$\partial_{\mu_{1}}\partial_{\mu_{2}}\tilde{\delta}x^{\nu}=\partial_{\mu_{1}}\partial_{\mu_{2}}\partial_{\mu_{3}}\tilde{\delta}x^{\nu}=\cdots=0$
and
$\partial_{\mu_{1}}\partial_{\mu_{2}}\partial_{\mu_{3}}\tilde{\delta}x^{\nu}=\partial_{\mu_{1}}\partial_{\mu_{2}}\partial_{\mu_{3}}\partial_{\mu_{4}}\tilde{\delta}x^{\nu}=\cdots=0$
requires the following equations to hold, respectively:

$\displaystyle 2g^{\mu\beta}\,{}^{\rm
EL}\\!\mathcal{D}_{g^{\beta\nu}}\sqrt{-g}\mathcal{F}-2B_{\nu\beta}\,^{\rm
EL}\\!\mathcal{D}_{B_{\mu\beta}}\sqrt{-g}\mathcal{F}-A_{\nu}\,^{\rm
EL}\\!\mathcal{D}_{A_{\mu}}\sqrt{-g}\mathcal{F}+u^{\mu}\,{}^{\rm
EL}\\!\mathcal{D}_{u^{\nu}}\sqrt{-g}\mathcal{F}$
$\displaystyle-\sqrt{-g}T^{\mu}_{\ \
\nu}[\sqrt{-g}\mathcal{F}]+\partial_{\rho}\Omega^{\rho\mu}_{\ \ \
\nu}[\sqrt{-g}\mathcal{F}]\equiv 0$ (362)
$\displaystyle\frac{1}{2}\Big{(}\Omega^{\mu_{1}\mu_{2}}_{\ \ \ \ \ \
\nu}[\sqrt{-g}\mathcal{F}]+\Omega^{\mu_{2}\mu_{1}}_{\ \ \ \ \ \
\nu}[\sqrt{-g}\mathcal{F}]\Big{)}+\partial_{\rho}\Xi^{\rho\mu_{1}\mu_{2}}_{\ \
\ \ \ \ \ \nu}[\sqrt{-g}\mathcal{F}]\equiv 0.$ (363)

Combining identities (358) and (362), the quantity
$\partial_{\mu}\partial_{\rho}\Omega^{\rho\mu}_{\ \ \
\nu}[\sqrt{-g}\mathcal{F}]$ identically vanishes:

$\displaystyle\partial_{\mu}\partial_{\rho}\Omega^{\rho\mu}_{\ \ \
\nu}[\sqrt{-g}\mathcal{F}]\equiv 0.$ (364)

Therefore, the quantity
$\partial_{\rho}\Omega^{\rho\mu}_{\nu}[\sqrt{-g}\mathcal{F}]$ can be regarded
as an energy-momentum peudotensor666The quantity
$\partial_{\rho}\Omega^{\rho\mu}_{\ \ \ \nu}[\sqrt{-g}\mathcal{F}]$ does not
transform as a tensor by coordinate transformations.  including the
gravitational field.

$\displaystyle M_{\rm Noerther}:=-\partial_{\rho}\Omega^{\rho 0}_{\ \ \
0}[\sqrt{-g}\mathcal{F}]$ (365) $\displaystyle(P_{\rm
Noether})_{a}:=-\partial_{\rho}\Omega^{\rho 0}_{\ \ \
a}[\sqrt{-g}\mathcal{F}]$ (366)

are conserved charges, which can be interpreted as energy and momentum of the
fields including the gravity, respectively.

Let us deform the identity (362) into the form used in section 4.1.
Contracting Eq. (362) with $u_{\mu}u^{\nu}$,

$\displaystyle 2u^{\beta}u^{\nu}\,{}^{\rm
EL}\\!\mathcal{D}_{g^{\beta\nu}}\sqrt{-g}\mathcal{F}+u_{\nu}\,^{\rm
EL}\\!\mathcal{D}_{u_{\nu}}\sqrt{-g}\mathcal{F}$
$\displaystyle-2u_{\mu}u^{\nu}B_{\nu\beta}\,^{\rm
EL}\\!\mathcal{D}_{B_{\mu\beta}}\sqrt{-g}\mathcal{F}-u^{\nu}A_{\nu}u_{\mu}\,^{\rm
EL}\\!\mathcal{D}_{A_{\mu}}\sqrt{-g}\mathcal{F}-\sqrt{-g}\mathcal{F}$
$\displaystyle\equiv$ $\displaystyle-
u_{\mu}u^{\nu}\partial_{\rho}\Omega^{\rho\mu}_{\ \ \
\nu}[\sqrt{-g}\mathcal{F}]+\sum_{k=0}^{\infty}u_{\mu}\,^{\rm
EL}\\!\mathcal{D}_{\partial_{\mu}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}\sqrt{-g}\mathcal{F}\cdot
u^{\nu}\partial_{\nu}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f.$

Using the relation (63), the first two terms of Eq. (B.2) are combined and
yield $\,{}^{\rm EL}\\!\bar{\mathcal{D}}_{g^{\rho\nu}}$, i.e.,

$\displaystyle 2u^{\rho}u^{\nu}\,{}^{\rm
EL}\\!\bar{\mathcal{D}}_{g^{\rho\nu}}\sqrt{-g}\mathcal{F}-2u_{\mu}u^{\nu}B_{\nu\beta}\,^{\rm
EL}\\!\mathcal{D}_{B_{\mu\beta}}\sqrt{-g}\mathcal{F}-u^{\nu}A_{\nu}u_{\mu}\,^{\rm
EL}\\!\mathcal{D}_{A_{\mu}}\sqrt{-g}\mathcal{F}-\sqrt{-g}\mathcal{F}$
$\displaystyle\equiv$ $\displaystyle-
u_{\mu}u^{\nu}\partial_{\rho}\Omega^{\rho\mu}_{\ \ \
\nu}[\sqrt{-g}\mathcal{F}]+\sum_{k=0}^{\infty}u_{\mu}\,^{\rm
EL}\\!\mathcal{D}_{\partial_{\mu}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}\sqrt{-g}\mathcal{F}\cdot
u^{\nu}\partial_{\nu}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f.$

###  B.3 Identities from the $U(1)$ gauge transformation

Next, consider the transformation,

$\displaystyle\varphi^{\prime}(x)$ $\displaystyle=$ $\displaystyle
e^{ie\Lambda}\varphi(x)$ (369) $\displaystyle A_{\mu}^{\prime}(x)$
$\displaystyle=$ $\displaystyle A_{\mu}(x)+\partial_{\mu}\Lambda(x),$ (370)

where $\Lambda(x)$ is an arbitrary function, and $e$ is the electric charge.
This is called $U(1)$ gauge transformation. In this subsection, let us derive
the identities that any $U(1)$ gauge symmetric function $\mathcal{L}$
satisfies.

The infinitesimal transformation of $U(1)$ gauge transformation is

$\displaystyle\tilde{\delta}x=0$ (371)
$\displaystyle\tilde{\delta}\varphi=ie\varphi\delta\Lambda$ (372)
$\displaystyle\tilde{\delta}A_{\mu}=\partial_{\mu}\delta\Lambda$ (373)
$\displaystyle\tilde{\delta}B_{\mu\nu}=0$ (374)
$\displaystyle\tilde{\delta}g^{\mu\nu}=0.$ (375)

Note that since the gauge transformation does not contain the coordinate
transformation, the variations $\tilde{\delta}$ and $\delta$ are equivalent:

$\tilde{\delta}=\delta.$

The variations of the derivatives of $\varphi$ and $A_{\mu}$ are

$\displaystyle\tilde{\delta}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}\varphi$
$\displaystyle=$ $\displaystyle ie\sum_{j=0}^{k}\bigg{(}\begin{array}[]{c}k\\\
j\end{array}\bigg{)}(\partial_{\mu_{j+1}}\cdots\partial_{\mu_{k}}\varphi)\partial_{\mu_{1}}\cdots\partial_{\mu_{j}}\delta\Lambda$
(378)
$\displaystyle\tilde{\delta}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}A_{\mu}$
$\displaystyle=$
$\displaystyle\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}\partial_{\mu}\delta\Lambda.$
(379)

Substituting Eqs. (378), (379), and $\tilde{\delta}x^{\mu}=0$ into Eq. (B.1),

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\int_{\mathcal{M}}{\mathrm{d}}^{n}x\bigg{[}\,^{\rm
EL}\\!\mathcal{D}_{f}\sqrt{-g}\mathcal{F}\cdot\tilde{\delta}f+\partial_{\mu}\bigg{(}\sum_{k=0}^{\infty}\,{}^{\rm
EL}\\!\mathcal{D}_{\partial_{\mu}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}\sqrt{-g}\mathcal{F}\cdot\tilde{\delta}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f\bigg{)}\bigg{]}$
(382) $\displaystyle=$
$\displaystyle\int_{\mathcal{M}}{\mathrm{d}}^{n}x\bigg{[}\,^{\rm
EL}\\!\mathcal{D}_{A_{\mu}}\sqrt{-g}\mathcal{F}\cdot\partial_{\mu}\delta\Lambda+ie\varphi\,^{\rm
EL}\\!\mathcal{D}_{\varphi}\sqrt{-g}\mathcal{F}\cdot\delta\Lambda$
$\displaystyle+\partial_{\mu}\bigg{(}\sum_{k=0}^{\infty}\,{}^{\rm
EL}\\!\mathcal{D}_{\partial_{\mu}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}\sqrt{-g}\mathcal{F}\cdot\tilde{\delta}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f\bigg{)}\bigg{]}$
$\displaystyle=$
$\displaystyle\int_{\mathcal{M}}{\mathrm{d}}^{n}x\bigg{\\{}\bigg{[}-\partial_{\mu}\Big{(}\,^{\rm
EL}\\!\mathcal{D}_{A_{\mu}}\sqrt{-g}\mathcal{F}\Big{)}+ie\varphi\,^{\rm
EL}\\!\mathcal{D}_{\varphi}\sqrt{-g}\mathcal{F}\bigg{]}\cdot\delta\Lambda$
$\displaystyle+\partial_{\mu}\bigg{(}\,^{\rm
EL}\\!\mathcal{D}_{A_{\mu}}\sqrt{-g}\mathcal{F}\cdot\delta\Lambda+\sum_{k=0}^{\infty}\,{}^{\rm
EL}\\!\mathcal{D}_{\partial_{\mu}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}\sqrt{-g}\mathcal{F}\cdot\tilde{\delta}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f\bigg{)}\bigg{\\}}.$

Since the gauge function $\Lambda$ is arbitrary, this integral is invariant
under the gauge transformation such that the variation of the gauge function
$\delta\Lambda$ is $0$ on $\partial\mathcal{M}$. Therefore, the square bracket
in Eq. (382) should be identically $0$, i.e.,

$\displaystyle\partial_{\mu}\Big{(}\,^{\rm
EL}\\!\mathcal{D}_{A_{\mu}}\sqrt{-g}\mathcal{F}\Big{)}-ie\varphi\,^{\rm
EL}\\!\mathcal{D}_{\varphi}\sqrt{-g}\mathcal{F}\equiv 0.$ (383)

Substituting the identity (383) into Eq. (382),

$\displaystyle 0$ $\displaystyle=$
$\displaystyle\int_{\mathcal{M}}{\mathrm{d}}^{n}x\partial_{\mu}\bigg{(}\,^{\rm
EL}\\!\mathcal{D}_{A_{\mu}}\sqrt{-g}\mathcal{F}\cdot\delta\Lambda+\sum_{k=0}^{\infty}\,{}^{\rm
EL}\\!\mathcal{D}_{\partial_{\mu}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}\sqrt{-g}\mathcal{F}\cdot\tilde{\delta}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f\bigg{)}$
(386) $\displaystyle=$
$\displaystyle\int_{\mathcal{M}}{\mathrm{d}}^{n}x\bigg{[}\Big{(}\partial_{\mu}\,^{\rm
EL}\\!\mathcal{D}_{A_{\mu}}\sqrt{-g}\mathcal{F}\Big{)}\cdot\delta\Lambda+\,^{\rm
EL}\\!\mathcal{D}_{A_{\mu}}\sqrt{-g}\mathcal{F}\cdot\partial_{\mu}\delta\Lambda$
$\displaystyle+\partial_{\mu}\bigg{(}\sum_{k=0}^{\infty}\,{}^{\rm
EL}\\!\mathcal{D}_{\partial_{\mu}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f}\sqrt{-g}\mathcal{F}\cdot\tilde{\delta}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}f\bigg{)}\bigg{]}$
$\displaystyle=$
$\displaystyle\int_{\mathcal{M}}{\mathrm{d}}^{n}x\bigg{\\{}\partial_{\mu}\bigg{(}\,^{\rm
EL}\\!\mathcal{D}_{A_{\mu}}\sqrt{-g}\mathcal{F}+ie\sum_{k=0}^{\infty}(\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}\varphi)\,^{\rm
EL}\\!\mathcal{D}_{\partial_{\mu}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}\varphi}\sqrt{-g}\mathcal{F}\bigg{)}\cdot\delta\Lambda$
$\displaystyle+\bigg{[}\,^{\rm
EL}\\!\mathcal{D}_{A_{\mu}}\sqrt{-g}\mathcal{F}+\partial_{\nu}\,^{\rm
EL}\\!\mathcal{D}_{\partial_{\nu}A_{\mu}}\sqrt{-g}\mathcal{F}+ie\sum_{k=0}^{\infty}(\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}\varphi)\,^{\rm
EL}\\!\mathcal{D}_{\partial_{\nu}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}\varphi}\sqrt{-g}\mathcal{F}$
$\displaystyle+ie\partial_{\nu}\bigg{(}\sum_{k=0}^{\infty}k(\partial_{\mu_{2}}\cdots\partial_{\mu_{k}}\varphi)\,^{\rm
EL}\\!\mathcal{D}_{\partial_{\nu}\partial_{\mu}\partial_{\mu_{2}}\cdots\partial_{\mu_{k}}\varphi}\sqrt{-g}\mathcal{F}\bigg{)}\bigg{]}\cdot\partial_{\mu}\delta\Lambda$
$\displaystyle+\cdots\bigg{\\}}.$

The integral (386) is $0$ no matter what the variation of the gauge function
$\delta\Lambda$, the derivatives $\partial_{\mu}\Lambda$,
$\partial_{\mu_{1}}\partial_{\mu_{2}}\delta\Lambda$, $\cdots$, and the domain
of the integration $\mathcal{M}$. Therefore, the coefficients of
$\delta\Lambda$, $\partial_{\mu}\delta\Lambda$,
$\partial_{\mu_{1}}\partial_{\mu_{2}}\delta\Lambda$, $\cdots$ should be
identically $0$. For example, the first two are

$\displaystyle\partial_{\mu}\bigg{(}\,^{\rm
EL}\\!\mathcal{D}_{A_{\mu}}\sqrt{-g}\mathcal{F}+ie\sum_{k=0}^{\infty}(\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}\varphi)\,^{\rm
EL}\\!\mathcal{D}_{\partial_{\mu}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}\varphi}\sqrt{-g}\mathcal{F}\bigg{)}\equiv
0$ (387)

and

$\,{}^{\rm
EL}\\!\mathcal{D}_{A_{\mu}}\sqrt{-g}\mathcal{F}+\partial_{\nu}\,^{\rm
EL}\\!\mathcal{D}_{\partial_{\nu}A_{\mu}}\sqrt{-g}\mathcal{F}+ie\sum_{k=0}^{\infty}(\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}\varphi)\,^{\rm
EL}\\!\mathcal{D}_{\partial_{\nu}\partial_{\mu_{1}}\cdots\partial_{\mu_{k}}\varphi}\sqrt{-g}\mathcal{F}$
$\displaystyle+ie\partial_{\nu}\bigg{(}\sum_{k=0}^{\infty}k(\partial_{\mu_{2}}\cdots\partial_{\mu_{k}}\varphi)\,^{\rm
EL}\\!\mathcal{D}_{\partial_{\nu}\partial_{\mu}\partial_{\mu_{2}}\cdots\partial_{\mu_{k}}\varphi}\sqrt{-g}\mathcal{F}\bigg{)}\equiv
0.$ (388)

## References

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  *

# A flexible modular all-fiber based photon pair source for quantum key
distribution in a network

Maximilian Tippmann, Erik Fitzke, Oleg Nikiforov, Philipp Kleinpaß, Till
Dolejsky, Maximilian Mengler and Thomas Walther  Institute for Applied
Physics, Technische Universität Darmstadt,
Schlossgartenstraße 7, 64289 Darmstadt, Germany<EMAIL_ADDRESS>darmstadt.de

###### Abstract

Entanglement-based QKD protocols require robust and stable photon pair sources
in terms of high heralding efficiencies or photon pair generation rates even
under harsh environmental conditions, e.g. when operated in the field. In this
paper, we report on a flexible, tunable, alignment-free, all-fiber coupled
photon source based on spontaneous parametric down-conversion in periodically
poled crystals. It can be operated in continuous-wave and pulsed modes,
respectively. Its rack-compatible and modular setup allows a straight forward
plug-and-play integration of coding-modules e.g. interferometers to enable
various QKD protocols such as phase or phase-time coding. We demonstrate
operation as a type-II and a type-0 SPDC stage proving the high flexibility of
our source. Furthermore, we demonstrate simultaneous operation of SHG and SPDC
in a double-pass configuration within the same nonlinear crystal further
simplifying the hardware requirements of our source. To evaluate the
conversion efficiencies of our modules, we employ data post-processing to
remove artefacts from detector afterpulsing and deadtimes of the detectors. We
investigate the source performance for various repetition rates.

Keywords: photon pair source, quantum key distribution, network, heralded
single photon source, double-pass setup

##  1 Introduction

Photon pair sources offer a large variety of applications such as quantum key
distribution (QKD) [1, 2, 3] and quantum computing [4]. Such sources can be
used as heralded single photon sources [5, 6, 7] or to provide entangled pairs
of photons [8, 9, 10, 11, 12, 13, 14]. Each application comes with a unique
set of source requirements in terms of photon production rates, noise
suppression and filtering as well as photon separation and coupling for the
experiments.

There are various methods to produce photon pairs. The most common is
spontaneous parametric down-conversion (SPDC) typically in crystal waveguides
employing periodic-poling for quasi-phase matching [13]. However, for typical
application scenarios in QKD setups the produced photon pairs are required to
be coupled to an optical fiber to bridge larger distances. Thus, some photon
pair sources use spontaneous four-wave-mixing in fibers to avoid coupling from
a waveguide to a fiber reducing losses. These sources have the drawback to
require cryogenic cooling of the setup to suppress Raman-noise photons which
otherwise would degrade the signal-to-noise ratio of the source [15, 16].

With our source, we demonstrate that a good performance can be achieved with
commercially available fiber-packaged crystal waveguides allowing a simple
setup while accessing fibers to provide a stable source for entanglement-based
QKD setups [17]. Many other sources are optimized towards a single purpose.
For instance, these use asymmetric filtering in signal/idler paths, fixed pump
powers, are not completely fiber-coupled or employ a seed laser at different
wavelengths [5, 18, 19, 20]. However, our source can be used for various
setups due to its plug-and-play design and high flexibility in terms of
continuous wave (cw) to high repetition rate pulsed pumping for SPDC photon
pairs. This allows an easy integration of coding modules such as
interferometers for QKD protocols like phase-time-coding [17, 21] or adaption
for phase-coding [18]. Producing photon pairs in the telecom C-band centered
at 1550 nm, our source is ideally suited for use in real-world fiber QKD
applications as this window offers lowest transmission losses in standard
optical fibers. The source is completely fiber-coupled, hence dropping the
need for any optical alignment.

##  2 Source Setup

Figure 1: Setup of the photon pair source. Figure (a) shows the complete
source with its modules. The interferometer (IF) in the dashed box depicts the
position where coding modules can be placed into the source setup. Figures
(b)-(e) show the setup of each module in more detail. Note that there are two
different setups for the SPDC-module, a type-II and a type-0 module.
Abbreviations: AM - Amplitude Modulator; BS - Beamsplitter; FI - Faraday
Isolator; FBG - Fiber Bragg Grating; Pulsgen. - Pulse Generator; Bias - Bias
Controller; WDM - Wavelength Division Multiplexer; LP-Filter - Long Pass
Filter; PBS - Polarization Beam Splitter; AWG - Arrayed Waveguide Grating;
C-B.-Filter - C-Band Filter.

The photon source features a modular design with six main components (cf.
figure 1(a)). The seed laser is a wavelength-stabilized DFB-laser (Wavelength
References Clarity NLL-1550-HP) operating at 1550.51 nm on the edges of two
adjacent 100 GHz ITU-DWDM grid channels. The laser is followed by an electro-
optic modulator shaping pulses of flexible duration and repetition rate from
the cw radiation of the seed laser. Then, the ensuing laser pulses are
amplified employing an in-house made erbium-doped fiber amplifier (EDFA). This
is followed by type-0 second harmonic generation (SHG) in a periodically poled
LiNbO3 crystal. In the final step, we use type-II or type-0 spontaneous
parametric down-conversion (SPDC) in a periodically poled LiNbO3 crystal
waveguide as well. Throughout the setup, several beamsplitters/tap couplers
are placed used either for diagnostic purposes or as feedback for power
stabilization. Depending on the intended use of the photon pairs we introduce
an imbalanced interferometer (IF) between the EDFA and the SHG step which acts
as a coding module for QKD applications e.g. employing a phase-time protocol
[17]. Notably, by starting with 1550 nm radiation, undergoing SHG followed by
SPDC we obtain several advantages: First, commercially available components
for telecom wavelengths can be used, hence dropping the need of expensive TiSa
lasers at 775 nm. Second, by placing the interferometer in front of the SHG it
operates at 1550 nm. Thus, it can be manufactured by the same method and with
the same components as receiver interferometers for a phase-time coding setup
like [17], drastically simplifying the precise fabrication. This proves
especially useful in these kinds of setups as all interferometers from source
and receivers have to be matched in their arm lengths with a precision of a
few hundred micrometers [17].

The electro-optic modulator shapes pulses followed by a beamsplitter where a
small portion (10 %) of the light is fed to the bias controller. This device
delivers a DC offset to the electro-optic modulator to either lock the
electro-optic modulator to minimum or maximum transmission. A pulse generator
(either HP8131A or HP8133A) generates the electrical pulses to switch the
electro-optical modulator to produce optical pulses. If cw operation is
required, the pulse generator can be switched off and the bias controller is
set to maximum transmission locking. Hence, no physical change of the setup is
required to switch from pulsed to cw operation.

Figure 2: Averaged EDFA output power for pulsed operation at different pump
power and pulse repetition rates. The pulse shapes are nearly rectangular with
estimated FWHM pulse widths of 433 ps for a repetition rate of 10 MHz, 396 ps
for 100 MHz and 501 ps for 300 MHz, respectively.

The in-house built double-pass EDFA incorporates fiber Bragg gratings serving
as pump and amplified spontaneous emission (ASE) filters achieving a
spectrally pure seed output with residual light suppressed by at least 80 dB.
The pump laser can deliver a power of up to 900 mW at 976 nm. This leads to a
large variety of accessible output powers for various repetition rates and
pulse lengths. Thus, average output powers of up to 100 mW at 300 MHz pulse
repetition rate with pulse durations of 501 ps have been observed (cf. figure
2).

Figure 3: Averaged output powers of the SHG crystal when pumped with
fundamental pulses of 501 ps FWHM at 300 MHz repetition rate. The SHG-power is
measured at the 99 % output of the tap coupler (cf. figure 1(d)).

For the SHG stage, the setup uses 5 meters of tightly coiled PM780 fiber
following the SHG crystal to filter out residual fundamental light at 1550 nm.
When tested with 2 meters of fiber, again, very high suppression of at least
79 dB is reached without causing significant insertion loss ($<1~{}$dB) for
the SHG light at 775 nm. The SHG crystal is a 34 mm long PPLN Type-0 crystal
waveguide from NTT Electronics. The crystal features conversion efficiencies
of up to 22 % at only 54 mW averaged fundamental power (equivalent to
approximately 360 mW pulse peak power) without using a resonator setup (cf.
figure 3). The pulse duty cycle is approximately 15 % which allows SHG pulse
peak powers of around 100 mW, when increasing the pump even further. Of
course, for photon pair generation aiming for QKD only fundamental powers of
less than 10 mW are necessary.

The curve of the generated SHG power can be explained considering a simple
pump depletion model following [22] and extending it by a constant to factor
in the coupling loss at the crystal’s facets,

$P_{\textrm{SHG}}=c^{2}P_{\textrm{Fundamental}}\cdot\tanh^{2}\left(\sqrt{\eta_{\textrm{BK}}\cdot
c\cdot P_{\textrm{Fundamental}}}\right)$ (1)

with the coupling ratio $c$, the Boyd-Kleinman conversion efficiency
$\eta_{\textrm{BK}}$, and the pulse peak power $P_{\textrm{Fundamental}}$
calculated from the average fundamental power $\bar{P}$, the pulse repetition
rate $r$ and pulse length $\tau$ according to:

$P_{\textrm{Fundamental}}=\frac{\bar{P}}{r\cdot\tau}\textrm{ .}$ (2)

For simplicity of the fit model the coupling ratio is set to be equal for both
facets. As demonstrated, our setup enables a wide range of available SHG
powers to ensure a large accessible region of different mean photon pair
numbers per pulse $\mu$ at the SPDC process.

Each SPDC module consists of a PPLN-crystal for photon pair generation
followed by two long pass filters to remove the remaining 775 nm light. The
subsequent 7.1 nm band pass (full-width half-maximum) or C-band filter (cf.
figure 1(e)) makes sure the photon pairs are within the operation ranges of
the photon separation filters to avoid undesired crosstalk between the
channels. To separate the photons, we employed a polarization beam splitter
(PBS) for the type-II process or a standard telecom C-band arrayed-waveguide-
grating (AWG) with 100 GHz channel separation for the type-0 process,
respectively. A comparison of both crystals or modules can be found in table
1. The crystals are usually operated at $43.56~{}^{\circ}$C in case of the
type-0 and $41.64~{}^{\circ}$C for the type-II crystal, respectively, via
package integrated thermo-electric coolers. The temperature is stabilized
within a 10 mK range of these temperatures.

The complete source is made of fiber-coupled components with the crystals
being fiber coupled and packaged as well. The modules are connected via
standard optical fibers. These connections allow flexible source configuration
e.g. bypassing of the amplitude modulator or integrating interferometers but
introduce additional losses because of the fiber connectors. However, this
does not degrade the performance of the source as all losses before the SPDC
process can be simply mitigated by increasing the pump power of the EDFA. The
setup is polarization maintaining until photon separation to ensure a stable
pump power at the SPDC crystals in the relevant polarization axis, thus
allowing stable photon pair generation rates. Further stabilization of pump
power can be added by adapting the driver current of the EDFA via a software-
based PID loop using the power measured at the beamsplitter after SHG as
feedback. If additional pump power is needed, the plug-and-play design allows
an easy integration of further EDFAs.

Table 1: Comparison of type-0 and type-II SPDC crystals (above horizontal
line) and modules (below horizontal line).

  * Property | Type-0 | Type-II
---|---|---
manufacturer | NTT Electronics | AdVR
material | ZnO:PPLN | MgO:PPLN
crystal length | 34 mm | 24 mm
fiber coupled | yes | yes
est. conversion efficiency | $(4.8\pm 0.2)\times 10^{-7}$ | $(7.6\pm 0.4)\times 10^{-10}$
| (within 7.1 nm band) |
| 5.1 THz usable width with |
| C-band filter |
Filters | longpass | longpass
| \+ C-Band filter | \+ 7.1 nm bandpass
Photon pair separation | wavelength | polarization
| (symmetric to center WL) | (slow/fast)

###  2.1 Double-pass type-0 configuration

Both, the SHG crystal and the type-0 SPDC crystal are 34-mm long PPLN-
Crystals. If the same crystal is used for both SHG and SPDC, the complexity
and cost of the photon pair source is considerably reduced. Cascaded SHG and
SPDC within a single crystal have been demonstrated [23] utilizing a single-
pass configuration. However, this can make filtering of noise photons tedious.
While there are setups utilizing a double-pass configuration with a single
crystal, these are usually designed such that polarization entangled photons
can be produced [13, 24, 25]. As our source does not produce polarization
entanglement, we have investigated an operation mode of the source where the
type-0 SPDC crystal was used to produce SHG light in forward direction and
photon pairs in backward direction. The setup is shown in figure 4.

Figure 4: Setup for photon pair production in double-pass configuration. The
type-0 SPDC module is used to produce SHG light in the forward direction and
photon pairs in the reverse direction. Abbreviations: PMR - Polarization
Maintaining Retroreflector; P.Filter - Pump light Filter; C-B.-Filter - C-Band
Filter.

To operate the photon pair source with a single crystal a polarization
maintaining circulator is placed in front of the type-0 module (without the
C-band filter), sending 1550-nm pump light through the long pass filters into
the crystal. After passing a wound fiber the produced 775-nm light is retro-
reflected back into the crystal. As in the SHG-module a 90:10 beam splitter is
introduced to monitor the back-propagating SHG power that now pumps the SPDC
process. Finally, the generated photon pairs are guided through a pump light
filter followed by the C-band filter before they are distributed via
wavelength-division demultiplexing. The pump light filter is used to remove
any remaining laser light and consists of two DWDM filters and two fiber Bragg
gratings. This filter as well as all other added optical components are
connected through standard mating-sleeves, thus rebuilding the source from
single-pass to double-pass configuration only requires reconnecting optical
fibers.

##  3 Experimental Details and Results

###  3.1 Heralding Efficiency

An important feature of photon sources is the heralding efficiency, i.e. the
probability that an idler photon is extracted when the signal has been
detected [26]. We estimated the heralding efficiency of our source for both
modules and for different channel combinations using the setup depicted in
figure 1(a). Both outputs (for the type-0 module two frequency conjugate
outputs of the AWG) are connected to a detection unit each. The photons are
detected with two single-photon avalanche detectors (ID Quantique ID220) with
set detection efficiencies of 10 % and dead times of 5 $\mu$s. The detection
events are time-tagged by an ID Quantique ID900 time controller. A pulse
repetition rate of 33 MHz was chosen. The heralding efficiency for channel 1
and 2 is given by [27] $\eta_{1/2}=C_{12}/(\eta_{\textrm{Det}}R_{2/1})$ with
the count rates in each channel $R_{i}$, the coincidence rate between both
channels $C_{12}$ and the detector efficiency $\eta_{\textrm{Det}}$ to account
for the losses due to imperfect photon detection. The actual efficiency for
our detectors is 10.6 % and 9.3 % [28]. The type-II module allows heralding
efficiencies of up to 36 % at 1.8 pJ per pump pulse. The heralding
efficiencies of the type-0 module are lower, reaching up to 11.3 % at best for
the channel at 191.7 THz while ranging between 8.7 % and 12.3 % for the other
tested DWDM channels at approximately 0.05 pJ per pump pulse. The pump powers
were chosen to achieve similar count rates at the detectors for the type-II
and type-0 process. The results for all channels are displayed in figure
11(a). The large difference between the type-II and type-0 modules can be
attributed to the different multiplexing technique accompanied by a higher
insertion loss of the AWG. Thus, the heralding efficiency can be significantly
improved by using low-loss filters and gratings for photon separation. For the
type-II module, we chose a symmetric filter configuration in contrast to [5]
allowing both fast and slow axis to achieve similar heralding efficiencies
e.g. enabling both axes for use as signal/herald arm. This is important as the
source is featured as a photon pair source and not solely a heralded single
photon source. We also note that the heralding efficiencies for both modules
can be improved by directly splicing the SPDC-crystals’ fibers to the
subsequent filters instead of using standard fiber connectors. The same holds
for the C-band filter in the type-0 module. The count rates yielded from the
heralding efficiency measurements can be utilized to estimate the source
brightness following the method from [5] with

$B=\frac{R_{1}R_{2}}{C_{12}P_{\textrm{Pump}}t\Delta\lambda}\textrm{ .}$ (3)

This yields brightnesses of $2.7\times 10^{6}$ pairs/(s$\cdot$mW$\cdot$nm) for
the type-II module (at 1.8 pJ per pulse) and $3.9\times 10^{8}$ to $4.4\times
10^{8}$ pairs/(s$\cdot$mW$\cdot$nm) (at $0.048-0.056$ pJ per pulse) for the
type-0 module, respectively. The values obtained via (3) do not take into
account count rate / coincidental rate distortions induced by the detector
dead times. This might be negligible for detectors with short dead times
compared to the count rates per second such as superconducting-nanowire
detectors [29]. For detectors where this is not the case, one detector being
in dead time will cause an underestimation of the coincidental rates leading
to an overestimation of the brightness.

###  3.2 Conversion Efficiencies

To overcome this issue of imperfect detectors the brightness can be determined
by evaluating the crystals’ conversion efficiencies with a slightly more
sophisticated model. To estimate the conversion efficiencies or the photon
pair production rates of our SPDC crystals we employed a setup shown in figure
5. The SHG module was followed by a 90:10 beam splitter to monitor the SHG
power impinging on the SPDC crystals. Either one of the type-0 or type-II
crystals had been connected to the 90 % output of the beam splitter and was
followed by the long pass filters to remove the SHG light. Contrary to the
setup of the SPDC modules in figure 1(e) a 7.1 nm band pass filter has been
placed into the setup for both types of crystals to allow for a better
comparison of the respective conversion efficiencies.

Figure 5: Setup to measure the photon pair production rates and estimate the
respective conversion efficiencies of both (type-0/type-II) waveguides. Only
one SPDC waveguide is inserted into the measurement setup at a time. Following
the filters, the setup is connected to two single-photon avalanche detectors
(ID Quantique ID220). Note that for the type-0 waveguide a 7.1 nm-band pass
filter was used instead of the C-band filter to allow a better comparison to
the type-II process in terms of conversion efficiency. The detection events
are time-tagged by an ID Quantique ID900 (not depicted). For the type-0
process a 50/50 beamsplitter is used instead of the polarization beam
splitter.

For both crystals several measurements at various input (SHG) powers have been
conducted. From the timestamps of both detectors a photon pair production rate
can be deduced applying equations for the coincidence and count rate
statistics following [30] to

$\mu_{\textrm{gen}}=\gamma\frac{\eta_{\textrm{pair}}}{\eta_{s}\eta_{i}}\frac{\left(\frac{N_{1}}{T}-r_{1}\right)\left(\frac{N_{2}}{T}-r_{2}\right)T}{N_{\mu}}\textrm{
.}$ (4)

$N_{i}$ is the number of photon counts detected at a single detector $i$
within measurement time $T$. The quantity $r_{i}$ is the measured dark count
rate of a detector. $N_{\mu}$ is the number of coincidence counts without
accidental counts. The factor $\gamma$ in (4) is $\gamma=1$ for deterministic
photon pair splitting, e.g. using a PBS for the type-II process, or
$\gamma=1/2$ for probabilistic splitting, e.g. using a 50/50 beam splitter in
case of the type-0 process. The factor 1/2 comes from the 50/50 beam splitter
guiding both photons into the same channel with 50% probability, so that the
measured coincidence rate is halved. The factors $\eta$ arise from the fact
that the losses are not frequency-independent for photon pair production i.e.
$\eta_{\textrm{pair}}\neq\eta_{i}\eta_{s}$. Due to the spectral correlation of
the signal and idler photons, the detection probabilities for signal and idler
photons are not independent. Hence, these factors are defined via the given
filters’ transmission functions $p(f)$ as

$\eta_{S}=\frac{1}{\Delta F}\int_{F}p_{S}(\nu_{0}+f)df\textrm{ and
}\eta_{I}=\frac{1}{\Delta F}\int_{F}p_{I}(\nu_{0}-f)df$ (5)
$\eta_{\textrm{pair}}=\frac{1}{\Delta
F}\int_{F}p_{S}(\nu_{0}+f)p_{I}(\nu_{0}+f)df$ (6)

where $F$ is a frequency interval with width $\Delta F$.

However, as shown in [28] the detectors are prone to afterpulsing and dead
time effects. One detector being in dead time while the other is not, will
lead to an underestimation of the ratio between coincidences to single counts.
Afterpulsing manifests itself as a detection event causing additional clicks
after the dead time ended although no photon was incident on the detector. For
the type-II measurement the detectors showed a maximum afterpulsing
probability of 5.11 % and 3.43 % respectively. In addition, this considerably
distorts the coincidental to single count rate ratio, hence corrupting the
estimate of the conversion efficiencies. In order to avoid this detection
drawback, post-processing the recorded timestamps to remove such effects from
the data has been introduced [31, 32]. As found in [28] the afterpulsing is
limited to several tens of $\mu s$ after the dead time ended. By choosing an
extended dead time of $\tau_{\textrm{sel}}=40~{}\mu s$ we post-select the data
as depicted in figure 6.

Figure 6: Post-selection of the crystal efficiency measurement data to remove
afterpulsing and dead time effects. Upper diagram: Correction for individual
count rates. Lower diagram: Correction for coincidence rates. If an event has
been registered the detector is considered to be not ready for time
$\tau_{\textrm{sel}}$ thus ignoring subsequent events within this timespan.
For the coincidence correction a subsequent event is kept if it occurs within
the coincidental time window.

In general, a detection event is only considered when outside the detectors
deadtime, i.e. when the event was detected after a sufficiently long period of
time after the previous event. To evaluate the single count rate of a detector
all timestamps within $\tau_{\textrm{sel}}$ after a previous detection are
removed. For coincidences both detectors have to be considered: Here,
$\tau_{\textrm{sel}}$ is applied to both detectors, although an event only has
been detected at one detector. However, if the second event is within the
coincidence time of the previous event, it is still kept as a valid detection.
To yield count rates, the number of counts and coincidences of the post-
selected data are divided by the total time a detector has been ready. Using
our previous results for the photon pair production rates and the post-
selected data we have calculated conversion efficiencies for every pump power.
The results are depicted in figure 7. An average weighted with the uncertainty
of the data points is estimated to state the respective conversion
efficiencies. The weighted average is indicated by a horizontal line in figure
7. This corresponds to conversion efficiencies of $(4.8\pm 0.2)\times 10^{-7}$
for the type-0 and $(7.6\pm 0.4)\times 10^{-10}$ for the type-II crystal in
terms of photon pairs produced per impinging photon at 775 nm. For the type-0
crystal the conversion efficiency is estimated using the 7.1-nm band pass
filter only cutting off large parts of the spectrum, as can be seen from the
spectrum in the following section. However, this method is convenient as it
allows a better comparison to the type-II process and avoids distortion of the
measurement introduced by components (e.g. 50/50 coupler) for wavelengths far
from the center wavelength. Nevertheless, the conversion efficiency of the
type-0 crystal is significantly higher than the estimated value when
considering a larger wavelength range.

The brightness of our source can be directly calculated from the conversion
efficiencies when considering the pump photon flux and the respective spectral
widths. For the type-0 spectrum a brightness of $B=3.32\times
10^{8}\textrm{~{}photon pairs}/(\textrm{s nm mW})$ is achieved. The type-II
module shows a brightness of $B=2.48\times 10^{6}\textrm{~{}photon
pairs}/(\textrm{s nm mW})$ when taking into account that the full-width-half-
maximum of the spectra is estimated to 1.2 nm (cf. figure 10), i.e. much
smaller than the 7.1 nm transmission window of the installed filter. Both
values are in close vicinity of the previously estimated values from (3),
confirming the applicability of our model.

For QKD applications the photon pairs will be separated and sent via different
channels to several parties, hence a good estimate of the mean photon number
produced per pump pulse $\mu$ per channel is required to control parameters
like time-base QBER e.g. in phase-time protocols avoiding multiple photon
pairs per detection cycle [17]. For the type-II process, we can separate the
photons via polarization into two channels, so the mean photon number per pump
pulse per channel can be calculated by

$\mu=\eta_{\textrm{conv}}\cdot N_{\textrm{Pump}}$ (7)

with the respective process’ conversion efficiency $\eta_{\textrm{conv}}$ and
the number of pump photons per pulse $N_{\textrm{Pump}}$. For the type-0
process, the formula has to be adapted, as the photon separation occurs via
wavelength multiplexing instead of polarization. Hence, the mean photon number
per wavelength channel has to be considered which strongly depends on
transmission characteristics of the installed photon separation device e.g. an
AWG. Thus, it is more convenient to state a spectral conversion efficiency
which is $\eta_{\nu}=6.81\times 10^{-7}/\textrm{THz}$ and calculating the mean
photon number from it employing the respective channel width. For further
evaluation of the photon transmissions it is important to take into account
the spectral dependence of the respective wavelength-multiplexer employing (5)
and (6).

###  3.3 Photon Pair spectra

As a next step, we investigated the spectra for both SPDC crystals for various
waveguide temperatures. Using the setup depicted in figure 8 each temperature
setting has been measured separately. The results have been merged into
figures 9 and 10 for better comparison of the tested temperatures. The
operating temperatures were chosen in such a way that the type-II process
generates degenerate photons while the type-0 process offers a wide, almost
flat spectrum in a region of high conversion efficiencies, allowing efficient,
controllable QKD operation as well as many accessible ITU-DWDM channels. The
latter case is especially important if the source shall be used in a setting
with several simultaneous pairwise key distribution as in [17].

(a)

(b)

Figure 7: SPDC conversion efficiency for various SHG powers impinging on the
type-0 waveguide (a) and type-II waveguide (b). The horizontal lines indicate
the weighted average conversion efficiencies of the crystals. The large error
bar for the first data point in (b) is due to low count rates at the
respective pump power. Figure 8: Setup for measurement of the photon pair
spectra. Only one SPDC waveguide is inserted into the measurement setup at a
time. For the type-0 waveguide, no band pass filter was used allowing
measurement of the full spectral width of the generated photon pairs.
Furthermore, a polarization beam splitter is not required for the type-0
spectrum measurements. The setup uses an Andor Shamrock 500i Czerny-Turner
spectrograph with a 600 lines/mm grating, 1900 nm blaze and an Andor iDus
InGaAs $1.7~{}\mu$m CCD detector. The spectrograph setup is placed inside a
box cooled down to close to $0~{}^{\circ}$C in order to further lower the
temperature of the camera chip and allow for single-photon detection. Figure
9: Photon pair spectra of the type-0 waveguide for various waveguide
temperatures. The spectra were linearly interpolated and displayed with
Gouraud shading for better visibility. Towards higher temperatures one can see
a broadening of the spectrum, while for lower temperatures the spectrum
narrows and the photon pair production rates are reduced. The tendency to
higher intensity towards the left half of the figure is attributed to a
slightly reduced detection efficiency of our spectrograph for higher
wavelengths. At temperatures of around 44 ∘C the spectrum is basically flat
and has a width of around 100 nm. Employing an arrayed waveguide grating, this
wide spectrum enables simultaneous QKD with dozens of receivers. Figure 10:
Photon pair spectra of the type-II waveguide for various waveguide
temperatures. Depicted is the sum of the fast and slow axis outputs. The
spectra were linearly interpolated and displayed with Gouraud shading for
better visibility. When tuning the temperature of the waveguide, one can
clearly observe the expected X-shaped structure, given by a wavelength
degenerate state at approximately $42~{}^{\circ}$C combined with the crossover
for both polarization states at this point. The doubling of the intensity at
the intersection compared to the exterior regions arises from adding the rates
of the two polarization states generated in the type-II process. The
relatively low intensities on the red side of the spectrum at $35~{}^{\circ}$C
are due to the transmission edge of the 7.1 nm band pass filter.

(a)

(b)

Figure 11: (11(a)) displays the heralding efficiencies for various tested
channels for both kinds of SPDC processes. (11(b)) shows the obtained values
for the second-order Glauber-correlation. Error-bars for the type-II process
are not displayed for visibility, as they are much smaller than the type-0
errors. Triangles depict the theoretical $g^{2}(0)$ values based on multi-pair
events.

To characterize the quantum character of our source, we have estimated the
heralded second-order Glauber correlation function $g^{(2)}_{h}(0)$ for both
SPDC processes. This value is a measure for the bunching character of our
source, which is $g^{(2)}_{h}(0)=0$ for an ideal single photon source/heralded
photon pair source. Values larger than zero indicate photon pair production
with multi-photon events or noise e.g. from non-correlated background photons.
SPDC is a statistical process sometimes producing multiple photon pairs per
pulse. The larger the pump power of the SPDC process the higher the share of
pulses containing multiple photon pairs after the SPDC. As a consequence the
choice of the pump power directly affects the minimal $g^{(2)}(0)$ value a
source can achieve. To setup the measurement of $g^{(2)}_{h}(0)$ one output of
the photon pair source is directly connected to a detector. The other output
is connected to a 50/50 beam splitter with both of these outputs being routed
to a detector each. In case of the type-II process the outputs are the
orthogonal polarization axes, while for the type-0 process pairs of channels
with entangled wavelengths have been connected. Measuring the dark count
corrected count and coincidental rates, $R_{i}$ and $C_{ijk}$,
$g^{(2)}_{h}(0)$ can be obtained by

$g^{(2)}_{h}(0)=\frac{C_{123}R_{3}}{C_{13}C_{23}}$ (8)

where $i=3$ indicates the heralding arm (detector being directly connected to
output) [33]. This gives $g^{(2)}_{h}(0)\approx(6\pm 0.8)\times 10^{-3}$ for
the fast axis of the type-II process and $(7.5\pm 0.9)\times 10^{-3}$ for the
slow axis (pulse energy 0.91 pJ and 1.06 pJ respectively, equivalent to
$\mu=2.7\times 10^{-3}$ to $3.1\times 10^{-3}$). Both values are close to
$g^{(2)}_{h}(0)=0$ displaying the good single photon character of our source.
For the type-0 process several channel combinations with pulse energies
ranging from $4.3\times 10^{-2}$ pJ to $5.5\times 10^{-2}$ pJ ($\mu=1.5\times
10^{-2}$ to $1.9\times 10^{-2}$) have been measured (cf. figure 1111(b)). The
observed $g^{(2)}_{h}(0)$ ranges from $2.0\times 10^{-2}$ to $3.6\times
10^{-2}$. We expect a large proportion of this value to be given by multi-
photon emissions which are dependent on the pulse energy. As an estimate we
calculate the $g^{(2)}_{h}(0)$ for a Poissonian distribution of multi-pair
events. Considering one-photon-pair-events as the main contribution for dual
coincidences and two-photon-pair-events for triple coincidences while
neglecting higher order terms,

$g^{(2)}(0)=\frac{P(I_{1},I_{2},S)P(S)}{P(I_{1},S)P(I_{2},S)}=\mu\left(2\frac{\eta_{S}\eta_{I}}{\eta_{\textrm{pair}}}-\eta_{S}T_{S}\right)\textrm{
.}$ (9)

While $T_{S}$ is the transmission probability of a signal photon, the factors
$\eta$ are defined by transmission functions of the AWG $p(f)$ following (5)
and (6). Consequently, $\eta_{S}\eta_{I}/\eta_{\textrm{pair}}$ gives a factor
of approximately 0.67 for a frequency interval of 130 GHz. Taking into account
the various losses on the path for a signal photon (crystal decoupling, filter
transmissions, detector efficiency, it follows that
$2\eta_{S}\eta_{I}/\eta_{\textrm{pair}}\gg\eta_{S}T_{S}$, hence

$g^{(2)}_{h}(0)\approx
2\frac{\eta_{S}\eta_{I}}{\eta_{\textrm{pair}}}\mu\textrm{ .}$ (10)

Employing the spectral $\mu$ density, found from the estimated conversion
efficiency, for 130 GHz one finds the estimates for $g^{(2)}_{h}(0)$ at our
pump energies. The values are depicted in figure 1111(b). We observe a good
agreement of our measured $g^{(2)}_{h}(0)$ values with the value predicted by
the number of multi-pair events. Hence, the emission characteristics of our
source can be mostly explained by considering multi-pair emissions.

The measurement was repeated with the photon source in double-pass
configuration (cf. figure 4). Although the $g^{(2)}_{h}(0)$ value is
independent of losses, the additional losses introduced by the circulator and
the pump-filter required a higher pump power in order to obtain suitable count
statistics. This resulted in a higher $\mu$-value of around $4\times 10^{-2}$
(with the exception for the 195.0 THz measurement at $\mu\approx 2.1\times
10^{-2}$) and therefore yielded higher $g^{(2)}_{h}(0)$ values between
$4.3\times 10^{-2}$ and $6.3\times 10^{-2}$. For both configurations, all
values are close to zero and way below the standard threshold for considering
a source as a heralded single photon source of $g^{(2)}_{h}(0)=0.5$. Again,
there is a good agreement with the theoretical predictions for multi-pair
emissions, when taking into account the measurement uncertainties.

To approximate the photon emission statistics of a photon pair source, we
consider the spectra of the photon pairs generated by the SPDC process. These
are determined by the joint spectral amplitude (JSA)

$\psi(\omega_{\mathrm{s}},\omega_{\mathrm{i}})=\alpha(\omega_{\mathrm{s}}+\omega_{\mathrm{i}})\Phi(\omega_{\mathrm{s}},\omega_{\mathrm{i}}),$
(11)

where $\alpha$ describes the pump pulse distribution and $\Phi$ is the phase-
matching function of the nonlinear crystal and
$\omega_{\mathrm{s}},\omega_{\mathrm{i}}$ are the angular frequencies of the
signal and idler photons with wavelengths
$\lambda_{\mathrm{s}},\lambda_{\mathrm{i}}$. Even for pump pulse durations as
short as $400\text{\,}\mathrm{ps}$, the bandwidth of the photon pair spectra
is approximately two orders of magnitude wider than the pump pulse spectrum.
Thus, we may assume that the phase-matching function is constant over the
narrow range of $\omega_{+}=\omega_{\mathrm{s}}+\omega_{\mathrm{i}}$ governed
by the pump pulse and solely depends on the difference frequency
$\omega_{-}=\omega_{\mathrm{s}}-\omega_{\mathrm{i}}$. This allows us to
reconstruct the JSA from measurements of the pump pulse shape and the SPDC
spectra acquired by cw-pumping. The resulting joint spectral density
$|\psi(\omega_{\mathrm{s}},\omega_{\mathrm{i}})|^{2}$ is presented in figure
12 a) for the type-II crystal at the degeneracy temperature and a pump
duration of approximately $400\text{\,}\mathrm{ps}$, for which the
corresponding spectrum is given in figure 12 b). The shape of the JSA in anti-
diagonal $\lambda_{-}=2\pi c/\omega_{-}$ direction is determined by the phase
matching function. For an ideal crystal with length $L$ and phase mismatch
$\Delta
k=[n(\omega_{+})\omega_{+}-n(\omega_{s})\omega_{s}-n(\omega_{i})\omega_{i}]/c$,
the phase matching function is given by
$\Phi(\omega_{\mathrm{s}},\omega_{\mathrm{i}})=\mathrm{sinc}^{2}[\Delta kL/2]$
[34]. Figure 12 b) shows different sizes of the side lobes at both sides of
the central wavelength. Although side lobes are expected from the
sinc2-function, its symmetry is not met, most likely due to imperfections of
the crystal.

Figure 12: (a) Joint spectral intensity for type-II SPDC at the degeneracy
temperature, pumped by a transform-limited pulse of approximately
$400\text{\,}\mathrm{ps}$ duration. The pump spectrum was obtained by fast
Fourier transform of a temporal pulse shape measurement and is presented in
(b), where $\lambda_{+}=2\pi c/\omega_{+}$. The corresponding Schmidt number
amounts to $K\approx 120$, classifying the generated state as highly entangled
with a large number of independent squeezers.

An important measure to quantify the entanglement between the signal and idler
photons is given by the Schmidt number $K=1/\sum_{j}\lambda_{j}^{2}$, where
the Schmidt coefficients $\lambda$ are given by the expansion coefficients of
the Schmidt decomposition

$\psi(\omega_{\mathrm{s}},\omega_{\mathrm{i}})=\sum_{j}\sqrt{\lambda_{j}}u_{j}(\omega_{\mathrm{s}})v_{j}(\omega_{\mathrm{i}})$
(12)

with sets of orthogonal function $\\{u(\omega_{s})\\}$ and
$\\{v(\omega_{i})\\}$ [35, 36, 37, 38].

Calculating the Schmidt-decomposition of the discretized joint spectral
amplitude of the type-II process presented in figure 12 via singular value
decomposition [37, 39, 40, 41] yields a Schmidt number of $K\approx 120$. For
the type-0 process even larger Schmidt numbers $K\gg 100$ are obtained,
classifying the two-photon states produced by our photon source as highly
entangled states compared to the regimes considered for example in [38, 40].
The unheralded second-order Glauber correlation function can be approximated
from the number of Schmidt modes via $g_{u}^{(2)}(0)=1+\frac{1}{K}$ [40, 42].
A thermal photon-number distribution for a single two-mode squeezer, resembled
by $K=1$, gives $g_{u}^{(2)}(0)=2$, while for an infinite number of two mode
squeezers it converges to a Poissonian photon number distribution. Hence,
small Schmidt numbers indicate a thermal photon distribution, while large
Schmidt numbers display a Poissonian photon emission [40, 38]. Following
$g_{u}^{(2)}(0)=1+\frac{1}{K}$, we obtain
$g_{u}^{(2)}(0)=1+\frac{1}{120}\approx 1.008$ which underlines the Poissonian
nature ($g_{u}^{(2)}(0)=1)$ of our SPDC source.

##  4 Discussion and Outlook

We demonstrated a versatile alignment-free modular photon-pair source being
able to host plug-and-play type-0 and type-II SPDC modules. Intrinsically, the
photon separation of the type-II process via polarization allows to connect
two receivers for the photons. If one aims for larger QKD networks to
demonstrate simultaneous pairwise key exchange with multiple pairs of
receivers [43, 17], this can be achieved by adding wavelength-division
multiplexing filters at each polarization channel. Another variant would be to
employ beamsplitters (e.g. 1x2 50:50 splitters) and implement time-division
multiplexing [44]. Of course, the latter comes with a trade-off in terms of
key rates when considering pairwise links as the routing of the photons is not
deterministic anymore.

The type-0 process with its wide spectrum is ideally suited for such types of
QKD networks, allowing many pairs of receivers being connected to a single
photon pair source. The AWG used in our setup in combination with the C-band
filter allows the use of 17 ITU-channel pairs (34 channels of 100 GHz). As the
C-band filter offers a usable range of $\pm 2.55$ THz around the center
frequency of our photon pair spectrum one could drastically increase the
number up to 102 users by employing a 50 GHz channel spacing AWG while
accessing the complete usable C-band filter range. The type-0 spectra is
approximately 9.3 THz wide. Hence, replacing the C-band filter can increase
the number of WDM-connected users even further.

The source has been tested for various operational modes from cw to pulsed
operation with several hundred of MHz repetition rates and pulses as short as
400 ps. In principle, even higher repetition rates up to a few GHz and pulse
durations down to 150 ps should be applicable as the amplitude modulator is
specified for frequencies up to 10 GHz. Hence, largely variable photon pair
production rates can be achieved. This is an asset, as there are many types of
single photon detectors with different detection efficiencies and dead times.
If the photon pair production rate does not match the detector properties and
experimental details, such as transmission losses e.g. in fiber QKD
experiments, the detector can be driven into saturation, thus being in dead
time for a large proportion of the measurement time. Considering entanglement-
based experiments this will cause a drop of the coincidence rates compared to
the count rates of the detectors, considerably reducing the signal-to-noise
ratio. This effect can be mitigated when adapting the photon pair production
rate with respect to the capabilities of the detectors.

Furthermore, the large tunability of our source is especially useful for QKD
protocols with interferometers such as phase-time coding. Here, the repetition
rate and pulse duration must be chosen with respect to the delay introduced by
the interferometers’ arm length differences to avoid pulse overlapping.
Consequently, our source can be employed in settings with various
interferometers without the need to change any hardware.

The EDFA and SHG modules generate pump pulses without background from
amplified spontaneous emission at a large range of powers enabling a wide
range of accessible mean photon numbers per pulse $\mu$ in the SPDC module.
Tuning $\mu$ can be important in various ways. First, it can affect the
detector saturation as described for the photon pair production rate above.
Second, in QKD protocols like phase-time coding it does yield a minimum bound
for the quantum bit error rate (QBER) of the setup. For our source, as the
Schmidt numbers prove, the SPDC is a Poissonian process. Hence, the number of
events where not only a single photon pair but multiple photon pairs are
emitted from a single pump pulse increases with higher $\mu$. These extra
photon pairs will cause additional clicks in the detectors not being
correlated to the clicks caused by the original photon pair, thus increase the
QBER. Here, the Poissonian nature of our source is advantageous, as the
probability for multi-pair events at low $\mu$ is lower than for a thermal
distribution, as can be shown by evaluating the multi-pair emission
probabilities for both types of photon number distributions. Consequently, by
tuning $\mu$ our source does allow to choose between higher raw key rates vs
lower QBER, depending on the applicable scenario in such QKD systems.

When considering the performance of our source it is important to stress that
the crystals are of-the-shelf components with no particular coupling
optimization. In terms of performance (e.g. heralding efficiencies, in-source
losses of photon pairs) all losses occurring after photon pair production are
particularly costly. Thus, an optimized coupling of the SPDC crystals could
yield a better source performance. However, this would require to step back
from standard products. As an easier point of improvement, we can take into
account any fiber-to-fiber connectors. These typically account for 0.3 dB of
loss each. We employed such connectors to attach the SPDC crystal’s output to
the subsequent filters. In case of the type-0 module the C-band filter
introduces another fiber connector. This accounts to a total of
$1.2~{}\textrm{dB}=2\times 0.6~{}\textrm{dB}$ of avoidable loss for the type-0
module when considering that both photons of a pair experience these losses.
To lift the attenuation one could replace the named connection by fiber
splices which hardly cause losses for telecom wavelengths standard fibers.

##  5 Conclusion

We report on two flexible single photon-pair sources. The system consists of
spectrally pure generation of 775 nm light, allowing subsequent SPDC modules
to generate photon pairs around 1550 nm. The first is based on a type-II SPDC
process and the second on a type-0 SPDC process. Their robust design makes
them ideal for many applications either as a heralded single-photon source or
as a photon pair source as basis e.g. for entanglement based QKD protocols.
Both can be operated either in cw or pulsed operation. Furthermore, we have
demonstrated that the source can produce photon pairs of high quality with a
single crystal operated in a double-pass configuration for both SHG and SPDC.
In case of cw operation, the source is ideally suited for phase-coding QKD
applications [18]. In pulsed operation, we have demonstrated a small QKD
network based on time-bin entanglement [17]. We report on a brightness of
$2.48\times 10^{6}~{}\textrm{photon pairs}/(\textrm{s nm mW})$ for the type-II
process and $3.32\times 10^{8}~{}\textrm{photon pairs}/\textrm{s nm mW}$ for
the type-0 process, with heralding efficiencies up to 36 % and from 9 % to 12
%, respectively. The heralded Glauber correlation function $g^{(2)}(0)$ takes
values of $<7.5\times 10^{-3}$ and $<3.6\times 10^{-2}$ for $\mu$ of $\approx
3.1\times 10^{-3}$ and $\approx 1.9\times 10^{-2}$, respectively, showcasing a
good photon pair emission behaviour.

## Acknowledgements

This research has been funded by the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation), under Grant No. SFB 1119–236615297. We thank Paul
Wagner from Deutsche Telekom Technik GmbH for lending us the AWG.

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