# Super-resolution of ultrafast pulses via spectral inversion

Michał Lipka  1,2,* and Michał Parniak  1,2  1Centre for Quantum Optical
Technologies, Centre of New Technologies, University of Warsaw, Banacha 2c,
02-097 Warsaw, Poland  2Faculty of Physics, University of Warsaw, Pasteura 5,
02-093 Warsaw, Poland<EMAIL_ADDRESS>
††journal: opticajournal††articletype: Research Article

The resolution limits of classical spectroscopy can be surpassed by quantum-
inspired methods leveraging the information contained in the phase of the
complex electromagnetic field. Their counterpart in spatial imaging has been
widely discussed and demonstrated; however, the spectral-domain
implementations are few and scarce. We experimentally demonstrate a
spectroscopic super-resolution method aimed at broadband light (10s to 100s of
GHz), and based on the spectral-domain analog of image inversion
interferometry. In a proof-of-principle experiment, we study the paradigmatic
problem of estimating a small separation between two incoherent spectral
features of equal brightness, with a small number of photons per coherence
time. On the grounds of asymptotic estimation theory, more than a $2$-fold
improvement over the spectral direct imaging is demonstrated in terms of
required resources (photons) for a given estimator variance. The setup is
based on an actively stabilized Mach-Zehnder-type interferometer with electro-
optic time lenses and passive spectral dispersers implementing the inversion.
As such, the method promises on-chip integration, good scalability, and
further applications e.g. for mode sorting.

##  1 Introduction

The concept of resolution inseparably accompanies any measurement technique.
While very intuitive, its rigorous description long remained surprisingly
elusive with a practical formulation dating back to Lord Rayleigh [1] and
known as the Rayleigh limit, together with its spectroscopic counterpart – the
Fourier limit [2]. The descriptions, similar in essence, referred to a minimal
separation between two identical features at which they could still be
distinguished. While a modern analysis on the grounds of asymptotic parameter
estimation theory brought quantitative answers [3], a truly groundbreaking
finding was how to overcome these limits.

Traditional direct imaging or spectroscopy employs quadratic-response
photodetectors which output is proportional to the image-plane intensity
$\propto\absolutevalue{E(f)}^{2}$. In a seminal paper Tsang et al. [4]
demonstrated the potential for using the information encoded in the phase of
the complex electric field $E(f)$, inherently lost in direct imaging, to
overcome the Rayleigh limit. This finding inspired numerous theoretical
proposals and experimental super-resolution demonstrations [5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25].

From a theoretical point of view, space-time analogy [26] makes the problem of
spectral resolution almost equivalent to 1-dimensional spatial imaging [27].
However, surprisingly, there are few experimental implementations of super-
resolution spectral imaging. These include a quantum-pulse-gate-based super-
resolution in the spectral [28] and temporal [29] domains, heterodyne
detection with post-processing [30], and a time-axis-inversion-based protocol
implemented in a quantum memory [31]. The latter is based on the same
principle as this work, namely of image inversion interferometry, however, the
implementation inherently targets tens of $\text{\,}\mathrm{kHz}$ bandwidth
pulses and hence differs in almost every aspect.

While we focus on weak and incoherent light, such as encountered in
fluorescence spectroscopy – a powerful and vastly employed tool of organic
chemistry, biology, and medicine [32], different spectroscopic settings merit
individual approach. For instance super-resolution has been demonstrated for
laser spectroscopy [33] and with controlled illumination spectroscopic methods
can be vastly enhanced with quantum light probes [34, 35, 36, 37].

In this work, we present a spectral-domain super-resolution method inspired by
the image inversion interferometry – a technique well described and
demonstrated in the context of super-resolution in spatial imaging [38, 7, 8,
11]. We call the method SUper-resolution via Spectral Inversion (SUSI). In an
analogy to its spatial counterpart, SUSI is based on interfering an inverted
copy of the input state with its original version. The inversion operator
$\mathscr{P}$ acts in the spectral degree of freedom, around an a priori known
centroid. The symmetric ($+$) and antisymmetric ($-$) parts of the state are
eigenvectors of $\mathscr{P}$ with $\pm 1$ eigenvalues, hence in the
interferometric setup the ($+$) and ($-$) parts can be spatially separated and
independently undergo photon counting. If the input state consists of an
incoherent mixture of closely separated equally bright pulses, almost all
information on their separation is contained in the number of photons counted
in the ($-$) port and there is no inherent background i.e. for 0 separation,
with an ideal setup, no photons are going to the ($-$) port. In this way SUSI
provides a sensitive and hence efficient estimation in the sub-Fourier regime.

The super-resolution methods essentially perform mode sorting in a scenario-
specific basis of modes. Often in practice, the basis is truncated to the
first two modes which are nevertheless sufficient in the regime of very small
separations. Promising multi-mode-sorting techniques have been described [39]
and demonstrated [40] in the spectral domain, with applications beyond
parameter estimation. SUSI could be in principle extended to multi-mode
sorting with a network of spectral-inversion interferometers and linear optics
– a promising scheme if combined with on-chip integration.

The article is organized as follows: in sec. 2 we briefly introduce the
problem and SUSI in mathematical terms and consider a simplified case of
Gaussian pulses; in sec. 3 experimental results are presented; sec. 4
describes the methodology, and finally sec. 5 and sec. 6 give a conclusion and
outlook.

Figure 1:  The idea of SUSI. An incoherent mixture of spectrally separated (by
$\varepsilon$) pulses, with a priori known centroid, enters a Mach-Zehnder-
type interferometer. The combined action of a Fourier transform $\mathscr{F}$
in one of its arms and an inverse Fourier transform $\mathscr{F}^{-1}$ in the
other is equivalent to a spectral inversion $\mathscr{P}$ in a single arm.
Interfering the signals from both arms separates the symmetric ($+$) and
antisymmetric ($-$) part of the input mode. Photon counting at the output
ports performs a more sensitive sensing of $\varepsilon$ (lower estimator
variance) than direct imaging (spectroscopy), hence constituting a super-
resolution method. 50/50 BS - balanced beam splitter, SPD - single photon
detector.

##  2 Theory

In a standard spectroscopic approach estimating the separation between two
spectral features would involve detecting a statistic of $N$ photons with a
spectrally-resolving detector. The number of photons in each spectral bin
would be used to construct e.g. a maximum likelihood estimator of the
separation under an assumed statistical model.

Increasing $N$ will lower the estimator variance. Hence, even for spectral
features of width $\sigma$ separated by $\delta f$ such that
$\varepsilon=\delta f/\sigma\ll 1$ a standard approach (direct imaging) is
still possible, if not practically feasible.

Inevitably the problem must be formulated in terms of required resources. Here
a meaningful choice is the number of observed photons $N$, which usually
corresponds to the measurement time. A super-resolution method can lower the
number of resources required for a given estimator variance, with all other
factors constant. In comparative terms, super-resolution methods excel for
$\varepsilon\ll 1$ and henceforth we shall assume this sub-Fourier regime.

In the paradigm of frequentist inference, a modern and convenient evaluation
of an estimation scheme comes from the Cramér–Rao bound and Fisher Information
[41]. The variance of any unbiased estimator $\hat{\varepsilon}$ follows the
Cramér–Rao bound:

$\Delta^{2}\hat{\varepsilon}\geq\frac{1}{F(\varepsilon)},$ (1)

where $F(\varepsilon)$ is the Fisher information of estimating a parameter
$\varepsilon$ in a measurement scheme given by a set of outcomes $\\{i\\}$
associated with conditional probabilities $\\{P(i|\varepsilon)\\}$.
$F(\varepsilon)$ is given by

$F(\varepsilon)=\sum_{i}\frac{\left(\partial_{\varepsilon}P(i|\varepsilon)\right)^{2}}{P(i|\varepsilon)}.$
(2)

Let us note that the problem of estimating the frequency separation of two
incoherent pulses is in many ways analogous to the well-studied case of the
spatial far-field imaging of two point-like sources. The main difference is as
follows. In our case, the width of the pulse is its intrinsic property. It is
also the scale parameter of the problem. However, the spectral resolution of
any instrument is assumed to be arbitrarily good. On the other hand, for
spatial imaging, the width of the imaging instrument’s point spread function
determines this scale. Of course, considering space-time duality, one can
imagine a temporal aperture that gives the spectral shape to the pulses. Note
that minute changes in the spectral shape of the pulses can drastically
broaden its temporal extent leaving the spectrum width $\sigma$ constant. A
limiting case is a perfectly square spectrum which would require an infinite
temporal width. Hence, under such an unconstrained formulation, the DI methods
will seemingly have a very good performance for super-Gaussian pulses.
Introducing a limited temporal aperture (or equivalently limited spectrometer
resolution) would restore a realistic DI behavior.

We shall follow a standard methodology for the spatial super-resolution
methods. Let us assume that the number of photons per coherence time is very
low. It is a well-grounded assumption, well discussed in ref. [4] and true for
fluorescence spectroscopy and most natural phenomena such as stellar emission
[42]. This way, the probabilities $\\{P(i|\varepsilon)\\}$ will be Poisson-
distributed with means $\mu_{i}(\varepsilon)=Np(i|\varepsilon)$, where
$p(i|\varepsilon)$ is the equivalent probability for a single photon. Poisson
statistic produces the shot-noise scaling of the estimator variance
$\Delta^{2}\hat{\varepsilon}\propto 1/N$ or $F\propto N$, hence we will
further consider the Fisher Information per photon $\mathcal{F}=F/N$, which is
given by Eq. (2) upon substituting $P(i|\varepsilon)\rightarrow
p(i|\varepsilon)$.

Furthermore, we shall assume fully incoherent and equally bright pulses.
Deviations from this simplest scenario are well studied in the spatial imaging
setting [43, 44, 45, 46, 47]. We will also assume an a priori known centroid
of the pulses joint spectrum. In practice, a part of the available photons (or
measurement time) would be sacrificed to estimate the centroid, possibly in an
adaptive strategy [48].

For notational convenience, we will use dimensionless quantities with the
frequency $\xi=f/\sigma$, and pulse separation $\varepsilon=\delta f/\sigma$.
For a single pulse a normalized slowly varying envelope of the electric field
will be denoted by $\mathcal{A}(\xi)$ obeying symmetry
$\mathcal{A}(\xi)=\mathcal{A}(-\xi)$ and normalization

$\int\differential{\xi}\absolutevalue{\mathcal{A}(\xi)}^{2}\xi^{2}=\int\differential{\xi}\absolutevalue{\mathcal{A}(\xi)}^{2}=1.$
(3)

A convenient feature of the incoherent mixture of two weak pulses
$\mathcal{A}(\xi\pm\varepsilon/2)$ is that each pulse can be considered
separately. In the end, the photon counts would be added together for
$\pm\varepsilon/2$ pulses.

###  2.1 Spectral inversion interferometry

A simplified setup for SUSI has been depicted in Fig. 1. The main part
consists of a Mach-Zehnder-type interferometer with a Fourier transform
$\mathscr{F}$ in one arm and an inverse Fourier transform $\mathscr{F}^{-1}$
in the other. Since all elements are linear and the Fourier transform does not
change the total energy (Parseval’s theorem) such a setup is equivalent to
placing two Fourier transforms in series in a single arm, and leaving the
other arm empty. The composition of two Fourier transforms has the property of
inversion $\mathscr{F}\circ\mathscr{F}\left[g(x)\right]=g(-x)$ which
implements the spectral equivalent of image inversion interferometry.
Distributing the transform into two arms has practical merits in ensuring
similar efficiency and higher-order imperfections in each arm.

Let us calculate the result of a SUSI measurement for a single shifted pulse
$\mathcal{A}(\xi-\varepsilon/2)$, in the regime of $\varepsilon\ll 1$. From
the pulse expansion

$\mathcal{A}(\xi-\frac{\varepsilon}{2})=\mathcal{A}(\xi)-\frac{\varepsilon}{2}\partial_{\xi}\mathcal{A}(\xi)+\frac{\varepsilon^{2}}{8}\partial^{2}_{\xi}\mathcal{A}(\xi)+\order{\xi^{3}}.$
(4)

we can see that the information about $\varepsilon$ is contained in the
antisymmetric part $\partial_{\xi}\mathcal{A}(\xi)$. Neglecting higher-order
terms and acting with the (inverse) Fourier transform
($\xi\xrightarrow{}\tau$) we get

$\displaystyle\mathscr{F}\left[\mathcal{A}(\xi-\frac{\varepsilon}{2})\right]$
$\displaystyle=\tilde{\mathcal{A}}(\tau)\left[1+i\tau\frac{\varepsilon}{2}-\frac{\varepsilon^{2}\tau^{2}}{8}\right],$
(5)
$\displaystyle\mathscr{F}^{-1}\left[\mathcal{A}(\xi-\frac{\varepsilon}{2})\right]$
$\displaystyle=\tilde{\mathcal{A}}(\tau)\left[1-i\tau\frac{\varepsilon}{2}-\frac{\varepsilon^{2}\tau^{2}}{8}\right],$
(6)

where $\tilde{\mathcal{A}}=\mathscr{F}\left[\mathcal{A}\right]$. Including the
effect of non-perfect visibility $0\leq\mathcal{V}<1$ and after the final
beamsplitter transformation, we get the following spectral intensities:

$I_{\pm}=\frac{1}{4}\left(\absolutevalue{\mathscr{F}\left[\mathcal{A}(\xi-\frac{\varepsilon}{2})\right]}^{2}+\absolutevalue{\mathscr{F}^{-1}\left[\mathcal{A}(\xi-\frac{\varepsilon}{2})\right]}^{2}\right)\\\
\pm\frac{\mathcal{V}}{2}\Re{\mathscr{F}\left[\mathcal{A}(\xi-\frac{\varepsilon}{2})\right]\mathscr{F}^{-1}\left[\mathcal{A}(\xi-\frac{\varepsilon}{2})\right]^{*}}.$
(7)

Substituting Eqs. (5) and (6) gives

$I_{\pm}(\tau)=\absolutevalue{\tilde{\mathcal{A}}(\tau)}^{2}\left[\frac{1\pm\mathcal{V}}{2}\mp\frac{\mathcal{V}}{4}\tau^{2}\varepsilon^{2}\right]+\order{\varepsilon^{3}}.$
(8)

Assuming bucket detectors, we integrate over $\tau$ and drop higher-order
$\varepsilon$ terms:

$\displaystyle n_{\pm}$
$\displaystyle=\int\differential{\tau}I_{\pm}(\tau)=\frac{1\pm\mathcal{V}}{2}\mp\frac{\mathcal{V}}{4}\varepsilon^{2}\mathit{\Delta},$
(9) $\displaystyle\mathit{\Delta}$
$\displaystyle=\int\differential{\tau}\absolutevalue{\tilde{\mathcal{A}}(\tau)}^{2}\tau^{2},$
(10)

where we used Parseval’s theorem and $\mathcal{A}(\xi)$ normalization. The
variance $\mathit{\Delta}$ depends on the spectrum shape. For instance for
Gaussian pulses in our dimensionless units, it reads
$\mathit{\Delta}_{\mathrm{Gauss}}=1/4$.

Summing $n_{\pm}$ for $\pm\varepsilon/2$ pulses and using Eq. (2) with
$P(i|\varepsilon)\rightarrow p(i|\varepsilon)$ and

$p(\pm|\varepsilon)=n_{\pm}/(n_{+}+n_{-})$ (11)

we get Fisher information per photon
$\mathcal{F}(\varepsilon)=F(\varepsilon)/N$ of

$\mathcal{F}(\varepsilon)=\frac{\mathcal{V}^{2}}{1-\mathcal{V}^{2}}\frac{\varepsilon^{2}\mathit{\Delta}^{2}}{\left(1-\frac{\varepsilon^{2}\mathit{\Delta}}{2}\right)^{2}}=\frac{\mathcal{V}^{2}\mathit{\Delta}^{2}}{1-\mathcal{V}^{2}}\varepsilon^{2}+\order{\varepsilon^{3}}.$
(12)

For identical incoherent pulses in an ideal scenario of perfect visibility
$\mathcal{V}=1$, the quantum Fisher Information (which ultimately bounds the
Fisher Information for any measurement scheme) remains constant regardless of
$\varepsilon$. However, it has been shown that any imperfections restore the
$\propto\varepsilon^{2}$ scaling [49, 50]. Hence, one meaningful way to
compare super-resolution protocols is via the ratio of their quadratic scaling
factors. Following ref. [31], the super-resolution parameter is defined as

$\mathbf{s}=\lim_{\varepsilon\rightarrow{}0}\frac{\mathcal{F}_{\mathrm{SUSI{}}}(\varepsilon)}{\mathcal{F_{\mathrm{DI}}}(\varepsilon)}$
(13)

and quantifies the reduction in the number of photons required for a given
estimation precision, compared to direct imaging.

For example an analytically tractable case of Gaussian pulses gives
$\mathcal{F}_{\mathrm{SUSI{}}}^{(\mathrm{Gauss})}=\varepsilon^{2}/16\times\mathcal{V}^{2}/(1-\mathcal{V}^{2})$
and $\mathcal{F}_{\mathrm{DI}}^{(\mathrm{Gauss})}=\varepsilon^{2}/8$, hence
$\mathbf{s}^{(\mathrm{Gauss})}=1/2\times\mathcal{V}^{2}/(1-\mathcal{V}^{2})$.
In this case, to perform better than DI, SUSI would require a visibility of
more than ca. $81.7\%$.

Notably, in the limit of $\varepsilon\rightarrow 0$ a fraction of
$(1+\mathcal{V})/2$ of total $\mathcal{F}_{\mathrm{SUSI{}}}$ comes from the
dark ($-$) port counts. For instance with our experimental visibility of ca.
$\mathcal{V}=93\%$ the dark port provides $96.5\%$ of information.

##  3 Results

Experimental results are depicted in Fig. 2. The effect of shifting a pulse
from the central frequency of the Fourier transformers can be observed in the
spectra, as demonstrated in Fig. 2 (a). Fig. 2 (b) depicts histograms of
photon counts at $\pm$ ports, collected for an input of a single pulse shifted
by $\varepsilon/2$, for clarity in the regime of large $\varepsilon$. In the
main measurement, for each $\varepsilon$, analogous histograms
$h_{\pm}(t,\varepsilon)$ are collected. Counts are then integrated over the
measurement period
$n_{\pm}(\varepsilon)=\int\differential{t}\>h_{\pm}(t,\varepsilon)$ and summed
over experiment repetitions. Finally, the conditional probability
$p(-|\varepsilon)$ is estimated as the fraction of photon counts in the dark
port. A quadratic model of Eqs. (9) and (11) is fitted to $p(-|\varepsilon)$
and the corresponding Fisher information is calculated from Eq. (12). Fig. 2
(c) depicts experimental $p(-|\varepsilon)$ alongside a quadratic fit and a
result from a SUSI simulation. Fisher information $\mathcal{F}(\varepsilon)$
for the quadratic fit, SUSI simulation, simulated direct imaging, and a
theoretical prediction of Eq. (12) has been depicted in Fig. 2 (d). Both SUSI
and DI simulations assume an experimentally measured pulse spectrum and in the
case of SUSI a visibility of $93\%$ (c.f. appendix sec. A, and B). For the
theoretical prediction $\mathit{\Delta}\approx 0.293$ is calculated from the
Fourier transform of the experimental spectrum.

The quadratic fit to experimental data and the simulated DI give a super-
resolution parameter of $\mathbf{s}=2.13$ The results and parameters have been
summarized in Tab. 1.

We observe a good agreement of the experimental data with both the theoretical
prediction and the numerical simulation. By constructing a measurement model,
super-resolution with respect to the spectral DI is demonstrated. The super-
resolution is quantified in terms of a higher Fisher Information per photon in
the regime of low separations $\varepsilon\ll 1$, regardless of $\varepsilon$.
Via the Cramér-Rao bound it can be understood as the equivalent reduction in
the required number of photons to reach a fixed estimator variance.

Table 1: Summary of results assuming $\mathcal{V}=93\%$.  | $\mathit{\Delta}$ | $\partial^{2}_{\varepsilon}\mathcal{F}(0)/2$ | $\mathbf{s}$
---|---|---|---
DI | $0.293$ | $0.24$ | 1
Theor | $0.55$ | $2.29$
Sim | $0.48$ | $2.01$
Fit | $0.51$ | $2.13$
DI Gauss | $1/4$ | $1/8$ | 1
Theor Gauss | $0.40$ | $1.60$
Figure 2:  (a) Spectra at the interferometer output for the constructive (max)
or destructive (min) interference, with an input pulse frequency-shifted by 0
or $\pm 42\text{\,}\mathrm{GHz}$. (b) Rescaled histograms of photon counts at
the $\pm$ interferometer outputs for the measurement period ($0\leq
t\leq$200\text{\,}\mathrm{\SIUnitSymbolMicro s}$$) and the beginning of the
interferometer-stabilization period
($t>$200\text{\,}\mathrm{\SIUnitSymbolMicro s}$$). Rows correspond to
subsequent shifts $\varepsilon/2$ of a single input pulse. (c) Fraction of
integrated photo counts in the ($-$) port $p(-|\varepsilon)$ during the
measurement period, for an incoherent mix of pulses spectrally separated by
$\varepsilon$. Experimental data (Exp) with a parabolic fit (Fit) and a result
simulated with the measured spectrum and visibility (Sim). (d) Fisher
information per photon $\mathscr{F}$ for $\varepsilon$ estimation. Fit to
experimental data (Fit), analytical prediction (Theor), simulation (Sim), and
simulated direct imaging (DI) using the measured spectrum.

##  4 Methods

The experimental implementation of SUSI is depicted in Fig. 3. The central
part of the setup consists of a Mach-Zehnder-type interferometer with electro-
optic (inverse) Fourier transformers in each arm, described in more detail in
ref. [51]. Each (inverse) Fourier transformer is a sequence of a pulse
stretcher with the group delay dispersion (GDD) of $\Phi_{2}$, a time lens
with a chirp rate of $K=1/\Phi_{2}$, and another pulse stretcher with a GDD of
$\Phi_{2}$. For the direct Fourier transform we have $K>0$, while for the
inverse $K<0$. The frequency of the shortest optical path in the pulse
stretchers determines the central frequency of the Fourier transformer.

The interferometer is actively phase-stabilized with an intermittent
continuous-wave (CW) reference beam. The CW beam and the signal pulses are
combined with a $\pi/2$ phase difference by employing the polarization degree
of freedom. This way, the interferometer can be stabilized for an equal number
of photo counts in each arm with CW, while having a dark port with the signal
light.

The input state is prepared by spectrally filtering $100\text{\,}\mathrm{fs}$
pulses from a Ti:Sapphire laser (SpectraPhysics MaiTai) using a custom pulse
shaper. A diffraction grating is followed by a folded unit-magnification
telescope with a regulated slit and a retro-reflecting mirror placed in the
far field of the grating. Motorized lateral movement of the slit alters
$\varepsilon$.

To obtain an incoherent mix of two spectrally separated pulses, partial
measurements are collected with a single pulse shifted by $\pm\varepsilon/2$
and the raw photo counts for opposite signs are combined in post-processing.
Pulses with a spectrum of $85\text{\,}\mathrm{GHz}$ full width at half maximum
are used.

Figure 3:  (a)-(j) Experimental setup and experiment sequence. (a) General
schematic. Setup comprises a Mach-Zehnder-type phase-stabilized interferometer
with electro-optic (inverse) Fourier transformers in each arm. (b) Single-
photon detection, time-tagging, and histogramming followed by FPGA (RedPitaya)
processing. FPGA-based feedback loop stabilizes the difference of photon
counts (interferometer phase) (c) Experimental sequence controlled via an FPGA
sequencer. (d) Quadruple-pass delay line. (e) General schematic of (an
inverse) a Fourier transformer $\mathscr{F}$ ($\mathscr{F}^{-1}$). (f)
Splitting of the signal pulses and combining the signal with a continuous wave
(CW) laser used for the feedback loop. (g) A femtosecond laser
($100\text{\,}\mathrm{fs}$) beam is split with a small power fraction
optically driving time lenses, while the rest undergoes spectral filtering
(4f) to produce the signal pulse (ca. $10\text{\,}\mathrm{ps}$). (h) Electro-
optic time lens. The signal from an optically excited photodiode is filtered
and amplified to drive an electro-optic modulator. (i) Pulse stretcher
producing group delay dispersion $\Phi_{2}$ for $\mathscr{F}$. (j) CW source
with a double-pass acousto-optical modulator (AOM) switch. (k) Synchronization
of a radio frequency (RF) driving signal with an optical pulse, for different
signs of the time lens chirp rate $K$. (N)PBS – (non) polarizing beamsplitter,
HWP (QWP) – half-(quarter-)waveplate, amp – amplifier, piezo – piezoelectric
actuator, BPF – bandpass filter, EOM – electro-optical modulator, pol ctrl –
fiber polarization controller, PID – proportional-integral controller, SPD –
single-photon detector, TDC – time-digital converter, HST – histogramming
module, LPF – low-pass filter, MMV – monostable multivibrator, ND – neutral-
density filter.

##  5 Discussion

Quantum-inspired super-resolution methods employ the information contained in
the phase of complex electric field, lost in the direct imaging with quadratic
detectors. Proposed and demonstrated implementations often rely on mode-
specific detection in a problem-dependent basis of modes. SUSI as a method
based on image inversion interferometry, fundamentally relies on the
demultiplexing of a symmetric and antisymmetric part of the input state. In
the regime of $\varepsilon\ll 1$ this is equivalent to a simple mode sorting
into 2 modes. However, SUSI can be extended to the mode sorting of multiple
modes. A network of SUSI interferometers joined with linear optics would
provide a spectral mode-sorting platform – a fundamental device for spectro-
temporal quantum information processing [52]. A similar interferometer network
with Fourier transformers has been recently proposed and theoretically
analyzed in this context [39].

An alternative approach to mode sorting and super-resolution based on spectro-
temporal shaping has been proposed by Shah et al. [53]. It incorporates a
cascade of electro-optic modulators with case-specific arbitrary temporal
phase modulation intertwined with spectral dispersers. In comparison with SUSI
the serial approach does not require a stabilized interferometer; however, the
requirement of high-bandwidth arbitrary electro-optic modulation is very
stringent and experimentally infeasible for tens-of-GHz pulses. In the task of
mode sorting, with the increasing number of modes the cascade approach
requires a growing number of elements in series, whereas SUSI would involve an
interferometer network with a logarithmically growing depth. Depending on the
implementation details these approaches could vastly differ in terms of the
total efficiency.

We note that as a general alternative to pure spectral-temporal domain
protocols, spectro-spatial mapping allows spatial techniques to be employed in
the spectral domain, including measurement [54] and mode sorting [55].

##  6 Conclusions

We have experimentally demonstrated SUSI – a spectral-domain super-resolution
method based on the principle of image inversion interferometry and compatible
with ultrafast pulses with tens-of-GHz bandwidth. The method is aimed at weak
($\ll 1$ photon per coherence time) incoherent light encountered in practical
scenarios such as fluorescence spectroscopy or stellar observations.

Building on the ideas of image inversion interferometry in the position-
momentum degree of freedom, our spectral implementation splits the inversion
task into two parts. A Fourier transform is placed in one arm of the
interferometer and an inverse Fourier transform in the other, ensuring a
simple setup and inherent efficiency balance between the arms. The
demonstrated experimental implementation requires simple components – passive
spectral dispersers, a quadratic electro-optic modulation, linear optics, and
an active interferometer stabilization.

The demonstrated SUSI setup outperforms an equivalent DI measurement with an
over two-fold reduction in the required resources for the same estimator
variance. SUSI comprises a simple alternative to fairly few other demonstrated
quantum-inspired super-resolution methods in the spectral domain. In
comparison, no non-linear optics or complex cold-atomic setups are required.
Furthermore, the design is compatible with on-chip components and fiber
optics, promising practical applications in super-resolution spectroscopy and
spectral mode sorting.

Funding Fundacja na rzecz Nauki Polskiej (MAB/2018/4 “Quantum Optical
Technologies”); European Regional Development Fund; Narodowe Centrum Nauki
(2021/41/N/ST2/02926);

Acknowledgments The “Quantum Optical Technologies” project is 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. This research was funded in whole or in part by National
Science Centre, Poland 2021/41/N/ST2/02926. ML was supported by the Foundation
for Polish Science (FNP) via the START scholarship. We would like to thank K.
Banaszek, W. Wasilewski, and M. Mazelanik for the support and discussions.

Disclosures The authors declare no conflicts of interest.

Data availability Data underlying the results presented in this paper are
available in ref. [56] (Harvard Dataverse).

##  Appendix A SUSI simulation

An idealized SUSI setup was simulated with a non-perfect visibility
$\mathcal{V}=93\%$ as the only imperfection. The input state was an incoherent
mixture of two pulses with a spectral intensity as measured in the experiment
and a flat spectral phase. The simulation proceeds as follows:

  1. 1.

The experimental spectrum has its background removed by fitting a mixture of a
Gaussian and linear functions, and subtracting the latter’s best fit.

  2. 2.

We take a square root of the spectrum to get an equivalent envelope of the
pulse’s electric field with a flat spectral phase. Rare small negative values
are replaced with zeros.

  3. 3.

Spectrum is shifted by $\varepsilon/2$.

  4. 4.

Such a signal is then padded with zeros. In parallel a fast Fourier transform
(FFT) and its inverse (iFFT) are computed.

  5. 5.

A beamsplitter transformation is performed between FFT ($a$) and iFFT ($b$)
arms, accounting for limited visibility

$\tilde{c}_{\pm}(\varepsilon)=\frac{1}{2}\left[\mathcal{V}\absolutevalue{a\pm
b}^{2}+\left(1-\mathcal{V}\right)\left(\absolutevalue{a}^{2}+\absolutevalue{b}^{2}\right)\right].$
(14)

  6. 6.

The results is symmetrized with respect to the change of the
$\pm\varepsilon/2$ shift sign

$c_{\pm}(\varepsilon)=\frac{1}{2}(\tilde{c}_{\pm}(\varepsilon)+\tilde{c}_{\pm}(-\varepsilon)),$
(15)

and the dark port fraction is obtained

$p_{\mathrm{sim}}(-|\varepsilon)=\frac{c_{-}(\varepsilon)}{c_{+}(\varepsilon)+c_{-}(\varepsilon)}.$
(16)

At this point, Fisher information can be calculated in the same way as from
experimental data by first fitting a quadratic function to
$p_{\mathrm{sim}}(-|\varepsilon)$.

##  Appendix B Direct imaging

For Gaussian pulses, Fisher information for direct imaging
$\mathcal{F}_{\mathrm{DI}}$ can be evaluated analytically and reads
$\varepsilon^{2}/8$. However, a Gaussian does not describe well our
experimental spectrum. We note that minute changes to the spectrum shape can
affect $\mathcal{F}_{\mathrm{DI}}$ drastically. This stems from an unlimited
temporal aperture (or equivalently spectral resolution) assumed in our
analysis for DI.

For a fair comparison, we use the experimentally measured spectrum and
calculate the Fisher information numerically. The procedure is as follows:

  * •

The spectrum has its background removed by fitting a mixture of a Gaussian and
linear functions, and subtracting the latter’s best fit.

  * •

Two copies of the spectrum are added together, each shifted by
$\pm\varepsilon/2$ with opposite signs.

  * •

Such a signal is normalized to a unit integral. Let us denote it by
$\rho(f|\varepsilon)$.

Photo counts density for each frequency $f$ will follow a Poisson distribution
with a mean $N\rho(f|\varepsilon)$, where $N$ is the total number of photons
and $\int\differential{f}\rho(f|\varepsilon)=1$. Fisher information per photon
for estimating $\varepsilon$ will be given by

$\mathcal{F}_{\mathrm{DI}}=\int\differential{f}\mathcal{R}_{\mathrm{DI}}(f),$
(17)

where the Fisher information density reads

$\mathcal{R}_{\mathrm{DI}}(f)=\frac{\left(\partial_{\varepsilon}\rho(f|\varepsilon)\right)^{2}}{\rho(f|\varepsilon)}.$
(18)

We evaluate this expression numerically and integrate
$\mathcal{R}_{\mathrm{DI}}$ to $\mathcal{F}_{\mathrm{DI}}$. The Fisher
information density has been depicted in Fig. 4. A quadratic fit gives
$\mathcal{F}_{\mathrm{DI}}=0.24\times\varepsilon^{2}$.

Figure 4:  Numerically calculated Fisher information spectral density for
$\varepsilon$ estimation with direct imaging.

##  Appendix C Setup details and calibration

###  C.1 Pulse preparation

Signal preparation begins with spectrally filtering $100\text{\,}\mathrm{fs}$
pulses at $798\text{\,}\mathrm{nm}$ from a Ti:Sapphire laser (Spectra-Physics
Mai Tai) with an average power of ca. $4\text{\,}\mathrm{W}$ and an
$80\text{\,}\mathrm{MHz}$ repetition rate. The filtering setup, depicted in
Fig. 3 (g), comprises a diffraction grating
($1800\text{\,}\mathrm{l}\mathrm{i}\mathrm{n}\mathrm{e}\mathrm{s}\mathrm{/}\mathrm{m}\mathrm{m}$)
far-field imaged onto a rectangular slit with a mirror placed just after the
slit, in its near field. The slit has a manually adjustable width and is
mounted on a motorized stage controlling the lateral position. The slit width
controls the signal bandwidth and the stage movement changes the center
frequency of the pulse. The reflected beam returns to the grating at a
slightly different height and leaves through a D-shaped mirror.

For the main experiment, the pulse is set to ca. $85\text{\,}\mathrm{GHz}$
full width at half maximum.

###  C.2 Interferometer stabilization

During the stabilization period [cf. Fig. 3 (c)] time lenses are switched off
with an acousto-optical modulator (AOM) deflecting off the optical driving
signal (K drv). In turn, a continuous-wave (CW) laser beam is switched on (via
an AOM) and probes the phase difference between the interferometer arms.

As depicted in Fig. 3 (a), (f), (j) the CW beam enters the main interferometer
together with the signal light. The CW laser (Toptica DL pro 780) matches the
central frequency of the pulse stretchers and signal pulses ($\varepsilon=0$).
The CW beam is combined with the signal on a non-polarizing beamsplitter.
Circular polarization of the CW and a diagonal polarization of the signal
ensures a $90\text{\,}\mathrm{\SIUnitSymbolDegree}$ phase delay between these
fields. This way, neglecting non-ideal visibility, when the CW light exits
through both output ports with equal intensity, the signal will leave only
through a single one (the bright port). Hence, we can stabilize the CW
intensity difference at the output ports with an unambiguous error signal and
high sensitivity. Equivalently, the CW probe is kept at the highest slope of
the error signal versus phase, while the signal stays at its extremum.

The CW laser is highly attenuated so that the output ports can be observed
with the same superconducting single-photon detectors (ID Quantique) as used
for the measurement period. As depicted in Fig. 3 (b), photo counts are
timestamped by a time-to-digital converter (ID Quantique ID900) which includes
a triggered generator (monostable multivibrator) outputting a single pulse of
constant width ($50\text{\,}\mathrm{ns}$) for each registered photo count.
These pulses are then processed in an FPGA-based (RedPitaya + custom system)
proportional-integral controller (PID).

Pulses are digitized and low-pass filtered. Their difference constitutes the
error signal. Through an amplifier, the output of the PID drives a mirror
mounted on a piezo-electric actuator [c.f. Fig. 3(f)].

Fine-tuning of the PID setpoint can correct a small efficiency imbalance in
the interferometer arms, slightly mismatched polarization of the CW and signal
beams, and imperfect splitting of the NPBS.

During the measurement period, the PID is stopped and holds the last output
value. During normal operation, the integral part will sometimes saturate and
automatically reset. Such a behavior is required since the stabilized quantity
– the optical phase difference is periodic.

Since only AOMs have to switch (which takes at most a few
$\text{\,}\mathrm{\SIUnitSymbolMicro s}$) between the measurement and
stabilization periods, we can use short measurement windows
($200\text{\,}\mathrm{\SIUnitSymbolMicro s}$). Which is a requirement with a
large, quickly fluctuating interferometer.

The experimental sequence is controlled by a sequencer implemented in the FPGA
and programmed from the PC.

###  C.3 Fourier transformer calibration

An ideal Fourier transformer requires a matched GDD of both pulse stretchers.
We cross-calibrated 4 pulse stretchers – 2 positive-dispersion for direct
Fourier transform and 2 negative-dispersion for the inverse. Each positive-
dispersion stretcher was put in series with each negative-dispersion stretcher
in a single arm of a Mach-Zehnder-type interferometer. The other arm consisted
of a regulated delay. A test pulse with a spectral width of a few hundred GHz
(maximal possible with our 4f pulse shaper) was sent into both arms. With a
spectrometer, we observed fringes and adjusted the delay to cancel a linear
component in the spectral phase difference between the arms. One of the
stretchers $\Phi_{2}$ was then adjusted by moving a platform with the grating,
depicted in Fig. 3 (i).

The second requirement $K=1/\Phi_{2}$ was accomplished by adjusting the time
lens chirp rate $K$ via a programmable attenuator controlling the amplitude of
the EOM driving signal. The time lens details are depicted in Fig. 3 (h). A
pair of temporally separated coherent pulses (produced with an auxiliary
interferometer with a delay line) was sent through the Fourier transformer
under calibration. The output was observed with a spectrometer. With $K=0$
spectral fringes can be observed. As $K$ is increased the pulses spectrally
separate. $K$ was set such that the observed separation matched a
theoretically calculated one for $K=1/\Phi_{2}$.

In turn, $\Phi_{2}$ measurements were carried with the stationary phase point
method [57]. Its implementation is described in detail in ref. [51].

###  C.4 Single-photon histograms

The TDC implements a histogramming module which is triggered from the FPGA
sequencer. Histograms are set to have $1\text{\,}\mathrm{\SIUnitSymbolMicro
s}$ wide bins and around $3.5\text{\,}\mathrm{ms}$ are collected in each shot.
The sequence is repeated at ca. $100\text{\,}\mathrm{Hz}$. PC controls the
collection and storage of histograms, settings of the FPGA sequencer, and
position of the stepper motor corresponding to $\varepsilon$. The latter via a
custom ARM-based controller (stm32f103c8t6).

## References

  * [1] Rayleigh, “XXXI. Investigations in optics, with special reference to the spectroscope,” The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 8, 261–274 (1879).
  * [2] K. D. Moller and W. G. Rothschild, _Far-Infrared Spectroscopy_ (John Wiley & Sons, 1971).
  * [3] E. Bettens, D. Van Dyck, A. den Dekker, _et al._ , “Model-based two-object resolution from observations having counting statistics,” Ultramicroscopy 77, 37–48 (1999).
  * [4] M. Tsang, R. Nair, and X.-M. Lu, “Quantum Theory of Superresolution for Two Incoherent Optical Point Sources,” Phys. Rev. X 6, 031033 (2016).
  * [5] M. Paúr, B. Stoklasa, Z. Hradil, _et al._ , “Achieving the ultimate optical resolution,” Optica 3, 1144–1147 (2016).
  * [6] F. Yang, A. Tashchilina, E. S. Moiseev, _et al._ , “Far-field linear optical superresolution via heterodyne detection in a higher-order local oscillator mode,” Optica 3, 1148–1152 (2016).
  * [7] R. Nair and M. Tsang, “Interferometric superlocalization of two incoherent optical point sources,” Optics Express 24, 3684–3701 (2016).
  * [8] Z. S. Tang, K. Durak, and A. Ling, “Fault-tolerant and finite-error localization for point emitters within the diffraction limit,” Opt. Express 24, 22004–22012 (2016).
  * [9] W.-K. Tham, H. Ferretti, and A. M. Steinberg, “Beating Rayleigh’s Curse by Imaging Using Phase Information,” Phys. Rev. Lett. 118, 070801 (2017).
  * [10] M. Parniak, S. Borówka, K. Boroszko, _et al._ , “Beating the Rayleigh Limit Using Two-Photon Interference,” Phys. Rev. Lett. 121, 250503 (2018).
  * [11] W. Larson, N. V. Tabiryan, and B. E. Saleh, “A common-path polarization-based image-inversion interferometer,” Optics Express 27, 5685–5695 (2019).
  * [12] C. Datta, M. Jarzyna, Y. L. Len, _et al._ , “Sub-Rayleigh resolution of two incoherent sources by array homodyning,” Phys. Rev. A 102, 063526 (2020).
  * [13] J. Frank, A. Duplinskiy, K. Bearne, and A. I. Lvovsky, “Passive superresolution imaging of incoherent objects,” Optica 10, 1147–1152 (2023).
  * [14] A. N. Jordan and J. C. Howell, “Fundamental Limits on Subwavelength Range Resolution,” Phys. Rev. Appl. 20, 064046 (2023).
  * [15] D. Triggiani, G. Psaroudis, and V. Tamma, “Ultimate Quantum Sensitivity in the Estimation of the Delay between two Interfering Photons through Frequency-Resolving Sampling,” Phys. Rev. Appl. 19, 044068 (2023).
  * [16] M. Paúr, B. Stoklasa, J. Grover, _et al._ , “Tempering Rayleigh’s curse with PSF shaping,” Optica 5, 1177–1180 (2018).
  * [17] J. Řeháček, M. Paúr, B. Stoklasa, _et al._ , “Intensity-Based Axial Localization at the Quantum Limit,” Phys. Rev. Lett. 123, 193601 (2019).
  * [18] P. Boucher, C. Fabre, G. Labroille, and N. Treps, “Spatial optical mode demultiplexing as a practical tool for optimal transverse distance estimation,” Optica 7, 1621–1626 (2020).
  * [19] S. De, J. Gil-Lopez, B. Brecht, _et al._ , “Effects of coherence on temporal resolution,” Phys. Rev. Res. 3, 033082 (2021).
  * [20] U. Zanforlin, C. Lupo, P. W. R. Connolly, _et al._ , “Optical quantum super-resolution imaging and hypothesis testing,” Nature Communications 13, 5373 (2022).
  * [21] J. Frank, A. Duplinskiy, K. Bearne, and A. I. Lvovsky, “Passive superresolution imaging of incoherent objects,” Optica 10, 1147–1152 (2023).
  * [22] S. A. Wadood, K. R. Sethuraj, K. Liang, _et al._ , “Experimental demonstration of quantum-inspired optical symmetric hypothesis testing,” Opt. Lett. 49, 750–753 (2024).
  * [23] A. Sajjad, M. R. Grace, Q. Zhuang, and S. Guha, “Attaining quantum limited precision of localizing an object in passive imaging,” Phys. Rev. A 104, 022410 (2021).
  * [24] G. Sorelli, M. Gessner, M. Walschaers, and N. Treps, “Optimal Observables and Estimators for Practical Superresolution Imaging,” Phys. Rev. Lett. 127, 123604 (2021).
  * [25] Y. Wang and V. O. Lorenz, “Fundamental limit of bandwidth-extrapolation-based superresolution,” Phys. Rev. A 108, 012602 (2023).
  * [26] V. Torres-Company, J. Lancis, and P. Andrés, “Chapter 1 - Space-Time Analogies in Optics,” (Elsevier, 2011), pp. 1–80.
  * [27] L. Rudnicki and T. Linowski, “Spectral and temporal metrology with bandlimited functions and finite-time measurements,” arXiv preprint arXiv:2402.12995 (2024).
  * [28] J. M. Donohue, V. Ansari, J. Řeháček, _et al._ , “Quantum-Limited Time-Frequency Estimation through Mode-Selective Photon Measurement,” Phys. Rev. Lett. 121, 090501 (2018).
  * [29] V. Ansari, B. Brecht, J. Gil-Lopez, _et al._ , “Achieving the Ultimate Quantum Timing Resolution,” PRX Quantum 2, 010301 (2021).
  * [30] W. Krokosz, M. Mazelanik, M. Lipka, _et al._ , “Beating the spectroscopic Rayleigh limit via post-processed heterodyne detection,” Opt. Lett. 49, 1001–1004 (2024).
  * [31] M. Mazelanik, A. Leszczyński, and M. Parniak, “Optical-domain spectral super-resolution via a quantum-memory-based time-frequency processor,” Nature Communications 13, 691 (2022).
  * [32] A. Bose, I. Thomas, and E. Abraham, “Fluorescence spectroscopy and its applications: A Review,” Int. J. Adv. Pharm. Res 8, 1–8 (2018).
  * [33] A. Boschetti, A. Taschin, P. Bartolini, _et al._ , “Spectral super-resolution spectroscopy using a random laser,” Nature Photonics 14, 177–182 (2020).
  * [34] S. Mukamel, M. Freyberger, W. Schleich, _et al._ , “Roadmap on quantum light spectroscopy,” J. Phys. B: At. Mol. Opt. Phys. 53, 072002 (2020).
  * [35] M. Lipka and M. Parniak, “Single-Photon Hologram of a Zero-Area Pulse,” Phys. Rev. Lett. 127, 163601 (2021).
  * [36] K. Hashimoto, D. B. Horoshko, M. I. Kolobov, _et al._ , “Fourier-transform infrared spectroscopy with undetected photons from high-gain spontaneous parametric down-conversion,” arXiv preprint arXiv:2403.05423 (2024).
  * [37] F. Grenapin, D. Paneru, A. D’Errico, _et al._ , “Superresolution Enhancement in Biphoton Spatial-Mode Demultiplexing,” Phys. Rev. Appl. 20, 024077 (2023).
  * [38] K. Wicker, S. Sindbert, and R. Heintzmann, “Characterisation of a resolution enhancing image inversion interferometer,” Opt. Express 17, 15491–15501 (2009).
  * [39] D. B. Horoshko and M. I. Kolobov, “Interferometric sorting of temporal Hermite-Gauss modes via temporal Gouy phase,” arXiv preprint arXiv:2310.11918 (2023).
  * [40] L. Serino, J. Gil-Lopez, M. Stefszky, _et al._ , “Realization of a multi-output quantum pulse gate for decoding high-dimensional temporal modes of single-photon states,” PRX quantum 4, 020306 (2023).
  * [41] S. M. Kay, _Fundamentals of Statistical Signal Processing: Estimation Theory_ (Prenitce Hall, 1993).
  * [42] Z. Huang, C. Schwab, and C. Lupo, “Ultimate limits of exoplanet spectroscopy: A quantum approach,” Phys. Rev. A 107, 022409 (2023).
  * [43] M. Tsang, “Resolving starlight: a quantum perspective,” Contemporary Physics 60, 279–298 (2019).
  * [44] L. S. Amato, F. Sgobba, and C. Lupo, “Single-photon sub-Rayleigh precision measurements of a pair of incoherent sources of unequal intensity,” arXiv preprint arXiv:2309.02295 (2023).
  * [45] L. Santamaria, D. Pallotti, M. S. de Cumis, _et al._ , “Spatial-mode demultiplexing for enhanced intensity and distance measurement,” Opt. Express 31, 33930–33944 (2023).
  * [46] A. B. Greenwood, R. Oulton, and H. Gersen, “On the impact of realistic point sources in spatial mode demultiplexing super resolution imaging,” Quantum Science and Technology 8, 015024 (2023).
  * [47] C. Hu, L. Xu, B. Wang, _et al._ , “Experimental 3D super-localization with Laguerre–Gaussian modes,” Quantum Frontiers 2, 20 (2023).
  * [48] M. R. Grace, Z. Dutton, A. Ashok, and S. Guha, “Approaching quantum-limited imaging resolution without prior knowledge of the object location,” JOSA A 37, 1288–1299 (2020).
  * [49] C. Oh, S. Zhou, Y. Wong, and L. Jiang, “Quantum limits of superresolution in a noisy environment,” Physical Review Letters 126, 120502 (2021).
  * [50] K. Schlichtholz, T. Linowski, M. Walschaers, _et al._ , “Practical tests for sub-Rayleigh source discriminations with imperfect demultiplexers,” Optica Quantum 2, 29–34 (2024).
  * [51] M. Lipka and M. Parniak, “Ultrafast electro-optic time-frequency fractional Fourier imaging at the single-photon level,” Opt. Express 32, 9573–9588 (2024).
  * [52] B. Brecht, D. V. Reddy, C. Silberhorn, and M. G. Raymer, “Photon Temporal Modes: A Complete Framework for Quantum Information Science,” Phys. Rev. X 5, 041017 (2015).
  * [53] M. Shah and L. Fan, “Frequency Superresolution with Spectrotemporal Shaping of Photons,” Physical Review Applied 15, 034071 (2021).
  * [54] M. Lipka and M. Parniak, “Fast imaging of multimode transverse–spectral correlations for twin photons,” Opt. Lett. 46, 3009–3012 (2021).
  * [55] M. Jastrzębski, S. Kurzyna, B. Niewelt, _et al._ , “Spectrum-to-position mapping via programmable spatial dispersion implemented in an optical quantum memory,” Phys. Rev. A 109, 012418 (2024).
  * [56] M. Lipka and M. Parniak, “Data for: Super-resolution of ultrafast pulses via spectral inversion,” Harvard Dataverse, https://doi.org/10.7910/DVN/ETXDRO (2024).
  * [57] C. Sainz, P. Jourdian, R. Escalona, and J. Calatroni, “Real time interferometric measurements of dispersion curves,” Opt. Commun. 110, 381–390 (1994).