# Effects of coherence on temporal resolution

S. De  Integrated Quantum Optics, Institute for Photonic Quantum Systems
(PhoQS), Warburger Straße 100, 33098 Paderborn, Germany  J. Gil-Lopez
Integrated Quantum Optics, Institute for Photonic Quantum Systems (PhoQS),
Warburger Straße 100, 33098 Paderborn, Germany  B. Brecht  Integrated Quantum
Optics, Institute for Photonic Quantum Systems (PhoQS), Warburger Straße 100,
33098 Paderborn, Germany  C. Silberhorn  Integrated Quantum Optics, Institute
for Photonic Quantum Systems (PhoQS), Warburger Straße 100, 33098 Paderborn,
Germany  L. L. Sánchez-Soto  Departamento de Óptica, Facultad de Física,
Universidad Complutense, 28040 Madrid, Spain  Max-Planck-Institut für die
Physik des Lichts, Staudtstraße 2, 91058 Erlangen, Germany  Z. Hradil
Department of Optics, Palacký University, 17. listopadu 12, 77146 Olomouc,
Czech Republic  J. Řeháček  Department of Optics, Palacký University, 17.
listopadu 12, 77146 Olomouc, Czech Republic

###### Abstract

Measuring small separations between two optical sources, either in space or in
time, constitute an important metrological challenge as standard intensity-
only measurements fail for vanishing separations. Contrarily, it has been
established that appropriate coherent mode projections can appraise
arbitrarily small separations with quantum-limited precision. However, the
question of whether the optical coherence brings any metrological advantage to
mode projections is still a point of debate. Here, we elucidate this problem
by experimentally investigating the effect of varying coherence on estimating
the temporal separation between two single-photon pulses. We show that, for an
accurate interpretation, special attention must be paid to properly normalize
the quantum Fisher information to account for the strength of the signal. Our
experiment demonstrates that coherent mode projections are optimal for any
degree of coherence.

_Introduction.—_ In numerous applications, including radar signal processing
[1], radio acoustic sounding [2], ultrasonic testing [3], and medical imaging
[4], one is faced with the challenge of determining the temporal delay between
two closely spaced, overlapping, ultrashort pulses. There are several
efficient techniques to estimate these small offsets, such as the cross-
correlation, phase-shift, and delay line methods [5, 6, 7]. However, all of
them conspicuously fail when the time delay is significantly shorter than the
pulse duration.

The same pitfall appears in the spatial domain: the resolution of an imaging
system is limited by the size of its point spread function (PSF) that
specifies the intensity response to a point source [8]. This gives an
intuitive picture of the mechanisms that obstruct resolution, but it is very
heuristic. For example, the Rayleigh limit [9] is defined as the distance from
the center to the first minimum of the PSF. Yet that can be made arbitrarily
small with ordinary linear optics, at the price of the side lobes becoming
much higher than the central maximum. This hints that estimating the
separation between two points becomes also a matter of photon statistics [10].

Lately, the resolution limits have been revisited from the alternative
perspective of quantum metrology. The idea is to use the quantum Fisher
information (FI) and the associated quantum Cramér-Rao bound (CRB) to assess
how well the separation between two point sources can be estimated [11, 12,
13, 14, 15]. For direct imaging, the classical FI drops to zero as the
separation between the sources decreases and the error with which we can
determine the separation diverges accordingly, which has been dubbed
Rayleigh’s curse [16]. Surprisingly, when the quantum FI (i.e., optimized over
all measurements allowed by quantum mechanics) is calculated, it stays
constant, evidencing that the Rayleigh limit is not essential, but linked to a
particular detection scheme.

These remarkable predictions have fuelled a number of experimental
implementations, both in the spatial [17, 18, 19, 20] and the time-frequency
[21] domains. The key behind these achievements is the use of phase-sensitive
projections onto optimal modes [22] instead of intensity measurements, as the
latter discard the phase information carried by the signal.

The approach has been generalized to more realistic scenarios, where the
signals may have different intensities. This involves the simultaneous
estimation of separation, centroid, and relative intensities [23, 24]. Still
in this multiparameter case [25, 26, 27, 28], optimal quantum-limited
measurements have been worked out [29] and experimentally demonstrated [30].

The discussion thus far assumes incoherence between the signals. This conforms
with the conditions underlying Rayleigh’s criterion. In the temporal domain,
this happens with, for example, remote clocks (e.g., GPS), incoherent
excitations in biological samples, condensed matter physics, and astronomical
observations [31].

A recent debate addressed the role of coherence in the resolution limits [32,
33, 34, 35, 36, 37, 38]. Any coherent superposition of two time-delayed pulses
can be decomposed in terms of in-phase and anti-phase combinations of the two
pulses. Yet, these two channels are not equivalent concerning the strength of
the signal: the anti-phase mode does carry the information about the temporal
separation, but the intensity in this mode vanishes as the time offset
decreases. Hence, each photon therein carries a huge amount of information.

However, this effect is not necessarily a metrological advantage, because the
input signal still contains many photons and thus, on average, the information
per photon of the signal is limited. Performing the decomposition in in-phase
and anti-phase modes is then simply a clever way of sorting the information
into a dark channel.

We stress that ignoring the resources required to generate the input signal
might lead one to false conclusions about the information content of the
measurement. In this sense, the limit of incoherent mixtures represents a
bound that cannot be overcome without prior information coded into the state
[39]. We confirm here these predictions with an experiment that benefits from
classical and quantum resources, both contained in the quantum FI of the
signal.

_Theoretical model.—_ Let us first set the stage for our model. To facilitate
possible generalizations, we phrase what follows in a quantum language, so
that a pulse waveform with complex temporal envelope $\psi(t)$ is assigned to
a ket $|\psi\rangle$, such that $\psi(t)=\langle t|\psi\rangle$.

We consider two pulses of identical shape but displaced by a time offset
$\tau$, whose magnitude we want to estimate. We define the time-shifted
versions, $\lvert\psi_{\pm}\rangle$, as $\psi_{\pm}(t)=\psi(t\pm\tau/2)$. In
addition, for convenience, we keep the total intensity normalized to unity:
$\int_{-\infty}^{\infty}dt\,[\lvert\psi_{+}(t)\rvert^{2}+\lvert\psi_{-}(t)\rvert^{2}]=1$.
In general, $\langle\psi_{-}|\psi_{+}\rangle\neq 0$, so these modes are not
orthogonal. This overlap is at the heart of all the difficulties of the
problem, for it implies that the two modes cannot be separated by independent
measurements.

To capture the essence of the problem we introduce symmetric and antisymmetric
(nonnormalized) coherent modes

$\psi_{\mathrm{s}}(t)=\frac{1}{\sqrt{2}}[\psi_{+}(t)+\psi_{-}(t)]\,,\qquad\psi_{\mathrm{a}}(t)=\frac{1}{\sqrt{2}}[\psi_{+}(t)-\psi_{-}(t)]\,.$
(1)

These coherent modes can be generated by a mode converter using linear optical
transformations. Such an operation is, for instance, readily implemented by
sending the two signals into different input ports of a balanced beam
splitter. The output ports of the beam splitter then contain the coherent
modes from above. The total intensity is conserved in this process.
Physically, the symmetric mode corresponds to an in-phase superposition of the
time-shifted components, whereas the antisymmetric one corresponds to an anti-
phase superposition.

To proceed further, we need to specify the explicit waveform of the pulse. For
simplicity, we will use a Gaussian
$\psi(t)=1/(2\pi\sigma_{t}^{2})^{1/4}\exp[-t^{2}/(4\sigma_{t}^{2})]$ of width
$\sigma_{t}$.

We first consider a fully incoherent mixture of the time-shifted components.
Equivalently, this can be prepared as an incoherent sum of the in-phase and
anti-phase channels; in the experimental realization this amounts to an equal
mixing of the measurement data for each coherent mode in post-processing. The
situation can be thus represented by the density matrix
$\varrho(\tau)=\lvert\psi_{+}\rangle\langle\psi_{+}\rvert+\lvert\psi_{-}\rangle\langle\psi_{-}\rvert=\lvert\psi_{\mathrm{s}}\rangle\langle\psi_{\mathrm{s}}\rvert+\lvert\psi_{\mathrm{a}}\rangle\langle\psi_{\mathrm{a}}\rvert$.
Now, we can directly apply quantum estimation theory. The pivotal quantity is
the quantum FI [40], which is a mathematical measure of the sensitivity of an
observable quantity (pulse waveform) to changes in its underlying parameters
(time delay). Replicating the calculations performed in the spatial domain
[17], one immediately gets that the quantum FI, denoted by $\mathcal{Q}$, is
constant, $\mathcal{Q}(\tau)=1/(4\sigma^{2}_{t})$. The associated quantum CRB
ensures then that the variance of any unbiased estimator $\widehat{\tau}$ of
the time delay $\tau$ is lower bounded by the reciprocal of the quantum FI
(per single detection event); viz,
$\mathop{\mathrm{Var}}\nolimits_{\varrho}(\widehat{\tau})\geq
1/\mathcal{Q}(\tau)=4\sigma^{2}_{t}$.

In what follows, it will prove convenient to look at the problem from a
slightly different perspective. As the optimal measurement attaining the CRB
is known, we can calculate the FI for such a measurement. In fact, the optimal
scheme involves projections onto the successive derivatives (properly
orthonormalized) of the pulse amplitude [22]. For our basic Gaussian waveform,
this reduces to the Hermite-Gauss temporal modes
$\mathop{\mathrm{HG}}\nolimits_{n}(t)=\langle
t|\mathop{\mathrm{HG}}\nolimits_{n}\rangle$, defined as
$\mathop{\mathrm{HG}}\nolimits(t)={(2\pi\sigma_{t}^{2})^{-1/4}}(2^{n}\,n!)^{-1/2}H_{n}({t}/{\sqrt{2}\sigma_{t}})\,\exp[-{t^{2}}/{(4\sigma_{t}^{2})}]$.

Then, we have the following detection probabilities

$\displaystyle
p_{s}(n|\tau)\equiv|\langle\mathop{\mathrm{HG}}\nolimits_{n}|\psi_{\mathrm{s}}\rangle|^{2}=\left\\{\begin{array}[]{ll}p_{n}(\tau)&n=0,2,4,\ldots\\\
0&n=1,3,5,\ldots\end{array}\right.$ (4) (5) $\displaystyle
p_{a}(n|\tau)\equiv|\langle\mathop{\mathrm{HG}}\nolimits_{n}|\psi_{\mathrm{a}}\rangle|^{2}=\left\\{\begin{array}[]{ll}0&n=0,2,4,\ldots\\\
p_{n}(\tau)&n=1,3,5,\ldots\end{array}\right.$ (8)

Here, $p_{\alpha}(n|\tau)$ ($\alpha\in\\{a,s\\}$) denotes the probability
density for a detection when projecting the symmetric (anti-symmetric)
coherent mode $|\psi_{\mathrm{s}}\rangle$ ($|\psi_{\mathrm{a}}\rangle$) onto
the mode $\mathop{\mathrm{HG}}\nolimits_{n}$, conditional on the value of the
time delay $\tau$ and we have defined
$p_{n}(\tau)={1}/(n!\,16^{n})({\tau}/{\sigma_{t}})^{2n}\exp[-{\tau^{2}}/{(16\sigma_{t}^{2})}]$.

Due to the limitations of the experimental setup described later, we cannot
generate incoherent signals directly. However, we can generate coherent in-
phase and anti-phase superpositions, which directly correspond to the output
ports of the beam splitter (1). Mixing the measurement data for both
superpositions in post-processing allows for realizing an arbitrary amount of
coherence between the two signal pulses. Note that measuring the output of the
interference between two signals is relevant for many applications, such as
stellar interferometry, and thus not just a convenient theoretical approach.
In consequence, we have now

$p_{\mathrm{incoh}}(n|\tau)=p_{s}(n|\tau)+p_{a}(n|\tau)=p_{n}(\tau)\,.$ (9)

The classical FI about $\tau$ from these mode projections on in-phase and
anti-phase states is

$F_{\mathrm{incoh}}(\tau)=\sum_{n}\frac{1}{p_{\mathrm{incoh}}(n|\tau)}\left[\frac{\partial
p_{\mathrm{incoh}}(n|\tau)}{\partial n}\right]^{2}\,.$ (10)

Since in-phase and anti-phase detection happen in even and odd mode
projections, respectively, no information is lost in this process and the
incoherent FI does saturate the quantum bound:
$F_{\mathrm{incoh}}(\tau)={1}/{(4\sigma^{2}_{t})}$.

The incoherent mixture is given either by a sum of time-delayed Gauss
components or by a sum of unnormalized in-phase and anti-phase superpositions
as in (1). The FI for the sum of probabilities (9) saturates the ultimate
limit for time-localization of input components $\psi_{\pm}$ and the beam
splitter action (1) is a unitary process, preserving information.
Consequently, the Hermite-Gauss projections saturate the quantum bound
simultaneously for both in-phase and anti-phase superpositions.

Let us now consider the opposite case of fully coherent signals. The
estimation of the time shift $\tau$ requires projections applied to in-phase
and anti-phase states; the results are

$\displaystyle F_{s}(\tau)$ $\displaystyle=$
$\displaystyle\frac{1}{8\sigma_{t}^{2}}-\left(\frac{1}{8\sigma_{t}^{2}}-\frac{\tau^{2}}{32\sigma_{t}^{4}}\right)\exp\left(-\frac{\tau^{2}}{8\sigma_{t}^{2}}\right)\,,$
$\displaystyle F_{a}(\tau)$ $\displaystyle=$
$\displaystyle\frac{1}{8\sigma_{t}^{2}}+\left(\frac{1}{8\sigma_{t}^{2}}-\frac{\tau^{2}}{32\sigma_{t}^{4}}\right)\exp\left(-\frac{\tau^{2}}{8\sigma_{t}^{2}}\right)\,.$

If we detect both outputs, we have
$F_{\mathrm{coh}}(\tau)={1}/({4\sigma^{2}_{t}})$, which also saturates the
quantum bound. Note, though, that for small separations, $\tau\rightarrow 0$,
we have $F_{s}(\tau)\simeq 0$ and $F_{a}(\tau)\simeq{1}/({4\sigma^{2}_{t}})$,
that is, almost all the information resides in the anti-phase channel,
$F_{a}$. However, in this limit the intensity available in this channel
becomes $\sum_{n}p_{a}(n|\tau)\rightarrow 0$. Nonetheless, we must remember
that a constant amount of input intensity is spent on generating the anti-
phase superposition for any separation. In a sense, the beam splitter acts as
an information sorter that directs the information about the timing separation
to the weak anti-phase channel. The majority of the signal intensity is sent
to the in-phase output, where other parameters (e.g. the timing centroid) can
be simultaneously accessed. In the case of complete incoherence, no
interference on the beam splitter occurs: the measured intensity in the anti-
phase output is half of the input intensity, regardless of the timing
separation. In this case, a simple intensity measurement is insufficient to
resolve the separation of the two signals. It is known, however, that mode
projections are ideal and, in fact, remain optimal for any degree of
coherence.

A PSF-independent formulation can be provided by defining a modified quantum
FI for states with parameter-dependent norm
$N(\tau)=\langle\psi(\tau)|\psi(\tau)\rangle$. In fact, it reads [39]

$\displaystyle\widetilde{\mathcal{Q}}(\tau)$ $\displaystyle=$ $\displaystyle
4\langle\partial_{\tau}\psi(\tau)|\partial_{\tau}\psi(\tau)\rangle$ (12)
$\displaystyle+$
$\displaystyle\frac{1}{N(\tau)}\left[\langle\psi(\tau)|\partial_{\tau}\psi(\tau)\rangle-\langle\partial_{\tau}\psi(\tau)|\psi(\tau)\rangle\right]^{2}\,.$

Applying this to the superpositions (1) confirms the optimality of Hermite-
Gauss projections.

As we said before, partial coherence has been the subject of a recent
controversy [32, 33, 34, 35, 36, 37]. Actually, we maintain that partial
coherence just redistributes the information into two (or more) interfering
channels. Information is carried both by the norm (i.e., intensity-modulated
by the estimated parameter) and the underlying normalized quantum state.
However total information is preserved and can be extracted with a suitable
measurement.

For all degrees of coherence and to remove the impact of an intensity-
dependent norm in the experiment, we realize partially coherent states as
mixtures of two coherent detection schemes with $s$ and $a$ channels
interchanged. We quantify coherence with the parameter $\gamma$ of such convex
combinations, where $\gamma=0$ means fully coherent and $\gamma=1/2$ means
fully incoherent. The projection on the same temporal Hermite-Gauss modes of
in-phase and anti-phase channels are

$\displaystyle{P}_{s}^{\gamma}(n|\tau)=\left\\{\begin{array}[]{ll}(1-\gamma)p(n|\tau)&n=0,2,4,\ldots\\\
\gamma\,p(n|\tau)&n=1,3,5,\ldots\end{array}\right.$ (15) (16)
$\displaystyle{P}_{a}^{\gamma}(n|\tau)=\left\\{\begin{array}[]{ll}\gamma\,p(n|\tau)&n=0,2,4,\ldots\\\
(1-\gamma)p(n|\tau)&n=1,3,5,\ldots\end{array}\right.$ (19)

Notice that for $\gamma=1/2$, we obtain two identical sets of probabilities
${P}^{1/2}_{a}(n|\tau)={P}^{1/2}_{s}(n|\tau)$, which upon adding we recover
the incoherent probabilities (9). Naturally, the total FI again saturates the
quantum bound for any $\gamma$ and all separations

$\displaystyle{F}_{s}^{\gamma}(\tau)$ $\displaystyle=$
$\displaystyle(1-\gamma)F_{s}(\tau)+\gamma F_{a}(\tau)\,,$
$\displaystyle{F}_{a}^{\gamma}(\tau)$ $\displaystyle=$ $\displaystyle\gamma
F_{s}(\tau)+(1-\gamma)F_{a}(\tau)\,,$

and therefore
${F}^{\gamma}(\tau)={F}_{s}^{\gamma}(\tau)+{F}_{a}^{\gamma}(\tau)=F_{\mathrm{incoh}}(\tau)=1/(4\sigma_{t}^{2})$.

From the discussion thus far, it should be clear that Hermite-Gauss temporal
modes are optimal for any degree of coherence. Of course, any other complete
mode decomposition with even/odd temporal symmetry will do the same job [22].
However, these projections require sophisticated equipment. Intensity
detection is still the cut-and-dried method used in the laboratory. As the
dominant part of the information about separation is contained in the norm of
the anti-phase superposition, total intensity represents a valuable source of
information.

For incoherent states, intensity detection leads to Rayleigh’s curse and it is
not optimal. For full coherence, however, intensity detection is one optimal
solution. This can be readily shown by calculating the FI for the intensity
profiles of in-phase and anti-phase channels, whose probabilities of detection
are $P_{\alpha}(t|\tau)=\lvert\psi_{\alpha}(t)\rvert^{2},$ with
$\alpha\in\\{a,s\\}$, whence we get

$F_{\alpha}^{\mathrm{int}}(\tau)=\int
dt\frac{1}{P_{\alpha}(t|\tau)}\left[\frac{\partial
P_{\alpha}(t|\tau)}{\partial\tau}\right]^{2}=F_{\alpha}(\tau)\,,$ (21)

with $F_{\alpha}(\tau)$ given by (LABEL:eq:Fas). Therefore,
$F_{s}^{\mathrm{int}}(\tau)+F_{a}^{\mathrm{int}}(\tau)=1/\sigma_{t}$.
Consequently, there is no need for sophisticated mode projections when working
with a fully coherent signal. However, the temporal resolution can be strongly
improved by these detections for partially coherent and incoherent signals,
especially in the limit $\tau\rightarrow 0$, which is precisely the regime of
interest.

_Experimental results.—_

Figure 1: Schematic of the experimental setup. Inset: in-phase and anti-phase
input signals. The in-phase and anti-phase input pulses with different time
shifts ($\tau$) are derived from a broadband OPO at 1540 nm and attenuated to
the few-photon levels using a commercial pulse shaper. Gating pulses at 860 nm
with Hermite-Gauss profiles are shaped with an in-house pulse shaper. The
input and gating pulses co-propagate through a PPLN waveguide. A sum-frequency
process generates green output photons at 551 nm which are bandpass filtered
and subsequently counted using a silicon avalanche photo-diode (SiAPD).

The key building block in our experiment for the implementation of the optimal
Hermite-Gauss temporal-mode projections is a quantum pulse gate (QPG) [41,
42]. It is based on group-velocity matched sum-frequency generation between a
strong gating pulse and a weak signal pulse in a nonlinear waveguide.
Detecting the upconverted photons then realizes projective measurements, in
which the temporal-mode projections are defined by the shapes of the gating
pulse.

The detailed scheme of our experimental setup is sketched in Fig. 1. A
titanium-sapphire (Ti:Sapph) laser and an optical parametric oscillator (OPO)
provide the gating and the input pulses with a repetition rate of
$80~{}\mathrm{MHz}$, respectively. The gating pulses are carved from a laser
spectrum centered at $860~{}\mathrm{nm}$ with a full-width at half-maximum
(FWHM) bandwidth of $7.25~{}\mathrm{nm}$. The gating pulses are shaped into
user-defined HG temporal modes with a home-built pulse shaper, comprised of a
spatial light modulator at the Fourier plane of a 4-f line arrangement.

The input pulses are derived from the OPO, delivering light at
$1540~{}\mathrm{nm}$ with an FWHM bandwidth of $23~{}\mathrm{nm}$. A
commercial fiber-coupled pulse shaper prepares the input signal that consists
of coherent superpositions of two time-shifted Gaussian pulses of
$1.26~{}\mathrm{ps}$ width with equal (in-phase) or opposite (anti-phase)
phase. As shown in the inset of Fig. 1, one receives a positive time shift of
$\tau/2$ and the other one receives a negative time shift of $-\tau/2$ with
respect to a reference that is set at zero without loss of generality. In our
experiment, seven different time shifts ranging from $0$ to $\sigma_{\tau}$
are realized for both in-phase and anti-phase inputs by programming the pulse
shaper. Moreover, the input pulses are attenuated to a few photons per pulse.

The shaped gating and the input pulses are combined on a dichroic mirror and
then coupled into a home-built QPG—a $3.5~{}\mathrm{cm}$ long periodically
poled lithium niobate (PPLN) waveguide with a poling period of
$4.4~{}\mu\mathrm{m}$. The waveguide is designed for spatially single-mode
propagation of the input signal, whereas the propagation of the gating beam in
the fundamental spatial mode is ensured via optical mode matching. The sum-
frequency process yields a green output at $551~{}\mathrm{nm}$ that undergoes
tight spectral filtering in a 4-f line to discard the phase-matching side-
lobes, resulting in an FWHM bandwidth of $40~{}\mathrm{pm}$. The filtered
output is detected with a fiber-coupled silicon avalanche photodiode (SiAPD).
Finally, we record the single counts using a commercial time-tagger. We record
data for $16~{}\textrm{ms}$ for each setting of the input and gating pulses,
and repeat the measurement 100 times for the statistical analysis of the data.

Figure 2: Estimates $\widehat{\tau}$ versus true separations $\tau$, both in
units of the pulse width $\sigma_{t}$ for the three values of the coherence
indicated in the inset.

In our experiment, different coherence strengths are achieved in post-
processing by controlled mixing of the measured data for the in-phase and the
anti-phase input signals—each of them separately corresponds to the fully
coherent case, their equal mixing leads to the incoherent case, and unequal
mixing corresponds to the partial coherence. Rather than using the theoretical
projections, we determine the actual input-output relations of the implemented
imperfect QPG device, enabling the construction of an unbiased estimator of
the time separation despite the limited selectivity of the device [21].

For each coherence setting, this is achieved by fitting the average responses
of the first four HG projections with fourth-order polynomials in $\tau$. The
resulting measurement matrix is then used to process the individual detections
taking the generalized least squares estimator constrained by the condition
$\hat{\tau}\geq 0$. Estimator statistics are calculated from $100$ repeated
measurements for each combination of $\tau$ and $\gamma$ parameters.

Figure 3: Variance of the estimator $\widehat{\tau}$ as a function of the true
time offset $\tau$ for several values of the coherence parameter $\gamma$: 1/2
(blue crosses), 3/8 (orange crosses), 1/4 (green triangles), 1/8 (red
squares), and 0 (purple circles). The ultimate limit given by the quantum CRB
is given by the red solid line, whereas the classical incoherent detection
limit is the broken blue line. The inset shows a zoomed version where we can
better appreciate the behavior.

During the experiment, we had to take special care to avoid unwanted temporal
drift between the signal and gating fields that mainly originates from thermal
fluctuations. To remove the effect of residual drift, the two fields are
recentered by software control after every 10 measurement runs of each
setting.

Figure 2 shows the statistics of the experimental estimates of $\tau$ for
incoherent, coherent, and partially coherent superpositions. Mean values are
plotted with standard deviation bars around. The true values are inside the
standard deviation intervals for all separations and the estimator bias is
negligible.

Figure 3 shows the estimation errors (quantified by the variance) for five
different mixtures ranging from fully coherent to incoherent. Coherent
estimates saturate the quantum bound for small separations: we experimentally
resolve temporal offsets ten times smaller than their pulse duration, with a
tenfold improvement in precision over the intensity-only CRB. When coherence
is reduced, we see an increase of the experimentally determined variances,
especially for the smallest measured time separations. This effect is a
consequence of a tiny, yet nonnegligible, crosstalk between the odd and even
Hermite-Gauss projections. Leakage of the strongly populated
$\mathop{\mathrm{HG}}\nolimits_{0}$ mode of the symmetric channel towards odd
projections, upon mixing the channels as in (15), degrades the information
carried by the much weaker odd modes (and, in particular, the mode
$\mathop{\mathrm{HG}}\nolimits_{1}$) of the antisymmetric channel. But even
with these imperfections, we are still much below the intensity-only CRB.

Figure 4: Estimation errors for fully coherent signal ($\gamma=0$) per single
total detection (purple circles, as in Fig. 3) and per single antisymmetric
($\psi_{a}$) detection (brown squares). The quantum (red solid line) and
classical incoherent (blue broken line) limits are also shown.

Finally, in Fig. 4 we show the same errors for fully coherent superposition
both per single total detection and per single detection in the antisymmetric
($\psi_{a}$) channel. Ignoring the cost of creating the antisymmetric
superposition, the information carried by one such copy seemingly diverges at
$\tau\rightarrow 0$, rendering a false impression that the anti-phase
coherence attains a sensitivity better than the quantum limit. However, a
proper resource counting, in this case the total single counts, leads to the
correct quantum-limited estimation error. This is another way of expressing
our previous statement that coherence acts like an information sorter
condensing information about separation in the dark antisymmetric channel.

_Concluding remarks.—_ Parameter estimation in the time-frequency domain
benefits from quantum-inspired techniques. By resorting to mode-selective
measurements, sub-pulse-width separations can be estimated with quantum-
limited precision for a full range of temporal coherence.

Seemingly, coherence by itself does not provide a direct metrological
advantage in time resolution; incoherent superpositions setting the ultimate
limits in all cases. However, we stress that coherence can be exploited as an
information sorter, distributing information about different parameters into
different channels. In our case, for small-time delays, all information is
accessible from vanishing intensity in the anti-symmetric channel while the
bulk of the intensity goes into the symmetric channel, left available for
measurements of other relevant physical parameters.

We thank V. Ansari and J. M. Donohue for discussions. We acknowledge the
European Union’s Horizon 2020 research and innovation program (Grants ApresSF
and STORMYTUNE), Deutsche Forschungsgemeinschaft (231447078–TRR 142), the
Grant Agency of the Czech Republic (18-04291S), and the Spanish Ministerio de
Ciencia e Innovación (PGC2018-099183-B-I00).

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