# Continuous variable multimode quantum states
via symmetric group velocity matching

V. Roman-Rodriguez  Sorbonne Université, CNRS, LIP6, 4 place Jussieu, F-75005
Paris, France  B. Brecht  Integrated Quantum Optics, Institute for Photonic
Quantum Systems (PhoQS), Paderborn University, Warburger Straße 100 33098,
Paderborn, Germany  S.Kaali  Laboratoire Kastler Brossel, Sorbonne Université,
CNRS, ENS-PSL Research University, Collège de France, 4 place Jussieu, F-75252
Paris, France  C. Silberhorn  Integrated Quantum Optics, Institute for
Photonic Quantum Systems (PhoQS), Paderborn University, Warburger Straße 100
33098, Paderborn, Germany  N. Treps  Laboratoire Kastler Brossel, Sorbonne
Université, CNRS, ENS-PSL Research University, Collège de France, 4 place
Jussieu, F-75252 Paris, France  E.Diamanti  Sorbonne Université, CNRS, LIP6, 4
place Jussieu, F-75005 Paris, France  V. Parigi  Laboratoire Kastler Brossel,
Sorbonne Université, CNRS, ENS-PSL Research University, Collège de France, 4
place Jussieu, F-75252 Paris, France<EMAIL_ADDRESS>
###### Abstract

Configurable and scalable continuous variable quantum networks for
measurement-based quantum information protocols or multipartite quantum
communication schemes can be obtained via parametric down conversion (PDC) in
non-linear waveguides. In this work, we exploit symmetric group velocity
matching (SGVM) to engineer the properties of the squeezed modes of the PDC.
We identify type _II_ PDC in a single waveguide as the best suited process,
since multiple modes with non-negligible amount of squeezing can be obtained.
We explore, for the first time, the waveguide dimensions, usually only set to
ensure single-mode guiding, as an additional design parameter ensuring
indistinguishability of the signal and idler fields. We investigate here
potassium titanyl phosphate (KTP), which offers SGVM at telecommunications
wavelengths, but our approach can be applied to any non-linear material and
pump wavelength. This work paves the way towards the engineering of future
large-scale quantum networks in the continuous variable regime.

##  I Introduction

Photonic quantum technologies are grounded on the remarkable ability of
engineering quantum states of light in different non-linear optical processes
Roslund  _et al._ (2014); Ansari  _et al._ (2018a). Besides the versatility in
producing different quantum states, optics has the advantage of offering
scalability in the number of systems by exploiting multimode electro-magnetic
fields. This is even more relevant when we consider a continuous variable (CV)
encoding of quantum information since it is possible to deterministically
generate large multimode entangled states, where entanglement is established
between amplitude and phase quadratures of different light modes Roslund  _et
al._ (2014); Chen  _et al._ (2014); Yokoyama  _et al._ (2013); Cai  _et al._
(2017); Asavanant  _et al._ (2019). CV multimode entangled states are
resources for measurement based quantum computing Menicucci  _et al._ (2006);
Gu  _et al._ (2009), quantum simulations Nokkala  _et al._ (2018), multi-party
quantum communication Cai  _et al._ (2017); Arzani  _et al._ (2019), and
quantum metrology Pinel  _et al._ (2012); Gessner  _et al._ (2018).

For all the applications that rely on the transfer of information over
appreciable distances, telecommunication wavelengths offer the most reliable
solution. In this regime single-mode quantum states with large amount of
squeezing Ast  _et al._ (2013); Gehring  _et al._ (2015), and on-chip few-
modes squeezed states have been reported Mondain  _et al._ (2019); Lenzini
_et al._ (2018). Second-order non-linear waveguides are promising candidates
for the controlled generation of both single- and multimode quantum states at
telecom wavelengths Eckstein  _et al._ (2011); Harder  _et al._ (2016); Brecht
_et al._ (2015); Ansari  _et al._ (2018b), but a direct characterization of CV
multimode quantum resources, like cluster states, in these systems is still
missing. Waveguides offer in fact large nonlinearities and long interaction
lengths, along with the possibility to tailor the modal decomposition of the
quantum state Dirmeier  _et al._ (2020), but losses and modal engineering have
to be carefully controlled in order to measure the desired quantum effects via
homodyne detection used in CV protocols.

In this work we study the generation of CV multimode entangled states at
telecom wavelengths via parametric down conversion (PDC) in non-linear
waveguides. We aim at obtaining a controlled number of modes with sufficient
squeezing, which can coherently interfere in order to generate the wanted
entangled states via schemes exploiting both the spectral Roslund  _et al._
(2014) and the temporal Yokoyama  _et al._ (2013) degrees of freedom of a
femtosecond laser source Volpe  _et al._ (2020). We identify type _II_ PDC in
periodically poled non-linear waveguides in the so-called symmetric group
velocity matching (SGVM) configuration U’Ren  _et al._ (2005) as the most
promising scheme.

The paper is structured as follows. In Section II, we set the theoretical
grounds, define the physical variables and discuss the optimal configuration
of the non-linear waveguide. In Section III, we revisit the concepts
underlying SGVM and show the properties of type _0_ , _I_ and _II_ PDC
processes, concluding that type _II_ is best suited for the target
application. Section IV applies the results to potassium titanyl phosphate
(KTP) and yields realistic fabrication parameters. Section V finishes with the
conclusions.

##  II PDC as a continuous variables quantum resource

###  II.1 Squeezed modes

The parametric interaction in a $\chi^{(2)}$ non-linear crystal involving a
pump field and two output fields, usually called signal and idler, can be
expressed in appropriate conditions (see for example Horoshko  _et al._
(2019)) via an effective hamiltonian $\hat{\mathrm{H}}$ that governs the
evolution of the quantum operators in the interaction picture. This effective
hamiltonian can be written (assuming a classical undepleted pump), as:

$\mathrm{\hat{H}}=\sum_{i,j}J_{i,j}\hat{a}^{\dagger}_{i}\hat{b}^{\dagger}_{j}+\mathrm{h.c.},$
(1)

where $J_{i,j}$ are the elements of the so-called joint two-photon amplitude
and $\hat{a}_{i}^{\dagger}$ and $\hat{b}_{j}^{\dagger}$ are creation operators
of, respectively, the signal and idler photons in spatio-temporal modes
$f_{i}(\mathbf{r},t)$ and $l_{j}(\mathbf{r},t)$, each forming an orthonormal
mode basis.

The discrete form of the hamiltonian in Eq. (1) is due to the implicit
discrete mode basis $\\{f_{i}(\mathbf{r},t),l_{j}(\mathbf{r},t)\\}$ in which
it is expressed. In this work, we will restrict ourselves to temporal modes,
assuming a single spatial mode common for all the fields involved, a
configuration that is typically realised in waveguides. Under these conditions
we write $f_{i}(\mathbf{r},t)\rightarrow
f_{i}(t)\xleftrightarrow{F.T.}\tilde{f}_{i}(\omega)$, where $\omega$ is the
frequency and $F.T.$ stands for Fourier Transform.

In the continuous frequency basis, the effective hamiltonian of Eq. (1) is
written as:

$\mathrm{\hat{H}}=\int\int{\mathrm{d}\omega_{s}\mathrm{d}\omega_{i}J(\omega_{s},\omega_{i})\hat{a}^{\dagger}(\omega_{s})\hat{b}^{\dagger}(\omega_{i})}+\mathrm{h.c.},$
(2)

where now the joint two-photon amplitude is a two-dimensional function of the
signal and idler frequencies $\omega_{s}$ and $\omega_{i}$, known as the joint
spectral amplitude Grice and Walmsley (1997), and the creation operators
create field excitations at different frequencies.

The joint spectral amplitude, $J(\omega_{s},\omega_{i})$, contains information
about the pump field that enters the waveguide and the linear and non-linear
optical properties of the waveguide itself. One can perform a decomposition of
this function in a set of modal functions $\\{h_{k}(\omega_{s})\\}$ and
$\\{g_{k}(\omega_{i})\\}$ Law  _et al._ (1997):

$J(\omega_{s},\omega_{i})=\sum_{k}\lambda_{k}h_{k}(\omega_{s})g_{k}(\omega_{i}).$
(3)

This decomposition is also known as the Schmidt decomposition, and the set of
complex values $\\{\lambda_{k}\\}$ the Schmidt coefficients, fulfilling
$\sum_{k}|\lambda_{k}|^{2}=1$.

By inserting Eq. (3) into Eq. (2) we obtain:

$\hat{\mathrm{H}}=\sum_{k}\lambda_{k}\hat{A}^{\dagger}_{k}\hat{B}^{\dagger}_{k}+\mathrm{h.c}\\\
$ (4) $\displaystyle\hat{A}_{k}^{\dagger}$
$\displaystyle=\int{\mathrm{d\omega_{s}}h_{k}(\omega_{s})\hat{a}^{\dagger}(\omega_{s})}$
(5) $\displaystyle\hat{B}_{k}^{\dagger}$
$\displaystyle=\int{\mathrm{d\omega_{i}}g_{k}(\omega_{i})\hat{b}^{\dagger}(\omega_{i})}$

The Hamiltonian of Eq. (4) generates two sets of eigenmodes that are EPR
entangled with each other (twin modes), where each couple is in the so called
two-mode squeezed state Migdał and Wasilewski (2010); Horoshko  _et al._
(2019); Christ  _et al._ (2013a), and which are defined spectrally by the same
sets $\\{h_{k}(\omega_{s})\\}$ and $\\{g_{k}(\omega_{i})\\}$ of the Schmidt
decomposition in Eq. (3). These eigenmodes are also called temporal modes
Brecht  _et al._ (2015). Eq. (5) is then simply a basis change from the
continuous frequency basis to the discrete Schmidt basis.

From the continuous variable picture of the process, we define the normalized
quadrature operators associated to the temporal modes as:

$\displaystyle\begin{split}\\{\hat{q}_{A,k},\hat{p}_{A,k}\\}=\\{\hat{A}_{k}^{{\dagger}}+\hat{A}_{k},i(\hat{A_{k}}^{{\dagger}}-\hat{A_{k}})\\}\\\
\\{\hat{q}_{B,k},\hat{p}_{B,k}\\}=\\{\hat{B}_{k}^{{\dagger}}+\hat{B}_{k},i(\hat{B_{k}}^{{\dagger}}-\hat{B_{k}})\\}\end{split}$
(6)

The EPR couples are then characterized by the following relations of
quadrature variances
$\Delta^{2}(\hat{q}_{A,k}-\hat{q}_{B,k})=2e^{-2\lambda_{k}},\Delta^{2}(\hat{q}_{v})=2e^{-2\lambda_{k}}$
and
$\Delta^{2}(\hat{p}_{A,k}+\hat{p}_{B,k})=2e^{-2\lambda_{k}},\Delta^{2}(\hat{p}_{v})=2e^{-2\lambda_{k}}$,
where $\Delta^{2}(\hat{q}_{v})$ and $\Delta^{2}(\hat{p}_{v})$ are the
variances of the quadratures in the case of vacuum states, that here are taken
equal to one according to the quadrature definition in Eq. (6). This means
that in the case of high gain, _i.e._ , high value for the Schmidt
coefficients $\lambda_{k}$, the quantum state of each EPR couple is an
approximation of the so called quadrature entangled EPR state Einstein  _et
al._ (1935), characterized by $\hat{q}_{A,k}=\hat{q}_{B,k}$ and
$\hat{p}_{A,k}=-\hat{p}_{B,k}$.

It can be shown Christ  _et al._ (2013b) that the effective number of modes,
$K$, _i.e._ , the effective number of pairs of functions from
$\\{h_{k}(\omega_{s}),g_{k}(\omega_{i})\\}$, can be calculated as

$K=\frac{1}{\sum_{k}|\lambda_{k}|^{4}}.$ (7)

This quantity is often called the Schmidt number, and it will be a figure of
merit in our analysis.

The temporal modes form eigenbases of the signal and idler parts of the total
field. One can show Horoshko  _et al._ (2019) that the eigenvalues of the
total joint spectral amplitude are doubly degenerate, with eigenvectors
$\\{s_{k}(\omega)\\}$ such that:

$s_{k}(\omega)=\frac{1}{\sqrt{2}}(h_{k}(\omega)\pm g_{k}(\omega)),$ (8)

which are mutually uncorrelated modes that present squeezing. These modes are
called squeezed modes (or “supermodes” Patera  _et al._ (2010)), and their
creation operators are defined as
$\hat{S}^{{\dagger}}_{k\pm}=\frac{1}{\sqrt{2}}(\hat{A}^{{\dagger}}_{k}\pm\hat{B}^{{\dagger}}_{k}$).
The effective hamiltonian in terms of the supermode creation operators is
then:

$\hat{\mathrm{H}}=\sum_{k}\frac{\lambda_{k}}{2}\left(\left(\hat{S}^{\dagger}_{k,+}\right)^{2}+\left(\hat{S}^{\dagger}_{k-}\right)^{2}\right)+\mathrm{h.c}\\\
$ (9)

The corresponding variance of the quadrature variables, with operators
$\\{\hat{q}_{S,k},\hat{p_{S,k}}\\}$, defined in the same way as in Eq. (6),
are $\Delta^{2}(\hat{q}_{S,k})=e^{-2\lambda_{k}}$ and
$\Delta^{2}(\hat{p}_{S,k})=e^{2\lambda_{k}}$. They are squeezed, and their
level of squeezing, $2\lambda_{k}$, is proportional to the corresponding
Schmidt eigenvalue $\lambda_{k}$.

Hence, in the temporal mode basis, the signal and idler parts of the total
field are individually diagonalized and there exists entanglement between
them, while in the squeezed mode basis, the modes are uncorrelated and carry
squeezing as a quantum resource.

It is important to note that the transformation of Eq. (8) physically
corresponds to the interference of the signal and idler modes on a balanced
beamsplitter.

In the following, we focus on the calculation and characterization of the
squeezed eigenmodes $s_{k}(\omega)$ and their squeezing eigenvalues
$\lambda_{k}$, since they constitute a natural multimode quantum state where
continuous variable quantum information can be encoded and processed. In
particular, the Bloch-Messiah reduction Bloch and Messiah (1962) ensures that
all multimode Gaussian states, _i.e._ , the states whose quadratures are
characterized by Gaussian statistics and that can be generated by hamiltonians
that are at most quadratic in the bosonic operators like the ones in Eqs. (1),
(2) (4) and (9), can be written as a set of single mode squeezed states and
two multiport interferometers. In our context, this means that we could
generate deterministically any highly entangled multimode Gaussian state if we
input the squeezed eigenmodes into an appropriate multiport interferometer,
generating a configurable source of cluster states that could then be used as
the resource for measurement based quantum computation, multiparty quantum
communication or quantum metrology.

###  II.2 Use of waveguides and mode detection

Until now we have considered the general case of non-degenerate parametric
down conversion and the emergence of the squeezed modes. We will now specify
and justify the configuration that will be treated in the rest of the paper.

We will consider the use of non-linear waveguides, instead of bulk crystals,
for their ability to confine light over a long length (in particular over the
coherence length of the pump laser) and their discretization in the number of
available spatial modes. Both effects enhance the non-linear interaction by
several orders of magnitude. This enables the generation of single-pass
squeezing Dirmeier  _et al._ (2020); no optical cavities are required. The use
of waveguides also means that we will work in the so-called collinear
configuration, where signal and idler fields exit the crystal in the same
beam.

Another important reason for the use of waveguides is that, as we will see
later, the engineering of the physical parameters of the waveguide opens up
new possibilities for engineering the PDC, and hence the generated modes,
which will benefit our target applications.

Furthermore, the measurement of the modes would be performed via coherent
detection, in particular homodyne detection, where the quadratures of the
targeted field can be accessed by interfering the field with a local
oscillator. Hence, in order to measure the quadratures of the different
squeezed modes, the local oscillator should be shaped spectrally to match the
corresponding modes $\\{s_{k}(\omega)\\}$. However, this becomes a difficult
task if the squeezed modes take arbitrary shapes. In Eq. (8) we mentioned that
the squeezed modes are obtained by the interference of signal and idler
temporal modes, which can themselves be approximated as Hermite Gauss modes
Ansari  _et al._ (2018b). In order for this interference to be efficient and
for the squeezed modes to inherit the approximate Hermite Gauss functional
form, both signal and idler temporal modes should be as similar as possible,
which translates in having indistinguishable fields after the non-linear
interaction in the waveguide.

The spatial indistinguishability is given due to the collinear configuration
imposed by the use of waveguides, while in the bulk crystal scenario a precise
alignment of the crystal would be needed. The spectral indistinguishability
constrains to work in the so-called degenerate case, where both central
frequencies for signal and idler are equal (and equal to half the pump central
frequency by conservation of energy):
$\omega_{s}^{0}=\omega_{i}^{0}=\omega_{p}^{0}/2$.

The final degree of freedom of the fields is their polarization. Depending on
the output polarization in the PDC process, three types of parametric
interactions can be found. In type _0_ and type _I_ , the polarization of
signal and idler fields is the same, while in type _II_ the fields have
orthogonal polarizations, making them distinguishable. However, we will show
in the next section that, due to their dispersion properties, both type _0_
and type _I_ prove unpractical. We will consequently restrict ourselves to
type _II_ processes. We note that the polarization distinguishability can be
erased by rotating the polarizations of the signal and idler by 45 degrees and
interfering them on a polarizing beamsplitter. At the output of the
beamsplitter, one then obtains the squeezed modes Dirmeier  _et al._ (2020).
We also note here that, due to the different polarizations of the signal and
idler modes, there may be a temporal walk-off after the waveguide, which can
be corrected for with appropriate compensation crystals.

Fig. 1 schematically shows the configuration discussed above: degenerate
collinear type _II_ parametric down conversion in a non-linear waveguide.

Figure 1: Periodically poled non-linear waveguide scheme picturing degenerate,
collinear, type _II_ parametric downconversion. The squeezed modes are
obtained after making the two sets of temporal (Schmidt) modes interfere. For
more details see text.

##  III Symmetric group-velocity matching

###  III.1 Joint spectral amplitude structure

The mathematical structure of the joint spectral amplitude, or JSA, of Eq.
(2), $J(\omega_{s},\omega_{i})$, can be expressed as:

$J(\omega_{s},\omega_{i})=\alpha_{p}(\omega_{s}+\omega_{i})\phi(\omega_{s},\omega_{i}),$
(10)

where the subscripts $s/i/p$ stand for signal, idler, and pump, and
$\omega_{p}=\omega_{s}+\omega_{i}$. The function
$\alpha_{p}(\omega_{s}+\omega_{i})$ is the pump envelope function in the
spectral domain, which we take to be a Gaussian with a certain width, denoted
by $w_{p}$ (see Fig. 1). Finally, $\phi(\omega_{s},\omega_{i})$ is the
phasematching function. It is governed by the waveguide’s optical properties
and the geometry of the problem. This function does not generally have
properties of symmetry.

The phasematching function can be written as:

$\phi(\omega_{s},\omega_{i})=\mathrm{sinc}\left(\frac{L}{2}\Delta
k(\omega_{s},\omega_{i})\right),$ (11)

where $L$ is the waveguide length and the function $\Delta
k(\omega_{s},\omega_{i})$ is the wavevector mismatch

$\Delta
k(\omega_{i},\omega_{s})=k_{p}(\omega_{i}+\omega_{s})-k_{s}(\omega_{s})-k_{i}(\omega_{i}),$
(12)

where the $k$ are the corresponding wavevectors. In general, Eq. (12) is a
vectorial equation, but we recover its scalar version because of the collinear
condition.

It is possible to match two arbitrary signal and idler frequencies so that
$\Delta k=0$ with a well-known technique known as quasi-phasematching Angell
_et al._ (1994), which introduces a new term to Eq. (12) by modulating the
$\chi^{(2)}$ coefficient along the waveguide with a poling period $\Lambda$.
Eq. (12) including the quasi-phasematching modulation can be written as:

$\Delta
k(\omega_{i},\omega_{s})=k_{p}(\omega_{i}+\omega_{s})-k_{s}(\omega_{s})-k_{i}(\omega_{i})-\frac{2\pi}{\Lambda},$
(13)

such that the selection of the poling period makes the mismatch zero for a
certain pair of desired frequencies. In this work, the poling period will
correspond to the case of matched signal/idler central frequencies.

###  III.2 Type _0_ and _I_ PDC

Indistinguishability of the signal and idler fields is required for their
proper interference after the non-linear interaction. This implies that the
joint spectral amplitude has to be symmetric under the exchange of the signal
and idler labels, _i.e._ ,

$J(\omega_{s},\omega_{i})=J(\omega_{i},\omega_{s}).$ (14)

Since the pump function $\alpha_{p}(\omega_{i}+\omega_{s})$ is naturally
symmetric under this exchange, the fields are indistinguishable if the
phasematching function shares this symmetry, _i.e._ , if
$\phi(\omega_{s},\omega_{i})=\phi(\omega_{i},\omega_{s})$. From Eq. (11) we
therefore require the condition on the wavevector mismatch: $\Delta
k(\omega_{s},\omega_{i})=\Delta k(\omega_{i},\omega_{s})$.

In the case of type _0_ and type _I_ PDC, it is easy to see that the symmetry
condition is fulfilled, since $k_{s}(\omega_{s})=k_{i}(\omega_{i})$ in these
cases.

As in U’Ren  _et al._ (2005), we write the Taylor series of the mismatch
function around the central frequencies
$\omega_{s}^{0}=\omega_{i}^{0}=\omega_{p}^{0}/2$ to second order:

$\displaystyle\Delta k=$ $\displaystyle\Delta
k^{0}+\gamma_{s}(\omega_{s}-\omega^{0}_{p}/2)+\gamma_{i}(\omega_{i}-\omega^{0}_{p}/2)+$
(15)
$\displaystyle\delta_{s}(\omega_{s}-\omega^{0}_{p}/2)^{2}+\delta_{i}(\omega_{i}-\omega^{0}_{p}/2)^{2}+$
$\displaystyle\delta_{p}(\omega_{s}-\omega^{0}_{p}/2)(\omega_{i}-\omega^{0}_{p}/2)+...$

The coefficients are related to the derivatives of the wavevector with respect
to the frequency as:

$\displaystyle\gamma_{s/i}$ $\displaystyle=\frac{\partial
k_{p}}{\partial\omega}\Bigr{|}_{\begin{subarray}{c}\omega_{p}\end{subarray}}-\frac{\partial
k_{s/i}}{\partial\omega}\Bigr{|}_{\begin{subarray}{c}\omega_{p}/2\end{subarray}}$
(16) $\displaystyle\delta_{s/i}$
$\displaystyle=\frac{\partial^{2}k_{p}}{\partial\omega^{2}}\Bigr{|}_{\begin{subarray}{c}\omega_{p}\end{subarray}}-\frac{\partial^{2}k_{s/i}}{\partial\omega^{2}}\Bigr{|}_{\begin{subarray}{c}\omega_{p}/2\end{subarray}}$
$\displaystyle\delta_{p}$
$\displaystyle=2\frac{\partial^{2}k_{p}}{\partial\omega^{2}}\Bigr{|}_{\begin{subarray}{c}\omega_{p}\end{subarray}},$

where we wrote $\omega_{p}$ instead of $\omega_{p}^{0}$ in the evaluation of
the derivatives for clarity in the notation. The constant term $\Delta
k^{0}=\Delta k(\omega_{s}^{0},\omega_{i}^{0})$ can be made zero via quasi-
phasematching, as described previously.

The coefficient $\gamma_{s}$ ($\gamma_{i}$) associated with the linear term in
the expansion is related to the pump and signal (idler) group velocities,
while the coefficients $\delta_{s}$ ($\delta_{i}$) associated with the
quadratic term is related to the pump and signal (idler) group velocity
dispersion.

In type _0_ and type _I_ PDC, $\gamma_{i}=\gamma_{s}\equiv\gamma$,
$\delta_{i}=\delta_{s}\equiv\delta$, etc. In addition, usually $w_{p}\leq
10\,$ nm, such that the linear term dominates. Hence, we can approximately
write:

$\Delta k(\mathrm{type\
\emph{0}/\emph{I}})=\gamma(\omega_{s}+\omega_{i}-\omega_{p}^{0})+\mathcal{O}(\omega_{s}^{2},\omega_{i}^{2},\omega_{s}\omega_{i})$
(17)

The phasematching can be approximated by a function of
$\omega_{s}+\omega_{i}$, and therefore so can the joint spectral amplitude
$J(\omega_{s}+\omega_{i})=\alpha_{p}(\omega_{s}+\omega_{i})\phi(\omega_{s}+\omega_{i})$.

Since both the pump envelope and the phasematching are functions of the sum of
the frequencies, they are aligned, making a $-$45∘ angle, in the 2-dimensional
coordinate system with axes ($\omega_{s}$,$\omega_{i}$). The JSA is then non
zero in a large spectral region, yielding spectrally broad temporal modes that
are not well-suited for homodyne detection, since one needs good spectral
overlap with a local oscillator. Fig. 2 a) shows this situation. Overcoming
this problem necessitates either very broad local oscillators and/or higher
order term contribution in the mismatch function, both of which is
unpractical.

Figure 2: Pump, phasematching and JSA functions in degenerate collinear PDC,
a) type _0_ /_I_ , b) type _II_.

###  III.3 Type _II_ PDC

For type _II_ PDC, the signal and idler polarizations are orthogonal, the
fields are distinguishable and hence their dispersion relations are different,
$k_{s}(\omega_{s})\neq k_{i}(\omega_{i})$.

However, looking at Eq. (15), we obtain identical modes if we manage to have
the condition $\gamma_{s}=-\gamma_{i}$, which is called the symmetric group
velocity matching (SGVM) condition U’Ren  _et al._ (2005). This translates
into having the following condition on the group velocities (recalling that
the group velocity is $v_{g}(\omega)=\partial\omega/\partial k$):

$\frac{1}{v_{g,p}(\omega_{p}^{0})}=\frac{1}{2}\left(\frac{1}{v_{g,s}(\omega_{s}^{0})}+\frac{1}{v_{g,i}(\omega_{i}^{0})}\right)$
(18)

In that case, the Taylor expansion of Eq. (15) gives:

$\Delta k(\mathrm{type\
\emph{II}})=\gamma(\omega_{s}-\omega_{i})+\mathcal{O}(\omega_{s}^{2},\omega_{i}^{2},\omega_{s}\omega_{i}),$
(19)

with $\gamma\equiv\gamma_{s}=-\gamma_{i}$. Hence, the mismatch function is
antisymmetric under the exchange of signal and idler labels: $\Delta
k(\omega_{s},\omega_{i})=-\Delta k(\omega_{i},\omega_{s})$. By Eq. (11) and
the even parity of the sinc function we obtain a symmetric phasematching
function, and hence a symmetric joint spectral amplitude (up to first order in
the Taylor series of Eq. (15)). Under this condition, signal and idler fields
are indistinguishable (except for their polarization).

From Eq. (19), the joint spectral amplitude is approximately of the form

$J(\omega_{i},\omega_{s})=\alpha_{p}(\omega_{i}+\omega_{s})\phi(\omega_{s}-\omega_{i}).$
(20)

In Fig. 2 b) we see that the phasematching function now makes an angle of
$+45^{\circ}$ in the ($\omega_{s}$,$\omega_{i}$) coordinate system, hence
making a $90^{\circ}$-angle with the pump envelope function. Therefore, the
JSA is non zero only in a small spectral region around the central
frequencies. In this situation, the coherent detection of the squeezed modes
is experimentally realizable and the interference of signal and idler fields
is approximately perfect after the change of polarization of one of the two.

In conclusion, all the reasoning exposed until now leads us to consider the
configuration of degenerate, collinear, type _II_ PDC in non-linear
waveguides. We will also consider being around the symmetric group velocity
matching condition, which means:

  * •

The poling period $\Lambda$ provides quasi-phasematching for the central
frequencies $\omega^{0}_{p}$ and
$\omega_{s}^{0}=\omega_{i}^{0}=\omega_{p}^{0}/2$.

  * •

The inverse group velocities at the central frequencies satisfy the condition
$\gamma_{s}=-\gamma_{i}$, where these coefficients are defined in Eq. (16).

  * •

The higher order terms in the Taylor expansion of Eq. (15) are sufficiently
small compared to the first order term in the entire frequency range
considered (see B for insights into this point).

We will also call the wavelength fulfilling these conditions as symmetric
group velocity matching wavelength $\lambda_{\mathrm{SGVM}}$.

###  III.4 Index of refraction

The dispersion relation

$k(\omega)=\frac{\omega}{c}n(\omega),$ (21)

where $c$ is the speed of light in vacuum and $n$ the index of refraction,
encapsulates the optical properties of the crystal and could depend on several
physical variables apart from frequency, such as temperature, $T$, and
waveguide characteristics. The latter include the waveguide section size, with
height $h$ and width $w$, and spatial mode order inside the waveguide, defined
by two integer numbers $n_{1}$ and $n_{2}$.

If one is interested in modeling the functional form of the index of
refraction with these waveguide parameters, approximations must be made. A
wide-spread approximation is the so-called metallic waveguide approximation
Brecht (2014), in which one assumes the waveguide to be surrounded by
perfectly conducting edges. This is the approximation that will be made in
this work. More refined modeling of the index of refraction involves more
advanced computational methods solving Maxwell equations inside and outside
the waveguide, like Mercatili’s method Marcatili (1969) or finite element
methods Fallahkhair  _et al._ (2008).

Using the metallic waveguide approximation, the index of refraction is written
as

$n(\lambda,T,n_{1},n_{2},w,h)=n(\lambda,T)+\left(\lambda\frac{n_{1}+1}{2h}\right)^{2}+\left(\lambda\frac{n_{2}+1}{2w}\right)^{2},$
(22)

where the function $n(\lambda,T)$ depends on the non-linear material under
consideration, and is given by the empirical Sellmeier equations (for e.g. KTP
see Kato and Takaoka (2002)).

In this work, we will restrict ourselves to the case in which only the
fundamental spatial mode travels through the waveguide, which translates into
$n_{1}=n_{2}=0$. This is exactly the case in a perfect single-mode waveguide
and, for shorter wavelengths, this can be achieved by careful mode-matching in
the input coupling.

Furthermore, we will have a different index of refraction of the form of Eq.
(22) for different light polarizations. For instance, in the case of uniaxial
crystals, we would have two indices of refraction (for the ordinary and
extraordinary axis), and in biaxial crystals, like KTP, we would have one
index of refraction for every spatial direction.

These indices define the different wavevectors for pump, signal and idler
fields in each case. We can therefore study the symmetric group velocity
matching condition via the dependence of the indices of refraction on
wavelength, temperature and waveguide dimensions.

###  III.5 Mode overlap

To wrap up this section, we define the quantity that will quantify the
similarity between signal and idler temporal modes as the overlap integral
$o_{n}$. The $n$-th order overlap between two modal functions, $h_{n}(\omega)$
and $g_{n}(\omega)$, is defined as:

$\displaystyle o_{n}$
$\displaystyle=\frac{1}{N}\Big{\lvert}\int_{-\infty}^{\infty}\mathrm{d}\omega
h^{*}_{n}(\omega)g_{n}(\omega)\Big{\rvert}$ (23) $\displaystyle N$
$\displaystyle=\sqrt{\int_{-\infty}^{\infty}\mathrm{d}\omega|h_{n}|^{2}(\omega)\int_{-\infty}^{\infty}{d}\mathrm{d}\omega|g_{n}|^{2}(\omega)},$

where $N$ is a normalization factor, such that the overlap is adimensional and
its maximum value is 1, if and only if $h_{n}(\omega)=g_{n}(\omega)$.

##  IV Results

In our analysis, we will focus on KTP as the non-linear material for the
waveguide. Lithium niobate (LN) is another important non-linear material,
since commercial LN waveguides already exist. However, calculations with LN in
our configuration lead to the conclusion that the waveguide width and height
should be in the order of 1 $\mu$m for the symmetric group velocity condition
to hold. This is smaller than the telecom wavelengths considered here (around
1.55 $\mu$m). The metallic waveguide approximation breaks down at this scale
and more elaborate techniques must be used in order to model the index of
refraction of these small structures. Therefore, we will only consider KTP in
this work, which naturally provides SGVM at telecom wavelenghts.

###  IV.1 SGVM wavelength

First, we performed calculations to compute the wavelengths fulfilling the
SGVM condition. This first section shows the existence of this wavelength for
KTP for reasonable physical parameters.

For this, we fixed the temperature and the waveguide dimensions to typical
values and computed the dispersion properties of the crystal, in particular
the different group velocities and group dispersions in KTP, from the indices
of refraction of Eq. (22), as a function of wavelength. Thanks to the metallic
waveguide approximation, the derivatives can be computed analytically in this
case. In this way the coefficients of Eq. (16), as well as the corresponding
poling period, are calculated as a function of wavelength.

Figure 3: Taylor coefficients for the linear term of Eq.(15), defined in
Eq.(16). The two curves cross at a certain wavelength where the SGVM condition
$\gamma_{s}=-\gamma_{i}$ is fulfilled.(KTP at $T=20\ ^{o}\mathrm{C}$ and
$w,h=6\ \mathrm{\mu m}$)

Fig. 3 shows the Taylor coefficients associated with the group velocities
($\gamma_{s}$ and $-\gamma_{i}$). They coincide at the SGVM phasematching
wavelength, $\lambda_{\mathrm{SGVM}}$, where $\gamma_{s}=-\gamma_{i}$.

Once $\lambda_{\mathrm{SGVM}}$ is obtained, it is important to check the
contribution of the higher order terms in Eq. (15). This is discussed in B.

###  IV.2 Telecom wavelength

In the calculations above, a wavelength fulfilling the SGVM condition was
proven to exist for KTP, Fig. 3. We will now be interested in varying the
other physical variables affecting the index of refraction in Eq. (22), namely
temperature and waveguide dimensions, to get the condition at a certain
desired wavelength.

In particular, we are interested in obtaining SGVM at 1550 nm, given that
telecom wavelengths present low losses in optical fibers, and there exist
optimized of-the-shelf components at this wavelength, opening the way to
practical applications.

It is important to remark the geometrical choice of the problem, because
results would be different depending on the orientation of the crystal’s
optical axes with respect to the pump field polarization. In particular, the
Kleinman symmetry Boyd (2011) provides the supported processes for all
combinations in propagation and polarization directions. The only
configuration in which we have results at wavelengths around 1550 nm is the
one that corresponds to a crystal cut (_i.e._ , a vertical polarization
direction) in the $z$ direction and propagation direction along the $x$
direction, where the indices of refraction for every spatial direction are
ordered like in Kato and Takaoka (2002). Therefore this is the case considered
in this work. More details can be found in A.

As announced above, we will now consider the dependence of the index of
refraction on temperature and waveguide dimensions.

#### Temperature:

Fig. 4 shows the temperature dependence of the SGVM wavelength for KTP. For
this calculation we fixed the waveguide dimensions such that
$\lambda_{\mathrm{SGVM}}\simeq$ 1550 nm.

Figure 4: Temperature dependence of the SGVM wavelength for KTP. The waveguide
dimensions were set to $w=h=9\ \mathrm{\mu m}$.

We obtain a change in the SGVM wavelength of $\Delta\lambda_{SGVM}\simeq 4$ nm
in a range of temperatures of $\Delta T\simeq 230$ oC. This result indicates
that temperature has a very limited effect on the change of
$\lambda_{\mathrm{SGVM}}$.

According to these results, it seems reasonable to fix the temperature and
focus on the tuning of the waveguide dimensions. We therefore fixed the
temperature to be room temperature for the rest of the work, which is natural
when considering future technology applications, where involved temperature
control may be undesirable.

#### Waveguide Dimensions:

Fig. 5 shows the SGVM wavelength as a function of the waveguide width and
height for KTP. As the waveguide size increases, the surface flattens, making
apparent the diminishing impact of the waveguide dispersion. We also find this
behaviour in Eq. (22), where the waveguide dimensions contribution to the
index of refraction disappears as $h,w\rightarrow\infty$. The green area in
Fig. 5 marks values of $\lambda_{\mathrm{SGVM}}$ that lie in the interval
between 1549 and 1551 nm. Therefore, we conclude that to work at 1550 nm in
KTP, the waveguide should have a size of about 9 by 9 $\mu$m in width and
height. From Fig. 4 we know that this is almost independent of temperature in
the temperature interval considered.

Figure 5: SGVM wavelength as a function of waveguide height and width for KTP.
The green area indicates the region where
$\lambda_{\mathrm{SGVM}}\in[1549-1551]$ nm. The temperature was set to $T=20\
^{o}\mathrm{C}$.

###  IV.3 Engineering of the quantum states

Next, we performed simulations of the PDC process for KTP to compute the
signal and idler modes of our system with the associated squeezing eigenvalues
under the SGVM condition. We remark that this exact procedure could be done
for any non-linear optical material if its dispersion properties are known.

Our pump function is modeled as a (square-normalized) Gaussian function in the
following way:

$\alpha_{p}(\omega_{s}+\omega_{i})=\frac{1}{\sqrt{\sqrt{\pi}w_{p}}}\exp\left(-\frac{(\omega_{p}^{0}-(\omega_{s}+\omega_{i}))^{2}}{2w_{p}^{2}}\right),$
(24)

where the parameter $w_{p}$ is the pump width. The pump central wavelength is
$\lambda_{p}^{0}=775$ nm, corresponding to a central frequency
$\omega_{p}^{0}=2\pi c/\lambda_{p}^{0}$. Furthermore, in our results, we will
express the pump width in length units (wavelength bandwidth instead of
frequency bandwidth), by redefining $w_{p}$ as $2\pi c/w_{p}$.

The phasematching function is that of Eq. (11), where we fix the waveguide
length. The wavevector mismatch is calculated using Eqs. (21) and (22), where
we also set values for the temperature and waveguide dimensions in each case.
The poling period is calculated for every case so that the signal/idler
central wavelengths, $\lambda_{s}^{0}=\lambda_{i}^{0}=2\lambda_{p}^{0}=1550$
nm, are perfectly phasematched.

With these two functions we compute the joint spectral amplitude, or JSA,
according to Eq. (10).

We then perform the Schmidt decomposition of Eq. (3) by computing the singular
value decomposition of the JSA, obtaining two matrices containing the signal
and idler temporal modes and a diagonal matrix with the Schmidt coefficients
$\\{\lambda_{k}\\}$. We recall that the distribution of the squeezing
eigenvalues is equivalent (except for a global factor of 2) to the
distribution of the Schmidt coefficients. From these coefficients we obtain
the Schmidt number according to Eq. (7) and we also compute the temporal
modes’ full width at half maximum (FWHM). This yields the spectral width of
the temporal modes, which is important for designing their coherent detection.
Finally, we use Eq. (23) to obtain the overlap between the signal and idler
modes in order to quantify their similarity.

The main goal of this analysis is to obtain the dependence of the quantum
fields on input parameters that will be specified in the following. For the
purpose of this work, we can see the computation just described as a set of
input variables that are experimentally controllable, and output variables
after the parametric interaction. The main input variables are the waveguide
length, the waveguide dimensions (height and width), and the pump width, and
the main output variables are the Schmidt number, the modes FWHM and their
overlap. We then perform distinct simulations to obtain the corresponding
dependencies. Note that the temperature could be taken as another input
variable. However, following the discussion earlier, we consider it to be a
fixed variable (at room temperature), since we have seen that it would have a
minor effect on the output modes.

For the sake of clarity, we construct Table 1, where each column represents an
input variable and each row a corresponding output variable. The table
elements show general dependencies that will be discussed in detail in the
following.

Table 1: Columns are input variables and rows are outputs. For each element,
the description expresses the output/input dependency in the PDC process.
| Pump Width | Waveguide Dimensions | Waveguide Length
---|---|---|---
Schmidt Number | Minimum and linear growth |  $\sim$ Constant | Grows
Mode FWHM | Grows | Inverse for | Decreases
| signal/idler |
Overlap | Constant | SGVM | Constant
| Analysis |

The range of values adopted for the input variables is the following: we
consider pump widths from 1 to 12 nm, waveguide dimensions between 4 and 10
$\mathrm{\mu m}$, and waveguide lengths between 5 and 30 mm. The pump width
can be modified by tuning the pump field that enters the waveguide, while the
waveguide dimensions and length are fixed during the waveguide fabrication
process.

The rest of this section will explore Table 1 in detail. As a general comment,
the dependencies are based on the following observations:

  * •

The waveguide length, $L$, changes the bandwidth of the phasematching function
as $\mathrm{sinc}\left({L^{-1}}\right)$.

  * •

The width of the pump, $w_{p}$, although self-explicative, changes the width
of the pump function.

  * •

The waveguide dimensions mainly change the angle between the pump function and
the phasematching function, because they change the optical properties (index
of refraction) of the material. This is why they are the critical parameter in
order to obtain the SGVM condition and why it is interesting considering them
as an engineering parameter for the quantum states. We have also seen in this
work that even though temperature also takes part in the index of refraction,
its contribution is negligible in the usual experimental range of values. We
remark that given fixed waveguide dimensions and temperature, the angle
between the phasematching function and the pump function is totally defined by
the choice of the non-linear material.

The results of this section can be understood keeping in mind these three
observations, which are schematically shown in Fig. 6.

Figure 6: Example of a JSA function, highlighting the impact of the waveguide
length $L$, pump width, $w_{p}$, and waveguide dimensions $(w,h)$. For more
details see text.

For the sake of clarity, we will refer to the elements of Table 1 like matrix
elements. For instance, the element 2.1 refers to the second row and first
column, which corresponds to the dependence of the mode FWHM on the pump
width.

#### First row of Table 1, the Schmidt number:

Fig. 7 shows the Schmidt number as a function of pump width for different
waveguide dimensions (elements 1.1 and 1.2 of Table 1). The Schmidt number,
and thus the number of modes, grows linearly with the pump width, while the
waveguide dimensions have almost no influence. The waveguide length was set to
$L=10$ mm for this simulation. This result can be interpreted in the following
way: as the pump width increases, the pump envelope does too, and hence its
intersection with the phasematching envelope. This creates frequency
correlations between signal and idler fields and hence increases $K$. This is
true because in this set-up the pump envelope is bigger at any time than the
phasematching envelope. In general, there would be a minimum value of $K$ when
both envelopes have equal widths.

Figure 7: Schmidt number as a function of pump width for different waveguide
dimensions. Corresponding to the elements 1.1 and 1.2 of Table 1.

Fig. 8 shows the Schmidt number as a function of the waveguide length for
different pump widths. In this case we are changing the phasematching
envelope, and we observe the minimum value of $K$ when both the pump
envelope’s and the phasematching envelope’s widths are equal. This is why we
see that the waveguide length in which the minimum value is reached depends on
the pump width. Also, the minimum value itself would approximately equal 1
when the intersection of $\alpha_{p}$ and $\phi$ was at $90^{\circ}$ that is,
at the SGVM condition. This is interesting for applications in single photon
sources, where $K=1$ ensures maximum purity of heralded single photons.

After the minimum point, increasing the waveguide length decreases the width
of the phasematching envelope, and so we are effectively in the same situation
as in Fig. 7, and we observe the same type of behaviour, which is a linear
increase in $K$. It is important to note that in Fig. 8 we see that one can
control the slope of the increasing linear regime by controlling the pump
width, which opens the possibility of modifying the number of modes just by
adjusting the pump field. In a realistic scenario, the maximum waveguide
length would be limited by the optical losses within the waveguide and the
size of the available substrate material. State of the art KTP waveguides have
losses of 0.25 dB/cm, which would point towards waveguide lengths in between
10 and 20 mm. This is in accordance with the maximum size of KTP substrates of
roughly $30\times 30\,$mm2.

Equivalently, the Schmidt number increases rapidly by decreasing the waveguide
length before the minimum, because in that case the 1/L dependence on the
phasematching width is more important, creating strong wiggles in the sinc
function in Eq. (11), and in turn producing strong correlations between the
output fields, hence increasing $K$.

Figure 8: Schmidt number as a function of waveguide length for different pump
functions. The waveguide dimensions were set to $w,h=6\ \mathrm{\mu m}$.
Element 1.3 from Table 1.

#### Second row of Table 1, the modes bandwidth:

We remark that, given the nature of the Hermite Gauss modes, the mode FWHM of
interest is only the first mode (that is approximately Gaussian), as the rest
would grow in width as $\sqrt{n}$, where $n$ is the mode order. Therefore
knowing the first FWHM is sufficient to approximately characterize all of
them.

Fig. 9 shows the width of this mode as a function of pump width, which
corresponds to the element 2.1 in Table 1. The waveguide dimensions were set
to $w,h=6\ \mathrm{\mu m}$, intentionally away from the SGVM condition. We can
see that indeed signal and idler modes have different widths for all pump
widths and hence their interference would not be perfect with these waveguide
dimensions, as expected from the results of the SGVM analysis.

Figure 9: First mode FWHM as a function of pump width for a waveguide of
$w,h=6\ \mathrm{\mu m}$. Element 2.1 in Table 1.

Fig.10 shows the modes’ FWHM as a function of the waveguide dimensions for
pump widths of 2 and 12 nm. Both signal and idler modes coincide at waveguide
dimensions near 9 $\mu$m (we considered square waveguides), as expected from
the SGVM analysis before. We show the plots for two different pump widths to
remark again that the similarity of signal and idler modes is independent of
the pump width, as it is related to the waveguide’s phasematching function.
These relations correspond to the element 2.2 in Table 1.

Figure 10: First mode FWHM as a function of waveguide dimensions for pump
widths of 2 and 12 nm. Elements 2.2 in Table 1.

Finally, Fig. 11 shows the modes FWHM as a function of waveguide length for
$w_{p}=3$ nm and $w,h=6\ \mathrm{\mu m}$, again intentionally away from the
SGVM condition. Both widths decrease with length, which would point towards
the use of the longest possible waveguide if we are to measure the fields by
coherent detection. This relation corresponds to element 2.3 in Table 1.

Figure 11: First mode FWHM as a function of crystal length with $w_{p}=3$ nm
and $w,h=6\ \mathrm{\mu m}$. Element 2.3 in Table 1.

#### Third row of Table 1, the modes overlap:

Fig. 12 shows the overlap of the first six temporal modes as a function of
pump width, leaving fixed the rest of the variables. We can see that the
overlap decreases with increasing mode order; the reason is discussed in
Appendix C, in which we showed that this effect is due to two factors: the
increasing width of the modes with the mode order, and the mathematical
properties of the Hermite Gauss modes themselves. The pump width does not
change the overlap, even though we have just seen that it changes the relative
modes’ FWHM by up to 25%. This counter-intuitive behaviour arises due to the
stability of the relative width of two Gaussian functions when their relative
widths change. The mathematical result is that for a relative width difference
of 25%, _i.e._ , if the width of one Gaussian is 1.25 times the width of the
other, then the change in their overlap does not even reach 1%. This explains
why we obtain a constant overlap for different pump widths, although the
modes’ widths change significantly.

In a similar way, Fig. 13 shows that the waveguide length has no effect on the
mode overlap. The Schmidt numbers, and hence effective mode numbers, for
different lengths are marked in the plot. We remark that the real number of
non zero modes (that we compute with the singular value decomposition of the
JSA), is always higher than the Schmidt number $K$. For example, in Fig. 12,
at a pump width of 4 nm, we have $K=2.30$ and 4 Schmidt modes, while when
$K=3.54$, at a pump width of 6 nm, we already have 6 Schmidt modes. It is
therefore worth noting that the Schmidt number $K$ is an indicative of the
number of Schmidt modes present, but not their exact quantity Christ  _et al._
(2013b). In particular, if $m$ is equal to the real number of Schmidt modes,
then $K\leq m$. These two figures correspond to the elements 3.1 and 3.3 in
Table 1.

Figure 12: Overlap of the first 6 Schmidt temporal modes as a function of pump
width. Element 3.1 in Table 1. Figure 13: Overlap of the first 6 Schmidt
temporal modes as a function of crystal. Element 3.3 in Table 1. Figure 14:
Overlap of the first 6 Schmidt temporal modes as a function of waveguide
dimensions. Element 3.2 in Table 1.

Completing Table 1, Fig. 14 shows the mode overlap as a function of the
waveguide dimensions. We observe that the individual temporal mode overlaps
have a common maximum at waveguide dimensions around 9 $\mu$m, as we computed
in the SGVM study. Also, for every waveguide dimension, the similarity between
modes decreases as the mode order increases (again the reason for this can be
found in Appendix C). This dependence corresponds to element 3.2 in Table 1.

##  V Conclusions

In this work, we have studied the generation of multimode squeezed states in
single-pass PDC in non-linear optical waveguides. Such states are appealing
candidates for the realization of numerous quantum information applications,
including measurement based quantum computation and multipartite quantum
communication. For long distances and integrated application, it is desirable
to generate these states at telecom wavelengths.

We have first set the theoretical grounds for the PDC process in a non-linear
waveguide, leading to the generation of uncorrelated squeezed states, the
“supermodes”, and we identify them as a natural platform for continuous
variable quantum information science. Then, we explored the different PDC
configurations.

In type _0_ and type _I_ PDC the generated signal and idler fields are
indistinguishable. Nevertheless we have shown that these schemes are not
practical for CV protocols. Although highly multimode states are generated,
the generated modes have indeed a too large spectrum to be accessible via
homodyne detection.

Consequently, we targeted type _II_ PDC as the suitable scheme. In this case,
the process generates a number of two-mode squeezed states, that, upon
interference on a beamsplitter, can be turned into the single-mode squeezed
resource states. For this to work efficiently, signal and idler fields have to
be spectrally indistinguishable. We showed that this requirement enforces the
use of SGVM in PDC.

We then investigated the use of KTP, a nonlinear material which naturally
provides SGVM for wavelengths around the telecommunications regime. We studied
the number of generated squeezed states and their similarity as function of
the PDC pump spectral width and waveguide length. In addition, we explored the
influence of the waveguide dimensions on the generated PDC state. Waveguide
dimensions are usually only set to ensure spatially single-mode operations.
However, it turns out that they play a crucial role for achieving spectrally
indistinguishable signal and idler fields at a desired wavelength.

Our calculations reveal that a waveguide width and height of $9\,\mu$m
facilitates SGVM at a signal and idler central wavelength of $1550\,$nm. In
this configuration, both the waveguide length and pump spectral bandwidth can
be used to tune the number of generated states, while having no detrimental
impact on signal-idler similarity whatsoever.

In conclusion we have explored symmetric group velocity matching in non-linear
waveguides for the tailored generation of scalable quantum resource for
continuous variable based technologies. Moreover we have identified the suited
processes and materials along with realistic parameters for their generation
at telecommunication wavelengths.

## Acknowledgments

This work has been supported by the European Research Council under the
Consolidator Grant COQCOoN (Grant No. 820079), by the QuantERA grant QuICHE,
and by the German BMBF. VRR acknowledges support from the DGA.

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##  Appendix A Kleinman’s Symmetry

When the nonlinear susceptibility tensor is frequency independent and the
material is lossless, one can reduce the third rank tensor $\chi^{(2)}$ to a 3
by 6 matrix Boyd (2011):

$d_{il}=\frac{1}{2}\chi^{(2)}_{ijk}$ (25)

This matrix tells how the components of the second order non-linear
polarization vector relates to the possible combinations of electric field
components of the form $E_{i}E_{j}$. If an entry of this matrix is zero, it
means that the non-linear process associated to the combination of those
components does not contribute to the component of the polarization vector. If
one works with linearly polarized fields where the polarization direction
coincides with one of the defined material’s optical axis, then the zero
entries of $d_{il}$ give the corresponding not allowed processes.

For KTP, the $d$ matrix is given by ind :

$d_{il}=\begin{pmatrix}0&0&0&0&-4.6&2.2\\\ -2.2&2.2&0&-4.6&0&0\\\
-4.6&-4.6&25&0&0&0\\\ \end{pmatrix}\ \mathrm{pm/V}$ (26)

In our context, each of the non-zero entries can be seen as a type _0_ , type
_I_ or type _II_ process, defined by the polarization (along $x$, $y$ or $z$)
of the pump, signal and idler fields.

In this respect, along the computation in the paper we took the propagation
direction (parallel to the waveguide) to be $x$ and the vertical polarization
(parallel to the crystal cut) to be $z$. Hence, the vertical polarization is
associated with the index of refraction $n_{z}$, and the horizontal
polarization with the index $n_{y}$. The type _II_ process was described in
terms of polarization as $(y\rightarrow z,y)$ (pump polarized in the $y$
direction, signal in the $z$ direction and idler also in the $y$ direction).
The corresponding second order susceptibility tensor element is then
$\chi^{(2)}_{232}$, which in the contracted notation of the matrix $d_{il}$
that corresponds to the element $d_{24}=-4.6$ pm/V. The fact that this element
is non-zero tells us that the process is allowed for KTP.

We tried all possible combinations giving an allowed type _II_ process in KTP,
finding that those combinations of polarizations were actually the only ones
giving symmetric group velocity matching conditions at $1550$ nm, and
therefore it is the one implicit in the manuscript. However, we remark that
with other polarization combinations or other non-linear materials, different
wavelengths fulfilling the symmetric group velocity matching condition could
be found even for the interesting case of waveguide dimensions compatible with
spatial single-mode propagation.

##  Appendix B Higher order contribution in the mismatch

We have seen that one of the conditions for being under SGVM is that the
second and higher order terms of Eq. (15) should be negligible compared to the
first order contribution. As we are performing a two-variable Taylor
expansion, the higher order contributions are expected to get bigger as we get
further away from the central frequencies $\omega_{s}^{0}=\omega_{i}^{0}$.

Inserting Eq. (22) in Eq. (21) we can numerically compute the wavevectors for
pump, signal and idler, hence allowing the calculation (up to numerical
error), of the wavector mismatch $\Delta k$, by Eq. (13), where the poling
period $\Lambda$ would be computed beforehand for phasematching the central
frequencies.

The first order term, that we will denote here as $F(\omega_{s},\omega_{i})$,
can be calculated as:

$F(\omega_{s},\omega_{i})=\gamma_{s}(\omega_{s}-\omega_{p}^{0}/2)+\gamma_{i}(\omega_{i}-\omega_{p}^{0}/2);$
(27)

where the coefficients have been defined in Eq. (16).

Therefore, the higher order terms, denoted as $O(\omega_{s},\omega_{i})$ can
be computed by:

$O(\omega_{s},\omega_{i})=\Delta
k(\omega_{s},\omega_{i})-F(\omega_{s},\omega_{i})$ (28)

In general, the higher order contributions depend on waveguide dimensions and
temperature. In our case, we are interested in seeing the contribution of
these terms under the phasematching conditions, _i.e._ , when
$\gamma_{s}=-\gamma_{i}$, so that our waveguide dimensions are fixed in order
to satisfy the condition.

As we have seen in the paper, temperature does not play an important role in
the wavelength where the first order term provides the phasematching
condition. We have checked that temperature does not change appreciably the
relative values of the higher order terms with respect to the first order term
either.

Also, we will show here only the case of KTP, although the same procedure
could be performed for any non-linear material.

Figure 15: Left to right: numerical mismatch, $\Delta
k(\lambda_{s},\lambda_{i})$, first order term, $F(\lambda_{s},\lambda_{i})$
and higher order terms, $O(\lambda_{s},\lambda_{i})$ as a function of signal
and idler wavelengths, $\lambda_{s},\lambda_{i}$.

Fig. 15 shows the three functions $\Delta k$, $F$ and $O=\Delta k-F$ under
phasematching conditions for KTP as a function of the signal and idler
wavelengths (recall that $\lambda=2\pi/\omega$). We can see that the mismatch
function is effectively well described by the first order term in the
wavelength region considered (that is bigger than the Schmidt modes bandwidth
calculated in the paper).

We therefore conclude that, for KTP, and under our configuration, the higher
order terms in the mismatch can be discarded in practice and the SGVM
condition holds.

###  B.1 Analitical mode overlap calculation

In Fig.12, Fig.13 and Fig.14, we can see that the overlap between the modes
decreases as the mode order increases.

The reason for this is two-folded: first, as the mode order increases, their
spectral width does too, reaching frequencies further away from the central
frequency, where the Taylor expansion of Eq.(15) is less precise and hence the
modes are expected to present more differences between them. Their overlap is
expected to be smaller, as we confirm.

The second reason resides in the mathematical structure of the modes itself.
If we approximate the phasematching function (sinc function), by a gaussian
function, the Schmidt modes can be calculated analytically, giving two sets of
Hermite-Gauss modes of the form:

$\mathrm{HG}_{n}(x)=\frac{1}{\sqrt{n!\sqrt{\pi}2^{n}w}}H_{n}(x)\exp(-(x-x_{0})^{2}/2w^{2})$
(29)

where $H_{n}(x)$ is the n-th order Hermite polynomial and $w$ is the
associated width of the zeroth-order function, which is a gaussian. Our
Schmidt modes, depicted in Fig.1, are not Hermite-Gauss, but they approximate
them fairly well.

If we try to compute the n-th order overlap defined in Eq.(23) from two sets
of Hermite-Gauss modes with different widths, $w$ and $w^{\prime}$, we find,
using Eq.(29):

$\displaystyle o_{n}$
$\displaystyle=\frac{\int_{-\infty}^{\infty}H^{2}_{n}(x)e^{-a_{1}x^{2}+b_{1}x}\mathrm{d}x}{\sqrt{\int_{-\infty}^{\infty}H^{2}_{n}(x)e^{-a_{2}x^{2}+b_{2}x}\mathrm{d}x\int_{-\infty}^{\infty}H^{2}_{n}(x)e^{-a_{3}x^{2}+b_{3}x}\mathrm{d}x}}$
(30) $\displaystyle a_{1}$
$\displaystyle=\frac{1}{2}\left(\frac{1}{w^{2}}+\frac{1}{w^{\prime
2}}\right)\qquad b_{1}=x_{0}\left(\frac{1}{w^{2}}+\frac{1}{w^{\prime
2}}\right)$ $\displaystyle a_{2}$
$\displaystyle=\frac{1}{w^{2}}\qquad\qquad\qquad\quad
b_{2}=\frac{2x_{0}}{w^{2}}$ $\displaystyle a_{3}$
$\displaystyle=\frac{1}{w^{\prime 2}}\qquad\qquad\qquad\quad
b_{3}=\frac{2x_{0}}{w^{\prime 2}}$

and therefore the problem is reduced to finding the integral:

$L\equiv\int_{-\infty}^{\infty}H^{2}_{n}(x)e^{-ax^{2}+bx}\mathrm{d}x$ (31)

for the solution of this integral one can define both the whole family of two
variable Hermite polynomials $H_{n}(x,y)$ and the family of two-indices
Hermite polynomials $H_{n,m}(x,y,w,z|\tau)$ Babusci  _et al._ (2012). By the
method of the generating function, one finds that the solution to the integral
$L$ in Eq.(31) is an evaluation of the two-index (with the same index $n$ and
$n$) Hermite polynomial in specific coordinates depending only on $a$ and $b$.

Figure 16: Schmidt modes overlap as a function of the mode order for two sets
of Hermite-Gauss modes with relative widths of $w/w^{\prime}=0.95$.

For typical values of relative widths for signal and idler Schmidt modes
computed in our simulations, the overlap between modes decreases slightly and
linearly. If we also model the fact that our modes are expected to be less
similar as the mode order increases, due to their larger spectral width, by
making their relative widths to increase with the mode order, then we obtain
Fig.16. In this Figure, the relative widths for the firsts modes (gaussians)
is set to $w/w^{\prime}=0.95$. From this value, we decrease their relative
widths by 2% for every higher mode order. We observe a decreasing overlap
function with the mode order, in very good accordance with the results from
the manuscript. We conclude that both the dispersion and the mathematical
nature of the modes are responsible for the observed behaviour of the overlap
as the mode order increases.