# Parameter estimation of time and frequency shifts
with generalized HOM interferometry

N<EMAIL_ADDRESS>Centre for Quantum Optical Technologies,
Centre of New Technologies, University of Warsaw, ul. Banacha 2c, 02-097
Warszawa, Poland  S. Felicetti  Istituto di Fotonica e Nanotecnologie,
Consiglio Nazionale delle Ricerche (CNR-IFN), Via Cineto Romano 42, I-00156
Rome, Italy

###### Abstract

Hong-Ou-Mandel interferometry takes advantage of the quantum nature of two-
photon interference to increase the resolution of precision measurements of
time-delays. Relying on few-photon probe states, this approach is applicable
also in cases of extremely sensible samples and it achieves attosecond
(nanometer path length) scale resolution, which is relevant to cell biology
and two-dimensional materials. Here, we theoretically analyze how the
precision of Hong-Ou-Mandel interferometers can be significantly improved by
engineering the spectral distribution of two-photon probe states. In
particular, we assess the metrological power of different classes of biphoton
states with non-Gaussian time-frequency spectral distributions, considering
the estimation of both time- and frequency-shifts. We find that grid states,
characterized by a periodic structure of peaks in the chronocyclic Wigner
function, can outperform standard biphoton states in sensing applications. The
considered states can be feasibly produced with atomic photon sources, bulk
non-linear crystals and integrated photonic waveguide devices.

##  I Introduction

Quantum technologies can reach an intrinsic advantage over classical devices
in sensing applications, by exploiting nonclassical features of photonic,
atomic and solid-state systems. A paradigmatic experiment unveiling
nonclassical features of light is given by Hong, Ou and Mandel (HOM)
interferometry Hong et al. (1987), which consists in the interference of two
identical particles onto a beam-splitter, followed by a coincidence
measurement. HOM interference has been exploited in a variety of applications
in quantum optics Ou (2007); Aspuru-Guzik and Walther (2012); Gianani et al.
(2020). The characteristic dip in the coincidence rate is a direct consequence
of the particle indistinguishability, and it has been exploited to assess the
purity of single photons Cassemiro et al. (2010); Mosley et al. (2008) and to
characterize solid-state photon sources Kim et al. (2016); Aharonovich et al.
(2016). Multi-dimensional HOM interference has also been demonstrated making
use of spatial-temporal Devaux et al. (2020) and spectral-temporal Gerrits et
al. (2015); Jin et al. (2015); Gianani et al. (2018); Thiel et al. (2020);
Chen et al. (2020); Zhang et al. (2021) modes. Of particular interest for the
present work, HOM interferometry makes it also possible to engineer the joint
spectrum of photon pairs. For example, fermionic or bosonic statistics can be
simulated generating entangled biphoton states Sansoni et al. (2012);
Francesconi et al. (2020). In the generalized HOM scheme an additional
frequency shift is applied on one of the two arms of the interferometer, while
the usual phase shift is applied on the other one. Generalized HOM
interferometry has been used to characterize the chronocyclic Wigner function
of photon pairs generated via parametric processes Douce et al. (2013);
Boucher et al. (2015). This paved the way to the generation and manipulation
of entangled grid states defined in the time-frequency space Fabre et al.
(2020a); Lu et al. (2003). Complex engineering in two-dimensional frequency
space has also been demonstrated with bulk Walborn et al. (2003); Jin et al.
(2018) and integrated devices Francesconi et al. (2020).

HOM interferometry is a compelling approach for sensing applications, as it is
robust to changes of the relative phase between the photons, and so it does
not require demanding stabilization stages that are usually needed in
classical interferometers. In this context, the HOM interference has been used
to measure polarization mode dispersion Branning et al. (2000), to perform
quantum optical coherence tomography Abouraddy et al. (2002); Nasr et al.
(2003, 2004); Lopez-Mago and Novotny (2012), and in nonlinear spectroscopy
Mukamel et al. (2020); Dorfman et al. (2021). These approaches are based on
the detection of the minimum of the coincidence rate, and so the ultimate
limit of the precision is given by the width of the HOM dip, which is set by
the photon coherence time. Very recently, a new approach has been proposed
applying the framework of quantum metrology in HOM interferometry. This
approach allows one to tune the path delay to the value that minimizes the
estimation error, and it made it possible to reach attosecond resolution in
precision measurements of time delays Lyons et al. (2018); Chen et al. (2019).
Such an interferometer was also used for polarization measurement Harnchaiwat
et al. (2020), and the effect of the time resolution of the photodetectors was
investigated in Scott et al. (2020), paving the way to calibration-free
protocols. Furthermore, it has been demonstrated Chen et al. (2019) that using
a frequency-entangled photon pair can result in a significant increase of the
estimation precision.

In this article, we show how the precision of quantum parameter-estimation
protocols based on generalized HOM interferometry can be improved by
engineering the the spectral distribution of photon pairs. We analyze the
estimation of time and frequency displacements considering biphoton states
that can be produced with different platforms, such as trapped atomic system,
bulk non-linear crystal and optical integrated waveguide devices. We derive
analytical expressions for the quantum Fisher information (QFI), in order to
assess the metrological power of different kinds of non-Gaussian continuous-
variable states defined in the time-frequency space. Then, to analyze the
estimation precision achievable with HOM interferometers, we derive the Fisher
information (FI) relative to the measurement of the coincidence number. In
particular, we compare the performances achievable using different classes of
biphoton states, considering (1) states with a Gaussian chronocyclic Wigner
distribution, (2) non-Gaussian coherent superpositions of two Gaussian states,
namely time-frequency cat-like states Fabre et al. (2020b), and (3) grid
states with a chronocyclic Wigner function characterized by a periodic
structure in both time and frequency axis. We also show how the addition of a
frequency or temporal chirp can improve the metrological power for
multiparameter estimation. Notably, we find that grid states can outperform
simpler biphoton states in both frequency- and temporal-displacement
estimation. Notice that Grid states represent a time-frequency analogue of
Gottesman, Kitaev and Preskill (GKP) states defined in the standard
quadrature-momentum phase space Gottesman et al. (2001), which also represent
a valuable resource for quantum sensing Duivenvoorden et al. (2017); Legero et
al. (2006). Thus, our analysis links the framework of HOM interferometry and
the field of quantum sensing with non-Gaussian continuous-variable states
Huang et al. (2018); Gessner et al. (2019).

Let us now briefly comment on the quantum nature of the considered protocols.
The common approach to assess quantum advantage in parameter estimation
protocols consists in analyzing the scaling of the achievable precision as a
function of the number $N$ of probes Giovannetti et al. (2006); Tóth and
Apellaniz (2014). The estimation precision is limited by the Cramer-Rao bound,
that is, the variance of any unbiased estimator is bounded by the reciprocal
of the FI. When $N$ uncorrelated states are used as a probe, the FI grows at
most as $~{}\sqrt{N}$, and so the optimal scaling of the estimation error
achievable with a classical protocol is given by $~{}1/\sqrt{N}$. This
fundamental limit can be overcome using quantum resources, such as squeezing
Kwon et al. (2019); Garbe et al. (2019) or quantum correlations between the
probe systems Pezzè et al. (2018); Hyllus et al. (2010), which make it
possible to reach a $~{}1/N$ scaling of the estimation error, known as
Heisenberg limit. However, in quantum estimation protocols based on HOM
interferometry Lyons et al. (2018); Chen et al. (2019); Scott et al. (2020);
Harnchaiwat et al. (2020) the input is composed of two-photon states, and so
for each run of the protocol the number of probes is fixed to $N=2$. In this
case, quantum metrological advantage is not coming from the number of
entangled photons, but it resides in the single-photon nature of the
considered states. Indeed, the HOM dip is limited to one half for classical
fields while it can reach zero for biphoton states, and so the visibility of
the interference signal is intrinsically higher when quantum resources are
available. This approach makes it in principle possible to achieve relevant
quantum advantage also when only a very limited number of photons can be used.
A compelling example where the intensity of the probe field can be relevant is
biological sensing Taylor and Bowen (2016), where a low photon flux must be
used to avoid high-power fields that can damage the sample. The proposed
quantum-sensing scheme can achieve frequency resolutions comparable to the
typical size of lipids $7.5$ nm and water molecules $0.27$ nm.

The paper is organized as follows. In Sec. II, we introduce the formalism and
the theoretical tools used in the rest of the manuscript. First, we describe
the generalized HOM experiment using the chronocyclic Wigner function
formalism; then, we introduce the FI and the QFI, and we derive general
expressions valid for HOM interferometry with biphoton states. Finally, we
analyze the metrological power of states with a Gaussian distribution in the
time-frequency space. In Sec. III, we evaluate the metrological power of
frequency and time cat-like biphoton states. We first derive the QFI, and then
the FI for generalized HOM measurement, and we analyze how the multi-
paramementer estimation precision can be improved adding a frequency or
temporal chirp. In Sec. IV, we derive the expression of the QFI of different
time-frequency grid states and show that it can reach higher values compared
to time-frequency Gaussian and cat-like states. Also in this case, we analyze
the FI for HOM measurement and the effect of frequency or temporal chirp.
Finally, in Sec. V we provide a brief summary of our results and comments on
future perspectives.

##  II Quantum metrology with time-frequency biphoton states

Let us review the description of the generalized HOM experiment Douce et al.
(2013) where, in addition to the usual time displacement applied in one of the
arms of the interferometer, a frequency shift is applied on the other arm,
before the recombination into the beam-splitter. We then introduce the Fisher
information (FI) and the Quantum Fisher Information (QFI) and proceed to the
evaluation of the QFI for a simple time-frequency Gaussian state.

###  II.1 Generalized HOM experiment and chronocyclic Wigner function
formalism

We consider a two-photon pair with orthogonal polarization, called signal and
idler photons, produced by a type-II SPDC process, by a three or four wave
mixing non-linear effect. Since the photons have opposite polarizations, they
can be separated deterministically with a polarizing beam-splitter into two
spatial paths labeled $a$ and $b$. After the PBS, the wavefunction of this
pure quantum state can be written as,

$\ket{\psi}=\iint
d\omega_{s}d\omega_{i}\text{JSA}(\omega_{s},\omega_{i})\hat{a}^{\dagger}(\omega_{s})\hat{b}^{\dagger}(\omega_{i})\ket{0},$
(1)

where $\ket{0}$ is the vacuum state, while
$\hat{a}^{\dagger},\hat{b}^{\dagger}$ are the creation operator of a single
photon in the spatial mode $a$ and $b$, respectively. We denote with
$\omega_{s/i}$ the frequency of the signal and idler photons. The joint
spectral amplitude (JSA) can be written as,

$\text{JSA}(\omega_{s},\omega_{i})=f_{+}(\omega_{+}-\omega_{p})f_{-}(\omega_{-}),$
(2)

where $f_{+,-}(\omega_{\pm})$ is the function characterizing the energy
conservation (resp. the phase matching condition), and is a function of the
collective variables $\omega_{\pm}=\omega_{s}\pm\omega_{i}$. For the sake of
simplicity, the degeneracy frequency $\omega_{p}$ of the SPDC process will be
set to zero with no loss of generality. The joint spectral intensity (JSI) is
defined as the absolute squared value of the JSA and it corresponds to the
joint probability of measuring a signal photon at frequency $\omega_{s}$ and
an idler photon at frequency $\omega_{i}$. Then, with the decomposition
defined by Eq. (2), the wave function can be written:

$\ket{\psi}=\iint
d\omega_{+}d\omega_{-}f_{+}(\omega_{+})f_{-}(\omega_{-})\ket{\omega_{s}}\ket{\omega_{i}}.$
(3)

The range of integration is $\mathds{R}$ because we are dealing with the
collective variables $\omega_{\pm}$. The normalization of the wave function
imposes that $\iint
d\omega_{s}d\omega_{i}\absolutevalue{\text{JSA}(\omega_{s},\omega_{i})}^{2}=\int
d\omega_{+}\absolutevalue{f_{+}(\omega_{+})}^{2}\int
d\omega_{-}\absolutevalue{f_{-}(\omega_{-})}^{2}$ =1 . We define the joint
temporal amplitude (JTA) as the Fourier transform of the JSA,
$\text{JTA}(t_{s},t_{i})=\iint
d\omega_{s}d\omega_{i}\text{JSA}(\omega_{s},\omega_{i})e^{i(\omega_{s}t_{s}+\omega_{i}t_{i})}$.
Similarly to the JSI, the joint temporal intensity (JTI) is defined as the
absolute square value of the JTA, and it corresponds to the probability of
measuring the signal photon at arrival time $t_{s}$ and the idler photon at
$t_{i}$. We will denote the Fourier transform of the functions
$f_{+}(\omega_{\pm})$ with a tilde as follows
$\tilde{f}_{\pm}(t_{\pm})=\int_{\mathds{R}}d\omega_{\pm}e^{i\omega_{\pm}t_{\pm}}f_{\pm}(\omega_{\pm})$
where we defined the temporal collective variables $t_{\pm}=t_{s}\pm t_{i}$.

In general, continuous-variable systems are conveniently described in terms of
the Wigner distribution. The analogue of the Wigner function defined for time-
frequency continuous variables is called chronocyclic Wigner function Fabre et
al. (2020a); Brecht and Silberhorn (2013), and for biphoton states it can be
written under the form,

$W_{\hat{\rho}}(\omega_{s},\omega_{i},t_{s},t_{i})=\frac{1}{\pi^{2}}\iint
d\omega_{s}^{\prime}d\omega_{i}^{\prime}e^{2i\omega_{s}^{\prime}t_{s}}e^{2i\omega^{\prime}_{i}t_{i}}\\\
\crossproduct\bra{\omega_{s}-\omega_{s}^{\prime},\omega_{i}-\omega^{\prime}_{i}}\hat{\rho}\ket{\omega_{s}+\omega_{s}^{\prime},\omega_{i}+\omega^{\prime}_{i}},$
(4)

where $\hat{\rho}$ is the density matrix and the expression is normalized to
one, $\iint\iint
d\omega_{s}dt_{s}d\omega_{i}dt_{i}W_{\hat{\rho}}(\omega_{s},\omega_{i},t_{s},t_{i})=1$.
The four marginals of the chronocyclic Wigner distribution correspond to
physical quantities that are experimentally accessible MacLean et al. (2019),

$\displaystyle\iint
W_{\hat{\rho}}(\omega_{s},\omega_{i},t_{s},t_{i})d\omega_{s}d\omega_{i}=\text{JTI}(t_{s},t_{i}),$
(5) $\displaystyle\iint
W_{\hat{\rho}}(\omega_{s},\omega_{i},t_{s},t_{i})dt_{s}dt_{i}=\text{JSI}(\omega_{s},\omega_{i}),$
(6) $\displaystyle\iint
W_{\hat{\rho}}(\omega_{s},\omega_{i},t_{s},t_{i})d\omega_{s/i}dt_{i/s}=\text{JTSI}(\omega_{s/i},t_{i/s}).$
(7)

We defined the joint temporal-spectral intensity (JTSI) that corresponds to
the probability of measuring one photon with arrival time $t$ and the other
with frequency $\omega$. Since we consider a pure state
$\hat{\rho}=\ket{\psi}\bra{\psi}$, and with the factorization defined in Eq.
(3), the chronocyclic Wigner distribution of the photon pair can be factorized
as follows,

$W_{\hat{\rho}}(\omega_{s},\omega_{i},t_{s},t_{i})=W_{+}(\omega_{+},t_{+})W_{-}(\omega_{-},t_{-}),$
(8)

where $W_{\pm}(\omega_{\pm},t_{\pm})=\frac{1}{\pi}\int
e^{2i\omega^{\prime}t_{\pm}}f_{\pm}(\omega_{\pm}-\omega^{\prime})f^{*}_{\pm}(\omega_{\pm}+\omega^{\prime})d\omega^{\prime}$.
As discussed in the following, the generalized HOM experiment corresponds to a
direct measurement of the phase-matching function $W_{-}(\omega_{-},t_{-})$.
The marginal of the chronocyclic Wigner distribution
$W_{\pm}(\omega_{\pm},t_{\pm})$ are respectively the squared absolute value of
the spectral and temporal distribution of the energy conservation (resp. phase
matching function) $\absolutevalue{f_{\pm}(\omega_{\pm})}^{2}$ and
$\absolutevalue{\tilde{f}_{\pm}(t_{\pm})}^{2}$. The inverse transform from the
chronocyclic Wigner distribution to the function $f_{\pm}$ can be cast in the
form:

$f_{\pm}(\omega_{\pm})=\frac{1}{f_{\pm}^{*}(0)}\int
W_{\pm}(\omega_{\pm}/2,t)e^{i\omega_{\pm}t}dt.$ (9)

Note that each individual chronocyclic Wigner distribution $W_{\pm}$ is
normalized to one. We can give some examples of the function $f_{\pm}$. The
time-frequency analogue of EPR states (i.e. two-mode squeezed state with
infinite squeezing parameter) will be defined as
$\absolutevalue{f_{+}(\omega_{+})}^{2}=\delta(\omega_{+})$ for anticorrelated
frequency photon pair, while for correlated frequency photon pair, we will
have $\absolutevalue{f_{-}(\omega_{-})}^{2}=\delta(\omega_{-})$. A physically-
relevant approximation to these states can be obtained replacing the Dirac
distribution with Gaussian functions.

Let us now describe the HOM interferometer using the formalism we have
introduced. After the separation of the photon pair generated by SPDC (see
Fig. 1), two unitary operations are performed: a frequency displacement in
path $a$ and a time displacement operation in path $b$. The wavefunction of
the photon pair after these displacements operations can be written as:

$\ket{\psi(\tau,\mu)}=\iint
d\omega_{s}d\omega_{i}e^{i\omega_{s}\tau}f_{+}(\omega_{+}+\mu)f_{-}(\omega_{-}+\mu)\\\
\hat{a}^{\dagger}(\omega_{s})\hat{b}^{\dagger}(\omega_{i})\ket{0}.$ (10)

After changing the polarization with a half-plate of one photon of the pair,
the two photons are then recombined into a balanced beam-splitter. By
selecting only the coincidence events, the post-selected wave function can be
written as,

$\ket{\psi(\tau,\mu)}=\frac{1}{2}[\iint
d\omega_{s}d\omega_{i}f_{+}(\omega_{+}+\mu)(e^{i\omega_{s}\tau}f_{-}(\omega_{-}+\mu)\\\
-e^{i\omega_{i}\tau}f_{-}(-\omega_{-}+\mu))\hat{a}^{\dagger}(\omega_{s})\hat{b}^{\dagger}(\omega_{i})\ket{0}].$
(11)

Assuming a flat frequency response for the detectors, the coincidence
probability is: $I(\tau,\mu)=\iint
d\omega_{s}d\omega_{i}\absolutevalue{\bra{\omega_{s},\omega_{i}}\ket{\psi(\tau,\mu)}}^{2}$
(see Fabre et al. (2020b) for a mathematical proof), and so we can write

$I(\mu,\tau)=\frac{1}{2}(1-\int
d\omega_{+}\absolutevalue{f_{+}(\omega_{+})}^{2}\int
d\omega_{-}e^{2i\omega_{-}\tau}\\\ \crossproduct
f_{-}(\omega_{-}+\mu)f^{*}_{-}(-\omega_{-}+\mu)).$ (12)

Normalization of the wavefunction sets $\int
d\omega_{+}\absolutevalue{f_{+}(\omega_{+})}^{2}=1$. The coincidence rate can
also be expressed in terms of the chronocyclic Wigner distribution Brecht and
Silberhorn (2013); Fabre et al. (2020a) $W_{-}$ of the phase matching function
of the SPDC process,

$I(\mu,\tau)=\frac{1}{2}(1-\pi W_{-}(\tau,\mu)).$ (13)

The last equation shows that the generalized HOM experiment allows one to
perform the tomography of the phase-matching function. Being $I_{B}$ the
probability of bunching, that is of two photons exiting the interferometer
through the same port, we have that $I_{B}(\mu,\tau)+I(\mu,\tau)=1$. When the
photons are fully distinguishable, the coincidence rate is given by 1/2, and
the bunching and coincidence events happen with the same probability.
Furthermore, when $I(\mu,\tau)>1/2$ the chronocyclic Wigner distribution is
negative, which is an entanglement witness Eckstein and Silberhorn (2008).

Figure 1:  (a) Schematic of the generalized HOM experiment. A monochromatic
pump crosses a non-linear optical medium generating a photon pair by a type-II
SPDC described by the wavefunction $\ket{\psi}$. The photon pair is then
spatially separated by a polarizing beam-splitter. A frequency $\mu$ (resp. a
time $\tau$) delay is applied in spatial port A and B. Finally, a non-time
resolved coincidence measurement is performed. (b) Modelisation of the photon
losses of the photo-detector with a beam-splitter with reflectivity $\gamma$
which quantifies the photon losses.

Let us now comment on how it is possible to perform measurements of time-
frequency continuous variables in practice. To our knowledge, the full
reconstruction of the chronocyclic Wigner distribution of the phase matching
function $W_{-}$ has not yet been experimentally implemented. Since the
spectral bandwidth in Orieux et al. (2011); Francesconi et al. (2020); Ferreri
et al. (2020) for the optical integrated systems is in the order of 50 GHz,
and in bulk system Chen et al. (2019) is 250 GHz, the technical challenge is
in the required sampling of such a distribution. The smallest frequency shift
achievable with nowadays electro-optic modulators is limited to 15 GHz in the
telecom band Chen et al. (2021). Another way for implementing such frequency
shift for wider spectral distributions consists in introducing a dynamical
shift of the eigenfrequencies or eigenmodes of optical resonators or
waveguides Kuramochi and Notomi (2016), which corresponds to a tuning of the
photon frequency. A waveguide-based optomechanical frequency shift can
displace the frequency of telecom wavelength single photon up to 150 GHz,
thanks to the piezoelectric effect (see Ref. Fan et al. (2016)). It has also
been demonstrated that such shift does not deteriorate the HOM visibility. The
experimental possibility of implementing frequency displacements motivates the
interest in a protocol that estimates the amount of frequency shifts. HOM
interferometry has already proven to be a valuable approach for temporal
displacement estimation Lyons et al. (2018); Chen et al. (2019), and so the
generalized HOM setting represents a promising method to simultaneously
estimate temporal $\tau$ and frequency $\mu$ shifts.

###  II.2 Classical and Quantum Fisher information

The precision of a parameter estimation protocol is quantified by the
covariance $\text{cov}(\vec{\alpha})$, where $\vec{\alpha}=(\tau,\mu)$ is a
vector of the parameters to be estimated. The optimal limit of achievable
precision is set by the Cramer-Rao (CR) bound
$\text{cov}(\vec{\alpha}_{\text{CR}})$, if only classical resources are
available. When quantum resources are available, the optimal measurement can
saturate the quantum Cramer-Rao bound (QCR)
$\text{cov}(\vec{\alpha}_{\text{QCR}})$. In particular we can write

$\text{cov}(\vec{\alpha})\geq\text{cov}(\vec{\alpha}_{\text{CR}})=\frac{1}{N}(F)^{-1}\geq\frac{1}{N}({\cal{F}}^{\text{tot}})^{-1}=\text{cov}(\vec{\alpha}_{\text{QCR}}),$
(14)

where $N$ is the number of repetitions of the experiments, $F$ is the Fisher
information and $\mathcal{F}^{\text{tot}}$ is the quantum Fisher information.
The covariance is defined as $\text{cov}(\omega t)=\langle\omega
t\rangle-\langle\omega\rangle\langle t\rangle$. Here, we consider the
estimation of time and frequency displacements defined as it follows. The time
(resp. frequency) displacement operation can be explicitly written Fabre et
al. (2020a) as
${\cal{\hat{D}}}(\tau)=\int\hat{a}^{\dagger}(t+\tau)\hat{a}(t)dt$,
$\hat{D}(\mu)=\int d\omega\hat{a}^{\dagger}(\omega+\mu)\hat{a}(\omega)$. After
the displacement operations, the wave function is described by Eq. (10). For
such a pure state $\ket{\psi(\mu,\tau)}\bra{\psi(\mu,\tau)}$ , the Quantum
Fisher information can be cast under the form Liu et al. (2020):

${\cal{F}}^{\text{tot}}=\begin{pmatrix}{\cal{F}}^{\text{tot}}_{\tau\tau}&{\cal{F}}^{\text{tot}}_{\mu\tau}\\\
{\cal{F}}^{\text{tot}}_{\mu\tau}&{\cal{F}}^{\text{tot}}_{\mu\mu}\end{pmatrix}$
(15)

where the coefficients are

${\cal{F}}^{\text{tot}}_{\mu\tau}=4(\bra{\frac{\partial\psi(\tau)}{\partial\tau}}\ket{\frac{\partial\psi(\mu)}{\partial\mu}}\\\
-\absolutevalue{\bra{\psi(\mu)}\ket{\frac{\partial\psi(\mu)}{\partial\mu}}\bra{\psi(\tau)}\ket{\frac{\partial\psi(\tau)}{\partial\tau}}}).$
(16)

The QFI matrix elements can be expressed in two ways. The first is:

$\displaystyle{\cal{F}}^{\text{tot}}_{\tau\tau}=$ $\displaystyle
4\text{Var}(\omega_{i})$ (17) $\displaystyle{\cal{F}}^{\text{tot}}_{\mu\mu}=$
$\displaystyle 4\text{Var}(t_{s})$ (18)
$\displaystyle{\cal{F}}^{\text{tot}}_{\mu\tau}=$ $\displaystyle
4\text{cov}(\omega_{i}t_{s})$ (19)

where for instance the variance is
$\text{Var}(\omega_{i})=\langle\omega^{2}_{i}\rangle-\langle\omega_{i}\rangle^{2}$
defined with a statistical average, taken with respect to the chronocyclic
Wigner distribution of the photon pair (see Eq. (4)). The second expression of
the QFI is relevant when we consider the factorization of the Wigner
distribution as in Eq. (8):

$\displaystyle{\cal{F}}^{\text{tot}}_{\tau\tau}=$ $\displaystyle
4(\text{Var}(\omega_{+})+\text{Var}(\omega_{-}))=4({\cal{F}}^{+}_{\tau\tau}+{\cal{F}}^{-}_{\tau\tau})$
(20) $\displaystyle{\cal{F}}^{\text{tot}}_{\mu\mu}=$ $\displaystyle
4(\text{Var}(t_{+})+\text{Var}(t_{-}))=4({\cal{F}}^{+}_{\mu\mu}+{\cal{F}}^{-}_{\mu\mu})$
(21) $\displaystyle{\cal{F}}^{\text{tot}}_{\mu\tau}=$
$\displaystyle-4\text{cov}(\omega_{-}t_{-})={\cal{F}}^{-}_{\mu\tau}$ (22)

the first two equations will be called the temporal and the frequency QFI (see
Appendix A for more details). The QFI part ${\cal{F}}_{-}$ which depends only
on the phase-matching function will be now written ${\cal{F}}$. Here the
variances denote a statistical average taken over the chronocyclic Wigner
distribution of the phase matching function:

$\displaystyle\langle\omega_{\pm}^{\alpha}t_{\pm}^{\beta}\rangle=\iint
d\omega_{\pm}dt_{\pm}\omega_{\pm}^{\alpha}t_{\pm}^{\beta}W_{\pm}(\omega_{\pm},t_{\pm}).$
(23)

Note that ${\cal{F}}$ does not depend on the parameters to estimate $\tau$ or
$\mu$, but only on the parameters of the input wave function. Besides, the
knowledge of one variance affects the knowledge of the other, which is a
consequence of the non-commutativity of time and frequency displacement
operators Fabre et al. (2020a). The origin of the non-commutativity of these
operators are the commutation relations of creation and annihilation
operators, and so it is a feature of few-photon states and it has no
counterpart for classical fields. The Cramer-Rao bound is written in terms of
the inverse of the QFI matrix,

$\displaystyle\tilde{{\cal{F}}}_{\tau\tau}=\frac{{\cal{F}}_{\mu\mu}}{{\cal{F}}_{\tau\tau}{\cal{F}}_{\mu\mu}-{\cal{F}}_{\mu\tau}^{2}}$
(24)
$\displaystyle\tilde{{\cal{F}}}_{\mu\mu}=\frac{{\cal{F}}_{\tau\tau}}{{\cal{F}}_{\tau\tau}{\cal{F}}_{\mu\mu}-{\cal{F}}_{\mu\tau}^{2}}$
(25)
$\displaystyle\tilde{{\cal{F}}}_{\mu\tau}=\frac{-{\cal{F}}_{\mu\tau}}{{\cal{F}}_{\tau\tau}{\cal{F}}_{\mu\mu}-{\cal{F}}_{\mu\tau}^{2}}.$
(26)

When ${\cal{F}}_{\mu\tau}=0$, the inverse of the QFI is simply a diagonal
matrix of the inverse of ${\cal{F}}_{\mu\mu}$ and ${\cal{F}}_{\tau\tau}$.

We then consider the Fisher information Liu et al. (2020), which is defined
for a fixed measurement basis, and which can be evaluated as

$F_{\mu\tau}(\alpha)=\sum_{y}\frac{\partial_{\mu}P(y|\alpha)\partial_{\tau}P(y|\alpha)}{P(y,\alpha)},$
(27)

where $P$ is the probability of getting the measurement result $y$ given the
parameter $\alpha$ verifying the condition $\sum_{\alpha}P(y|\alpha)=1$. For
the generalized HOM experiment, the sum in Eq. (27) is taken with respect to
the zero, bunching and coincidence events Lyons et al. (2018); Chen et al.
(2019):

$F_{\mu\tau}(\tau,\mu)=\frac{\partial_{\tau}P_{2}(\tau,\mu)\partial_{\mu}P_{2}(\tau,\mu)}{P_{2}(\tau,\mu)}+\frac{\partial_{\tau}P_{1}(\tau,\mu)\partial_{\mu}P_{1}(\tau,\mu)}{P_{1}(\tau,\mu)}\\\
+\frac{\partial_{\tau}P_{0}(\tau,\mu)\partial_{\mu}P_{0}(\tau,\mu)}{P_{0}(\tau,\mu)},$
(28)

where $P_{2}$ is the coincidence rate, $P_{1}$ the single count probability,
and $P_{0}$ the no-count probability. The finite efficiency of photodetectors
is modelized introducing two beam-splitters of reflectivity $\gamma$ before
the photo-detectors Lyons et al. (2018); Chen et al. (2019) (see Fig. 1):

$\begin{pmatrix}P_{0}(\tau,\mu)\\\ P_{1}(\tau,\mu)\\\
P_{2}(\tau,\mu)\end{pmatrix}=\begin{pmatrix}\gamma^{2}&\gamma^{2}\\\
2\gamma(1-\gamma)&1-\gamma^{2}\\\
1-2\gamma(1-\gamma)-\gamma^{2}&0\end{pmatrix}.\begin{pmatrix}I(\tau,\mu)\\\
I_{B}(\tau,\mu)\end{pmatrix}.$ (29)

Therefore, the probabilities can be expressed as:

$\displaystyle P_{0}(\tau,\mu)=$ $\displaystyle\ \gamma^{2},$ (30)
$\displaystyle P_{1}(\tau,\mu)=$
$\displaystyle\frac{1}{2}(1-\gamma)^{2}(\frac{1+3\gamma}{1-\gamma}+\pi
W_{-}(\tau,\mu)),$ (31) $\displaystyle P_{2}(\tau,\mu)=$
$\displaystyle\frac{1}{2}(1-\gamma)^{2}(1-\pi W_{-}(\tau,\mu)).$ (32)

Then, we can evaluate the Fisher information, defined by Eq. (28):

$F_{\mu\tau}(\mu,\tau)=\frac{\pi^{2}}{2}(1-\gamma)^{2}\partial_{\tau}W_{-}(\mu,\tau)\partial_{\mu}W_{-}(\mu,\tau)\\\
\crossproduct(\frac{1}{1-\pi
W_{-}(\mu,\tau)}+\frac{1}{\frac{1+3\gamma}{1-\gamma}+\pi W_{-}(\mu,\tau)}).$
(33)

For single parameter estimation, we are interested in the diagonal element of
the Fisher information matrix. For time-displacement estimation, when the
frequency shift is set to zero, the FI reads as:

$F_{\tau\tau}(0,\tau)\equiv
F_{\tau\tau}(\tau)=\frac{\pi^{2}}{2}(1-\gamma)^{2}(\partial_{\tau}W_{-}(0,\tau))^{2}\\\
\crossproduct(\frac{1}{1-\pi
W_{-}(0,\tau)}+\frac{1}{\frac{1+3\gamma}{1-\gamma}+\pi W_{-}(0,\tau)}).$ (34)

In the limit of zero losses $\gamma\rightarrow 0$ and when the parameters
$\tau,\mu\rightarrow 0$, the FI $F_{\tau\tau}(\tau)$ reaches the variance
${\cal{F}}^{-}_{\tau\tau}=\text{Var}(\omega_{-})$ for an even phase matching
function, which means that the measurement is optimal (see Appendix B). In
particular, it is the case for a Gaussian phase matching function Lyons et al.
(2018), or for the frequency cat-like state Chen et al. (2019), which is
discussed in more details in Sec.II.3 and Sec. III. A measurement strategy
that saturates the Cramer-Rao bound $\delta\tau=\delta\tau_{\text{CR}}$ in Eq.
(14) has been introduced in Lyons et al. (2018), here we extend it to the
multi-parameter case in Appendix E. Notice that, by taking into account the
losses only at the detection stage, we have neglected the photon losses
between the generation of the pair and the beam-splitter, which would have as
consequence to create a mixed state in the particle-number degree of freedom.
Also, in such a case, the QFI would be no longer the variances of the
chronocyclic Wigner distribution. Notice that one could increase the total QFI
by increasing $\text{Var}(\omega_{+})$ with the same strategies that we are
going to develop in this paper for increasing the term
$\text{Var}(\omega_{-})$. However, we will be interested in this paper only in
the $f_{-}$ part of the biphoton wavefunction because the information of
$f_{+}$ is not accessible with generalized HOM interferometry. Accordingly,
here we will consider only the QFI corresponding to the covariances of the
anti-correlated frequency and time variables.

###  II.3 Quantum Fisher information of a time-frequency Gaussian state

In order to evaluate the performance of various time-frequency entangled
photon pair, we will use as benchmark the precision achievable with a time-
frequency Gaussian state, that is a state with a Gaussian chronocyclic Wigner
distribution $W_{-}$. The wavefunction of a time-frequency Gaussian state can
be written as $\ket{\psi}=\iint
d\omega_{s}d\omega_{i}f_{+}(\omega_{+})f^{\sigma}_{-}(\omega_{-}-\Delta)\ket{\omega_{s}}\ket{\omega_{i}}$.
The part of the spectrum depending on $f_{+}$ is not exploitable within our
metrological protocol since the generalized HOM interferometer does not give
any information about this function. The function $f^{\sigma}_{-}$ is a
Gaussian function of width $\sigma$ and centered at
$\Delta=\omega_{1}-\omega_{2}$. The chronocyclic Wigner distribution of the
associated phase-matching function is also Gaussian and can be written as:

$W_{-}(\omega,t)=\text{exp}(-\frac{(\omega-\Delta)^{2}}{\sigma^{2}})\text{exp}(-t^{2}\sigma^{2}).$
(35)

The quantum Fisher information matrix as defined in Eq. (15) is given by,

${\cal{F}}=\frac{1}{2}\begin{pmatrix}\sigma^{2}&0\\\
0&1/\sigma^{2}\end{pmatrix}.$ (36)

Then, the higher (lower) the frequency bandwidth of the phase matching
function is, the better the temporal (frequency) estimation of a parameter is.

Let us now comment the sensitivity that could be achieved with such a state
using state-of-the-art photon sources. A Gaussian phase-matching function of
SPDC process can be engineered in optical integrated circuit as in Francesconi
et al. (2020); Ferreri et al. (2020), or by four-wave mixing in atomic media
in the group delay regime where the projected pump field is transferred onto
the bandwidth with a scaling factor $v_{g}$, which is the group speed. Typical
value of $\sigma$ is $2\pi 10.9$ THz in AlGaAs optical integrated devices
Fabre et al. (2020a), or in the order of $0.1$ THz in Francesconi et al.
(2020). For periodically-poled potassium titanyl phosphate (PPKTP) crystal, it
is possible to engineer the phase-matching function by modulating the duty
cycle of the grating structure, which is centered at 1582 nm with a width of 5
nm (0.5 THz) in Dixon et al. (2013). This engineering was first pointed out in
Brańczyk et al. (2011), where a Gaussian phase-matching function has been
implemented with an indirect modulation of the crystal nonlinearity. Such a
technique, called Gaussian apodization is also applicable for KTP crystal van
der Meer et al. (2020). For photons produced by SPDC with laser-cooled atomic
system (see for instance Zhao et al. (2015, 2014)), the typical frequency
width is $\sigma\sim 10^{6}$ Hz. In Table. 1, we present the QCR bound for a
given number of repetitions of experiment $N$ for all these sources with
Gaussian phase-matching function. From these references values, the aim is to
engineer the phase matching function to beat these values, for these two
ranges of parameters.

By adding a quadratic frequency phase, the wavefunction becomes:
$\ket{\psi}=\iint
d\omega_{s}d\omega_{i}f_{+}(\omega_{+})f^{\sigma}_{-}(\omega_{-}-\Delta)f^{iC}_{-}(\omega_{-}-\Delta)\ket{\omega_{s}}\ket{\omega_{i}}$,
where $f^{iC}_{-}(\omega_{-}-\Delta)=\text{exp}(\pm
i(\omega-\Delta)^{2}/2C^{2})$. The corresponding chronocyclic Wigner
distribution is:

$W_{-}(\omega,t)=\text{exp}(-\frac{(\omega-\Delta)^{2}}{\sigma^{2}})\text{exp}(-t^{2}\sigma^{2})\text{exp}(\mp\frac{\omega
t\sigma^{2}}{C^{2}}).$ (37)

Then the QFI has non zero anti-diagonal elements:

${\cal{F}}=\frac{1}{2}\begin{pmatrix}\sigma^{2}&\mp\sigma^{2}/C^{2}\\\
\mp\sigma^{2}/C^{2}&1/\sigma^{2}+\sigma^{2}/C^{4}\end{pmatrix}.$ (38)

A chirp adds a time-frequency correlation ($\langle\omega_{-}t_{-} \rangle\neq
0$) and increases the temporal variance and can be created by engineering a
temporal lens $C^{2}=40$ kHz/$\mu$s in Mazelanik et al. (2020). In this last
example, we point out that $\sigma^{2}/C^{4}=6.25\ 10^{-10}\text{Hz}^{-2}$
while $1/\sigma^{2}\sim 10^{-12}\text{Hz}^{-2}$ then the frequency chirp used
in laser-cooled atomic system is indeed beneficial for frequency estimation
parameter. Then, the inverse of the QFI is:

$\displaystyle\tilde{{\cal{F}}}_{\tau\tau}=$ $\displaystyle\
1/\sigma^{2}+\sigma^{2}/C^{4}$ (39) $\displaystyle\tilde{{\cal{F}}}_{\mu\mu}=$
$\displaystyle\ \sigma^{2}$ (40) $\displaystyle\tilde{{\cal{F}}}_{\mu\tau}=$
$\displaystyle\ \sigma^{2}/C^{2}.$ (41)

The corresponding QCR bound are indicated with a red color in Table. 1. The
use of such a chirp does not improve temporal estimation but it is beneficial
for simultaneous measurement of a time and frequency parameter. Alternatively,
by using a temporal chirp (noted $\tilde{C}$ in unit of time), the inverse of
the QFI is:

$\displaystyle\tilde{{\cal{F}}}_{\tau\tau}=$ $\displaystyle\ 1/\sigma^{2}$
(42) $\displaystyle\tilde{{\cal{F}}}_{\mu\mu}=$ $\displaystyle\
\sigma^{2}+1/(\sigma^{2}\tilde{C}^{4})$ (43)
$\displaystyle\tilde{{\cal{F}}}_{\mu\tau}=$ $\displaystyle\
\sigma^{2}\tilde{C}^{2}.$ (44)

Typical value of the chirp of a pump $\tilde{C}^{2}\sim 10^{5}fs^{2}$ in bulk
platform is for instance Brecht and Silberhorn (2013), with $\sigma\sim 1.5$
THz, then $\sigma^{2}\gg 1/\sigma^{2}\tilde{C}^{4}$ which means that the
typical value of temporal chirp is not useful for temporal estimation. The
corresponding QCR bound are indicated with a blue color in Table. 1. For
integrated optics, such as AlGaAs devices Francesconi et al. (2020), the phase
matching function of the photon pair is the spatial profile of the pump, the
quadratic chirp could then be engineered with a spatial light modulator.

The expression of the FI can also be handled. In particular, the expression of
the temporal FI for zero chirp $C=0$ has been given in Lyons et al. (2018).
Note that the frequency FI has the same expression as the temporal one in that
case, just by inverting $\sigma$ to $1/\sigma$ in the expression of interest.
When the frequency chirp is different than zero, $1/\sigma^{2}$ would be
replaced by $1/\sigma^{2}+\sigma^{2}/C^{4}$, and it is then possible to reach
a lower value of the Cramer-Rao bound. Finally, even if there are
photodetectors losses, the choice of parameters that increase the QFI, also
increase the FI.

|  $\Delta\tau\sqrt{N}$ (s) |  $\Delta\mu\sqrt{N}$ (Hz) |  $\Delta(\mu\tau)\sqrt{N}$
---|---|---|---
AlGaAs device Fabre et al. (2020a) | $10^{-14}$ | $10^{14}$ |  /
AlGaAs device Orieux et al. (2011) and PPKTP device Dixon et al. (2013) | $10^{-12}$ | $10^{12}$ |  /
Laser Cooled atomic system Zhao et al. (2015, 2014) |  $10^{-6}$,  $10^{-5}$ |  $10^{6}$,  $10^{6}$ |  25
Bulk system Brecht and Silberhorn (2013) |  $10^{-13}$, $10^{-13}$ |  $10^{13}$, $10^{13}$ |  $10^{9}$
Table 1: Typical values of the QCR bound for Gaussian phase matching function,
considering physical parameters of nowadays photon sources. In red, values
reachable in our knowledge by using a frequency chirp for bulk system. In
blue, value reachable in our knowledge with a temporal chirp for the laser
cooled atomic platform. In the left column, we have also indicated examples of
experimental set-ups. N stands for the number of repetitions of the
experiment.

##  III Parameter estimation with cat-like states

In this section we analyze estimation precision achievable with time-frequency
cat-like states Fabre et al. (2020b), which are biphoton states with a non-
Gaussian chronocyclic Wigner function.

###  III.1 Estimation of temporal or frequency shifts

The phase-matching function of a frequency cat-like state is given by the sum
of two Gaussian functions centered at different frequencies. Accordingly, the
chronocyclic Wigner function of a time-frequency cat-like state is of the same
form of the phase-space Wigner distribution of a standard Schroedinger cat
state. In Ref. Fabre et al. (2020a), we explained why the time-frequency
continuous variables of single photon are mathematically analogous to the
quadrature position-momentum variables, owing to the underlying non-
commutative algebra of creation and annihilation operators corresponding to
frequency or arrival-time degrees of freedom. The wave function of the
frequency cat-like state can be written as:

$\ket{\psi}=N_{+}\iint\text{d}\omega_{s}d\omega_{i}f_{+}(\omega_{+})(f^{\sigma}_{-}(\omega_{-}-\Delta)+f^{\sigma}_{-}(\omega_{-}+\Delta))\\\
\crossproduct\ket{\omega_{s}}\ket{\omega_{i}}\\\
=N_{+}(\ket{\psi_{\Delta}}+\ket{\psi_{-\Delta}}),$ (45)

where $N_{+}$ is a normalization factor given by
$N_{+}^{2}(1+\text{exp}(-\Delta^{2})/\sigma^{2})=1$. The Fourier transform of
the frequency cat-like state Eq. (45) is:

$\tilde{f}_{-}(t_{-})=2N_{+}\text{cos}(\Delta t_{-})e^{-t_{-}^{2}\sigma^{2}}.$
(46)

The Chronocyclic Wigner distribution of the phase-matching function is:

$W_{-}(\omega_{-},\tau)=N_{+}\big{[}e^{-\tau^{2}\sigma^{2}}e^{-(\omega_{-}+\Delta)^{2}/\sigma^{2}}+e^{-\tau^{2}\sigma^{2}}\\\
\crossproduct
e^{-(\omega_{-}-\Delta)^{2}/\sigma^{2}}+2e^{-\omega_{-}^{2}/\sigma^{2}}e^{-\tau^{2}\sigma^{2}}\text{cos}(2\Delta\tau)\big{]},$
(47)

and is represented in Fig. 2 (b). The distribution is composed of two auto-
terms, which correspond to the two peaks centered at $(\pm\Delta,0)$ and an
interference pattern along the time axis. The presence of the oscillations
indicates that the state is indeed a coherent superposition of two Gaussian
functions. We remind that the usual HOM experiment when the frequency shift is
set to zero gives the coincidence probability
$I(0,\tau)=1/2(1-W_{-}(0,\tau))$, which is represented in Fig. 5 (a).

Figure 2:  Chronocyclic Wigner distribution of the time cat-like state (left)
and frequency cat-like state (right). On the left, the units of the horizontal
axis (resp. vertical) is the $\mu$s (resp. MHz) for atomic laser-cooled
system. On the right, the units of the horizontal (resp. vertical) axis is ps
(resp. 1 THz) with the examples given bulk, but for the integrated sources it
is $10^{-1}$ ps (resp. 100 GHz).

An evaluation of the Quantum Fisher information for the frequency cat-like
state gives,

$\displaystyle{\cal{F}}_{\tau\tau}=$ $\displaystyle\
\frac{\sigma^{2}(1+e^{-\Delta^{2}/\sigma^{2}})+2\Delta^{2}}{1+e^{-\Delta^{2}/\sigma^{2}}}=\sigma^{2}+\frac{2\Delta^{2}}{1+e^{-\Delta^{2}/\sigma^{2}}},$
(48) $\displaystyle{\cal{F}}_{\mu\mu}=$ $\displaystyle\
\frac{1}{\sigma^{2}}(1-2\frac{\Delta^{2}}{\sigma^{2}}\frac{e^{-\Delta^{2}/\sigma^{2}}}{1+e^{-\Delta^{2}/\sigma^{2}}}),$
(49) $\displaystyle{\cal{F}}_{\mu\tau}=$ $\displaystyle\ 0.$ (50)

These results are represented in Fig. 3 and 4. The temporal QFI of the
frequency cat-like state Eq. (48) with different notation has been provided in
Chen et al. (2019), while our formalism allows us to extend this result to
multiparameter estimation of time and frequency displacements. The QFI of the
coherent superposition of frequency cat-like state is indeed greater than the
one of a coherent state $\sigma^{2}$. In general, the frequency peaks are well
separated, which means $\Delta=10\sigma$, and then the denominator of the
normalization factor is equal to $1+e^{-\Delta^{2}/\sigma^{2}}\sim 1$. Hence,
the variance $\Delta\tau$ is reduced by a factor 10 compared to the time-
frequency Gaussian state. For instance, for the AlGaAs waveguide Orieux et al.
(2011), the typical separation between the peaks is 5 nm, while the width of
each peak is $0.5$ nm. The fact that the state is better suited for temporal
estimation and not for frequency estimation is reminiscent of the Heisenberg’s
inequality. In addition, the non-diagonal quantum Fisher matrix element is
zero ${\cal{F}}_{\mu\tau}=0$, and so this state is not suited for simultaneous
frequency and arrival-time measurement.

Figure 3:  Expression of the temporal QFI normalized by the width of each peak
as function of the ratio $(\Delta/\sigma)$ for the frequency cat-like state
(in blue) and for the time-frequency Gaussian state (in red). Figure 4:
Expression of the frequency QFI normalized by the width of each peak as
function of the ratio $(\Delta/\sigma)$ for the frequency cat-like state (in
blue) and for the time-frequency Gaussian (in red).

Note that a statistical mixture of two colors
$\hat{\rho}=1/2(\ket{\psi_{\Delta}}\bra{\psi_{\Delta}}+\ket{\psi_{-\Delta}}\bra{\psi_{-\Delta}})$
gives a lower QFI than the state in Eq. (45), as we obtain

${\cal{F}}_{\tau\tau}(\tau)=8e^{-2\tau^{2}\sigma^{2}}(\Delta\text{sin}(\Delta\tau)+\tau\sigma^{2}\text{cos}(\Delta\tau))^{2}\\\
+8\tau^{2}\sigma^{4}e^{-2\tau^{2}\sigma^{2}}e^{-2\Delta^{2}/\sigma^{2}}.$ (51)

For mixed states the QFI is not given by the variance, the derivation is shown
in the Appendix C. The QFI of the mixed state depends on the parameter to
estimate $\tau$ and is lower than the QFI of the pure state. The first term
corresponds to the square of the derivative of the Wigner distribution, and it
gives the structure of double peaks. As $\tau\rightarrow 0$, the QFI of this
mixed state goes to 0, which is the main difference compared to the pure case.

It is possible to exchange the role of the frequency and arrival-time
variables. We define a time cat-like state, the state corresponding to the
$\pi/2$ rotation of the chronocyclic Wigner distribution of the frequency cat-
like state (see right side Fig. 2), whose wavefunction is given by,

$\ket{\psi}=N_{+}\iint
dt_{s}dt_{i}\tilde{f}_{+}(t_{+})(\tilde{f}_{-}^{1/\sigma}(t_{-}-\tilde{\Delta})+\tilde{f}_{-}^{1/\sigma}(t_{-}+\tilde{\Delta}))\\\
\crossproduct\ket{t_{s}}\ket{t_{i}},$ (52)

the normalization factor being
$N_{+}^{2}(1+\text{exp}(-\tilde{\Delta}^{2}\sigma^{2}))=1$, and where
$\tilde{\Delta}=t_{1}-t_{2}$, as it is for the frequency cat-like state. The
chronocyclic Wigner distribution of such a state is

$W_{-}(\omega_{-},\tau)=N_{+}(e^{-\frac{\omega_{-}^{2}}{\sigma^{2}}}e^{-(\tau-\tilde{\Delta})^{2}\sigma^{2}}+e^{-\frac{\omega_{-}^{2}}{\sigma^{2}}}e^{-(\tau+\tilde{\Delta})^{2}\sigma^{2}}\\\
+2e^{-\omega_{-}^{2}/\sigma^{2}}e^{-\tau^{2}\sigma^{2}}\text{cos}(2\omega_{-}\tilde{\Delta})),$
(53)

and it is represented in Fig. 2(a). The QFI has the same form as for the
frequency cat-like state and can be written as:

$\displaystyle{\cal{F}}_{\mu\mu}=$
$\displaystyle\frac{1}{\sigma^{2}}+\frac{2\tilde{\Delta}^{2}}{1+e^{-\tilde{\Delta}^{2}\sigma^{2}}},$
(54) $\displaystyle{\cal{F}}_{\tau\tau}=$ $\displaystyle\
\frac{1}{\sigma^{2}}(1-2\frac{\tilde{\Delta}^{2}}{\sigma^{2}}\frac{e^{-\tilde{\Delta}^{2}/\sigma^{2}}}{1+e^{-\tilde{\Delta}^{2}/\sigma^{2}}}).$
(55)

These functions are represented respectively in Fig. 3 and Fig. 4 since the
frequency and time variables have been exchanged. Laser-cooled atomic systems
represent the most suitable platform to generate broader temporal
distributions of single-photon wave-packets to be applied in frequency
parameter estimation protocols. The frequency-estimation error is improved by
a factor $10$ with respect to a Gaussian phase-matching function, by assuming
that the difference between the arrival times of the photon pairs can reach
the value $\tilde{\Delta}\sim 10\sigma$, which corresponds to two well-
separated temporal wave packets.

###  III.2 Fisher information for frequency and time cat-like states

The temporal Fisher information of the frequency cat-like state [see Eq. (34)]
can be cast as,

$F_{\tau\tau}(\tau)=\frac{1}{2}(1-\gamma)^{2}(2\Delta\text{sin}(2\Delta\tau)+\\\
+2\tau\sigma^{2}\text{cos}(2\Delta\tau))^{2}e^{-2\tau^{2}\sigma^{2}}\\\
\times[\frac{1}{1-e^{-\tau^{2}\sigma^{2}}\text{cos}(2\Delta\tau)}+\frac{1}{\frac{1+3\gamma}{1-\gamma}+e^{-\tau^{2}\sigma^{2}}\text{cos}(2\Delta\tau)}]$
(56)

while for frequency estimation we obtain,

$F_{\mu\mu}(\mu)=\frac{1}{2}(1-\gamma)^{2}\frac{4}{N_{+}^{2}\sigma^{2}}((\mu+\Delta)e^{-(\mu+\Delta)^{2}/\sigma^{2}}\\\
+(\mu-\Delta)e^{-(\mu-\Delta)^{2}/\sigma^{2}}+2\mu
e^{-\mu^{2}/\sigma^{2}})^{2}\\\
\crossproduct(\frac{1}{1-\frac{1}{N_{+}}(e^{-(\mu+\Delta)^{2}/\sigma^{2}}+e^{-(\mu-\Delta)^{2}/\sigma^{2}}+2e^{-\mu^{2}/\sigma^{2}}}\\\
+\frac{1}{\frac{1+3\gamma}{1-\gamma}+\frac{1}{N_{+}}(e^{-(\mu+\Delta)^{2}/\sigma^{2}}+e^{-(\mu-\Delta)^{2}/\sigma^{2}}+2e^{-\mu^{2}/\sigma^{2}})})).$
(57)

We represent Eq. (56) in Fig. 5 (b) and (d) for different values of the losses
($\gamma=0,0.3$ and with $\Delta/\sigma=10$, with the corresponding
coincidence probability $I(0,\tau)$ in Fig. 5 (a) and (c). For instance, in
Ref. Chen et al. (2019), $\sigma=250$ GHz and $\Delta=2.5$ THz, while in Ref.
Francesconi et al. (2020), $\sigma\sim 50$ GHz and for a feasible spectral
interval $\Delta=500$ GHz. For zero losses, as $\tau\rightarrow 0$ and
$\mu\rightarrow 0$, the Fisher information reaches the Quantum Fisher
information (see the proof in the Appendix B) because the phase matching is an
even function, which means again that the measurement is optimal. The
expression of Eq. (57) is given for the sake of completeness given that, even
if it does not lead to a useful estimation strategy, it shows the wide
applicability of the chronocyclic Wigner function formalism. Notice that if
the photons have another source of distinguishability as in Chen et al.
(2019), the FI is not maximum at $\tau=0$, even if there are no photodetector
losses. In this case, the maximum of the FI must be found in a post-processing
calculation, which allows finding the optimal point for measuring a time
parameter.

Note that the temporal QFI Eq. (56) is equivalent to the frequency QFI of the
time cat-like state Eq. (52). In other words, since the performance for
temporal and frequency estimation is inverted when a $\pi/2$ rotation is
processed, then the FI of the time cat-like state will also have the form Eq.
(56), by replacing $\tau$ by $\mu$ and $\Delta$ by $\tilde{\Delta}$. Hence,
the analysis presented in Chen et al. (2019) holds for frequency estimation
with a time cat-like state. The saturation of the Cramer-Rao bound can be done
with the procedure explained in Appendix E, in the case of simultaneous
estimation of a frequency and time parameter.

Figure 5: (a) (c) Coincidence probability $I(0,\tau)$ (see Eq. (13)) of the
frequency cat-like state for $\Delta/\sigma=10$ and for different values of
photodetector losses $\gamma=0,0.3$. (b), (d) Expression of the FI as function
of the time delay $\tau$ for the frequency cat-like state for different values
of losses $\gamma=0,0.3$. The maximum of the FI is at $\tau=0$ but the FI does
not reach the QFI in red (equals to $2\Delta^{2}+\sigma^{2}$) unless the
photodetector losses are set to zero. The units depends on the used platform.
We have chosen the units as in the experimental platform Chen et al. (2019).

###  III.3 Effect of a temporal and frequency chirp

Let us now discuss the addition of a chirp on the time or frequency cat-like
state for simultaneous estimation. We can apply a frequency or a time chirp on
a temporal or frequency cat-like state. They will both give the same off-
diagonal elements of the QFI matrix. We then limit ourselves to present the
case of the wave function of the time cat-like state. When a temporal chirp is
added, the wave function is given by the convolutional product of the temporal
spectrum Eq. (52), and the chirp function $g(t)=e^{-i\pi/4}e^{it^{2}C^{2}}$,

$\ket{\psi}=N_{+}(\iint
dt_{s}dt_{i}\tilde{f}(t_{+})(\tilde{f}_{-}^{1/\sigma}(t_{-}-\tilde{\Delta})\circledast
g(t_{-}-\tilde{\Delta})\\\
+\tilde{f}_{-}^{1/\sigma}(t_{-}+\tilde{\Delta})\circledast
g(t_{-}+\tilde{\Delta}))\ket{t_{s}}\ket{t_{i}}),$ (58)

where $N_{+}$ is the normalization factor again given by
$N_{+}^{2}(1+\text{exp}(-\Delta^{2})/\sigma^{2})=1$. The associated
chronocyclic Wigner distribution can be calculated (see for instance Matz and
Hlawatsch (2003)) as:

$W_{-}(\omega_{-},\tau)=N_{+}(e^{-(\tau-\tilde{\Delta})^{2}\sigma^{2}}e^{-(\omega_{-}+(\tau-\tilde{\Delta})C^{2})^{2}/\sigma^{2}}\\\
+e^{-(\tau+\tilde{\Delta})^{2}\sigma^{2}}e^{-(\omega_{-}+(\tau+\tilde{\Delta})C^{2})^{2}/\sigma^{2}}\\\
+2e^{-\tau^{2}\sigma^{2}}e^{-(\omega_{-}+\tau
C^{2})^{2}/\sigma^{2}}\text{cos}(2\tilde{\Delta}\omega_{-})).$ (59)

The QFI for such a state is then:

$\displaystyle{\cal{F}}_{\mu\mu}=$ $\displaystyle\
\frac{1}{\sigma^{2}}+\frac{2\tilde{\Delta}^{2}}{1+e^{-\tilde{\Delta}^{2}\sigma^{2}}},$
(60) $\displaystyle{\cal{F}}_{\tau\tau}=$ $\displaystyle\
\sigma^{2}+\frac{C^{4}}{\sigma^{2}}-\frac{\tilde{\Delta}^{2}\sigma^{4}}{2}\frac{e^{-\tilde{\Delta}^{2}\sigma^{2}}}{1+e^{-\tilde{\Delta}^{2}\sigma^{2}}},$
(61) $\displaystyle{\cal{F}}_{\mu\tau}=$ $\displaystyle\
\frac{(C/\sigma)^{2}}{1+e^{-\tilde{\Delta}^{2}\sigma^{2}}}.$ (62)

Then, the inverse of the QFI in the case where $\tilde{\Delta}\sigma\gg 1$:

$\displaystyle\tilde{{\cal{F}}}_{\mu\mu}=\frac{\sigma^{2}+C^{4}/\sigma^{2}}{1+2\tilde{\Delta}^{2}(\sigma^{2}+C^{4}/\sigma^{2})},$
(63)
$\displaystyle\tilde{{\cal{F}}}_{\tau\tau}=\frac{1/\sigma^{2}+2\tilde{\Delta}^{2}}{1+2\tilde{\Delta}^{2}(\sigma^{2}+C^{4}/\sigma^{2})},$
(64)
$\displaystyle\tilde{{\cal{F}}}_{\mu\tau}=\frac{(C/\sigma)^{2}}{1+2\tilde{\Delta}^{2}(\sigma^{2}+C^{4}/\sigma^{2})}.$
(65)

The corresponding state with a Gaussian phase-matching function would give an
off-diagonal QFI: $\tilde{{\cal{F}}}_{\mu\tau}=(C/\sigma)^{2}$. Hence, since
$\tilde{\Delta}\sigma=10$, $\tilde{{\cal{F}}}_{\mu\tau}$ is reduced from a
factor 100 with a cat-like state. For the other terms, their expressions show
that in the same limit, there is no advantage for temporal and frequency
estimation of using such a chirp.

The next step is to evaluate the Fisher information $F_{\mu\tau}(\mu,\tau)$.
For the sake of brevity we do not show its expression here, the analysis is
the same as for the temporal estimation using a frequency cat-like state.
Since the phase-matching function is an even one, the FI converges to the QFI
when $\gamma,\mu,\tau\rightarrow 0$. In such a limit, the generalized HOM
interferometer allows one to fully benefit from the cat structure for
simultaneous measurement of the time and frequency parameters.

##  IV Parameter estimation with time-frequency grid states

In this section, we quantify the performance of estimating time and frequency
parameters with a photon pair possessing a phase matching function with a
periodic grid structure, using the time-frequency Gaussian state as a
benchmark. We consider states that are generated when the non-linear medium
producing a photon pair is placed inside an optical cavity. In this case, the
Fabry-Perot structure of peaks Olindo et al. (2006) will give the spectral
distribution of the generated biphoton state.

###  IV.1 Quantum Fisher information for temporal grid states

For anti-correlated photons, the frequency phase-matching function of what we
will call a time-grid state can be written as:

$f_{-}(\omega_{-})=Ae^{-\frac{(\omega_{-}-\omega_{0})^{2}}{2\sigma^{2}}}T\sum_{n=0}^{\infty}R^{n}e^{2in(\omega_{-}-\omega_{0})\overline{\tau}},$
(66)

with the normalization factor $A$ given by
$1=\absolutevalue{A}^{2}\sqrt{\pi\sigma^{2}}\sum_{n,m}R^{n+m}e^{-(n-m)^{2}\overline{\tau}^{2}\sigma^{2}}$.
We denote with $\overline{\tau}=2\pi/\Delta\omega$ the round trip of the
cavity, being $\Delta\omega$ the free spectral range. $T$ and $R$ are the
transmission and the reflexion intensity coefficient of the optical cavity,
respectively, while $\omega_{0}$ is the central frequency of the distribution.
Beware that the sum is running over the positive integers, contrary to the
Gaussian model (presented in the Appendix F). Such a function is represented
in Fig. 6 (b) for $R=0.9$. We define the temporal phase-matching function as
the Fourier transform of the phase matching function Eq. (66) is:

$\tilde{f}_{-}(t_{-})=Ae^{i\omega_{0}t_{-}}T\sum_{n=0}^{\infty}R^{n}e^{-(t_{-}+2n\overline{\tau})^{2}\sigma^{2}/2},$
(67)

this function is represented in Fig. 6(c). Note the absence of reflection
symmetry with respect to the origin, which is a consequence of the sum running
over the positive integer values. A frequency anti-correlated state can be
produced by an AlGaAs optical integrated waveguide working at room temperature
Autebert et al. (2016); Fabre et al. (2020a). But many others experimental
devices satisfy this condition, which justify the interest of the following
application. Note that, we are dealing with time-frequency non-Gaussian state,
such that even the envelop function is not necessary a Gaussian one, contrary
to Sec. II.3 where we only consider Gaussian phase matching function.

The chronocyclic Wigner distribution corresponding to Eq. (66) is given by

$W_{-}(\omega_{-},\tau)=\frac{{\cal{N}}}{\pi}\sum_{n,m}R^{n+m}e^{\frac{-(\omega_{-}-\omega_{0})^{2}}{\sigma^{2}}}e^{2i(n-m)(\omega_{-}-\omega_{0})\overline{\tau}}\\\
\crossproduct e^{-(\tau-(n+m)\overline{\tau})^{2}\sigma^{2}},$ (68)

which is represented in Fig. 6(a) for $R=0.9$ and $\sigma\overline{\tau}=20$.
The normalization factor in this case is given by
${\cal{N}}=\sum_{n,m}R^{n+m}e^{-\overline{\tau}^{2}\sigma^{2}(n-m)^{2}}$. The
distribution is not symmetric along the two orthogonal directions, and we
anticipate different performance for frequency and temporal parameter
estimation. A coincidence measurement when the frequency shift is zero
$I(\tau)$ corresponds to a measure of a cut of the Wigner distribution at zero
frequency. All the peaks from this cut of the Wigner distribution are positive
Lu et al. (2003); Fabre et al. (2020a). Negative peaks of the Wigner
distribution, which can lead to a coincidence probability larger than $1/2$,
can be observed only with finite frequency shifts. In the analytical
expression of the chronocyclic Wigner distribution with the Airy function
cavity model, the alternance of the positive and negative peaks is not
analytically clear, as it is in the Dirac comb model. This last case was
already studied Testorf and Ojeda-Castañeda (1996) for a classical field and
it also appears in Gottesman et al. (2001) for the calculation of the Wigner
distribution of the quadrature position-momentum GKP state. More details are
provided in Appendix (F.2).

Figure 6:  (a) Numerical Chronocyclic Wigner distribution $R=0.9$,
$\sigma\overline{\tau}=20$. (b),(c) Marginals of the chronocyclic Wigner
distribution, which are respectively the absolute value of the frequency and
temporal phase matching function. The absence of reflection symmetry with
respect to the origin to the temporal phase matching function is visible on
the Wigner distribution. The peak on the left of (a) corresponds to the peak
at zero time of the temporal phase matching function. The units of the
chronocyclic Wigner distribution are normalized with respect to the sampling
frequency, while the ones of the marginals with respect to the periodicity of
the comb. Concrete examples of units is given in Sec. IV.3.

For such a grid state Eq. (68), the quantum Fisher information matrix can be
written as:

$\displaystyle{\cal{F}}_{\tau\tau}=$ $\displaystyle\
\frac{\sigma^{2}}{2}[1-2\overline{\tau}^{2}\sigma^{2}\text{Var}(n-m)],$ (69)
$\displaystyle{\cal{F}}_{\mu\mu}=$ $\displaystyle\
\frac{1}{2\sigma^{2}}[1+2\overline{\tau}^{2}\sigma^{2}\text{Var}(n+m)],$ (70)
$\displaystyle{\cal{F}}_{\mu\tau}=$ $\displaystyle\ 0,$ (71)

where the variance $\text{Var}(n\pm m)=\langle(n\pm m)^{2}\rangle-(\langle
n\pm m\rangle)^{2}$ is taken with respect to:

$\langle(n\pm m)^{\alpha}\rangle=\frac{\sum_{n,m}(n\pm
m)^{\alpha}R^{n+m}e^{-(m-n)^{2}\overline{\tau}^{2}\sigma^{2}}}{\sum_{n,m}R^{n+m}e^{-(m-n)^{2}\overline{\tau}^{2}\sigma^{2}}},$
(72)

where $\alpha=1,2$. The off-diagonal term of the QFI matrix
${\cal{F}}_{\mu\tau}$ can be different from zero if we add a quadratic phase
in the phase-matching function as we shall see in Sec. IV.4. Note that the
average $\langle n-m\rangle$ is equal to zero. Since there is a minus in the
expression Eq. (69), there is no advantage to use such a state for temporal
estimation, but for a certain parameter regime $R,\sigma,\overline{\tau}$, the
variance can be negligible $\langle(m-n)^{2}\rangle-\langle n-m\rangle^{2}$.
The fact that the precision of frequency estimation increases while the
precision of the temporal estimation decreases is reminiscent of the
Heisenberg inequality for time and frequency variables (see the case of the
inequality based on the sum of variances Maccone and Pati (2014)). The fact
that such a grid state is useful for frequency estimation can be seen directly
from the Chronocyclic Wigner distribution, which contains in the exponential
imaginary linear terms in frequency but not in time variable.

In Fig. 7(a), we represent
$2\sigma^{2}{\cal{F}}_{\mu\mu}=[1-2\overline{\tau}^{2}\sigma^{2}\text{Var}(n-m)]$
for different values of the reflectivity and $\overline{\tau}\sigma$. For
simplicity, $\omega_{0}$ is set to zero. In general, as $R$ and
$\overline{\tau}\sigma$ increase, the variance $\text{Var}(n+m)$ increases,
and so it does the frequency QFI. The gain from the time-frequency grid state
through the factor $2\sigma^{2}\overline{\tau}^{2}\text{Var}(n+m)$ compared to
the one gained by time cat-like state
$\frac{2\tilde{\Delta}^{2}}{1+e^{-\tilde{\Delta}^{2}/\sigma^{2}}}$ (see Eq.
54) is remarkable and, as it can be seen in Fig. 7, it is relevant even for
low reflectivity of the cavity mirrors. In the experimental implementations
Autebert et al. (2016); Fabre et al. (2020a), the free spectral range is $2\pi
19.2$ GHz, which gives the product $\sigma\overline{\tau}=3.51.10^{3}$,
corresponding to approximately 500 frequency peaks. In general, the product
$\sigma\overline{\tau}$ approximately enumerates the number of peaks for each
experimental platform. For instance, the device presented in Imany et al.
(2018), produced qudits with 40 frequency modes and presents a good
performance for the estimation of a frequency parameter according to Fig.
7(a). Also, the optical device in Chang et al. (2021), producing 648 modes
presents a great interest.

Figure 7:  (a) Quantum Fisher information
$2\sigma^{2}{\cal{F}}_{\mu\mu}=1+2\sigma^{2}\overline{\tau}^{2}\text{Var}(n+m)$
with respect to $R$ is the reflectivity of the cavity, for different value of
the product $\sigma\overline{\tau}$. (logarithmic scale) (b) Quantum Fisher
information
$2{\cal{F}}_{\tau\tau}/\sigma^{2}=1-2\sigma^{2}\overline{\tau}^{2}\text{Var}(n-m)$
with respect to $R$ is the reflectivity of the cavity, for different values of
the product $\sigma\overline{\tau}$. The red, yellow and purple curves
coincide.

On the contrary, the temporal QFI is lower than the time-frequency Gaussian
state, and is represented in Fig. 7(b). When the temporal QFI is equal to one,
there is no benefit of using such a time-frequency state compared to a
Gaussian one. But then, as soon as $R\rightarrow 1$, the temporal QFI
decreases, as a balance of the increasing of the frequency QFI.

Finally, we perform in the Appendix F the calculation of the QFI for a
Gaussian micro-comb, which is the case in which the envelope is given by a
Gaussian function. It allows one evaluating the performance of the time-
frequency GKP state introduced in Fabre et al. (2020a), which is actually
quite similar to the state considered in this section, but with a different
envelop function. In addition, this result could also be applied for
quadrature position-momentum GKP states Gottesman et al. (2001) and gives
perspective for quantum metrology with a specific non-Gaussian quadrature
state. This case is also relevant given that the phase-matching function is
even, and so the FI can converge to the QFI.

###  IV.2 Quantum Fisher information for frequency grid states

In order to outperform a Gaussian or a frequency cat-like state for temporal
estimation, we have to invert the role of the frequency and the time
parameters of the grid state Eq. (66). We call such a state a frequency grid
state, since it is a $\pi/2$ rotation of the temporal grid state in the
chronocyclic phase space. We propose to engineer the phase-matching function
of the form:

$\tilde{f}_{-}(t_{-})=e^{-t_{-}^{2}\sigma^{2}/2}\sum_{n=0}^{\infty}R^{n}e^{2int_{-}/\overline{\tau}},$
(73)

which can be seen as a frequency cat-like state (see Eq. (46)) with a higher
number of temporal peaks. The associated chronocyclic Wigner distribution is
indeed a $\pi/2$ rotation of Eq. (68), which corresponds to an exchange
between the frequency and the time variable. In this case, we obtain opposite
performances for frequency and time parameter estimation. Indeed, the QFI
matrix is:

$\displaystyle{\cal{F}}_{\mu\mu}=$ $\displaystyle\
\frac{1}{2\sigma^{2}}(1-2\overline{\tau}^{2}\sigma^{2}\text{Var}(n-m)),$ (74)
$\displaystyle{\cal{F}}_{\tau\tau}=$ $\displaystyle\
\frac{\sigma^{2}}{2}(1+2\overline{\tau}^{2}\sigma^{2}\text{Var}(n+m)),$ (75)
$\displaystyle{\cal{F}}_{\mu\tau}=$ $\displaystyle\ 0.$ (76)

The QFI matrix for the state Eq. (74) is the same as in the previous section,
we have just to exchange the role of ${\cal{F}}_{\mu\mu}$ by
${\cal{F}}_{\tau\tau}$ in Fig. (7). Such phase-matching function could be
obtained by adding a SLM before a semiconductor AlGaAs device generating THz
photon pairs Francesconi et al. (2020), where the spatial pump profile
$\tilde{f}$ is the phase-matching function
$\tilde{f}(z)=\tilde{f}_{-}(t_{-}v_{g})$, where the scaling factor $v_{g}$ is
the group speed velocity. We assume also that there is no cavity, and the full
device and the detection is integrated, to avoid the natural Fabry-Perot
effect when the semiconductor device is placed into the air [see Eq. (66)]. In
addition, for GHz photons produced by laser-cooled atoms system in the group
delay regime, the phase matching function is also the spatial profile of the
pump with the same rescaled group velocity factor and our results also apply
for this measurement setup.

Figure 8:  Metrological performance of Eq. (69), with $R=0.9$ and
$\overline{\tau}\sigma=10$. (a) In blue, analytical temporal Fisher
information $F_{\tau\tau}(\tau)$ (in $\mu\text{s}^{-2}$ for atomic cooled
laser system) for $\gamma=0$ (no losses) with respect to the temporal delay
(in $\mu$s). In red, Cut of the Wigner distribution at zero frequency
(proportional to the coincidence probability of the HOM experiment). In
yellow, QFI of the grid state Eq. (69). (b) In blue, analytical frequency
Fisher information $F_{\mu\mu}(\mu)$ ($\text{MHz}^{-2}$) for $\gamma=0$ (no
losses) with respect to the frequency delay (MHz). In yellow, QFI of the grid
state (see Eq. (70)) which is equal to 9000 $\text{MHz}^{-2}$ . Even if the FI
does not converge towards the QFI, we gain an huge factor compare to the cat-
like state. By figure of comparison, the QFI of the time cat-like state is 40
for the parameters $\tilde{\Delta}\sigma\sim 10$ for the same total bandwidth
with the time-frequency grid state.

###  IV.3 Fisher information matrix for time-frequency grid states

The Fisher information for the estimation of a temporal parameter with the
grid state Eq. (66) $F_{\tau\tau}(\tau)$ is represented in Fig. 8(a), where we
have chosen $\overline{\tau}\sigma=10$ and $R=0.9$. In order to compare with
the performance of the frequency (or time) cat-like state described in Chen et
al. (2019), we can choose $\sigma=1$ THz so that to have the same total
bandwidth corresponding to the spectral separation of the two frequencies of
the frequency cat-like state. The round-trip of the cavity will be then
$\overline{\tau}\sim 10$ ps. The Fisher information presents a periodic
structure with a maximum at $\overline{\tau}=100$ps, maximum which depends on
the number of modes and the reflectivity $R$. The presence of double peaks in
the FI is due to the presence of the derivative of the cut of Wigner
distribution, which is also the case for the cat-like state. Note that, the FI
does not saturate the QFI [the yellow line in Fig. 8(a)] because the phase-
matching function is not even. As $\gamma$ increases, the entire curve
shrinks. The choice of the value of the parameter $\overline{\tau}\sigma=10$
and $R=0.9$ is taken to show that there is no need to have a very large number
of frequency peaks (or in other words entangled qudits) to achieve good
performance for frequency estimation.

The Fisher information for the estimation of a frequency parameter with the
grid state Eq. (66) is given in Fig. 8(b), for $\gamma=0$. Again, the FI does
not reach the QFI even for zero photodetector losses. The optimal points for
frequency estimation are slightly on the left or the right of the origin and
can be found numerically after the measurement of the coincidence and the
single count. Even if the FI does not converge to the QFI, we obtain better
performance compared to what it is possible to achieve with reasonable
parameters with a time cat-like state $\sigma\tilde{\Delta}=10$ (see Sec.
III.1) and with $\tilde{\Delta}$ equals to the temporal envelope width of the
grid state. Indeed, since we have fixed the number of photons, we compare the
metrological performance of different biphoton spectral distributions
according to their total bandwidth. Then, there is a gain for the chosen
parameters in the FI of approximately $10^{2}$ (see Fig. 7), which leads to an
improvement of the frequency resolution by a factor 10. Notice that by using
the grid state Eq. (73), the graph Fig. 8 (a) and (b) are inverted, a relevant
situation for both bulk and integrated optical devices. By consequence, the
estimation of the temporal parameter would be sub-attosecond. However, we
considered conservative physical parameters, a higher reflectivity of the
cavity and higher product $\overline{\tau}\sigma$ would lead to better
performance for both frequency (with state Eq. (66) ) and temporal estimation
(with state Eq. (73)). We point out that for $\gamma=0$, we have
$\delta\alpha=1/\sqrt{F_{\mu,\tau}}$ (see Eq. (111)), which means that we
saturate the Cramer-Rao bound. Besides, if there is a source of
indistinguishability of the photon pair, the maximum of the FI is no longer at
the points indicated in Fig. 8(b). Again, for time parameter estimation, the
state Eq. (73) is the most suitable.

|  $\Delta\tau\sqrt{N}$ (s) |  $\Delta\mu\sqrt{N}$ (Hz) |  $\Delta\mu\tau\sqrt{N}$
---|---|---|---
Gaussian state | $10^{-14}$ | $10^{6}$ |  25
Cat-like state | $10^{-15}$ | $10^{5}$ |  2.5
Grid state | $[10^{-14},10^{-17}]$ | $[10^{3},10^{6}]$ |  $[10^{-2},10^{1}]$
Table 2:  For a fixed total bandwidth, comparison of the reachable temporal
and frequency estimation for the Gaussian, time or frequency cat-like state
and the time and frequency grid state. The bracket in the grid state line
refers to all possible values obtained by varying $R$ and $\tau$, according to
Fig. 7. N is the number of repetitions of the experiment.

###  IV.4 Frequency chirp and multiparameter estimation with temporal grid
states

In this section, we analyze the QFI for grid states when a chirp is applied.
In particular, we consider a quadratic temporal chirp on a frequency grid
state and, respectively, a frequency chirp on a temporal grid state. Let us
consider a phase-matching function which has a grid structure (see Eq. (66))
but with an additional frequency chirp:

$f_{-}(\omega_{-})=Ae^{-\frac{(\omega_{-}-\omega_{0})^{2}}{2\sigma^{2}}}e^{-i\frac{(\omega_{-}-\omega_{0})^{2}}{2C^{2}}}T\sum_{n=0}^{\infty}R^{n}e^{2in(\omega_{-}-\omega_{0})\overline{\tau}},$
(77)

where $C$ is the frequency chirp parameter as in Sec. II.3. The chronocyclic
Wigner distribution is this case can be cast under the form

$W_{-}(\omega,\tau)=\frac{{\cal{N}}}{\pi}\sum_{n,m=0}^{\infty}R^{n+m}e^{-(\omega-\omega_{0})^{2}/\sigma^{2}}e^{2i(n-m)(\omega-\omega_{0})\overline{\tau}}\\\
\crossproduct
e^{-(\tau-(n+m)\overline{\tau}-\frac{(\omega-\omega_{0})}{C^{2}})^{2}\sigma^{2}}.$
(78)

where the normalization constant is again given by
${\cal{N}}=\sum_{n,m}R^{n+m}e^{-\overline{\tau}^{2}\sigma^{2}(n-m)^{2}}$. The
frequency $\omega_{0}$ will be set to zero for simplicity. The QFI matrix is
given by (see Appendix D),

$\displaystyle{\cal{F}}_{\mu\mu}=$
$\displaystyle\frac{1}{2\sigma^{2}}(1+2\overline{\tau}^{2}\sigma^{2}\text{Var}(n+m))$
$\displaystyle+$
$\displaystyle\frac{\sigma^{2}}{2C^{4}}(1-2\overline{\tau}^{2}\sigma^{2}\text{Var}(n-m)),$
(79) $\displaystyle{\cal{F}}_{\tau\tau}=$ $\displaystyle\
\frac{\sigma^{2}}{2}(1-2\overline{\tau}^{2}\sigma^{2}\text{Var}(n-m)),$ (80)
$\displaystyle{\cal{F}}_{\mu\tau}=$
$\displaystyle\frac{\sigma^{2}}{2C^{2}}(1-2\overline{\tau}^{2}\sigma^{2}\text{Var}(n-m)).$
(81)

This expression is composed of the same terms as the one of the time-frequency
Gaussian state Eq. (38) but with different multiplicative factors coming from
the grid state. The inverse of the QFI matrix is given by:

$\displaystyle\tilde{{\cal{F}}}_{\mu\mu}=$
$\displaystyle\frac{2\sigma^{2}}{1+2\overline{\tau}^{2}\sigma^{2}\text{Var}(n+m)},$
(82) $\displaystyle\tilde{{\cal{F}}}_{\tau\tau}=$
$\displaystyle\frac{2/\sigma^{2}}{1-2\overline{\tau}^{2}\sigma^{2}\text{Var}(n-m)}+\frac{2\sigma^{2}/C^{4}}{1+2\overline{\tau}^{2}\sigma^{2}\text{Var}(n+m)},$
(83) $\displaystyle\tilde{{\cal{F}}}_{\mu\tau}=$
$\displaystyle-\frac{2\sigma^{2}/C^{2}}{1+2\overline{\tau}^{2}\sigma^{2}\text{Var}(n+m)}.$
(84)

With the parameters $R=0.9$, $\overline{\tau}\sigma\gg 10$ the variance
$\text{Var}(n-m)\sim 10^{-44}$ is negligible. The inverse of the temporal QFI
is in practice the same as if there was no chirp
$\tilde{{\cal{F}}}_{\tau\tau}=2/\sigma^{2}$. The grid structure of the state
allows one to drastically reduce $\tilde{{\cal{F}}}_{\mu\tau}$ compared to the
Gaussian case through the term
$1+2\overline{\tau}^{2}\sigma^{2}\text{Var}(n+m)\sim 500$ according to Fig.
7(a) where $\overline{\tau}\sigma=10$, for instance. In Table. 2, we finally
recap the temporal and frequency precision reachable by the state presented in
this paper.

##  V Conclusions

We have analyzed the performance of frequency and temporal parameter
estimation protocols that make use of photon pairs as probe states, and which
are based on generalized HOM interferometry. In particular, we have compared
various classes of phase-matching functions that can be engineered with
nowadays technology, considering continuous variables defined as the frequency
and arrival time of biphoton states. Our results show that the estimation
precision can be substantially improved by engineering time-frequency grid
states, whose chronocyclic Wigner function is composed of a superposition of
equally-spaced peaks. We have also shown that the optimal limit of precision,
the quantum Cramer-Rao bound, can be achieved using a HOM measurement on
states with even phase-matching functions. A quantitative estimation of the
precision achievable with state-of-the-art devices is presented, comparing the
expected performances of specific sources of biphoton states. Our results pave
the way to novel applications of quantum photonic devices in sensing
applications, in particular in cases where intense light fields can not be
used, such as in biological sensing Taylor and Bowen (2016).

Our results have been obtained working with the chronocyclic Wigner function
formalism, but they could be generalized to other continuous variable systems.
Indeed, the considered time-frequency grid state are the analogue of the
Gottesman, Kitaev and Preskill states Gottesman et al. (2001), which can be
applied for displacement estimation Duivenvoorden et al. (2017); Legero et al.
(2006) and for continuous-variable quantum error correction Barros et al.
(2017); Terhal and Weigand (2016); Farías et al. (2015).

As a future perspective, the non-diagonal elements of the QFI matrix, relevant
for multiparameter estimation, could be improved generalizing our results to
include continuous rotations in the chronocyclic phase space. Furthermore,
other degrees of freedom such as polarization D’Ambrosio et al. (2013) could
be simultaneously exploited to enhance the measurement precision. Finally,
multiphoton interference could be used in order to combine in a single
protocol the improvement coming from the engineering of the phase-matching
function, with the metrological advantage coming from multi-photon
entanglement.

## ACKNOWLEDGMENT

N.Fabre acknowledges support from the project “Quantum Optical Technologies”
carried out within the International Research Agendas programme of the
Foundation for Polish Science co-financed by the European Union under the
European Regional Development Fund. We acknowledge useful discussion with
Perola Milman, Arne Keller, Sara Ducci, Florent Baboux, Arnault Raymond, Louis
Garbe and Roberto Di Candia.

##  Appendix A Quantum Fisher information

In this section, we present the expression of the QFI for a frequency
anticorrelated biphoton state. For a pure state the QFI is given by
${\cal{F}}^{\text{tot}}_{\mu\tau}=4\left(\bra{\frac{\partial\psi(\tau)}{\partial\tau}}\ket{\frac{\partial\psi(\mu)}{\partial\mu}}-\absolutevalue{\bra{\psi(\mu)}\ket{\frac{\partial\psi(\mu)}{\partial\mu}}\bra{\psi(\tau)}\ket{\frac{\partial\psi(\tau)}{\partial\tau}}}\right)$,
where the wavefunction of the biphoton state is written as:
$\ket{\psi(\mu,\tau)}={\cal{\hat{D}}}(\tau)\hat{D}(\mu)\ket{\psi}=\iint
d\omega_{s}d\omega_{i}e^{i\omega_{i}\tau}f_{+}(\omega_{+}+\mu)f_{-}(\omega_{-}+\mu)\ket{\omega_{s},\omega_{i}}$.
The QFI for temporal parameter estimation is,

${\cal{F}}^{\text{tot}}_{\tau\tau}=4(\iint
d\omega_{s}d\omega_{i}\omega_{i}^{2}\absolutevalue{f_{+}(\omega_{+})}^{2}\absolutevalue{f_{-}(\omega_{-})}^{2}-\\\
\absolutevalue{\iint
d\omega_{s}d\omega_{i}\omega_{i}\absolutevalue{f_{+}(\omega_{+})}^{2}\absolutevalue{f_{-}(\omega_{-}})^{2}}^{2}),$
(85)

the temporal QFI can be written as the variance with respect to the frequency
$\omega_{i}$ variable:

${\cal{F}}^{\text{tot}}_{\tau\tau}=4(\langle\omega^{2}_{i}\rangle-\langle\omega_{i}\rangle^{2})=4\text{Var}(\omega_{i}).$
(86)

The average is evaluated with respect to the chronocyclic Wigner distribution
of the photon pair: $\langle\omega^{2}_{i}\rangle=\iint\iint
d\omega_{s}d\omega_{i}dt_{s}dt_{i}W_{\hat{\rho}}(\omega_{s},\omega_{i},t_{s},t_{i})\omega_{i}^{2}$.
By performing the change of variable $\omega_{i}=(\omega_{+}-\omega_{-})$ and
assuming a factorization as in Eq. (8), the temporal QFI becomes:

${\cal{F}}^{\text{tot}}_{\tau\tau}=4(\langle(\omega_{+}-\omega_{-})^{2}\rangle-\langle\omega_{+}-\omega_{-}\rangle^{2})=4(\text{Var}(\omega_{+})+\text{Var}(\omega_{-})).$
(87)

For the other elements of the QFI matrix, we have an analogous expression:

$\displaystyle{\cal{F}}^{\text{tot}}_{\mu\mu}=4(\langle
t_{s}^{2}\rangle-\langle
t_{s}\rangle^{2})=4(\text{Var}(t_{+})+\text{Var}(t_{-}))$ (88)
$\displaystyle{\cal{F}}^{\text{tot}}_{\tau\mu}=4(\langle\omega_{s}t_{i}\rangle-\langle\omega_{s}\rangle\langle
t_{i}\rangle)=-4(\langle\omega_{-}t_{-}\rangle-\langle\omega_{-}\rangle\langle
t_{-}\rangle),$ (89)

We now mention two particular cases where the QFI matrix element depends only
to the phase-matching part. The first is the case of frequency anti-correlated
photon
$W_{+}(\omega_{+},t_{+})=e^{-\omega^{2}_{+}/\sigma^{2}}e^{-t_{+}^{2}\sigma^{2}}$
where $\text{Var}(\omega_{-})\gg\sigma$, the QFI reduces to:

$\displaystyle{\cal{F}}^{\text{tot}}_{\mu\mu}=4(\frac{1}{2\sigma^{2}}+\text{Var}(t_{-}))$
(90)
$\displaystyle{\cal{F}}^{\text{tot}}_{\tau\tau}=4(\sigma^{2}+\text{Var}(\omega_{-})\sim\text{Var}(\omega_{-})).$
(91)

In this case, the resolution of a temporal parameter depends only on the
phase-matching function $f_{-}$ of the biphoton wavefunction. When the
photodetectors losses are set to zero and when the phase-matching function is
even, for a delay converging towards zero, the FI obtained from the HOM
interferometer converges to the QFI. Nevertheless, the factor $1/2\sigma^{2}$
in the frequency QFI Eq. (90) is an improvement of the frequency estimation
which is not reachable with the HOM interferometer. The second case is for
frequency correlated photon pair, where the role of the frequency and the time
variable can be inverted. Notice that the moments are written as a statistical
average, but it is also possible to write them in terms of quantum average
with time and frequency operators by using the De-Wooters theory. It is
possible to define such operators by adding a quantum clock as explained in
Maccone and Sacha (2020-03-16); Fabre (2020).

##  Appendix B Optimal measurement strategy

In this section, we show that the FI for the HOM interferometer converges to
the QFI when the photon detectors losses are zero and when the displacement
parameters converge to zero. For simplicity, we consider here only temporal
parameter estimation. First, we write the expression of the FI for lossless
detectors,

$F_{\tau\tau}(\tau)=\frac{1}{2}[\frac{(\partial_{\tau}W_{-}(\tau))^{2}}{1-W_{-}(\tau)}+\frac{(\partial_{\tau}W_{-}(\tau))^{2}}{1+W_{-}(\tau)}].$
(92)

We perform a Taylor expansion of the cut of the Wigner distribution
$W_{-}(0,\tau)\equiv W_{-}(\tau)$ at zero frequency and zero delay,

$\displaystyle W_{-}(\tau)=W_{-}(\tau=0)+\partial_{\tau}W_{-}(\tau=0)\tau+$
$\displaystyle+\partial^{2}_{\tau}W_{-}(\tau=0)\frac{\tau^{2}}{2}+O(\tau^{2})$
(93)

For an even phase-matching function $f_{-}(\omega_{-})=f_{-}(-\omega_{-})$, we
obtain that $\partial_{\tau}W_{-}(\tau=0)=2i\langle\omega_{-}\rangle$, where
the average value is again taken with respect to $W_{-}$ (or after integration
on the time variable, with respect to $\absolutevalue{f_{-}}^{2}$). Also, we
have $W_{-}(\tau=0)=1$, owing to the normalization condition of the biphoton
wavefunction and because the phase matching is an even function. Then we
obtain the following expression for different terms which intervene in the
expression of the FI:

$\displaystyle
1-W_{-}(\tau)=-\sum_{k=1}^{\infty}\frac{(2i\tau)^{k}}{k!}\langle\omega^{k}\rangle,$
(94) $\displaystyle
1+W_{-}(\tau)=2+\sum_{k=1}^{\infty}\frac{(2i\tau)^{k}}{k!}\langle\omega^{k}\rangle,$
(95)
$\displaystyle\partial_{\tau}W_{-}(\tau)=\sum_{k=1}^{\infty}\frac{(2i)^{k}\tau^{k-1}}{(k-1)!}\langle\omega^{k}\rangle$
(96)

For an even state centered at the origin, we have that
$\langle\omega_{-}\rangle=0$, and so at the first order in $\tau$

$\displaystyle\frac{(\partial_{\tau}W_{-}(\tau))^{2}}{P_{2}(\tau)}=8\langle\omega_{-}^{2}\rangle,$
(97)
$\displaystyle\frac{(\partial_{\tau}W_{-}(\tau))^{2}}{P_{1}(\tau)}\rightarrow
0.$ (98)

Then, the FI converges to $4\langle\omega_{-}^{2}\rangle$ when
$\tau\rightarrow 0$ which is indeed the QFI $\mathcal{F}$ which depends only
on the phase-matching function. The previous mathematical proof can also be
applied to show that the frequency FI for $\gamma,\mu\rightarrow 0$, converges
to the frequency QFI, by using the fact that the Fourier transform of a
complex even function is also complex and even. The frequency cat-like state,
time cat-like state with or without chirp and also for the Gaussian comb (see
Appendix F) has indeed an even phase-matching function. It is not the case for
the Airy function model (see Eq. (66)), since we have:

$f_{-}(\omega_{-})\pm
f_{-}(-\omega_{-})=\left\\{\begin{array}[]{ll}e^{-\omega_{-}^{2}/\sigma^{2}}\frac{2-2R\text{cos}(\omega_{-}\tau)}{1-2R\text{cos}(\omega_{-}\tau)+R^{2}}\\\
e^{-\omega_{-}^{2}/\sigma^{2}}\frac{2i\text{sin}(\omega_{-}\tau)}{1-2R\text{cos}(\omega_{-}\tau)+R^{2}}\end{array}\right.$
(99)

which is not an even or an odd function. In such a case, a different
interferometric scheme must be devised to obtain an optimal measurement that
saturates the QFI.

##  Appendix C QFI of mixture of two-colors state

We start by the coherent mixture of a biphoton state in two frequencies:

$\hat{\rho}=\frac{1}{2}(\ket{\psi_{\Delta}}\bra{\psi_{\Delta}}+\ket{\psi_{-\Delta}}\bra{\psi_{-\Delta}}).$
(100)

After time and frequency displacement, the state is:
$\hat{\rho}(\mu,\tau)=\hat{D}(\tau){\cal{\hat{D}}}(\mu)\hat{\rho}\hat{\hat{D}}^{\dagger}(\tau){\cal{\hat{D}}}^{\dagger}(\mu)$.
The QFI matrix can be written Liu et al. (2020) using the spectral
decomposition of the density matrix
$\hat{\rho}=\sum_{i=1}^{d}\lambda_{i}\ket{\lambda_{i}}\bra{\lambda_{i}}$ as,

${\cal{F}}_{\mu\tau}=\sum_{i,j=0}^{d-1}\frac{2\text{Re}(\bra{\lambda_{i}}\partial_{\mu}\hat{\rho}\ket{\lambda_{j}}\bra{\lambda_{j}}\partial_{\tau}\hat{\rho}\ket{\lambda_{i}})}{\lambda_{i}+\lambda_{j}}.$
(101)

The QFI for such coherent mixtures of frequency cat-like states
($\lambda_{i}=1/2$) can be cast as:

${\cal{F}}_{\tau\mu}=2\text{Re}(\bra{\psi_{\Delta}}\partial_{\tau}\hat{\rho}\ket{\psi_{\Delta}}\bra{\psi_{\Delta}}\partial_{\mu}\hat{\rho}\ket{\psi_{\Delta}}\\\
+\bra{\psi_{\Delta}}\partial_{\tau}\hat{\rho}\ket{\psi_{-\Delta}}\bra{-\psi_{\Delta}}\partial_{\mu}\hat{\rho}\ket{\psi_{\Delta}}\\\
+\bra{\psi_{-\Delta}}\partial_{\tau}\hat{\rho}\ket{\psi_{\Delta}}\bra{\psi_{\Delta}}\partial_{\mu}\hat{\rho}\ket{\psi_{-\Delta}}\\\
+\bra{\psi_{-\Delta}}\partial_{\tau}\rho\ket{\psi_{-\Delta}}\bra{\psi_{-\Delta}}\partial_{\mu}\hat{\rho}\ket{\psi_{-\Delta}}).$
(102)

Where, for instance,

$\partial_{\tau}\hat{\rho}=\frac{1}{2}(\ket{\partial_{\tau}\psi_{\Delta}}\bra{\ket{\psi_{\Delta}}}+\ket{\psi_{\Delta}}\bra{\partial_{\tau}\psi_{\Delta}}\\\
+\ket{\partial_{\tau}\psi_{-\Delta}}\bra{\ket{\psi_{-\Delta}}}+\ket{\psi_{-\Delta}}\bra{\partial_{\tau}\psi_{-\Delta}}).$
(103)

We start evaluating the diagonal terms. We obtain the sum of three terms,

${\cal{F}}_{\tau\tau}(\tau)=4(\int
d\omega_{-}\absolutevalue{f_{-}^{\Delta}(\omega_{-})}^{2}\omega_{-}\text{sin}(2\omega_{-}\tau))^{2}\\\
+4(\int
d\omega_{-}\absolutevalue{f_{-}^{-\Delta}(\omega_{-})}^{2}\omega_{-}\text{sin}(2\omega_{-}\tau))^{2}\\\
+4\absolutevalue{\int
d\omega_{-}f^{\Delta}(\omega_{-})f_{-}^{-*\Delta}(\omega_{-})\omega_{-}\text{sin}(2\omega_{-}\tau)}^{2}\\\
+4(\absolutevalue{\int
d\omega_{-}f^{*\Delta}(\omega_{-})f_{-}^{-\Delta}(\omega_{-})\omega_{-}\text{sin}(2\omega_{-}\tau)}^{2}).$
(104)

The last expression can be written by using statistical average of the
chronocyclic Wigner distribution of the phase matching function of both peak:

${\cal{F}}_{\tau\tau}(\tau)=4\sum_{i,j=1}^{2}(\iint d\omega_{-}dt\
\omega_{-}\text{sin}(2\omega_{-}\tau)W^{i,j}_{-}(\omega_{-},t))^{2},$ (105)

where $i,j=\pm\Delta$, and for instance $W^{\Delta,\Delta}_{-}(\omega,t)=\int
d\omega_{-}e^{2i\omega^{\prime}t}f^{\Delta}_{-}(\omega-\omega^{\prime})f^{*\Delta}_{-}(\omega+\omega^{\prime})$
and the cross Wigner distribution: $W^{\Delta,-\Delta}_{-}(\omega,t)=\int
d\omega^{\prime}e^{2i\omega^{\prime}t}f^{\Delta}_{-}(\omega-\omega^{\prime})f^{*-\Delta}_{-}(\omega+\omega^{\prime})$.
This formula could be generalized for others terms, such as the grid mixed
state. Then, we recover Eq. (51) after calculation. Note that the frequency
QFI ${\cal{F}}_{\mu\mu}(\mu)$ will have a similar expression as Eq. (105).

##  Appendix D Example of calculation of variances for grid states

In this section, we show the expression of the QFI for grid states modified by
a chirp (see Eq. (IV.4)). The covariance matrix of such a state can be written
as:

$\langle\omega_{-}t_{-}\rangle=\iint d\omega_{-}d\tau\ \omega_{-}\tau
W_{-}(\omega_{-},\tau)\\\ ={\cal{N}}\sum_{n,m}\iint d\omega_{-}d\tau\
\omega_{-}e^{-\omega_{-}^{2}/\sigma^{2}}e^{2i(n-m)\omega_{-}\overline{\tau}}e^{-\tau^{2}\sigma^{2}}\\\
\crossproduct(\tau+(n+m)\overline{\tau}+\frac{\omega_{-}}{C^{2}}),$ (106)

where ${\cal{N}}$ is the denominator of the chronocyclic Wigner distribution
(see Eq. (78)). The first term is zero, because there is a temporal
integration of the product of an odd and an even function. The second term,
which is imaginary, is also zero after summation on $n$ and $m$. Only the
third term is different from zero and by using the Gaussian integral Eq.
(126), we obtain:

${\cal{N}}\sum_{n,m}R^{n+m}\iint d\omega_{-}d\tau
e^{-\omega_{-}^{2}/\sigma^{2}}e^{2i(n-m)\omega_{-}\overline{\tau}}e^{-\tau^{2}\sigma^{2}}\frac{\omega_{-}^{2}}{C^{2}}\\\
={\cal{N}}\sum_{n,m}R^{n+m}\frac{\sigma^{2}}{2C^{2}}(1-(n-m)^{2}\overline{\tau}^{2}\sigma^{2}/2)e^{-(n-m)^{2}\overline{\tau}^{2}\sigma^{2}},$
(107)

which gives indeed Eq. (IV.4). This example of calculation can be applied for
the other expression of the QFI for the time-frequency grid state.

Figure 9:  (a) Frequency $2{\cal{F}}_{\mu\mu}\sigma^{2}$ (in log scale) and
temporal (b) Quantum Fisher information $2{\cal{F}}_{\tau\tau}/\sigma^{2}$
with respect to $\Delta\omega/\overline{\omega}$ for different value of the
product $\sigma\overline{\tau}$.

##  Appendix E Saturation of the Cramer-Rao bound

###  E.1 Variances and saturation of Cramer-Rao

The measure for the confidence interval in the determination of $\alpha$ is
given by :

$\text{cov}(\alpha)=\int\prod_{k}dx_{k}p(x_{k}|\alpha)(\tilde{\alpha}-\alpha)^{2},$
(108)

which is bounded by the Fisher information as: $\text{cov}(\alpha)\geq
1/NF_{\alpha}(\alpha)$. We will look for the optimal estimation strategy using
a maximum-likelihood estimator. The likelihood estimator is a multinomial
distribution $L(N_{0},N_{1},N_{2})=P_{0}^{N_{0}}P_{1}^{N_{1}}P_{2}^{N_{2}}$
and which is extremized as:

$0=\frac{N_{0}\partial_{\tau}P_{0}}{P_{0}}+\frac{N_{1}\partial_{\tau}P_{1}}{P_{1}}+\frac{N_{2}\partial_{\tau}P_{2}}{P_{2}}.$
(109)

This extremization leads to an analytic expression of the optimal estimator
$\tilde{\alpha}=(\tilde{\tau},\tilde{\mu})$ given by
$N_{1}P_{2}(\tilde{\tau})=N_{2}P_{1}(\tilde{\tau})$. By using Eq. (30), we
find:

$W_{-}(\tilde{\tau},\tilde{\mu})=\frac{N_{2}-N_{1}\frac{1+3\gamma}{1-\gamma}}{N_{1}+N_{2}},$
(110)

where $N_{1}$ and $N_{2}$ are the number of single detections and coincidence
counts, respectively. This procedure allows one to obtain an analytic
expression of the optimal estimator
$\tilde{\alpha}=(\tilde{\tau},\tilde{\mu})$. Indeed, in the limit of large
$N$, the probability can be written as a normal distribution, the maximum
likelihood estimator is the optimal one since it saturates the Cramer Rao
bound, $\Delta^{2}\alpha=1/NF_{\alpha}(\tilde{\alpha})$. Notice that it also
permits to find the photodetector losses $\gamma$ Lyons et al. (2018), which
depends on $N_{1}$ and $N_{2}$.

Figure 10:  (a) Frequency $2{\cal{F}}_{\mu\mu}\sigma^{2}$ (in log scale) and
temporal (b) Quantum Fisher information $2{\cal{F}}_{\tau\tau}/\sigma^{2}$
with respect to $\Delta\omega/\overline{\omega}$ for different value of the
product $\sigma\overline{\tau}$.

We discuss now the temporal estimation, i.e. we set the frequency shift to
zero $W_{-}(\tilde{\tau},0)$. For the case where the phase-matching function
is Gaussian (see Eq. (35)), an analytic expression of $\tilde{\tau}$ with
respect to $N_{1,2}$ and $\gamma$ can be found in Lyons et al. (2018). The
resolution of Eq. (110) can be found only numerically when the biphoton state
is a micro-comb. The estimator is said to be efficient when it saturates the
Cramer-Rao bound for a large number of repetitions of the experiment. For
large photon losses, namely when $N_{2}<N_{1}\frac{1+3\gamma}{1-\gamma}$ the
right term of Eq. (110) becomes negative, the maximum likelihood technique
does not allow to find $\tilde{\tau}$ for certain biphoton states which have
positive Wigner distribution. In that case, other estimation techniques such
as the mean square method can be applied.

In the HOM experiment, one measures the coincidence probability to obtain an
estimation of $\tau$ and $\mu$. The time or frequency precision of the non-
resolved HOM experiment is given by,

$\Delta\alpha=\frac{\sqrt{\langle\Delta
P_{2}(\alpha,0)\rangle}}{\absolutevalue{\frac{\partial\langle
P_{2}(\alpha,0)\rangle}{\partial\alpha}}},$ (111)

where the variance of the non-resolved time coincidence measurement is (see
Ref. Chen et al. (2019)) $\Delta
P_{2}(\alpha,0)=P_{2}(\alpha,0)(1-P_{2}(\alpha,0))$ where again
$\alpha=\tau,\mu$ and the denominator depends on the sensitivity of the mean
value of the operators to the variables $\tau$ or $\mu$. Finally, for zero
losses the FI is under the form,

$F_{\tau\tau}(\tau)=\frac{1}{2}\frac{(\partial_{\tau}W_{-}(0,\tau))^{2}}{1-W_{-}^{2}(0,\tau)},$
(112)

which indeed converges towards the estimator Eq. (111).

##  Appendix F Quantum Fisher information of Gaussian and Dirac grid states

###  F.1 Quantum Fisher information of the Gaussian grid state

Let us now consider a state whose phase matching function can be cast as the
product of an envelop of width $\sigma$ and the sum of Gaussian functions of
width $\Delta\omega$,

$f_{-}(\omega_{-})=Ae^{-\omega_{-}^{2}/2\sigma^{2}}\sum_{n=-\infty}^{\infty}e^{-(\omega_{-}-n\overline{\omega})^{2}/2\Delta\omega^{2}}.$
(113)

Here $A$ is a normalisation factor which can be expressed as
$1=A^{2}\sum_{n,m}\sqrt{\frac{\pi}{1/\sigma^{2}+1/\Delta\omega^{2}}}\text{exp}(-(n-m)^{2}\overline{\omega}^{2}/4\Delta\omega^{2})$.
Then, the phase-matching function is even and the QFI converges to the FI, for
again zero photodetectors losses and $\tau,\mu\rightarrow 0$. By using the
Poisson’s summation formula (valid since
$\absolutevalue{f_{-}(\omega_{-})}\leq
C/(1+\absolutevalue{\omega_{-}}^{\alpha}))$, we can sum over $\mathds{Z}$ and
the phase-matching function can be rewritten as,

$f_{-}(\omega_{-})=\frac{A}{\overline{\omega}}e^{-\omega_{-}^{2}/2\sigma^{2}}\sum_{n\in\mathds{Z}}e^{-\frac{n^{2}}{2}(\frac{\Delta\omega}{\overline{\omega}})^{2}}e^{2in\pi\omega/\overline{\omega}}.$
(114)

We directly see the resemblance with the Fabry-Perot cavity model of Eq. (66),
where $R^{n}=\text{exp}(n\text{ln}(R))\leftrightarrow
e^{-\frac{n^{2}}{2}(\frac{\Delta\omega}{\overline{\omega}})^{2}}$. We can then
express the Fourier transform of the phase-matching function as,

$\tilde{f}_{-}(t_{-})=\frac{A}{\overline{\omega}}\sum_{n\in\mathds{Z}}e^{-\frac{n^{2}}{2}(\frac{\Delta\omega}{\overline{\omega}})^{2}}e^{-(t_{-}+\frac{2\pi
n}{\overline{\omega}})^{2}\sigma^{2}/2}.$ (115)

We will define $\overline{\tau}=2\pi/\overline{\omega}$. In the case where
$\Delta\omega\rightarrow 0$ and $\sigma\rightarrow\infty$, then the function
becomes a Dirac Comb of period $\overline{\omega}$, presented in the next
section. The Chronocyclic Wigner distribution can be cast as follows:

$\displaystyle
W_{-}(\omega,\tau)={\cal{N}}_{G}\sum_{n,m\in\mathds{Z}^{2}}e^{-\frac{n^{2}}{2}(\frac{\Delta\omega}{\overline{\omega}})^{2}}e^{-\frac{m^{2}}{2}(\frac{\Delta\omega}{\overline{\omega}})^{2}}$
(116) $\displaystyle\crossproduct
e^{2i\pi(n-m)(\omega-\omega_{0})/\overline{\omega}}e^{-(\tau+\frac{2\pi(n+m)}{\overline{\omega}})^{2}\sigma^{2}/2},$
(117)
$\displaystyle{\cal{N}}_{G}=\sum_{n,m\in\mathds{Z}^{2}}e^{-\frac{n^{2}}{2}(\frac{\Delta\omega}{\overline{\omega}})^{2}}e^{-\frac{m^{2}}{2}(\frac{\Delta\omega}{\overline{\omega}})^{2}}e^{-(\frac{2\pi(n+m)}{\overline{\omega}})^{2}\sigma^{2}/2}.$
(118)

Hence, the Quantum Fisher information has the same expression as in Eq. (69),
but with the moments are,

$\displaystyle\langle(n\pm
m)^{\alpha}\rangle={\cal{N}}_{G}\sum_{n,m\in\mathds{Z}^{2}}(n\pm
m)^{\alpha}e^{-\frac{n^{2}}{2}(\frac{\Delta\omega}{\overline{\omega}})^{2}}$
(119) $\displaystyle\times
e^{-\frac{m^{2}}{2}(\frac{\Delta\omega}{\overline{\omega}})^{2}}e^{-(m-n)^{2}\tau^{2}\sigma^{2}}.$
(120)

We represent the temporal and the frequency QFI in Fig. 9(a),(b) in Fig.
10(a),(b) as function of the ratio $\Delta\omega/\overline{\omega}$ for
different values of the product $\overline{\tau}\sigma$. The QFI increases as
$\Delta\omega/\overline{\omega}$ is closer to 0, namely that when each peak is
narrower. As
$e^{-\frac{n^{2}}{2}(\frac{\Delta\omega}{\overline{\omega}})^{2}}\sim 1$ the
state is no longer periodic.

###  F.2 Dirac Comb model

To understand the succession of positive and negative peaks in the
chronocyclic Wigner distribution (see Fig. 6), we remind the Dirac comb case
$f_{-}(\omega_{-})=\sum_{n\in\mathds{Z}}\delta(\omega_{-}-n\overline{\omega})$
from Ref.Testorf and Ojeda-Castañeda (1996), which present the chronocyclic
Wigner distribution of a classical field with such a grid-structured spectrum.
The chronocyclic Wigner distribution of the Dirac comb is,

$W_{-}(\omega,\tau)=\sum_{n,m\in\mathds{Z}^{2}}e^{2i\pi\omega/\overline{\omega}(n-m)}\delta(\tau-\frac{(n+m)}{2\overline{\omega}}).$
(121)

We perform the change of variable: $n^{\prime}=n+m$,

$W_{-}(\omega,\tau)=\sum_{n\in\mathds{Z}}\delta(\omega-n\overline{\omega}/2)\sum_{n^{\prime}\in\mathds{Z}}e^{2i\pi\omega
n^{\prime}/\overline{\omega}}\delta(\tau-\frac{n^{\prime}}{2\overline{\omega}})$
(122)

and we finally obtain

$W_{-}(\omega,\tau)=\sum_{n,n^{\prime}\in\mathds{Z}^{2}}(-1)^{nn^{\prime}}\delta(\omega-n\overline{\omega}/2)\delta(\tau-\frac{n^{\prime}}{2\overline{\omega}}).$
(123)

From this expression, the succession of positive and negative peaks is
clearer. For even values of $n,n^{\prime}$, the chronocyclic Wigner
distribution is positive and the negative peaks do not correspond to the peaks
of the marginals of the distribution.

##  Appendix G Gaussian integrals

We remind the following Gaussian integrals, which are used all along this
paper.

$\int_{\mathds{R}}e^{-\alpha\omega^{2}}e^{\beta\omega}d\omega=\sqrt{\frac{\pi}{\alpha}}e^{\beta^{2}/4\alpha}$
(124)

$\int_{\mathds{R}}\omega
e^{-\omega^{2}\alpha}e^{\omega\beta}=\frac{\beta}{2\alpha}\sqrt{\frac{\pi}{\alpha}}e^{\beta^{2}/4\alpha}$
(125)

$\int_{\mathds{R}}\omega^{2}e^{-\omega^{2}\alpha}e^{\omega\beta}=\frac{1}{2\alpha}\sqrt{\frac{\pi}{\alpha}}(1+\frac{\beta^{2}}{2\alpha})e^{\beta^{2}/4\alpha}$
(126)

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