Foundations

Spontaneous-parametric down-conversion is a nonlinear optical process in which
correlated photon pairs are created under energy conservation after a high-
intensity pump beam illuminates a crystal with second-order susceptibility
$\chi^{(2)}$. The simultaneously emitted photons of a pair are called signal
and idler. The momentum conservation
$\bm{k}_{\operatorname{\mathrm{p}}}=\bm{k}_{\operatorname{\mathrm{s}}}+\bm{k}_{\operatorname{\mathrm{i}}}$
ensures the constructive interference between the propagating waves inside the
crystal. This wave-vector equation with $\bm{k}_{j}$ defines the so-called
phase-matching condition, where
$j\in\\{\operatorname{\mathrm{p}},\operatorname{\mathrm{s}},\operatorname{\mathrm{i}}\\}$
refer to pump, signal, and idler, respectively.

###  II.1 Geometry and assumptions

In our study, we follow typical quantum optical assumptions for the SPDC
process to obtain the bipartite quantum state of signal and idler, the so-
called biphoton state.

Since the intensity of the pump beam exceeds the photon pair intensity several
orders of magnitude, we assume the pump as undepleted and neglect further
quantum effects. We fix the phase-matching relation as type II-SPDC [46], such
that signal and idler have orthogonal polarization. We choose the pump beam to
propagate along the $z$-axis focused in the center of the crystal at $z=0$.
The created signal and idler photons are assumed to propagate almost parallel
(quasi-collinear) to the pump direction. Since most optical elements only
support paraxial waves, we apply the paraxial approximation
$|\bm{q}|\ll|\bm{k}|$ to separate the wave vector $\bm{k}$ into a transverse
momentum vector $\bm{q}$ and the longitudinal component

$k_{z}=\sqrt{k(\omega)^{2}-|\bm{q}|^{2}}\approx
k(\omega)-\frac{|\bm{q}|^{2}}{k(\omega)},$ (1)

where the wave number is given by $k=\frac{n\omega}{c}$. In this regime
$\bm{q}$ is referred to as the spatial and $\omega$ as the spectral DOF.
Furthermore, when the transverse size of the crystal is large, compared to the
pump beam waist, the conservation of the transverse momentum
$\bm{q}_{\operatorname{\mathrm{p}}}=\bm{q}_{\operatorname{\mathrm{s}}}+\bm{q}_{\operatorname{\mathrm{i}}}$
is fulfilled. In our framework, all photons are assumed to be monochromatic
and to fulfill the energy conservation
$\omega_{\operatorname{\mathrm{p}}}=\omega_{\operatorname{\mathrm{s}}}+\omega_{\operatorname{\mathrm{i}}}$
exactly. This condition is experimentally enforced with frequency filters in
the set-up and is known as the narrowband regime. These approximations account
for a broad class of experimental scenarios that are based on the entanglement
of OAM light modes [15, 47, 48, 18, 49].

Under the mentioned assumptions and approximations, the biphoton state of a
signal and idler pair with well-defined polarization and frequencies in
momentum representation reads [47, 50]

$\displaystyle\ket{\psi}=\mathcal{N}$ $\displaystyle\iint
d\bm{q}_{\operatorname{\mathrm{s}}}\>d\bm{q}_{\operatorname{\mathrm{i}}}\>V_{\operatorname{\mathrm{p}}}(\bm{q}_{\operatorname{\mathrm{s}}}+\bm{q}_{\operatorname{\mathrm{i}}})\>\Phi_{\text{PM}}(\Delta
k_{z})$ (2)
$\displaystyle\times\>\hat{a}_{\operatorname{\mathrm{s}}}^{\dagger}(\bm{q}_{\operatorname{\mathrm{s}}})\>\otimes\>\hat{a}_{\operatorname{\mathrm{i}}}^{\dagger}(\bm{q}_{\operatorname{\mathrm{i}}})\ket{\text{vac}},$

where $\mathcal{N}$ is a normalization constant and $\Delta
k_{z}=k_{\operatorname{\mathrm{p}},z}-k_{\operatorname{\mathrm{s}},z}-k_{\operatorname{\mathrm{i}},z}$
the longitudinal wave vector mismatch. Evidently, the state is obtained by the
integration over all possible spatial states of signal and idler generated
from the vacuum by creation operators $\hat{a}^{\dagger}(\bm{q})$.

The angular spectrum
$V_{\operatorname{\mathrm{p}}}(\bm{q}_{\operatorname{\mathrm{p}}})$ gives rise
to the transverse spatial distribution of the pump beam, whereas the phase-
matching function (PMF) $\Phi_{\text{PM}}(\Delta k_{z})$ depends on the
crystal structure. The phase mismatch along the $z$-axis follows from Eq. (1)
as

$\displaystyle\Delta k_{z}$
$\displaystyle=k_{\operatorname{\mathrm{p}}}(\omega_{\operatorname{\mathrm{s}}}+\omega_{\operatorname{\mathrm{i}}})-k_{\operatorname{\mathrm{s}}}(\omega_{\operatorname{\mathrm{s}}})-k_{\operatorname{\mathrm{i}}}(\omega_{\operatorname{\mathrm{i}}})$
(3)
$\displaystyle-\frac{|\bm{q}_{\operatorname{\mathrm{s}}}+\bm{q}_{\operatorname{\mathrm{i}}}|^{2}}{k_{\operatorname{\mathrm{p}}}(\omega_{\operatorname{\mathrm{s}}}+\omega_{\operatorname{\mathrm{i}}})}-\frac{|\bm{q}_{\operatorname{\mathrm{s}}}|^{2}}{k_{\operatorname{\mathrm{s}}}(\omega_{\operatorname{\mathrm{s}}})}-\frac{|\bm{q}_{\operatorname{\mathrm{i}}}|^{2}}{k_{\operatorname{\mathrm{i}}}(\omega_{\operatorname{\mathrm{i}}})}.$

The minimization of $\Delta k_{z}$ guarantees an efficient down-conversion
process and is realized by techniques like angle tuning in birefringent
crystals [51] or by quasi-phase-matched nonlinear crystals [37]. The
expression of the PMF in Eq. (2) is similar to a Fourier transform of the
nonlinear susceptibility profile $\chi^{(2)}(z)$ [43, 41], which we assume to
be a general function along the $z$-axis (i.e. an one-dimensional profile):

$\Phi_{\text{PM}}(\Delta
k_{z})=\int_{-\frac{L}{2}}^{\frac{L}{2}}dz\>\chi^{(2)}(z)\>\mathrm{e}^{i\Delta
k_{z}z},$ (4)

where $L$ is the length of the crystal.

###  II.2 Decomposition in LG basis

The discussion on maximizing the entanglement in OAM by shaping the pump
requires a general spatial profile for the pump beam. Within the paraxial
regime, any transverse profile can be decomposed into the Laguerre Gaussian
(LG) basis since they form a complete and orthogonal set of optical modes.
Therefore, we initially model the pump beam as an LG mode

$V_{\operatorname{\mathrm{p}}}(\bm{q}_{\operatorname{\mathrm{p}}})=\text{LG}^{\ell_{\operatorname{\mathrm{p}}}}_{p_{\operatorname{\mathrm{p}}}}(\bm{q}_{\operatorname{\mathrm{p}}},w_{\operatorname{\mathrm{p}}}),$
(5)

and later generalize it to a superposition of LG beams. The choice of LG modes
is particularly advantageous because they are eigenstates of the orbital
angular momentum operator, enabling easy control over the OAM amplitudes of
the biphoton state. LG beams used for SPDC are mostly generated with hologram-
displaying spatial light modulators (SLMs) [52].

The angular spectrum of a LG mode in the momentum space at $z=0$ is given by

$\displaystyle\text{LG}^{\ell}_{p}(\bm{q},w)$
$\displaystyle=\sqrt{\frac{w^{2}p!}{2\pi(p+|\ell|)!}}(-1)^{p+\frac{1}{2}}\left(\frac{|\bm{q}|}{\sqrt{2}w}\right)^{|\ell|}$
(6)
$\displaystyle\times\mathrm{e}^{-\frac{|\bm{q}|^{2}w^{2}}{4}}\>\text{L}_{p}^{|\ell|}\left(\frac{|\bm{q}|^{2}w^{2}}{2}\right)\mathrm{e}^{i\ell\text{Arg}(\bm{q})},$

where $\text{L}_{p}^{|\ell|}\left(\cdot\right)$ are the generalized Laguerre
polynomials and the continuous parameter $w$ is the beam waist. The radial
index $p\in\mathbb{N}_{0}$ refers to the radial nodes observed in the
transverse plane. The spiral index $\ell\in\mathbb{Z}$ characterizes the
amount of OAM (projected on the z-axis, in units of $\hbar$) each photon
carries. $|\ell|$ also dictates how often the phase changes from $-\pi$ to
$\pi$ on a closed path around the propagation axis and is associated with a
prominent spatially dependent phase structure [53].

In the next step, we discretize the continuous spatial space of signal and
idler in Eq. (2) with LG modes to investigate the OAM mode distribution of the
biphoton state. Mathematically, this corresponds to the basis expansion

$\ket{\psi}=\sum_{p_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{s}}}}\sum_{p_{\operatorname{\mathrm{i}}},\ell_{\operatorname{\mathrm{i}}}}C_{p_{\operatorname{\mathrm{s}}},p_{\operatorname{\mathrm{i}}}}^{\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}}\biggl{(}\ket{p_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{s}}}}\otimes\ket{p_{\operatorname{\mathrm{i}}},\ell_{\operatorname{\mathrm{i}}}}\biggl{)},$
(7)

where $\ket{p,\ell}=\int
d\bm{q}\>\text{LG}_{p}^{\ell}(\bm{q},w)\>\hat{a}^{\dagger}(\bm{q})\ket{\text{vac}}$
is the single-photon state representation of a LG mode. The expansion
coefficients of the LG basis
$C_{p_{\operatorname{\mathrm{s}}},p_{\operatorname{\mathrm{i}}}}^{\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}}$
for an LG pump beam are given by [50, 54]

$\displaystyle
C^{\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}}_{p_{\operatorname{\mathrm{s}}},p_{\operatorname{\mathrm{i}}}}$
$\displaystyle\bigl{(}\underbrace{p_{\operatorname{\mathrm{p}}},\ell_{\operatorname{\mathrm{p}}},w_{\operatorname{\mathrm{p}}}}_{\text{pump}};\underbrace{\Phi_{\text{PM}},L}_{\text{crystal}};\underbrace{w_{\operatorname{\mathrm{s}}},w_{\operatorname{\mathrm{i}}}}_{\text{collection}}\bigl{)}=\biggl{(}\bra{p_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{s}}}}\otimes\bra{p_{\operatorname{\mathrm{i}}},\ell_{\operatorname{\mathrm{i}}}}\biggl{)}\ket{\psi}$
(8) $\displaystyle=\mathcal{N}\iint
d\bm{q}_{\operatorname{\mathrm{s}}}\>d\bm{q}_{\operatorname{\mathrm{i}}}\>\text{LG}_{p_{\mathrm{p}}}^{\ell_{\operatorname{\mathrm{p}}}}(\bm{q}_{\operatorname{\mathrm{s}}}+\bm{q}_{\operatorname{\mathrm{i}}},w_{\operatorname{\mathrm{p}}})\>\Phi_{\text{PM}}(\Delta
k_{z})$ $\displaystyle\hskip
28.45274pt\times\left[\text{LG}_{p_{\operatorname{\mathrm{s}}}}^{\ell_{\operatorname{\mathrm{s}}}}(\bm{q}_{\operatorname{\mathrm{s}}},w_{\operatorname{\mathrm{s}}})\right]^{*}\left[\text{LG}_{p_{\operatorname{\mathrm{i}}}}^{\ell_{\operatorname{\mathrm{i}}}}(\bm{q}_{\operatorname{\mathrm{i}}},w_{\operatorname{\mathrm{i}}})\right]^{*}.$

The joint probability of a projection into the LG modes
$\ket{p_{\operatorname{\mathrm{s}}},\ell_{\mathrm{s}}}$ for signal and
$\ket{p_{\operatorname{\mathrm{i}}},\ell_{\mathrm{i}}}$ for idler is
$P_{p_{\operatorname{\mathrm{s}}},p_{\operatorname{\mathrm{i}}}}^{\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}}=|C_{p_{\operatorname{\mathrm{s}}},p_{\operatorname{\mathrm{i}}}}^{\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}}|^{2}$.
In addition to the pump beam and phase-matching inside the crystal, the
expansion amplitude depends on the collection optics used in the projection
scheme [55]. The spatial state of the signal and idler is often examined
trough mode converters [56, 49].

For instance, SLMs display computer-generated holograms that convert an
incident LG mode, based on the encoded phase profile, into the fundamental
Gaussian mode (FGM), $p=\ell=0$. The FGM can be efficiently coupled into a
single-mode fiber (SMF) and registered at a photodetector. If other orthogonal
LG modes are incident on the SLM, the same hologram produces a light mode
orthogonal to the FGM, preventing a coupling into the SMF. Through this
measurement procedure, the collected signal and idler modes possess defined
beam sizes $w_{\operatorname{\mathrm{s}}}$ and
$w_{\operatorname{\mathrm{i}}}$.

The accurate measurement of the radial part of LG modes was demonstrated to be
experimentally challenging since the transverse structure and the
orthogonality necessitate an amplitude-sensitive detection [57, 58].
Consequently, most set-ups concentrate on the OAM part for spatial
entanglement and collect only modes with
$p_{\operatorname{\mathrm{s}}}=p_{\operatorname{\mathrm{i}}}=0$ for
experimental feasibility. This corresponds to the projection

$\displaystyle\ket{\psi^{\prime}}$
$\displaystyle=\biggl{(}\ket{0,\ell_{\operatorname{\mathrm{s}}}}\otimes\ket{0,\ell_{\operatorname{\mathrm{i}}}}\biggl{)}\biggl{(}\bra{0,\ell_{\operatorname{\mathrm{s}}}}\otimes\bra{0,\ell_{\operatorname{\mathrm{i}}}}\biggl{)}\ket{\psi}$
(9)
$\displaystyle=\sum_{\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}}C^{\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}}_{0,0}\ket{\ell_{\operatorname{\mathrm{s}}}}\otimes\ket{\ell_{\operatorname{\mathrm{i}}}}:=\sum_{\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}}C^{\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}}\ket{\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}}.$

We abbreviate
$\ket{\ell_{\operatorname{\mathrm{s}}}}\otimes\ket{\ell_{\operatorname{\mathrm{i}}}}:=\ket{\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}}$
as the biphoton OAM mode, meaning signal and idler are in the radial projected
LG modes
$\ket{\ell_{\operatorname{\mathrm{s}}}}:=\ket{0,\ell_{\operatorname{\mathrm{s}}}}$
and
$\ket{\ell_{\operatorname{\mathrm{i}}}}:=\ket{0,\ell_{\operatorname{\mathrm{i}}}}$.
Thus, we define the joint collection probability of the biphoton OAM mode as
$P^{\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}}=\Bigl{|}C^{\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}}\Bigl{|}^{2}$.
The probability distribution in terms of the discrete integers
$\ell_{\operatorname{\mathrm{s}}}$ and $\ell_{\operatorname{\mathrm{i}}}$
describes the OAM spectrum. Theoretical and experimental studies [23, 59]
showed, that the OAM conservation law
$\ell_{\operatorname{\mathrm{p}}}=\ell_{\operatorname{\mathrm{s}}}+\ell_{\operatorname{\mathrm{i}}}$
is fulfilled in quasi-collinear regimes. All generated biphoton OAM modes
$\ket{\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}}$ obey
this conservation rule but have in general different probabilities
$P^{\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}}$.

Limiting the sum in Eq. (9) to a finite cut-off provides a practical setting
for the experimental implementation of OAM qudits. Most applications utilize
only modes projected into a specific subspace, representing a fraction of the
infinite-dimensional Hilbert space. In this analysis, we also restrict
ourselves to equal subspaces for the signal and idler, i.e., we investigate
the single-photon Hilbert spaces
$\mathcal{H}_{\operatorname{\mathrm{s}}}=\mathcal{H}_{\operatorname{\mathrm{i}}}=\mathbb{C}^{d}$
of dimension $d$. The total bipartite system is described in a two-photon
Hilbert space formed by the tensor product
$\mathcal{H}_{\operatorname{\mathrm{s}}}\otimes\mathcal{H}_{\operatorname{\mathrm{i}}}$
with dimensionality $d^{2}$.

Figure 1:  Normalized OAM spectra within the range
$\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}=-3,...,3$
for different pump beams. a) & b) Depending on the pump OAM index of the
$\text{LG}_{0}^{\ell_{\operatorname{\mathrm{p}}}}$ mode, biphoton OAM modes
$\ket{\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}}$ with
different probabilities
$P^{\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}}$ are
generated along a single anti-diagonal obeying
$\ell_{\operatorname{\mathrm{p}}}=\ell_{\operatorname{\mathrm{s}}}+\ell_{\operatorname{\mathrm{i}}}$.
c) A superposition of multiple LG modes nests several anti-diagonals in the
OAM spectrum. Here, we show an equal-weighted superposition of LG modes with
$\ell_{\operatorname{\mathrm{p}}}=-2,0,2$.

###  II.3 Maximal entanglement

A pure quantum state of a bipartite system that cannot be factored as a tensor
product of states of its local constituents is called entangled. It has to be
considered as an inseparable whole.

A particularly convenient measure of entanglement is the Schmidt number $K$
obtained by the composition of a bipartite state into its Schmidt form [60]

$\ket{\Psi}=\sum_{k=1}^{r}\lambda_{k}\ket{\operatorname{\mathrm{s}}_{k}}\otimes\ket{\operatorname{\mathrm{i}}_{k}}.$
(10)

Here, $r$ is the Schmidt rank, $\lambda_{k}$ are the Schmidt coefficients, and
$\ket{\operatorname{\mathrm{s}}_{k}},\ket{\operatorname{\mathrm{i}}_{k}}$ are
the Schmidt modes for the subsystems signal and idler, respectively. The
normalization condition is $\sum_{k}\lambda_{k}^{2}=1$. The Schmidt
decomposition represents the joint basis that maximizes the mutual dependence
of two variables in the two subsystems [61]. Because of the single sum, any
entangled state is characterized by a rank $r>1$ and allows the definition of
the Schmidt number $K=1/\sum_{k}\lambda_{k}^{4}$. A normalized state is fully
separable for $K=1$ and indicates the absence of entanglement. The more the
Schmidt coefficients are distributed in the sum of Eq. (10), the more
entangled the state is, and $K$ increases.

The Schmidt number for a state of two qudits ($\mathcal{H}=\mathbb{C}^{d}$)
has an upper bound of $K=d$, implying maximal entanglement. Maximally
entangled states (MESs) have a maximal rank $r=d$ and are characterized by
equal singular values $\lambda_{k}=1/\sqrt{d}$, leading to complete non-
separability. In the OAM basis, MESs are represented by a matrix in Eq. (9)
that is proportional to a permutation matrix, where each column and row
contains a single entry of $1/\sqrt{d}$.

##  III Quantum state engineering

We conduct our proof-of-principle investigation for degenerate, collinear type
II SPDC in the narrowband regime with
$k_{\operatorname{\mathrm{p}}}=2k_{\operatorname{\mathrm{s}}}=2k_{\operatorname{\mathrm{i}}}$.
In this setup, a
$\lambda_{\operatorname{\mathrm{p}}}=$405\text{\,}\mathrm{nm}$$ pump with a
beam waist
$w_{\operatorname{\mathrm{p}}}=$25\text{\,}\mathrm{\SIUnitSymbolMicro m}$$
produces photon pairs in a $L=$15\text{\,}\mathrm{mm}$$ long KTP crystal. The
signal and idler photons are assumed to be collected in modes with sizes
$w_{\operatorname{\mathrm{s}}}=w_{\operatorname{\mathrm{i}}}=$33\text{\,}\mathrm{\SIUnitSymbolMicro
m}$$. The experimental parameters are adapted from Ref. [34].

###  III.1 Pump Engineering

The ability to experimentally modulate the amplitude and phase of spatial
light modes offered new perspectives to engineer an arbitrary superposition of
Laguerre-Gauss (LG) modes as a pumping source for SPDC. Each LG mode in this
superposition can be manipulated independently, and based on OAM conservation
$\ell_{\operatorname{\mathrm{p}}}=\ell_{\operatorname{\mathrm{s}}}+\ell_{\operatorname{\mathrm{i}}}$,
we can predict which signal and idler modes are generated. All modes of non-
zero detection probability for a single pump OAM index are positioned along
anti-diagonal lines, like shown in Fig. 1 for
$\ell_{\operatorname{\mathrm{p}}}=0$ in a) and for
$\ell_{\operatorname{\mathrm{p}}}=1$ in b). When considering a pump
superposition of LG modes, the OAM spectrum is nested with modes along several
anti-diagonal lines.

Fig. 1 c) displays a typical visualization of the OAM spectrum for an equal-
weighted pump superposition of radial-projected LG modes with
$\ell_{\operatorname{\mathrm{p}}}=-2,0,2$, where only the most significant
contributions of OAM biphoton modes
$\ket{\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}}$ in a
range of
$\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}=-3,...,3$
are shown. Note that setting
$k_{\operatorname{\mathrm{p}}}=2k_{\operatorname{\mathrm{s}}}=2k_{\operatorname{\mathrm{i}}}$
introduces interchangeability of the OAM indices for the expansion amplitudes
$C^{\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}}=C^{\ell_{\operatorname{\mathrm{i}}},\ell_{\operatorname{\mathrm{s}}}}$,which
leads to equal probabilities
$P^{\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}}=P^{\ell_{\operatorname{\mathrm{i}}},\ell_{\operatorname{\mathrm{s}}}}$
and a symmetric OAM spectrum with respect to
$\ell_{\operatorname{\mathrm{s}}}$ and $\ell_{\operatorname{\mathrm{i}}}$.

Experimentally, we are not restricted to equal weightings, as we can encode
corresponding holograms for any superposition on the SLM. Thus, we introduce
free pump coefficients $a_{\ell_{\operatorname{\mathrm{p}}}}\in\mathbb{C}$
into the updated angular spectrum from Eq. (5):

$V_{\operatorname{\mathrm{p}}}(\bm{q}_{\operatorname{\mathrm{p}}})=\sum_{\ell_{\operatorname{\mathrm{p}}}}a_{\ell_{\operatorname{\mathrm{p}}}}\>\text{LG}_{0}^{\ell_{\operatorname{\mathrm{p}}}}(\bm{q}_{\operatorname{\mathrm{p}}},w_{\operatorname{\mathrm{p}}}).$
(11)

The OAM spectrum can be reconfigured by adjusting the values of
$a_{\ell_{\operatorname{\mathrm{p}}}}$. Suitable pump coefficients for OAM
modes of equal probability were determined using simultaneous perturbation
stochastic approximation algorithms [34]. In addition, Ref. [50] showed that
with the expression for
$C^{\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}}$, the
pump coefficients can be calculates analytically.

By inserting the LG superposition (11) into Eq. (8) and (9), we can write the
state because of OAM conservation as

$\displaystyle\ket{\psi^{\prime}}$
$\displaystyle=\sum_{\ell_{\operatorname{\mathrm{p}}},\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}}\>a_{\ell_{\operatorname{\mathrm{p}}}}\>\delta_{\ell_{\operatorname{\mathrm{p}}},\ell_{\operatorname{\mathrm{s}}}+\ell_{\operatorname{\mathrm{i}}}}\>C^{\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}}\ket{\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}}$
(12)
$\displaystyle=\sum_{\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}}\>a_{\ell_{\operatorname{\mathrm{s}}}+\ell_{\operatorname{\mathrm{i}}}}\>C^{\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}}\ket{\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}}.$

As an example, we study the generation of the MES
$\ket{\Psi_{1}}=\frac{1}{\sqrt{3}}(\ket{-1,-1}+\ket{0,0}+\ket{1,1})$ like in
Refs. [34, 35]. We can represent the modified amplitudes of the state (12) in
a matrix form with matrix elements
$A_{\ell_{s},\ell_{i}}=a_{\ell_{\operatorname{\mathrm{s}}}+\ell_{\operatorname{\mathrm{i}}}}\>C^{\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}}$
and compare it with a similar matrix for the state $\ket{\Psi_{1}}$. If we
restrict the summations in Eq. (12) to OAM indices
$\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}=-1,0,1$ ,
we end up with matrices:

$\begin{pmatrix}a_{-2}\>C^{-1,-1}&a_{-1}\>C^{0,-1}&a_{0}\>C^{1,-1}\\\
a_{-1}\>C^{-1,0}&a_{0}\>C^{0,0}&a_{1}\>C^{1,0}\\\
a_{0}\>C^{-1,1}&a_{1}\>C^{0,1}&a_{2}\>C^{1,1}\\\
\end{pmatrix}\overset{!}{=}\frac{1}{\sqrt{3}}\begin{pmatrix}1&0&0\\\ 0&1&0\\\
0&0&1\\\ \end{pmatrix}.\\\ $

The comparison of coefficients directly indicates $a_{-1}=a_{1}=0$ as we
clearly do not intend to generate modes with
$\ell_{\operatorname{\mathrm{s}}}+\ell_{\operatorname{\mathrm{i}}}=\pm 1$.
Furthermore, it follows $a_{-2}\propto 1/C_{-1,-1},a_{0}\propto
1/C_{0,0},a_{2}\propto 1/C_{1,1}$.

The OAM spectrum with the pump coefficients determined above can be seen in
Fig. 2. The biphoton OAM modes $\\{\ket{-1,-1},\ket{0,0},\ket{1,1}\\}$ have
equal expansion amplitudes and projection probabilities, as desired. However,
this condition is not sufficient for maximal entanglement. Since
$C_{-1,1}=C_{1,-1}\neq 0$ and $a_{0}\neq 0$, the modes $\ket{-1,1}$ and
$\ket{1,-1}$ will have non-vanishing probabilities leading to a decrease of
entanglement. In order to analyze the entanglement, we consider the Hilbert
space
$\mathcal{H}_{\operatorname{\mathrm{s}}}=\mathcal{H}_{\operatorname{\mathrm{i}}}=\\{\ket{-1},\ket{0},\ket{1}\\}$,
resulting in a nine-dimensional two-photon subspace

$\displaystyle\mathcal{H}_{\operatorname{\mathrm{s}}}\otimes\mathcal{H}_{\operatorname{\mathrm{i}}}=$
$\displaystyle\Bigl{\\{}\ket{-1,-1},\ket{-1,0},\ket{-1,1},\ket{0,-1},$
$\displaystyle\ket{0,0},\ket{0,1},\ket{1,-1},\ket{1,0},\ket{1,1}\Bigl{\\}}.$

We denote this subspace as $S_{3\times 3}$ because each subsystem has three
levels.

Figure 2:  The pump-engineered OAM spectrum from Fig. 1 c) within the range
$\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}=-3,...,3$.
The target state
$\ket{\Psi_{1}}=\frac{1}{\sqrt{3}}\bigl{(}\ket{-1,-1}+\ket{0,0}+\ket{1,1}\bigl{)}$
is not maximally entangled in the subspace $S_{3\times 3}$ (corresponding to
$\ell_{\operatorname{\mathrm{s}}}=\ell_{\operatorname{\mathrm{i}}}=-1,0,1$),
due to the presence of the unintended modes $\ket{1,-1}$ (yellow circle) and
$\ket{-1,1}$ (not visible). The Schmidt number is $K_{3\times 3}=2.76<3$.

The OAM spectrum in Fig. 2 analyzed in this subspace reveals two types of
modes. On the one hand, we have target modes, which have equal probability and
contribute to desired MESs like $\ket{-1,-1},\ket{0,0},\ket{1,1}$ in our
example. On the other hand, we find unintended modes, which are still present
despite pump engineering due to OAM conservation, like $\ket{-1,1}$ and
$\ket{1,-1}$. If we calculate the Schmidt number in $S_{3\times 3}$, denoted
by $K_{3\times 3}$, we get $K_{3\times 3}=2.76$ due to non-vanishing
contributions of the unintended modes. An actual MES would require $K_{3\times
3}=3$. The suppression of the unwanted expansion amplitudes $C^{-1,1}$ and
$C^{1,-1}$ is not possible with further pump engineering.

Figure 3:  Conceptual sketch of different crystal-domain configurations to
achieve nontrivial quasi-phase-matching. a) Periodic poling: domains of equal
width are poled in an alternating pattern of a positive ($\uparrow$) or
negative ($\downarrow$) effective nonlinear susceptibility. b) Domain-width
annealing: use of a periodically alternating pattern, but each domain width is
individually adjusted. c) Costume poling: the domain width is kept fixed, but
the orientation pattern is customized as desired.

###  III.2 Tailored nonlinear crystals

####  III.2.1 Enginnering the phase-matching function

The idea of tailoring the nonlinearity of crystals used in SPDC caught much
attention lately due to novel fabrication methods [41]. Instead of a single
nonlinear material with susceptibility $\chi^{(2)}$, we consider a medium
decomposed in thin layers of the birefringent material, so-called domains.
Each domain is poled with an either up ($+\chi_{\text{eff}}$) or down
($-\chi_{\text{eff}}$) oriented effective susceptibility, so the nonlinearity
profile along the propagation axis of the beams $\chi^{(2)}(z)$ results in a
discontinuous function. However, the corresponding PMF for the entire crystal
is the linear superposition of the PMF of each individual domain [43].

Figure 4:  a) The normalized OAM spectrum for the target state
$\ket{\Psi_{1}}=\frac{1}{\sqrt{3}}\bigl{(}\ket{-1,-1}+\ket{0,0}+\ket{1,1}\bigl{)}$
within the range
$\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}=-3,...,3$.
The state is maximally entangled in the subspace $S_{3\times 3}$ by combined
pump and crystal engineering. The unintended modes $\ket{1,-1}$ and
$\ket{-1,1}$ as observed in Fig. 2 are suppressed. b) The corresponding
intensity and phase profile of the LG superposition with pump coefficients
$a_{-2}=a_{2}=2.46$ and $a_{0}=1.73$. c) The customized PMF
$\Phi_{\text{PM}}(\Delta k)$ with crystal coefficients $c_{0}=1.045$ and
$c_{1}=-0.865$ (green) and for comparison, a typical sinc-like PMF of a
periodically poled crystal (red).

Periodically poled crystals, with alternating domains of uniform length as in
Fig. 3 a), emerged as highly effective in maintaining quasi-phase-matching
(QPM) for a wide range of wavelengths in different nonlinear materials [62,
63]. This configuration results in an approximate sinc-like PMF. Since
periodically poled crystals are designed to optimize the brightness of photon
pair sources, alternative PMFs realized with the crystal would automatically
generate fewer pairs with the same pump beam.

However, the implementation of different PMFs has been investigated lately and
tested experimentally. The PMF can be shaped using nontrivial QPM, where the
constant poling pattern is maintained, but the domain width is individually
adapted [39, 64] (so-called domain width-anneling, see Fig. 3 b)) or different
domains exhibit different values for $\chi_{\text{eff}}$ [38]. Alternatively,
each domain orientation can be regulated individually [40] so that the
effective PMF can match nearly any desired target function by customizing the
poling configuration [43] like in Fig. 3 c). All approaches successfully
demonstrated high-purity single-photon generation via SPDC without post-
filtering due to minimized spectral [44, 42] or spatial [45] correlations
between signal and idler photons. Custom poled structures in waveguides like
the multi-output quantum pulse gate (MGPQ) [65] also proved beneficial for
decoding high-dimensional single-photon temporal modes. While higher pump
powers can adjust the reduced MQPG efficiency [66], this may amplify multi-
photon effects [67].

Here, we leverage the ability to tailor PMFs. Similar to the pump beam
construction from the previous section, we assume a general function for
susceptibility function $\chi^{(2)}(z)$ by expanding it into a cosine Fourier
series like in Ref. [45]:

$\chi^{(2)}(z)=\sum_{n=0}^{N}c_{n}\cos{\Bigl{(}\frac{nz}{\sigma}\Bigl{)}},$
(13)

where the crystal coefficients $c_{n}$ are undetermined. The scaling factor
$\sigma=L/4$ insures high single-photon purity without significant reduction
in the source brightness [40, 41, 45]. In analog to pump engineering, we treat
$c_{n}$ as free parameters and tune them to equalize or suppress expansion
amplitudes in Eq. (8). The according PMF when inserting Eq. (13) in Eq. (4)
reads

$\displaystyle\Phi_{\text{PM}}(\Delta k)=\frac{L}{2}\>\sum_{n=0}^{N}$
$\displaystyle c_{n}\Biggl{(}\>\text{sinc}\Bigl{(}\frac{\Delta
kL}{2}+2n\Bigl{)}$ (14) $\displaystyle+\>\text{sinc}\Bigl{(}\frac{\Delta
kL}{2}-2n\Bigl{)}\Biggl{)},$

so we fully parameterize the PMF by the crystal coefficients $c_{n}$. The
familiar sinc-PMF is regained for $c_{0}=1$ while $c_{n>0}=0$.

####  III.2.2 Constraints by the relative mode number

We assume, for simplicity, that
$w_{\operatorname{\mathrm{p}}}\approx\sqrt{\frac{L}{k_{\operatorname{\mathrm{p}}}}}$
and $\sqrt{2}w_{\operatorname{\mathrm{p}}}\approx
w_{\operatorname{\mathrm{s}}}=w_{\operatorname{\mathrm{i}}}$, which can
equivalently be enforced by assuming degenerate SPDC
$k_{\mathrm{p}}=2k_{\mathrm{s}}=2k_{\mathrm{i}}$ and demanding equal Rayleigh
lengths for pump, signal, and idler. Experimentally, this setting is favored
to concentrate the spread of OAM modes along the anti-diagonal
$\ell_{\operatorname{\mathrm{p}}}=0$ to the mode $\ket{0,0}$ [34]. Under this
consideration, we find a explicit expression for Eq. (13) from Ref. [50] as

$C^{\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}}(N_{R})\propto\int_{-\frac{L}{2}}^{\frac{L}{2}}\>dz\>c_{n}\>\underbrace{\frac{(k_{\operatorname{\mathrm{p}}}w_{\operatorname{\mathrm{p}}}^{2}+2iz)^{N_{R}}}{(k_{\operatorname{\mathrm{p}}}w_{\operatorname{\mathrm{p}}}^{2}-2iz)^{N_{R}+1}}}_{:=\xi(N_{R},z)}\cos{\Bigl{(}\frac{nz}{\sigma}\Bigl{)}},$
(15)

where the relative mode number (RMN) $N_{R}$ appears in the $z$-dependent
function $\xi(N_{R},z)$.

The RMN was already discussed in the context of decoupling the spatial and
spectral DOFs in the SPDC process [50]. In our investigation the value is
$N_{R}=|\ell_{\operatorname{\mathrm{s}}}+\ell_{\operatorname{\mathrm{i}}}|-|\ell_{\operatorname{\mathrm{s}}}|-|\ell_{\operatorname{\mathrm{i}}}|$.

Figure 5:  Example for a maximally entangled two-ququint state ($d=5$). a) The
normalized OAM spectrum for the target state
$\ket{\Psi_{3}}=\frac{1}{\sqrt{5}}\bigl{(}\ket{-2,-1}+\ket{-1,-2}+\ket{0,0}+\ket{1,2}+\ket{2,1}\bigl{)}$
within the range
$\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}=-3,...,3$.
The state is maximally entangled in the subspace $S_{5\times 5}$. b) The
required pump superposition is given by the pump coefficients
$a_{-3}=a_{3}=3.68$ and $a_{0}=2.24$. c) The corresponding PMF
$\Phi_{\text{PM}}(\Delta k)$ with $c_{0}=1.604,c_{1}=-1.665$, and
$c_{2}=0.912$ (green) and for comparison, a typical sinc-like PMF of a
periodically poled crystal (red). Note that here more crystal coefficients are
needed to find the optimal PMF than in Fig. 4.

Eq. (15) implies that the expansion amplitudes of biphoton OAM modes are
proportional when their RMNs are equal. This means that OAM modes with the
same RMN cannot be manipulated independently, which limits our state
engineering capabilities. Specifically, if $N_{R}=N_{R}^{\prime}$, we cannot
find a set of crystal coefficients $c_{n}$ to suppress
$C^{\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}}(N_{R})$
while ensuring
$C^{\ell_{\operatorname{\mathrm{s}}}^{\prime},\ell_{\operatorname{\mathrm{i}}}^{\prime}}(N_{R}^{\prime})\neq
0.$

Consider for example the generation of the target state
$\ket{\Psi_{2}}=\frac{1}{\sqrt{3}}\bigl{(}\ket{-1,0}+\ket{0,1}+\ket{1,-1}\bigl{)}$,
which is a MES in $S_{3\times 3}$. The existence of the desired mode
$\ket{1,-1}$ governs the pump superposition to have a pump coefficient
$a_{0}\neq 0$, which in turn means the unintended mode $\ket{0,0}$ appears in
the OAM spectrum. Eq. (15) suggests that if we can find crystal coefficients
$c_{n}$ to suppress $C^{0,0}\approx 0$, we will also suppress the target mode
$C^{0,1}\approx 0$, as both modes fulfill RMN $N_{R}=0$.

Therefore, crystal engineering can optimize the generation of MESs when all
unintended modes in the considered subspace have a different RMN than the
target modes. More details on RMN and general guidelines for selecting MESs
that follow this rule can be found in the Appendix A.

##  IV Results

###  IV.1 Combined pump and crystal engineering

As we have discussed before, the target state
$\ket{\Psi_{1}}=\frac{1}{\sqrt{3}}\bigl{(}\ket{-1,-1}+\ket{0,0}+\ket{1,1}\bigl{)}$
is not maximally entangled in the subspace $S_{3\times 3}$ (see Fig. 2) if we
only apply pump engineering. Since all three target modes have the RMN
$N_{R}=0$ and the unintended modes $\ket{-1,1}$ und $\ket{1,-1}$ have
$N_{R}=-2$, we can utilize crystal engineering to suppress $\ket{-1,1}$ und
$\ket{1,-1}$. Thus we like to engineer $C^{-1,1}=C^{1,-1}=0$ while
$C^{-1,-1}=C^{0,0}=C^{1,1}=1/\sqrt{3}$. The latter condition can be reached by
pure pump engineering. The first condition results in a linear system of
equations obtained by Eq. (15) (see Appendix A):

$\displaystyle\frac{1}{\sqrt{3}}\begin{pmatrix}1\\\ 0\\\
\end{pmatrix}\overset{!}{=}\begin{pmatrix}C^{0,0}(N_{R}=0)\\\
C^{-1,1}(N_{R}=-2)\\\ \end{pmatrix}\propto$ $\displaystyle\begin{pmatrix}\int
dz\>\xi(0,z)&\int dz\>\xi(0,z)\cos{(\frac{z}{\sigma})}\\\ \int
dz\>\xi(-2,z)&\int
dz\>\xi(-2,z)\cos{(\frac{z}{\sigma})}\end{pmatrix}\cdot\begin{pmatrix}c_{0}\\\
c_{1}\end{pmatrix}$ (16)

with solution $c_{0}=1.045$ and $c_{1}=-0.865$. The condition
$C^{1,-1}=C^{-1,1}$ is automatically fulfilled due to
$k_{\operatorname{\mathrm{s}}}=k_{\operatorname{\mathrm{i}}}$. The modified
OAM spectrum by the adapted PMF using the calculated crystal coefficients is
displayed in Fig. 4 a). The engineered state is now a MES with Schmidt number
$K_{3\times 3}=3$.

Additionally, Fig. 4 plots the corresponding spatial distribution of the pump
beam and the normalized PMF in b) and c), respectively. The pump configuration
remains unchanged as determined in Section III.1. The four intensity clumps
are in agreement with experimental demonstrations [34, 35]. The equal weights
$a_{-2}=a_{2}$ in the LG superposition can be explained by the property
$C^{\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}}=(C^{-\ell_{\operatorname{\mathrm{s}}},-\ell_{\operatorname{\mathrm{i}}}})^{*}$
[50]: the anti-diagonals $\ell_{\operatorname{\mathrm{p}}}=-2$ and
$\ell_{\operatorname{\mathrm{p}}}=2$ are adjusted similarly to equalize
$C^{-1,-1}$ and $C^{1,1}$. Therefore, the superimposed phases of
$\text{LG}_{0}^{-2}$ and $\text{LG}_{0}^{2}$ results in a phase profile of
$\pi$ and $-\pi$ only. The calculated crystal coefficients $c_{0}$ and $c_{1}$
suggest that the best choice to generate the target state differs from a
periodically poled crystal profile. For comparison, we show both PMFs for a
crystal of length $L=$15\text{\,}\mathrm{mm}$$. The customized function for
maximal entanglement (green) has more pronounced outer sidelobes compared to
the sinc-function of a periodically poled crystal (red).

Besides the MES
$\ket{\Psi_{1}}=\frac{1}{\sqrt{3}}\bigl{(}\ket{-1,-1}+\ket{0,0}+\ket{1,1}\bigl{)}$,
an infinite number of states can be inferred which fulfill $K_{3\times 3}=3$
when taking the relative phases between the modes into account:
$\ket{\Psi_{1}^{\prime}}=\frac{1}{\sqrt{3}}\bigl{(}\ket{-1,-1}+\mathrm{e}^{i\kappa_{1}}\ket{0,0}+\mathrm{e}^{i\kappa_{2}}\ket{1,1}\bigl{)}$
with arbitrary $\kappa_{1},\kappa_{2}\in\mathbb{R}$. Such relative phases can
be inherited by the LG superposition of the pump beam [35] due to the
linearity in Eq. (12), where all biphoton OAM modes
$\ket{\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}}$ with
$\ell_{\operatorname{\mathrm{p}}}=\ell_{\operatorname{\mathrm{s}}}+\ell_{\operatorname{\mathrm{i}}}$
obtain the same prefactor from $a_{\ell_{\operatorname{\mathrm{p}}}}$. We can
introduce arbitrary $\mathrm{e}^{i\kappa_{1,2}}$ for the mode
$\ket{\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}}$ by
crystal engineering as well, but the mode
$\ket{\ell_{\operatorname{\mathrm{i}}},\ell_{\operatorname{\mathrm{s}}}}$
simultaneously acquire the same prefactor.

Figure 6:  a) The normalized OAM spectrum for the target state
$\ket{\Psi_{4}}=\frac{1}{\sqrt{3}}\bigl{(}\ket{-1,1}+\ket{0,0}+\ket{1,-1}\bigl{)}$
within the range
$\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}=-3,...,3$.
The state is maximally entangled in the subspace $S_{3\times 3}$. This example
demonstrates that crystal engineering also enables to equalize OAM mode
amplitudes. b) The pump is not a superposition of LG modes, but simply the
fundamental Gaussian mode. c) The customized PMF $\Phi_{\text{PM}}(\Delta k)$
with $c_{0}=3.555$ and $c_{1}=-5.506$ (green) and for comparison, a typical
sinc-like PMF of a periodically poled crystal (red).

We extend our method to $d>3$ and investigate the generation of a two-ququint
state ($d=5$) in the well-defined subspace $S_{5\times 5}$. In this case, the
subspace refers to
$\mathcal{H}_{\operatorname{\mathrm{s}}}\otimes\mathcal{H}_{\operatorname{\mathrm{i}}}$
where
$\mathcal{H}_{\operatorname{\mathrm{s}}}=\mathcal{H}_{\operatorname{\mathrm{i}}}=\bigl{\\{}\ket{-2},\ket{-1},\ket{0},\ket{1},\ket{2}\bigl{\\}}$.
A neat example of combined pump and crystal engineering is shown in Fig. 5 a)
with the target state
$\ket{\Psi_{3}}=\frac{1}{\sqrt{5}}\bigl{(}\ket{-2,-1}+\ket{-1,-2}+\ket{0,0}+\ket{1,2}+\ket{2,1}\bigl{)}$.
Pump engineering enables to equalize the target mode amplitudes
$C^{-1,-2}=C^{0,0}=C^{1,2}$. The conditions $C^{-1,-2}=C^{-2,-1}$ and
$C^{1,2}=C^{2,1}$ are automatically fulfilled by degeneracy
$k_{\operatorname{\mathrm{s}}}=k_{\operatorname{\mathrm{i}}}$. The modes
generated by OAM conservation with non-zero projection probability of
$\ell_{\operatorname{\mathrm{p}}}=-2$ and $\ell_{\operatorname{\mathrm{p}}}=2$
lie outside of the considered subspace. Thus, all target modes inside
$S_{5\times 5}$ have the RMN $N_{R}=0$ while the RMN of the unintended modes
are $N_{R}=-2$ (modes $\ket{-1,1},\ket{1,-1}$) or $N_{R}=-4$ (modes
$\ket{-2,2},\ket{2,-2}$). So we can apply crystal engineering to suppress the
expansion amplitudes $C^{-2,2}=C^{-1,1}=C^{1,-1}=C^{2,-2}=0$ of the unintended
modes.

The intensity distribution of the pump beam features six peaks, see Fig. 5 b).
Given that $a_{-3}$ and $a_{3}$ are equally weighted, the calculated spatial
profile is similar to Fig. 4 b). The corresponding PMF in c) is determined by
three crystal coefficients $c_{0},c_{1},c_{2}$. The number of crystal
coefficients has increased because we now have more unintended modes to
suppress than before. The calculated PMF shows two maxima enclosing three
suppressed lobes around $\Delta k_{z}=0$.

###  IV.2 Crystal engineering with a Gaussian pump

Finally, we examine a special case where only the crystal is engineered and
the pump beam is left in the fundamental Gaussian mode. This means that the
biphoton state is given by

$...+C^{-1,1}\ket{-1,1}+C^{0,0}\ket{0,0}+C^{1,-1}\ket{1,-1}+...\>.$ (17)

This state becomes maximally entangled if we can equalize the OAM amplitudes.
We restrict the discussion to the subspace $S_{3\times 3}$ and consider the
target state
$\ket{\Psi_{4}}=\frac{1}{\sqrt{3}}\bigl{(}\ket{-1,1}+\ket{0,0}+\ket{1,-1}\bigl{)}$.
Similar to Eq. (16), but with $C^{0,0}=\frac{1}{\sqrt{3}}$ instead of
$C^{1,-1}=0$, we construct linear system of equations and find the solution
$c_{0}=3.555$ and $c_{1}=-5.506$. The OAM spectrum is shown in Fig. 6 a).
Since all target modes lie along the anti-diagonal
$\ell_{\operatorname{\mathrm{p}}}=0$, pump engineering is not required and the
spatial pump profile corresponds to a Gaussian beam. The OAM spectrum without
crystal engineering corresponds to Fig. 1 a), whose Schmidt number is
$K_{3\times 3}=1.14$ indicating an almost separable state. The crystal-
engineered state has $K_{3\times 3}=3$ and is maximally entangled. This
highlights that crystal engineering not only enables the suppression of OAM
modes, but also allows to equalize the mode amplitudes. The customized PMF to
achieve this OAM spectrum is displayed in green in Fig. 6 c) and compared to
the course to a periodically poled crystal (red). Similar to before, the
calculated PMF has two symmetric maxima at $\Delta k_{z}\neq 0$ while the lobe
at $\Delta k_{z}=0$ is suppressed.

In principle, this MES can be generalized for odd $d>3$, like in $d=5$ to the
MES
$\frac{1}{\sqrt{5}}\bigl{(}\ket{-2,2}+\ket{-1,1}+\ket{0,0}+\ket{1,-1}+\ket{2,-2}\bigl{)}$,
where the calculated crystal coefficients lead to more sophisticated PMFs than
in Fig. 6 c).

##  V Conclusion

In this paper, we have analyzed the entanglement of high-dimensional quantum
states in OAM between photon pairs generated from SPDC.

We confirm that, although pump engineering methods can generate equally
probable biphoton OAM modes, these do not represent high-dimensional maximally
entangled states. OAM spectra generated solely by pump-engineered sources
generally have unintended OAM modes present that obey the OAM conservation law
for collinear SPDC. This results in a Schmidt numbers $K_{d\times d}<d$. We
resolved this issue by demonstrating theoretically that suitable PMF
modifications of the crystal can actually enhance the entanglement between
signal and idler photons. We have presented PMF that enables the generation of
MES with $K_{d\times d}=d$.

Our findings are relevant for implementing multidimensional quantum
information processing applications, which aim to utilize genuine, well-
defined two-qudit MES generated directly from SPDC.

While we have improved MES generation in specific OAM subspaces, post-
selection and filtering remain necessary, particularly in the spectral domain
or the radial DOF of LG modes. Future investigations should explore fully
controllable generation techniques that completely eliminate the need for
post-selection.

###### Acknowledgements.

The authors thank René Sondenheimer and Fabian Steinlechner for useful
discussions regarding this work.

##  Appendix A  Details on the relative mode number

###  A.1 Advice to find tunable MES

As discussed in the main text, the relative mode number (RMN) primarily
determines the expansion amplitude in our discussion. If we want to suppress
the probability of unintended modes, this only works if their RMN differs from
the RMN of the target modes.

Fig. 7 examines the RMN in more detail. By definition, most modes
$\ket{\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}}$ have
$N_{R}=0$ (dark blue color). RMNs $N_{R}<0$ in steps of negative even integers
appear in a regular pattern only if either $\ell_{\operatorname{\mathrm{s}}}$
or $\ell_{\operatorname{\mathrm{i}}}$ is negative. In detail, when fixing a
signal OAM index $\ell_{\operatorname{\mathrm{s}}}\geq 0$, three horizontal
areas can be distinguished when varying the idler OAM index:

$\displaystyle\ell_{\operatorname{\mathrm{i}}}\geq 0:\>$ $\displaystyle
N_{R}=0$
$\displaystyle-\ell_{\operatorname{\mathrm{s}}}<\ell_{\operatorname{\mathrm{i}}}<0:\>$
$\displaystyle N_{R}=-2,-4,\ldots,-2|\ell_{\operatorname{\mathrm{s}}}|$
$\displaystyle\ell_{\operatorname{\mathrm{i}}}<-\ell_{\operatorname{\mathrm{s}}}:\>$
$\displaystyle N_{R}=-2|\ell_{\operatorname{\mathrm{s}}}|$

For $\ell_{\operatorname{\mathrm{s}}}<0$, the same applies, but with reversed
inequality symbols. This behavior is also observed when swapping the roles of
$\ell_{\operatorname{\mathrm{s}}}$ and $\ell_{\operatorname{\mathrm{i}}}$,
except that vertical stripes of the same RMN can be seen.

Figure 7:  The relative mode number (RMN)
$N_{R}=|\ell_{\operatorname{\mathrm{s}}}+\ell_{\operatorname{\mathrm{i}}}|-|\ell_{\operatorname{\mathrm{s}}}|-|\ell_{\operatorname{\mathrm{i}}}|$
of OAM modes
$\ket{\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}}$
colorized within the range
$\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}=-10,...,10$.
Most modes have $N_{R}=0$ (dark blue). If either
$\ell_{\operatorname{\mathrm{s}}}$ or $\ell_{\operatorname{\mathrm{i}}}$ is
negative, the RMN can be a mutiple of $-2$. OAM modes of the same $N_{R}<0$
are grouped along vertical and horizontal strips. This pattern continues to
infinity.

We should construct MES in a subspace, in which different RMNs appear to
effectively make use of crystal engineering. The most straightforward way is
to generalize $S_{3\times 3}$ to the subspace $S_{d\times d}$, where $d$ is an
odd number. Analogous, this denotes the two-photon Hilbert space
$\mathcal{H}_{\operatorname{\mathrm{s}}}\otimes\mathcal{H}_{\operatorname{\mathrm{i}}}$
where
$\mathcal{H}_{\operatorname{\mathrm{s}}}=\mathcal{H}_{\operatorname{\mathrm{i}}}=\bigl{\\{}\ket{-\frac{d-1}{2}},...,\ket{0},...,\ket{\frac{d-1}{2}}\bigl{\\}}$.
The subspace is symmetric around the mode $\ket{0,0}$. The dimension of the
subspace $S_{d\times d}$ is $d^{2}$. In this well-defined two-photon subspace
MESs are characterized with an Schmidt number $K_{d\times d}=d$. The RMN in
this subspace ranges from $N_{R}=0,-2,...,-d+1$.

The RMN constraints compel us to choose MES with target modes, whose RMN are
in the range $N_{R}=0,-2,...,m$ (with $m<0$), and the unintended modes created
due to OAM conservation either

  1. (i)

lie outside of the considered subspace $S_{d\times d}$ if their RMN is
$N_{R}=0,-2,...,m$ (i.e. we cut desirable OAM modes out) or

  2. (ii)

have a RMN $N_{R}<m$.

When these conditions are met, quantum state engineering for a MES with
optimal crystal coefficients is possible.

###  A.2 Determining suitable crystal coefficients

In the subspace $S_{d\times d}$, we can tune only expansion amplitudes by
crystal engineering that lie along a single anti-diagonal in the OAM spectrum.
If we have several anti-diagonals, we would introduce separable modes or have
several modes with $N_{R}=0$ (which all cannot be equalized by crystal
engineering). We can equalize at most two modes with RMN $N_{R}=0$ along an
anti-diagonal.

Effectively, we are thus restricted to apply crystal engineering along
$\ell_{\operatorname{\mathrm{p}}}=0$ (with expansion amplitude $C^{0,0}$),
along $\ell_{\operatorname{\mathrm{p}}}=-1$ (with $C^{-1,0}=C^{0,-1}$) or
along $\ell_{\operatorname{\mathrm{p}}}=1$ (with $C^{0,1}=C^{0,1}$). These
conditions are automatically fulfilled due to degeneracy
$k_{\operatorname{\mathrm{s}}}=k_{\operatorname{\mathrm{i}}}$.

Along one of these anti-diagonals, crystal engineering allows us to equalize
(suppress) the amplitudes of the target (unintended) modes with different RMN.
We find one condition for every RMN and their corresponding expansion
amplitude. The expansion amplitude of all target modes have to be equal to
$C^{\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}}\propto
1/\sqrt{d}$ and expansion amplitude of unintended modes
$C^{\ell_{\operatorname{\mathrm{s}}},\ell_{\operatorname{\mathrm{i}}}}\approx
0$. In the subspace $S_{d\times d}$, we find $\frac{d+1}{2}$ different RMNs.
Due to the ansatz Eq. (13), the conditions can directly be related to a linear
system of $\frac{d+1}{2}$ equation to be solved for the crystal coefficients
$c_{n}$, where $n$ is ranging from $0,...,\frac{d-1}{2}$:

$\frac{1}{\sqrt{d}}\begin{pmatrix}1\\\\[2.70003pt] 1\\\\[2.70003pt]
\vdots\\\\[2.70003pt] 0\\\\[2.70003pt]
\end{pmatrix}\overset{!}{=}\begin{pmatrix}C^{\ell_{s},\ell_{i}}({N_{R}=0})\\\\[2.70003pt]
C^{\ell_{s},\ell_{i}}({N_{R}=-2})\\\\[2.70003pt] \vdots\\\\[2.70003pt]
C^{\ell_{s},\ell_{i}}({N_{R}=-d+1})\\\\[2.70003pt]
\end{pmatrix}\overset{\eqref{RMNinExpansionAmplitude}}{\propto}\begin{pmatrix}\int
dz\>\xi(0,z)\cos{\Bigl{(}\frac{0\cdot z}{\sigma}\Bigl{)}}&&...&\int
dz\>\xi(0,z)\cos{\Bigl{(}\frac{(d-1)z}{2\sigma}\Bigl{)}}\\\\[2.70003pt] \int
dz\>\xi(-2,z)\cos{\Bigl{(}\frac{0\cdot z}{\sigma}\Bigl{)}}&&...&\int
dz\>\xi(-2,z)\cos{\Bigl{(}\frac{(d-1)z}{2\sigma}\Bigl{)}}\\\\[2.70003pt]
\vdots&&\ddots&\vdots\\\\[2.70003pt] \int
dz\>\xi(-d+1,z)\cos{\Bigl{(}\frac{0\cdot z}{\sigma}\Bigl{)}}&&...&\int
dz\>\xi(-d+1,z)\cos{\Bigl{(}\frac{(d-1)z}{2\sigma}\Bigl{)}}\\\\[2.70003pt]
\end{pmatrix}\cdot\begin{pmatrix}c_{0}\\\ c_{1}\\\ \vdots\\\
c_{\frac{d-1}{2}}\\\ \end{pmatrix}$

The rows increase in steps of $-2$ from $N_{R}=0$ to $N_{R}=-d+1$, while the
columns increase with the cosine argument in the range
$n=0,1,...,\frac{d-1}{2}$. The number of $0$s and $1$s on the very left side
depends on the explicit target state, i.e. how many modes have to be
suppressed or equalized along the main anti-diagonal. By solving the system of
equations for $\\{c_{0},...,c_{\frac{d-1}{2}}\\}$, we have found the optimized
crystal coefficients for the MES.

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