# Optical-domain spectral super-resolution via a quantum-memory-based time-
frequency processor

Mateusz Mazelanik<EMAIL_ADDRESS>Centre for Quantum Optical
Technologies, Centre of New Technologies, University of Warsaw, Banacha 2c,
02-097 Warsaw, Poland  Faculty of Physics, University of Warsaw, Pasteura 5,
02-093 Warsaw, Poland  Adam Leszczyński  Centre for Quantum Optical
Technologies, Centre of New Technologies, University of Warsaw, Banacha 2c,
02-097 Warsaw, Poland  Faculty of Physics, University of Warsaw, Pasteura 5,
02-093 Warsaw, Poland  Michał Parniak<EMAIL_ADDRESS>Centre for
Quantum Optical Technologies, Centre of New Technologies, University of
Warsaw, Banacha 2c, 02-097 Warsaw, Poland  Niels Bohr Institute, Universtiy of
Copenhagen, Blegdamsvej 17, 2100 Copenhagen, Denmark

###### Abstract

Existing super-resolution methods of optical imaging hold a solid place as an
application in natural sciences, but many new developments allow for beating
the diffraction limit in a more subtle way. One of the recently explored
strategies to fully exploit information already present in the field is to
perform a quantum-inspired tailored measurements. Here we exploit the full
spectral information of the optical field in order to beat the Rayleigh limit
in spectroscopy. We employ an optical quantum memory with spin-wave storage
and an embedded processing capability to implement a time–inversion
interferometer for input light, projecting the optical field in the
symmetric–antisymmetric mode basis. Our tailored measurement achieves a
resolution of 15 kHz and requires 20 times less photons than a corresponding
Rayleigh-limited conventional method. We demonstrate the advantage of our
technique over both conventional spectroscopy and heterodyne measurements,
showing potential for application in distinguishing ultra-narrowband emitters,
optical communication channels, or signals transduced from lower-frequency
domains.

Optical spectroscopy is an indispensable tool in the studies of matter and
light, including chemistry [1, 2], astronomy [3], biology and medicine [4],
metrology, and more. Yet, the resolution of all state-of-the-art methods, such
as grating-based [5, 6] and Fourier spectrometers [7, 8, 9], is subject to the
Fourier limit. Methods of beating analogous Rayleigh limit are widely known in
the context of imaging and include modifying, or exploiting very specific
properties of the source or illumination [10, 11, 12, 13], which is often
impossible to implement. Furthermore, even though the Rayleigh limit has been
originally formulated in the context of a spectroscope [14], super-resolution
methods of spectroscopy are scarce and limited to laser spectroscopy [15]. The
task and challenges of fluorescence spectroscopy are starkly different: a
typical emitter provides only a small photon number per single spectro-spatial
mode, which is a strong incentive to seek quantum-enhanced protocols.

Figure 1: The idea for time-inverting interferometer for superresolving
spectroscopy. The task is to measure a separation between two spectral lines.
Separation is smaller than the linewidths, which are determined by the
properties of the source itself or by the temporal aperture of the
spectroscopic measurement device. As shown in the inset, photon shot noise
(stochastic counts) prevent accurate estimate of separation when to two lines
overlap strongly. For small separations, however, the information about the
separation is all contained in the antisymmetric part of the signal pulse. To
extract this information, the signal pulse is sent to the pulse-division time-
axis-inversion (PuDTAI) interferometer that performs decomposition onto
symmetric and antisymmetric components. First, a time-dependent-transmission
mirror splits the pulse in half between the two interferometer arms. One of
the arms includes a time inversion device that mirrors the first half of the
pulse. Finally, the two components are combined on a beamsplitter, and the
resulting symmetric ($+$) and antisymmetric ($-$) projections are detected.

In imaging, the modern understanding of the Rayleigh limit is formulated as a
vanishing information about separation distance between two sources for small
separations, termed the “Rayleigh curse” [16]. Tsang et al. [16] have noticed
that the quantum Cramér-Rao bound (Q-CRB), which identifies maximum
information available in the optical field, is not saturated by traditional
methods. This work inspired new methods that allow for reaching the quantum
limit [17, 18, 19, 20, 21, 22, 23], most recently also in the time domain [24,
25]. The idea to apply an analogous protocol for spectroscopy faces
challenges, especially if operation with narrowband optical signal is desired.
Yet, the inspiration can be drawn from another domain: in the nuclear-magnetic
resonance (NMR) spectroscopy a quantum memory can lead to increased resolution
when sensing with color centers in diamond [26, 27, 28]. A general framework
for those experiments has been formulated by Gefen et al. [29] who identified
specific quantum measurements that facilitate super-resolution. A resulting
idea is therefore to use a quantum memory to achieve spectral super-resolution
in the optical domain.

Here we bring the quantum-inspired super-resolution methods to the spectral
domain and demonstrate a device that can resolve frequency differences of two
emitters with precision below the Fourier limit. Here, by analogy to modern
works on imaging, we understand this limit as a vanishing information about
frequency separation between two spectral lines. Our method utilizes a
Gradient Echo Memory (GEM) with built-in time-frequency processing
capabilities which we program to realize a pulse-division time-axis-inversion
(PuDTAI) interferometer. Our protocol operates in the ultra-narrowband domain,
achieving a resolution of 15 kHz with simultaneous super-resolution
enhancement factor of $\mathfrak{s}=20\pm 0.5$ which means that about 20 times
less photons are needed to achieve the same resolution as direct spectroscopy
under the same experimental conditions. Our work not only establishes a new
super-resolution spectroscopy method, but also provides an high spectral
resolution in absolute terms. Fundamentally, this super-resolving method
exploits the spectral information already present in light, not requiring
specific properties of source or illumination. Such performance is achieved by
employing a spin-wave quantum memory to fully extract both phase and amplitude
from the optical field. This is enabled by the the long coherence time of the
memory that allow us to capture, process and release the light thus allowing
optimal detection.

## Results

### Information in spectral resolution

Let us first set the stage of the problem and introduce the theoretical
framework to evaluate the super-resolution enhancement. Two mutually
incoherent sources with equal brightness emit photons with the same spectral
envelope $\tilde{\psi}(\omega)$ that is assumed to be known, and is the
Fourier transform of the time-domain envelope $\psi(t)$. Such scenario is
relevant both for the cases of fluorescence (with photon pulses being
determined by spontaneous emission, assuming fast excitation and no
inhomogenous broadening) and scattering (such as Raman scattering, where we
may need to consider relative phases of ground-state coherences). It is also
an optical analogue of the radio-frequency-domain physics present in spectrum
analyzers and nano-NMR [29]. A more conventional, but equally relevant
scenario is when the linewidths of two sources are very narrow, and the
resolution is limited by an aperature, which sets the transfer function
$\tilde{\psi}(\omega)$.

The two sources in question have slightly different central frequencies
$\omega_{-}=\omega_{0}-\delta\omega/2$ and
$\omega_{+}=\omega_{0}+\delta\omega/2$, where the frequency separation
$\delta\omega=\sigma\varepsilon$ is smaller than the width $\sigma$ of the
spectral envelope $\tilde{\psi}(\omega)$ that for the normalized separation
parameter $\varepsilon$ translates to $\varepsilon\ll 1$. The spectroscopist’s
task is to estimate the separation $\varepsilon$ with maximum efficacy - to
obtain maximum information about the separation per collected photon.

Figure 2: Experimental setup and sequence. a The Gradient Echo Memory (GEM)
device enhanced with the spatial AC-Stark modulation, used to store, process
and release the signal pulses. The same setup is used to implement the simple
GEM, the temporal imaging (TI) and the PuDTAI protocol, for which the pulses
are shown here. In the PuDTAI interferometer implementation a symmetrically-
chirped control field maps the signal pulse onto atomic ground-state coherence
of 87Rb atoms and simultaneously implements the time-dependent-transmission
mirror and time-inversion transformation. AC-Stark phase modulation with
chirped square wave gratings (AC-S Gratings) is used to implement the
beamsplitter operation and sequentially read-out the resulting interferometer
ports ($\mathrm{Out}_{-}$, $\mathrm{Out}_{+}$). b Experimental sequences and
AC-Stark beam patterns (G0-G2) used to implement the basic (pass through) GEM
protocol, TI spectrometer and the PuDTAI interferometer. The full sequence for
PuDTAI protocol and relevant atomic levels are shown in Figure 7 in the
Methods section.

In the conventional approach a spectrometer that measures the normalized
spectral intensity
$\tilde{I}(\omega)=\frac{1}{2}\left(|\tilde{\psi}(\omega-\delta\omega/2)|^{2}+|\tilde{\psi}_{-}(\omega+\delta\omega/2)|^{2}\right)$
is used to collect data that is then processed to estimate $\delta\omega$, for
example by fitting the theoretical curve. Similarly, a Fourier spectrometer
measures the second-order autocorrelation function, which is then Fourier-
transformed to yield $\tilde{I}(\omega)$. Notably, from the informational
point of view one may obtain precision of $\delta\omega$ much greater than
$\sigma$, simply by collecting enough statistics. Therefore, to quantify the
metrological advantage, we need to consider photon shot noise resulting from
finite statistics, for which the typical asymptotic behaviour is
$\Delta^{2}\hat{\varepsilon}\propto 1/N$ with $N$ being the number of
collected photons. This behaviour is captured by the Cramér-Rao bound, which
limits the maximum achievable precision for any locally unbiased estimator of
the normalized separation $\hat{\varepsilon}$ [30]:

$\Delta^{2}\hat{\varepsilon}\geq\frac{1}{N\mathcal{F}_{\varepsilon}},\mathcal{F}_{\varepsilon}=\int\frac{1}{p_{\varepsilon}(x)}\left(\frac{\partial}{\partial\varepsilon}p_{\varepsilon}(x)\right)^{2}\mathrm{d}x,$
(1)

where $\Delta^{2}\hat{\varepsilon}$ is the variance of the estimator,
$\mathcal{F}_{\varepsilon}$ is the Fisher information, $N$ is the number of
independent single photons used, and $p_{\varepsilon}(x)$ represents the
measurement outcome distribution parametrized by the true separation value
$\varepsilon$. For Gaussian envelopes

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(2)

the Fisher information for DI in case of small separations $\varepsilon\ll 1$
can be approximated as $\mathcal{F}_{\mathrm{DI}}\approx\varepsilon^{2}/8$.
This unfavourable scaling has been termed the Rayleigh curse, therefore we
consider DI to be a Rayleigh-limited method. Physically, this reflects the
fact that it is extremely hard to estimate the separation of two noisy,
overlapping spectral lines as shown in the inset in Fig. 1. With more
statistics, one may be able to do it, but the per-photon information
$\mathcal{F}_{\mathrm{DI}}$ is small. At the same time the quantum Fisher
information (QFI) [31] that yields a measurement-strategy-independent
precision bound for given estimation task turns out to have constant value
$\mathcal{F}_{\mathrm{Q}}=1/4$, independent of $\varepsilon$. Such analysis
leads to the conclusion that the direct spectroscopy scenario, which is an
analogy for real space direct imaging (DI) of two incoherent sources, is not
optimal [16]. Hence, any strategy with Fisher information above
$\mathcal{F}_{\mathrm{DI}}$ can be considered a super-resolving method beating
the Rayleigh limit, or equivalently the Fourier limit in spectroscopy.

These observations have led to extensive search for different measurement
schemes that will approach the ultimate bound given by Q-CRB and will not
suffer from Rayleigh curse. Most of those super-resolution measurement schemes
are based on the following observation: a shifted Gaussian mode function
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}\definecolor[named]{pgffillcolor}{rgb}{1,0,0}\pgfsys@moveto{-14.22638pt}{0.0pt}\pgfsys@lineto{-14.22638pt}{0.01563pt}\pgfsys@lineto{-13.0407pt}{0.0521pt}\pgfsys@lineto{-11.85503pt}{0.15543pt}\pgfsys@lineto{-10.66934pt}{0.41895pt}\pgfsys@lineto{-9.48367pt}{1.01549pt}\pgfsys@lineto{-8.29799pt}{2.2181pt}\pgfsys@lineto{-7.11232pt}{4.36542pt}\pgfsys@lineto{-5.92664pt}{7.74141pt}\pgfsys@lineto{-4.74097pt}{12.36993pt}\pgfsys@lineto{-3.55528pt}{17.81206pt}\pgfsys@lineto{-2.36961pt}{23.10788pt}\pgfsys@lineto{-1.18393pt}{27.01266pt}\pgfsys@lineto{0.00172pt}{28.45276pt}\pgfsys@lineto{1.18741pt}{27.00441pt}\pgfsys@lineto{2.37308pt}{23.09442pt}\pgfsys@lineto{3.55876pt}{17.7947pt}\pgfsys@lineto{4.74443pt}{12.35516pt}\pgfsys@lineto{5.93011pt}{7.72968pt}\pgfsys@lineto{7.11578pt}{4.3576pt}\pgfsys@lineto{8.30147pt}{2.21332pt}\pgfsys@lineto{9.48714pt}{1.01288pt}\pgfsys@lineto{10.67282pt}{0.41765pt}\pgfsys@lineto{11.85849pt}{0.15543pt}\pgfsys@lineto{13.04417pt}{0.0521pt}\pgfsys@lineto{14.22638pt}{0.0pt}\pgfsys@closepath\pgfsys@fill\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{}{}{}{}{}{}{}
{}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{}
{}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{}
{}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{}
{}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{}
{}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{}
{}{{}{}{}}{}{}{} {}{{}{}{}} {}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{1.42264pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.22638pt}{0.01563pt}\pgfsys@curveto{-14.22638pt}{0.01563pt}{-13.3697pt}{0.0327pt}{-13.0407pt}{0.0521pt}\pgfsys@curveto{-12.71169pt}{0.07149pt}{-12.18404pt}{0.10454pt}{-11.85503pt}{0.15543pt}\pgfsys@curveto{-11.52602pt}{0.20631pt}{-10.99835pt}{0.29962pt}{-10.66934pt}{0.41895pt}\pgfsys@curveto{-10.34033pt}{0.53827pt}{-9.81268pt}{0.76587pt}{-9.48367pt}{1.01549pt}\pgfsys@curveto{-9.15466pt}{1.2651pt}{-8.627pt}{1.75331pt}{-8.29799pt}{2.2181pt}\pgfsys@curveto{-7.96898pt}{2.68288pt}{-7.44133pt}{3.59908pt}{-7.11232pt}{4.36542pt}\pgfsys@curveto{-6.78331pt}{5.13176pt}{-6.25565pt}{6.6308pt}{-5.92664pt}{7.74141pt}\pgfsys@curveto{-5.59763pt}{8.85202pt}{-5.06998pt}{10.97266pt}{-4.74097pt}{12.36993pt}\pgfsys@curveto{-4.41196pt}{13.76721pt}{-3.8843pt}{16.32219pt}{-3.55528pt}{17.81206pt}\pgfsys@curveto{-3.22627pt}{19.30193pt}{-2.69862pt}{21.83131pt}{-2.36961pt}{23.10788pt}\pgfsys@curveto{-2.0406pt}{24.38445pt}{-1.51294pt}{26.27109pt}{-1.18393pt}{27.01266pt}\pgfsys@curveto{-0.85492pt}{27.75424pt}{-0.32729pt}{28.4539pt}{0.00172pt}{28.45276pt}\pgfsys@curveto{0.33073pt}{28.45161pt}{0.8584pt}{27.74786pt}{1.18741pt}{27.00441pt}\pgfsys@curveto{1.51642pt}{26.26096pt}{2.04407pt}{24.37224pt}{2.37308pt}{23.09442pt}\pgfsys@curveto{2.70209pt}{21.8166pt}{3.22975pt}{19.28474pt}{3.55876pt}{17.7947pt}\pgfsys@curveto{3.88777pt}{16.30464pt}{4.41542pt}{13.75166pt}{4.74443pt}{12.35516pt}\pgfsys@curveto{5.07344pt}{10.95866pt}{5.6011pt}{8.83931pt}{5.93011pt}{7.72968pt}\pgfsys@curveto{6.25912pt}{6.62004pt}{6.78677pt}{5.12299pt}{7.11578pt}{4.3576pt}\pgfsys@curveto{7.4448pt}{3.59222pt}{7.97246pt}{2.67738pt}{8.30147pt}{2.21332pt}\pgfsys@curveto{8.63048pt}{1.74925pt}{9.15813pt}{1.26202pt}{9.48714pt}{1.01288pt}\pgfsys@curveto{9.81615pt}{0.76373pt}{10.34381pt}{0.5366pt}{10.67282pt}{0.41765pt}\pgfsys@curveto{11.00183pt}{0.29869pt}{11.52948pt}{0.20613pt}{11.85849pt}{0.15543pt}\pgfsys@curveto{12.1875pt}{0.10472pt}{13.04417pt}{0.0521pt}{13.04417pt}{0.0521pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{p}}(\omega)$ with a
small correction of the first Hermite-Gaussian function:

$\displaystyle\tilde{\psi}_{\scalerel*{\leavevmode\hbox to29.16pt{\vbox
to29.86pt{\pgfpicture\makeatletter\hbox{\hskip 14.9377pt\lower-0.6957pt\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}
{} {}{} {}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{}{}{}{}{}{}{}
{}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{}
{}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{}
{}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{}
{}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{}
{}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{}
{}{{}{}{}}{}{}{} {}{{}{}{}} {}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{0,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{0,0,0}\pgfsys@color@gray@stroke{0}\pgfsys@invoke{
}\pgfsys@color@gray@fill{0}\pgfsys@invoke{
}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{1.42264pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.22638pt}{0.01563pt}\pgfsys@curveto{-14.22638pt}{0.01563pt}{-13.3697pt}{0.0327pt}{-13.0407pt}{0.0521pt}\pgfsys@curveto{-12.71169pt}{0.07149pt}{-12.18404pt}{0.10454pt}{-11.85503pt}{0.15543pt}\pgfsys@curveto{-11.52602pt}{0.20631pt}{-10.99835pt}{0.29962pt}{-10.66934pt}{0.41895pt}\pgfsys@curveto{-10.34033pt}{0.53827pt}{-9.81268pt}{0.76587pt}{-9.48367pt}{1.01549pt}\pgfsys@curveto{-9.15466pt}{1.2651pt}{-8.627pt}{1.75331pt}{-8.29799pt}{2.2181pt}\pgfsys@curveto{-7.96898pt}{2.68288pt}{-7.44133pt}{3.59908pt}{-7.11232pt}{4.36542pt}\pgfsys@curveto{-6.78331pt}{5.13176pt}{-6.25565pt}{6.6308pt}{-5.92664pt}{7.74141pt}\pgfsys@curveto{-5.59763pt}{8.85202pt}{-5.06998pt}{10.97266pt}{-4.74097pt}{12.36993pt}\pgfsys@curveto{-4.41196pt}{13.76721pt}{-3.8843pt}{16.32219pt}{-3.55528pt}{17.81206pt}\pgfsys@curveto{-3.22627pt}{19.30193pt}{-2.69862pt}{21.83131pt}{-2.36961pt}{23.10788pt}\pgfsys@curveto{-2.0406pt}{24.38445pt}{-1.51294pt}{26.27109pt}{-1.18393pt}{27.01266pt}\pgfsys@curveto{-0.85492pt}{27.75424pt}{-0.32729pt}{28.4539pt}{0.00172pt}{28.45276pt}\pgfsys@curveto{0.33073pt}{28.45161pt}{0.8584pt}{27.74786pt}{1.18741pt}{27.00441pt}\pgfsys@curveto{1.51642pt}{26.26096pt}{2.04407pt}{24.37224pt}{2.37308pt}{23.09442pt}\pgfsys@curveto{2.70209pt}{21.8166pt}{3.22975pt}{19.28474pt}{3.55876pt}{17.7947pt}\pgfsys@curveto{3.88777pt}{16.30464pt}{4.41542pt}{13.75166pt}{4.74443pt}{12.35516pt}\pgfsys@curveto{5.07344pt}{10.95866pt}{5.6011pt}{8.83931pt}{5.93011pt}{7.72968pt}\pgfsys@curveto{6.25912pt}{6.62004pt}{6.78677pt}{5.12299pt}{7.11578pt}{4.3576pt}\pgfsys@curveto{7.4448pt}{3.59222pt}{7.97246pt}{2.67738pt}{8.30147pt}{2.21332pt}\pgfsys@curveto{8.63048pt}{1.74925pt}{9.15813pt}{1.26202pt}{9.48714pt}{1.01288pt}\pgfsys@curveto{9.81615pt}{0.76373pt}{10.34381pt}{0.5366pt}{10.67282pt}{0.41765pt}\pgfsys@curveto{11.00183pt}{0.29869pt}{11.52948pt}{0.20613pt}{11.85849pt}{0.15543pt}\pgfsys@curveto{12.1875pt}{0.10472pt}{13.04417pt}{0.0521pt}{13.04417pt}{0.0521pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{p}}(\omega\pm\varepsilon\sigma/2)\approx\tilde{\psi}_{\scalerel*{\leavevmode\hbox
to29.16pt{\vbox to29.86pt{\pgfpicture\makeatletter\hbox{\hskip
14.9377pt\lower-0.6957pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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}\pgfsys@color@rgb@fill{0}{0}{0}\pgfsys@invoke{
}\pgfsys@setlinewidth{0.4pt}\pgfsys@invoke{ }\nullfont\hbox
to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}
{} {}{} {}\pgfsys@beginscope\pgfsys@invoke{
}\color[rgb]{1,0,0}\definecolor[named]{pgfstrokecolor}{rgb}{1,0,0}\pgfsys@color@rgb@stroke{1}{0}{0}\pgfsys@invoke{
}\pgfsys@color@rgb@fill{1}{0}{0}\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{}{}{}{}{}{}{}
{}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{}
{}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{}
{}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{}
{}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{}
{}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{}
{}{{}{}{}}{}{}{} {}{{}{}{}} {}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
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}\definecolor[named]{pgffillcolor}{rgb}{0,0,0}\pgfsys@setlinewidth{1.42264pt}\pgfsys@invoke{
}{}\pgfsys@moveto{-14.22638pt}{0.01563pt}\pgfsys@curveto{-14.22638pt}{0.01563pt}{-13.3697pt}{0.0327pt}{-13.0407pt}{0.0521pt}\pgfsys@curveto{-12.71169pt}{0.07149pt}{-12.18404pt}{0.10454pt}{-11.85503pt}{0.15543pt}\pgfsys@curveto{-11.52602pt}{0.20631pt}{-10.99835pt}{0.29962pt}{-10.66934pt}{0.41895pt}\pgfsys@curveto{-10.34033pt}{0.53827pt}{-9.81268pt}{0.76587pt}{-9.48367pt}{1.01549pt}\pgfsys@curveto{-9.15466pt}{1.2651pt}{-8.627pt}{1.75331pt}{-8.29799pt}{2.2181pt}\pgfsys@curveto{-7.96898pt}{2.68288pt}{-7.44133pt}{3.59908pt}{-7.11232pt}{4.36542pt}\pgfsys@curveto{-6.78331pt}{5.13176pt}{-6.25565pt}{6.6308pt}{-5.92664pt}{7.74141pt}\pgfsys@curveto{-5.59763pt}{8.85202pt}{-5.06998pt}{10.97266pt}{-4.74097pt}{12.36993pt}\pgfsys@curveto{-4.41196pt}{13.76721pt}{-3.8843pt}{16.32219pt}{-3.55528pt}{17.81206pt}\pgfsys@curveto{-3.22627pt}{19.30193pt}{-2.69862pt}{21.83131pt}{-2.36961pt}{23.10788pt}\pgfsys@curveto{-2.0406pt}{24.38445pt}{-1.51294pt}{26.27109pt}{-1.18393pt}{27.01266pt}\pgfsys@curveto{-0.85492pt}{27.75424pt}{-0.32729pt}{28.4539pt}{0.00172pt}{28.45276pt}\pgfsys@curveto{0.33073pt}{28.45161pt}{0.8584pt}{27.74786pt}{1.18741pt}{27.00441pt}\pgfsys@curveto{1.51642pt}{26.26096pt}{2.04407pt}{24.37224pt}{2.37308pt}{23.09442pt}\pgfsys@curveto{2.70209pt}{21.8166pt}{3.22975pt}{19.28474pt}{3.55876pt}{17.7947pt}\pgfsys@curveto{3.88777pt}{16.30464pt}{4.41542pt}{13.75166pt}{4.74443pt}{12.35516pt}\pgfsys@curveto{5.07344pt}{10.95866pt}{5.6011pt}{8.83931pt}{5.93011pt}{7.72968pt}\pgfsys@curveto{6.25912pt}{6.62004pt}{6.78677pt}{5.12299pt}{7.11578pt}{4.3576pt}\pgfsys@curveto{7.4448pt}{3.59222pt}{7.97246pt}{2.67738pt}{8.30147pt}{2.21332pt}\pgfsys@curveto{8.63048pt}{1.74925pt}{9.15813pt}{1.26202pt}{9.48714pt}{1.01288pt}\pgfsys@curveto{9.81615pt}{0.76373pt}{10.34381pt}{0.5366pt}{10.67282pt}{0.41765pt}\pgfsys@curveto{11.00183pt}{0.29869pt}{11.52948pt}{0.20613pt}{11.85849pt}{0.15543pt}\pgfsys@curveto{12.1875pt}{0.10472pt}{13.04417pt}{0.0521pt}{13.04417pt}{0.0521pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
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to29.16pt{\vbox to26.01pt{\pgfpicture\makeatletter\hbox{\hskip
14.9377pt\lower-13.00546pt\hbox to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
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{} {}{} {}\pgfsys@beginscope\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
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{} {}{} {}\pgfsys@beginscope\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{}{}{}{}{}{}{}
{}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{}
{}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{}
{}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{}
{}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{}
{}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{}
{}{{}{}{}}{}{}{} {}{{}{}{}} {}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
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}{}\pgfsys@moveto{-14.22638pt}{-0.04297pt}\pgfsys@curveto{-14.22638pt}{-0.04297pt}{-13.3697pt}{-0.08731pt}{-13.0407pt}{-0.13068pt}\pgfsys@curveto{-12.71169pt}{-0.17404pt}{-12.18404pt}{-0.2543pt}{-11.85503pt}{-0.35556pt}\pgfsys@curveto{-11.52602pt}{-0.45682pt}{-10.99835pt}{-0.65262pt}{-10.66934pt}{-0.86049pt}\pgfsys@curveto{-10.34033pt}{-1.06836pt}{-9.81268pt}{-1.48163pt}{-9.48367pt}{-1.85384pt}\pgfsys@curveto{-9.15466pt}{-2.22604pt}{-8.627pt}{-2.97112pt}{-8.29799pt}{-3.54312pt}\pgfsys@curveto{-7.96898pt}{-4.11513pt}{-7.44133pt}{-5.24269pt}{-7.11232pt}{-5.97656pt}\pgfsys@curveto{-6.78331pt}{-6.71043pt}{-6.25565pt}{-8.09525pt}{-5.92664pt}{-8.83243pt}\pgfsys@curveto{-5.59763pt}{-9.56961pt}{-5.06998pt}{-10.82387pt}{-4.74097pt}{-11.28975pt}\pgfsys@curveto{-4.41196pt}{-11.75563pt}{-3.8843pt}{-12.29414pt}{-3.55528pt}{-12.19019pt}\pgfsys@curveto{-3.22627pt}{-12.08623pt}{-2.69862pt}{-11.3776pt}{-2.36961pt}{-10.5404pt}\pgfsys@curveto{-2.0406pt}{-9.70322pt}{-1.51294pt}{-7.62009pt}{-1.18393pt}{-6.15631pt}\pgfsys@curveto{-0.85492pt}{-4.69254pt}{-0.32729pt}{-1.70103pt}{0.00172pt}{0.00954pt}\pgfsys@curveto{0.33073pt}{1.72011pt}{0.8584pt}{4.70996pt}{1.18741pt}{6.17236pt}\pgfsys@curveto{1.51642pt}{7.63477pt}{2.04407pt}{9.71445pt}{2.37308pt}{10.54951pt}\pgfsys@curveto{2.70209pt}{11.38458pt}{3.22975pt}{12.08908pt}{3.55876pt}{12.19106pt}\pgfsys@curveto{3.88777pt}{12.29303pt}{4.41542pt}{11.75168pt}{4.74443pt}{11.28455pt}\pgfsys@curveto{5.07344pt}{10.81741pt}{5.6011pt}{9.56174pt}{5.93011pt}{8.82419pt}\pgfsys@curveto{6.25912pt}{8.08664pt}{6.78677pt}{6.70232pt}{7.11578pt}{5.96875pt}\pgfsys@curveto{7.4448pt}{5.23518pt}{7.97246pt}{4.10858pt}{8.30147pt}{3.53705pt}\pgfsys@curveto{8.63048pt}{2.96552pt}{9.15813pt}{2.22115pt}{9.48714pt}{1.84949pt}\pgfsys@curveto{9.81615pt}{1.47783pt}{10.34381pt}{1.06578pt}{10.67282pt}{0.85832pt}\pgfsys@curveto{11.00183pt}{0.65086pt}{11.52948pt}{0.45522pt}{11.85849pt}{0.35426pt}\pgfsys@curveto{12.1875pt}{0.25331pt}{13.04417pt}{0.13068pt}{13.04417pt}{0.13068pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{p}}(\omega)$ (3)
$\displaystyle\tilde{\psi}_{\scalerel*{\leavevmode\hbox to29.16pt{\vbox
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to0.0pt{\pgfsys@beginscope\pgfsys@invoke{
}{}{{}}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}{}
{} {}{} {}\pgfsys@beginscope\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope
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{} {}{} {}\pgfsys@beginscope\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{}{}{}{}{}{}{}
{}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{}
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{}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{}
{}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{}
{}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{} {}{{}{}{}}{}{}{}
{}{{}{}{}}{}{}{} {}{{}{}{}} {}{}{}{}\pgfsys@beginscope\pgfsys@invoke{
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{p}}(\omega)=\frac{\omega}{\sigma}\left(\sqrt{2\pi}\sigma\right)^{-1/2}\exp\left(-\frac{\omega^{2}}{4\sigma^{2}}\right).$
(4)

From this we see that only the antisymmetric component
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
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the information about the separation, thus filtering out the (orthogonal)
symmetric mode $\tilde{\psi}_{\scalerel*{\leavevmode\hbox to29.16pt{\vbox
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope{{}}{}{}{}{}{}{}{}{}
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
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to massive improvement of signal-to-noise ratio and boosts up the separation
estimation precision.

### Protocol

Figure 3: Temporal imaging in GEM. a Simulation of GEM protocol for signal
($A_{\mathrm{in}}$) with two spectral components with finite bandwidth. The
maps show evolution of the atomic coherence ($\rho_{\mathrm{hg}}$) in the
wavevector (top) and real space (bottom). The frequency-to-space and time-to-
wavevector mapping feature of GEM protocol is visible on the crossections
(taken along the dashed lines) of the coherence maps that are compared to the
temporal and spectral shape of the stored optical field. b Temporal far field
imaging using two time-lenses separated by temporal propagation. The output
signal is a Fourier transform of the input signal. The temporal aperture clips
the input signal and sets the frequency resolution limit. The inset i shows
propagation of the same input signal through similar TI setup but with a lens
with negative focal length followed by backward temporal propagation,
resulting in the inverted spectrum at the output. c Phase shapes that are
sequentially imprinted on the input signal to implement the lens-propagation-
lens TI setup. d Time-frequency and space-wavevector Wigner representation
($\mathcal{W}$) of the input signal at each stage of the TI setup. The arrows
represents the transformations of the phase-space upon each transformation
(lens or propagation).

The idea for achieving sub-Fourier performance in our spectrometer via the
PuDTAI protocol relies on engineering a measurement that can split the signal
pulse into symmetric and antisymmetric combinations with respect to the mean
time or frequency. The protocol that realizes this projective measurement is
inspired by a technique known from conventional (real space) super-resolved
imaging called SLIVER (superlocalization via image-inversion interferometry)
[32, 33, 21, 34, 22], where an image is inverted in one arm of a Mach-Zhender
interferometer. The Fisher information associated with photon detection
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extraordinary sensitivity for small ($\varepsilon\ll 1$) separation estimation
$\mathcal{F}_{\mathrm{SLIVER}}\approx\frac{1}{4}-\frac{\varepsilon^{2}}{32}$
and for $\varepsilon\to 0$ approaches $\mathcal{F}_{\mathrm{Q}}$.

In our protocol, schematically depicted in Fig. 1 we divide the signal in
half, rather than splitting the signal to prepare two identical copies. This
method while being slightly different from SLIVER, achieves the same
sensitivity to source separation and inherently, as an additional feature,
allows use with $N$-photon states [35]. This framework assumes prior knowledge
of the mean frequency, also known as the source centroid. This assumption is
practically valid, as an adaptive strategy can be used where the centroid is
first estimated with desired precision using direct detection [36].

### Temporal Imaging in GEM

Figure 4: Step-by-step realization of the protocol for symmetric and
antisymmetric input signals. a Schematic representation of the PuDTAI
interferometer superimposing the divided signal in the Fourier space. The
color fill represents the phase (Arg) of the signal field envelope, BS -
beamsplitter. b  The corresponding spin-wave evolution in $(z,k_{z})$ space is
represented by Wigner function maps ($\mathcal{W}$). The time-frequency domain
Wigner maps are shown for reference. Thanks to the symmetrically-chirped
control field used in GEM protocol (Storage) the two halves of the signal
pulse are mapped in a symmetric manner on separate Fourier ($k_{z}$)
components while overlapping in the longitudinal coordinate $z$. The black
dashed lines represent the temporal aperture that removes the central part of
the singnal pulse during the mapping. The spatially-resolved AC-Stark phase
modulation (AC-S) with chirped square grating splits the spin-waves in the
$k_{z}$ direction and makes the two halves to interfere at central $k_{z}$
coordinate \- the antisymmetric port of the interferometer which is then read-
out by control field pulse (Out-). The second (symmetric) port is restored by
modulating the spin waves again with a second chirped square wave grating with
opposite phase (AC-S). The symmetric port is finally retrieved (Out+).

Before introducing the details of the PuDTAI protocol, let us first discuss a
simpler case of DI spectrometer based on an atomic Gradient Echo Memory (GEM)
[37, 38, 39] that is realized in the same GEM setup schematically depicted in
Fig. 2a. In this setup a signal pulse with a slowly varying amplitude
$\mathcal{A}(t)$ enters the atomic cloud placed in the magnetic field gradient
along the propagation axis $g=\partial_{z}B_{z}$. At the same time the cloud
is illuminated by a strong, co-propagating control field $\mathcal{C}$ that
due to two-photon interaction maps the signal field onto atomic coherence
described by the off diagonal terms $\varrho_{hg}(z)$ of the spatially
dependent $\\{|g\rangle,|h\rangle\\}$ two-level atom density matrix
$\hat{\varrho}(z)$. To restore the signal light, the gradient is switched to
the opposite $g\to-g$ and the control field pulse is applied again, as shown
in the corresponding sequence in Fig. 2b. In Fig. 3a we show numerical
simulation of the GEM protocol for a signal pulses with two spectral
components of different amplitudes. The density maps below the time trace show
the evolution of the atomic coherence in the real space and wave vector
coordinates. The presence of the magnetic field gradient enables the crucial
feature of the GEM protocol which is the spectro-spatial mapping that links
spectral components of the signal pulse with the amplitude of the atomic
coherence along the ensemble
$\mathcal{\tilde{A}}(\omega)\leftrightarrow\varrho_{hg}(z)$. At the same time,
in time domain, the pulse shape is transferred to wavevector-space components
of the coherence $\mathcal{A}(t)\leftrightarrow\tilde{\rho}_{hg}(k_{z})$,
where $\tilde{\varrho}_{hg}(k_{z})$ is the Fourier transform of the real space
coherence. This correspondence is directly visible on the cross sections of
the atomic coherence maps (right panels) taken along the dashed lines where we
also show the optical field magnitudes in the time and spectral domain.

The DI imaging spectrometer utilizes a temporal imaging (TI) technique to
perform a Fourier transform of the signal pulse and thus map a spectrum of the
signal pulse onto temporal shape, as illustrated in Fig. 3b. This is achieved
by applying sequentially temporal and spectral phase modulations corresponding
to lens-propagation-lens components. A time lens corresponds to applying a
quadratic phase in time, while propagation is achieved by imprinting a
quadratic phase in spectrum onto the signal pulse as shown in Fig. 3c [40]. In
GEM, the temporal phase modulation is achieved by chirping the control field
that controls the light-coherence mapping process. The temporal phase
modulation is this way reflected in the phase of the coherence in the
wavevector space ($k_{z}$). On the other hand the phase modulation in the
spectral domain is achieved by Spatial Spin-wave Modulation (SSM) [41, 42,
43], that utilizes spatially varying ac-Stark shifts to imprint arbitrary
phase profiles onto the coherence in the real space coordinate ($z$). The ac-
Stark shifts are caused by an intensity-shaped beam that illuminates the atoms
from the side (see Fig. 2 for the geometry and the actual sequence of the
experiment) during the storage time.

The most convenient way to understand evolution of the optical field or the
corresponding atomic coherence at each step of the protocol is to investigate
a phase-space quasi-probability given by the Wigner function

$\mathcal{W}_{\mathcal{Q}}(q,p)=\frac{1}{\sqrt{2\pi}}\int\mathcal{Q}(q+\xi/2)\mathcal{Q}^{*}(q-\xi/2)\exp(-ip\xi)\mathrm{d}\xi$
(5)

where the quantity of interest $\mathcal{Q}$ and the conjugate variables
$\\{q,p\\}$ are either $\mathcal{A},\ \\{t,\omega\\}$ for which it becomes a
chronocyclic Wigner function [44] describing the signal pulse or
$\varrho_{hg},\ \\{k_{z},z\\}$ for the atomic coherence.

In Fig. 3d we show evolution of the chronocyclic Wigner function for the input
signal propagating through the TI setup from Fig. 3b. A temporal phase
modulations $\mathcal{A}(t)\to\mathcal{A}(t)\exp\left(i\phi(t)\right)$
reshapes the chronocyclic Wigner function
$\mathcal{W}_{\mathcal{A}}(t,\omega)$ in the $\omega$ direction, at the same
time the corresponding coherence quasiprobability is reshaped in the
$z$-direction
$\mathcal{W}(k_{z},z)\xrightarrow{\phi(t)}\mathcal{W}(k_{z},z^{\prime})$. For
example, in Fig. 3d the quadratic phase profile that implements the time-lens
stretches the $\omega$ axis on the $\mathcal{W}_{\mathcal{A}}(t,\omega)$ map
linearly with $t$ as marked by the arrows.

The spectro-spatial mapping enables implementation of spectral phase
operations by phase-modulating the coherence in real-space coordinates
$\varrho_{hg}(z)\to\varrho_{hg}(z)\exp(i\chi(z))$. These are linked to
$k_{z}$-direction reshaping of the quasiprobability
$\mathcal{W}(k_{z},z)\xrightarrow{\chi(z)}\mathcal{W}(k_{z}^{\prime},z$) and
$t$-axis transformation of the chronocyclic Wigner function. This is visible
in the third panel of Fig. 3d corresponding to propagation transformation
realized by a quadratic spectral phase profile. In the last panel, the second
time-lens operation completes the $90^{\circ}$ rotation of the phase - the
Fourier transform of the input signal amplitude $\mathcal{A}(t)$ is complete.
In practice, however one may omit the last step as it only corrects the phase
of the resulting output signal that is (the phase) irrelevant in case of phase
insensitive detection as photon counting.

The discussed TI setup not only serves as introductory example but is
implemented experimentally in our memory [37] to serve as DI reference to our
superresolution protocol which we describe below.

### PuDTAI interferometer

Finally, within the introduced framework, we are able to realize both the
time-dependent-transmission mirror that divides the pulse and performs a time-
inversion as well as beamsplitter that combines the two components and
performs the final projection. The sequence of operations performed using the
quantum memory is shown in Fig. 2b.

The step-by-step evolution of the quasiprobability $\mathcal{W}(z,k_{z})$ in
the PuDTAI protocol for two input pulse shapes
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Fig. 4b. First, similarly to TI, a temporal phase modulation with
$\phi(t)=-\alpha t|t|/2$, corresponding to $z\to z^{\prime}=z+\alpha|k_{z}|$
transformation in the phase space implements the time-dependent-transmission
beamsplitter and simultaneously performs the time-inversion of the first part.
In fact, such modulation represents a dual time-lens that is half convex, half
concave i.e. the focal length of the first half ($t<0$) is positive when for
the second part ($t>0$) is negative. The Wigner function maps (dark
background) illustrate that the two parts of the pulse are mapped
symmetrically on the opposite sides of the $k_{z}$ space. For practical
reasons the temporal phase modulation is accompanied with Cassegrain-type
temporal aperture to prevent mapping of the central part of the pulse (around
$t=0$), the aperture is represented by a black dashed line on the signal
pulses time traces (first row of Fig. 4b). The (minor) influence of the
aperture on the protocol’s performance is discussed in the Methods section.

Next, by performing $k_{z}$-direction splitting transformation in the phase
space we overlap the two parts and make them interfere. This is achieved by
imprinting a $\pi$-depth square-wave grating (G1) with linearly increasing
grating wavevector $k_{g}=\kappa z$ (see Fig. 2b). The chirped grating
transfers the spin wave into diffraction orders with $z$-dependent wavevector
spacing $k_{z}\to k_{z}^{\prime}=k_{z}\pm\kappa z$. Let us now focus on the
relevant diffraction orders: the minus-first in the negative time part and the
first one in the positive time part. Those two orders in the language of
temporal imaging are corresponding to forward and backward propagation
respectively. If we combine this with the dual-lens operation implemented in
the first step we achieve dual-far-field temporal imaging setup that performs
forward Fourier transform of the first half of the pulse and backward Fourier
transform of the second half. Moreover, as the resulting parts overlap in the
Fourier domain they interfere as represented in Fig. 4 a. For this to happen
the propagation distance must be equal the focal length of the time lens, the
condition is met for the grating chirp parameter $\kappa$ equal to the inverse
of the storage chirp $\kappa=\nicefrac{{1}}{{\alpha}}$. The third row of Fig.
4b represents the Wigner function after application of the grating. Here we
see that the two signal pulse components are forced by the grating (G1) to
interfere at $k_{z}=0$ coordinate (the symmetry axis of the quasiprobalitlity
distribution). By taking a $z$-direction integral of the quasiprobability
around $k_{z}=0$ we retrieve light intensity at the output of the memory.
Moreover, the phase between the two interfering components can be adjusted by
shifting the grating phase by $\zeta$: $\mathrm{sq}(\kappa
z^{2})\to\mathrm{sq}(\kappa z^{2}+\zeta)$. For $\zeta$=0 the interference for
symmetric input shapes as $\mathcal{G}$ is destructive, while for
antisymmetric $\mathcal{HG}$ it becomes constructive, as can be seen in Fig.
4a. To access the symmetric port the coherence is modulated again with similar
square-wave grating (G2), but with doubled chirp parameter $k_{g}=2\kappa
z=\frac{2}{\alpha}z$, and shifted in phase by $\pi/2$. With this modulation we
simply reverse previous temporal propagation step, and make it again but with
opposite phase between the two components. This is illustrated in the fifth
row of Fig. 4, where we observe constructive interference for the symmetric
input shape, and destructive for antisymmetric one. Finally, the area around
$k_{z}=0$ is remapped to light which constitutes the output from the symmetric
port of the interferometer (last row of Fig. 4). Notably, the Wigner space
domain transformations are the same for both input shapes, and the different
results at the output are caused solely by the difference in the input shape.
In Table 1 we provide a survey of the phase operations needed to implement the
PuDTAI protocol. To summarize, the protocol realizes a $\pi/2$ ($-\pi/2$)
rotation of the Wigner function of the first (second) half oh the signal
pulse, and interferes the two parts in the Fourier domain. Additionally, as at
the output have the the Fourier domain mapped to the time axis, a longer
signal pulses give shorter pulses at the output of the interferometer that
leads to higher signal-to-noise ratio when using noisy detectors. This
provides an additional practical advantage while dealing with very narrowband
states of light.

Function | Type | Formula | Wigner space
---|---|---|---
lens | $t\leftrightarrow k_{z}$ | $\phi(t)=\frac{\alpha}{2}t^{2}$ | $z\to z-\alpha k_{z}$
propagation | $z$ | $\chi(z)=\frac{\beta}{2}z^{2}$ | $k_{z}\to k_{z}+\beta z$
dual lens | $t\leftrightarrow k_{z}$ | $\phi(t)=-\frac{\alpha}{2}t|t|$ | $z\to z+\alpha|k_{z}|$
| bidirectional
---
propagation
$z$ | $\chi(z)=\mathrm{sq}(\kappa z^{2})$ | $k_{z}\to k_{z}\pm\kappa z$
Table 1: Summary of the temporal and spectral phase modulations used to
implement both PuDTAI and DI spectrometer. The $t\leftrightarrow k_{z}$ type
of transformation means that it is implemented during the light-to-atoms
mapping stage by temporal phase modulation of the control field, while the $z$
stands for the real-space phase modulation of the atomic coherence. The
formula for the square wave pattern is
$\mathrm{sq}(\xi)=\pi\left((-1)^{\left\lfloor{\xi/\pi}\right\rfloor}+1\right)/2$.
Each modulation has its corresponding Wigner space transformation that can be
expressed in the $\\{k_{z},z\\}$ or $\\{t,\omega\\}$ coordinates.

### Experimental Results

Figure 5: Separation estimation using QMTI and PuDTAI protocols. a Raw
$\varepsilon$ estimates obtained from DI approach (QMTI ) and PuDTAI protocol.
The shaded regions represent estimation uncertainty given by the square root
of the estimator variance (SEM) normalized for 10 processed photons. b
Estimator’s biases for both schemes. c Estimation precision compared with
idealized and real world CRBs given by corresponding Fisher information
($\mathcal{F}_{\mathrm{Q}}$\- Quantum Fisher information,
$\mathcal{F}_{\mathrm{SLIVER}}$ \- ideal SLIVER protocol,
$\mathcal{F}_{\mathrm{DI}}$ \- ideal DI spectrometer,
$\mathcal{F}_{\mathrm{QMTI}}$ \- Quantum memory temporal imaging,
$\mathcal{F}_{\mathrm{PuDTAI}}$ \- PuDTAI protocol). d Frequency-labeled
single photon counts distributions obtained with QMTI spectrometer for
different true separation values $\varepsilon$. e Contributions to the
antisymmetric ($N_{-}$) and symmetric ($N_{+}$) port of the PuDTAI device and
the ratio $N_{-}/N_{+}$ along with calibrated theoretical model.

To benchmark our protocol we artificially prepare signal pulses composed of
two mutually incoherent spectral components

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}{}\pgfsys@moveto{-14.22638pt}{0.01563pt}\pgfsys@curveto{-14.22638pt}{0.01563pt}{-13.3697pt}{0.0327pt}{-13.0407pt}{0.0521pt}\pgfsys@curveto{-12.71169pt}{0.07149pt}{-12.18404pt}{0.10454pt}{-11.85503pt}{0.15543pt}\pgfsys@curveto{-11.52602pt}{0.20631pt}{-10.99835pt}{0.29962pt}{-10.66934pt}{0.41895pt}\pgfsys@curveto{-10.34033pt}{0.53827pt}{-9.81268pt}{0.76587pt}{-9.48367pt}{1.01549pt}\pgfsys@curveto{-9.15466pt}{1.2651pt}{-8.627pt}{1.75331pt}{-8.29799pt}{2.2181pt}\pgfsys@curveto{-7.96898pt}{2.68288pt}{-7.44133pt}{3.59908pt}{-7.11232pt}{4.36542pt}\pgfsys@curveto{-6.78331pt}{5.13176pt}{-6.25565pt}{6.6308pt}{-5.92664pt}{7.74141pt}\pgfsys@curveto{-5.59763pt}{8.85202pt}{-5.06998pt}{10.97266pt}{-4.74097pt}{12.36993pt}\pgfsys@curveto{-4.41196pt}{13.76721pt}{-3.8843pt}{16.32219pt}{-3.55528pt}{17.81206pt}\pgfsys@curveto{-3.22627pt}{19.30193pt}{-2.69862pt}{21.83131pt}{-2.36961pt}{23.10788pt}\pgfsys@curveto{-2.0406pt}{24.38445pt}{-1.51294pt}{26.27109pt}{-1.18393pt}{27.01266pt}\pgfsys@curveto{-0.85492pt}{27.75424pt}{-0.32729pt}{28.4539pt}{0.00172pt}{28.45276pt}\pgfsys@curveto{0.33073pt}{28.45161pt}{0.8584pt}{27.74786pt}{1.18741pt}{27.00441pt}\pgfsys@curveto{1.51642pt}{26.26096pt}{2.04407pt}{24.37224pt}{2.37308pt}{23.09442pt}\pgfsys@curveto{2.70209pt}{21.8166pt}{3.22975pt}{19.28474pt}{3.55876pt}{17.7947pt}\pgfsys@curveto{3.88777pt}{16.30464pt}{4.41542pt}{13.75166pt}{4.74443pt}{12.35516pt}\pgfsys@curveto{5.07344pt}{10.95866pt}{5.6011pt}{8.83931pt}{5.93011pt}{7.72968pt}\pgfsys@curveto{6.25912pt}{6.62004pt}{6.78677pt}{5.12299pt}{7.11578pt}{4.3576pt}\pgfsys@curveto{7.4448pt}{3.59222pt}{7.97246pt}{2.67738pt}{8.30147pt}{2.21332pt}\pgfsys@curveto{8.63048pt}{1.74925pt}{9.15813pt}{1.26202pt}{9.48714pt}{1.01288pt}\pgfsys@curveto{9.81615pt}{0.76373pt}{10.34381pt}{0.5366pt}{10.67282pt}{0.41765pt}\pgfsys@curveto{11.00183pt}{0.29869pt}{11.52948pt}{0.20613pt}{11.85849pt}{0.15543pt}\pgfsys@curveto{12.1875pt}{0.10472pt}{13.04417pt}{0.0521pt}{13.04417pt}{0.0521pt}\pgfsys@stroke\pgfsys@invoke{
} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope{}{}{}\hss}\pgfsys@discardpath\pgfsys@invoke{\lxSVG@closescope
}\pgfsys@endscope\hss}}\lxSVG@closescope\endpgfpicture}}}{p}}(\omega+\sigma\varepsilon/2)\right)$
(6)

where the random phase $\varphi\in[0,2\pi[$ is drawn from uniform
distribution. The corresponding chronocyclic Wigner function [44] of such
signal distribution reads:

$\mathcal{W}_{\mathcal{S}}(t,\omega)=\psi_{\scalerel*{\leavevmode\hbox
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} \pgfsys@invoke{\lxSVG@closescope }\pgfsys@endscope
\pgfsys@invoke{\lxSVG@closescope
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We send the pulses to the memory where they are processed by our pulse-
division time-inversion interferometer. To read-out the contributions in
symmetric and antisymmetric ports we use two sequentially applied pulses of
the control field that are interleaved with the second AC-Stark modulation.
The processed signal light coming from the memory is detected using single-
photon counting module (SPCM) and photo-counts are time-tagged to identify the
output ports (see Fig. 9a in the Methods section for the time-bin histogram).
The mean single-shot signal contribution summed over the symmetric and
antisymmetric port was set to be around $\bar{n}\approx 0.69$. After many
experimental repetitions we count the total contributions $N_{-}$, $N_{+}$ in
the antisymmetric and symmetric port respectively and calibrate the maximum-
likelihood estimator $\hat{\varepsilon}(N_{-}/N_{+})$ based on theoretically
expected counts ratio $N_{-}/N_{+}$. The estimator is then used to estimate
the value of the separation $\varepsilon$. For this, for each $\varepsilon$
value we collected approx. $1.5\times 10^{5}$ counts. To estimate the
estimator variance we follow the bootstrapping technique: for each
$\varepsilon$ setting we randomly prepare $10^{3}$ sets of samples, each
containing $1.5\times 10^{5}$ total counts. The estimator is then evaluated on
each set and the mean and variance of $\hat{\varepsilon}$ is calculated. In
Fig. 5a we show raw estimation results for PuDTAI protocol and DI approach on
a common $\langle\hat{\varepsilon}\rangle$ plot. The filled regions
corresponds to the estimation uncertainty given by the square root of the
estimator variance and normalized to 10 processed photons. Both methods give
similar values of the separation parameter, but as we expect the PuDTAI
protocol vastly outperforms the DI over the whole $\varepsilon<1$ range. The
improvement expressed in variance ratio for the same experimental conditions
is most prominent for small separation values and reaches about 20 for
$\varepsilon=0.08$. In other words using PuDTAI protocol we need, 20 times
less photons to achieve the same precision.

The measurements in DI scheme are obtained by using ultranarrowband far-field
temporal imaging technique which we call quantum-memory temporal imaging
(QMTI) [37]. The protocol is implemented in the same GEM device following the
recipe from the introductory section. It performs a $90^{\circ}$ rotation of
the signal pulse Wigner space which results in spectrum-to-time mapping
$\omega\to 2\alpha_{\mathrm{DI}}t$, where $\alpha_{\mathrm{DI}}$ is the
control field chirp parameter corresponding to the time-lens magnitude. The
output signal is detected using the same SPCM as in the case of PuDTAI, where
counts corresponding to distinct spectral components have different arrival
time (timetags). The counts distributions with already calibrated frequency
labels are presented in Fig. 5d. From those, we estimate the separation using
maximum-likelihood separation estimator $\hat{\varepsilon}_{\mathrm{DI}}$. The
variance of this estimator is calculated by following the same bootstrapping
technique as in the case of PuDTAI protocol. Figure 5b portrays the
estimators’ biases $\langle\hat{\varepsilon}\rangle-\varepsilon$ proving
unbiasedness of both estimators even for small $\varepsilon$ and validating
the results for variances.

In Fig. 5c we compare the achieved precision with CRB for both schemes. Here
we also show the ultimate bound set by the QFI ($\mathcal{F}_{\mathrm{Q}}$) as
well as bounds for ideal SLIVER protocol ($\mathcal{F}_{\mathrm{SLIVER}}$) and
idealized DI ($\mathcal{F}_{\mathrm{DI}}$). The real bounds are placed by FIs
that take into account experimental imperfection such as detection noise and
finite bandwidth of the ensemble - $\mathcal{F}_{\mathrm{QMTI}}$, as well as
finite interferometer visibility and interferometer losses in the case of
PuDTAI protocol - $\mathcal{F}_{\mathrm{PuDTAI}}$ (see Methods). We see that
the PuDTAI protocol vastly outperforms the QMTI for normalized separations
$\varepsilon<1$ with maximum improvement in terms of variance of about 30 for
$\varepsilon=0.4$. Finally, we show the data used for estimation, to better
illustrate the origin of sensitivity enhancement. In Fig. 5d the results of
direct imaging are presented, and we observe that for $\varepsilon<1$ the