# Plug-and-play measurement of chromatic dispersion by means of two-photon
interferometry

Romain Dalidet1, Anthony Martin1, Mattis Riesner1, Sidi-Ely Ahmedou2, Romain
Dauliat2, Baptiste Leconte2, Guillaume Walter3, Grégory Sauder1, Jean-
Christophe Delagnes3, Guy Millot4,5, Philippe Roy2, Raphaël Jamier3, Sébastien
Tanzilli1, Laurent Labonté1<EMAIL_ADDRESS>1Université Côte
d’Azur, CNRS, Institut de physique de Nice, France
2Université de Limoges, CNRS, XLIM, UMR 7252, F-87000 Limoges, France
3CELIA, Centre Lasers Intenses et Applications, Université de Bordeaux-CNRS-
CEA, UMR 5107, F-33405 Talence Cedex, France
4ICB, Université de Bourgogne, CNRS, UMR 6303, F-21078 Dijon, France
5Institut Universitaire de France (IUF), 1 Rue Descartes, Paris, France

###### Abstract

Since the first proof-of-principle experiments 25 years ago, quantum metrology
has matured from fundamental concepts to versatile and powerful tools in a
large variety of research branches, such as gravitational-wave detection,
atomic clocks, plasmonic sensing, and magnetometry. At the same time, two-
photon interferometry, which underpins the possibility of entanglement to
probe optical materials with unprecedented levels of precision and accuracy,
holds the promise to stand at the heart of innovative functional quantum
sensing systems. We report a novel quantum-based method for measuring the
frequency dependence of the velocity in a transparent medium, i.e, the
chromatic dispersion (CD). This technique, using energy-time entangled
photons, allows straightforward access to CD value from the visibility of two-
photon fringes recorded in a free evolution regime. In addition, our quantum
approach features all advantages of classical measurement techniques, i.e,
flexibility and accuracy, all in a plug-and-play system.

††preprint: APS/123-QED

##  I Introduction

Quantum metrology is one of the most advanced applications of quantum
technologies exploiting quantum physics foundations towards real-world
applications. While the fragility of quantum states poses stringent
constraints to the development of quantum computer and quantum communication
systems [1, 2], it allows unexpected possibilities for measurement systems.
Several physical platforms can be employed to develop quantum sensors as cold-
atoms, nitrogen-vacancy centers or superconducting circuits, making precise
measurements of time, accelerations, rotations, gravity, and magnetic fields
[3, 4, 5, 6, 7], respectively. They share as common feature on quantum
properties, standing as a distinct and rapidly growing branch of research
within the area of quantum science and technology. Another appealing quantum
system are photons, enabled by their inherent properties such as high
mobility, together with the available technology for their generation,
manipulation, and detection. Recent development of platforms and techniques to
generate suitable quantum photonic states able to provide quantum-enhancement
in different metrology tasks, such as biological systems [8], optical coherent
tomography [9], microscopy and imaging [10], is of significant importance.
Photonic quantum correlations represent a remarkable resource allowing to
enhance the performance of quantum sensors [11].

Since many physical problems can be regarded as phase estimation processes,
two-photon interferometry represents one of the most relevant approach,
notably for the measurement of dispersive properties such as the chromatic
dispersion (CD) [12]. The measurement of CD holds significant relevance in
numerous applications within the field of photonics, particularly in classical
[13] and quantum communications [14], as well as the generation of nonlinear
effects [13], encompassing platforms based on $\chi^{(2)}$ and $\chi^{(3)}$
nonlinearities. To date, measuring CD has been achieved using two main
categories of techniques, relying on spectral or temporal properties of the
light probe. On one hand, temporal CD measurements exhibit a great flexibility
but requires using a spectrally broad light source and a picosecond time
resolution detector over a kilometer-length, resulting in a moderate accuracy
[15]. On the other hand, interferometric techniques enable measurements with
an excellent accuracy over widest spectral range, at the price of (i) complex
experimental setup including spectrometer (ii) systematic errors, and (iii)
moderate signal-to-noise ratio [16]. Even if quantum-enhancement of accuracy
has recently been demonstrated, still, process is rather tedious, time-
consuming and difficult to automate [17].

In this work, we propose a disruptive approach among current CD measurement
methods merging the best aspects of classical techniques with unique quantum
features, in a plug-and-play fashion system associated with a 1% accuracy.
This method relies on the peculiar properties of photonic entanglement through
two-photon interference. We highlight 3 advantages : i) the method lies in
recording the free evolution of the interference fringes, neither spectrometer
nor stabilisation system are required; ii) the value of CD is directly
obtained thanks to an elegant formalism inferring the CD coefficient directly
from the visibility of the two-photon interference; iii) this free-alignment
method is definitively oriented towards real-world applications.

##  II Theoretical framework

Frequency entangled photons generated via spontaneous parametric
downconversion (SPDC) can be described by the 2-photon state:

$|\psi(t)\rangle\propto\iint
d\omega_{s}d\omega_{i}\alpha(\omega_{s}+\omega_{i})\Gamma(\omega_{s},\omega_{i})\hat{a}^{+}_{s}\hat{a}^{+}_{i}|0\rangle,$
(1)

where subscripts s,i denote signal and idler photons, respectively. $\eta$,
$\alpha$, $\Gamma$ represent the strength of the non-linear interaction, the
complex amplitude of the pump spectrum (approximated by a Dirac function in
the continuous regime), and the spectral distribution of the photon pair, i.e
the phase matching function, respectively. The revelation of the entanglement
carried by this bi-photon state lies in the two-photon interference through an
appropriate unbalanced interferometer, often referred to a Franson
configuration [18, 19]. It should be noted that in our case, we regard the
two-photon state propagating within the interferometer as equivalent to an
N00N state, achieved through post-selection during detection [20, 21]. The
visibility of the two-photon interference depends on the indistinguishability
of the two paths in term of losses, polarization, spatial modes, as well as
chromatic dispersion. More specifically, considering Taylor expansion of the
wave-vector around the degeneracy wavelength, odd-order terms of the
dispersion vanishes thanks to energy conservation [22] associated with SPDC.
The phase accumulated by the N00N state propagating within the interferometer
can be expressed:

$\begin{split}\phi_{N00N}&=L\sum_{j=s,i}\sum_{n=0}^{\infty}\frac{\Delta\omega^{n}_{j}}{n!}\beta^{(n)}_{j}\\\
&=L\Big{(}2\beta^{(0)}+\beta^{(2)}\Delta\omega^{2}\Big{)}+O(\Delta\omega^{4}),\end{split}$
(2)

where $\beta^{(n)}=\left.\frac{\partial
k}{\partial\omega}\right|_{\omega_{0}}$, $\Delta\omega$ and $L$ stand as the
detuning from the center frequency $\omega_{0}$ and the length of the sample
under test (SUT), respectively. The first term, $\beta^{(0)}$, represents a
simple phase shift while the second term, $\beta^{(2)}$, is the definition of
the CD. By performing a careful analysis of the evolution of the photon pair
within the interferometer, using Eq.1 and 2, the visibility of the fringes
reads :

$\begin{split}V&=\Bigg{[}\Big{(}\int_{-\infty}^{\infty}d\Delta\omega|\Gamma(\Delta\omega)|^{2}cos(\beta^{(2)}\Delta\omega^{2})\Big{)}^{2}\\\
&+\Big{(}\int_{-\infty}^{\infty}d\Delta\omega|\Gamma(\Delta\omega)|^{2}sin(\beta^{(2)}\Delta\omega^{2})\Big{)}^{2}\Bigg{]}^{1/2},\end{split}$
(3)

(see appendix A for details). The evolution of the visibility highly depends
on the spectral distribution shape of the photon pairs, as shown in Figure1
where gaussian and square shapes are considered. In the case of a Gaussian
distributed spectrum, i.e
$|\Gamma(\Delta\omega)|^{2}=\frac{1}{\sigma\sqrt{2\pi}}e^{\frac{-1}{2}(\frac{\Delta\omega}{\sigma})^{2}}$,
with spectral width $\sigma$, the visibility can be written under the analytic
form:

$V=\frac{1}{\sqrt[4]{\gamma^{2}+1}},$ (4)

with $\gamma=2\sigma^{2}\beta^{(2)}L$ represents an adimensional parameter
carrying all the information on dispersion (sample length $L$, spectral
bandwidth $\sigma$, and dispersion coefficient value $\beta^{(2)}$). This
simple form indicates a straightforward relation between the visibility and
the CD parameter. In other words, when photon pair probes an object,
information about the dispersive behavior of this object is imprinted on the
two-photon state. This direct access to CD appears as an unique feature as
temporal based CD measurements usually give access to the group delay before
inferring the CD parameter. It is important to note that the sensitivity to
$\gamma$ depends on the spectral profile of the photon pairs, as also shown in
Figure1. The highest sensitivity is obtained when the leading coefficient of
the tangent is maximum, namely the inflexion point. Considering only positive
values of $\gamma$, the latter is located at the (only) inflexion of the curve
where the second order derivative is null :

$\frac{\partial^{2}V}{\partial\gamma^{2}}=\frac{3\gamma^{2}-2}{4(\gamma^{2}+1)^{9/4}}\Leftrightarrow\gamma=\sqrt{\frac{2}{3}},$
(5)

corresponding to a visibility of $V=(\frac{5}{3})^{-1/4}\approx 0.88$.
In summary, assuming a Gaussian spectral distribution, a direct access to the
CD parameter is granted thanks to the precise knowledge of the visibility,
which stands as a parameter easily accessible experimentally, without the need
for spectrometer nor active stabilization system.

Figure 1: Visibility of the two-photon interference (V) as a function of the
parameter $\gamma$, assuming Square/Gaussian functions represented by the
red/blue curves, respectively. The red dot shows the inflexion point from Eq.
5.

##  III Two-photon interference visibility estimation

Figure 2: A photon pair experiences a Mach-Zenhder interferometer in which a
phase shifter is introduced. At the output beam splitter both beams interfere
creating interference patterns  Figure 3: Top: Simulated Eq. 6 with $V=0.75$
and $P_{c}(max)=100$, (a) and $P_{c}(max)=1000$ (b). Filled curves represent
the statistical fluctuation due to the Poissonian distribution of the
detection. Bottom: theoretical probability density function (red curve).
Histograms are generated using Eq. 6 for $N=10^{4}$ occurrences, with
$\phi\in[0,2\pi]$ randomly distributed. The orange (c) and the blue (d)
histograms refer to (a) and (b), respectively

In most of optical experiments, stabilisation of the interferometric system is
of utmost importance in order to acquire precise and accurate measurements.
However, many environmental parameters (e.g. temperature, pressure, vibration)
impact the final result, requiring a fine control. In interferometric systems,
both passive and active stabilisation are often considered. For passive
stabilisation, interferometers are placed in an hermetic case with a thermal
feedback. While this method is easy to implement, short- and long-term thermal
drift cannot be managed in the same way, excluding high-degree of accuracy and
long runs. On the other hand, active stabilisation, commonly based on an error
signal generated from the output of the interferometer and fed back to a
phase-compensating mechanism in one of its arms, is more robust against any
kind of variations, at the price of more expensive and complex apparatus (e.g
narrow-linewidth and stable laser, wavelength demultiplexer). Here, we show
theoretically and experimentally an interferometric method based on the free-
evolution of two-photon interference, avoiding the use of spectrometer and
stabilization systems.
FIG. 2 shows a diagram of the general principle of our method. The coincidence
probability at the output of the interferometer is given by:

$P_{c}(\phi)\propto\frac{1}{2}\big{[}1+Vcos(\phi+\phi_{0})\big{]},$ (6)

where $V$ represents the visibility of the two-photon interference:

$V=\frac{C_{max}-C_{min}}{C_{max}+C_{min}},$ (7)

with $C_{max}$ and $C_{min}$ the maximum and minimum coincidence rates,
respectively. The cumulative distribution function (CDF) is defined as the
cumulative density of the variable $\phi$, $\Psi(x)=P_{c}(\phi)<P_{c}(x)$:

$\Psi(x)=\frac{asin(\frac{2}{V}(x-\frac{1}{2}))}{\pi}+\frac{1}{2}.$ (8)

This function monotonically increases and is related to the probability of the
parameter $\phi$ to take a value below $x$. The probability density function
(PDF), $\xi(x)$, which therefore sets the visibility likelihood for a
distribution of coincidences, is defined as the derivative of the
aforementioned function $\Psi(x)$:

$\xi(x)=\frac{2}{\pi V\sqrt{1+\frac{-4x^{2}+4x-1}{V^{2}}}}.$ (9)

In order to illustrate this formalism, we simulate the effect of poissonian
statistic on the parameter estimation V and the PDF through Eq. 9. As shown in
Figure3, we set a two-photon interference visibility to 0.75 with, on average,
100 (a) and 1000 (b) photon-pairs. The red curves represent the theoretical
prediction without finite statistic and the yellow and blue filled curves
represent the spread due to the Poissonian distribution for 100 and 1000
photon-pairs, respectively. The PDF functions, presented in Figure3 (c,d),
show that the probabilities are very close to the extreme values of 0.125 and
0.875 (corresponding to a visibility V$=$0.75) as most of the possible values
spread over the extremes. The Poissonian statistic of the detection induce
modifications of the PDF: i) the edges spread over the two extreme values, ii)
an asymmetry between these two extreme values arises. The first behavior
naturally emerges from the Poissonian noise while the second one comes from
the assymetry of the Poissonian distribution $\gamma=\lambda^{\frac{1}{2}}$,
where $\lambda$ stands as the parameter of Poissonian distribution. In other
words, the relative error of the distribution is inversely proportional to the
number of events. This model clearly shows the necessity of considering high
coincidence rate ($>10^{3}$) to mitigate the impact of the statistic.

Our approach based on the exploitation of the two-photon interference is
reminiscent of the seminal violation of Bell’s inequality applied to
interferometric, or also called Franson, configuration in the 1980’s [18].
This interferometric technique used to measure quantum correlation induced by
energy-time entanglement [19], has opened the way to a wide range of
applications such as non-local dispersion cancellation to ensure the security
of Franson-based quantum key distribution protocols [23, 24, 25]. All cases
require a precise knowledge of the visibility. Two strategies are primarily
exploited. The first one relies on Eq. 7, where the extreme points $C_{max}$
and $C_{min}$ are inferred from the data analysis. On the other hand, the
interference pattern can be fitted by Eq. 6. As the offset always fluctuates
over the acquisition time, only the first oscillations are considered. None of
the methods were convincingly enough as they consider only a part of the
statistics. By contrast, we highlight the originality and the relevance of our
approach which relies on the acquisition of long sets of data, each one being
part of the PDF function. By the way, all the statistic contribute to the
precise knowledge of the V parameter.

##  IV Experimental results

Figure 4: Experimental setup. A CW laser pumps a periodically poled lithium
niobate waveguide (ppLN) from which correlated photon pairs are emitted via
SPDC. These pairs are spectrally shaped with a filter before passing trough a
Michelson interferometer. Coincidence counts are registered using
superconducting nanowire single photon detectors (SNSPD) and time digital
converter (TDC). BS: 50/50 beam-splitter, PC: polarization controller

A CW laser (Toptica TA Pro) at 780.23 nm is sent in a polarisation controller
followed by a pigtailed type-0 periodically poled lithium niobate (PPLN)
waveguide in order to generate frequency correlated photon pairs, through
spontaneous down-conversion (SPDC) process, symmetrically thanks a type-0
around the degenerated wavelength $\lambda_{0}=1560.46$ nm, as shown in
Figure4. The phase matching condition of the non-linear interaction is
adjusted to generate photon-pairs with a flat-top spectrum apart the
degeneracy wavelength. Then the pairs are shaped thanks a filter to obtain a
Gaussian spectrum centered around twice the pump laser wavelength (see Figure
5). Photon pairs travel along a fiber-based interferometer where the sample
under test (SUT) is a standard SMF-28 fiber. Faraday mirrors are placed at the
output of each arm, ensuring mode polarisation indistinguishability. Finally,
coincidences are recorded using superconducting nanowire single photon
detectors (SNSPD) and a Time-to-Digital Converter (TDC). As our method relies
on the acquisition of the interferences operating in a free-running regime, a
careful analysis has to be driven to ensure that the bandwidth of the
detection system is higher than the phase drift.

Figure 5: Spectrum of the photon pairs. The black dotted curve correspond to
the natural spectrum of the pairs. The blue dotted curve is the spectrum
measured after the band-pass filter. The red curve is a fit of the latter,
from which we extract a width of $\sigma=4.57\,nm$. SPDC: pontaneous down-
conversion Figure 6: Top (a): Acquired coincidences (central peak of the
Franson histogram) as a function of time. Bottom left (b): Extracted histogram
from acquired data. Bottom right (c): fitted curve (red) of the left part of
(b), using equation 9.

From the temperature-dependent refractive index of fused silica corresponding
to $\dfrac{\Delta n}{n}\approx 4.8.10^{-6}.K^{-1}.\Delta T$, with $\Delta T$
represents the temperature shift in K [26], the sensitivity for a standard
fiber can be extracted and is $0.2$ K/m per fringes, which corresponds to a
variation of 45 mK in case of a 2.4 m long fiber. By acquiring the temperature
of the experiment room for 10 hours, the Fourier transform of the latter shows
that typical variations have an amplitude of $\approx 35$ mK and a frequency
$f\in[0.5,2]$ Hz, which is consistent with a 100 ms acquisition time.

We set two different strategies for measuring the CD parameter. We extract the
visibility: i) at a fixed operating point $\gamma$, corresponding to the
inflexion point (see Figure3 (a), and then Eq 4 is reverted, ii) for different
values of $\gamma$ corresponding to different spectral bandwidth of the
filter, fitted by Eq. 4.
For both approaches, a prerequisite has to be fulfilled to guarantee that the
CD is the only factor impacting the visibility, ruling out other
distinguishability criterion as differential losses, spatial, time and
polarization modes. A calibration is fulfilled by measuring a 100% visibility
by considering narrow filtering, ensuring all the latter criterion are
satisfied.
Method based on the inflexion point. Regarding Figure3, we need to make a
trade-off concerning the pump power regime: a high number of coincidences
ensuring that the statistics tends to the PDF, while maintaining a double-pair
generation rate low enough to not deteriorate the visibility (typically $<$
0.01 pairs/window of interest), due to accidental coincidences. The pump power
is set to generate a maximum coincidence rate of $P_{c}\approx 10^{4}$ Hz,
while the FWHM of the Gaussian filter is set to $4.5\,nm$, giving a visibility
close to 0.88, in accordance with the inflexion point of Eq. 4. The raw
spectrum out of the PPLN and the Gaussian-like filtered spectrum are shown in
Figure5.

Typical two-photon fringes drifting in a free-running regime are shown in
Figure6(a). An histogram is built from a set of 500 points (50s) in
Figure6(b), and then is fitted thanks to Eq. 9. As expected, the PDF is shaped
by the inherent Poissonian statistic. While the relative error induced by a
Poissonian distribution is lower for high values, the absolute error is
smaller for low values. In order to minimize the fitting error induced by the
distribution, only the left part of the histogram is used (see Figure6(c)),
where the absolute error is the smallest. Then, Eq. 4 is reversed to extract
the CD. We obtain after 200 measurements (Figure8(a)), a CD parameter of
$17,1(2)$ ps/(nm.km).

Figure 7: Chromatic dispersion measurement from 200 data samples (blue
histogram) and its Gaussian fit (red).

Method based on multiple operating points. The CD parameter is extracted by
fitting Eq. 4. Two scenarii are possible to tune the parameter $\gamma$ over
the x-axis in Figure1, either the length of the sample or the spectral
bandwidth of the filter. This latter parameter is easily accessible as most of
band-pass filter can be tunable, strengthening our vision of user-friendly
demonstrator. The length of the sample shall be such that the visibility curve
can be described with standard tunable filters, whose its spectral bandwidth
ranges from $\sim$0.1 nm to $\approx$ 10 nm. According Figure1, the length of
the sample is set to 4.5 m. As the previous method, a calibration is proceeded
prior to any measurement ensuring 100% visibility by considering narrow
filtering. To keep the same poissonian statistic with respect to the different
spectral bandwidth, the pump power increases as the bandwidth of the filter
reduces so that the maximum number of coincidence remains the same
$P_{c}\approx 10^{4}$ Hz. The number of points acquired as well as the
procedure of the visibility extraction remains identical to the first method,
described in Sect.III.

Figure 8: Extracted visibility as a function of the filter bandwidth. The
number of measurements for each points is 200.

Figure8 shows the visibility values associated with each filter spectral
width. The graph is fitted using Eq. 4 with only a free parameter being the CD
parameter, leading to a CD value equal to 17.2(2) ps/(km.nm).
The values of the CD parameter are 17.1(2) ps/(km.nm) and 17.2(2) ps/(km.nm)
for both methods. Both approaches give results that are consistent each of
them within 1% of accuracy, and also with the manufacturer’s data that predict
a value $\leq$ 18 ps/(km.nm)[27]. We would have expected that the first method
would have given a better accuracy as it relies on the highest sensitive
point, whereas the second method implements points with different
sensitivities. This intuition has to be mitigated by a most favourable
statistic because a higher number of runs for the second method has been
performed, 3000 instead of 200 measurements.

We summarize in Table 1 the main characteristics for different CD measurement
methods, including both classical and quantum approaches. Even if the accuracy
appear as a primary importance for a measurement technique, other criteria
should be considered. Exhibiting a precision on the CD parameter of sub % is
requiring for non-linear effects which are sensitive to higher-order of
dispersion $\beta_{i}$ (with $i\geq 2$), as four-wave mixing or modulation
instability. But in most cases as telecommunication networks, sub % accuracy
is not necessary whereas friendly-user, practical and intuitive techniques are
preferred, explaining the prevalence of CD apparatus based on temporal method
in the industry. Our approach stands a trade-off among the different methods,
allowing to exhibit a high accuracy while keeping a simple implementation with
a direct access to the dispersion thanks to unique features of quantum
photonics. We emphasize the novelty of our approach based on free running
acquisition combining the simplicity of temporal methods with the accuracy of
spectral methods, leading the potential to compete with well-established
conventional techniques. Beyond the measurement of CD, this method opens a new
application path, that of probing the dispersive properties of materials, in
the same way as distance measurement for optical coherence tomography or
transmission/absorption for spectroscopy.

Method | CD accuracy | Sample length | Active stabilization | Dedicated equipment
---|---|---|---|---
Phase shift [28] | 0.1% | km | No | Standard
Time of flight [29] | 3% | m | No | Standard
WLI [30] | 1.5% | [cm; m] | Yes | Spectrometer + balanced interferometer
QWLI [17] | 0.02% | [cm; m] | Yes | Spectrometer
Our work | 1% | [cm; m] | No | Standard
Table 1: Comparison between different CD measurement technique. WLI and QWLI
denotes white-light interferometry and quantum white-light interferometry,
respectively

##  V Discussion

We have implemented a new technique for measuring the CD through a direct
relation between the visibility of the two-photon interferences and the CD
parameter, based on the assumption of a gaussian profile of the two-photon
states. This model goes beyond the scope of probing dispersive properties in
material, and could be useful for a wide variety of applications requiring an
accurate knowledge of the visibility as for ensuring security for quantum key
distribution protocols. The key point of our technique gathers the simplicity
of implementation of temporal technique associated to high accuracy of
spectral ones without the requirement of active stabilization. We have shown
that our technique is compliant with two complementary variations resulting to
a similar 1% of accuracy. This work represents a step towards a realistic and
a friendly-user quantum enhanced demonstrator, outperforming classical
counterparts (accuracy, reproducibility and immunity to environment)
demonstrated so far for the last 20 years. Our strategy focuses on a specific
use-case in this present work, but has to be included in a more general
framework, within the acceleration of quantum technologies, of emerging from
the laboratory new generation of quantum-enhanced sensors.

## Acknowledgment

This work has been conducted within the framework of the project OPTIMAL
granted by the European Union by means of the Fond Européen de développement
regional (FEDER). The authors also acknowledge financial support from the
Agence Nationale de la Recherche (ANR) through the projects METROPOLIS, the
CNRS through its program “Mission interdisciplinairité” under project labeled
FINDER, and the French government through its Investments for the Future
programme under the Université Côte d’Azur UCA-JEDI project (Quantum@UCA)
managed by the ANR (ANR-15-IDEX-01).

## Author information

M.R. established the theory, R.D. and G.S. performed the experiments under the
supervision of A.M.
All the authors contributed to the the paper.

## Competing interests

The authors declare that there are no competing interests.

## Data Availability

Data are available from the authors on reasonable request.

##  VI Supplementary Data

###  VI.1 Evolution of input mode a through Michelson interferometer

The creation operator $a^{\dagger}$ at the input of the interferometer evolve
as $a^{\dagger}\rightarrow\frac{1}{\sqrt{2}}(a^{\dagger}+b^{\dagger})$ at the
first beamsplitter, $a^{\dagger}$ and $b^{\dagger}$ representing the lower and
upper path of the interferometer respectively. The operators $c^{\dagger}$ and
$d^{\dagger}$ are the two output modes of the interferometer, i.e. at the
output of the second beamsplitter. The phase that each photon is accumulating
when passing either trough the upper or lower arm, is noted
$\Psi_{a,b}(\omega)$. This function can be developed around a central
frequency $\omega_{0}=\frac{\omega_{pump}}{2}$ :

$\Psi_{i}(\omega)=L_{i}\sum\frac{\partial^{n}k_{i}}{\partial\omega^{n}}\frac{(\omega_{0}-\omega)^{n}}{n!}=L_{i}\sum\beta_{i}^{(n)}\frac{\Omega^{n}}{n!},$
(10)

Where L is the length of the chosen arm. $\beta^{(0)},\beta^{(1)},\beta^{(2)}$
represents a simple phase shift, the inverse of group velocity and the group
velocity dispersion, respectively. The evolution inside the interferometer is
given by :

$\displaystyle a_{in}^{\dagger}$
$\displaystyle\rightarrow\frac{1}{\sqrt{2}}(a^{\dagger}+b^{\dagger})$ (11)
$\displaystyle\rightarrow\frac{1}{\sqrt{2}}(e^{i\Psi_{a}(\omega)}a^{\dagger}+e^{i\Psi_{b}(\omega)}b^{\dagger})$
(12)
$\displaystyle\rightarrow\frac{1}{2}(e^{i\Psi_{a}(\omega)}(c^{\dagger}+d^{\dagger})+e^{i\Psi_{b}(\omega)}(c^{\dagger}-d^{\dagger}))$
(13)
$\displaystyle=\frac{1}{2}\left[(e^{i\Psi_{a}(\omega)}+e^{i\Psi_{b}(\omega)})c^{\dagger}+(e^{i\Psi_{a}(\omega)}-e^{i\Psi_{b}(\omega)})d^{\dagger}\right]$
(14)

###  VI.2 2-photon state through the interferometer

Photon pairs are created by a down-conversion source e.g. type 0 periodically
poled lithium niobate crystal. These pairs are spectrally filtered and enter
the interferometer via input mode a. They can be described by the 2-photon
state:

$|\psi_{in}\rangle=\eta\iint
d\omega_{s,i}\alpha(\omega_{s}+\omega_{i})\Gamma(\omega_{s},\omega_{i})\hat{a}^{+}_{s,i}|0\rangle$
(16)

where subscripts s,i denote signal/idler photons.$\alpha$, $\Gamma$ represent
the complex amplitude of the pump spectrum (approximated by a Dirac function
in the continuous regime) and the spectral distribution of the photon pair,
i.e the phase matching function, respectively. In the following lines, the
evolution through the interferometer is regarded :

$\displaystyle|\psi_{in}\rangle$ $\displaystyle=\iint
d\omega_{s,i}\Gamma(\omega_{s},\omega_{i})\hat{a}^{+}_{s,i}|0\rangle$ (17)
$\displaystyle=\frac{1}{4}\iint
d\omega_{s,i}\Gamma(\omega_{s},\omega_{i})\left[(e^{i\Psi_{a}(\omega_{s})}+e^{i\Psi_{b}(\omega_{s})})c^{\dagger}_{s}+(e^{i\Psi_{a}(\omega_{s})}-e^{i\Psi_{b}(\omega_{s})})d^{\dagger}_{s}\right]$
(18)
$\displaystyle\qquad\qquad\qquad\qquad\quad\times\left[(e^{i\Psi_{a}(\omega_{i})}+e^{i\Psi_{b}(\omega_{i})})c^{\dagger}_{i}+(e^{i\Psi_{a}(\omega_{i})}-e^{i\Psi_{b}(\omega_{i})})d^{\dagger}_{i}\right]|0\rangle$
(19) $\displaystyle|\psi_{out}\rangle$ $\displaystyle=\frac{1}{4}\iint
d\omega_{s,i}\Gamma(\omega_{s},\omega_{i})\left[(e^{i\Psi_{a}(\omega_{s})}+e^{i\Psi_{b}(\omega_{s})})(e^{i\Psi_{a}(\omega_{i})}+e^{i\Psi_{b}(\omega_{i})})c^{\dagger}_{s}c^{\dagger}_{i}\right.$
(20)
$\displaystyle\qquad\qquad\qquad\qquad\quad+(e^{i\Psi_{a}(\omega_{s})}-e^{i\Psi_{b}(\omega_{s})})(e^{i\Psi_{a}(\omega_{i})}-e^{i\Psi_{b}(\omega_{i})})d^{\dagger}_{s}d^{\dagger}_{i}$
(21)
$\displaystyle\qquad\qquad\qquad\qquad\quad+(e^{i\Psi_{a}(\omega_{s})}+e^{i\Psi_{b}(\omega_{s})})(e^{i\Psi_{a}(\omega_{i})}-e^{i\Psi_{b}(\omega_{i})})c^{\dagger}_{s}d^{\dagger}_{i}$
(22)
$\displaystyle\qquad\qquad\qquad\qquad\quad\left.+(e^{i\Psi_{a}(\omega_{s})}-e^{i\Psi_{b}(\omega_{s})})(e^{i\Psi_{a}(\omega_{i})}+e^{i\Psi_{b}(\omega_{i})})d^{\dagger}_{s}c^{\dagger}_{i}\right]|0\rangle$
(23)

###  VI.3 Coincidence Probability

The coincidence probability $P_{c}$ is given by the projection of the output-
state $|\psi_{out}\rangle$ onto the state
$c^{\dagger}_{\omega^{\prime}}d^{\dagger}_{\omega^{\prime\prime}}|0\rangle=|{\omega_{c}\omega^{\prime}_{d}}\rangle$,
that corresponds to detecting one photon at frequency $\omega$ in output-mode
c and one photon at frequency $\omega^{\prime}$ in output-mode d. Assuming a
photo detector measuring photon absorption without distinguishing between
different frequency components, the coincidence probability is equivalent to :

$\displaystyle P_{c}$ $\displaystyle=\iint d\omega
d\omega^{\prime}\left|\langle{\omega_{c}\omega^{\prime}_{d}}|{\Phi_{out}\rangle}\right|^{2}$
(24)

where the integration is over the monochromatic modes that enter the
detectors. Since
$[c_{\omega},c^{\dagger}_{\omega^{\prime}}]=\delta(\omega-\omega^{\prime})$
and due to the symmetry of
$G=G(\omega_{s},\omega_{i})=G(\omega_{i},\omega_{s})$, equation(15) simplifies
to :

$\displaystyle P_{c}$ $\displaystyle=\frac{1}{8}\iint d\omega
d\omega^{\prime}|\Gamma(\omega,\omega^{\prime})|^{2}\left|(e^{i\Psi_{a}(\omega)}+e^{i\Psi_{b}(\omega)})(e^{i\Psi_{a}(\omega^{\prime})}-e^{i\Psi_{b}(\omega^{\prime})})\right|^{2}\quad$
(25)

by posing $\omega=\omega_{0}+\Omega$ and $\omega^{\prime}=\omega_{0}-\Omega$,
rearranging the phase functions $\Psi_{a,b}(\omega)$ to form a total phase
term , equation (16) can be written as :

$\displaystyle P_{c}$ $\displaystyle=\frac{1}{8}\int
d\Omega|\Gamma(\Omega)|^{2}\left|(1+e^{i\Psi(\Omega)})(1-e^{i\Psi(-\Omega)})\right|^{2}$
(26)
$\displaystyle=\frac{1}{8}\int^{\infty}_{-\infty}d\Omega|\Gamma(\Omega)|^{2}(1+e^{i\Psi(\Omega)})(1+e^{-i\Psi(\Omega)})(1-e^{i\Psi(-\Omega)})(1-e^{-i\Psi(-\Omega)})$
(27)
$\displaystyle=\frac{1}{2}\int^{\infty}_{-\infty}d\Omega|\Gamma(\Omega)|^{2}\left[1+\cos(\Psi(\Omega))\right]\left[1-\cos(\Psi(-\Omega))\right]$
(28)
$\displaystyle=\frac{1}{2}\int^{\infty}_{-\infty}d\Omega|\Gamma(\Omega)|^{2}\left[1+\cos(\Psi(\Omega))-\cos(\Psi(-\Omega))-\cos(\Psi(\Omega))\cos(\Psi(-\Omega))\right]$
(29)

The term $\cos(\Psi(\Omega))-\cos(\Psi(-\Omega))$ is an odd function,
regardless the exact form of $\Psi(\Omega)$ while $|\Gamma(\Omega)|^{2}$ is a
even (symmetric) function. Therefore,
$|G^{\prime}(\Omega)|^{2}[(\cos(\Psi(+\Omega))-\cos(\Psi(-\Omega)]$ is odd as
well and the integral over the whole frequency-space equals zero. The term
$\cos(\Psi(\Omega))\cos(\Psi(-\Omega))$ in contrast is even and thus
contributes to $P_{c}$. It can be rewritten using product-to-sum identities
for trigonometric functions as
$\cos(\Psi(\Omega))\cos(\Psi(-\Omega))=\frac{1}{2}{\cos[\Psi(\Omega)+\Psi(-\Omega)]}+\frac{1}{2}{\cos[\Psi(\Omega)-\Psi(-\Omega)]}$
. This separates the even ($\frac{\Psi(\Omega)+\Psi(-\Omega)}{2}$) and odd
($\frac{\Psi(\Omega)-\Psi(-\Omega)}{2}$)) parts of $\Psi(\Omega)$. equation
(20) then becomes :

$\displaystyle P_{c}$ $\displaystyle=\frac{1}{4}\int
d\Omega|\Gamma(\Omega)|^{2}\left\\{2-{\cos[\Psi(\Omega)+\Psi(-\Omega)]}-{\cos[\Psi(\Omega)-\Psi(-\Omega)]}\right\\}$
(30) $\displaystyle=\frac{1}{4}\left\\{2-\int
d\Omega|\Gamma(\Omega)|^{2}\cos[\Psi(\Omega)+\Psi(-\Omega)]-\int
d\Omega|\Gamma(\Omega)|^{2}\cos[\Psi(\Omega)-\Psi(-\Omega)]\right\\}$ (31)

###  VI.4 Evaluation and interpretation

In order to understand what each term in equation (22) is describing, we
primarily neglect the second and higher order terms of the dispersion :

$\displaystyle P_{c}$ $\displaystyle=\frac{1}{4}\left\\{2-\int
d\Omega|\Gamma(\Omega)|^{2}\cos[2\beta^{(0)}L]-\int
d\Omega|\Gamma(\Omega)|^{2}\cos[2\Omega\beta^{(1)}L]\right\\}$ (32)
$\displaystyle=\frac{1}{4}\left\\{2-\cos(2\beta^{(0)}L)-\hat{G^{\prime}}\star\hat{G}(2\beta^{1}L)\right\\}$
(33)

where $\hat{G(t)}=\int^{\infty}_{-\infty}d\Omega\Gamma(\Omega)\cos[\Omega
t]=\int^{\infty}_{-\infty}d\Omega\Gamma(\Omega)e^{i\Omega t}$ stands for the
Fourier transform of $\Gamma(\Omega)$ and
$\hat{G}\star\hat{G}(t)=\int^{\infty}_{-\infty}ds\hat{G(s)}\hat{G^{*}(s-t)}$
for the autocorrelation of $\hat{G}$.

This is the well known two-photon-state-interferogram, where the first term
originates from all the possible distinguishable paths, the second term comes
from two photons (NOON-state, with N=2)) traveling along the same path,
resulting in a Franson-type oscillation and the third term corresponds to the
interference of two identical monochromatic modes , equivalent to the Hong-ou-
Mandel (HOM) effect.

Including now second and third-order dispersion in our model, integration of
equation (22) is not trivial anymore. The The Term corresponding to the
Franson-Type-Oscillation becomes :

$\displaystyle F$ $\displaystyle=\int
d\Omega|\Gamma(\Omega)|^{2}\cos[\Psi(\Omega)+\Psi(-\Omega)]$ (34)
$\displaystyle=\int
d\Omega|\Gamma(\Omega)|^{2}\cos[2\beta^{(0)}L+\beta^{(2)}\Omega^{2}L]$ (35)
$\displaystyle=\int
d\Omega|\Gamma(\Omega)|^{2}\left[\cos(2\beta^{(0)}L)\cos(\beta^{(2)}\Omega^{2}L)-\sin(2\beta^{(0)}L)\sin(\beta^{(2)}\Omega^{2})L\right]$
(36) $\displaystyle=\cos(2\beta^{(0)}L)\cdot\int
d\Omega|\Gamma(\Omega)|^{2}\cos(\beta^{(2)}\Omega^{2}L)-\sin(2\beta^{(0)}L)\cdot\int
d\Omega|\Gamma(\Omega)|^{2}\sin(\beta^{(2)}\Omega^{2}L)$ (37)
$\displaystyle=V_{D}\cdot\cos(2\beta^{(0)}L+\psi_{D})\quad,$ (38)

where

$\psi_{D}=\tan^{-1}\left(\frac{\int
d\Omega|\Gamma(\Omega)|^{2}\sin(\beta^{(2)}\Omega^{2}L)}{\int
d\Omega|\Gamma(\Omega)|^{2}\cos(\beta^{(2)}\Omega^{2}L)}\right)$ (39)

and

$V_{D}=\sqrt{\left(\int
d\Omega|\Gamma(\Omega)|^{2}\cos(\beta^{(2)}\Omega^{2}L)\right)^{2}+\left(\int
d\Omega|\Gamma(\Omega)|^{2}\sin(\beta^{(2)}\Omega^{2}L)\right)^{2}}$ (40)

are the phase and the visibility of the oscillation. Using the Cauchy–Schwarz
inequality one can find that $V_{D}\leq 1$, but the exact value is highly
depending on the spectrum $\Gamma(\Omega)$.

Assuming a Gaussian distributed spectrum of bandwidth $\sigma$, the visibility
and the phase of the oscillation can be calculated as a function of
$\gamma=2\sigma^{2}\beta^{(2)}L$, a constant that combines the spectral width
and chromatic dispersion :

$\int
d\Omega|\Gamma(\Omega)|^{2}\cos(\beta^{(2)}\Omega^{2}L)=\sqrt{\frac{1+\sqrt{1+\gamma^{2}}}{2+2\gamma^{2}}}$
(41) $\int
d\Omega|\Gamma(\Omega)|^{2}\sin(\beta^{(2)}\Omega^{2}L)=\frac{\gamma}{\sqrt{2(1+\gamma^{2})(1+\sqrt{1+\gamma^{2}}}}$
(42)

Equation (31) thus become :

$V_{D}(d)=\frac{1}{\sqrt[4]{\gamma^{2}+1}}$ (43)

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