††thanks: These two authors contributed equally††thanks: These two authors
contributed equally

# Fundamental limitations of time measurement precision in Hong-Ou-Mandel
interferometry

O. Meskine1 E. Descamps1,2 A. Keller1,3 A. Lemaître4 F. Baboux1 S. Ducci1
corresponding author, experiments<EMAIL_ADDRESS>P. Milman1
corresponding author, theory<EMAIL_ADDRESS>1Laboratoire Matériaux
et Phénomènes Quantiques, Université Paris Cité, CNRS UMR 7162, 75013, Paris,
France  2Département de Physique de l’Ecole Normale Supérieure - PSL, 45 rue
d’Ulm, 75230, Paris Cedex 05, France  3Department de Physique, Université
Paris-Saclay, 91405 Orsay Cedex, France  4Univ. Paris-Saclay, CNRS, Centre de
Nanosciences et de Nanotechnologies, 91120 Palaiseau, France

###### Abstract

In quantum mechanics, the precision achieved in parameter estimation using a
quantum state as a probe is determined by the measurement strategy employed.
The ultimate quantum limit of precision is bounded by a value set by the state
and its dynamics. Theoretical results have revealed that in interference
measurements with two possible outcomes, this limit can be reached under ideal
conditions of perfect visibility and zero losses. However, in practice, this
cannot be achieved, so precision never reaches the quantum limit. But how do
experimental setups approach precision limits under realistic circumstances?
In this work we provide a general model for precision limits in two-photon
Hong-Ou-Mandel interferometry for non-perfect visibility. We show that the
scaling of precision with visibility depends on the effective area in time-
frequency phase space occupied by the state used as a probe, and we find that
an optimal scaling exists. We demonstrate our results experimentally for
different states in a set-up where the visibility can be controlled and
reaches up to $99.5\%$. In the optimal scenario, a ratio of $0.97$ is observed
between the experimental precision and the quantum limit, establishing a new
benchmark in the field.

The Hong-Ou-Mandel (HOM) interferometer is currently used to demonstrate the
phenomenon of bunching of two identical, independent bosonic quantum
particles, such as single photons [1] (see Fig. 1). In this setup, photons are
made to interfere on a balanced beam-splitter (BS) and their detection in
coincidence at the output indicates whether they have bunched together or not.
To control the distinguishability of the two paths of the interferometer, a
time delay can be introduced for one of the input photons, consequently
changing the coincidence detection probability. Despite its seemingly
straightforward operating principles, the HOM interferometer has found diverse
applications beyond its original scope [2]. For instance, the coincidence
detection signal at the BS output has been demonstrated to serve as an
entanglement witness [3, 4, 5], to provide phase space information about the
spectral function [6, 7, 8], and to enable the simulation of different quantum
exchange statistics [9, 10], among other applications [11, 12, 13]. In
particular, the HOM interferometer is a valuable apparatus for quantum
parameter estimation [14, 15, 16, 17, 18, 19, 20]: its low-intensity regime
opens the possibility of applying the tools of quantum metrology to small and
fragile probes, as biological ones [21]; since the HOM effect is based on two-
photon interference, it is robust against background noise and group velocity
dispersion [22]. Last but not least, theoretical and experimental results
indicate that it can arbitrarily approach the quantum ultimate precision limit
for time delay (or path difference) estimation. However, in spite of the
recent experiments reaching up to attosecond precision on time delay
estimations [17, 18, 20], the exact mechanisms determining the limits and
limitations of time measurement precision using the HOM are unknown. In
particular, a curious result observed not only for HOM interferometers but for
any parameter estimation dichotomic measurement (see also [23, 24], for
instance) concerns the behavior of precision with the visibility $V$ at the
point where, in the ideal case, the former is expected to saturate the quantum
limit. For instance in the HOM set-up, if $V=1$, one can attain the ultimate
precision limit at the point photons perfectly bunch or anti-bunch.
Nevertheless, in the experimentally realistic case where the visibility $V<1$,
total bunching or anti-bunching is no longer observed and precision drops down
to zero at this same point, suffering a discontinuity. A consequence of this
is that for finite visibility, the ultimate quantum precision limit can only
be approached, and a way to circumvent this in the context of a Mach-Zender
interferometer was studied in [25] using a mode engineering-based strategy.
Nevertheless, it is not clear, from a fundamental point of view, what
regulates the achievable precision in a HOM experiment and why some spectral
functions seem to present a better performance than others [15, 17, 18, 19].

Figure 1: Experimental setup for investigating the metrological performance of
the Hong Ou Mandel (HOM) experiment, showing the generation, joint spectral
amplitude (JSA) engineering and HOM interferometer stages

In the present Letter, we provide a theoretical model together with its
experimental demonstration explaining both qualitatively and quantitatively
the performance of the HOM experiment as a quantum metrological tool. Our
results explain why certain wavefunctions exhibit higher precision performance
than others in the regime of finite visibility and we have exactly
theoretically predicted and experimentally confirmed the wavefunction-
dependency of the scaling of precision with the visibility for different
wavefunctions. For some configurations, we reach the highest ratio between the
achieved precision and the maximum possible one to date, thereby setting a new
benchmark in this field.

In a typical metrological protocol, a probe is prepared in an initial state,
and undergoes a dynamical evolution depending on a parameter to be estimated,
$\theta$. The probe is measured providing an outcome $k$, which is used to
estimate $\theta$. By associating the function $p_{k}(\theta)$ to the
probability of obtaining an outcome $k$, the precision on the estimation of
$\theta$ is bounded by the relation $\delta\theta\geq 1/\sqrt{\nu F(\theta)}$,
where $F(\theta)=\sum_{k}\frac{1}{p_{k}(\theta)}\left(\frac{\partial
p_{k}(\theta)}{\partial\theta}\right)^{2}$ is the Fisher information (FI) and
$\nu$ is the number of repetitions of the experiment. When using quantum
mechanical resources - as individual photons, which is the case in a HOM
experiment - one can define the quantum Fisher information (QFI) by using a
quantum state as a probe [26]. The probe’s evolution depends on the parameter
$\theta$, and for pure states and unitary evolutions, this dependency can be
expressed as $\ket{\psi(\theta)}=e^{i\hat{H}\theta}\ket{\psi}$, where
$\hat{H}$ is the Hamiltonian generating the dynamical evolution. Precision is
thus limited by the relation $\delta\theta\geq 1/\sqrt{\nu{\cal F}}$, where
${\cal F}$ is the QFI, obtained by maximizing $F(\theta)$ over all possible
measurements on $\ket{\psi(\theta)}$. In the case discussed above, ${\cal
F}=4\Delta^{2}\hat{H}$, where the variance is taken with respect to the
initial state $\ket{\psi}$ [27]. The bound $1/\sqrt{\nu{\cal F}}$ is called
the quantum Cramér-Rao bound (QCR) [28].

The mathematical procedure to determine ${\cal F}$ is known, but the problem
of finding an experimental measurement strategy where $F(\theta)={\cal F}$
remains. There is no general rule for this, even though optimization
procedures can be applied to particular states [29, 30] and symmetry arguments
can be evoked in specific situations [31, 32], as in the HOM experiment [16,
14] that we’ll discuss in details. As can be found in the literature, time
precision in this type of interferometer with perfect visibility can reach the
QCR bound. For this reason, several attempts have been made to reach this
bound, obtaining astonishing precision on the estimation of time delays in HOM
experiments [17, 18, 19, 20]. To understand these results, we consider as
initial state (probe) a photon pair prepared in an arbitrary pure state, that
enters the two input arms $1,2$ of a perfectly balanced BS:

$\ket{\psi}=\int\int
d\omega_{1}\omega_{2}f(\omega_{1},\omega_{2})\hat{a}_{1}^{\dagger}(\omega_{1})\hat{a}_{2}^{\dagger}(\omega_{2})\ket{0},$
(1)

where $f(\omega_{1},\omega_{2})$ is the complex-valued normalized joint
spectral amplitude (JSA). Before impinging the balanced BS, one of the photons
of the state (1) can be subjected to a time delay $\tau$, the parameter to be
estimated. This delay is described by a unitary evolution, and the associated
Hamiltonian is $\hat{H}=\hbar\int
d\omega\omega\hat{a}_{1}^{\dagger}(\omega)\hat{a}_{1}(\omega)=\hbar\hat{\omega}_{1}$
(we have supposed, without loss of generality, that the arm labeled $1$ is
delayed), and state (1) is transformed as
$\ket{\psi(\tau)}=e^{i\hat{H}\tau/\hbar}\ket{\psi}=\hat{U}\ket{\psi}$. If we
now consider the action of the BS and compute the probability of detecting
both photons in coincidence $P_{c}(\tau)$, we obtain [14]
$P_{c}(\tau)=\frac{1}{2}(1-\bra{\psi}\hat{U}^{\dagger}\hat{S}\hat{U}\ket{\psi})$,
where
$\hat{S}\hat{a}_{1}^{\dagger}(\omega_{1})\hat{a}_{2}^{\dagger}(\omega_{2})\hat{S}^{\dagger}=\hat{a}_{1}^{\dagger}(\omega_{2})\hat{a}_{2}^{\dagger}(\omega_{1})$
corresponds to the swap of spatial modes. $P_{c}(\tau)$ is typically directly
obtained from the recorded experimental data. As for $P_{a}(\tau)$,
probability of anti-coincidences, it can also be directly detected [20] or
inferred from the relation $P_{a}(\tau)=1-P_{c}(\tau)$ [33]. Finally, the QFI
can be expressed as [34] ${\cal F}=4\Delta^{2}\hat{\omega}_{1}$. Notice that
for perfectly symmetric (S) (anti-symmetric (AS)) states $\ket{\psi}_{S(AS)}$
with respect to the exchange of spatial modes, we have that
$\hat{S}\ket{\psi}_{S(AS)}=\pm\ket{\psi}_{S(AS)}$ and $P_{c}(\tau=0)=0(1)$.
These states correspond to the situations where the HOM has perfect visibility
and the ultimate quantum precision limit can be reached.

The HOM experiment provides information about the collective variables
$\omega_{-}=\omega_{1}-\omega_{2}$ [16]. We’ll now suppose that the input
photons of the interferometer are generated by Spontaneous Parametric Down
Conversion (SPDC) and that
$f(\omega_{1},\omega_{2})=f_{-}(\omega_{-})f_{+}(\omega_{+})$ in (1), with
$\omega_{\pm}=\omega_{1}\pm\omega_{2}$. Functions $f_{-}$ and $f_{+}$ are
normalized functions related to the phase matching condition and to the energy
conservation, respectively. Notice that this separability hypothesis has no
impact on our results but simplifies their presentation. If one is using the
HOM with metrological purposes [14, 16, 15], the best configuration is the one
where $f_{+}$ is a Dirac function centered on $\omega_{p}$ (the pump’s
frequency) and $\omega_{+}$ is close to constant (strict energy conservation),
a situation that maximizes frequency correlation between photons [35]. In this
case ${\cal F}=4\Delta^{2}\omega_{1}=\Delta^{2}\omega_{-}$, where we have
supposed for simplicity that both photons have the same spectral variance.

Still using the proposed factorization in collective variables $\omega_{\pm}$,
in the case of perfect visibility (so $f_{-}(\omega_{-})$ is, for instance, a
perfectly even function of $\omega_{-}$ [36]), we have that
$\bra{\psi}\hat{U}^{\dagger}\hat{S}\hat{U}\ket{\psi}=\int
d\omega_{-}e^{i\omega_{-}\tau}f_{-}(\omega_{-})f_{-}^{*}(-\omega_{-})=W(0,\tau)$.
Here, $W(\mu,\tau)$ denotes the chronocyclic Wigner function associated with
$f_{-}(\omega_{-})$ on the time-frequency phase space (TFPS), specifically on
the axis $\mu=0$ while $\tau$, the time delay, is variable. In this context,
$\mu$ represents the phase space variable associated with $\omega_{-}$. It has
been demonstrated in [6] and experimentally validated in [7, 8] that the HOM
experiment directly measures the Wigner function points $W(0,\tau)$, i.e.,
along the axis $\mu=0$ of TFPS. Adopting this representation facilitates an
intuitive understanding of the factors that dictate the limitations imposed by
non-perfect visibility, which is the issue studied in the following.

To this aim, we model the dependency of $P_{c}(\tau)$ with the visibility $V$
as follows [34]:

$P_{c}(\tau)=\frac{1}{2}-\frac{V}{2}W(0,\tau),$ (2)

where $0\leq V\leq 1$ and $W(0,\tau)$ is the Wigner function of the ideal
state, i.e. a perfectly symmetric one, that leads to a perfect visibility.
Since $W(0,0)=1$, $P_{c}(0)=(1-V)/2$ (this is simply the normalization
condition). In the present model, we are not considering explicitly the role
of experimental noise or losses in the measurement, evolution or preparation
steps: these effects can either be included in the QFI - which is consequently
modified - or in the state’s purity [37, 38]. Including noise in the state
corresponds to considering a different (non-pure) state as a probe, so a
different function $W(0,\tau)$. Thus, state noise or measurement losses have
no incidence on the model (2), that remains valid. However, even in the case
where a state is pure and is measured in a lossless configuration it can lead
to a non-perfect visibility due to state preparation imperfections that cannot
be circumvented [34] and this is why we focus on the visibility in the present
work. Using (2), the FI at point $\tau$ for a general value of $V$ is given by

$F(V,\tau)=V^{2}\frac{(W^{\prime}(0,\tau))^{2}}{(1-V^{2}W^{2}(0,\tau))},$ (3)

where the superscript ′ denotes the time derivative. By defining ${\rm
max}_{\tau}F(V,\tau)\overset{\text{\tiny def}}{=}\widetilde{F}_{V}$, we
obtain, for $V=1$ (perfectly S or AS states), $\widetilde{F}_{1}=F(1,0)={\cal
F}$ [16, 14, 32]. This shows that the quantum precision limit can be achieved
for perfect visibility. For $V\neq 1$ we can still compute
$\widetilde{F}_{V}$, which is obtained at a point $\tau=\tau_{M}\neq 0$. As a
matter of fact, for $\tau=0$, the function $F(V,0)$, when considered as a
function of $V$, exhibits a discontinuity at $V=1$: indeed, $F(V<1,0)=0$,
while $F(1,0)=\widetilde{F}_{1}=\mathcal{F}$, as previously shown. For $V<1$,
the maximal values of $\widetilde{F}_{V<1}$ satisfy the condition
$\widetilde{F}_{V<1}=-W^{\prime\prime}(0,\tau_{M})/W(0,\tau_{M})$.
Experimental investigations of these results were conducted in [20, 17, 19],
and [23], but a general theory explaining the overall behavior of the
attainable values of $\widetilde{F}_{V}$ and their limitations remains
unknown. We’ll now elucidate how $\widetilde{F}_{V}$ approaches ${\cal F}$.
Importantly, we find that this approach depends not only on the visibility but
we also identify a relation with the effective phase space occupation of the
state’s Wigner function $W(\mu,\tau)$ [39]. In other words, the scaling of
$\widetilde{F}_{V}$ with $V$ is connected to how far the quantum state is from
saturating the time-frequency Heisenberg uncertainty principle [40, 41]. A
first remark is that since $V\leq 1$, using (3), we have that $F(V,\tau)\leq
V^{2}F(1,\tau)$, so $\widetilde{F}_{V}=V^{2}{\cal F}$ is the best possible
scaling of the FI with $V$. This is a proof that for $V<1$ the HOM can never
reach the QCR bound, even in the absence of losses. In addition, using (3) we
can see that the best possible scaling is obtained when $W(0,\tau_{M})=0$
(i.e., $P_{c}(\tau_{M})=1/2$) and $W^{\prime}(0,\tau_{M})\neq 0$. A sinusoidal
function of frequency $\sqrt{{\cal F}}$ satisfies these conditions (a solution
also leading to a constant FI in $\tau$ for $V=1$ [42, 34]). Although this
corresponds to an unphysical state, it represents the limit situation of
Schrödinger cat (SC)-like states [43, 20, 18] with
$\Delta^{2}\hat{\omega}_{-}\Delta^{2}\hat{t}\gg 1$, so occupying a large
effective area in the time-frequency phase space (TFPS) [34]. Surprisingly,
SC-like states exhibit remarkable robustness in the presence of decreased
visibility, making them the most resilient states in HOM-based quantum
metrology, which is yet another interesting quality of these states in quantum
metrology [44].

To identify the states leading to the worst possible scaling of
$\widetilde{F}_{V}$ with the visibility, we should identify the states that
minimize $\widetilde{F}_{V<1}$, and a trivial solution similar to the one
discussed in [45, 46] is a constant function with $W(0,\tau)=1$, so that
$\widetilde{F}_{1}={\cal F}$. This is of course an unphysical solution, and
the physical states which are the closest to this solution are Gaussian
states, which saturate the Heisenberg’s relation
$\Delta^{2}\hat{\omega}_{-}\Delta^{2}\hat{t}=1$. In conclusion, Gaussian
states have the worst rate of distinguishability [46] and exhibit the worst
scaling with $V$, even though they can have the same limit value for the FI as
SC states for $V=1$ and $\tau=0$, i.e., the QFI. In addition, for Gaussian
states, the scaling with $V$ does not depend on the exact values of
$\Delta\hat{\omega}_{-}$ or $\Delta\hat{t}$, but on the associated function
(Gaussian), which is univocally determined by the product of the two
quantities. As states’ phase space occupation change from the Gaussian to the
sinusoidal behavior, their scaling with visibility improves. A remarkable
consequence of the above discussion is that while the wavefunction shape do
not play a role in the value of ${\cal F}$ [47, 34] it does play a role in the
scaling of $\widetilde{F}_{V}$ with $V$, in a way that is related to the
effective occupation of the TFPS. Previous works [39] established the
connection between metrological properties and the state’s occupation of the
quadrature phase space [48, 49], where small structures determine quantum
properties such as the QFI. Our analysis indicate that the small structures
and phase space occupation contribute to the optimization of the scaling of
precision with visibility, which is a distinct physical property. We emphasize
that our analysis is applicable to various experimental setups where the
visibility model (2) holds [23, 24], such as experiments with more than two
photons [50], where the scaling with $V$ is important to determine the
tolerance of the sub-shot noise region to visibility decrease (see [23, 34]).

We now validate our model using an experiment allowing to engineer two-photon
states described by different functions $f_{-}(\omega_{-})$ exhibiting diverse
scaling behaviors with the visibility $V$. The quantum source consists of an
AlGaAs Bragg reflector waveguide, generating polarization-entangled photon
pairs via type II Spontaneous Parametric Down-Conversion (SPDC) at telecom
wavelengths and operating at room temperatures [51]. A sketch of the
experimental setup is provided in Fig. 1. A continuous-wave laser having a
wavelength $\lambda_{pump}=$772.42\text{\,}\mathrm{nm}$$ is coupled into the
waveguide using a microscope objective (MO). The output signal is collected by
a second microscope objective, and the pump beam is filtered out using a long-
pass filter (LPF). The generated photon pairs are then collected in a single
mode fiber and possibly directed to a programmable filter (PF, Finisar 4000s),
enabling the JSA engineering. When the filter is not inserted, the state
generated by the source is described by
$f_{-}(\omega_{-})=\operatorname{sinc}(a\omega_{-}^{2}+b\omega_{-}+c)$ where
the coefficients $a$, $b$ and $c$ are related to optical properties of the
material such as birefringence and chromatic dispersion [52, 34]. In addition
to the study of this case, three different filter shapes are used: a
$15\text{\,}\mathrm{nm}$-wide rectangular filter centered on the degeneracy
wavelength $\lambda_{deg}=$1544.8\text{\,}\mathrm{nm}$$, a Gaussian filter of
identical width centered at the same wavelength, and a combination of two
$5\text{\,}\mathrm{nm}$-wide rectangular filters centered at
$\lambda_{1}=$1560\text{\,}\mathrm{nm}$$ and
$\lambda_{2}=$1530\text{\,}\mathrm{nm}$$ corresponding to energy-matched
channels and allowing to create a SC-like state, analogously to as described
in [53, 54]. The functions can be classified according to the parameter ${\cal
S}=\Delta^{2}\hat{\omega}_{-}\Delta^{2}\hat{t}$, which determines the scaling
with respect to $V$ (see [34] for details). At the output of the filtering
process, the photon pairs are separated by a polarizing beam splitter (PBS).
The $H$ ($V$)-polarized photon enters the HOM interferometer through the arm 1
(2). Precise control over the polarization distinguishability, and thus the
HOM visibility, is enabled by two fibered polarization controllers, one in
each arm (FPC1 and FPC2). The temporal delay between the two photons is
controlled by a motorized optical delay line (MDL). The two paths are
recombined and separated by a 50/50 BS, then directed to superconducting
nanowires single photon detectors (SNSPD). Temporal correlations between the
detected photons are analyzed by a time-to-digital converter (TDC).

We perform a series of measurements on the four states, systematically varying
$V$ to investigate the scaling of the ratio $\widetilde{F}_{V}/{\cal F}$. Fig.
2 illustrates the results obtained. The coincidence counts data (red points)
are fitted (red lines) using $P_{c}(\tau)$ of Eq. (2) and the theoretical
expression of each wavefunction [34]. The FI $F(V,\tau)$ (blue lines) is then
computed using Eq. (3).

Figure 2: Left column: Joint Spectral Amplitude of the four different states
analyzed in this work. Central and right columns: corresponding Hong-Ou-Mandel
coincidence probability $P_{c}(t)$ and FI for different values of visibility
$V$.

Firstly, we notice that a reduction in visibility directly leads to a decrease
in $F(V,\tau)$. As expected, with finite visibility, the value of $F(V,\tau)$
drops to zero at $\tau=0$. Remarkably high visibilities exceeding $99\%$ are
achieved with the Gaussian, rectangular and SC-like states. Due to a small
modal birefringence of the AlGaAs source, the maximum visibility attainable
with the full state is $94,9\%$, still an excellent value given the broad
spectral width it covers, approximately $100\text{\,}\mathrm{nm}$. This broad
spectrum results in a narrow HOM curve $P_{c}(t)$, leading to a high FI value
of $2100\text{\,}\mathrm{p}\mathrm{s}^{-2}$, which is two orders of magnitude
higher than those obtained with the Gaussian and rectangular states. While
filtering the quantum state does increase the FI, it also decreases the number
of detected photons, therefore influencing the overall performance of the
metrological protocol in a given integration time. Nevertheless, in this
proof-of-principle experiment, we are mainly interested in testing the scaling
of the ratio $\widetilde{F}_{V}/{\cal F}$ to demonstrate our model, which can
serve as a guideline to other experiments using different strategies. In Fig.
3, the evolution of this ratio with respect to visibility for the four
engineered states is reported, both for experiments (points) and theory
(lines). We clearly observe that the SC-like state exhibits the most favorable
scaling behavior, in contrast to the Gaussian state, which displays a less
optimal one. For instance, at a visibility level of around $83\%$, the ratio
drops from 0.64 for the SC-like state to 0.35 for the Gaussian state.

Figure 3: Scaling of the ratio $\widetilde{F}_{V}/{\cal F}$ for different
biphoton states with respect to the HOM visibility $V$.

In conclusion, we have presented a comprehensive theoretical model and
experimentally confirmed how precision limits scale with both visibility and
the state’s wavefunction in practical metrological protocols employing the HOM
effect. The very good agreement between the experimental results and the
simulations supports the validity of the model given by Eq. (2). Our findings
show that reaching the precision limits in realistic conditions presents
challenges that depend on the particular state under consideration. Moreover,
our theoretical and experimental analysis establishes a general framework for
interpreting previous experimental results [17, 18, 19, 20]. Our work holds
significant implications, particularly in aiding the identification of optimal
conditions to advance HOM-based quantum metrology protocols, leading to
enhanced precision in measurements while minimizing the number of repetitions.
Finally, some aspects of the presented results can be readily extended to
other experiments where parity measurements are employed for quantum parameter
estimation [55, 56, 6, 57, 58].

## Acknowledgements

We acknowledge funding from the Plan France 2030 through the project
ANR-22-PETQ-0006, N. Fabre and G. Bié Alves for fruitful discussions and M.
Karr-Ducci for Fig. 1. O.M. acknowledges Labex SEAM (Science and Engineering
for Advanced Materials and devices), ANR-10-LABX-0096 and ANR-18-IDEX-0001 for
financial support.

## References

  * Hong  _et al._ [1987] C. K. Hong, Z. Y. Ou, and L. Mandel, Measurement of subpicosecond time intervals between two photons by interference, Phys. Rev. Lett. 59, 2044 (1987).
  * Fabre  _et al._ [2022a] N. Fabre, M. Amanti, F. Baboux, A. Keller, S. Ducci, and P. Milman, The Hong-Ou-Mandel experiment: from photon indistinguishability to continuous-variable quantum computing, Eur. Phys. J. D 76, 196 (2022a).
  * Brecht and Silberhorn [2013] B. Brecht and C. Silberhorn, Characterizing entanglement in pulsed parametric down-conversion using chronocyclic Wigner functions, Phys. Rev. A 87, 053810 (2013), publisher: American Physical Society.
  * Eckstein and Silberhorn [2008] A. Eckstein and C. Silberhorn, Broadband frequency mode entanglement in waveguided parametric downconversion, Opt. Lett. 33, 1825 (2008).
  * Ray and van Enk [2011] M. R. Ray and S. J. van Enk, Verifying entanglement in the Hong-Ou-Mandel dip, Phys. Rev. A 83, 042318 (2011).
  * Douce  _et al._ [2013] T. Douce, A. Eckstein, S. P. Walborn, A. Z. Khoury, S. Ducci, A. Keller, T. Coudreau, and P. Milman, Direct measurement of the biphoton Wigner function through two-photon interference, Scientific Reports 3, 3530 (2013).
  * Tischler  _et al._ [2015] N. Tischler, A. Büse, L. G. Helt, M. L. Juan, N. Piro, J. Ghosh, M. J. Steel, and G. Molina-Terriza, Measurement and shaping of biphoton spectral wave functions, Phys. Rev. Lett. 115, 193602 (2015).
  * Kurzyna  _et al._ [2022] S. Kurzyna, M. Jastrzebski, N. Fabre, W. Wasilewski, M. Lipka, and M. Parniak, Variable electro-optic shearing interferometry for ultrafast single-photon-level pulse characterization, Opt. Express 30, 39826 (2022).
  * Francesconi  _et al._ [2021] S. Francesconi, A. Raymond, N. Fabre, A. Lemaître, M. I. Amanti, P. Milman, F. Baboux, and S. Ducci, Anyonic two-photon statistics with a semiconductor chip, ACS Photonics 8, 2764 (2021), https://doi.org/10.1021/acsphotonics.1c00901 .
  * Francesconi  _et al._ [2020] S. Francesconi, F. Baboux, A. Raymond, N. Fabre, G. Boucher, A. Lemaître, P. Milman, M. I. Amanti, and S. Ducci, Engineering two-photon wavefunction and exchange statistics in a semiconductor chip, Optica 7, 316 (2020), 1907.07935 .
  * Dorfman  _et al._ [2021] K. Dorfman, S. Asban, B. Gu, and S. Mukamel, Hong-Ou-Mandel interferometry and spectroscopy using entangled photons,  4, 49 (2021).
  * Ndagano  _et al._ [2022] B. Ndagano, Defienne, D. H., Branford, Y. Shah, A. Lyons, N. Westerberg, E. M. Gauger, and D. Faccio, Quantum microscopy based on Hong–Ou–Mandel interference, Commun. Phys. 16, 384 (2022).
  * Lee and Nha [2012] S.-Y. Lee and H. Nha, Second-order superposition operations via hong-ou-mandel interference, Phys. Rev. A 85, 043816 (2012).
  * Descamps  _et al._ [2023a] E. Descamps, A. Keller, and P. Milman, Time-frequency metrology with two single-photon states: Phase-space picture and the hong-ou-mandel interferometer, Phys. Rev. A 108, 013707 (2023a).
  * Fabre and Felicetti [2021] N. Fabre and S. Felicetti, Parameter estimation of time and frequency shifts with generalized Hong-Ou-Mandel interferometry, Phys. Rev. A 104, 022208 (2021).
  * Jordan  _et al._ [2022] K. M. Jordan, R. A. Abrahao, and J. S. Lundeen, Quantum metrology timing limits of the Hong-Ou-Mandel interferometer and of general two-photon measurements, Phys. Rev. A 106, 063715 (2022).
  * Lyons  _et al._ [2018] A. Lyons, G. C. Knee, E. Bolduc, T. Roger, J. Leach, E. M. Gauger, and D. Faccio, Attosecond-resolution Hong-Ou-Mandel interferometry, Sci. Adv. 4, eaap9416 (2018).
  * Chen  _et al._ [2019] Y. Chen, M. Fink, F. Steinlechner, J. P. Torres, and R. Ursin, Hong-Ou-Mandel interferometry on a biphoton beat note,  5, 43 (2019).
  * Chen  _et al._ [2023] C. Chen, Y. Chen, and L. Chen, Spectrally resolved Hong-Ou-Mandel interferometry with discrete color entanglement, Phys. Rev. Appl. 19, 054092 (2023).
  * Johnson  _et al._ [2023] S. J. Johnson, C. P. Lualdi, A. P. Conrad, N. T. Arnold, M. Vayninger, and P. G. Kwiat, Toward vibration measurement via frequency-entangled two-photon interferometry, in  _Quantum Sensing, Imaging, and Precision Metrology_, Vol. 12447, edited by J. Scheuer and S. M. Shahriar, International Society for Optics and Photonics (SPIE, 2023) p. 124471C.
  * Taylor and Bowen [2016] M. A. Taylor and W. P. Bowen, Quantum metrology and its application in biology, Physics Reports 615, 1 (2016), quantum metrology and its application in biology.
  * Steinberg  _et al._ [1992] A. M. Steinberg, P. G. Kwiat, and R. Y. Chiao, Dispersion cancellation and high-resolution time measurements in a fourth-order optical interferometer, Phys. Rev. A 45, 6659 (1992).
  * Penasa  _et al._ [2016] M. Penasa, S. Gerlich, T. Rybarczyk, V. Métillon, M. Brune, J. M. Raimond, S. Haroche, L. Davidovich, and I. Dotsenko, Measurement of a microwave field amplitude beyond the standard quantum limit, Phys. Rev. A 94, 022313 (2016).
  * Dalvit  _et al._ [2006] D. A. R. Dalvit, R. L. de Matos Filho, and F. Toscano, Quantum metrology at the Heisenberg limit with ion trap motional compass states, New Journal of Physics 8, 276 (2006).
  * Jachura  _et al._ [2016] M. Jachura, R. Chrapkiewicz, R. Demkowicz-Dobrzański, W. Wasilewski, and K. Banaszek, Mode engineering for realistic quantum-enhanced interferometry, Nat. Comm. 7, 11411 (2016).
  * Giovannetti  _et al._ [2006] V. Giovannetti, S. Lloyd, and L. Maccone, Quantum metrology, Phys. Rev. Lett. 96, 010401 (2006).
  * Boixo  _et al._ [2007] S. Boixo, S. T. Flammia, C. M. Caves, and J. Geremia, Generalized limits for single-parameter quantum estimation, Phys. Rev. Lett. 98, 090401 (2007).
  * Cramer [1946] H. Cramer,  _Mathematical methods of statistics_ (Princeton University Press Princeton, 1946) pp. xvi, 575 p.
  * Pinel  _et al._ [2012] O. Pinel, J. Fade, D. Braun, P. Jian, N. Treps, and C. Fabre, Ultimate sensitivity of precision measurements with gaussian quantum light : a multi-modal approach, Phys. Rev. A 85, 010101 (2012), 1105.2644 .
  * Gessner  _et al._ [2022] M. Gessner, N. Treps, and C. Fabre, Quantum limits on mode parameter estimation (2022), arXiv:2201.04050 [quant-ph] .
  * Hofmann [2009] H. F. Hofmann, All path-symmetric pure states achieve their maximal phase sensitivity in conventional two-path interferometry, Phys. Rev. A 79, 033822 (2009).
  * Seshadreesan  _et al._ [2013] K. P. Seshadreesan, S. Kim, J. P. Dowling, and H. Lee, Phase estimation at the quantum cramér-rao bound via parity detection, Phys. Rev. A 87, 043833 (2013).
  * [33] In some situations, $P_{a}(\tau)$ can be directly measured, as in the case of [20] where an entangled state of non-degenerate photons is used as a probe.
  * [34] See supplementary material [url], with references [52, Taylor, boucher_toolbox_2015, 25].
  * Giovannetti  _et al._ [2001] V. Giovannetti, S. Lloyd, and L. Maccone, Quantum-enhanced positioning and clock synchronization, Nature 412, 417 (2001).
  * Com [a] We could have chosen it to be an odd function as well, and in the discussion that follows we would work with an anti-symmetric function rather than a symmetric one. (a).
  * Escher  _et al._ [2001] B. Escher, R. de Matos Filho, and L. Davidovich, General framework for estimating the ultimate precision limit in noisy quantum-enhanced metrology, Nature Physics 7, 406 (2001).
  * Len  _et al._ [2022] Y. L. Len, T. Gefen, A. Retzker, and J. Koł odyński, Quantum metrology with imperfect measurements, Nat. Comm. 13, 6971 (2022).
  * Zurek [2001] W. H. Zurek, Sub-Planck structure in phase space and its relevance for quantum decoherence, Nature 412, 712 (2001).
  * Fadel and Maccone [2021] M. Fadel and L. Maccone, Time-energy uncertainty relation for quantum events, Phys. Rev. A 104, L050204 (2021).
  * Fabre  _et al._ [2022b] N. Fabre, A. Keller, and P. Milman, Time and frequency as quantum continuous variables, Phys. Rev. A 105, 052429 (2022b).
  * Cafaro and Alsing [2018] C. Cafaro and P. M. Alsing, Decrease of Fisher information and the information geometry of evolution equations for quantum mechanical probability amplitudes, Phys. Rev. E 97, 042110 (2018).
  * Ramelow  _et al._ [2009] S. Ramelow, L. Ratschbacher, A. Fedrizzi, N. K. Langford, and A. Zeilinger, Discrete tunable color entanglement, Phys. Rev. Lett. 103, 253601 (2009).
  * Com [b] Entangled non-degenerate photons are also useful to produce broadband states in the collective variable $\omega_{-}$ using single photons with relatively narrow spectrum (before being entangled with one another). (b).
  * Rohde  _et al._ [2005] P. P. Rohde, T. C. Ralph, and M. A. Nielsen, Optimal photons for quantum-information processing, Phys. Rev. A 72, 052332 (2005).
  * Töppel  _et al._ [2012] F. Töppel, A. Aiello, and G. Leuchs, All photons are equal but some photons are more equal than others, New Journal of Physics 14, 093051 (2012).
  * Descamps  _et al._ [2023b] E. Descamps, N. Fabre, A. Keller, and P. Milman, Quantum metrology using time-frequency as quantum continuous variables: Resources, sub-shot-noise precision and phase space representation, Phys. Rev. Lett. 131, 030801 (2023b).
  * Kwon  _et al._ [2019] H. Kwon, K. C. Tan, T. Volkoff, and H. Jeong, Nonclassicality as a quantifiable resource for quantum metrology, Phys. Rev. Lett. 122, 040503 (2019).
  * Yadin  _et al._ [2018] B. Yadin, F. C. Binder, J. Thompson, V. Narasimhachar, M. Gu, and M. S. Kim, Operational resource theory of continuous-variable nonclassicality, Phys. Rev. X 8, 041038 (2018).
  * Ou [2007] Z. Y. Ou, Multi-photon interference and temporal distinguishability of photons, International Journal of Modern Physics B 21, 5033 (2007).
  * Appas  _et al._ [2022] F. Appas, O. Meskine, A. Lemaître, M. Morassi, F. Baboux, M. I. Amanti, and S. Ducci, Nonlinear quantum photonics with algaas bragg-reflection waveguides, Journal of Lightwave Technology 40, 7658 (2022).
  * Maltese [2019] G. Maltese,  _Generation and manipulation of high-dimensional photonics states with AlGaAs chips_ , Ph.D. thesis (2019), thèse de doctorat dirigée par Ducci, Sara Physique. Optique Quantique Sorbonne Paris Cité 2019.
  * Ou and Mandel [1988] Z. Y. Ou and L. Mandel, Observation of spatial quantum beating with separated photodetectors, Phys. Rev. Lett. 61, 54 (1988).
  * Fabre  _et al._ [2020] N. Fabre, J. Belhassen, A. Minneci, S. Felicetti, A. Keller, M. I. Amanti, F. Baboux, T. Coudreau, S. Ducci, and P. Milman, Producing a delocalized frequency-time Schrödinger-cat-like state with Hong-Ou-Mandel interferometry, Phys. Rev. A 102, 023710 (2020), publisher: American Physical Society.
  * Lutterbach and Davidovich [1997] L. G. Lutterbach and L. Davidovich, Method for direct measurement of the Wigner function in cavity QED and ion traps, Phys. Rev. Lett. 78, 2547 (1997).
  * Bertet  _et al._ [2002] P. Bertet, A. Auffeves, P. Maioli, S. Osnaghi, T. Meunier, M. Brune, J. M. Raimond, and S. Haroche, Direct measurement of the Wigner function of a one-photon fock state in a cavity, Phys. Rev. Lett. 89, 200402 (2002).
  * Shalibo  _et al._ [2013] Y. Shalibo, R. Resh, O. Fogel, D. Shwa, R. Bialczak, J. M. Martinis, and N. Katz, Direct wigner tomography of a superconducting anharmonic oscillator, Phys. Rev. Lett. 110, 100404 (2013).
  * Winkelmann  _et al._ [2022] F.-R. Winkelmann, C. A. Weidner, G. Ramola, W. Alt, D. Meschede, and A. Alberti, Direct measurement of the Wigner function of atoms in an optical trap, Journal of Physics B: Atomic, Molecular and Optical Physics 55, 194004 (2022).