# Beating the spectroscopic Rayleigh limit via post-processed heterodyne
detection

Wiktor Krokosz  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  Mateusz
Mazelanik  Centre for Quantum Optical Technologies, Centre of New
Technologies, University of Warsaw, Banacha 2c, 02-097 Warsaw, Poland  Michał
Lipka  Centre for Quantum Optical Technologies, Centre of New Technologies,
University of Warsaw, Banacha 2c, 02-097 Warsaw, Poland  Marcin Jarzyna
Centre for Quantum Optical Technologies, Centre of New Technologies,
University of Warsaw, Banacha 2c, 02-097 Warsaw, Poland  Wojciech Wasilewski
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  Konrad Banaszek
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

###### Abstract

Quantum-inspired superresolution methods surpass the Rayleigh limit in
imaging, or the analogous Fourier limit in spectroscopy. This is achieved by
carefully extracting the information carried in the emitted optical field by
engineered measurements. An alternative to complex experimental setups is to
use simple homodyne detection and customized data analysis. We experimentally
investigate this method in the time-frequency domain and demonstrate the
spectroscopic superresolution for two distinct types of light sources: thermal
and phase-averaged coherent states. The experimental results are backed by
theoretical predictions based on estimation theory.

##  I Introduction

Spectroscopic measurements serve as an indispensable tool for studying the
properties of matter and light in a wide range of scientific disciplines, such
as physics, chemistry [1], astronomy [2], biology [3], or metrology [4].
However, the achievable resolution of state-of-the-art spectrometers,
including grating-based and Fourier spectrometers, is fundamentally
constrained by the Fourier limit. This is especially important for emissive
spectroscopy, where the properties of the light source can not be modified. In
such scenario, when one deals with complex spectral distributions (e.g.
multiple overlapping spectral lines of different widths), this limit is hard
to overcome. However, there are cases in which specific characteristics of the
light sources under study can be exploited to tailor an optimal measurement
and beat this restriction by achieving a sub-Fourier resolution [5, 6, 7, 8].
This is analogous to a similar effect present in optical imaging, where a
Rayleigh criterion, formulated in a modern approach using tools from
estimation theory, limits the fundamental spatial resolution of an image [9,
10]. The basic example is a system of two identical and mutually incoherent
point sources separated by some unknown distance that is much smaller than the
spatial spread of each source. Many approaches to measuring the separation
with sub-Rayleigh precision have been proposed [11, 12, 13, 14, 15] and
experimentally demonstrated [16, 17, 18, 19, 20, 21, 22, 23, 24]. Such methods
utilize shot-to-shot coherence and focus on information encoded within the
electromagnetic (EM) field of light rather than its measured intensity as in
traditional techniques. However, all of them require intricate and specialized
equipment crafted to specific experimental conditions, as they rely on
projecting the field on a specific set of modes. Another recently proposed
approach, which also yields superior results compared to traditional
intensity-based methods, is to use homodyne or heterodyne sensing with
tailored digital postprocessing [25, 26]. A prominent advantage of such
technique is a significantly simplified measurement setup, as the complexity
is shifted to the data analysis step.

In this study, we present an experimental demonstration of spectral
superresolution accomplished via heterodyne sensing, utilizing previously
proposed theoretical frameworks [25, 26, 27]. It is important to note that our
approach differs from [18] as we do not utilize the shaping of the local
oscillator beam. By leveraging the advantages of heterodyne sensing, our
method provides a more accessible and practical means of achieving spectral
superresolution in cases where the signal is not dominated by the shot noise,
with applications in domains where fine spectral discrimination plays a
pivotal role. Potential candidates are lasing structures [28], optomechanics
[29, 30, 31] or quantum transducers (e.g. based on Rydberg atoms) [32, 33, 34]
and sensors [35]. Our analysis method leverages the limitations of a typical
approach, reminiscent of direct imaging, where only a power spectral density
of the signal is used to analyze the spectral lines. At the same time, the
experimental setup remains essentially the same, providing a prospect for
immediate applications.

##  II Heterodyne superresolving estimation

Our experimental demonstration is based on simulating and measuring the light
source using an experimental setup presented in Fig. 1. By employing an
acousto-optic modulator (AOM) together with an arbitrary signal generator, we
are able to encode the desired EM field envelope described below, which is
later measured by the time-resolved heterodyne sensor. In this section we will
describe our estimation approach. The detailed theoretical analysis of our
method can be found in [25, 26].

Figure 1: The experimental setup takes in a laser beam, which is then split in
two using a half-wave plate $\left(\frac{\lambda}{2}\right)$ and a polarizing
beam splitter (PBS). Both beams are then transmitted into one of the acousto-
optic modulators (AOMs), which encode the local oscillator (LO) frequency as
well as the programmatically set signal. The quarter-wave plate
$\left(\frac{\lambda}{4}\right)$, convex lens, beam stop and mirror constitute
a double pass AOM setup. Once the laser light is properly modulated, the two
signals are then joined together and directed into a differential photodiode
(DPD). Finally, the heterodyne signal is collected and I/Q demodulated.

Let us consider two spectral lines, each emitting light in a Gaussian temporal
mode. The time envelope of such EM field, assuming the lines are separated by
$\varepsilon\sigma$, is given by

$\begin{split}A(t+t_{c})=&\;G(t)e^{i\omega_{c}t}\\\
&\times\left(A_{1}e^{i\phi_{1}}e^{-i\frac{\sigma\varepsilon
t}{2}}+A_{2}e^{-i\phi_{2}}e^{i\frac{\sigma\varepsilon
t}{2}}\right),\end{split}$ (1)

where $A_{i},\,\phi_{i}$ for $i=1,2$ are respectively modulus value and phase
of complex amplitude emitted by each line, $\omega_{c}$ and $t_{c}$ are
displacements in frequency and time domains and $\sigma$ is the characteristic
spread of the Gaussian temporal mode

$G(t)=\sqrt[4]{\frac{2\sigma^{2}}{\pi}}e^{-\sigma^{2}t^{2}}.$ (2)

We will specifically consider the cases of incoherent thermal radiation and
coherent light averaged over the relative phase. In the former scenario, one
has to use Eq. (1) with amplitudes and phases $(A_{i},\phi_{i})$ randomly
chosen in each realization from a complex Gaussian distribution for each line,
that is, $A_{i}e^{i\phi_{i}}\sim\mathcal{CN}(0,\mathrm{A_{0}^{2}/2})$. The
latter scenario, involves taking $A_{1}=A_{2}=A_{0}/\sqrt{2}$ and
$\phi_{1}=\phi_{2}=\phi_{0}$ in Eq. (1) with the phase $\phi_{0}$ randomly
chosen from a flat distribution on the $2\pi$ period in each realization while
$A_{0}$ is kept fixed i.e. only the relative phase is changing as the global
phase referred to LO is unknown and changes in time. The resulting envelope or
a particular realization reads

$A(t+t_{c})=A_{0}\sqrt{2}e^{i\omega_{c}(t)}G(t)\cos\left(\frac{\sigma\varepsilon
t}{2}-\phi_{0}\right).$ (3)

Let us now consider the time-resolved heterodyne detection of an optical field
described above. The analog heterodyne signal is digitalized and in-phase (I)
and $\pi/2$-quadrature (Q) demodulated in real time and stored as complex
signal traces $Z(t_{i})=I(t_{i})+iQ(t_{i})$. Our estimation procedure begins
by calculating the projections of the $Z(t_{i})$ onto the first and second
Hermite-Gauss (HG) modes:

$\displaystyle u^{(0)}(t)$
$\displaystyle=G(t)=\sqrt[4]{\frac{2\sigma^{2}}{\pi}}e^{-\sigma^{2}t^{2}},$
(4) $\displaystyle u^{(1)}(t)$ $\displaystyle=2\sigma
tG(t)=\sqrt[4]{\frac{32\sigma^{6}}{\pi}}te^{-\sigma^{2}t^{2}},$ (5)

where the Gaussian width $\sigma$ is set the same as in the input signal.
Crucially, the second HG mode, in contrast to the first, is antisymmetric and
therefore sensitive to discrepancies of the signal from $u^{(0)}(t)$, which
describes a single line. For very small separations between the pulses, we can
assume that almost all of the signal is distributed between these two modes
[27]. The projections are calculated using a simple discrete sum as

$z^{(k)}(t_{r},\omega_{r})=\sum_{i}Z(t_{i})^{*}u^{(k)}(t_{i}-t_{r})e^{i\omega_{r}t}\Delta
t,$ (6)

where $\omega_{r},\,t_{r}$ are some spectral and temporal displacements.

Next, for each set of projections specified by the chosen values of
$(\omega_{r},t_{r})$, we compute a variance
$V^{(k)}=\mathrm{Var}[z^{(k)}(t_{r},\omega_{r})]$. The variances represent the
amount of signal in a given spatial mode, but also contain a shot-noise
contribution and are in measurement units (e.g. Volts at photodetector). To
recover the dimensionless signal contribution in the $u^{(1)}$ mode relative
to the contribution in the fundamental $u^{(0)}$ mode, we calculate a minimal
normalized and noise-subtracted variance:

$V_{\varepsilon}=\min_{t_{r},\omega_{r}}\frac{V^{(1)}-V^{(1)}_{\mathrm{noise}}}{V^{(0)}-V^{(0)}_{\mathrm{noise}}},$
(7)

where $V^{(k)}_{\mathrm{noise}}$ represents variance from a projection of a
shot-noise calculated for measurement without the input signal. The values
obtained from $t_{r},\omega_{r}$ that minimize the variance serve as the
centroid estimates that preserve high precision even for small separations
[25]. Finally, the variance $V_{\varepsilon}$ can be used to estimate the
separation value:

$\hat{\varepsilon}=4\sqrt{\max\\{V_{\varepsilon},0\\}}.$ (8)

Note that negative $V_{\varepsilon}$ values may arise when the signal
contribution to $V^{(1)}$ is almost negligible. The estimation error may be
found by employing the statistical bootstrapping method [36], described in
Sec. IV.

Finally, to estimate the average number of registered photons we calculate the
signal-to-noise ratio $\mathcal{S}$ by comparing the variances in the
fundamental mode for signal and noise:

$\mathcal{S}=\frac{V^{(0)}-V^{(0)}_{\mathrm{noise}}}{V^{(0)}_{\mathrm{noise}}},$
(9)

for $(t_{r},\omega_{r})$ obtained from Eq. (7). This approach is valid for
small $\varepsilon$ and for $\varepsilon>1$ can be extended by including
higher order modes in the formula. The signal-to-noise ratio is directly
related to the number of photons $\bar{n}$ in the experiment by
$\bar{n}=\mathcal{S}$. Notably, by following the above procedure, from a
single measurement we estimate the centroids, separation, and average photon
flux.

##  III Precision limits

The maximum precision limit of any estimation method based on a given
measurement scheme is specified by the Cramér-Rao bound (CRB) [37]. Assuming
the outcomes $z$ are distributted according to a probability distribution
$P(z|\varepsilon)$ the limit is equal to

$(\Delta\varepsilon)^{2}\geq 1/F,\quad
F=\int\frac{1}{P(z|\varepsilon)}\left[\frac{\partial
P(z|\varepsilon)}{\partial\varepsilon}dz\right],$ (10)

where $F$ is Fisher information (FI).

The CRB for scenario described in the previous section Sec. II can be derived
from the measured quadratures statistic for both phase-averaged coherent and
thermal states. However, direct calculation may be not be straightforward,
particularly in the former case as the resulting quantum state is non-Gaussian
[38, 39]. For this reason, we rather decompose the signal into HG modes and
deal with each component separately.

The amplitude contribution from each HG mode can be calculated directly from
the source spectral envelope as

$c_{k}(\varepsilon)=\int\mathrm{d}\omega\tilde{G}\left(\omega-\frac{\sigma\varepsilon}{2}\right)\tilde{u}^{(k)}(\omega)=\frac{e^{-\frac{\varepsilon^{2}}{32}}}{\sqrt{k!}}\left(\frac{\varepsilon}{4}\right)^{k},$
(11)

for $k\in\\{0,1\\}$ and where $\tilde{G}$ and $\tilde{u}$ denote the time-
domain Fourier transform of $G$ and $u$ respectively. Each source contributes
with this amplitude to the measured signal in a given mode, and as a result,
they interfere. For thermal sources, the result of the interference is still a
thermal state with the mean photon number rescaled by
$|c_{k}(\varepsilon)|^{2}$. Therefore, the quadrature distribution in the
$u^{(1)}$ mode for the received state with $\bar{n}$ photons on average is
given by:

$P_{\mathrm{th}}(z|\varepsilon)=\frac{1}{\pi(2|\alpha(\varepsilon)|^{2}+1)}\exp\left(-\frac{|z|^{2}}{2|\alpha(\varepsilon)|^{2}+1}\right),$
(12)

where $z$ is a complex variable and
$\alpha(\varepsilon)=\sqrt{\bar{n}/2}c_{1}(\varepsilon)$ represents the
average amplitude with $|\alpha(\varepsilon)|^{2}$ being average number of
photons in the mode. The factor of $2$ that divides the number of photons is
the result of using a heterodyne measurement [27] in which each quadrature is
measured on half of the signal and is accompanied by its individual shot
noise. In the case of phase-averaged coherent states, the output state is more
complex with the quadrature distribution in the following form:

$P_{\mathrm{coh}}(z|\varepsilon)=\int\displaylimits_{0}^{2\pi}\int\displaylimits_{0}^{2\pi}\frac{\mathrm{d}\phi_{1}\mathrm{d}\phi_{2}}{4\pi^{3}}e^{-|z-\alpha(\varepsilon)(e^{i\phi_{1}}-e^{i\phi_{2}})|^{2}}.$
(13)

The precision bounds may be found by plugging Eqs. (12) and (13) into the CRB
in Eq. (10). Importantly, one often speaks about superresolution when the
precision of separation estimation outperforms what can be achieved with
direct sensing (DS). The bound for latter can be found by normalizing the
modulus square of the EM field envelope and treating it as a probability
distribution $P_{\mathrm{DS}}\sim|\tilde{A}(\omega)|^{2}$ for which one can
evaluate the corresponding CRB.

Using the outcome distributions derived above, the per-photon FI for each mode
can be calculated as:

$\mathcal{F}_{\mathrm{th/coh}}=\frac{1}{\bar{n}}F\left[P_{\mathrm{th/coh}}(z|\varepsilon)\right].$
(14)

For small separations ($\varepsilon\ll 1$) the $u^{\mathrm{(1)}}$ mode
contains almost all the information, and the other modes can be neglected. The
formula for FI for thermal states can be evaluated analytically and yields:

$\mathcal{F}_{\mathrm{th}}=\frac{\bar{n}\left(\varepsilon/4\right)^{2}(\left(\varepsilon/4\right)^{2}-1)^{2}}{4\left(e^{\left(\varepsilon/4\right)^{2}}+\bar{n}\left(\varepsilon/4\right)^{2}\right)^{2}}.$
(15)

In the case of phase-averaged coherent states the FI from Eq. (14) is
evaluated numerically to give the CRB for particular experimental conditions.
The average number of photons $\bar{n}$ in the experiment is simply
$\mathcal{S}$.

##  IV Results

We conducted the experiment for two different cases: phase-averaged coherent
states and thermal states. For both scenarios, we generated Gaussian pulses
with small spectral separations and measured them using the heterodyne
detector. The envelope for thermal states was given by Eq. (1), which for a
single realization of a phase-averaged coherent state simplifies to Eq. (3).

In the case of thermal states, two numbers were randomly generated from a
complex normal distribution. Their modulus and argument were used as the
amplitude and phase shift respectively, thus emulating a thermal light source.
For phase-averaged coherent states, the amplitude was fixed and the phases
were selected to uniformly cover the entire $2\pi$ period. To ensure the
reliability and statistical significance of our results, we performed
measurements for different mean signal intensities and varied the separations
$\varepsilon$. For each parameter set, approximately $N\approx 10^{6}$ pulse
repetitions were collected, providing a robust statistical sample.

Since we want to verify the saturability of the CRB, we need to calculate the
variance of the estimator, given by Eq. (8). To do that, we employ the
statistical bootstrapping method, where we generate 1000 ensembles of size $N$
from the calculated projections. Once we get the estimator values from each
sample, we can calculate the precision by taking the inverse of the variance
of the set multiplied by $\mathcal{S}$ and $N$.

Figure 2: Plots representing the measured separation estimation precision in
the case of thermal states. Each of the a-f plots corresponds to a single
$\mathcal{S}$ value, shown in ascending order. The data points are compared
with the CRB given by the calculated FI for heterodyne sensing (solid curves)
and DS (dashed curves). The error bars correspond to 2 standard deviations of
the evaluated measured precision. The insets represent the estimator bias for
each data point in the main plot. Figure 3: Plots representing the measured
separation estimation precision in the case of phase-averaged coherent states.
Each of the a-f plots corresponds to a single $\mathcal{S}$ value, shown in
ascending order. The data points are compared with the CRB given by FI for
heterodyne sensing (solid curves) and DS (dashed curves). The error bars
correspond to 2 standard deviations of the evaluated measure precision. The
insets represent the estimator bias for each data point in the main plot.

The results obtained for the thermal states are shown in Fig. 2. We can
clearly observe that the obtained precision taken as the inverse of the
estimator’s variance reliably follows the CRB (solid curve). For reference, we
also show the CRB for DS (dashed line). From the plots, we see that the
obtained precision beats the limit for DS proving the effectiveness of our
method for achieving spectral superresolution in the case of thermal states.

For low $\mathcal{S}$, we observe the effect of noise that increased the
estimator bias for the smallest separations [25]. Additionally, for larger
separations and $\mathcal{S}$, we observe that our estimator does not reach
the CRB. We attribute this to fluctuations of the estimated $\mathcal{S}$, Eq.
(9), that also influence the normalized variance, Eq. (7), used in the
estimator. This is limited by the statistics from the shot-noise measurement,
which could be improved. Currently, the number of shot-noise and signal
measurements is equal, but one can increase the number of shot-noise
measurements per experiment realization to estimate $\mathcal{S}$ more
precisely. We can draw similar conclusions from the results obtained for
phase-averaged coherent states depicted in Fig. 3. In this case, the precision
for $\varepsilon\ll 1$ follows CRB, and for larger separations, we observe a
larger decrease in precision. This is most likely a result of not sufficient
sampling of the phase space and use of the simple estimator that does not
grasp the correlations between the two quadratures that become important for
larger separations, as captured by the FI which is slightly larger when
compared to FI for thermal states.

##  V Summary

In this study, we have experimentally demonstrated the super-resolving
measurement of two spectral lines for two different optical states: thermal
and phase-averaged coherent. Our experimental setup, which involved the use of
an AOM and an arbitrary signal generator, allowed us to encode the desired EM
field envelope, which was subsequently measured by a simple heterodyne sensor.

Our data analysis, based on numerical decomposition of measured quadrature
distributions into first Hermite-Gauss modes, enables estimating frequency
separations and temporal and spectral centroid positions as well as average
photon numbers from a single measurement set. The achieved precision of
estimation was found to be in line with CRB derived for this measurement
scheme, validating the effectiveness of our method for achieving spectral
superresolution.

It is important to note that our method requires many photons per mode. While
this may limit its applicability in certain scenarios, there are cases where
this requirement can be met, and superresolution can be effectively achieved.
These include optomechanical systems where mechanical modes are usually probed
with a coherent light [40] or lasing microstructures [28].

In conclusion, our work contributes to ongoing efforts to overcome the
fundamental constraints of state-of-the-art spectrometers, offering a more
accessible and practical means of achieving spectral superresolution. This has
potential applications in various scientific disciplines where fine spectral
discrimination plays a pivotal role. In particular, our data analysis method
may be readily applied in heterodyne spectroscopy setups which have so far
used power spectral densities to recover properties of relevant spectral
lines.

## Data availability

Data underlying the results presented in this paper are available in the Ref.
[41].

## Code availability

The codes used for the calculations and the analysis of experimental data are
available from the authors upon request.

###### Acknowledgements.

The “Quantum Optical Technologies” (MAB/2018/4) 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 the Office
of Naval Research Global grant no. N62909-19-1-2127 and by National Science
Centre, Poland 2021/41/N/ST2/02926. ML was supported by the Foundation for
Polish Science (FNP) via the START scholarship

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