# Spectrally resolved Franson interference

Rui-Bo Jin1 Zi-Qi Zeng1 Dan Xu1 Chen-Zhi Yuan1<EMAIL_ADDRESS>Bai-
Hong Li2<EMAIL_ADDRESS>You Wang3 Ryosuke Shimizu4 Masahiro Takeoka5 Mikio
Fujiwara6 Masahide Sasaki6 Pei-Xiang Lu1 1Hubei Key Laboratory of Optical
Information and Pattern Recognition, Wuhan Institute of Technology, Wuhan
430205, China  2 Department of Physics, Shaanxi University of Science and
Technology, Xi’an 710021, China  3 Southwest Institute of Technical Physics,
Chengdu 610041, China  4 University of Electro-Communications, 1-5-1
Chofugaoka, Chofu, Tokyo 182-8585, Japan  5 Keio University, 3-14-1 Hiyoshi,
Kohoku, Yokohama, Kanagawa 223-8522, Japan  6 National Institute of
Information and Communications Technology , 4-2-1 Nukui-Kitamachi, Koganei,
Tokyo 184-8795, Japan

###### Abstract

Franson interference can be used to test the nonlocal features of energy-time
entanglement and has become a standard in quantum physics. However, most of
the previous Franson interference experiments were demonstrated in the time
domain, and the spectral properties of Franson interference have not been
fully explored. Here, we theoretically and experimentally demonstrate
spectrally resolved Franson interference using biphotons with different
correlations, including positive correlation, negative correlation, and non-
correlation. It is found that the joint spectral intensities of the biphotons
can be modulated along both the signal and idler directions, which has
potential applications in generating high-dimensional frequency entanglement
and time-frequency grid states. This work may provide a new perspective for
understanding the spectral-temporal properties of the Franson interferometer.

##  I Introduction

Franson interference was proposed in 1989 to test the Bell inequality for
position or time, specifically to explore the feasibility of local hidden-
variable models using a new optical interferometer Franson (1989). In a
typical configuration for the Franson interferometer, the signal and idler
photons, generated simultaneously, are distributed to different terminals
while passing through unbalanced Mach-Zehnder interferometers (UMZIs) inserted
in their paths. The signal and idler photons can choose either short or long
pathways within the UMZIs. In delayed coincidence measurements, it is
convenient to consider only events where both signal and idler photons select
either the short or long pathways. Since these two cases can be
indistinguishable, they can interfere with each other. Interference fringes
can be observed in the coincidence measurements when the optical path-length
difference in the UMZI is shorter than the two-photon coherence length of the
signal and idler photons. The interference from single photons can be
eliminated by setting the optical path-length difference longer than the
coherence length of either the signal or idler photons. Several experiments
have modified the Franson interferometer from its original configuration for
different purposes Cabello  _et al._ (2009); Kwiat  _et al._ (1990); Mittal
_et al._ (2021). For instance, hug-type configurations have been invented to
remove the post-selection loophole in the original configurationCabello  _et
al._ (2009), and single Michelson configurations have been used to make the
setup more compactKwiat  _et al._ (1990); Mittal  _et al._ (2021).

Figure 1:  (a) The model of the traditional unfolded Franson interference. (b)
The Mach–Zehnder-type folded Franson interference. (c) The Michelson-type
folded Franson interference. (d) The experimental setup based on (c). LPFs =
long-pass filters, PZT = piezoelectric motor, BS = beam splitter, FBS = fiber
beam splitter, D = detector, TIA = time interval analyzer.

Numerous experiments have been conducted to observe Franson interference,
which has become a standard tool in quantum optics for verifying energy-time
or time-bin entanglement Ou  _et al._ (1990); Brendel  _et al._ (1991). In
these experiments, various mechanisms have been employed to generate photon
pairs, including spontaneous parametric down-conversion (SPDC) processes in
bulk crystals with $\chi^{(2)}$ nonlinearity Ou  _et al._ (1990); Brendel  _et
al._ (1991), SPDC or spontaneous four-wave mixing (SFWM) processes in
waveguides or microresonators with $\chi^{(2)}$ or $\chi^{(3)}$ nonlinearities
Sanaka  _et al._ (2001); Ma  _et al._ (2020); Grassani  _et al._ (2015), SFWM
in atomic ensembles Park  _et al._ (2018), and cascaded emission in quantum
dots (QDs) Jayakumar  _et al._ (2014). The applications of Franson
interference range from testing fundamental physical principles Tittel  _et
al._ (1998); Stefanov  _et al._ (2002) to quantum cryptography Ali-Khan  _et
al._ (2007), entanglement-based quantum networks Sun  _et al._ (2017), and
quantum imaging Gao  _et al._ (2019). However, most of the previous Franson
interference experiments have focused on time-resolved measurements, and it is
expected that a spectrally-resolved configuration would provide new
capabilities.

Spectrally-resolved interferometers create interference fringes with different
frequency components separated spatially or temporally. These interferometers
have already been employed in measuring the linear and nonlinear dielectric
properties of materials Tokunaga  _et al._ (1992), coherently controlling
ultrafast carrier dynamics in semiconductor nanostructures Heberle  _et al._
(1995), measuring laser-generated shock waves in metal thin films Gahagan  _et
al._ (2000), and studying the dynamics of ultrashort laser-produced plasma
Salières  _et al._ (1999). In the field of quantum optics, frequency-resolved
Hong-Ou-Mandel (HOM) interference has been demonstrated Jin  _et al._ (2015,
2016); Orre  _et al._ (2019); Yepiz-Graciano  _et al._ (2020); Merkouche  _et
al._ (2022) and used in entanglement swapping of energy-time entanglement
Merkouche  _et al._ (2022) and quantum optical coherence tomography Yepiz-
Graciano  _et al._ (2020).

In this article, we theoretically and experimentally demonstrate a spectrally
resolved Franson interferometer. In theory, we confirm that a folded Franson
interferometer can achieve the same performance as the original Franson
interference. We compare time-resolved and spectrally resolved interferograms
for biphotons with positive correlation, negative correlation, and non-
correlation. In the experiment, we measure the spectrally resolved
interferograms of biphotons generated by SPDC under different time delays. We
find that the joint spectral intensities of the biphoton can be modulated
along both the signal and idler directions. Additionally, we observe that the
spectrally resolved interferograms remain clear even when the time-resolved
interferogram disappears.

##  II Theory and simulation

Figure 2:  The first, second, and third rows display the simulation results of
spectrally non-correlated, positively correlated, and negatively correlated
biphotons, respectively. (a1, b1, c1) are the simulated coincidence
probability $P(\tau)$ as a function of the delay $\tau$. The insets in (a1,
b1, c1) show $P(\tau)$ within a time duration of 0 to 15.84 fs, corresponding
to a phase delay of 0-6$\pi$. The spectral distributions
$S(\omega_{s},\omega_{i},\tau)$ at different time delays are represented by
(a2-a5), (b2-b5), and (c2-c5) for 0 ps, 5 ps, 10 ps, and 15 ps, respectively.
On the other hand, (d1-d5) illustrate $S(\omega_{s},\omega_{i},\tau)$ for
spectrally non-correlated biphotons with a time delay ranging from 5.00364 ps
to 5.00892 ps, corresponding to phases from 0 to $2\pi$.

The two-photon state from an SPDC process can be described as:

$\left|\psi\right\rangle=\int_{0}^{\infty}{\int_{0}^{\infty}{d\omega_{s}d\omega_{i}}}f(\omega_{s},\omega_{i})\hat{a}_{s}^{\dagger}(\omega_{s})\hat{a}_{i}^{\dagger}(\omega_{i})\left|{00}\right\rangle,$
(1)

where $\omega$ is the angular frequency, $\hat{a}^{\dagger}$ is the creation
operator, and the subscripts $s$ and $i$ denote the signal and idler photons
from SPDC, respectively. $f(\omega_{s},\omega_{i})$ represents the joint
spectral amplitude of the signal and idler photons.

As calculated in the Appendix, the coincidence probability $P_{0}(\tau)$ in
the traditional unfolded Franson interference (with the setup in Fig. 1(a)) is
given by:

$\begin{array}[]{lll}P_{0}(\tau)&=&\int_{0}^{\infty}\int_{0}^{\infty}d\omega_{s}d\omega_{i}\left|f\left(\omega_{s},\omega_{i}\right)\right|^{2}\\\
&&\times\left[1+\cos\left(\omega_{s}\tau\right)\right]\left[1+\cos\left(\omega_{i}\tau\right)\right],\\\
\end{array}$ (2)

where $\tau$ is the optical path delay between the long and the short arms.
For a folded Franson interference with the setup in Fig. 1(b) or (c), the
coincidence probability $P(\tau)$ is given by:

$\begin{array}[]{lll}P(\tau)&=&\frac{1}{8}\int_{0}^{\infty}\int_{0}^{\infty}d\omega_{s}d\omega_{i}\left|f\left(\omega_{s},\omega_{i}\right)\right|^{2}\\\
&&\times\left[1+\cos\left(\omega_{s}\tau\right)\right]\left[1+\cos\left(\omega_{i}\tau\right)\right].\\\
\end{array}$ (3)

The joint spectral correlation $S(\omega_{s},\omega_{i},\tau)$ at different
delay positions can be calculated as:

$\displaystyle
S(\omega_{s},\omega_{i},\tau)=\left|f\left(\omega_{s},\omega_{i}\right)\right|^{2}\left[1+\cos\left(\omega_{s}\tau\right)\right]\left[1+\cos\left(\omega_{i}\tau\right)\right].$
(4)

Eq. (2) and Eq. (3) have a similar form, indicating that the folded Franson
interference can achieve the same performance as the original Franson
interference.

By using Eq. (3) and Eq. (4) , we can simulate $P(\tau)$ and
$S(\omega_{s},\omega_{i},\tau)$ at different delays $\tau$ and with different
spectral distributions $f\left(\omega_{s},\omega_{i}\right)$, as shown in Fig.
2. The first row presents the case of spectrally non-correlated biphotons.
This calculation is performed using a 30-mm-long PPKTP crystal and a pump
laser with a Gaussian distribution. The laser has a center wavelength of 792
nm and a full width at half maximum (FWHM) of 0.40 nm. The second row is the
case of positively correlated biphotons, which are calculated using a 50-mm-
long PPKTP crystal and a pump laser with an FWHM of 2.35 nm. The third row
shows the case of negatively correlated biphotons, which are calculated using
a 10-mm-long PPKTP crystal and a pump laser with an FWHM of 0.12 nm.

Fig. 2(a1, b1, c1) displays the coincidence probability $P(\tau)$ as a
function of delay, while the corresponding $|f(\omega_{1},\omega_{2})|^{2}$ is
shown in Fig. 2(a2, b2, c2), respectively. Within the range of -20 ps to 20
ps, the envelope of the coincidence probability exhibits distinct variations
for biphotons with different correlations. However, between 0 and 15.84 fs,
the interference patterns remain consistent, as illustrated by the insets in
Fig. 2(a1, b1, c1).

The spectral distribution $S(\omega_{s},\omega_{i},\tau)$ with different
correlations at 0 ps, 5 ps, 10 ps, and 15 ps are depicted in Fig. 2(a2-a5),
(b2-b5), and (c2-b5). Notably, an increase in delay leads to a greater
separation of spectral modes into multiple components. This phenomenon can be
effectively explained by Eq. (3). To facilitate a comparison of the spectral
distribution at different phases, Fig. 2(d1-d5) illustrates
$S(\omega_{s},\omega_{i},\tau)$ at phase differences of 0, $\pi/2$, $\pi$,
$3\pi/2$, and $2\pi$. We can observe that the mode number changes gradually
from 1 mode to 4 modes, and then returns to 1 mode.

##  III Experiment and results

Figure 3:  Experimental results. (a1, b1) The measured coincidence (single)
counts as a function of the time delay scanned with a stepping motor, with a
step of 4 $\mu$m. The insets show the measured coincidence (single) counts by
scanning a PZT with a step of 40 nm. (a2-a5) The measured JSIs at the delay
position of 0 ps, 1.33 ps, 4.00 ps, and 5.33 ps, respectively. The
accumulation time is 10 seconds for each figure. (b2-b4) The time-of-arrival
measurement for single count of channel 1 (SC1, in red), single count of
channel 2 (SC2, in blue), and coincidence counts (CC, in black) at 0 ps, 1.33
ps, and 4.00 ps. (b5) is an enlarged view of the center section of (a5).

The experimental setup is shown in Fig. 1(d). Laser pulses with a temporal
width of around 2 ps and a center wavelength of 792 nm were utilized to pump a
30-mm-long periodically poled KTiOPO4 (PPKTP) crystal. The PPKTP crystal was
type-II phase matched (y$\to$y+z), and the signal and idler photons generated
from the SPDC process were orthogonally polarized Jin  _et al._ (2022). After
filtering by the long-path filters, the biphotons were sent to a time delay
system, which consisted of a beamsplitter (BS), a PZT, and a stepping motor.
Then, the photons were coupled into a fiber beamsplitter, which was connected
to a fiber spectrometer . The fiber spectrometer consisted of two 7.5-km-long
SMFs, two SNSPDs, one synchronization signal from the laser, and a TIA Jin
_et al._ (2016). The dispersion of the SMFs was calibrated as 27.3 ps/km/nm at
1584 nm. Considering an estimated 100 ps FWHM jitter of the detection system,
the resolution of this fiber spectrometer was calculated to be 0.5 nm.

The measured coincidence counts as a function of optical path delay are shown
in Fig. 3(a1). The main figure was obtained by scanning the stepping motor
with a step length of 4 $\mu$m. The FWHM of the upper envelope is 0.56 ps The
insert in Fig. 3(a1) was obtained by scanning a PZT with a step length of 40
nm. The visibility is 99.90% $\pm$ 0.00%, indicating a high
indistinguishability of the signal and idler photons. The main figure and
insert in Fig. 3(a1) are consistent with the simulation results in Fig. 2(b1).

Fig. 3(a2-a5) shows the measured JSI at 0 ps, 1.33 ps, 4 ps, and 5.33 ps
respectively. It can be observed that with the increase of time delay, the
mode number increases. The mode numbers in Fig. 3(a2-a5) are 1, 3, 6, and 15,
respectively. Fig. 3(b5) is an enlarged view of (a5), and it is clear that the
modes are separated in both the horizontal and vertical directions..

We also measured the single counts at the same time as the coincidence
measurement, as shown in Fig. 3(b1). The single counts have a constant
baseline, which is different from the varying baseline in the coincidence
counts in Fig. 3(a1). The insert in Fig. 3(b1) shows the single counts,
obtained by scanning the PZT, and the visibility is 79.16% $\pm$ 0.05%. We
also measured the time-of-arrival (TOA) of channel 1 (ch1) and channel 2 (ch2)
in Fig. 3(b2-b4). It can be observed that with the increase of time delay, the
single peak in the single counts evolves into multiple peaks. However, the
peak in the TOA of the coincidence counts remains a single peak. This is
caused by the fact that the TOA of single counts is obtained by projecting the
JSI data onto the horizontal and vertical axes, while the TOA of the
coincidence counts is obtained by projecting the JSI data onto the anti-
diagonal line, i.e., the line of $\omega_{s}-\omega_{i}$.

##  IV Discussion

There are two types of coherence time for biphotons: the sum-frequency
coherence time and the difference-frequency coherence time Jin  _et al._
(2018); MacLean  _et al._ (2018). The sum-frequency coherence time is
determined by the pump laser and can be tested in the NOON state interference.
The difference-frequency coherence time is determined by the phase-matching
condition of the nonlinear crystal and can be tested in the HOM interference
Jin and Shimizu (2018). For single photons, the coherence time is determined
by the projection of joint temporal distributions onto the signal or idler
direction. In traditional Franson interference, the sum-frequency coherence
time is much longer than the coherence time of single photons.

The spectrally resolved measurement has been previously investigated in HOM
interference Gerrits  _et al._ (2015); Jin  _et al._ (2015); Chen  _et al._
(2023), modified HOM interferenceLi  _et al._ (2023), NOON state interference
Jin  _et al._ (2021), and also demonstrated in the characterization of time-
energy entangled state MacLean  _et al._ (2018, 2019). It can be observed that
even when there are no interference patterns in the time domain, the
interference patterns in the spectral domain are still very clear. The
spectral measurement in interference can be fundamentally understood as a tool
for temporal filtering, which increases the coherence time of the photons by
filtering. The JSI measured in quantum interference is helpful and is
complementary to the measurement of temporal interference.

Since the joint spectral intensities of the biphotons can be modulated along
both the signal and idler directions, it is possible to generate high-
dimensional entangled states (entangled qudits) and time-frequency grid states
using spectrally resolved Franson interference. As demonstrated in Fig. 3(a4),
this is indeed a kind of entangled qudits Yang  _et al._ (2023). For example,
the state generated in Fig. 2(a5) is a time-frequency grid state Fabre  _et
al._ (2020), which can be used to implement measurement-based quantum error
correction in fault-tolerant quantum computing using time-frequency continuous
variables Gottesman  _et al._ (2001); Menicucci (2014); Baragiola  _et al._
(2019).

##  V Conclusion

In summary, we have theoretically and experimentally demonstrated spectrally
resolved Franson interference using biphotons with different correlations. The
joint spectral intensities of the biphotons were measured at different delay
positions in an Franson interference. It can be observed that even when there
are no interference patterns in the time domain, the interference patterns in
the spectral domain are still very clear. This work provides a new perspective
by considering the joint spectral distribution to understand the spectral-
temporal properties in Franson interference. Furthermore, this approach can be
used to generate high-dimensional entangled states and time-frequency grid
states.

## Acknowledgments

This work was supported by the National Natural Science Foundations of China
(Grant Numbers 92365106, 12074309, and 12074299), and the Natural Science
Foundation of Hubei Province (2022CFA039).

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### Appendix 1: Calculation of standard Franson interference

Figure A1:  (a) The setup of an unfolded Franson interferometer. (b) The setup
of a folded Franson interferometer.

In this section, we deduce the equations for the Franson interference using
multi-mode theory. The setup of the Franson interference is shown in Fig. A1
(a). The two-photon state from a spontaneous parametric down-conversion (SPDC)
process can be described as

$\left|\psi\right\rangle=\int_{0}^{\infty}{\int_{0}^{\infty}{d\omega_{s}d\omega_{i}}}f(\omega_{s},\omega_{i})\hat{a}_{s}^{\dagger}(\omega_{s})\hat{a}_{i}^{\dagger}(\omega_{i})\left|{00}\right\rangle,$
(A1)

where $\omega$ is the angular frequency; $\hat{a}^{\dagger}$ is the creation
operator and the subscripts $s$ and $i$ denote the signal and idler photons
from SPDC, respectively; $f(\omega_{s},\omega_{i})$ is the joint spectral
amplitude of the signal and idler photons.

The detection field operators of detector 1 (D1) and detector 2 (D2) are

$\hat{E}_{1}^{(+)}(t_{1})=\frac{1}{{2}}\int_{0}^{\infty}{d\omega_{1}}\hat{a}_{1}(\omega_{1})e^{-i\omega_{1}t_{1}},$
(A2)
$\hat{E}_{2}^{(+)}(t_{2})=\frac{1}{{2}}\int_{0}^{\infty}{d\omega_{2}\hat{a}_{2}(\omega_{2})}e^{-i\omega_{2}t_{2}},$
(A3)

where the subscripts $1$ and $2$ denote the photons detected by D1 and D2,
respectively. The transformation rule after the delay times $T_{1}$ and
$T_{2}$ is

$\hat{a}_{1}\left(\omega_{1}\right)=\frac{1}{\sqrt{2}}\left[\hat{a}_{s}\left(\omega_{1}\right)+\hat{a}_{s}\left(\omega_{1}\right)e^{-i\omega_{1}T_{1}}\right],$
(A4)
$\hat{a}_{2}\left(\omega_{2}\right)=\frac{1}{\sqrt{2}}\left[\hat{a}_{i}\left(\omega_{2}\right)+\hat{a}_{i}\left(\omega_{2}\right)e^{-i\omega_{2}T_{2}}\right].$
(A5)

So, we can rewrite the field operators as

$\begin{array}[]{lll}\hat{E}_{1}^{(+)}\left(t_{1}\right)=\frac{1}{2\sqrt{2\pi}}\int_{0}^{\infty}d\omega_{1}\left[\hat{a}_{s}\left(\omega_{1}\right)+\hat{a}_{s}\left(\omega_{1}\right)e^{-i\omega_{1}T_{1}}\right]e^{-i\omega_{1}t_{1}},\\\
\end{array}$ (A6)

and

$\begin{array}[]{lll}\hat{E}_{2}^{(+)}\left(t_{2}\right)=\frac{1}{2\sqrt{2\pi}}\int_{0}^{\infty}d\omega_{2}\left[\hat{a}_{i}\left(\omega_{2}\right)+\hat{a}_{i}\left(\omega_{2}\right)e^{-i\omega_{2}T_{2}}\right]e^{-i\omega_{2}t_{2}}.\\\
\end{array}$ (A7)

The coincidence probability $P(\tau)$, which is also the second-order
correlation function $G_{2}(\tau)$, can be expressed as

$P(\tau)\equiv
G_{2}(\tau)=\int_{-\infty}^{\infty}{\int_{-\infty}^{\infty}{dt_{1}dt_{2}}}\left\langle{\psi\left|{\hat{E}_{1}^{(-)}\hat{E}_{2}^{(-)}\hat{E}_{2}^{(+)}\hat{E}_{1}^{(+)}}\right|\psi}\right\rangle.$
(A8)

First of all, consider $\hat{E}_{2}^{(+)}\hat{E}_{1}^{(+)}$,

$\begin{array}[]{l}\begin{aligned}
\hat{E}_{2}^{(+)}\hat{E}_{1}^{(+)}&=\frac{1}{8\pi}\int_{0}^{\infty}\int_{0}^{\infty}d\omega_{1}d\omega_{2}\left[\hat{a}_{s}\left(\omega_{1}\right)+\hat{a}_{s}\left(\omega_{1}\right)e^{-i\omega_{1}T_{1}}\right]\left[\hat{a}_{i}\left(\omega_{2}\right)+\hat{a}_{i}\left(\omega_{2}\right)e^{-i\omega_{2}T_{2}}\right]e^{-i\omega_{1}t_{1}}e^{-i\omega_{2}t_{2}}\\\
&=\frac{1}{8\pi}\int_{0}^{\infty}\int_{0}^{\infty}d\omega_{1}d\omega_{2}\hat{a}_{s}\left(\omega_{1}\right)\hat{a}_{i}\left(\omega_{2}\right)\left[1+e^{-i\omega_{1}T_{1}}+e^{-i\omega_{2}T_{2}}+e^{-i\left(\omega_{1}T_{1}+\omega_{2}T_{2}\right)}\right]e^{-i\omega_{1}t_{1}}e^{-i\omega_{2}t_{2}}\\\
&=\frac{1}{8\pi}\int_{0}^{\infty}\int_{0}^{\infty}d\omega_{1}d\omega_{2}\hat{a}_{s}\left(\omega_{1}\right)\hat{a}_{i}\left(\omega_{2}\right)\left(1+e^{-i\omega_{1}T_{1}}\right)\left(1+e^{-i\omega_{2}T_{2}}\right)e^{-i\omega_{1}t_{1}}e^{-i\omega_{2}t_{2}}.\end{aligned}\\\
\end{array}$ (A9)

Then, consider $\hat{E}_{2}^{(+)}\hat{E}_{1}^{(+)}|\psi\rangle$

$\begin{array}[]{l}\begin{aligned}
\hat{E}_{2}^{(+)}\hat{E}_{1}^{(+)}|\psi\rangle&=\frac{1}{8\pi}\int_{0}^{\infty}\int_{0}^{\infty}d\omega_{1}d\omega_{2}\hat{a}_{s}\left(\omega_{1}\right)\hat{a}_{i}\left(\omega_{2}\right)\left(1+e^{-i\omega_{1}T_{1}}\right)\left(1+e^{-i\omega_{2}T_{2}}\right)e^{-i\omega_{1}t_{1}}e^{-i\omega_{2}t_{2}}\\\
&\times\int_{0}^{\infty}d\omega_{s}\int_{0}^{\infty}d\omega_{i}f\left(\omega_{s},\omega_{i}\right)\hat{a}_{s}^{\dagger}\left(\omega_{s}\right)\hat{a}_{i}^{\dagger}\left(\omega_{i}\right)|00\rangle\\\
&=\frac{1}{8\pi}\int_{0}^{\infty}\int_{0}^{\infty}\int_{0}^{\infty}\int_{0}^{\infty}d\omega_{1}d\omega_{2}d\omega_{s}d\omega_{i}f\left(\omega_{s},\omega_{i}\right)\delta\left(\omega_{1}-\omega_{s}\right)\delta\left(\omega_{2}-\omega_{i}\right)\\\
&\times\left(1+e^{-i\omega_{1}T_{1}}\right)\left(1+e^{-i\omega_{2}T_{2}}\right)e^{-i\omega_{1}t_{1}}e^{-i\omega_{2}t_{2}}|00\rangle\\\
&=\frac{1}{8\pi}\int_{0}^{\infty}\int_{0}^{\infty}d\omega_{1}d\omega_{2}f\left(\omega_{1},\omega_{2}\right)\left(1+e^{-i\omega_{1}T_{1}}\right)\left(1+e^{-i\omega_{2}T_{2}}\right)e^{-i\omega_{1}t_{1}}e^{-i\omega_{2}t_{2}}|00\rangle.\end{aligned}\\\
\end{array}$ (A10)

In the above calculation, the equations of
$\hat{a}_{s}(\omega_{1})\hat{a}_{s}^{\dagger}(\omega_{s})\left|{0}\right\rangle=\delta(\omega_{1}-\omega_{s})\left|{0}\right\rangle$
and
$\hat{a}_{i}(\omega_{2})\hat{a}_{i}^{\dagger}(\omega_{i})\left|{0}\right\rangle=\delta(\omega_{2}-\omega_{i})\left|{0}\right\rangle$
are used.

Then,

$\begin{array}[]{lll}\begin{aligned}
\left\langle\psi\left|\hat{E}_{1}^{(-)}\hat{E}_{2}^{(-)}\hat{E}_{2}^{(+)}\hat{E}_{1}^{(+)}\right|\psi\right\rangle&=\frac{1}{64\pi^{2}}\int_{0}^{\infty}\int_{0}^{\infty}d\omega_{1}^{\prime}d\omega_{2}^{\prime}f^{*}\left(\omega_{1}^{\prime},\omega_{2}^{\prime}\right)\left(1+e^{i\omega_{1}^{\prime}T_{1}}\right)\left(1+e^{i\omega_{2}^{\prime}T_{2}}\right)e^{i\omega_{1}^{\prime}t_{1}}e^{i\omega_{2}^{\prime}t_{2}}\\\
&\times\int_{0}^{\infty}\int_{0}^{\infty}d\omega_{1}d\omega_{2}f\left(\omega_{1},\omega_{2}\right)\left(1+e^{-i\omega_{1}T_{1}}\right)\left(1+e^{-i\omega_{2}T_{2}}\right)e^{-i\omega_{1}t_{1}}e^{-i\omega_{2}t_{2}}.\end{aligned}\\\
\end{array}$ (A11)

Finally,

$\begin{array}[]{lll}\begin{aligned}
P(\tau)&=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}dt_{1}dt_{2}\left\langle\psi\left|\hat{E}_{1}^{(-)}\hat{E}_{2}^{(-)}\hat{E}_{2}^{(+)}\hat{E}_{1}^{(+)}\right|\psi\right\rangle\\\
&=\frac{1}{64\pi^{2}}\int_{0}^{\infty}\int_{0}^{\infty}dt_{1}dt_{2}\int_{0}^{\infty}\int_{0}^{\infty}d\omega_{1}^{\prime}d\omega_{2}^{\prime}f^{*}\left(\omega_{1}^{\prime},\omega_{2}^{\prime}\right)\left(1+e^{i\omega_{1}^{\prime}T_{1}}\right)\left(1+e^{i\omega_{2}^{\prime}T_{2}}\right)e^{i\omega_{1}^{\prime}t_{1}}e^{i\omega_{2}^{\prime}t_{2}}\\\
&\times\int_{0}^{\infty}\int_{0}^{\infty}d\omega_{1}d\omega_{2}f\left(\omega_{1},\omega_{2}\right)\left(1+e^{-i\omega_{1}T_{1}}\right)\left(1+e^{-i\omega_{2}T_{2}}\right)e^{-i\omega_{1}t_{1}}e^{-i\omega_{2}t_{2}}.\end{aligned}\\\
\end{array}$ (A12)

By utilizing
$\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-i(\omega-\omega^{\prime})t}dt=\delta(\omega-\omega^{\prime})$,
the above equation can be further simplified as

$\begin{array}[]{lll}\begin{aligned}
P(\tau)&=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}dt_{1}dt_{2}\left\langle\psi\left|\hat{E}_{1}^{(-)}\hat{E}_{2}^{(-)}\hat{E}_{2}^{(+)}\hat{E}_{1}^{(+)}\right|\psi\right\rangle\\\
&=\frac{1}{16}\int_{0}^{\infty}\int_{0}^{\infty}\int_{0}^{\infty}\int_{0}^{\infty}d\omega_{1}d\omega_{2}d\omega_{1}^{\prime}d\omega_{2}^{\prime}\delta\left(\omega_{1}-\omega_{1}^{\prime}\right)\delta\left(\omega_{2}-\omega_{2}^{\prime}\right)f\left(\omega_{1},\omega_{2}\right)\\\
&\times\left(1+e^{-i\omega_{1}T_{1}}\right)\left(1+e^{-i\omega_{2}T_{2}}\right)f^{*}\left(\omega_{1}^{\prime},\omega_{2}^{\prime}\right)\left(1+e^{i\omega_{1}^{\prime}T_{1}}\right)\left(1+e^{i\omega_{2}^{\prime}T_{2}}\right)e^{i\omega_{1}^{\prime}t_{1}}e^{i\omega_{2}^{\prime}t_{2}}\\\
&=\frac{1}{16}\int_{0}^{\infty}\int_{0}^{\infty}d\omega_{1}d\omega_{2}f\left(\omega_{1},\omega_{2}\right)f^{*}\left(\omega_{1},\omega_{2}\right)\left(1+e^{-i\omega_{1}T_{1}}\right)\left(1+e^{-i\omega_{2}T_{2}}\right)\left(1+e^{i\omega_{1}T_{1}}\right)\left(1+e^{i\omega_{2}T_{2}}\right)\\\
&=\frac{1}{4}\int_{0}^{\infty}\int_{0}^{\infty}d\omega_{1}d\omega_{2}\left|f\left(\omega_{1},\omega_{2}\right)\right|^{2}\left[1+\operatorname{cos}\left(\omega_{1}T_{1}\right)\right]\left[1+\operatorname{cos}\left(\omega_{2}T_{2}\right)\right].\end{aligned}\\\
\end{array}$ (A13)

If the delay of the two paths is now equal, then:

$\begin{array}[]{lll}\begin{aligned}
P(\tau)=\frac{1}{4}\int_{0}^{\infty}\int_{0}^{\infty}d\omega_{1}d\omega_{2}\left|f\left(\omega_{1},\omega_{2}\right)\right|^{2}\left[1+\operatorname{cos}\left(\omega_{1}T\right)\right]\left[1+\operatorname{cos}\left(\omega_{2}T\right)\right]\\\
\end{aligned}\end{array}.$ (A14)

Next, calculate the count probability of the single count and take detector 1
as an example. Assuming that the biphoton state produced in the SPDC process
is separable, then the single photon state passing through the path of T1 is:

$\begin{array}[]{lll}\begin{aligned}
\left|\psi\right\rangle_{s}=\int_{0}^{\infty}{d{\omega_{s}}}f({\omega_{s}})\hat{a}_{s}^{\dagger}({\omega_{s}})\left|0\right\rangle.\end{aligned}\end{array}$
(A15)

Similarly, the detector operator is:

$\begin{array}[]{lll}\begin{aligned}
\hat{E}_{1}^{(+)}({t_{1}})=\frac{1}{{\sqrt{2\pi}}}\int_{0}^{\infty}{d{\omega_{1}}}{\hat{a}_{1}}({\omega_{1}}){e^{-i{\omega_{1}}{t_{1}}}}.\end{aligned}\end{array}$
(A16)

The transformation rule after the delay time $T_{1}$ is

$\begin{array}[]{lll}\begin{aligned}
{\rm{}}{\hat{a}_{1}}({\omega_{1}})=\frac{1}{2}\left[{{{\hat{a}}_{s}}({\omega_{1}})+{{\hat{a}}_{s}}({\omega_{1}}){e^{-i{\omega_{1}}{T_{1}}}}}\right].\end{aligned}\end{array}$
(A17)

So, we can rewrite the field operators as

$\begin{array}[]{lll}\begin{aligned}
\hat{E}_{1}^{(+)}({t_{1}})=\frac{1}{{2\sqrt{2\pi}}}\int_{0}^{\infty}{d{\omega_{1}}}[{\hat{a}_{s}}({\omega_{1}})+{\hat{a}_{s}}({\omega_{1}}){e^{-i{\omega_{1}}{T_{1}}}}]{e^{-i{\omega_{1}}{t_{1}}}}\\\
\end{aligned}.\end{array}$ (A18)

The single count probability $P_{SC}(\tau)$, can be expressed as

$P_{SC}(\tau)={\int_{-\infty}^{\infty}{d{t_{1}}}}\left\langle{\psi\left|{\hat{E}_{1}^{(-)}\hat{E}_{1}^{(+)}}\right|\psi}\right\rangle.$
(A19)

Firstly, considering $\hat{E}_{1}^{(+)}\left|\psi\right\rangle$

$\begin{array}[]{l}\begin{aligned}
\hat{E}_{1}^{(+)}\left|\psi\right\rangle&=\frac{1}{{2\sqrt{2\pi}}}\int_{0}^{\infty}{d{\omega_{1}}{{\hat{a}}_{s}}({\omega_{1}})}[1+{e^{-i{\omega_{1}}{T_{1}}}}]{e^{-i{\omega_{1}}{t_{1}}}}\times\int_{0}^{\infty}{d{\omega_{s}}}f({\omega_{s}})\hat{a}_{s}^{\dagger}({\omega_{s}})\left|0\right\rangle\\\
&=\frac{1}{{2\sqrt{2\pi}}}\int_{0}^{\infty}{d{\omega_{1}}}f({\omega_{1}})[1+{e^{-i{\omega_{1}}{T_{1}}}}]{e^{-i{\omega_{1}}{t_{1}}}}\left|0\right\rangle.\end{aligned}\end{array}$
(A20)

In the above calculation, the equations of
$\hat{a}_{s}(\omega_{1})\hat{a}_{s}^{\dagger}(\omega_{s})\left|{0}\right\rangle=\delta(\omega_{1}-\omega_{s})\left|{0}\right\rangle$
are used.

Then,

$\begin{array}[]{lll}\left\langle{\psi\left|{\hat{E}_{1}^{(-)}\hat{E}_{1}^{(+)}}\right|\psi}\right\rangle=\frac{1}{{8\pi}}\int_{0}^{\infty}{d\omega_{1}^{,}}\mathop{f}\nolimits^{*}(\omega_{1}^{,})[1+{e^{i\omega_{1}^{,}{T_{1}}}}]{e^{i\omega_{1}^{,}{t_{1}}}}\times\int_{0}^{\infty}{d{\omega_{1}}}f({\omega_{1}})[1+{e^{-i{\omega_{1}}{T_{1}}}}]{e^{-i{\omega_{1}}{t_{1}}}}.\end{array}$
(A21)

Finally,

$\begin{array}[]{lll}\begin{aligned}
P_{SC}(\tau)&=\int{d{t_{1}}}\left\langle{\psi\left|{\hat{E}_{1}^{(-)}\hat{E}_{1}^{(+)}}\right|\psi}\right\rangle\\\
&=\frac{1}{{8\pi}}\int_{-\infty}^{\infty}{d{t_{1}}}\int_{0}^{\infty}{d\omega_{1}^{,}}\mathop{f}\nolimits^{*}(\omega_{1}^{,})[1+{e^{i\omega_{1}^{,}{T_{1}}}}]{e^{i\omega_{1}^{,}{t_{1}}}}\times\int_{0}^{\infty}{d{\omega_{1}}}f({\omega_{1}})[1+{e^{-i{\omega_{1}}{T_{1}}}}]{e^{-i{\omega_{1}}{t_{1}}}}\\\
&=\frac{1}{4}\int_{0}^{\infty}{\int_{0}^{\infty}{d{\omega_{1}}d\omega_{1}^{,}}}f({\omega_{1}})\mathop{f}\nolimits^{*}(\omega_{1}^{,})[1+{e^{-i{\omega_{1}}{T_{1}}}}][1+{e^{i\omega_{1}^{,}{T_{1}}}}]\delta(\omega-\omega^{\prime})\\\
&=\frac{1}{4}\int_{0}^{\infty}{{d{\omega_{1}}}}{\left|{f({\omega_{1}})[1+{e^{-i{\omega_{1}}{T_{1}}}}]}\right|^{2}}\\\
&=\frac{1}{2}\int_{0}^{\infty}{{d{\omega_{1}}}}{\left|{f({\omega_{1}})}\right|^{2}}[1+\cos({\omega_{1}}{T_{1}})].\\\
\end{aligned}\end{array}$ (A22)

In the above calculation, the relationship of
$\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-i(\omega-\omega^{\prime})t}dt=\delta(\omega-\omega^{\prime})$
is utilized.

### Appendix 2: Calculation of folded Franson interference

Then, we deduce the equations for the folded Franson interference using multi-
mode theory. The setup of the Franson interference is shown in Fig. A1(b). The
two-photon state from a spontaneous parametric down-conversion (SPDC) process
can be described as

$|\psi\rangle=\int_{0}^{\infty}d\omega_{s}\int_{0}^{\infty}d\omega_{i}f\left(\omega_{s},\omega_{i}\right)\hat{a}_{sH}^{\dagger}\left(\omega_{s}\right)\hat{a}_{iV}^{\dagger}\left(\omega_{i}\right)|00\rangle,$
(A23)

where $\omega$ is the angular frequency; $\hat{a}^{\dagger}$ is the creation
operator and the subscripts $s$ and $i$ denote the signal and idler photons
from SPDC, respectively; $H$ and $V$ represent the polarization of signal and
idler photons; $f(\omega_{s},\omega_{i})$ is the joint spectral amplitude of
the signal and idler photons. The detection field operators of detector 1 (D5)
and detector 2 (D6) are

$\hat{E}_{5}^{(+)}(t_{5})=\frac{1}{{\sqrt{2\pi}}}\int_{0}^{\infty}{d\omega_{5}}\hat{a}_{5}(\omega_{5})e^{-i\omega_{5}t_{5}},$
(A24)
$\hat{E}_{6}^{(+)}(t_{6})=\frac{1}{{\sqrt{2\pi}}}\int_{0}^{\infty}{d\omega_{6}\hat{a}_{6}(\omega_{6})}e^{-i\omega_{6}t_{6}},$
(A25)

where the subscripts $5$ and $6$ denote the photons detected by D5 and D6
respectively. The transformation rule after the delay time $\tau$ is

$\begin{aligned}
&\hat{a}_{5}\left(\omega_{5}\right)=\frac{1}{\sqrt{2}}\hat{a}_{4}\left(\omega_{5}\right)=\frac{1}{2}\left[\hat{a}_{3}\left(\omega_{5}\right)e^{-i\omega_{5}\tau}+\hat{a}_{2}\left(\omega_{5}\right)\right]=\frac{1}{2\sqrt{2}}\left[\hat{a}_{1}\left(\omega_{5}\right)e^{-i\omega_{5}\tau}+\hat{a}_{1}\left(\omega_{5}\right)\right]=\frac{1}{2\sqrt{2}}\left(e^{-i\omega_{5}\tau}+1\right)\hat{a}_{1}\left(\omega_{5}\right)\\\
\end{aligned},$ (A26) $\begin{aligned}
&\hat{a}_{6}\left(\omega_{6}\right)=\frac{1}{\sqrt{2}}\hat{a}_{4}\left(\omega_{6}\right)=\frac{1}{2}\left[\hat{a}_{3}\left(\omega_{6}\right)e^{-i\omega_{6}\tau}+\hat{a}_{2}\left(\omega_{6}\right)\right]=\frac{1}{2\sqrt{2}}\left[\hat{a}_{1}\left(\omega_{6}\right)e^{-i\omega_{6}\tau}+\hat{a}_{1}\left(\omega_{6}\right)\right]=\frac{1}{2\sqrt{2}}\left(e^{-i\omega_{6}\tau}+1\right)\hat{a}_{1}\left(\omega_{6}\right)\\\
\end{aligned}.$ (A27)

So, we can rewrite the field operators as

$\begin{aligned}
&\hat{E}_{5}^{(+)}\left(t_{5}\right)=\frac{1}{\sqrt{2\pi}}\int_{0}^{\infty}d\omega_{5}\hat{a}_{5}\left(\omega_{5}\right)e^{-i\omega_{5}t_{5}}=\frac{1}{4\sqrt{\pi}}\int_{0}^{\infty}d\omega_{5}\left(e^{-i\omega_{5}\tau}+1\right)\hat{a}_{1}\left(\omega_{5}\right)e^{-i\omega_{5}t_{5}}\\\
\end{aligned},\\\ $ (A28)

and

$\begin{aligned}
&\hat{E}_{6}^{(+)}\left(t_{6}\right)=\frac{1}{\sqrt{2\pi}}\int_{0}^{\infty}d\omega_{6}\hat{a}_{6}\left(\omega_{6}\right)e^{-i\omega_{6}t_{6}}=\frac{1}{4\sqrt{\pi}}\int_{0}^{\infty}d\omega_{6}\left(e^{-i\omega_{6}\tau}+1\right)\hat{a}_{1}\left(\omega_{6}\right)e^{-i\omega_{6}t_{6}}\\\
\end{aligned}.\\\ $ (A29)

Consider the polarization:

$\hat{E}_{5}^{(+)}\left(t_{5}\right)\hat{E}_{6}^{(+)}\left(t_{6}\right)=\hat{E}_{5H}^{(+)}\left(t_{5}\right)\hat{E}_{6V}^{(+)}\left(t_{6}\right)+\hat{E}_{5V}^{(+)}\left(t_{5}\right)\hat{E}_{6H}^{(+)}\left(t_{6}\right)+\hat{E}_{5H}^{(+)}\left(t_{5}\right)\hat{E}_{6H}^{(+)}\left(t_{6}\right)+\hat{E}_{5V}^{(+)}\left(t_{5}\right)\hat{E}_{6V}^{(+)}\left(t_{6}\right).$
(A30)

In the above equation, only 2 out of 4 terms exist:

$\hat{E}_{5}^{(+)}\left(t_{5}\right)\hat{E}_{6}^{(+)}\left(t_{6}\right)=\hat{E}_{5H}^{(+)}\left(t_{5}\right)\hat{E}_{6V}^{(+)}\left(t_{6}\right)+\hat{E}_{5V}^{(+)}\left(t_{5}\right)\hat{E}_{6H}^{(+)}\left(t_{6}\right).$
(A31)

The coincidence probability $P(\tau)$, which is also the second-order
correlation function $G_{2}(\tau)$, can be expressed as

$\displaystyle P(\tau)\equiv G_{2}(\tau)$
$\displaystyle=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}dt_{5}dt_{6}\left\langle\psi\left|\hat{E}_{6}^{(-)}\hat{E}_{5}^{(-)}\hat{E}_{5}^{(+)}\hat{E}_{6}^{(+)}\right|\psi\right\rangle$
(A32)
$\displaystyle=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}dt_{5}dt_{6}\left\langle\psi\left|\hat{E}_{6V}^{(-)}\hat{E}_{5H}^{(-)}\hat{E}_{5H}^{(+)}\hat{E}_{6V}^{(+)}\right|\psi\right\rangle+\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}dt_{5}dt_{6}\left\langle\psi\left|\hat{E}_{6H}^{(-)}\hat{E}_{5V}^{(-)}\hat{E}_{5V}^{(+)}\hat{E}_{6H}^{(+)}\right|\psi\right\rangle$
$\displaystyle=P_{HV}(\tau)+P_{VH}(\tau).$

First of all, consider
$P_{HV}(\tau)=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}dt_{5}dt_{6}\left\langle\psi\left|\hat{E}_{6V}^{(-)}\hat{E}_{5H}^{(-)}\hat{E}_{5H}^{(+)}\hat{E}_{6V}^{(+)}\right|\psi\right\rangle$.
In this equation:

$\displaystyle\hat{E}_{5H}^{(+)}\left(t_{5}\right)\hat{E}_{6V}^{(+)}\left(t_{6}\right)$
$\displaystyle=\frac{1}{4\sqrt{\pi}}\int_{0}^{\infty}d\omega_{5}\left(e^{-i\omega_{5}\tau}+1\right)\hat{a}_{1H}\left(\omega_{5}\right)e^{-i\omega_{5}t_{5}}\times\frac{1}{4\sqrt{\pi}}\int_{0}^{\infty}d\omega_{6}\left(e^{-i\omega_{6}\tau}+1\right)\hat{a}_{1V}\left(\omega_{6}\right)e^{-i\omega_{6}t_{6}}$
(A33)
$\displaystyle=\frac{1}{16\pi}\int_{0}^{\infty}\int_{0}^{\infty}d\omega_{5}d\omega_{6}\left(e^{-i\omega_{5}\tau}+1\right)\left(e^{-i\omega_{6}\tau}+1\right)\hat{a}_{1H}\left(\omega_{5}\right)\hat{a}_{1V}\left(\omega_{6}\right)e^{-i\omega_{5}t_{5}}e^{-i\omega_{6}t_{6}}.$

Then, consider
$\hat{E}_{5H}^{(+)}\left(t_{5}\right)\hat{E}_{6V}^{(+)}\left(t_{6}\right)|\psi\rangle$

$\displaystyle\hat{E}_{5H}^{(+)}\left(t_{5}\right)\hat{E}_{6V}^{(+)}\left(t_{6}\right)|\psi\rangle$
$\displaystyle=\frac{1}{16\pi}\int_{0}^{\infty}\int_{0}^{\infty}d\omega_{5}d\omega_{6}\left(e^{-i\omega_{5}\tau}+1\right)\left(e^{-i\omega_{6}\tau}+1\right)\hat{a}_{1H}\left(\omega_{5}\right)\hat{a}_{1V}\left(\omega_{6}\right)e^{-i\omega_{5}t_{5}}e^{-i\omega_{6}t_{6}}$
(A34)
$\displaystyle\times\int_{0}^{\infty}d\omega_{s}\int_{0}^{\infty}d\omega_{i}f\left(\omega_{s},\omega_{i}\right)\hat{a}_{sH}^{\dagger}\left(\omega_{s}\right)\hat{a}_{iV}^{\dagger}\left(\omega_{i}\right)|00\rangle$
$\displaystyle=\frac{1}{16\pi}\int_{0}^{\infty}\int_{0}^{\infty}d\omega_{5}d\omega_{6}\int_{0}^{\infty}\int_{0}^{\infty}d\omega_{s}d\omega_{i}\delta\left(\omega_{5}-\omega_{s}\right)\delta\left(\omega_{6}-\omega_{i}\right)$
$\displaystyle\times
f\left(\omega_{s},\omega_{i}\right)\left(e^{-i\omega_{5}\tau}+1\right)\left(e^{-i\omega_{6}\tau}+1\right)e^{-i\omega_{5}t_{5}}e^{-i\omega_{6}t_{6}}|00\rangle$
$\displaystyle=\frac{1}{16\pi}\int_{0}^{\infty}\int_{0}^{\infty}d\omega_{5}d\omega_{6}f\left(\omega_{5},\omega_{6}\right)\left(e^{-i\omega_{5}\tau}+1\right)\left(e^{-i\omega_{6}\tau}+1\right)e^{-i\omega_{5}t_{5}}e^{-i\omega_{6}t_{6}}|00\rangle.$

In the above calculation, the equations of
$\hat{a}_{1H}\left(\omega_{5}\right)\hat{a}_{sH}^{\dagger}\left(\omega_{s}\right)|0\rangle=\delta\left(\omega_{5}-\omega_{s}\right)|0\rangle$
and
$\hat{a}_{1V}\left(\omega_{6}\right)\hat{a}_{iV}^{\dagger}\left(\omega_{i}\right)|0\rangle=\delta\left(\omega_{6}-\omega_{i}\right)|0\rangle$
are used. $\hat{a}_{sH}^{\dagger}\left(\omega_{s}\right)$ and $\hat{a}_{1H}$
are both acting on H photons in path 1, so
$\hat{a}_{sH}^{\dagger}\equiv\hat{a}_{1H}$.

Then,

$\displaystyle\left\langle\psi\left|\hat{E}_{6V}^{(-)}\hat{E}_{5H}^{(-)}\hat{E}_{5H}^{(+)}\hat{E}_{6V}^{(+)}\right|\psi\right\rangle$
$\displaystyle=\frac{1}{16\pi}\int_{0}^{\infty}\int_{0}^{\infty}d\omega_{5}^{\prime}d\omega_{6}^{\prime}f^{*}\left(\omega_{5}^{\prime},\omega_{6}^{\prime}\right)\left(e^{i\omega_{5}^{\prime}\tau}+1\right)\left(e^{i\omega_{6}^{\prime}\tau}+1\right)e^{i\omega_{5}^{\prime}t_{5}}e^{i\omega_{6}^{\prime}t_{6}}$
(A35)
$\displaystyle\times\frac{1}{16\pi}\int_{0}^{\infty}\int_{0}^{\infty}d\omega_{5}d\omega_{6}f\left(\omega_{5},\omega_{6}\right)\left(e^{-i\omega_{5}\tau}+1\right)\left(e^{-i\omega_{6}\tau}+1\right)e^{-i\omega_{5}t_{5}}e^{-i\omega_{6}t_{6}}$
$\displaystyle=\frac{1}{256\pi^{2}}\int_{0}^{\infty}\int_{0}^{\infty}d\omega_{5}d\omega_{6}\int_{0}^{\infty}\int_{0}^{\infty}d\omega_{5}^{\prime}d\omega_{6}^{\prime}f\left(\omega_{5},\omega_{6}\right)f^{*}\left(\omega_{5}^{\prime},\omega_{6}^{\prime}\right)$
$\displaystyle\times\left(e^{-i\omega_{5}\tau}+1\right)\left(e^{-i\omega_{6}\tau}+1\right)e^{-i\omega_{5}t_{5}}e^{-i\omega_{6}t_{6}}\left(e^{i\omega_{5}^{\prime}\tau}+1\right)\left(e^{i\omega_{6}^{\prime}\tau}+1\right)e^{i\omega_{5}^{\prime}t_{5}}e^{i\omega_{6}^{\prime}t_{6}}.$

Finally,

$\displaystyle P_{HV}(\tau)$
$\displaystyle=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}dt_{5}dt_{6}\left\langle\psi\left|\hat{E}_{6V}^{(-)}\hat{E}_{5H}^{(-)}\hat{E}_{5H}^{(+)}\hat{E}_{6V}^{(+)}\right|\psi\right\rangle$
(A36)
$\displaystyle=\frac{1}{256\pi^{2}}\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}dt_{5}dt_{6}\int_{0}^{\infty}\int_{0}^{\infty}d\omega_{5}d\omega_{6}\int_{0}^{\infty}\int_{0}^{\infty}d\omega_{5}^{\prime}d\omega_{6}^{\prime}f\left(\omega_{5},\omega_{6}\right)f^{*}\left(\omega_{5}^{\prime},\omega_{6}^{\prime}\right)$
$\displaystyle\times\left(e^{-i\omega_{5}\tau}+1\right)\left(e^{-i\omega_{6}\tau}+1\right)e^{-i\omega_{5}t_{5}}e^{-i\omega_{6}t_{6}}\left(e^{i\omega_{5}^{\prime}\tau}+1\right)\left(e^{i\omega_{6}^{\prime}\tau}+1\right)e^{i\omega_{5}^{\prime}t_{5}}e^{i\omega_{6}^{\prime}t_{6}}.$

By utilizing
$\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{-i(\omega-\omega^{\prime})t}dt=\delta(\omega-\omega^{\prime})$,
the above equation can be further simplified as

$\displaystyle P_{HV}(\tau)$
$\displaystyle=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}dt_{5}dt_{6}\left\langle\psi\left|\hat{E}_{6V}^{(-)}\hat{E}_{5H}^{(-)}\hat{E}_{5H}^{(+)}\hat{E}_{6V}^{(+)}\right|\psi\right\rangle$
(A37)
$\displaystyle=\frac{1}{64}\int_{0}^{\infty}\int_{0}^{\infty}\int_{0}^{\infty}\int_{0}^{\infty}d\omega_{5}d\omega_{6}d\omega_{5}^{\prime}d\omega_{6}^{\prime}\delta\left(\omega_{5}-\omega_{5}^{\prime}\right)\delta\left(\omega_{6}-\omega_{6}^{\prime}\right)f\left(\omega_{5},\omega_{6}\right)$
$\displaystyle\times\left(e^{-i\omega_{5}\tau}+1\right)\left(e^{-i\omega_{6}\tau}+1\right)f^{*}\left(\omega_{5}^{\prime},\omega_{6}^{\prime}\right)\left(e^{i\omega_{5}^{\prime}\tau}+1\right)\left(e^{i\omega_{6}^{\prime}\tau}+1\right)$
$\displaystyle=\frac{1}{64}\int_{0}^{\infty}\int_{0}^{\infty}d\omega_{5}d\omega_{6}f\left(\omega_{5},\omega_{6}\right)f^{*}\left(\omega_{5},\omega_{6}\right)\left(e^{-i\omega_{5}\tau}+1\right)\left(e^{-i\omega_{6}\tau}+1\right)\left(e^{i\omega_{5}^{\prime}\tau}+1\right)\left(e^{i\omega_{6}^{\prime}\tau}+1\right)$
$\displaystyle=\frac{1}{64}\int_{0}^{\infty}\int_{0}^{\infty}d\omega_{5}d\omega_{6}\left|f\left(\omega_{5},\omega_{6}\right)\right|^{2}\left|\left(e^{-i\omega_{5}\tau}+1\right)\left(e^{-i\omega_{6}\tau}+1\right)\right|^{2}$
$\displaystyle=\frac{1}{16}\int_{0}^{\infty}\int_{0}^{\infty}d\omega_{5}d\omega_{6}\left|f\left(\omega_{5},\omega_{6}\right)\right|^{2}\left[1+\operatorname{cos}\left(\omega_{5}\tau\right)\right]\left[1+\operatorname{cos}\left(\omega_{6}\tau\right)\right].$

Similarly,

$\displaystyle P_{VH}(\tau)$
$\displaystyle=\int_{-\infty}^{\infty}\int_{-\infty}^{\infty}dt_{5}dt_{6}\left\langle\psi\left|\hat{E}_{6H}^{(-)}\hat{E}_{5V}^{(-)}\hat{E}_{5V}^{(+)}\hat{E}_{6H}^{(+)}\right|\psi\right\rangle$
(A38)
$\displaystyle=\frac{1}{64}\int_{0}^{\infty}\int_{0}^{\infty}\int_{0}^{\infty}\int_{0}^{\infty}d\omega_{5}d\omega_{6}d\omega_{5}^{\prime}d\omega_{6}^{\prime}\delta\left(\omega_{5}-\omega_{5}^{\prime}\right)\delta\left(\omega_{6}-\omega_{6}^{\prime}\right)f\left(\omega_{6},\omega_{5}\right)$
$\displaystyle\times\left(e^{-i\omega_{5}\tau}+1\right)\left(e^{-i\omega_{6}\tau}+1\right)f^{*}\left(\omega_{6}^{\prime},\omega_{5}^{\prime}\right)\left(e^{i\omega_{5}^{\prime}\tau}+1\right)\left(e^{i\omega_{6}^{\prime}\tau}+1\right)$
$\displaystyle=\frac{1}{64}\int_{0}^{\infty}\int_{0}^{\infty}d\omega_{5}d\omega_{6}f\left(\omega_{6},\omega_{5}\right)f^{*}\left(\omega_{6},\omega_{5}\right)\left(e^{-i\omega_{5}\tau}+1\right)\left(e^{-i\omega_{6}\tau}+1\right)\left(e^{i\omega_{5}^{\prime}\tau}+1\right)\left(e^{i\omega_{6}^{\prime}\tau}+1\right)$
$\displaystyle=\frac{1}{64}\int_{0}^{\infty}\int_{0}^{\infty}d\omega_{5}d\omega_{6}\left|f\left(\omega_{6},\omega_{5}\right)\right|^{2}\left|\left(e^{-i\omega_{5}\tau}+1\right)\left(e^{-i\omega_{6}\tau}+1\right)\right|^{2}$
$\displaystyle=\frac{1}{16}\int_{0}^{\infty}\int_{0}^{\infty}d\omega_{5}d\omega_{6}\left|f\left(\omega_{6},\omega_{5}\right)\right|^{2}\left[1+\operatorname{cos}\left(\omega_{5}\tau\right)\right]\left[1+\operatorname{cos}\left(\omega_{6}\tau\right)\right].$

Finally, the coincidence probability $P(\tau)$ is:

$\displaystyle P(\tau)$ $\displaystyle=P_{HV}(\tau)+P_{VH}(\tau)$ (A39)
$\displaystyle=\frac{1}{16}\int_{0}^{\infty}\int_{0}^{\infty}d\omega_{5}d\omega_{6}\left|f\left(\omega_{5},\omega_{6}\right)\right|^{2}\left[1+\operatorname{cos}\left(\omega_{5}\tau\right)\right]\left[1+\operatorname{cos}\left(\omega_{6}\tau\right)\right]$
$\displaystyle+\frac{1}{16}\int_{0}^{\infty}\int_{0}^{\infty}d\omega_{5}d\omega_{6}\left|f\left(\omega_{6},\omega_{5}\right)\right|^{2}\left[1+\operatorname{cos}\left(\omega_{5}\tau\right)\right]\left[1+\operatorname{cos}\left(\omega_{6}\tau\right)\right]$
$\displaystyle=\frac{1}{16}\int_{0}^{\infty}\int_{0}^{\infty}d\omega_{5}d\omega_{6}\left[\left|f\left(\omega_{5},\omega_{6}\right)\right|^{2}+\left|f\left(\omega_{6},\omega_{5}\right)\right|^{2}\right]\left[1+\operatorname{cos}\left(\omega_{5}\tau\right)\right]\left[1+\operatorname{cos}\left(\omega_{6}\tau\right)\right].$

If $f\left(\omega_{5},\omega_{6}\right)=f\left(\omega_{5},\omega_{6}\right)$,
then the coincidence probability $P(\tau)$ is

$\displaystyle
P(\tau)=\frac{1}{8}\int_{0}^{\infty}\int_{0}^{\infty}d\omega_{5}d\omega_{6}\left|f\left(\omega_{5},\omega_{6}\right)\right|^{2}\left[1+\operatorname{cos}\left(\omega_{5}\tau\right)\right]\left[1+\operatorname{cos}\left(\omega_{6}\tau\right)\right].$
(A40)

### Appendix 3: Calculation single counts of folded Franson interference

In this section, we deduce the single counts equations for the Franson
interference using multi-mode theory. The setup of the Franson interference is
shown in Fig. A1. The two-photon state from a spontaneous parametric down-
conversion (SPDC) process can be described as

$\left|\psi\right\rangle=\int_{0}^{\infty}{\int_{0}^{\infty}{d\omega_{s}d\omega_{i}}}f(\omega_{s},\omega_{i})\hat{a}_{s}^{\dagger}(\omega_{s})\hat{a}_{i}^{\dagger}(\omega_{i})\left|{00}\right\rangle,$
(A41)

where $\omega$ is the angular frequency; $\hat{a}^{\dagger}$ is the creation
operator and the subscripts $s$ and $i$ denote the signal and idler photons
from SPDC, respectively; $f(\omega_{s},\omega_{i})$ is the joint spectral
amplitude of the signal and idler photons. where $\omega$ is the angular
frequency; $\hat{a}^{\dagger}$ is the creation operator and the subscripts $s$
and $i$ denote the signal and idler photons from SPDC, respectively; $H$ and
$V$ represent the polarization of signal and idler photons; and
$f(\omega_{s},\omega_{i})$ is the joint spectral amplitude of the signal and
idler photons. The detection field operators of detector 1 (D5) and detector 2
(D6) are

$\hat{E}_{5}^{(+)}(t_{5})=\frac{1}{{\sqrt{2\pi}}}\int_{0}^{\infty}{d\omega_{5}}\hat{a}_{5}(\omega_{5})e^{-i\omega_{5}t_{5}},$
(A42)
$\hat{E}_{6}^{(+)}(t_{6})=\frac{1}{{\sqrt{2\pi}}}\int_{0}^{\infty}{d\omega_{6}\hat{a}_{6}(\omega_{6})}e^{-i\omega_{6}t_{6}},$
(A43)

where the subscripts $5$ and $6$ denote the photons detected by D5 and D6
respectively. The transformation rule after the delay time $T$ is (take D5 for
example)

$\displaystyle\hat{a}_{5}\left(\omega_{5}\right)=\frac{1}{\sqrt{2}}\hat{a}_{4}\left(\omega_{5}\right)=\frac{1}{2}\left[\hat{a}_{3}\left(\omega_{5}\right)e^{-i\omega_{5}\tau}+\hat{a}_{2}\left(\omega_{5}\right)\right]=\frac{1}{2\sqrt{2}}\left[\hat{a}_{1}\left(\omega_{5}\right)e^{-i\omega_{5}\tau}+\hat{a}_{1}\left(\omega_{5}\right)\right]=\frac{1}{2\sqrt{2}}\left(e^{-i\omega_{5}\tau}+1\right)\hat{a}_{1}\left(\omega_{5}\right).$
(A44)

So, we can rewrite the field operators as

$\displaystyle\hat{E}_{5}^{(+)}\left(t_{5}\right)=\frac{1}{\sqrt{2\pi}}\int_{0}^{\infty}d\omega_{5}\hat{a}_{5}\left(\omega_{5}\right)e^{-i\omega_{5}t_{5}}=\frac{1}{4\sqrt{\pi}}\int_{0}^{\infty}d\omega_{5}\left(e^{-i\omega_{5}\tau}+1\right)\hat{a}_{1}\left(\omega_{5}\right)e^{-i\omega_{5}t_{5}}.$
(A45)

The single counts’ probability $P(\tau)$, can be expressed as

$P(\tau)=P_{H}(\tau)+P_{V}(\tau)=\int_{-\infty}^{\infty}dt_{5}\left\langle\psi\left|\hat{E}_{5H}^{(-)}\left(t_{5}\right)\hat{E}_{5H}^{(+)}\left(t_{5}\right)\right|\psi\right\rangle+\int_{-\infty}^{\infty}dt_{5}\left\langle\psi\left|\hat{E}_{5V}^{(-)}\left(t_{5}\right)\hat{E}_{5V}^{(+)}\left(t_{5}\right)\right|\psi\right\rangle.$
(A46)

First of all, consider$\hat{E}_{5H}^{(+)}\left(t_{5}\right)|\psi\rangle$

$\displaystyle\hat{E}_{5H}^{(+)}\left(t_{5}\right)|\psi\rangle=\frac{1}{4\sqrt{\pi}}\int_{0}^{\infty}d\omega_{5}\left(e^{-i\omega_{5}\tau}+1\right)\hat{a}_{1H}\left(\omega_{5}\right)e^{-i\omega_{5}t_{5}}\times\int_{0}^{\infty}d\omega_{s}\int_{0}^{\infty}d\omega_{i}f\left(\omega_{s},\omega_{i}\right)\hat{a}_{sH}^{\dagger}\left(\omega_{s}\right)\hat{a}_{iV}^{\dagger}\left(\omega_{i}\right)|00\rangle$
(A47)
$\displaystyle=\frac{1}{4\sqrt{\pi}}\int_{0}^{\infty}d\omega_{5}\int_{0}^{\infty}d\omega_{s}\int_{0}^{\infty}d\omega_{i}f\left(\omega_{s},\omega_{i}\right)\left(e^{-i\omega_{5}\tau}+1\right)e^{-i\omega_{5}t_{5}}\delta\left(\omega_{s}-\omega_{5}\right)\hat{a}_{iV}^{\dagger}\left(\omega_{i}\right)|00\rangle$
$\displaystyle=\frac{1}{4\sqrt{\pi}}\int_{0}^{\infty}d\omega_{s}\int_{0}^{\infty}d\omega_{i}f\left(\omega_{s},\omega_{i}\right)\left(e^{-i\omega_{s}\tau}+1\right)e^{-i\omega_{s}t_{5}}\hat{a}_{iV}^{\dagger}\left(\omega_{i}\right)|00\rangle,$
$\displaystyle\left\langle\psi\left|\hat{E}_{5H}^{(-)}\left(t_{5}\right)\hat{E}_{5H}^{(+)}\left(t_{5}\right)\right|\psi\right\rangle$
$\displaystyle=\frac{1}{16\pi}\int_{0}^{\infty}d\omega_{s}^{\prime}\int_{0}^{\infty}d\omega_{i}^{\prime}f^{*}\left(\omega_{s}^{\prime},\omega_{i}^{\prime}\right)\left(e^{i\omega_{s}^{\prime}\tau}+1\right)e^{i\omega_{s}^{\prime}t_{5}}\hat{a}_{iV}\left(\omega_{i}^{\prime}\right)\times\int_{0}^{\infty}d\omega_{s}\int_{0}^{\infty}d\omega_{i}f\left(\omega_{s},\omega_{i}\right)\left(e^{-i\omega_{s}\tau}+1\right)e^{-i\omega_{s}t_{5}}\hat{a}_{iV}^{\dagger}\left(\omega_{i}\right)$
$\displaystyle=\frac{1}{16\pi}\int_{0}^{\infty}d\omega_{s}^{\prime}\int_{0}^{\infty}d\omega_{i}^{\prime}f^{*}\left(\omega_{s}^{\prime},\omega_{i}^{\prime}\right)\left(e^{i\omega_{s}^{\prime}\tau}+1\right)e^{i\omega_{s}^{\prime}t_{5}}\times\int_{0}^{\infty}d\omega_{s}\int_{0}^{\infty}d\omega_{i}f\left(\omega_{s},\omega_{i}\right)\left(e^{-i\omega_{s}\tau}+1\right)e^{-i\omega_{s}t_{5}}\delta\left(\omega_{i}-\omega_{i}^{\prime}\right)$
$\displaystyle=\frac{1}{16\pi}\int_{0}^{\infty}d\omega_{s}^{\prime}f^{*}\left(\omega_{s}^{\prime},\omega_{i}\right)\left(e^{i\omega_{s}^{\prime}\tau}+1\right)e^{i\omega_{s}^{\prime}t_{5}}\times\int_{0}^{\infty}d\omega_{s}\int_{0}^{\infty}d\omega_{i}f\left(\omega_{s},\omega_{i}\right)\left(e^{-i\omega_{s}\tau}+1\right)e^{-i\omega_{s}t_{5}}$
$\displaystyle=\frac{1}{16\pi}\int_{0}^{\infty}d\omega_{s}\int_{0}^{\infty}d\omega_{i}\int_{0}^{\infty}d\omega_{s}^{\prime}f^{*}\left(\omega_{s}^{\prime},\omega_{i}\right)f\left(\omega_{s},\omega_{i}\right)\left(e^{i\omega_{s}^{\prime}\tau}+1\right)\left(e^{-i\omega_{s}\tau}+1\right)e^{i\omega_{s}^{\prime}t_{5}}e^{-i\omega_{s}t_{5}}.$

Then,

$\displaystyle
P_{H}(\tau)=\int_{-\infty}^{\infty}dt_{5}\left\langle\psi\left|\hat{E}_{5H}^{(-)}\left(t_{5}\right)\hat{E}_{5H}^{(+)}\left(t_{5}\right)\right|\psi\right\rangle$
(A48)
$\displaystyle=\frac{1}{8}\int_{0}^{\infty}d\omega_{s}\int_{0}^{\infty}d\omega_{i}\int_{0}^{\infty}d\omega_{s}^{\prime}f^{*}\left(\omega_{s}^{\prime},\omega_{i}\right)f\left(\omega_{s},\omega_{i}\right)\left(e^{i\omega_{s}^{\prime}\tau}+1\right)\left(e^{-i\omega_{s}\tau}+1\right)\delta\left(\omega_{s}-\omega_{s}^{\prime}\right)$
$\displaystyle=\frac{1}{8}\int_{0}^{\infty}d\omega_{s}\int_{0}^{\infty}d\omega_{i}f^{*}\left(\omega_{s},\omega_{i}\right)f\left(\omega_{s},\omega_{i}\right)\left(e^{i\omega_{s}\tau}+1\right)\left(e^{-i\omega_{s}\tau}+1\right)$
$\displaystyle=\frac{1}{8}\int_{0}^{\infty}d\omega_{s}\int_{0}^{\infty}d\omega_{i}\left|f\left(\omega_{s},\omega_{i}\right)\left(e^{-i\omega_{s}\tau}+1\right)\right|^{2}.$

Similarly,

$\displaystyle
P_{V}(\tau)=\int_{-\infty}^{\infty}dt_{5}\left\langle\psi\left|\hat{E}_{5V}^{(-)}\left(t_{5}\right)\hat{E}_{5V}^{(+)}\left(t_{5}\right)\right|\psi\right\rangle$
(A49)
$\displaystyle=\frac{1}{8}\int_{0}^{\infty}d\omega_{i}^{\prime}\int_{0}^{\infty}d\omega_{s}\int_{0}^{\infty}d\omega_{i}f^{*}\left(\omega_{s},\omega_{i}^{\prime}\right)f\left(\omega_{s},\omega_{i}\right)\left(e^{i\omega_{i}^{\prime}\tau}+1\right)\left(e^{-i\omega_{i}\tau}+1\right)\delta\left(\omega_{i}^{\prime}-\omega_{i}\right)$
$\displaystyle=\frac{1}{8}\int_{0}^{\infty}d\omega_{i}^{\prime}\int_{0}^{\infty}d\omega_{s}f^{*}\left(\omega_{s},\omega_{i}\right)f\left(\omega_{s},\omega_{i}\right)\left(e^{i\omega_{i}\tau}+1\right)\left(e^{-i\omega_{i}\tau}+1\right)$
$\displaystyle=\frac{1}{8}\int_{0}^{\infty}d\omega_{s}\int_{0}^{\infty}d\omega_{i}\left|f\left(\omega_{s},\omega_{i}\right)\left(e^{-i\omega_{i}\tau}+1\right)\right|^{2}.$

Finally, the single counts’ probability $P(\tau)$ is:

$\displaystyle
P(\tau)=P_{H}(\tau)+P_{V}(\tau)=\frac{1}{8}\int_{0}^{\infty}d\omega_{s}\int_{0}^{\infty}d\omega_{i}\left|f\left(\omega_{s},\omega_{i}\right)\left(e^{-i\omega_{s}\tau}+1\right)\right|^{2}+\frac{1}{8}\int_{0}^{\infty}d\omega_{s}\int_{0}^{\infty}d\omega_{i}\left|f\left(\omega_{s},\omega_{i}\right)\left(e^{-i\omega_{i}\tau}+1\right)\right|^{2}$
(A50)
$\displaystyle=\frac{1}{4}\int_{0}^{\infty}d\omega_{s}\int_{0}^{\infty}d\omega_{i}\left|f\left(\omega_{s},\omega_{i}\right)\right|^{2}\left[1+\operatorname{cos}\left(\omega_{s}\tau\right)+1+\operatorname{cos}\left(\omega_{i}\tau\right)\right].$