# Deep learning enhanced Rydberg multifrequency microwave recognition

Zong-Kai Liu  Key Laboratory of Quantum Information, University of Science and
Technology of China, Hefei, Anhui 230026, China  Synergetic Innovation Center
of Quantum Information and Quantum Physics, University of Science and
Technology of China, Hefei, Anhui 230026, China  Li-Hua Zhang  Key Laboratory
of Quantum Information, University of Science and Technology of China, Hefei,
Anhui 230026, China  Synergetic Innovation Center of Quantum Information and
Quantum Physics, University of Science and Technology of China, Hefei, Anhui
230026, China  Bang Liu  Key Laboratory of Quantum Information, University of
Science and Technology of China, Hefei, Anhui 230026, China  Synergetic
Innovation Center of Quantum Information and Quantum Physics, University of
Science and Technology of China, Hefei, Anhui 230026, China  Zheng-Yuan Zhang
Key Laboratory of Quantum Information, University of Science and Technology of
China, Hefei, Anhui 230026, China  Synergetic Innovation Center of Quantum
Information and Quantum Physics, University of Science and Technology of
China, Hefei, Anhui 230026, China  Guang-Can Guo  Key Laboratory of Quantum
Information, University of Science and Technology of China, Hefei, Anhui
230026, China  Synergetic Innovation Center of Quantum Information and Quantum
Physics, University of Science and Technology of China, Hefei, Anhui 230026,
China  Dong-Sheng Ding<EMAIL_ADDRESS>Key Laboratory of Quantum Information,
University of Science and Technology of China, Hefei, Anhui 230026, China
Synergetic Innovation Center of Quantum Information and Quantum Physics,
University of Science and Technology of China, Hefei, Anhui 230026, China
Bao-Sen Shi<EMAIL_ADDRESS>Key Laboratory of Quantum Information,
University of Science and Technology of China, Hefei, Anhui 230026, China
Synergetic Innovation Center of Quantum Information and Quantum Physics,
University of Science and Technology of China, Hefei, Anhui 230026, China

(August 27, 2024)

Recognition of multifrequency microwave (MW) electric fields is challenging
because of the complex interference of multifrequency fields in practical
applications. Rydberg atom-based measurements for multifrequency MW electric
fields is promising in MW radar and MW communications. However, Rydberg atoms
are sensitive not only to the MW signal but also to noise from atomic
collisions and the environment, meaning that solution of the governing
Lindblad master equation of light-atom interactions is complicated by the
inclusion of noise and high-order terms. Here, we solve these problems by
combining Rydberg atoms with deep learning model, demonstrating that this
model uses the sensitivity of the Rydberg atoms while also reducing the impact
of noise without solving the master equation. As a proof-of-principle
demonstration, the deep learning enhanced Rydberg receiver allows direct
decoding of the frequency-division multiplexed (FDM) signal. This type of
sensing technology is expected to benefit Rydberg-based MW fields sensing and
communication.

Introduction

The strong interaction between Rydberg atoms and microwave (MW) fields that
results from their high polarizability means that the Rydberg atom is a
candidate medium for MW fields measurement, e.g. using electromagnetically
induced absorption (Liao  _et al._ , 2020), electromagnetically induced
transparency (EIT) (Fleischhauer  _et al._ , 2005; Holloway  _et al._ , 2017)
and the Autler–Townes effect (Sedlacek  _et al._ , 2012; Autler and Townes,
1955; Holloway  _et al._ , 2017; Abi-Salloum, 2010). The amplitudes (Meyer
_et al._ , 2018; Jiao  _et al._ , 2019; Gordon  _et al._ , 2019; Jing  _et
al._ , 2020), phases (Simons  _et al._ , 2019; Jing  _et al._ , 2020) and
frequencies (Gordon  _et al._ , 2019; Jing  _et al._ , 2020) of MW fields
could then be measured with high sensitivity. Based on this measurement
sensitivity for MW fields, the Rydberg atom has been used in communications
(Holloway  _et al._ , 2019a; Meyer  _et al._ , 2018; Jiao  _et al._ , 2019;
Holloway  _et al._ , 2019b) and radar (Robinson  _et al._ , 2021) as an atom-
based radio receiver. In the communications field, the Rydberg atom replaces
the traditional antenna with superior performance aspects that include sub-
wavelength size, high sensitivity, system international (SI) traceability to
Planck’s constant, high dynamic range, self-calibration and an operating range
that spans from MHz to THz frequencies (Bason  _et al._ , 2010; Zou  _et al._
, 2020; Gordon  _et al._ , 2019; Meyer  _et al._ , 2018; Jing  _et al._ ,
2020). One application is analogue communications, e.g. real-time recording
and reconstruction of audio signals (Holloway  _et al._ , 2019b). Another
application is digital communications, e.g. phase-shift keying and quadrature
amplitude modulation (Holloway  _et al._ , 2019a; Meyer  _et al._ , 2018; Jiao
_et al._ , 2019). The channel capacity of MW-based communications is limited
by the standard quantum limited phase uncertainty (Meyer  _et al._ , 2018).
Furthermore, a continuously-tunable radio-frequency carrier has been realized
based on Rydberg atoms (Song  _et al._ , 2019), thus paving the way for
concurrent multichannel communications. Detection and decoding of
multifrequency MW fields are highly important in communications for
acceleration of information transmission and improved bandwidth efficiency.
Additionally, MW fields recognition enables simultaneous detection of multiple
targets with different velocities from the multifrequency spectrum induced by
the Doppler effect. However, because of the sensitivity of Rydberg atoms, the
noise is superimposed on the message, meaning that the message cannot be
recovered efficiently. Additionally, it is difficult to generalize and scale
the band-pass filters to enable demultiplexing of multifrequency signals with
more carriers (Zou  _et al._ , 2020).

To solve these problems, we use a deep learning model for its accurate signal
prediction capability and its outstanding ability to recognize complex
information from noisy data without use of complex circuits. The deep learning
model updates the weights via backpropagation and then extracts features from
massive data without human intervention or prior knowledge of physics and the
experimental system. Because of these advantages, physicists have constructed
complex neural networks to complete numerous tasks, including far-field
subwavelength acoustic imaging (Orazbayev and Fleury, 2020), value estimation
of a stochastic magnetic field (Khanahmadi and Mølmer, 2021), vortex light
recognition (Giordani  _et al._ , 2020; Liu  _et al._ , 2019), demultiplexing
of an orbital angular momentum beam (Doster and Watnik, 2017; da Silva  _et
al._ , 2021) and automatic control of experiments (Wigley  _et al._ , 2016;
Tranter  _et al._ , 2018; Mukherjee  _et al._ , 2020; Mills  _et al._ , 2020;
Wang  _et al._ , 2020; Bukov  _et al._ , 2018).

Figure 1: Illustration of the setup. (a) Overview of experimental energy
diagram. Probe and coupling laser beams excite the atoms at ground state
$\left|5S_{1/2}\right\rangle$ to the Rydberg state
$\left|51D_{3/2}\right\rangle$. Multifrequency microwave (MW) electric fields
couple the Rydberg states $\left|51D_{3/2}\right\rangle$ and
$\left|50F_{5/2}\right\rangle$. (b) Schematic of Rydberg atom-based antenna
and mixer interacting with multifrequency signals. A 795 nm laser beam is
split into two beams, which then propagate in parallel through a heated Rb
cell (length: 10 cm, temperature: $44.6^{\circ}\mathrm{C}$, atomic density:
$9.0\times 10^{10}$ $\mathrm{cm^{-3}}$ (Šibalić  _et al._ , 2017)). One is the
probe beam, which counterpropagates with the coupling laser beam exciting
atoms to Rydberg states to reduce Doppler broadening. The other is the
reference beam, which does not counterpropagate with the coupling laser beam.
The beams are detected using a differencing photodetector (DD) to obtain the
probe transmission spectrum (inset). Multifrequency MW fields transmitted by a
horn are applied to the atoms, with a radiated direction that is perpendicular
to the laser beam propagation direction. The multifrequency MW fields are
modulated using a phase signal such that the phase differences between the
reference bin and the other bins carry the messages. The probe transmission
spectrum is fed into a well-trained neural network to retrieve the variations
of the phases with time. (c-e) Schematics of the neural network. The network
consists of (c) a one-dimensional convolution layer, (d) a bi-directional
long-short term memory layer and (e) a dense layer; for further details about
these layers, see the Methods section.

Here, we demonstrate a deep learning enhanced Rydberg receiver for frequency-
division multiplexed digital communication. In our experiment, the Rydberg
atoms act as a sensitive antenna and a mixer to receive multifrequency MW
signals and extract information (Gordon  _et al._ , 2019; Simons  _et al._ ,
2019; Holloway  _et al._ , 2019a). The modulated signal frequency is reduced
from several gigahertz to several kilohertz via the interaction between the
Rydberg atoms and the microwaves, thus allowing the information to be
extracted using simple apparatus. These interference signals are then fed into
a well-trained deep learning model to retrieve the messages. The deep learning
model extracts the multifrequency microwave signal phases, even without
knowing anything about the Lindblad master equation, which describes the
interactions between atoms and light beams in an open system theoretically.
The solution of the master equation is often complex because the higher-order
terms and the noises from the environment and from among the atoms are taken
into consideration. However, the deep learning model is robust to the noise
because of its generalization ability, which takes advantage of the
sensitivity of the Rydberg atoms while also reducing the impact of the noise
that results from this sensitivity. Our deep learning model is scalable,
allowing it to recognize the information carried by more than 20 MW bins.
Additionally, when the training is complete, the deep learning model extracts
the phases more rapidly than via direct solution of the master equation.

Figure 2: Flow chat of recognizing the multifrequency MWs and results. (a)
Process of encoding and decoding frequency-division multiplexed binary phase-
shift keying (FDM-2PSK) signal with Rydberg atoms and the deep learning model.
(b-c) Loss curve evolution with epochs for the training set (blue) and the
validation set (orange) during training with different training data and
validation data. The training data sizes are (b) 393 and (c) 1194. The
validation data sizes are (b) 131 and (c) 398; more details about the data set
split are presented in the Methods section. The loss curves for training and
validation converge at (b) 140, and (c) 30 epochs, where an epoch is a time
unit during which the model iterates once over the complete data set; see
Methods section. (d) Confusion matrix for a test set (the number of the
testing set is 160 and the labels are uniformly distributed) after training in
case (c). The accuracy reaches 99.38% after a 70-epoch training period. (e)
Deep learning model accuracy on the noisy test set after training on the noisy
training set. The x- and y-axes represent the standard deviations of the
additional white noise added to the test set and the training set,
respectively. The colorbar represents the accuracy of the model on the noisy
test set. The results were obtained by averaging five sets of predictions. The
diagonal (red line) indicates the accuracy of the model on a test set in which
the noise distribution is the same as that of the training set; more details
about the noise are shown in supplementary materials.

Results

Setup. We adapt a two-photon Rydberg-EIT scheme to excite atoms from a ground
state to a Rydberg state. A probe field drives the atomic transition
$\left|5S_{1/2},F=2\right\rangle\rightarrow\left|5P_{1/2},F^{\prime}=3\right\rangle$
and a coupling light couples the transition
$\left|5P_{1/2},F^{\prime}=3\right\rangle\rightarrow\left|51D_{3/2}\right\rangle$
in rubidium 85, as shown in Fig. 1(a). Multifrequency MW fields drive a radio-
frequency (RF) transition between the two different Rydberg states
$\left|51D_{3/2}\right\rangle$ and $\left|50F_{5/2}\right\rangle$. The energy
difference between these states is $17.62\,\mathrm{GHz}$. The multifrequency
MW fields consist of multiple MW bins (more than three bins) with frequency
differences of several kilohertz from the resonance frequency. The amplitudes,
frequencies and phases of the multiple MW bins can be adjusted individually
(further details are provided in the Methods section). The detunings of the
probe, coupling and MW fields are $\Delta_{\text{p}}$, $\Delta_{\text{c}}$ and
$\Delta_{\text{s}}$, respectively. The Rabi frequencies of the probe, coupling
and MW fields are $\Omega_{\text{p}}$, $\Omega_{\text{c}}$, and
$\Omega_{\text{s}}$, respectively. The experimental setup is depicted in Fig.
1(b). We use MW fields to drive the Rydberg states constantly, producing
modulated EIT spectra, i.e. the probe transmission spectra, as shown in the
inset of Fig. 1(b). The phases of the MW fields correlate with the modulated
EIT spectra and can be recovered from these spectra with the aid of deep
learning. Specifically, the probe transmission spectra are fed into a well-
trained deep learning model that consists of a one-dimensional convolution
layer (1D CNN), a bi-directional long-short term memory layer (Bi-LSTM) and a
dense layer to extract the phases of the MW fields. Figure 1(c-e) show these
components of the neural network (further details are presented in the Methods
section). Finally, the bin phases are recovered and the data are read out.

Figure 3: Reconstruction of the QR code after different epochs of training of
the deep learning model. After (a) 3 epochs, (b) 4 epochs and (c) 35 epochs of
training, the accuracies of our deep learning model are
$51.02\%,\,76.87\%,\,99.32\%$, respectively, as determined from the confusion
matrices (last column). The $y$-axis is the ground truth and the $x$-axis
represents the predictions of the deep learning model. The colour bar and the
numbers indicate the counts. If the prediction is correct, then the
corresponding diagonal term in the confusion matrix increases by 1. If
incorrect, then the nondiagonal element increases by 1.

FDM signal encoding and receiving. In the experiments, we use a four-bin
frequency-division multiplexing (FDM) MW signal for demonstration, where one
of the four MW bins is used as the reference bin. The relative phase
differences between the reference bin and the other bins are modulated by the
message signal. Specifically, for the four-bin MW signal,
$E=A_{1}\cos[(\omega_{0}+\omega_{1})t+\varphi_{1}]+A_{2}\cos[(\omega_{0}+\omega_{2})t+\varphi_{2}]+A_{3}\cos[(\omega_{0}+\omega_{3})t+\varphi_{3}]+A_{4}\cos[(\omega_{0}+\omega_{4})t+\varphi_{4}],$
where $\omega_{0}$ is the resonant frequency, $\omega_{1,2,3}$ are the
relative frequencies, the carrier frequencies are
$2\pi(\omega_{0}+\omega_{1})=17.62\,\mathrm{GHz}-3\,\mathrm{kHz}$,
$2\pi(\omega_{0}+\omega_{2})=17.62\,\mathrm{GHz}-1\,\mathrm{kHz}$,
$2\pi(\omega_{0}+\omega_{3})=17.62\,\mathrm{GHz}+1\,\mathrm{kHz}$ and
$2\pi(\omega_{0}+\omega_{4})=17.62\,\mathrm{GHz}+3\,\mathrm{kHz}$, the
frequency difference between two frequency-adjacent bins is $\Delta
f=2\,\mathrm{kHz}$ and the message signal is
$\varphi_{1,2,3}=0\,\mathrm{or}\,\pi$, standing for 3 bits (0 or 1), and the
reference phase is $\varphi_{4}=0$ (which remains unchanged). The phase list
$\left(\varphi_{1},\,\varphi_{2},\,\varphi_{3},\,\varphi_{4}\right)$ is a bit
string for time $t_{0}$. By varying the phase of $\varphi_{1,2,3}$ with time,
we then obtain the FDM signal for binary phase-shift keying (2PSK).
Additionally, the amplitudes of the four bins are $0.1A_{4}=A_{1,2,3}$ to
solve the problem that results from the nonlinearity of the atom, where the
probe transmission spectra of two different bit strings, e.g.
$(0,\,0,\,\pi,\,0)$ and $(0,\,\pi,\,0,\,0)$, are the same (further details are
presented in the Methods section). By increasing the frequency difference
$\Delta f$, we can obtain higher information transmission rates. For four bins
with $\Delta f=2\,\mathrm{kHz}$, the information transmission rate is
$n_{b}\times\Delta f=(4-1)\times 2\times
10^{3}\,\mathrm{bps}=6\,\mathrm{kbps}$, where $n_{b}$ is the number of bits.
In the experiments, disturbances originate from the environment and atomic
collisions. Because of the sensitivity of Rydberg atoms to MW fields, the
resulting noise submerges our signal. To use the sensitivity of the Rydberg
atoms and simultaneously minimize the effects of noise, the deep learning
model is used to extract the relative phases
$\left(\varphi_{1},\,\varphi_{2},\,\varphi_{3}\right)$.

Deep learning. To improve the robustness and speed of our receiver, we use a
deep learning model to decode the probe transmission signal. The complete
encoding and decoding process is illustrated in Fig. 2(a). The Rydberg antenna
receives the FDM-2PSK signal and down-converts this signal into the probe
transmission spectrum. The information is then retrieved from the spectrum
using the deep learning model. The precondition is that the different bit
strings correspond to distinct probe spectra; this is resolved by setting
$0.1A_{4}=A_{1,2,3}$, as discussed earlier. Then, we combine the 1D CNN layer,
the Bi-LSTM layer and the dense layer to form the deep learning model (See the
Methods section for further details) (Chollet  _et al._ , 2015; Pedregosa  _et
al._ , 2011). One of the reasons for using the 1D CNN layer and the Bi-LSTM
layer is that the data sequences are long, which means that prediction of the
phases
$\overrightarrow{\varphi}=\left(\varphi_{1},\,\varphi_{2},\,\varphi_{3},\,0\right)$
from the spectrum is a regression task and requires a long-term memory for our
model. Another reason is to combine the convolution layer’s speed with the
sequential sensitivity of the Bi-LSTM layer (Chollet, 2017). The input
sequence is first processed by the 1D CNN to extract the features, meaning
that a long sequence is converted into a shorter sequence with higher-order
features. This process is visualized to show how the deep learning model
treats the transmission spectrum; more details are presented in the
supplementary materials. The shorter sequence is then fed into the Bi-LSTM
layer and resized by the dense layer to match the label size (see the Methods
section for further details). Specifically, the probe spectrum
$\mathbf{T}=\left\\{T_{0},\,T_{\tau},\,T_{2\tau},\,\cdots,\,T_{i\cdot\tau}\cdots,\,T_{N\cdot\tau}\right\\}$
and the corresponding phases
$\overrightarrow{\varphi}=\left(\varphi_{1},\,\varphi_{2},\,\varphi_{3},\,\varphi_{4}=0\right)$
are collected to form the data set, where $T_{i\cdot\tau}$ is the i-th data
point of a probe spectrum and the fourth bit $\varphi_{4}=0$ is the reference
bit. Both the spectra and the phases are 1D vectors with dimensions of $N+1$
and 4, respectively. These independent, identically distributed data
{{$\mathbf{T}$}, {$\overrightarrow{\varphi}$}} are fed into our model as a
data set. By shuffling this data set and splitting it into three sets, i.e. a
training set, a validation set and a test set, we train our model on the
training set (feeding both the waveforms and labels {{$\mathbf{T}$},
{$\overrightarrow{\varphi}$}}), validate and test our model on the validation
and test sets, respectively (by feeding waveforms without labels and comparing
the predictions with ground truth labels). The validation set is used to
determine whether there is either overfitting or underfitting during training.
Finally, the performance (i.e. accuracy) of the model is estimated by
predicting the test set.

Figure 4: Prediction accuracy of deep learning model and the master equation
on the noisy test set. The noise is white noise with mean $\mu=0$ and a
standard deviation $\sigma$. The noise is added quantitatively using different
$\sigma$ values. Before and after addition of the noise, the data are scaled
between 0 and 1 using their maximum and minimum. The deep learning model is
trained in a training set without additional noise. In addition, we do not
involve noise spectrum when solving the master equation. Each point is
obtained after averaging of five predictions on the noisy data.

The performance of our deep learning model is affected by the training epochs
and the training and validation set sizes. The training curves on different
training sets and validation sets are shown in Figs. 2(b-c). Initially, our
model performs well on the training set only, implying overfitting. The curves
then converge (dashed line) and our model performs well on both the training
set and the validation set. The sudden jump in the loss curve in Fig. 2(c) is
caused by the change in the learning rate (further details are presented in
the Methods section). Use of more training and validation data causes the
curves to converge more quickly. The deep learning model performs well after
these few-sample training. In Fig. 2(d), we show a confusion matrix for
prediction of a uniformly distributed test set, which demonstrates accuracy of
99.38%.

The “noise” shown in Fig. 2(a) refers to two kinds of noises. One comes from
atoms and the external environment (systematic noise). The other comes from
the noise added on purpose (additional noise). The systematic noise cannot be
adjusted quantitatively and is discussed with its noise spectrum in the
supplementary materials. Because the noise on the data set is independent and
is distributed identically (i.i.d.), i.e., the entire data set is shuffled
before being split into the training and test sets, the systematic noise
pattern is almost the same in both the training set and the test set. The deep
learning model has already learned the systematic noise pattern during the
training process, which is one of the major advantages of use of deep learning
against systematic noise. However, there is a case where the noise is not
i.i.d. (i.e., the case where a specific noise occurs during testing only).
This problem can be solved by online learning and addition of prior knowledge
as new features into the data, e.g., data for the temperature, the weather,
and other factors (Sahoo  _et al._ , 2017). Here for simplicity, we talk about
the i.i.d. case only and add the white noise with a mean $\mu$ and a standard
deviation $\sigma$. We ignore the 1/f noise in this case because it decays
rapidly in the low frequency range and the signal with which it would
interfere is located within the $2\sim 200$ kHz range. The additional noise is
added both on the training set and the test set of the deep learning model in
Fig. 2(e), which demonstrates the performance of the deep learning model when
used on a data set with biased or unbalanced noise. The results below the red
line show the performance of the model after training on a weaker-noise
training set when predicting based on a stronger-noise test set, i.e.,
generalization for a stronger noise case. These results indicate that the deep
learning model has the generalization ability required to adapt to stronger
noise. In the area above the red line, there is more noise in the training set
than in the test set. Theoretically, a small amount of additional noise in the
training set will increase the robustness of the deep learning model. However,
when the noise increases, it affects the accuracy, which decays rapidly. Next,
the well-trained model is used to reconstruct the QR code. In Figs. 3(a-c),
the results and the corresponding confusion matrices with their epochs are
shown. First, the information is encoded into a QR code. After the code is
transmitted, received and decoded using the Rydberg atoms and the deep
learning model, the information is then reconstructed successfully using the
35-epoch training model in Fig. 3(c), but is not reconstructed in parts (a)
and (b). The accuracy is defined by the number of correctly predicted bit
strings divided by the total number of bit strings (147 bit strings). After 35
epochs, the accuracy reaches $99.32\%$ and the message is reconstructed
successfully from the QR code received.

Comparison between deep learning method and the master equation. In our case,
the master equation that we employed is the commonly used one without
considering the noise spectrum. The accuracies of the deep learning model and
the master equation fitting on noisy data are different. Fig. 4 shows the
accuracies obtained by the two methods. The deep learning model is trained on
a training set without additional noise, and tested on a test set with
additional white noise whose standard deviation is $\sigma$ (the transmission
spectra with noise are given in supplementary materials). Here for simplicity,
the data set is composed of the transmission of four MW bins only (one of them
is reference bin) and the frequency difference between the adjoin bins is
$\Delta f=2\mathrm{kHz}$. On the other hand, the result of the master equation
is given based on the same test set as that of the deep learning model. The
deep learning method outperforms the fitting of the master equation on the
noisy data set.

Figure 5: Loss curves and confusion maps of the deep learning method and the
master equation fitting method for MWs with 20 bins and frequency difference
$\Delta f$. (a) Loss vs. epoch curves for training and validation of a 20-bin
carrier with frequency difference $\Delta f=2\,\mathrm{kHz}$. (b) Confusion
matrix for epoch=78 on the test data set. Only the first 3 bits of the 20 bits
are selected to carry the messages and all other bits are 0. The accuracy is
100% for 123 testing spectra. (c) Loss vs. epoch curves for training and
validation of the four MW bins (one bin is the reference) with frequency
difference $\Delta f=200\,\mathrm{kHz}$. (d) Confusion matrix for epoch=83 on
the test data set. The accuracy is 98.83% for 171 testing spectra. (e)
Predicted solution of the master equation without consideration of higher-
order terms and noise on a 20-bin carrier with frequency difference $\Delta
f=2\,\mathrm{kHz}$. The number of spectra is 160, where the spectra were
sampled uniformly from eight categories. The accuracy is 20.63%. (f) Predicted
solution of the master equation without consideration of higher-order terms
and noise on four-bin carrier with frequency difference $\Delta
f=200\,\mathrm{kHz}$. The number of spectra is 160, which were sampled
uniformly from eight categories. The accuracy is 60.00%. In (b) and (e), only
the first 3 bits of a total of 20 bits (including a reference) are labelled.

Apart from the robutness to the noise, when the transmission rate is increased
by increasing the number of MW bins or the frequency difference $\Delta f$,
the deep learning model performs well, while it is difficult to retrieve the
messages with high accuracy using the master equation. Specifically, to
increase the bandwidth efficiency and the transmission rate, the number of MW
bins used to carry the messages must be increased, but the information is
still recognizable because of the scalability of the deep learning model. For
20 MW bins, the number of bits is $(20-1)$ with one reference bit, giving a
$\left(20-1\right)\times 2\,\mathrm{kbps}=38\,\mathrm{kbps}$ transmission
rate. The number of combinations of these bits is $2^{19}$, which increases
exponentially as the number of MW bins increases. Here, for demonstration
purposes, only the first 3 bits of the total of 19 bits carry the messages and
the other bits, including the reference, are set to be 0. To show how well our
model performs, we train, validate and test the model on this new data set
without varying the other parameters, with the exception of the training
epochs of our model. The loss curves for training and validation are shown in
Fig. 5(a). A confusion matrix for epoch 78 is shown in Fig. 5(b). The model
performs well on this new test set, which was sampled uniformly from eight
categories with accuracy of 100%. Another method that can be used to increase
the information transmission rate involves increasing the frequency
difference. In our case, the frequency difference is increased from $\Delta
f=2\,\mathrm{kHz}$ to $\Delta f=200\,\mathrm{kHz}$. The transmission rate
increases correspondingly, from $(4-1)\times
2\,\mathrm{kbps}=6\,\mathrm{kbps}$ to $(4-1)\times
200\,\mathrm{kbps}=0.6\,\mathrm{Mbps}$. To detect the high-speed signal, the
DD bandwidth is increased, which inevitably leads to increased noise. After
the model is trained on this new data set, the training and validation loss
curves are as shown in Fig. 5(c). A confusion matrix for epoch 83 is shown in
Fig. 5(d). Increasing the number of training epochs allows the model to
perform well on this new data set, with accuracy of 98.83% on a uniformly
sampled test set.

To compare the performances of the deep learning model and the master
equation, we fitted the probe spectra for 20 bins with a frequency difference
$\Delta f=2\,\mathrm{kHz}$ and four bins with a frequency difference $\Delta
f=200\,\mathrm{kHz}$ by solving the master equation without considering the
higher-order terms and the effects of noise. In each case, 160 probe spectra
were fitted that were sampled uniformly from every category. The prediction
results are shown in Fig. 5(e) and (f). The prediction accuracy of the master
equation is lower than that of the deep learning model. In our case, the
impact of increasing the number of bins is greater than increasing the DD
bandwidth for high-speed signals on the fitting accuracy. The prediction
accuracy for a 20-bin carrier with frequency difference $\Delta
f=2\,\mathrm{kHz}$ is 20.63%, which is like to the accuracy of guessing, i.e.,
$1/8$. This implies that there is a disadvantage that comes from the fitting
method itself, i.e., it can easily become trapped by local minima. Some type
of prior knowledge is required to overcome this disadvantage, e.g., provision
of the initial values of the phases before fitting. In contrast, the deep
learning model is data-driven and does not require any prior knowledge. The
local minima problem of deep learning can be overcome using some well-known
techniques, including learning rate scheduling and design of a more effective
optimizer (Chollet, 2017). Additionally, the accuracy difference for the
200-kHz-difference MW bins between the deep learning model and the master
equation means that the deep learning model is more robust to noise.
Furthermore, the prediction time for the master equation is $25\,\mathrm{s}$
per spectrum, while the time for the deep learning model is $1.6\mathrm{\,ms}$
per spectrum. The master equation is solved by “FindFit” function in
Mathematica 11.1 with both “AccuracyGoal” and “PrecisionGoal” default, while
the deep learning code is written in Python 3.7.6. These codes are run on the
same computer with NVIDIA GTX 1650 and Intel®$\mathrm{Core}^{\mathrm{TM}}$
i7-9750H.

Another method to decode the signal is available that uses an in-phase and
quadrature (I-Q) demodulator or a lock-in amplifier (Holloway  _et al._ ,
2019a; Meyer  _et al._ , 2018). However, the carrier frequency must be given
when decoding the signal in this case. Additionally, for multiple MW bins,
numerous bandpass filters are required. The deep learning method is thus much
more convenient.

Discussion

We report a work on Rydberg receiver enhanced via deep learning to detect
multifrequency MW fields. The results show that the deep learning enhanced
Rydberg mixer receives and decodes multifrequency MW fields efficiently; these
fields are often difficult to decode using theoretical methods. Using the deep
learning model, the Rydberg receiver is robust to noise induced by the
environment and atomic collisions and is immune to the distortion that results
from the limited bandwidths of the Rydberg atoms (from dipole-dipole
interactions and the EIT pumping rate, as studied in Ref. (Meyer  _et al._ ,
2018)) for high-speed signals ($\Delta f=200\,\mathrm{kHz}$). In addition to
increasing the transmission speed of the signals, further increments in the
information transmission rate are achieved by using more bins, which is
feasible because of the scalability of our model. Besides the transmission
rate, this deep learning enhanced Rydberg system promises for use in studies
of the channel capacity limitations. Because spectra that are difficult for
humans to recognize as a result of noise and distortion are distinguishable
when using the deep learning model, Rydberg systems enhanced by deep learning
could take steps toward the realization of the capacity limit proposed in the
literature Ref. (Cox  _et al._ , 2018). To obtain high performance (i.e. high
signal-to-noise ratio, information transmission rate, channel capacity and
accuracy), the training epochs and training set must be extended and enlarged.

In summary, we have demonstrated the advantages of receiving and decoding
multifrequency signals using a deep learning enhanced Rydberg receiver. In a
multifrequency signal receiver, rather than using multiple band-pass filters,
lock-in amplifier (Holloway  _et al._ , 2019a; Meyer  _et al._ , 2018) and
other complex circuits, signals can be decoded using the extremely sensitive
Rydberg atoms and the deep learning model at high speed and with high accuracy
without solving the Lindblad master equation. One of the advantages of use of
the Rydberg atom is that the accuracy of the Rydberg atom approaches the
photon shot noise limit (Kumar  _et al._ , 2017). In principle, the accuracy
of the Rydberg atom is higher than that of the classical antenna. According to
recent work based on the atomic superheterodyne method, ultrahigh sensitivity
can be obtained (Jing  _et al._ , 2020). However, in this proof-of-principle
demonstration, there is considerable room for the optimization required to
reach that limit (e.g., stabilization of the laser, narrowing the laser
linewidth, and temperature stabilization). The sensitivity of the Rydberg
atoms is a double-edged sword because it also involves noise. The deep
learning model restricts this side effect while taking full advantage of the
Rydberg atoms’ sensitivity to the signal. Using the automatic feature
extraction processes of the neural networks, the spectra are classified in a
supervised manner. If the features (e.g. mean value, variance, frequency
spectrum) are extracted manually, the spectra are then clustered by
unsupervised learning methods such as t-distributed stochastic neighbour
embedding (t-SNE) or the density-based spatial clustering of applications with
noise (DBSCAN) method (Pedregosa  _et al._ , 2011), without training on the
training set. Our work will be useful in fields including high-precision
signal measurement and atomic sensors. Additionally, this decoding ability can
be generalized further to decode other signals that are encoded by different
encoding protocols, e.g., frequency division multiplexing amplitude shift
keying (FDM-ASK), frequency division multiplexing quadrature amplitude
modulation (FDM-QAM), and IEEE 802.11ac WLAN standard signals for a 5 GHz
carrier. The frequency of carrier to be decoded covers from several hertz to
terahertz since for Rydberg atoms to receive MW with different wavelengths,
the only part of the system that needs to be tuned is the frequency of the
laser, while in classical receivers, the wavelength of the received MW is
limited by the size of the antenna (Meyer  _et al._ , 2021, 2020; Wade  _et
al._ , 2018; Holloway  _et al._ , 2014). In addition to communications, our
receiver can be used to detect multiple targets from multifrequency signals
caused by the Doppler effect.

Methods

Generation and calibration of MW fields. The MW fields used in our experiments
were synthesized by the signal generator (1465F-V from Ceyear) and a frequency
horn. Each bin in the multifrequency MW field is tunable in terms of
frequency, amplitude and phase. The RF source operates in the range from DC to
40 GHz. The frequency horn is located close to the Rb cell. We used an antenna
and a spectrum analyzer (4024F from Ceyear) to receive the MW fields and then
calibrated the amplitudes of the MW fields at the centre of the Rb cell.

The probe transmission spectrum in the time domain when $\Delta_{\text{p}}=0$,
$\Delta_{\text{c}}=0$ and $\Delta_{\text{s}}=0$ reflects the interference
among the multifrequency MW bins, which results from the beat frequencies of
the bins that occur through the interaction between the atoms and light. The
Rydberg atoms receive the MW bins by acting as an antenna and a mixer (Gordon
_et al._ , 2019; Simons  _et al._ , 2019; Holloway  _et al._ , 2019a). After
reception by the atoms, the frequency spectrum of the probe transmission shows
that we can obtain the frequency differential signal from the probe
transmission spectrum. This represents an application of our atoms to reduce
the modulated signal frequency (from terahertz to kilohertz magnitude), which
allows the signal to be received and decoded using simple apparatus. In our
experiment, more than 20 frequency bins can be added to the atoms, for which
the dynamic range is greater than $30\,\mathrm{dBm}$. The amplitudes, phases
and frequencies of these bins can be tuned individually. When the bandwidth is
increased to detect an increasing frequency difference $\Delta f$ signal, more
noise is involved, but this noise is suppressed by the deep learning model. In
other words, the signal can be recognized using the deep learning model when
the information transmission rate is increased by raising the frequency
difference $\Delta f$. These bins are used to send FDM-PSK signals in the “FDM
signal encoding and receiving ” section of the main text.

Master equation. The Lindblad master equation is given as follows:
$\text{d}\rho/\text{d}t=-i\left[H,\rho\right]/\hbar+L/\hbar$, where $\rho$ is
the density matrix of the atomic ensemble and $H=\sum_{k}H[\rho^{(k)}]$ is the
atom-light interaction Hamiltonian when summed over all the single-atom
Hamiltonians using the rotating wave approximation. This Hamiltonian has the
following matrix form:

$H=\hbar\left(\begin{array}[]{cccc}0&-\frac{\Omega_{\text{p}}}{2}&0&0\\\
-\frac{\text{$\Omega_{\text{p}}$
}}{2}&\Delta_{\text{p}}&-\frac{\Omega_{\text{c}}}{2}&0\\\
0&-\frac{\text{$\Omega_{\text{c}}$}}{2}&\text{$\Delta_{\text{c}}$}+\text{$\Delta_{\text{p}}$}&-\frac{\Omega_{\text{s}}(t)}{2}\\\
0&0&-\frac{\Omega_{\text{s}}(t)}{2}&\text{$\Delta_{\text{c}}$}+\text{$\Delta_{\text{p}}$}+\Delta_{\text{s}}\end{array}\right),$
(1)

where for the MW signal
$E=A_{1}\cos[\left(\omega_{0}+\omega_{1}\right)t+\varphi_{1}]+A_{2}\cos[\left(\omega_{0}+\omega_{2}\right)t+\varphi_{2}]+A_{3}\cos[\left(\omega_{0}+\omega_{3}\right)t+\varphi_{3}]+A_{4}\cos[\left(\omega_{0}+\omega_{4}\right)t+\varphi_{4}]$,
we have the Rabi frequency $\Omega_{\text{s}}(t)=\sqrt{E_{1}^{2}+E_{2}^{2}}$ ,
where
$E_{1}=A_{1}\sin[\omega_{1}t+\varphi_{1}]+A_{2}\sin[\omega_{2}t+\varphi_{2}]+A_{3}\sin[\omega_{3}t+\varphi_{3}]+A_{4}\sin[\omega_{4}t+\varphi_{4}]$
and
$E_{2}=A_{1}\cos[\omega_{1}t+\varphi_{1}]+A_{2}\cos[\omega_{2}t+\varphi_{2}]+A_{3}\cos[\omega_{3}t+\varphi_{3}]+A_{4}\cos[\omega_{4}t+\varphi_{4}]$.
The Rabi frequency can be derived as follows:

$\displaystyle E$
$\displaystyle=\sum_{i=1}^{4}A_{i}\cos\left[\left(\omega_{0}+\omega_{i}\right)t+\varphi_{i}\right]$
$\displaystyle=\sqrt{E_{1}^{2}+E_{2}^{2}}\cos\left(\omega_{0}t+\arctan\frac{\sum_{i=1}^{4}A_{i}\sin\left(\omega_{i}t+\varphi_{i}\right)}{\sum_{i=1}^{4}A_{i}\cos\left(\omega_{i}t+\varphi_{i}\right)}\right)$
(2)

where the second term (which resonates with the energy levels of the Rydberg
atoms) induces the normal EIT spectrum and the first term modulates that
spectrum. In the interaction between the atoms and the MW fields, the atoms
act as a mixer such that the output signal frequency ($\omega_{1}$,
$\omega_{2}$, $\omega_{3}$) is less than the input signal frequency
($\omega_{0}+\omega_{1}$, $\omega_{0}+\omega_{2}$, $\omega_{0}+\omega_{3}$).
The modulation signal’s nonlinearity is reduced by setting the reference and
increasing its amplitude as shown in equation 3, which is a precondition for
recognition of these phases via deep learning.

$\displaystyle\sqrt{E_{1}^{2}+E_{2}^{2}}$ $\displaystyle\approx
A_{4}\sqrt{1+2\sum_{i=1}^{3}\frac{A_{i}}{A_{4}}\cos\left[\left(\omega_{4}-\omega_{i}\right)t+\left(\varphi_{4}-\varphi_{i}\right)\right]}$
$\displaystyle\approx
A_{4}+\sum_{i=1}^{3}A_{i}\cos\left[\left(\omega_{4}-\omega_{i}\right)t+\left(\varphi_{4}-\varphi_{i}\right)\right]$
(3)

where the condition for the approximations on the second line and the third
line is $A_{4}\gg A_{1,2,3}$.

The Lindblad superoperator $L=\sum_{k}L[\rho^{(k)}]$ is composed of single-
atom superoperators, where $L[\rho^{(k)}]$ represents the Lindbladian and has
the following form:
$\frac{L[\rho^{(k)}]}{\hbar}=-\frac{1}{2}\sum_{m}\left(C_{m}^{\dagger}C_{m}\rho+\rho
C_{m}^{\dagger}C_{m}\right)+\sum_{m}C_{m}\rho C_{m}^{\dagger}$ where
$C_{1}=\sqrt{\Gamma_{\text{e}}}\left|g\right\rangle\left\langle e\right|$,
$C_{2}=\sqrt{\Gamma_{\text{r}}}\left|e\right\rangle\left\langle r\right|$ and
$C_{3}=\sqrt{\Gamma_{\text{s}}}\left|r\right\rangle\left\langle s\right|$ are
collapse operators that stand for the decays from state $\left|e\right\rangle$
to state $\left|g\right\rangle$, from state $\left|r\right\rangle$ to state
$\left|e\right\rangle$ and from state $\left|s\right\rangle$ to state
$\left|r\right\rangle$ with rates $\Gamma_{\text{e}}$, $\Gamma_{\text{r}}$ and
$\Gamma_{\text{s}}$, respectively. Because we are only concerned with the
steady state here, i.e. $t\rightarrow\infty$, the Lindblad master equation can
be solved using $\text{d}\rho/\text{d}t=0$. The complex susceptibility of the
EIT medium has the form
$\chi(v)=(\lvert\mu_{ge}\rvert^{2}/\epsilon_{0}\hbar)\rho_{eg}$ , where
$\rho_{eg}$ is the element of density matrix solved using the master equation.
The spectrum of the EIT medium can be obtained from the susceptibility using
$T\sim e^{-{\rm Im}[\chi]}.$

Deep learning layers. Our deep learning model consists of a 1D CNN layer, a
Bi-LSTM layer and a dense layer. The mathematical sketches for these layers
are given as follows.

The 1D CNN layer is illustrated in Fig. 1(c). The input signal convolutes the
kernel in the following form:

$f\otimes g=\sum_{m=0}^{N-1}f_{m}g_{(n-m)}.$ (4)

where $f$ represents the input data, $g$ is the convolution kernel, $m$ is the
input data index and $n$ is the kernel index. The 1D CNN extracts the higher-
order features from the input data to reduce the lengths of the sequences fed
into the Bi-LSTM layer. Before flowing into the Bi-LSTM layer, the data pass
through the batch normalization layer, the ReLU activation layer and the max-
pooling layer, in that sequence. For a mini-batch
$\mathcal{B}=\left\\{x_{1\cdots m}\right\\}$, the output from the batch
normalization layer is $y_{i}=BN_{\gamma,\beta}(x_{i})$ and the learning
parameters are $\gamma$ and $\beta$ (Ioffe and Szegedy, 2015). The update
rules for the batch normalization layer are:

$\displaystyle\mu_{\mathcal{B}}$ $\displaystyle\leftarrow$
$\displaystyle\frac{1}{m}\sum_{i=1}^{m}x_{i}$ (5)
$\displaystyle\sigma_{\mathcal{B}}^{2}$ $\displaystyle\leftarrow$
$\displaystyle\frac{1}{m}\sum_{i=1}^{m}\left(x_{i}-\mu_{\mathcal{B}}\right)^{2}$
(6) $\displaystyle\hat{x}_{i}$ $\displaystyle\leftarrow$
$\displaystyle\frac{x_{i}-\mu_{\mathcal{B}}}{\sqrt{\sigma_{\mathcal{B}}^{2}+\epsilon}}$
(7) $\displaystyle y_{i}$ $\displaystyle\leftarrow$
$\displaystyle\gamma\hat{x}_{i}+\beta\equiv BN_{\gamma,\beta}(x_{i})$ (8)

where equations 5 and 6 evaluate the mean and the variance of the mini-batch,
respectively; the data are normalized using the mean and the variance in
equation 7 and the results are then scaled and shifted in equation 8. The
training is accelerated using the batch normalization layer and the
overfitting is also weakened by this layer. The output then passes through the
ReLU activation layer. The activation function of this layer is
$f_{\mathrm{ReLU}}(x)=\max(x,0)$. The vanishing gradient problem is diminished
by this activation function. Next, the inputs are downsampled in a max-pooling
layer (Chollet  _et al._ , 2015).

Figure 6: Building blocks for LSTM layer and dense layer. The cells of a long-
short term memory (LSTM) layer and dense layer are presented in (a) and (b),
respectively, where $\sigma$ represents the sigmoid layer and $\tanh$
represents the tanh layer.

The LSTM layer and an LSTM cell are shown schematically in Fig. 1(d) and 6(a),
respectively. The equations for the LSTM are shown as equations 9-14 (Chollet,
2017; Hochreiter and Schmidhuber, 1997). At a time $t$, the input $x_{t}$ and
two internal states $C_{t-1}$and $h_{t-1}$ are fed into the LSTM cell. The
first thing to be decided by the LSTM cell is whether or not to forget in
equation 9, which outputs a number between 0 and 1 that represents retaining
or forgetting. Next, an input gate (equation 10) decides which values are to
be updated from a vector of new candidate values created using equation 11.
The new value is then added to the cell state and the old value is forgotten
in equation 12. Finally, the cell decides what to output using equations 13
and 14.

$\displaystyle f_{t}$ $\displaystyle=$
$\displaystyle\sigma\left(W_{f}\cdot\left[h_{t-1},x_{t}\right]+b_{f}\right)$
(9) $\displaystyle i_{t}$ $\displaystyle=$
$\displaystyle\sigma\left(W_{i}\cdot\left[h_{t-1},x_{t}\right]+b_{i}\right)$
(10) $\displaystyle\tilde{C}_{t}$ $\displaystyle=$
$\displaystyle\tanh\left(W_{C}\cdot\left[h_{t-1},x_{t}\right]+b_{C}\right)$
(11) $\displaystyle C_{t}$ $\displaystyle=$ $\displaystyle f_{t}\times
C_{t-1}+i_{t}\times\tilde{C}_{t}$ (12) $\displaystyle o_{t}$ $\displaystyle=$
$\displaystyle\sigma\left(W_{o}\left[h_{t-1},x_{t}\right]+b_{o}\right)$ (13)
$\displaystyle h_{t}$ $\displaystyle=$ $\displaystyle
o_{t}\times\tanh\left(C_{t}\right)$ (14)

where $\sigma(x)=1/(1+e^{-x})$ is the sigmoid function. The sigmoid and
$\tanh$ functions are applied in an element-wise manner. The LSTM is followed
by a time-reversed LSTM to constitute a Bi-LSTM layer that improves the memory
for long sequences.

Figure 7: Structure of our deep learning model and size of the data. The ReLU
activation function is $f(x)=max(0,x)$. After the data output from the max-
pooling layer, the visualizations are performed; see supplementary materials
for more details.

The dense layer and a neuron are drawn in Fig. 1(e) and 6(b), respectively,
and the corresponding equations are

$\displaystyle\mathbf{a}$ $\displaystyle=$
$\displaystyle\mathbf{w}\cdot\mathbf{x}+b$ (15) $\displaystyle\mathbf{y}$
$\displaystyle=$ $\displaystyle g(\mathbf{a})$ (16)

where $\mathbf{w}$ is the vector of weights, $b$ is the bias, $x$ represents
the input data, $g(\mathbf{a})=1/(1+e^{-\mathbf{a}})$ is the sigmoid
activation function used to limit the output values to between 0 and 1, and
$y$ is the output. The dense layer resizes the shape of the data obtained from
the Bi-LSTM to match the size of the label.

The training consists of both forward and backward propagation. A batch of
probe spectra propagates through the 1D CNN layer, the Bi-LSTM layer and dense
layer during the forward training process. The differentiable loss function is
then calculated. In our case, the differentiable loss function is the mean
squared error (MSE) between the predictions and the ground truth, which is
used widely in the regression task (Chollet, 2017). The equation for the MSE
is

$L_{MSE}=\frac{1}{m\cdot
n}\sum_{j=1}^{n}\sum_{i=1}^{m}\left(\varphi_{i,j}-f(T_{i,j})\right)^{2},$ (17)

where $m$ is the number of data points in one spectrum, $n$ is the mini-batch
size, $\varphi_{i}$ is the ground truth and $f(T_{i})$ is the model
prediction. In backpropagation, the trainable weights of each layer are
updated based on the learning rate and the derivative of the MSE loss function
with respect to the weights to minimize the loss $L_{MSE}$, such that

$W\leftarrow W-\eta\frac{\partial L_{MSE}}{\partial W},$ (18)

where $\eta$ is the learning rate and $W$ is the trainable weight for each
layer. The weights of each layer are then updated according to the RMSprop
optimizer(Geoffrey  _et al._ , 2012).

The network is implemented using the Keras 2.3.1 framework on Python 3.6.11
(Chollet  _et al._ , 2015). All weights are initialized with the Keras
default. The hyper-parameters of the deep learning model (including the
convolution kernel length, the number of hidden variables and the learning
rate) are tuned using Optuna (Akiba  _et al._ , 2019).

Deep learning pipeline. To obtain better fitting results, the data are scaled
based on their maximum and minimum values, i.e.
$T_{i}^{\prime}=(T_{i}-min(T))/(max(T)-min(T))$. The labels are encoded in
dense vectors with four elements rather than in one-shot encoding vectors to
save space (Chollet, 2017). Each of these elements is either 0 or 1,
representing the relative phase 0 or $\pi$ of each bin, respectively.

A one-dimensional convolution layer (1D CNN), a bidirectional long short-term
memory layer (Bi-LSTM) and a dense layer are used in our deep learning model.
The deep learning model structure is shown in Fig. 7. The data size for the
input layer is given in the form (batch size, length of probe spectrum, number
of features). The batch size is 64 in our case. Because the duration of the
spectrum ranges from $t=0$ to $t=0.999\,\mathrm{ms}$ with a time difference of
$\tau=1\,\mathrm{\mu s}$, the spectrum length is 1000\. For a one-dimensional
input, the number of features is 1. Therefore, the data size for the input
layer is (64, 1000, 1).

Figure 8: Data partition during training, validating, and testing. First, the
data are split into two sets. The first is the test set. The remaining data
set is copied four times and is then split into four sets, each with different
parts that act as training sets and validation sets for the training of four
deep learning models.

During training of this model, four-fold cross-validation is used to save the
amount of training data.The data set is split as shown in Fig. 8. First, the
data set is split into two parts. The first is the test set (red), which
remains untouched during training. The second (purple) is used to train the
model. In the cross-validation process, the rest data set (purple) is copied
four times and is divided equally into four parts each. One of these parts is
the validation data set (green) and the others are used as training sets
(blue). Four models are trained on the different training sets and validation
sets. Then the best model is chosen according to the validation set and is
tested on the test set. After splitting, the training set, the validation set,
and the test set all remain unchanged. In every epoch, each model iterates the
training set only once. There is no new set being taken; instead, the same
training set is iterated once each epoch.

The computational graph is cleared before each training sequence to prevent
leakage of the validation data. Gaussian noise (where the mean is 0 and the
standard deviation is 0.5) is added to the training data to increase the
robustness of the proposed model. In addition, the learning rate is adjusted
during training to jump out of the local minimum, which results in the jump in
Figure 2(c) in the main text. The initial learning rate is 0.001. If the loss
(mean-square error) of the validation set does not decrease over 10 epochs,
the learning rate is multiplied by 0.1. The RMSprop optimizer is used to
update the weight of each layer during training (Geoffrey  _et al._ , 2012).

The bidirectional LSTM layer can be replaced with the well-known self-
attention layer to improve the memory of our proposed model further (Vaswani
_et al._ , 2017). However, this would require more training time and increased
GPU memory. The current model has been able to meet our requirements to date.

Data availability. The data that support this study are available in Github
(Liu, 2022) (https://github.com/ZongkaiLiu/Deep-learning-enhanced-Rydberg-
multifrequency-microwave-recognition).

Code availability. The deep learning code and Mathematica code are provided in
Github (Liu, 2022) (https://github.com/ZongkaiLiu/Deep-learning-enhanced-
Rydberg-multifrequency-microwave-recognition).

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Acknowledgments. Zong-Kai Liu gratefully acknowledges the instructive
discussion about deep learning with Yue Chen at the National Engineering
Laboratory for Speech and Language Information Processing and the
enlightenment of leraning machine learning during studying with Dr. Lei Gong
at Department of Optics and Optical Engineering at USTC. D.-S.D acknowledges
funding from the National Key R&D Program of China (Grant No. 2017YFA0304800),
the National Natural Science Foundation of China (Grant Nos. U20A20218,
61525504, 61435011), the Anhui Initiative in Quantum Information Technologies
(Grant No. AHY020200), and the Youth Innovation Promotion Association of the
Chinese Academy of Sciences (Grant No. 2018490). B.-S.S acknowledges funding
from the National Natural Science Foundation of China (Grant No. 11934013). We
thank David MacDonald, MSc, from Liwen Bianji, Edanz Editing China
(www.liwenbianji.cn/ac), for editing the English text of a draft of this
manuscript.

Author contributions. D.-S.D. conceived the idea for the study. Z.-K.L.
implemented the physical experiments and designed the deep learning model.
Z.-K.L. wrote the manuscript. D.-S.D, B.-S.S. and, G.-C.G. supervised the
research. All authors contributed to discussions regarding the results and
analysis contained in the manuscript.

Competing financial interests. The authors declare no competing financial
interests.