# A scheme for fully programmable linear quantum networks
based on frequency conversion

Patrick Folge<EMAIL_ADDRESS>Michael Stefszky  Benjamin Brecht
Christine Silberhorn  Paderborn University, Integrated Quantum Optics,
Institute for Photonic Quantum Systems (PhoQS), Warburgerstr. 100, 33098
Paderborn, Germany

###### Abstract

Linear optical quantum networks, consisting of a quantum input state and a
multi-port interferometer, are an important building block for many quantum
technological concepts, e.g., Gaussian boson sampling. Here, we propose the
implementation of such networks based on frequency conversion by utilising a
so called multi-output quantum pulse gate (mQPG). This approach allows the
resource efficient and therefore scalable implementation of frequency-bin
based, fully programmable interferometers in a single spatial and polarization
mode. Quantum input states for this network can be provided by utilising the
strong frequency entanglement of a type-0 parametric down conversion (PDC)
source. Here, we develop a theoretical framework to describe linear networks
based on a mQPG and PDC and utilize it to investigate the limits and
scalabilty of our approach.

††preprint: APS/123-QED

##  I Introduction

Linear optical quantum networks (LOQN), which we consider as a multi-port
interferometer with a quantum input state and followed by photon counting or
homodyne detection, have become an increasingly relevant platform and building
block for many quantum technological applications. These include (Gaussian)
boson sampling [1, 2, 3], measurement-based quantum computation [4, 5],
quantum teleportation [6, 7], quantum walks [8, 9, 10], and quantum
simulations[11, 12]. However, to enable useful applications of these concepts,
which extend beyond proof of principle demonstrations, the underlying LOQN
have to reach sufficiently high dimensionality in terms of both contributing
modes and photons. Recent implementations of high dimensional LOQN were
achieved in both the spatial [13] and temporal [14] degrees of freedom and
were able to prove quantum computational advantages. However, these approaches
require many optical components as well as synchronisation and phase stable
implementation of large experimental setups. Thus, scaling these approaches is
a challenging technical task.

LOQNs can also be implemented using spectral encodings and have been explored
by using electro optical modulators (EOMs) [15, 16, 17, 18, 19, 20] or
spectrally multimode homodyne detection [21, 22, 23]. However, the EOM based
approach requires active spectral shaping of the input quantum state which can
result in significant losses and the implementation of arbitrary LOQNs
requires complex pulse shapes of the electrical radio frequency signals. On
the other hand, the homodyne based approach faces the challenge of introducing
non-Gaussian elements, which are a crucial requirement for many of the above
mentioned applications, and require a phase stable implementation.

Figure 1: Schematic depiction of a LOQN. The multi-port interferometer is
characterized by a unitary matrix U, describing how input and output modes are
connected. A quantum state is used as the quantum resource of the system.

In this paper we explore an alternative approach for LOQNs in the spectral
domain which is based on frequency conversion. This introduces a new platform
for photonic quantum information processing and offers a highly efficient
implementation of intrinsically phase stable quantum networks with full
programmability. The general concept of a LOQN is depicted in Fig. 1 and
illustrates the main requirements; controlled preparation of input quantum
states, a stable but reconfigurable multi-port interferometer and detection.
At the core of our approach lies a multi-output quantum pulse gate [24],
allowing one to implement fully programmable frequency bin interferometer. In
combination with a highly multi-mode type-0 parametric down conversion (PDC)
source, one can realise a high dimensional LOQN in one spatial mode by using
only two non-linear waveguides. Note, that if used together with detection in
the photon number basis, our scheme does not require active phase
stabilisation.

This work is organized as follows. First, we introduce the theoretical
modelling of the mQPG, and discuss how it can be utilized to implement
interferometers based on frequency bins. Next, we introduce type-0 PDC as an
appropriate source of input quantum states for the LOQN and theoretically
model the combined system of PDC and mQPG. For this we derive a formalism
which allows us to investigate the quality of the frequency conversion based
LOQN via the squeezing strength and purity of the output state. As an
instructive example, we apply our framework to simulate a minimal example of
an LOQN comprised of a frequency bin beamsplitter and squeezed input states.
Finally, we investigate the fundamental limits of our scheme and explore its
scalability to higher numbers of contributing modes.

##  II Theoretical model

In this work, we assume that all fields are in the form of optical pulses,
which are described by a complex spectral amplitude $F(\omega)$. Such modes
are usually labeled temporal modes (TM)[25]. Further, we assume for simplicity
that all fields are in one spatial and polarisation mode. The creation
operator of a photon in such a TM is given by [25, 26]

$\displaystyle\hat{F}^{\dagger}=\int\text{d}\omega
F^{*}(\omega)\hat{a}^{\dagger}(\omega).$ (1)

We will label operators associated with a TM $F(\omega)$ with the same capital
letter and a hat $\hat{F}$.

###  II.1 Frequency bin Interferometer

At the heart of a general LOQN lies a mulit-port interferometer, preferably
programmable, which allows one to interfere and process the input states. Such
an interferometer (e.g. based on spatial modes) is characterized by a unitary
matrix $U_{kl}$, which describes how the (spatial) input modes $\hat{f}_{l}$
are connected to the (spatial) output modes $\hat{h}_{k}$ via the operator
transformation

$\displaystyle\hat{h}_{k}=\sum_{l=1}^{N_{in}}U_{kl}\hat{f}_{l}.$ (2)

Here, $N_{in}$ is the number of input modes and therefore also the size of the
unitary matrix. In other words Eq. (2) implies that the interferometer’s
outputs correspond to different superpositions of the inputs, while
maintaining energy conservation.

In this work, we will present a scheme to implement such an interferometer on
the basis of a set of $N_{in}$ separated frequency bins $A_{l}(\omega_{in})$,
where $l$ labels the individual bins at central frequency
$\overline{\omega}^{in}_{l}$ and the $\omega_{in}$-dependence encodes the
spectral profile of the bins (e.g. Gaussian). We first define a set of
superposition modes

$\displaystyle
S_{k}(\omega_{in}):=\sum_{l=1}^{N_{in}}U_{kl}A_{l}(\omega_{in}),$ (3)

which correspond to the outputs of the interferometer. The mode operators of
these then take the form $\hat{S}_{k}=\sum_{l=1}^{N_{in}}U_{kl}\hat{A}_{l}$
and contain the operators $\hat{A}_{l}$ pertaining to the individual bins. To
implement an interferometer on the frequency bin basis, we now design a
process which is capable of operating on the superposition modes $\hat{S}_{k}$
given by Eq. (3). In the following we present the details for an experimental
implementation of this task, which utilises the so called multi-output quantum
pulse gate.

###  II.2 The mQPG as an Interferometer

Figure 2: Schematic depiction of the transfer function of a two-output mQPG,
implementing a frequency bin beam splitter. The transfer function (red and
blue) is given as the product of the phase matching function (green) and the
pump spectrum (grey). Imprinting specific amplitudes and phases onto the pump
allows one to program different transfer functions.

A multi-output quantum pulse gate (mQPG) is a specially designed sum-frequency
generation (SFG) process in a periodically poled non-linear waveguide [27,
24]. As an SFG process, it is characterized by a transfer function (TF)

$\displaystyle
G_{SFG}(\omega_{in},\omega_{out})=P(\omega_{P}=\omega_{out}-\omega_{in})\cdot\Phi(\omega_{in},\omega_{out})$
(4)

which is the product of the phase-matching function
$\Phi(\omega_{in},\omega_{out})$ of the nonlinear process and the complex
spectrum $P(\omega_{P})$ of the pump [28]. This TF describes how the
amplitudes at input frequencies $\omega_{in}$ are converted to the output
frequencies $\omega_{out}$. The distinct property of a mQPG, setting it apart
from general SFG, is group velocity matching of the pump and signal fields,
which can be achieved by dispersion engineering of the waveguides [27].
Because of this, the PM-function of a mQPG is oriented perpendicular to the
output-axis, which leads to a situation where the output frequency does not
change for a broad input frequency range.

Note, that the original quantum pulse gate [27, 29] had only one output, but
recently the concept has been expanded for multiple outputs making it ideal
for network applications [24]. The mQPG combines multiple spectrally separated
phasematching peaks within one device, by modulating the periodic poling with
a superstructure. The PM function of such an mQPG with $N_{out}$ peaks then
has the form

$\displaystyle\Phi(\omega_{in},\omega_{out})\approx\sum_{m=1}^{N_{out}}O_{m}(\omega_{out}),$
(5)

where $O_{m}(\omega_{out})$ describes the peak’s spectral profile (typically
sinc-shape) and $m$ labels the different central positions
$\overline{\omega}^{out}_{m}$ of the peaks. The PM function of such an mQPG is
depicted in Fig. 2, where we sketch the mQPG’s general working principle for
two inputs and outputs.

The mQPG allows us to perform operations on arbitrarily chosen superposition
modes of frequency bins. This works under the assumption that the pump
structures (here frequency bins with spectral profile $B(\omega_{P})$) are
spectrally broader than the individual phasematching peaks
$O_{m}(\omega_{out})$ 111This assumption ensures a single mode character of
the conversion process eliminating frequency correlations[31]. Since the mQPG
is an SFG process such a pump bin with a central frequency of
$\overline{\omega}^{pump}$ addresses an input frequency bin with a central
frequency of
$\overline{\omega}^{in}_{m}=\overline{\omega}^{out}_{m}-\overline{\omega}^{pump}$
and converts it to the $m-$th output with a central frequency
$\overline{\omega}^{out}_{m}$. In more detail this means that conversion is
achieved at the intersection of the bins’s pump function $B(\omega_{P})$ and
the PM function, hence, an input bin
$A_{m}(\omega_{in})=B(\overline{\omega}^{out}_{m}-\omega_{in})$ is converted
to the output mode $O_{m}$. Note that this input mode has the same complex
spectral profile as the corresponding pump bin, but is frequency shifted.
Furthermore, due to the orientation of the PM-function, the shape and position
of the output modes do not change when the pump bin is shifted. This crucial
feature allows for the necessary multi-path interference of interferometers,
since multiple input modes can be coherently mapped to the same output by
utilising multiple pump bins (compare Fig. 2). Since the phase and amplitude
of the pump bins also determines the phase and amplitude of the conversion, we
can implement the mapping of one of the superposition modes $S_{k}$ to one of
the output modes $O_{m}$. This is done by appropriately choosing the pump bins
so that all outputs address the same input bins at centers
$\overline{\omega}^{in}_{l}$. With this it is possible to realize a multi-port
interferometer, by programming a pump spectrum of the form

$\displaystyle
P(\omega_{P})=\sum_{m=1}^{N_{out}}\sum_{l=1}^{N_{in}}U_{ml}\cdot
B(\overline{\omega}^{out}_{m}-\overline{\omega}^{in}_{l}-\omega_{P}).$ (6)

Here, $P(\omega_{P})$ is the complete pump spectrum, which is composed of
individual frequency bins labeled by the corresponding frequencies of the
input and output bins and weighted by the corresponding entry $U_{ml}$ of the
unitary matrix describing the network. Using this yields a TF

$\displaystyle G_{U}(\omega_{in},\omega_{out})$
$\displaystyle=\sum_{m=1}^{N_{out}}\sum_{l=1}^{N_{in}}U_{ml}\cdot
A_{l}(\omega_{in})\cdot O_{m}(\omega_{out})$
$\displaystyle=\sum_{m=1}^{N_{out}}S_{m}(\omega_{in})\cdot
O_{m}(\omega_{out}).$ (7)

One simple example of this scheme is depicted in Fig. 2, namely the
implementation of the TF for a balanced beamsplitter
($U_{BS}=((1,1),(1,-1))/\sqrt{2}$) on the freqeuncy bin basis. The TF in this
case is given by

$\displaystyle\begin{split}G_{BS}(\omega_{in},\omega_{out})=(A_{1}(\omega_{in})+A_{2}(\omega_{in}))\cdot
O_{1}(\omega_{out})/\sqrt{2}\\\ +(A_{1}(\omega_{in})-A_{2}(\omega_{in}))\cdot
O_{2}(\omega_{out})/\sqrt{2}.\end{split}$ (8)

To understand the action of such a mQPG on a quantum input state we can
consider the problem within the Heisenberg picture, where a general SFG
process is described via the Bogoliubov transformations [28]:

$\displaystyle\hat{b}^{\prime\prime}(\omega_{in})$
$\displaystyle=\int\text{d}\omega_{in}^{\prime}\;U^{Q}_{b}(\omega_{in},\omega_{in}^{\prime})\hat{b}^{\prime}(\omega_{in}^{\prime})$
$\displaystyle\qquad+\int\text{d}\omega_{out}^{\prime}\;V^{Q}_{b}(\omega_{in},\omega_{out}^{\prime})\hat{a}^{\prime}(\omega_{out}^{\prime})$
(9) $\displaystyle\hat{a}^{\prime\prime}(\omega_{out})$
$\displaystyle=\int\text{d}\omega_{out}^{\prime}\;U^{Q}_{a}(\omega_{out},\omega_{out}^{\prime})\hat{a}^{\prime}(\omega_{out}^{\prime})$
$\displaystyle\qquad-\int\text{d}\omega_{in}^{\prime}\;V^{Q}_{a}(\omega_{out},\omega_{in}^{\prime})\hat{b}^{\prime}(\omega_{in}^{\prime}).$
(10)

Here, the operators representing the fields in front of the SFG are labeled by
a single dash (′) and fields after the SFG by a double dash (′′) (compare Fig.
1a). We consider two different monochromatic operators $\hat{a}$ and $\hat{b}$
for input and output modes to account for the possibility of having orthogonal
polarizations and for the two separated frequency ranges of $\omega_{in}$ and
$\omega_{out}$. The functions $U_{a}^{Q},V_{a}^{Q},U_{b}^{Q},V_{b}^{Q}$ can be
calculated directly from the TF, when time ordering effects are neglected (see
Appendix D). Eq. (10) for an mQPG with a TF (7) simplifies to

$\displaystyle\hat{S}^{\prime\prime}_{m}=\cos(\theta_{m})\hat{S}^{\prime}_{m}+sin(\theta_{m})\hat{O}^{\prime}_{m},$
(11)
$\displaystyle\hat{O}^{\prime\prime}_{m}=\cos(\theta_{m})\hat{O}^{\prime}_{m}-sin(\theta_{m})\hat{S}^{\prime}_{m}.$
(12)

These are the Heisenberg operator transformations for the superposition modes
of the mQPG. The parameter $\theta_{m}$ defines the conversion efficiency
$\sin(\theta_{m})^{2}$ of the $m$-th mode. It can be adjusted with the pump
power and can in principle reach unity [32]. In this case ($\theta_{m}=\pi/2$)
Eq. (12) takes the form

$\displaystyle\hat{O}^{\prime\prime}_{m}=-\hat{S}^{\prime}_{m}=-\sum_{l=1}^{N_{in}}U_{ml}\hat{A}^{\prime}_{i},$
(13)

which is equivalent to relation (2), characterizing the multi-port
interferometer. Note however that Eq. (13) is formulated in terms of frequency
bins which are connected via frequency conversion.

The action of a mQPG can also be interpreted as a coherent filtering of a
superposition mode $S_{m}$ and the simultaneous quantum transduction to an
output mode $O_{m}$. We call this process coherent filtering, because it is
sensitive to the spectral phase of the considered modes. In the next section,
we will describe a source of input states that are naturally compatible with
the mQPG.

###  II.3 Spectrally multimode squeezing source

One desirable set of input states for LOQNs are squeezed states, for example
in Gaussian boson sampling, which we consider here. An optimal source for our
frequency bin based network, would deliver squeezed states in the input bins
$A_{k}(\omega_{in})$. However, such sources are challenging to engineer and
would require a sophisticated control of the PDC process, e.g. by utilising
resonators [33]. Therefore, we consider the use of well established degenerate
type-0 PDC sources, which in the high gain regime generate squeezed states in
many TMs [21, 34]. Such PDC sources are characterized by their joint spectral
amplitude (JSA)

$\displaystyle
f(\omega_{in},\omega_{in}^{\prime})=P(\omega_{P}=\omega_{in}+\omega_{in}^{\prime})\cdot\Phi(\omega_{in},\omega_{in}^{\prime})$
(14)

which is given as the product of pump amplitude spectrum and phase matching
function [35]. Note, that since signal and idler are indistinguishable in
type-0 PDC the JSA has to fulfil
$f(\omega_{in},\omega_{in}^{\prime})=f(\omega_{in}^{\prime},\omega_{in})$. The
evolution of an input state (here vacuum) passing through the PDC is given by
the unitary operator

$\displaystyle\hat{U}_{PDC}=\exp\left(-\frac{i}{\hbar}\int\text{d}\omega_{in}\,\text{d}\omega_{in}^{\prime}f(\omega_{in},\omega_{in}^{\prime})\hat{b}^{\dagger}(\omega_{in})\hat{b}^{\dagger}(\omega_{in}^{\prime})\right.$
$\displaystyle\left.+\quad\text{h.c.}\vphantom{\int_{1}^{2}}\right).\qquad$
(15)

For a type-0 PDC source the JSA is given as a narrow stripe oriented along the
anti-diagonal (as illustrated in Fig. 3b). This results from the orientation
of the pump function $P$ and the phase matching $\phi$ along this axis [36].
For a very narrow pump the JSA can be approximated by a $\delta$-function

$\displaystyle f(\omega_{in},\omega_{in}^{\prime})$
$\displaystyle\propto\cdot\delta(\omega_{in}+\omega_{in}^{\prime}-2\omega_{0})$
$\displaystyle\propto\sum_{k}\phi_{k}(\omega_{in}-\omega_{0})\phi_{k}^{*}(-(\omega_{in}^{\prime}-\omega_{0}))$
(16)

which can be decomposed into any orthonormal basis $\\{\phi_{k}\\}$ fulfilling
the completeness relation
$\delta(\omega-\omega^{\prime})=\sum_{k}\phi_{k}(\omega)\phi^{*}_{k}(\omega^{\prime})$.
Note that in Eq. (16) the paired functions are mirrored around the degeneracy
point $\omega_{0}$, e.g. a bin $A_{1}$ at a central frequency
$\omega_{0}+\Delta$ is paired with a bin $A_{2}$ centered at
$\omega_{0}-\Delta$. Since these bins are part of an orthonormal basis the
unitary (15) takes on the form
$\hat{U}_{PDC}=\hat{U}_{12}\otimes\hat{U}_{rest}$ where the unitary describing
the subspace of the bins is

$\displaystyle\hat{U}_{12}=\exp(\alpha\hat{A}_{1}^{\dagger}\hat{A}_{2}^{\dagger}-\alpha^{*}\hat{A}_{1}\hat{A}_{2})$
(17)

and is independent of the unitary $\hat{U}_{rest}$ which describes the
remaining space. Note that Eq. (17) has the form of the well known two-mode
squeezing (TMS) operator [37]. This shows that such a PDC source provides TMS
states between pairs of frequency bins. The parameter $\alpha$ combines
multiple constants, including the pump strength, and determines the squeezing
strength.

Figure 3: a) Schematic depiction of the combined system of Type-0 PDC source
and mQPG. The transfer function of the mQPG can be programmed to implement an
arbitrary interferometer by shaping the pump b) left: schematic depiction of
the JSA in black. The blue areas highlight the effective JSA which is
coherently filtered from the PDC state by the mQPG. The dashed arrows
highlight different two-mode squeezed states. right: the transfer function of
the mQPG which maps the coherently filtered bins into different superpositions
to different output channels c) analogous interferometer in the spatial domain

However, in reality the JSAs of physical PDC sources have a finite width and
the approximation of Eq. (16) is not valid. Therefore, we consider a general
description of type-0 PDC in our model, which allows us to consider any shape
of the JSA. This will enable us to study the influences of it’s non-negligible
width in later sections. We model the PDC in the Heisenberg picture where Eq.
(15) takes the form of the Bogoliubov transformation

$\displaystyle\hat{b}^{\prime}(\omega_{in})=\int\text{d}\omega^{\prime}_{in}\;U^{P}(\omega_{in},\omega^{\prime}_{in})\hat{b}(\omega^{\prime}_{in})\quad\quad$
$\displaystyle+\int\text{d}\omega^{\prime}_{in}\;V^{P}(\omega_{in},\omega^{\prime}_{in})\hat{b}^{{\dagger}}(\omega^{\prime}_{in}).$
(18)

Here, fields after the PDC are labeled with a dash (’) while fields in front
of the PDC do not have an additional label (compare Fig. 3a). Eq. (18) is
similar to (10) of the SFG process, however only one set of monochromatic
operators $\hat{b}$ is considered here, since signal and idler field have the
same polarization and central frequency. The functions $U^{P}$ and $V^{P}$ can
be derived from the JSA (see Appendix C).

###  II.4 Describing the complete LOQN

In summary, our scheme to implement LOQNs reads as follows: A type-0 PDC
generates TMS states between pairs of frequency bins, which are subsequently
coherently filtered and superimposed in the output modes of a mQPG. The
resulting quantum state in the outputs is then analogous to the output state
of a spatial interferometer with TMS states in the input. In Fig. 3 we
illustrate our proposed scheme for a specific example network. We depict the
required experimental components of our specific PDC source and a fully
programmable mQPG. To model this combined system we adapt the theory of
intensity filtered type-2 PDC presented in Ref. [38] to include the coherent
filtering by the mQPG. This enables us to describe the frequency converted
quantum state $\rho_{out}$ in the mQPG’s output in the continuous variable
picture via it’s covariance matrix $\sigma_{kl}$. This is possible since we
consider only Gaussian transformations (squeezing and beam splitters)[39, 40].
Due to the fact that the mQPG’s output only consist of the modes $O_{K}$ we
can describe the full output state on the basis of the operators
$\hat{O}_{k}$. The quadrature operators
$\hat{X}_{k}=\frac{1}{\sqrt{2}}(\hat{O}_{k}+\hat{O}_{K}^{\dagger})$ and
$\hat{Y}_{k}=\frac{1}{i\sqrt{2}}(\hat{O}_{k}-\hat{O}_{k}^{\dagger})$
corresponding to the different output modes can be arranged in the vector

$\displaystyle\vec{\hat{R}}=(\hat{X}_{1},\hat{Y}_{1},\hat{X}_{2},\hat{Y}_{2},...).$
(19)

Then the individual elements of the covariance matrix can be expressed as

$\displaystyle\sigma_{kl}=\frac{1}{2}\left\langle\hat{R}_{k}\hat{R}_{l}+\hat{R}_{l}\hat{R}_{k}\right\rangle-\left\langle\hat{R}_{k}\right\rangle\left\langle\hat{R}_{l}\right\rangle.$
(20)

In the following we neglect the last term because we assume vacuum states in
all fields in front of the non-linear elements. Note, however, that this is
not a necessity and that our framework can readily be adapted to include other
input states. We describe the evolution of the states in the Heisenberg
picture, by successively applying the transformations (10) and (18) to the
operators $\hat{O}^{\prime\prime}_{k}$ which results in the expression

$\displaystyle\hat{O}^{\prime\prime}_{k}$
$\displaystyle=\int\;\text{d}\omega_{out}\;H^{1}_{k}(\omega_{out})\hat{a}^{\prime}(\omega_{out})$
$\displaystyle\qquad+\int\;\text{d}\omega_{in}\;H^{2}_{k}(\omega_{in})\hat{b}(\omega_{in})+\;H^{3}_{k}(\omega_{in})\hat{b}^{\dagger}(\omega_{in})$
(21)

where the amplitude functions take the form

$\displaystyle H^{1}_{k}(\omega_{out})$
$\displaystyle=\int\text{d}\omega^{\prime}_{out}\;O_{k}(\omega^{\prime}_{out})U^{Q}_{a}(\omega^{\prime}_{out},\omega_{out})$
$\displaystyle H^{2}_{k}(\omega_{in})$
$\displaystyle=-\int\text{d}\omega^{\prime}_{out}\text{d}\omega^{\prime}_{in}\;O_{k}(\omega^{\prime}_{out})V^{Q}_{a}(\omega^{\prime}_{out},\omega^{\prime}_{in})U^{P}(\omega^{\prime}_{in},\omega_{in})$
$\displaystyle H^{3}_{k}(\omega_{in})$
$\displaystyle=-\int\text{d}\omega_{out}\text{d}\omega^{\prime}_{in}\;O_{k}(\omega^{\prime}_{out})V^{Q}_{a}(\omega^{\prime}_{out},\omega^{\prime}_{in})V^{P}(\omega^{\prime}_{in},\omega_{in}).$
(22)

Inserting these operators into Eq. (20) then allows one to calculate the
covariance matrix for any given JSA and TF, by evaluating the vacuum
expectation values. The resulting form of $\sigma_{kl}$ is derived in Appendix
E. We would like to point out that our scheme, despite our description in the
framework of continuous variable quantum optics, does not assume any
particular detection method. Experimentally it is fully compatible with
detection in the photon number basis after separating the different output
channels by frequency filtering. While simulating this scenario is
computationally demanding since it is effectively a GBS system, the photon
number distributions can in principle be derived from the covariance matrix
[41].

##  III Frequency beam splitter

Figure 4: Simulation of a frequency beamsplitter, mapping the bins $A_{1}$ and
$A_{2}$ to bins $O_{1}$ and $O_{2}$. a) Analogous spatial domain scenario b)
Joint spectral amplitude (JSA) of the PDC. Green dots show the perfect two-
mode squeezed JSA between bins $A_{1}$ and $A_{2}$ c) Transfer function of the
mQPG. d) Absoulute value of covariance matrix between bins $A_{1}$ and $A_{2}$
after PDC, e) and between bins $O_{1}$ and $O_{2}$ after mQPG.

As an instructive example of our scheme we simulate the implementation of a
simple LOQN, namely the interference of both modes from a two mode squeezed
state (TMS) on a balanced beamsplitter. For this we expect two independent
single mode squeezed (SMS) states in the output, since this scenario is the
reverse of the well known generation of TMS states by interfering SMS states
on a beamsplitter [37].

The scenario is depicted in Fig. 4, where we summarise the simulation by
displaying the JSA and TF utilised as input for the calculation together with
the resulting covariance matrices both after the PDC and at the output of the
LOQN. To keep the results as general as possible we define the spectral
dimensions (bin width, positions etc.) in terms of the simulation’s input
range $\Delta\omega_{in}$, which bounds the simulation area. In an
experimental setting, this range can be understood as the bandwidth over which
our scheme can operate and which is limited, for example, by the limited pump
spectrum of the mQPG. To highlight the experimental feasibility of our scheme
we provide simulations of realistically achievable non linear processes in
periodically poled LiNbO3 waveguides in Appendix A, according to which we
model our idealised simulations presented here. This results in a JSA which is
approximated as a Gaussian cross-section of width
$\text{FWHM}_{JSA}=0.05\cdot\Delta\omega_{in}$ oriented along the anti-
diagonal (compare Fig 4b). We normalize this JSA to a mean photon number of
$\overline{n}=1$ within the simulation region, to obtain experimentally
realistic squeezing values. The frequency bin beamsplitter on the other hand
is modeled by considering a TF of the form (8), where we consider Gaussian
shapes for all modes ($A_{1}$,$A_{2}$,$O_{1}$,$O_{2}$). The input bins were
chosen to have a width $\text{FWHM}_{bin}=0.1\cdot\Delta\omega_{in}$, larger
than $\text{FWHM}_{JSA}$.

First, we only consider the PDC and calculate the covariance matrix between
two bins $A_{1}$ and $A_{2}$ which are placed symmetrically around the
degeneracy point at $\omega_{0}$. For this we apply (18) to the broadband
operators $\hat{A}_{1}$ and $\hat{A}_{2}$ and then evaluate (20) for the
corresponding quadrature operators. As expected from the discussion above, the
resulting covariance matrix (compare Fig. 4c) represents a TMS state. This is
evident from the sub-matrices of the individual modes, which show noise above
the vacuum level of 0.5 (as one would expect from a thermal state), while
being correlated when considered as as a joint system.

The covariance matrix between the output modes $O_{1}$ and $O_{2}$ after the
mQPG is derived by applying our theoretical model of the complete LOQN to
discretized versions (1500x1500 points) of the JSA and TF. The resulting
covariance matrix (depicted in Fig. 4d) is showing two independent SMS states,
which becomes apparent from the two quadrature variances (diagonal elements)
which are squeezed below the vacuum level. As previously discussed, this is
the expected result for the interference of a TMS state on a beamplitter and
therefore establishes the capability of our scheme to implement LOQN, even
when realist PDC sources with a finite JSA width are considered.

To better understand the limits of our scheme, we explore the quality of the
output state for varying widths of the input bins $A_{k}$. Here, we only
consider the even output ($A_{1}+A_{2}$) of the mQPG. We quantify the quality
of the output state by calculating the purity and squeezing strength of this
state from the resulting covariance matrix. The purity is given by
$\gamma=\text{tr}(\rho_{out}^{2})=1/(2^{N}\sqrt{\text{det}(\sigma)})$ [39] and
the squeezing strength in $dB$ as $S=-10\cdot\log(2\cdot a)$ where $a$ is the
minimal eigenvalue of $\sigma$ [42]. We simulate these quantities for input
bins in a range from $\text{FWHM}_{bin}\approx 0$ to
$\text{FWHM}_{bin}=0.15\Delta\omega_{in}$ and for three different
normalizations of the JSA. These normalizations correspond to different pump
strengths of the PDC process and are chosen to represent JSAs with mean photon
numbers of 0.25, 1 and 2. The results are depicted in Fig. 5.

Figure 5: Squeezing and purity calculated from the covariance matrix after the
freuqency beam splitter for different input bin width $\text{FWHM}_{bin}$. The
different line types correspond to different normalizations of the JSA, which
is proportional to the pump strength.

One can immediately sees from Fig. 5 that a minimum in purity can be observed
for bins which are smaller than the width of the JSA. This can be explained by
strong edge effects during the coherent filtering. Further, no clear optimal
regime for operating the LOQN is observable, instead in the limit of larger
bins purity and squeezing continuously improve. This result is in contrast to
heralded single photon sources from type-2 PDC, where strong spectral
intensity filtering on the herald results in highly pure heralded states [38],
and showcases the fundamentally different behaviour of a coherent filter.

Figure 6: Investigation of squeezing an purity of the single mode squeezed
state in the output channel of a mQPG after filtering from a type-0 PDC state,
for varying frequency bin with and number. The white area is inaccessible,
because neighboring bins overlap. We investigate the cases of equal and
alternating (0 and $\pi$) phase. Top: Purity and Bottom: Squeezing in the
output channel. Three different JSAs are considered with
$FWHM_{JSA}=0.05,0.02,0.01\Delta\omega_{in}$.The dashed white line corresponds
to an threshold purity pf $\gamma_{0}=0.99$ and the black dashed lines
correspond to a squeezing value of $S_{0}=3dB$.

##  IV Scaling

We argue that our scheme is an excellent candidate for the resource efficient
scaling of fully programmable LOQN to higher numbers of contributing modes,
since the complete network can be achieved in only two non-linear waveguides.
To understand the fundamental limits and get an estimate of achievable
dimensionality of the systems we perform simulations to investigate how many
bins can be implemented within the given spectral window $\Delta\omega_{in}$.
For this, we consider a single output mQPG with $N$ input bins. Here, we use
box shaped bins $A_{k}$ with a width of $D$, to make use of the complete
spectral range. The bins are positioned maximally spaced, equally distributed
and symmetrically placed around the degeneracy point. For the PDC we consider
the JSA from the previous section, with different widths $\text{FWHM}_{JSA}$,
all normalized to a mean photon number of 2. To account for different
programmings of the LOQN we consider the two extremal cases of equal and
alternating phases ($0$ and $\pi$) between neighboring bins for which SMS
states are expected in the output. Purity and squeezing strength are depicted
in Fig. 6 for varying bin widths and number of used bins.

The upper left corner, representing big bins with sufficient separation, is
expectedly the only area providing good purity and squeezing values for both
cases. Therefore, the LOQN can only operate in this specific region. However,
it also becomes apparent that for thinner JSAs the usable area becomes larger
and more homogeneous, thereby demonstrating that the dimensionality of LOQN
reachable with our approach goes well beyond the two modes of the frequency
bin beamsplitter.

We also want to highlight that the investigated widths of the JSAs are well
achievable with state of the art LiNbO3 waveguides. The thinnest JSA, with
$\text{FWHM}_{JSA}=0.01\Delta\omega_{in}$. for example well approximates the
JSA achievable in a 4cm long waveguide on an input window $\Delta\omega_{in}$
corresponding to 50 nm centered at 1550 nm. In Appendix B we display a rough
estimate of the accessible dimensionalities of our scheme. We find that, with
state of the art mQPGs, input numbers in the hundreds could be expected. One
limitation of our scheme is the realization of high numbers of output modes,
owing to the fact that the different outputs have to share the same pump
bandwidth. However, it is possible to cascade multiple mQPGs, since all
superposition modes which are not addressed are passing the device
unconverted. These modes can therefore be accessed by a consecutive mQPG
corresponding to different outputs, albeit at the cost of increasing the
number of required waveguides needed for implementation.

##  V Discussion

The scheme presented in this work considers frequency bins as a basis for the
LOQN, since these are relatively easy to shape and control. But in principle
the scheme can be implemented in many other TM bases, e.g., Hermite-Gaussian
modes. For these we have found similar results, with the difference that for
centered HG modes the input states of the LOQN are SMS instead of TMS. Further
we want to highlight, again, that our scheme does not assume any specific
detection method and even the use of different detection methods in different
output channels can be imagined. When only detection in the photon number
basis is considered our scheme does not require any phase stability between
PDC and mQPG. This is because both non-linear processes are intrinsically
phase stable and a relative phase between them only results in an unknown
global phase of the output modes, which is not detectable in the photon number
basis. In this case two repetition rate locked pump laser sources for PDC and
mQPG are sufficient for a implementation of the LOQN.

Moreover, we want to mention that we assume perfect mQPGs (unity conversion
efficiency and perfect mapping of modes) throughout this work, because we want
to focus on the fundamental limits of the presented scheme. However, our
theoretical framework also allows to study more complicated scenarios
including imperfections, since it only considers a general TF and JSA as
input. This for example allows to include multi-mode effects in the outputs of
the mQPG, which can occur for imperfect PM functions. In this case one output
of the mQPG is described by a bigger covariance matrix, which describes all
modes contributing to said output.

##  VI Conclusion

In this work we have presented a novel scheme for the implementation of LOQNs
based on frequency conversion, which utilises so-called multi-output quantum
pulse gates. This approach allows one to construct fully programmable and
inherently phase stable multi-port interferometer on a frequency bin basis. We
demonstrate the feasibility of this approach and its natural compatibility
with broadband squeezing sources, by performing simulations based on a
detailed theoretical model in the continuous variable picture.

A potential experimental implementation of LOQNs based on this approach
requires only two-nonlinear waveguides for the very multi-mode input state
generation and the programmable interferometer. In contrast to other encodings
(e.g. spatial or temporal domain) the achievable dimensionality of this LOQN
is mainly limited by spectral shaping resolution and not by the number of
utilised components (e.g. beamsplitters). Due to this, the relatively low
demand on required components and the inherent compatibility with integrated
optical platforms we believe, that this approach is a promising candidate for
scaling up LOQNs towards practical applications. We find that with state-of-
the art mQPGs a few hundred input modes are feasible. However, reducing the
phasematching width of mQPGs, by for example utilising resonators, could allow
for much larger networks. We expect our approach to become an enabling
platform for future quantum technologies thanks to its inherent scalability,
full programmability, and ease of experimental implementation.

##  VII Acknowledgement

The Autors thank J. Sperling and M. Santandrea for helpful discussions. This
work was supported in part by the European Commission H2020- FET-OPEN-RIA,
(STORMYTUNE) under Grant 899587

##  VIII Comments

During the preparation of the manuscript we became aware of similar work [43]

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##  Appendix A Simulated experiment

Figure 7: Simulations of left: joint spectral amplitude from a type-0 PDC
process in LiNbO3 waveguide. right: the transfer function of a two-output mQPG
implementing a frequency beamsplitter (represented by the dotted box)

In the main text we consider idealised systems, however to demonstrate the
feasibility of the proposed systems we here provide simulations of
realistically achievable nonlinear processes. These simulations are based on
the Sellmeier-equations of titanium in-diffused LiNbO3 waveguides. In Fig. 7a
the joint spectral amplitude of a 1cm long waveguide, pumped with a 3 ps long
pulsed laser at 775 nm, is depicted. To achieve degeneracy at 1550 nm a poling
period of 16.93 $\mu$m is considered. Note, that no sinc-sidelobes are
visible, since the pump width of 0.3 nm is narrower than the phasematching.
For the simulation of the transfer function of a two-output mQPG (depicted in
Fig. 7b) we consider a poling period of 4.33 nm, an 1cm long waveguide and the
superstructure presented in Ref. [44]. To simulate a frequency bin
beamsplitter as discussed in the main text we consider a pump which is
composed of four 3 nm wide bins. The bins are centered around a central
wavelength of 860 nm and could for example be carved out from a 100 fs long
pulse. Note, that these simulations utilise conservative assumptions for the
design parameters, e.g. mQPG waveguides with length around 7cm are obtainable.

##  Appendix B Scalability of the Approach

Here we estimate the scalability of our approach to higher dimensions. We
measure this dimensionality in terms of the number of achievable input bins
$N_{in}$. This number is fundamentally limited by four factors: 1) the
spectral range $\Delta\omega_{in}$ over which the type-0 PDC can provide TMS
states between the frequency bins. 2) the pump bandwidth $\Delta\omega_{pump}$
of the mQPG which also limits the available input range. This bandwidth also
has to be divided by the number of output bins $N_{out}$, since each output
requires an equally broad pump region. 3) the phasematching width
$\delta_{mQPG}$ of the mQPG because the mQPG is working under the assumption
that the PM is narrower than the pump structure (bins). 4) the PM width
$\delta_{PDC}$ of the PDC, since the LOQNs operation is limited by this number
as discussed in the main text.

In this the first two points limit the available input range while the latter
two limit the minimal bin size, therefore we estimate the amount of available
input bins by

$\displaystyle N_{in}$ $\displaystyle=\frac{\text{available input
range}}{\text{minimal bin size}}$
$\displaystyle=\frac{\text{min}(\Delta\omega_{in},\Delta\omega_{pump}/N_{pump})}{\text{max}(\delta_{PDC},\delta_{mQPG})}.$
(23)

The results of this estimation are depicted in Fig. 8, together with the
limits set by experimentally demonstrated mQPGs.[24, 45]. Considering a 7 cm
long mQPG together with a 4 THz pump spectrum for example could allow for
systems with 200 input bins.

Figure 8: Estimation of the achievable number of input bins for different
parameters of the available bandwidth of the network (vertical axis) and for
different phasematching widths (horizontal axis). The horizontal white lines
correspond to a mQPG with a pump bandwidth of 4 THz and different numbers of
outputs. The vertical lines correspond to mQPGs with different lengths.

##  Appendix C Theory of Type-0 PDC

A type-0 PDC in a single spatial mode and polarization (e.g. in wavguides) can
be described by the unitary operator [35]

$\displaystyle\hat{U}_{PDC}=\exp\left(-\frac{i}{\hbar}\int\text{d}\omega_{i}\text{d}\omega^{\prime}_{i}\;\text{f}(\omega_{i},\omega^{\prime}_{i})\hat{b}^{\dagger}(\omega_{i})\hat{b}^{\dagger}(\omega^{\prime}_{i})\right.$
$\displaystyle\left.\;+\;\text{h.c.}\right).$ (24)

Therein, $\text{f}(\omega_{i},\omega^{\prime}_{i})$ is the joint spectral
amplitude (JSA) of the process. Here, we neglect time ordering effects, which
become relevant for very strong pump fields [28]. In a type-0 PDC signal and
idler are indistinguishable and therefore the JSA has to fulfil
$f(\omega_{i},\omega^{\prime}_{i})=f(\omega^{\prime}_{i},\omega_{i})$. A
common approach in describing PDC states is by performing a Schmidt
decomposition of the JSA

$\displaystyle-\frac{i}{\hbar}\text{f}(\omega_{i},\omega^{\prime}_{i})$
$\displaystyle=\sum_{k}r_{k}^{P}\phi_{k}^{P*}(\omega_{i})\phi_{k}^{P*}(\omega^{\prime}_{i})$
(25)

which results in a set of orthogonal Schmidt-modes
$\left\\{\phi_{k}^{P}(\omega_{i})\right\\}$ with Schmidt-coefficients
$r_{k}^{P}$ [46]. These modes are equal for signal and idler because they are
indistinguishable. By defining the operators
$\hat{\phi}_{k}^{\dagger}:=\int\text{d}\omega_{i}\;\phi_{k}^{P*}(\omega_{i})\hat{b}^{\dagger}(\omega_{i})$,
the Schmidt-decomposition allows to rewrite the unitary (24) as

$\displaystyle\hat{U}_{PDC}=\bigotimes_{k}\exp\left[r_{k}^{P}(\hat{\phi}_{k}^{\dagger})^{2}\;+\;\text{h.c.}\right]=\bigotimes_{k}\hat{S}_{k}^{(SMS)}(r_{k}^{P}),$
(26)

which corresponds to multiple independent single mode squeezing operators on
the different Schmidt modes. However, besides this fundamental structure of
type-0 PDC sources we show in the main text, that in the case of very multi-
mode PDC, also two-mode squeezed states can be extracted from such a source.

In the Heisenberg picture, the unitary (24) takes the form of a linear
Bogoliobov transformation [28]:

$\displaystyle\hat{b}^{\prime}(\omega_{i})=\int\text{d}\omega^{\prime}_{i}\;U^{P}(\omega_{i},\omega^{\prime}_{i})\hat{b}(\omega^{\prime}_{i})+\int\text{d}\omega^{\prime}_{i}\;V^{P}(\omega_{i},\omega^{\prime}_{i})\hat{b}^{\dagger}(\omega^{\prime}_{i}).$
(27)

Here, $U^{P}$ and $V^{P}$ can be expressed with help of the Schmidt modes
$\phi_{k}^{P}(\omega_{i})$ and can therefore be directly obtained from the
JSA. They have the form [28]

$\displaystyle U^{P}(\omega_{i},\omega^{\prime}_{i})$
$\displaystyle=\sum_{k}\phi_{k}^{P*}(\omega_{i})\cosh(r_{k}^{P})\phi_{k}^{P}(\omega^{\prime}_{i})$
$\displaystyle V^{P}(\omega_{i},\omega^{\prime}_{i})$
$\displaystyle=\sum_{k}\phi_{k}^{P*}(\omega_{i})\sinh(r_{k}^{P})\phi_{k}^{P*}(\omega^{\prime}_{i}).$
(28)

##  Appendix D Theory of SFG

Because the multi-output quantum pulse gate is based on a sum frequency
generation (SFG) process, it can be described by the unitary operator of a
general SFG process [28]

$\displaystyle\hat{U}_{SFG}=\exp\left(-\frac{i}{\hbar}\int\text{d}\omega_{i}\text{d}\omega_{o}\;\text{G}(\omega_{i},\omega_{o})\hat{a}^{\dagger}(\omega_{o})\hat{b}(\omega_{i})\right.$
$\displaystyle\left.\;+\;\text{h.c.}\right).$ (29)

Here, $G(\omega_{i},\omega_{o})$ is the transfer function (TF) of the process,
which describes, how the input frequencies $\omega_{i}$ are converted the the
output frequencies $\omega_{o}$. Note, that we choose one of the input fields
of the mQPG to be represented by the same operators $\hat{b}(\omega_{i})$ as
the field of the PDC process.

In the Heisenberg picture the SFG process takes the form of the Bogoliobov
transformations [28]

$\displaystyle\hat{b}^{\prime\prime}(\omega_{i})$
$\displaystyle=\int\text{d}\omega^{\prime}_{i}\;U^{Q}_{b}(\omega_{i},\omega^{\prime}_{i})\hat{b}^{\prime}(\omega^{\prime}_{i})$
$\displaystyle\qquad+\int\text{d}\omega^{\prime}_{o}\;V^{Q}_{b}(\omega_{i},\omega^{\prime}_{o})\hat{a}^{\prime}(\omega^{\prime}_{o})$
$\displaystyle\hat{a}^{\prime\prime}(\omega_{o})$
$\displaystyle=\int\text{d}\omega_{o}^{\prime}\;U^{Q}_{a}(\omega_{o},\omega^{\prime}_{o})\hat{a}^{\prime}(\omega^{\prime}_{o})$
$\displaystyle\qquad-\int\text{d}\omega^{\prime}_{i}\;V^{Q}_{a}(\omega_{o},\omega^{\prime}_{i})\hat{b}^{\prime}(\omega^{\prime}_{i}).$
(30)

The functions U and V can again be calculated by performing a Schmidt
decomposition of the TF which takes the form

$\displaystyle-\frac{i}{\hbar}\text{G}(\omega_{i},\omega_{o})=-\sum_{k}r_{k}^{Q}\phi_{k}^{Q}(\omega_{i})\psi_{k}^{Q*}(\omega_{o})$
(31)

and results in the two orthonormal bases
$\left\\{\phi_{k}^{Q}(\omega_{i})\right\\}$ and
$\left\\{\psi_{k}^{Q}(\omega_{o})\right\\}$. This then allows to connect the
Schmidt-modes to the Bogoliobov transformations via [28]

$\displaystyle U^{Q}_{b}(\omega_{i},\omega^{\prime}_{i})$
$\displaystyle=\sum_{k}\phi_{k}^{Q*}(\omega_{i})\cos(r_{k}^{Q})\phi_{k}^{Q}(\omega^{\prime}_{i})$
$\displaystyle V^{Q}_{b}(\omega_{i},\omega^{\prime}_{o})$
$\displaystyle=\sum_{k}\phi_{k}^{Q*}(\omega_{i})\sin(r_{k}^{Q})\psi_{k}^{Q}(\omega^{\prime}_{o})$
$\displaystyle U^{Q}_{a}(\omega_{o},\omega^{\prime}_{o})$
$\displaystyle=\sum_{k}\psi_{k}^{Q*}(\omega_{o})\cos(r_{k}^{Q})\psi_{k}^{Q}(\omega^{\prime}_{o})$
$\displaystyle V^{Q}_{a}(\omega_{o},\omega^{\prime}_{i})$
$\displaystyle=\sum_{k}\psi_{k}^{Q*}(\omega_{o})\sin(r_{k}^{Q})\phi_{k}^{Q}(\omega^{\prime}_{i}).$
(32)

Defining the broadband operators
$\hat{R}_{k}=\int\text{d}\omega_{o}\psi_{k}^{Q}(\omega_{o})\hat{a}(\omega_{o})$
and
$\hat{H}_{k}=\int\text{d}\omega_{i}\phi_{k}^{Q}(\omega_{i})\hat{b}(\omega_{i})$
corresponding to the Schmidt modes allows to simply the transformation (30) to

$\displaystyle\hat{H}^{\prime}_{k}$
$\displaystyle=\cos(r_{k}^{Q})\hat{H}_{k}+\sin(r_{k}^{Q})\hat{R}_{k}$ (33)
$\displaystyle\hat{R}^{\prime}_{k}$
$\displaystyle=\cos(r_{k}^{Q})\hat{R}_{k}-\sin(r_{k}^{Q})\hat{H}_{k}.$ (34)

These equations have the same structure as (12), however since we are
considering general SFG the modes ($\hat{H}_{k}$ and $\hat{R}_{k}$) can
spectrally overlap and are therefore not separately detectable via spectral
multiplexing. This is one of the features enabled via considering a TF of form
(7), realizable in mQPGs, which converts to well separated output modes
$O_{k}$. In other words, the Schmidt modes of the mQPG with a TF (7) are the
superposition modes $S_{k}$ and the output modes $O_{k}$, with degenerate
(equal weights) Schmidt coefficients.

##  Appendix E Combining PDC and mQPG

The goal of our model is the description of the output quantum state of the
mQPG. Since each output channel corresponds to one mode $O_{k}$, this output
state can be characterised in terms of the density matrix $\sigma$ on the
basis of the output modes $O_{k}$ (compare Eq. (20)). We describe the dynamics
of the two non-linear processes in the Heisenberg picture by consecutively
applying (18) and (10) to the output operators

$\displaystyle\hat{O}^{\prime\prime}_{k}=\int\text{d}\omega_{o}O_{k}(\omega_{o})\hat{a}^{\prime\prime}(\omega_{o})$
(35)

and obtain

$\displaystyle\hat{O}^{\prime\prime}_{k}$
$\displaystyle=\int\text{d}\omega_{o}\;H^{1}_{k}(\omega_{o})\hat{a}^{\prime}(\omega_{o})$
$\displaystyle\qquad+\int\text{d}\omega_{i}H^{2}_{k}(\omega_{i})\hat{b}(\omega_{i})+H^{3}_{k}(\omega_{i})\hat{b}^{\dagger}(\omega_{i})$
(36)

where we have defined the functions

$\displaystyle H^{1}_{k}(\omega_{o})$
$\displaystyle=\int\text{d}\omega^{\prime}_{o}\;O_{k}(\omega^{\prime}_{o})U^{Q}_{a}(\omega^{\prime}_{o},\omega_{o})$
$\displaystyle H^{2}_{k}(\omega_{i})$
$\displaystyle=-\int\text{d}\omega^{\prime}_{o}\text{d}\omega^{\prime}_{i}\;O_{k}(\omega^{\prime}_{o})V^{Q}_{a}(\omega^{\prime}_{o},\omega^{\prime}_{i})U^{P}(\omega^{\prime}_{i},\omega_{i})$
$\displaystyle H^{3}_{k}(\omega_{i})$
$\displaystyle=-\int\text{d}\omega^{\prime}_{o}\text{d}\omega^{\prime}_{i}\;O_{k}(\omega^{\prime}_{o})V^{Q}_{a}(\omega^{\prime}_{o},\omega^{\prime}_{i})V^{P}(\omega^{\prime}_{i},\omega_{i}).$
(37)

These functions can be derived from a given JSA and TF by utilising (28) and
(32). To now describe the output state of the mQPG, we first observe that the
we can neglect displacement (second term of (20)), since we assume vacuum
states in front of the non-linear elements and do not consider seeding. By
considering the operator order of (19) the covariance matrix can be
constructed from the 2x2 submatrices

$\displaystyle\widetilde{\sigma}_{kl}=\begin{pmatrix}\left\langle\hat{X}_{k}\hat{X}_{l}\right\rangle+\left\langle\hat{X}_{l}\hat{X}_{k}\right\rangle,&\left\langle\hat{X}_{k}\hat{Y}_{l}\right\rangle+\left\langle\hat{Y}_{l}\hat{X}_{k}\right\rangle\\\
\left\langle\hat{Y}_{k}\hat{X}_{l}\right\rangle+\left\langle\hat{X}_{l}\hat{Y}_{k}\right\rangle,&\left\langle\hat{Y}_{k}\hat{Y}_{l}\right\rangle+\left\langle\hat{Y}_{l}\hat{Y}_{k}\right\rangle\end{pmatrix},$
(38)

where $k$ and $l$ label two modes from $\\{\hat{O}_{k}\\}$. The submatrices
for $k=l$ describe the substates in the individual channels and for for $k\neq
l$ it describes the the quadrature covariances between two different output
modes. To calculate these submatrices we first express the individual elements
in terms of the output operators and obtain

$\displaystyle\left\langle\hat{X}_{k}\hat{X}_{l}\right\rangle$
$\displaystyle=\frac{1}{2}\left\langle\
\hat{O}_{k}\hat{O}_{l}+\hat{O}_{k}\hat{O}_{l}^{\dagger}+\hat{O}_{k}^{\dagger}\hat{O}_{l}+\hat{O}_{k}^{\dagger}\hat{O}_{l}^{\dagger}\right\rangle$
$\displaystyle\left\langle\hat{X}_{k}\hat{Y}_{l}\right\rangle$
$\displaystyle=\frac{1}{2i}\left\langle\
\hat{O}_{k}\hat{O}_{l}-\hat{O}_{k}\hat{O}_{l}^{\dagger}+\hat{O}_{k}^{\dagger}\hat{O}_{l}-\hat{O}_{k}^{\dagger}\hat{O}_{l}^{\dagger}\right\rangle$
$\displaystyle\left\langle\hat{Y}_{k}\hat{X}_{l}\right\rangle$
$\displaystyle=\frac{1}{2i}\left\langle\
\hat{O}_{k}\hat{O}_{l}+\hat{O}_{k}\hat{O}_{l}^{\dagger}-\hat{O}_{k}^{\dagger}\hat{O}_{l}-\hat{O}_{k}^{\dagger}\hat{O}_{l}^{\dagger}\right\rangle$
$\displaystyle\left\langle\hat{Y}_{k}\hat{Y}_{l}\right\rangle$
$\displaystyle=\frac{-1}{2}\left\langle\
\hat{O}_{k}\hat{O}_{l}-\hat{O}_{k}\hat{O}_{l}^{\dagger}-\hat{O}_{k}^{\dagger}\hat{O}_{l}+\hat{O}_{k}^{\dagger}\hat{O}_{l}^{\dagger}\right\rangle.$
(39)

By assuming vacuum input states and inserting (37), we are then able to
calculate the terms in (39) which results in

$\displaystyle\bra{0}\hat{O}_{k}\hat{O}_{l}\ket{0}$
$\displaystyle=\int\text{d}\omega_{i}\;H^{2}_{k}(\omega_{i})H^{3}_{l}(\omega_{i})$
(40) $\displaystyle\bra{0}\hat{O}_{k}\hat{O}_{l}^{\dagger}\ket{0}$
$\displaystyle=\int\text{d}\omega_{o}\;H^{1}_{k}(\omega_{o})H_{l}^{1*}(\omega_{o})$
$\displaystyle\qquad+\int\text{d}\omega_{i}\;H^{2}_{k}(\omega_{i})H_{l}^{2*}(\omega_{i})$
$\displaystyle\bra{0}\hat{O}_{k}^{\dagger}\hat{O}_{l}\ket{0}$
$\displaystyle=\int\text{d}\omega_{i}\;H_{k}^{3*}(\omega_{i})H^{3}_{l}(\omega_{i})$
$\displaystyle\bra{0}\hat{O}_{k}^{\dagger}\hat{O}_{l}^{\dagger}\ket{0}$
$\displaystyle=\int\text{d}\omega_{i}\;H_{k}^{3*}(\omega_{i})H_{l}^{2*}(\omega_{i}).$
(41)

This now allows to calculate the complete covariance matrix at the output of
the mQPG. We want to mention that this approach can be applied to describe
general systems comprised of a type-0 PDC and a SFG processes, since it only
requires a JSA and TF as input. The output modes then take the form of the
output Schmidt basis ($\psi_{k}^{Q}(\omega_{o})$) of the TF. This for example
allows to study multi-mode effects occurring in imperfect mQPGs.


max}}dt\langle\Sigma_{0,1}^{(0,2)}(t)|B_{t}(V_{t})\varphi_{\rm o},$ (B.40)

where the surface state is defined as

$\langle\Sigma_{0,1}^{(0,2)}(t)|\Psi\coloneqq\langle
F_{t}\circ\Psi(0)\rangle_{C_{(\pi,2\pi t)}}.$ (B.41)

The local coordinate map $F_{t}(w)$ will be fixed shortly by imposing BRST
decoupling (which is equivalent to the homotopy relations). $V_{t}$ is the
Schiffer vector and $B_{t}(V_{t})$ the Beltrami form which are defined as
follows

$B_{t}(V_{t})=\oint_{0}\frac{dw}{2\pi i}b(w)V_{t}(w),\qquad\qquad
V_{t}(w)=\frac{\partial F^{-1}_{t}}{\partial t}\left(F_{t}(w)\right),$ (B.42)

where the contour surrounds the puncture in the local coordinate frame, see
[38] and [22] for more details.
Crucially, the surface state satisfies

$\frac{d}{dt}\langle\Sigma_{0,1}^{(0,2)}(t)|=-\langle\Sigma_{0,1}^{(0,2)}(t)|B_{t}(V_{t})Q_{\rm
o}.$ (B.43)

With these premises the full annulus amplitude (B.2) becomes

$\begin{split}A_{0;1}^{0,2}(\varphi_{\rm o})=&-\frac{1}{2\pi
i}\int_{0}^{1}\frac{dq}{q}\langle\Sigma_{1,1}^{(0,1)}|\left(1_{\mathcal{H}_{\rm
o}}\otimes b_{0}^{+}q^{L_{0}^{+}}\right)\left(\varphi_{\rm o}\otimes
l_{0,0}^{(0,1)}\right)\\\ &+\int_{t_{\rm min}}^{t_{\rm
max}}dt\langle\Sigma_{0,1}^{(0,2)}(t)|B_{t}(V_{t})\varphi_{\rm o}\\\
&-\int_{0}^{1}\frac{dq}{q}\langle\Sigma_{0,3}^{(0,1)}|\left(1_{\mathcal{H}_{\rm
o}}\otimes 1_{\mathcal{H}_{\rm o}}\otimes
b_{0}q^{L_{0}}\right)\left(o^{i}\otimes\varphi_{\rm o}\otimes
o_{i}\right).\end{split}$ (B.44)

Now, we want to verify if and how BRST exact states decouple. To do so let us
consider $\varphi_{\rm o}=Q_{\rm o}\Lambda$

$\begin{split}A_{0;1}^{0,2}(Q_{\rm o}\Lambda)=&-\frac{1}{2\pi
i}\int_{0}^{1}\frac{dq}{q}\langle\Sigma_{1,1}^{(0,1)}|\left(1_{\mathcal{H}_{\rm
o}}\otimes b_{0}^{+}q^{L_{0}^{+}}\right)\left(Q_{\rm o}\Lambda\otimes
l_{0,0}^{(0,1)}\right)\\\ &+\int_{t_{\rm min}}^{t_{\rm
max}}dt\langle\Sigma_{0,1}^{(0,2)}(t)|B_{t}(V_{t})Q_{\rm o}\Lambda\\\
&-\int_{0}^{1}\frac{dq}{q}\langle\Sigma_{0,3}^{(0,1)}|\left(1_{\mathcal{H}_{\rm
o}}\otimes 1_{\mathcal{H}_{\rm o}}\otimes
b_{0}q^{L_{0}}\right)\left(o^{i}\otimes Q_{\rm o}\Lambda\otimes
o_{i}\right),\end{split}$ (B.45)

by using the fact that the surface states are BRST invariant, the property
(B.43), and the relations $\bm{Q}_{\rm o}\bm{U}_{\rm o}=0$ and $Q_{\rm
c}l_{0,0}^{(0,1)}=0$ we get

$\begin{split}A_{0;1}^{0,2}(Q_{\rm o}\Lambda)&=+\frac{1}{2\pi
i}\int_{0}^{1}\frac{dq}{q}\langle\Sigma_{1,1}^{(0,1)}|\left(1_{\mathcal{H}_{\rm
o}}\otimes\left[b_{0}^{+}q^{L_{0}^{+}},Q_{\rm
c}\right]\right)\left(\Lambda\otimes l_{0,0}^{(0,1)}\right)\\\
&\quad-\int_{t_{\rm min}}^{t_{\rm
max}}dt\frac{d}{dt}\langle\Sigma_{0,1}^{(0,2)}(t)|\Lambda\\\
&\quad+\int_{0}^{1}\frac{dq}{q}\langle\Sigma_{0,3}^{(0,1)}|\left(1_{\mathcal{H}_{\rm
o}}\otimes 1_{\mathcal{H}_{\rm o}}\otimes\left[b_{0}q^{L_{0}},Q_{\rm
o}\right]\right)\left(o^{i}\otimes\Lambda\otimes o_{i}\right)\\\
&=-\frac{1}{2\pi
i}\int_{0}^{\infty}ds\frac{d}{ds}\langle\Sigma_{1,1}^{(0,1)}|\left(1_{\mathcal{H}_{\rm
o}}\otimes e^{-sL_{0}^{+}}\right)\left(\Lambda\otimes
l_{0,0}^{(0,1)}\right)\\\ &\quad+\langle\Sigma_{0,1}^{(0,2)}(t_{\rm
min})|\Lambda-\langle\Sigma_{0,1}^{(0,2)}(t_{\rm max})|\Lambda\\\
&\quad-\int_{0}^{\infty}ds\frac{d}{ds}\langle\Sigma_{0,3}^{(0,1)}|\left(1_{\mathcal{H}_{\rm
o}}\otimes 1_{\mathcal{H}_{\rm o}}\otimes
e^{-sL_{0}}\right)\left(o^{i}\otimes\Lambda\otimes o_{i}\right)\\\
&=\frac{1}{2\pi i}\langle\Sigma_{1,1}^{(0,1)}|\left(\Lambda\otimes
l_{0,0}^{(0,1)}\right)+\langle\Sigma_{0,1}^{(0,2)}(t_{\rm
min})|\Lambda-\langle\Sigma_{0,1}^{(0,2)}(t_{\rm
max})|\Lambda+\langle\Sigma_{0,3}^{(0,1)}|\left(o^{i}\otimes\Lambda\otimes
o_{i}\right),\end{split}$ (B.46)

where we made the change of variable $q=e^{-s}$ and we ignored the
contributions at the boundary of moduli space (open and closed string
degeneration). Notice that the last line is equivalent to the homotopy
relation (3.19). Therefore, the amplitude vanishes for all $\Lambda$ if we
impose

$\displaystyle\langle\Sigma_{0,3}^{(0,1)}|\left(o^{i}\otimes
1_{\mathcal{H}_{\rm o}}\otimes o_{i}\right)=\langle\Sigma_{0,1}^{(0,2)}(t_{\rm
max})|,$ (B.47) $\displaystyle-\frac{1}{2\pi
i}\langle\Sigma_{1,1}^{(0,1)}|\left(1_{\mathcal{H}_{\rm o}}\otimes
l_{0,0}^{(0,1)}\right)=\langle\Sigma_{0,1}^{(0,2)}(t_{\rm min})|.$ (B.48)

These two relations allow us to determine the local coordinate $F_{t}$ and
thus to fully define the fundamental vertex. Notice that the surface states in
the l.h.s are obtained respectively trough the open and closed plumbing
fixture, with $q=1$. Similarly, the surface states associated to the plumbing
fixture for generic $q$ can be written as

$\displaystyle\langle\Sigma_{\rm open}^{\rm
p.f.}(q)|=\langle\Sigma_{0,3}^{(0,1)}|\left(o^{i}\otimes 1_{\mathcal{H}_{\rm
o}}\otimes q^{L_{0}}o_{i}\right),$ (B.49) $\displaystyle\langle\Sigma_{\rm
closed}^{\rm p.f.}(q)|=-\frac{1}{2\pi
i}\langle\Sigma_{1,1}^{(0,1)}|\left(1_{\mathcal{H}_{\rm o}}\otimes
q^{L_{0}^{+}}l_{0,0}^{(0,1)}\right)$ (B.50)

and we have

$\displaystyle\langle\Sigma_{\rm open}^{\rm p.f.}(q)|\Psi=\langle
f_{0}\circ\Psi(0)\rangle_{\Sigma_{\rm open}^{\rm p.f.}(q)}=\langle g_{\rm
o}\circ f_{0}\circ\Psi(0)\rangle_{C_{\pi,2\pi t_{\rm o}}}=\langle G_{\rm
o}\circ\Psi(0)\rangle_{C_{\pi,2\pi t_{\rm o}}},$ (B.51)
$\displaystyle\langle\Sigma_{\rm closed}^{\rm p.f.}(q)|\Psi=\langle d\circ
f_{\rm o}\circ\Psi(0)\rangle_{\Sigma_{\rm closed}^{\rm p.f.}(q)}=\langle
g_{\rm c}\circ d\circ f_{\rm o}\circ\Psi(0)\rangle_{C_{\pi,2\pi t_{\rm
c}}}=\langle G_{\rm c}\circ\Psi(0)\rangle_{C_{\pi,2\pi t_{\rm c}}}.$ (B.52)

Therefore, considering the $b$-ghost insertions the amplitude becomes

$\begin{split}A_{0;1}^{0,2}(\varphi_{\rm o})=&+\int_{0}^{1}\frac{dq}{q}\langle
d\circ f_{\rm o}\circ\varphi_{\rm o}(0)d\circ f_{\rm c}\circ
b_{o}\rangle_{\Sigma_{\rm closed}^{\rm p.f.}(q)}\\\ &+\int_{t_{\rm
min}}^{t_{\rm max}}dt\langle B_{t}(V_{t})F_{t}\circ\varphi_{\rm
o}(0)\rangle_{C_{\pi,2\pi t}}\\\ &-\int_{0}^{1}\frac{dq}{q}\langle
f_{0}\circ\varphi_{\rm o}(0)f_{\xi}\circ b_{0}\rangle_{\Sigma_{\rm open}^{\rm
p.f.}(q)}\end{split}$ (B.53)

Finally, let us focus on the local coordinate map $F_{t}$. In particular, to
ensure (B.47) and (B.48), we need

$\begin{cases}&\langle\Sigma^{\rm p.f}_{\rm
open}(q=1)|\Psi=\langle\Sigma^{(0,2)}_{0,1}(t_{\rm max})|\Psi\\\
&\langle\Sigma^{\rm p.f}_{\rm
closed}(q=1)|\Psi=\langle\Sigma^{(0,2)}_{0,1}(t_{\rm
min})|\Psi\end{cases}\longrightarrow\begin{cases}&G_{\rm
o}(w)|_{q=1}=F_{t}(w)|_{t=t_{\rm max}}\\\ &G_{\rm
c}(w)|_{q=1}=F_{t}(w)|_{t=t_{\rm min}}\end{cases}$ (B.54)

thus $F_{t}$ can be any holomorphic function which continuously interpolates
$G_{\rm c}(w)|_{q=1}$ and $G_{\rm o}(w)|_{q=1}$ in the interval $t\in[t_{\rm
min},t_{\rm max}]$.

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