# Analytic continuation via domain-knowledge free machine learning

Hongkee Yoon  Department of Physics, KAIST, 291 Daehak-ro, Yuseong-gu, Daejeon
34141, Republic of Korea  Jae-Hoon Sim  Department of Physics, KAIST, 291
Daehak-ro, Yuseong-gu, Daejeon 34141, Republic of Korea  Myung Joon Han
<EMAIL_ADDRESS>Department of Physics, KAIST, 291 Daehak-ro, Yuseong-gu,
Daejeon 34141, Republic of Korea

###### Abstract

We present a machine-learning approach to a long-standing issue in quantum
many-body physics, namely, analytic continuation. This notorious ill-
conditioned problem of obtaining spectral function from imaginary time Green’s
function has been a focus of new method developments for past decades. Here we
demonstrate the usefulness of modern machine-learning techniques including
convolutional neural networks and the variants of stochastic gradient descent
optimiser. Machine-learning continuation kernel is successfully realized
without any ‘domain-knowledge’, which means that any physical ‘prior’ is not
utilized in the kernel construction and the neural networks ‘learn’ the
knowledge solely from ‘training’. The outstanding performance is achieved for
both insulating and metallic band structure. Our machine-learning-based
approach not only provides the more accurate spectrum than the conventional
methods in terms of peak positions and heights, but is also more robust
against the noise which is the required key feature for any continuation
technique to be successful. Furthermore, its computation speed is 104–105
times faster than maximum entropy method.

##  I Introduction

Matsubara Green’s function method is a useful theoretical tool for quantum
many-body problems. While the calculation often becomes much more tractable in
the imaginary time (or equivalently, frequency) domain, working with Matsubara
function inevitably introduces other theoretical difficulties. One of the most
typical cases happens when one tries to obtain a spectral function (or any
other measurable quantity), which is defined in real frequency space, from
imaginary Green’s function. This procedure is known as ‘analytic continuation’
and poses a notorious ill-conditioned inverse problem. The severe noise
sensitivity significantly undermines the predictability and the usefulness of
theoretical methods such as quantum Monte Carlo (QMC). Many different
approaches have been suggested to solve this problem including Pade
approximation Vidberg and Serene (1977); Gunnarsson  _et al._ (2010a), maximum
entropy (MEM) Haule  _et al._ (2010); Bergeron and Tremblay (2016); Jarrell
and Gubernatis (1996); Gunnarsson  _et al._ (2010b), and stochastic method
Sandvik (1998). All these methods are based on the physical knowledge or
utilize the pre-understanding of the problem which are expressed in their own
assumptions and fitting parameters. In other words, all these methods heavily
rely on ‘domain knowledge’.

Machine-learning (ML) approach is based on a different philosophy. The ML
procedure is to develop a machinery which can self-learn the governing rule or
the proper representation of a given problem through the massive dataset
‘training’ Rifai  _et al._ (2011); Goodfellow  _et al._ (2016); Zhu  _et al._
(2016); LeCun  _et al._ (2015). Due to the remarkable progress in both
hardware and software engineering, ML technique overwhelms the state-of-the-
art human-designed algorithms in many different areas Silver  _et al._ (2016);
LeCun  _et al._ (2015). Recently, it becomes more and more popular in physics
research. ML proves its capability in many different fields ranging from
materials science Behler and Parrinello (2007); Faber  _et al._ (2016); Seko
_et al._ (2017); Kolb  _et al._ (2017); Behler (2016); Artrith  _et al._
(2017) and statistical physics Torlai and Melko (2016); Huang and Wang (2017);
Carrasquilla and Melko (2017) to quantum many-body problems Arsenault  _et
al._ (2014); Li  _et al._ (2016); Wang (2016); Carleo and Troyer (2017); Ch’ng
_et al._ (2017); Wang  _et al._ (2017); Zhang and Kim (2017) and quantum
informations Cai  _et al._ (2015); Dunjko  _et al._ (2016); Lau  _et al._
(2017); Torlai and Melko (2017); Tubiana and Monasson (2017).

In this paper, we apply modern ML techniques Bengio  _et al._ (2007); Montufar
_et al._ (2014) to the long-standing physics problem of analytic continuation.
By using convolutional neural network (CNN) Kalchbrenner  _et al._ (2014);
LeCun  _et al._ (2015) and stochastic gradient descent based optimiser (i.e.,
stochastic gradient descent, Adadelta, Adagrad) Zeiler (2012); Kingma and Ba
(2014); Duchi  _et al._ (2011), we successfully construct the ML kernel which
can generate the real frequency space spectral function from imaginary Green’s
function. We emphasise that our method does not require any ‘domain-knowledge’
which is a distinctive feature from early-stage ML methods such as statistical
learning Arsenault  _et al._ (2017). In comparison to the conventional
techniques, ML-based algorithm demonstrates its superiority in terms of
accuracy and computation speed. The spectral weights and peak positions are in
better agreement, and the computation speed is 104–105 faster. Further, ML-
based method is more robust against the noise which is inevitably introduced
in Monte-Carlo calculation for example. Our results show that the domain-
knowledge free ML approach can be a new promising way to solve the long-
standing physics problem that has not been well understood based on the
currently available techniques.

##  II Method

###  II.1  Description of the problem

Matsubara frequency Green’s function $G(i\omega_{n})$ is analytically
continued to the real frequency $G(\omega)$. For a given $G(i\omega_{n})$, the
spectral function is $A(\omega)=-\frac{1}{\pi}\Im G(\omega+i0^{+})$. Note that
calculating the Green’s function for a given spectral function is
straightforward, not ill-conditioned. On the other hand, the spectral function
is obtained by inverting the integral equation

$\displaystyle G(i\omega_{n})$ $\displaystyle=\int
d\omega\frac{A(\omega)}{i\omega_{n}-\omega}$ (1) $\displaystyle=\int
d\omega{\bf K}(i\omega_{n},\omega){A}(\omega)$ (2)

where the kernel ${\bf K}(i\omega_{n},\omega)$ has different forms for
different problems. This continuation process is an ill-posed problem, and the
direct minimizing $\chi^{2}=\sum_{i\omega_{n}}^{N_{{\rm freq}}}||G-{\bf
K}A||^{2}$ is hardly feasible due to the high condition number. The key
question is how to deal with intrinsic noises.

###  II.2 Description of the machine learning

Here we note that many techniques to handle this kind of ill-posed problems
have been actively developed in the ML field of research for more than last
two decades Lecun  _et al._ (1998); Vapnik (1999); Pillonetto  _et al._
(2014). The early stage ML was basically rule-based, and many details of the
problem representation were implemented through handcrafted algorithms. On the
other hand, the modern ML algorithms automatically capture the representations
via training, which is often called as ‘self learning’ LeCun  _et al._ (2015);
Rifai  _et al._ (2011). Since any human knowledge is not directly implemented
in the kernel construction, this type of approach is called as domain-
knowledge-free ML. In this modern approach, crucially required are the
efficient data representation in high-dimensional space and the practical
algorithm to optimize massive variables in deep neural networks. In spite of
the challenging features of the problems, modern ML has dramatically surpassed
the other state-of-the-art technologies in many areas such as image
recognition Krizhevsky  _et al._ (2012); Farabet  _et al._ (2013); Szegedy
_et al._ (2015); Tompson  _et al._ (2014); Kalchbrenner  _et al._ (2014),
speech recognition Mikolov  _et al._ (2011); Hinton  _et al._ (2012); Sainath
_et al._ (2013), language processing Collobert  _et al._ (2011) and
translation Cho  _et al._ (2014); Jean  _et al._ (2014); Sutskever  _et al._
(2014).

In the current study, we adopted ‘fully connected layer (FCL)’ and CNN
Krizhevsky  _et al._ (2012); Kalchbrenner  _et al._ (2014); LeCun  _et al._
(2015); Carleo and Troyer (2017), and try to perform analytic continuation
within high-dimensional space. The CNN is one of the main players in the high-
dimensional data processing for images Ciresan  _et al._ (2011); Krizhevsky
_et al._ (2012); Simonyan and Zisserman (2014); Szegedy  _et al._ (2015); He
_et al._ (2016); LeCun  _et al._ (2015) and sound/video data LeCun  _et al._
(2015); Masci  _et al._ (2011); Donahue  _et al._ (2018); Karpathy  _et al._
(2014). We investigated both FCL only and FCL+CNN ML for the analytic
continuation problem without using domain-knowledge. As a modern domain-
knowledge-free ML technique, our approach is well distinguished from the
conventional rule-based regression methods Arsenault  _et al._ (2017). It is
noted that our neural networks self learn the ‘rule’ or ‘knowledge’ from the
massive training with a well prepared extensive data sets.

###  II.3 Training

In order to systematically check the input noise dependence, we considered
several different sets of random noise inputs and examined the output spectra.
The Gaussian random noise $N(iw_{n})$ is used for our main presentation with
the noise strength $\sigma$ (width of Gaussian distribution) varied from 0 to
0.01. The noised input is then defined as $G(i\omega_{n})^{\rm in}_{\rm
noise}=G(i\omega_{n})+N(i\omega_{n})$. We also considered the other types of
noise character. In particular, the frequency-dependent $\lambda(i\omega_{n})$
has been carefully investigated since it is often the case of QMC-DMFT
(dynamical mean-field theory) calculations. We also considered the uniformly
distributed random noise. We found that the results of the uniform noise
($N(iw_{n})\in[-0.05,0.05]$) are comparable with Gaussian noise $\sigma$=0.01.
While we mainly present the Gaussian random noise, any part of our conclusion
is not changed by this choice of noise type.

We constructed the ML-based analytic continuation kernel by using widely-
adopted open-source deep-learning framework, namely, ‘keras’ Chollet  _et al._
(2015) with ‘tensorflow’ Abadi  _et al._ (2015) backend. For continuation
problem, the training process is straightforward since the calculation of
$G(i\omega_{n})$ from a given $A(\omega)$ is not ill-conditioned. Our training
sets consist of $\sim$106,000 different combinations of peak numbers, heights
and positions. We generated 18,000 different training data sets for single-
peak spectra, 18,000 for double-peak, 20,000 for triple, and 10,000 for each
of 4$-$8 peak spectra. For each set of peak numbers the position, height, and
width of the peak are randomly generated in the range of [$-$10, 10], [0.2,
1.0], and [0.3, 1.2], respectively. It is straightforward to extend the number
of training sets to an arbitrary number. The validation set consists of 10,000
different types of peaks with different random sequences. For all cases, the
normalization condition of $\int A(w)dw=1$ was imposed. It should be noted
that, while this particular physical knowledge of normalization is implemented
in the training sets, our neural networks is constructed as ‘domain-knowledge-
free’ and the kernel should learn the knowledge from the training.

For training, we used ‘Adadelta’ Zeiler (2012) optimiser. We found that
‘stochastic gradient descent (SGD)’ and even ‘RMSprop’ Tieleman and Hinton
(2012) optimiser quite often suffer from the ‘gradient vanishing problem’;
i.e., all variables of a neural net are quickly set to zero. On the other
hand, the recently-developed adaptive stochastic variant optimisers (such as
‘Adadelta’, ‘Adagrad’ Duchi  _et al._ (2011), ‘Adam’ Kingma and Ba (2014), and
‘Adamax’ Kingma and Ba (2014)) produce the reliable results. We eventually
chose ‘Adadelta’ as it clearly exhibits the best performance. For the
activation function, we chose a combination of rectified linear unit (ReLU)
Hahnloser  _et al._ (2000); LeCun  _et al._ (2015); Ramachandran  _et al._
(2017) and scaled exponential linear unit (SeLU)Klambauer  _et al._ (2017). It
is found that $\sim$8000 epochs are mostly enough for neural network training
which corresponds to 16 hours ($\sim$ 7 sec/epoch) at the single desktop PC
level (we used one Nvidia 1080 GTX card).

##  III Result and discussion

###  III.1 Fully connected layers

Figure 1:  (Color online) FCL neural network. An illustration of our neural
network architecture which consists only of FCLs. The input $G(i\omega_{n})$
is an array of complex numbers. $\Re[G(i\omega_{n})]$ and
$\Im[G(i\omega_{n})]$ are arranged as a 1D array to be inserted into the
neural-net input. The dropout layers are located in between all the FCLs to
reduce the overfitting of neural networks (not shown). The green box
represents the output layer of $A(\omega)$. The blue neural network lines are
the schematic representation of activated connections.

As the first step toward ML-based analytic continuation, we consider the
neural network composed of FCLs which may be regarded as an early-stage ML
approach Frean (1990); Miller  _et al._ (1990); Ripley (1994). Roughly, the
use of single FCL can be regarded as one multiplication process of an
inversion matrix to the input Green’s function Kung (2009). Having more FCLs
thus corresponds to the increased number of matrix multiplications to
represent the inversion. Practically it is not expected to achieve a notable
improvement just by increasing the number of hidden layers Mühlenbein (1990);
Lecun  _et al._ (1998); Dean  _et al._ (2012); He  _et al._ (2016). After
testing many different numbers of hidden layer sets, we indeed found that the
performance is not much enhanced. Thus, in the below, we focus on the results
of three layers (Fig.1).

Figure 2:  (Color online) Analytic continuation result of FCL. The input
Green’s functions and the output spectra calculated by the FCL neural network
kernel (without CNN layer). (a, b) $\Re[G(i\omega_{n})]$ and
$\Im[G(i\omega_{n})]$ are presented in (a) and (b), respectively. The blue
curves are generated from the ideal spectrum shown by the black line in (c).
The purple lines show the noised input $G(i\omega_{n})^{\rm{in}}_{\rm{noise}}$
with $\sigma=0.005$ (see the main text for more details). (c) The calculated
spectral functions are presented by blue-solid ($\sigma$=0; noise-free) and
purple-dashed curve ($\sigma$=0.005) in comparison to the ideal spectrum
(black-solid line).

Figure 2 presents the result of analytic continuation by using FCLs neural
network. The black line in Fig. 2(c) is the spectrum from which imaginary
Green’s functions of Fig. 2(a) and (b) (blue lines) are generated. Therefore,
if the continuation procedure is perfect, the continued spectrum should be
identical with the black line in Fig. 2(c). Note that the process of obtaining
$G(i\omega_{n})$ from $A(\omega)$ is not ill-conditioned. Once
$G(i\omega_{n})$ is calculated, one can perform the continuation and compare
the result of $A(\omega)$ with the original one, namely the ideal spectrum.

The FCL continuation results are shown in Fig. 2(c). The blue-solid and
purple-dashed line corresponds to $\sigma$=0 and $\sigma$=0.005, respectively.
It is clearly noted that the continued spectra are not smooth and
significantly deformed in comparison to the ideal black line. This result
demonstrates the challenging nature of the problem. At the same time, however,
we also note that the overall shape of spectrum is captured by our FCL neural
network although the unexpected wriggles are found, and they become worse as
the noise level increases. We emphasize that this level of performance is
hardly achievable through the direct matrix inversion of Eq. (2). This
promising aspect is largely attributed to the ‘dropout’ and the regularisation
procedure which prevent overfittings Srivastava  _et al._ (2014). In this
regard, while not satisfactory at all, our FCL result shows a possibility of
neural network approach for the analytic continuation.

Figure 3:  (Color online) CNN neural network. An illustration of the neural
network architecture which consists of both FCLs and CNN. Two convolution
layers (blue and red squares) and one pooling layer (green squares) are placed
in between the FCLs. Coloured neural network lines are the schematic
representations of activated connections.

###  III.2 Convolutional neural network

Many techniques have been suggested to overcome the deficiency of FCL. One key
idea is to identify the essential features of a problem and to reconstruct
them in a higher dimensional space Montufar  _et al._ (2014); Pillonetto  _et
al._ (2014). Principal component analysis (PCA) Shlens (2014); Hall and
Hosseini-Nasab (2006) is an example which proved to be powerful for data
compression and dimensionality reduction. Unfortunately, however, PCA can only
be used in rank 1 (vector) and rank 2 (matrix) for most of the cases. While
some techniques for tensor PCA have been proposed, they seem to need further
developments Richard and Montanari (2014); Inoue (2016); Zhang and Xia (2018);
Li  _et al._ (2015). A typical fundamental limitation of PCA is that each
principal component is given by a linear combination of original variables
whereas non-linearity is essential for ill-posed problems Archambeau and Bach
(2009). In this regard, CNN is a useful advanced technique leading the modern
machine-learning era Ciresan  _et al._ (2011); Krizhevsky  _et al._ (2012);
Simonyan and Zisserman (2014); Szegedy  _et al._ (2015); He  _et al._ (2016);
LeCun  _et al._ (2015). The performance of CNN image processing surpasses the
human-designed algorithms based on ‘domain knowledge’ Ciresan  _et al._
(2011); Krizhevsky  _et al._ (2012). Due to its outstanding feature selection
in tensor space, CNN is widely adopted by high-dimensional noise filters for
auto-encoder and sound/video data Masci  _et al._ (2011); Donahue  _et al._
(2018); Karpathy  _et al._ (2014).

In analytic continuation, input/output data are represented by a certain set
of numbers. Thus it can be regarded as an inverse problem that has to be
performed within a dimension corresponding to those numbers. With this
observation, we applied CNN technique to the long-standing ill-posed problem
of analytic continuation. Figure 3 shows our neural network structure. We aim
to create a minimal model with the smallest possible number of layers. Thus
our neural network is designed to contain CNN layers in between two FCLs since
we learned in the above that three FCLs could capture the basic features of
spectra. While it is conventional to have CNN layers just next to the input
layer in the image processing (e.g., AlexNet Krizhevsky  _et al._ (2012), VGG
Simonyan and Zisserman (2014), GoogleNet Szegedy  _et al._ (2015), and ResNet
He  _et al._ (2016)), we take a different strategy of inserting the CNN layer
after the matrix operation through FCL. It is because the full information of
input Green’s function needs to be utilized in our problem. The total number
of parameters in our neural network is $\sim$600,000 and $\sim$500,000 for
including and excluding CNN, respectively. It is noted that the network size
is not much increased by having CNN layers. We have adopted modern
optimization algorithm, namely ‘Adadelta’ Zeiler (2012), to optimize this
large number of neural networks parameters.

Figure 4: (Color online) Analytic continuation result of CNN. (a, b)
$\Re[G(i\omega_{n})]$ and $\Im[G(i\omega_{n})]$ are presented in (a) and (b),
respectively. The light-blue curves are generated from the ideal spectrum
shown by the black line in (c). Five different noise levels are presented in
light-blue ($\sigma$=0), magenta ($\sigma$=0.003), purple ($\sigma$=0.005) and
red lines ($\sigma$=0.01). (c) Analytic continuation result of CNN-ML kernel
for $\sigma$=0 (noise-free; light-blue-solid), $\sigma$=0.003 (magenta-
dashed), $\sigma$=0.005 (purple-dashed) and $\sigma$=0.01 (red-dashed) along
with the ideal spectrum (black-solid line).

Figure 4 shows the continuation result of using CNN. The model spectrum (black
line in Fig. 4(c)) is designed to mimic a Mott-Hubbard insulator consisted of
two distinct Hubbard bands with different peak heights. The outstanding
performance of CNN can clearly be seen from that the continuation results are
significantly improved in comparison to the FCL-only data in Fig. 2. The
overall shape, peak positions and relative peak heights are well reproduced
without any undesirable wriggle. Importantly, the reproducibility remains
quite robust against the noise even if the deviation from the ideal spectrum
(black) becomes noticeable as the noise level increases (from light-blue-solid
lines to red-dashed). We also checked that the reconstructed
${\tilde{G}}(i\omega_{n})=({\rm\bf{K}}A)(i\omega_{n})$ is consistent with
$G(i\omega_{n})$ within the noise level (not shown). For example, the
calculated ${\tilde{\chi}}$ = 0.0044 for the case of $\sigma=0.003$ where
$\tilde{\chi}\equiv(\chi^{2}/N_{\rm freq})^{-1/2}$.

Figure 5: (Color online) Analytic continuation result of MEM. (a, b) Input
Green’s function of $\Re[G(i\omega_{n})]$ and $\Im[G(i\omega_{n})]$ are
presented in (a) and (b), respectively. The light blue curves are generated
from the ideal spectrum shown by the black line in (c). Four different noise
levels are presented in light-blue ($\sigma$=0), light-pink ($\sigma$=0.001),
magenta ($\sigma$=0.003) and purple lines ($\sigma$=0.005). (c) Analytic
continuation result of conventional MEM for$\sigma$=0 (noise-free; light-blue-
solid), $\sigma$=0.001 (light-pink-dashed), $\sigma$=0.003 (magenta-dashed)
and $\sigma$=0.005 (purple-dashed) along with the ideal spectrum (black-solid
line).

The robustness against the input noise is a crucially required feature for the
reliable analytic continuation since the noise is unavoidably present in
stochastic approaches. As shown in Fig. 4(c), the overall features and the
detailed shapes of the spectrum are well maintained even for the case of
significant noise levels. This result shows the powerfulness of ML-based
analytic continuation kernel.

The performance of ML kernel is further demonstrated by the comparison to the
conventional continuation technique, namely MEM. The details of our MEM
algorithms can be found in Ref. Bergeron and Tremblay, 2016; Sim and Han,
2018. Fig. 5 shows the result of MEM which is one of the most widely-used
methods for analytic continuation Haule  _et al._ (2010); Bergeron and
Tremblay (2016); Jarrell and Gubernatis (1996); Gunnarsson  _et al._ (2010b).
It is clearly noted that, even at a significantly lower noise level, the MEM
result is markedly deviated from the ideal spectrum in terms of peak position
and height. It is in a sharp contrast to the ML-based result of Fig. 4 in
which the spectrum is well preserved even at $\sigma=0.01$.

Figure 6: (Color online) Analytic continuation result of CNN for metallic
spectrum. (a, b) Input Green’s function of $\Re[G(i\omega_{n})]$ and
$\Im[G(i\omega_{n})]$ are presented in (a) and (b), respectively. The light-
blue curves are generated from the ideal spectrum shown by the black line in
(c). Five different noise levels are presented in light-blue ($\sigma$=0),
magenta ($\sigma$=0.003), purple ($\sigma$=0.005) and red lines
($\sigma$=0.01). (c) Analytic continuation result of CNN-ML kernel
for$\sigma$=0 (noise-free; light-blue-solid), $\sigma$=0.003 (magenta-dashed),
$\sigma$=0.005 (purple-dashed) and $\sigma$=0.01 (red-dashed) along with the
ideal spectrum (black-solid line).

Figure 6 shows the result of ML kernel for metallic spectrum having coherent
as well as incoherent peaks. Once again, our machine-learning kernel well
reproduces the original spectrum. The robustness against noise is also
excellent as in the insulating case. In particular, the coherent peak is
considerably well reproduced while the incoherent states are moderately
affected by the noises. It is a good indication for predicting the phase from
a given Green’s function.

As the last example, we present the results of a real material, namely SrVO3
monolayer. While the bulk SrVO3 is a correlated metal, it becomes an insulator
in the monolayer limit Yoshimatsu  _et al._ (2010); Schüler  _et al._ (2018).
The Green’s function is obtained from DFT+DMFT calculation Lichtenstein and
Katsnelson (1998); Georges and Kotliar (1992) combined with the hybridization
expansion continuous-time quantum Monte Carlo algorithm Werner  _et al._
(2006); Werner and Millis (2006); Haule (2007); com . The spectra obtained by
our ML kernel is reasonably well compared with those of MEM; see Fig. 7. While
the peak widths are slightly narrower in MEM, it should be noted that the
direct use of MEM for Green’s function tends to broaden the spectra Wang  _et
al._ (2009).

Of particular interest is the performance of our ML continuation kernel for
the cases that were not included in the training sets. Although the systematic
investigation of the training-set dependence is not the main interest of the
current study, we obtained some meaningful results. In terms of peak numbers,
the quality of continuation gets gradually worse as the number of peaks goes
out of the training range. However, its performance is still quite decent (we
tried up to the 25-peak case) and at least comparable with that of MEM. The
similar feature is found for the peak width. For the peak of width=0.1 and
0.2, the ML kernel produces the spectrum whose width is 0.22 and 0.24,
respectively. Considering that the conventional continuation techniques suffer
from the same problems in describing sharp peaks, we concluded that the ML
exhibits reasonably good performance. Finally, we emphasize the efficiency of
ML-based analytic continuation. Once the ML kernel is well trained, the
continuation process can be performed at a speed of $\sim$10,000 Green’s
functions/sec, which is at least $10^{4}$–$10^{5}$ faster than the
conventional MEM.

Figure 7:  (Color online) The calculated DFT+DMFT spectral function of
monolayer SrVO3 by (a) CNN and (b) MEM. The magenta and blue colors refer to
the $d_{xy}$ and $d_{yz,xz}$ orbital character, respectively. Comparison
between ML and MEM results for DFT+DMFT spectral function of the SrVO3
monolayer The input Green’s function is obtained from DFT+DMFT with CT-QMC
solver. Analytic continuation for the spectral function was performed using
(a) ML-CNN and (b) MEM.

##  IV Conclusion

Modern ML technique proves its usefulness for a long-standing physics problem
of analytic continuation. Its superiority over the conventional technique is
demonstrated in terms of the accuracy, speed, and the robustness against
noise. For both insulating and metallic spectrum, our CNN-based ML kernel
gives the better agreement with the ideal spectrum in terms of peak position
and height. Up to the high level of random noises, at which MEM fails to
produce the reliable results, ML technique retains its accuracy. In terms of
computation speed, the trained kernel is 104–105 times faster than the
conventional method. Our result suggests that ‘domain-knowledge’ free ML can
be used as an alternative tool for the physics problems where the conventional
methods have been struggling. We also note the possibility of certain types of
hybrid methods in which a part of physics intuitions would be combined with ML
approach.

##  V Acknowledgments

This work was supported by Basic Science Research Program through the National
Research Foundation of Korea (NRF) funded by the Ministry of Education
(2017R1D1A1B03032082) and Creative Materials Discovery Program through the NRF
funded by Ministry of Science and ICT (2018M3D1A1059001).

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# Image-charge detection of the Rydberg transition of electrons on superfluid
helium confined in a microchannel structure

S Zou, D Konstantinov  Quantum Dynamics Unit, Okinawa Institute of Science and
Technology (OIST) Graduate University, Okinawa 904-0495, Japan<EMAIL_ADDRESS>
###### Abstract

The image-charge detection provides a new direct method for the detection of
the Rydberg transition in electrons trapped on the surface of liquid helium.
The interest in this method is motivated by the possibility to accomplish the
spin state readout for a single trapped electron, thus opening a new pathway
towards using electron spins on liquid helium for quantum computing. Here, we
report on the image-charge detection of the Rydberg transition in a many-
electron system confined in an array of 20-$\mu$m wide and 4-$\mu$m deep
channels filled with superfluid helium. Such detection is made possible
because of a significant enhancement of the image-charge signal due to close
proximity of trapped electrons to the electrodes embedded in the microchannel
structure. The transition frequency of electrons in the range of 400-500 GHz
is highly controllable by the dc bias voltages applied to the device and is in
a good agreement with our calculations. This work demonstrates that
microchannel structures provide a suitable platform for electron manipulation
and their quantum state detection, with a feasibility of scaling the detection
method to a single electron.

  * June 2022

Keywords: surface-state electrons on liquid helium, superfluid helium, two-
dimensional electron systems, Rydberg states of orbital motion, image-charge
detection, electron spin qubits

##  1 Introduction

Electrons bound to the surface of liquid helium present an extremely clean
quantum system. The bound states of such surface electrons (SE) are formed due
to a weak attractive force from the image charge inside the dielectric liquid
and a strong repulsive barrier at the vapor-liquid interface. In such an image
potential, the quantized (Rydberg) states of the electron motion perpendicular
to the surface have a hydrogen-like spectrum, with an effective Rydberg
constant on the order of 1 meV, while the motion parallel to the surface is
free. The system is free of disorder and impurities [1]. At sufficiently low
temperatures below 1 K, the scattering of electrons is predominantly from the
thermal oscillations (ripplons) of the liquid helium surface [2]. This leads
to extremely high mobility of SE and relatively long coherence time of their
motional states [3, 4]. Moreover, the very weak intrinsic spin-orbit (Rashba)
interaction and the magnetically-clean environment of superfluid helium-4
result in extremely long coherence time of the spin states of SE [5]. This
presents SE on superfluid helium-4 as a very attractive system for realizing
qubits for quantum computing, with several theoretical proposals presented
over the past years [3, 4, 5, 6].

While unique transport properties of a many-electron system on liquid helium
had been extensively investigated in the past [7], methods for the quantum
state control of SE, in particular at the level of a single electron, were not
sufficiently developed. The lack of such methods presented the main obstacle
towards the realization of the above proposals for qubit implementation. Very
recently, a readout of the quantized states of the trapped lateral motion was
demonstrated for a single electron on superfluid helium [8] and on solid neon
[9]. In these experiments, the electrons were confined in an electrostatic
trap integrated into a circuit quantum electrodynamic (cQED) architecture
[10]. The above works present the first significant advance towards qubit
implementation using SE, with a further promise to realize coupling between
the motional and spin states of an electron, thus a readout of its spin states
[6].

An alternative approach for the spin-state readout was recently suggested by
Kawakami et al. who proposed to use a coupling between the spin states of an
electron and the Rydberg states of its vertical motion [11]. The idea of
readout is based on a relatively large difference between the spatial
extension of the electron wavefunction above the liquid surface for an
electron in the ground state and the first excited Rydberg state,
respectively. Thus, in the presence of a sufficiently large gradient of an
applied magnetic field, for example created by a magnet placed in proximity to
SE, an electron in different Rydberg states will experience different values
of the mean magnetic field, resulting in distinct Zeeman splitting of their
spin states. Consequently, the transition frequency between the ground state
and the first excited Rydberg state of an electron will depend on its spin
state. Then, the Rydberg transition can be excited spin-selectively by an
applied resonant microwave (MW) radiation providing that the above difference
in the Zeeman splitting exceeds the Rydberg transition linewidth. In other
words, the spin state of an electron can be determined by detecting its
Rydberg transition.

For this purpose, a new method of direct detection of the Rydberg transition
in SE, the image-charge detection, was developed [11]. The idea of this method
is also based on the difference in the spatial extension of the electron
wavefunction above the surface for different Rydberg states. A change in the
Rydberg state of an electron causes a corresponding change in the image
charges induced by the electron in surrounding metal electrodes. Thus, the
Rydberg transition in SE can be detected by measuring the image current at one
of the electrodes. The proposed method was successfully demonstrated by
placing a large number of SE in the middle between two plates of a parallel-
plate capacitor separated by a distance $D=2$ mm and measuring the image
current induced in the plates by SE in response to a pulse-modulated MW
excitation at the modulation frequency $\sim$100 kHz. For liquid helium-4, the
mean distance between an electron and the liquid surface is equal to 11.4 and
45.6 nm for the electron occupying the ground state and the first excited
Rydberg state, respectively, with the corresponding difference $\Delta
z\approx 34$ nm [1]. Thus, each excited electron causes a change in the image
charge at the bottom and top plates of the capacitor equal to $\delta q=\mp
e\Delta z/D$, respectively, where $e>0$ is the elementary charge. This amounts
to $|\delta q|\sim 10^{-5}e$ for the setup used in [11]. For about $10^{8}$
electrons used in the experiment, this produced a current on the order 10 pA,
which was directly measured using a lock-in amplifier.

The ultimate goal is the image-charge detection of the Rydberg transition of a
single electron. The simplest improvement towards increasing the sensitivity
of the proposed method is immediately seen, that is increasing the change in
the image charge $|\delta q|$. As follows from the above discussion, this can
be done by decreasing the distance $D$ between the capacitor plates.
Unfortunately, for a parallel-plate capacitor partially filled with liquid
helium this distance can not be made less than $\sim 1$ mm, the capillary
length of liquid helium, otherwise the capacitance will be completely filled
with the liquid by the capillary action [12]. For this reason, other electrode
structures have to be sought, for example, a coplanar capacitor covered with a
superfluid helium film, as was suggested in [11]. An alternative approach is
to use the so called microchannel devices, which have been very useful to
study the transport properties of SE on superfluid helium [13, 14, 15, 16].
Such a device consists of an array of microchannels, with a typical channel
depth $d\sim 1$ $\mu$m, which can be filled with the superfluid helium-4 by
the capillary action. SE floating on the surface of superfluid helium inside a
channel are capacitively coupled to the metal electrodes comprising the top
and bottom of the channel, therefore they are expected to produce a change in
the image charge induced at one of the electrodes equal to $|\delta q|=\kappa
e\Delta z/d\sim 10^{-2}e$, where $\kappa$ is a numerical factor order of
unity, which is determined by the ratio of the capacitive coupling of SE to
each of the two channel electrodes. Thus, the sensitivity of the image-charge
detection method is expected to enhance thousandfold for SE in a microchannel
device comparing with the previous experiment [11]. In addition, the
microchanel devices present several other advantages towards spin-state
control and quantum computing. The micromagnets [17], which are required to
introduce a sufficiently strong magnetic field gradient, can be easily
integrated to the microchannel structure, therefore can be placed in close
proximity to SE. It was also demonstrated that, for the sake of increasing
functionality of a device, the microchannel geometry can be easily integrated
with other useful mesoscopic devices, for example, the single-electron
transistor (SET) [18, 19], point-contact constriction [20], superconducting
coplanar waveguide (CPW) resonator [8, 9], and charge-coupled device (CCD)
[21]. The latter is particularly attractive towards realization of a quantum-
CCD architecture for scalable quantum computing [22, 23]. Finally, the Rydberg
resonance of SE confined in a single 4-$\mu$m deep channel was recently
observed by an indirect resistive detection method, thus showing feasibility
of the Rydberg state detection in such a system [24].

Here, we report on the first image-charge detection of the Rydberg transition
in SE on superfluid helium confined in a microchannel structure consisting of
an array of 20-$\mu$m wide and 4-$\mu$m deep microchannels. An image-current
signal from approximately $10^{5}$ electrons, whose Rydberg resonance is
excited by a pulse-modulated MW radiation in the frequency range 400-500 GHz,
is converted to voltage by a two-stage cryogenic amplifier and recorded at
room temperature by a lock-in amplifier. The microchannel device used in our
experiment shows high-level control of the transition frequency of confined SE
via the Stark shift of the Rydberg energy levels. Further improvement in the
image-current detection circuit promises the feasibility to enhance
sensitivity of our method to the detection of a single electron. Thus, our
work demonstrates that SE on superfluid helium confined in a microchanel
device present a suitable platform for realizing the quantum state detection
in electrons on helium for their qubit implementation.

##  2 Method

###  2.1 Image-charge detection

Figure 1:  Schematic view of the image-charge detection method (a) and its
equivalent Norton and Thevenin electric circuits, (b) and (c), respectively.
The image current $i_{out}$ at the channel electrode is generated by the
pulse-modulated MW excitation of SE in the channel with the modulation
frequency $\omega_{m}$.

A schematic view of the image-charge detection method, as well as its
equivalent electric circuit, for SE on superfluid helium confined in a channel
are shown in figure 1. Figure 1(a) schematically shows the cross-section of a
channel of depth $d$ filled with superfluid helium. Electrons trapped on the
surface of helium are capacitively coupled to the channel and guard electrodes
at the bottom and top of the channel, respectively. A positive dc bias voltage
$V_{\rm ch}$ is applied to the channel electrode to create an electrostatic
confining potential for SE. For the sake of simplicity, we assume that the
guard electrode is grounded. Excitation of SE by the pulse-modulated MW
radiation with the modulation frequency $\omega_{\rm m}$ causes an ac current
$i_{\rm out}$ in the channel electrode due to the periodic variation, also
with the frequency $\omega_{\rm m}$, of the electric charge induced in the
electrode. This current flows to the detection circuit via a coupling
capacitance $C_{1}$, which forms a bias-T with a resistor at the dc voltage
source. Thus, the voltage at the channel electrode is given by $V_{\rm
ch}+u_{\rm out}$, where $u_{\rm out}$ is the ac voltage induced by $i_{\rm
out}$.

The total electric charge at the channel electrode is given by $(Q+q)$, where
$Q=C_{0}(V_{\rm ch}+u_{\rm out})$ and $C_{0}$ is the capacitance between the
channel and guard electrodes determined by the channel geometry, while $q$ is
the image charge induced by SE. Thus, the output current is given by

$i_{\rm out}=-\frac{d(Q+q)}{dt}=-C_{0}\frac{du_{\rm out}}{dt}-\frac{dq}{dt}.$
(1)

The variation of the image charge at the channel electrode due to excitation
of SE can be estimated as $\Delta q=\rho_{22}N_{\rm e}\delta
q=-\rho_{22}N_{\rm e}\kappa e\Delta z/d$, where $N_{\rm e}$ is the total
number of SE in the channel and $\rho_{22}$ is the fraction of these electrons
that is excited by the MW radiation. For the microchannel geometry used in our
experiment, we estimate $\kappa=0.8$ [24]. For the pulse-modulated MW
excitation, the corresponding current is given by $i=-(dq/dt)=-\Delta
q\omega_{\rm m}$. Thus, as follows from (1), an equivalent electric circuit of
our setup can be represented as the Norton circuit with an ideal current
source $i$, see figure 1(b). Here, in addition to the coupling capacitance
$C_{1}$, we included a parasitic capacitance $C_{\rm p}$ which in our
experiment is mostly determined by the capacitance of a cryogenic cable
connecting the microchanel device to the detection circuit.

###  2.2 Detection circuit

In the experiment reported here, we used a two-stage cryogenic amplifier that
detected the voltage signal $u_{\rm out}$ due to the image current $i_{\rm
out}$, with a subsequent detection using a room-temperature lock-in amplifier
referenced at the modulation frequency $\omega_{\rm m}$. A detailed
description of the amplifier circuit is given elsewhere [25]. The first stage
of the amplifier was located in close proximity to the experimental cell, with
an estimated parasitic capacitance of the connecting cable $C_{\rm p}\approx
20$ pF. The two-stage amplifier provided a total voltage gain of $\sim 100$.

For the sake of comparison between our theoretical estimations and
experimental results, it is convenient to transform the Norton circuit given
in figure 1(b) to its Thevenin equivalent shown in figure 1(c), with an ideal
voltage source $u=i/(\omega_{\rm m}C_{0})=-\Delta q/C_{0}$. Assuming that the
first stage of the cryogenic amplifier has an infinitely large input impedance
[25], the voltage signal $u_{\rm out}$ at the input of the amplifier is given
by $u_{\rm out}=uC_{0}/(C_{0}+C_{\rm p})=-\Delta q/(C_{0}+C_{\rm p})$. Note
that the same result can be also obtained from (1). Thus, we obtain

$u_{\rm out}=\frac{\rho_{22}N_{\rm e}\kappa e\Delta z}{d(C_{0}+C_{\rm p})}.$
(2)

To estimate the magnitude of the expected measured signal, we take $\Delta
z/d=10^{-2}$, $N_{\rm e}=10^{5}$, $\rho_{22}=0.1$ (10% population of the
excited Rydberg state), and $C_{0}=1.3$ pF, which is estimated from the device
geometry, thus obtaining $u_{\rm out}~{}\approx 5$ $\mu$V. This corresponds to
an expected lock-in output signal of 0.5 mV.

###  2.3 Microchannel device

Figure 2:  Schematic view of the microchannel device (a) and its optical
microscope image showing the bonding pads (b). The inset is SEM micrograph
zooming the middle segment of the microchannel array.

A schematic view of the microchanal device used in our experiment, as well as
its optical and SEM micrographs, are shown in figure 2. The device consists of
an arrays of 47 connected channels, each 20-$\mu$m wide, arranged in parallel.
The whole structure is fabricated on a silicon substrate using UV lithography
methods and is composed of two thin patterned gold layers separated by an
insulating layer of hard-baked photoresist with a thickness $d=4$ $\mu$m,
which defines the depth of the channels. The channel array is divided into
three segments, each having a length of 1.7, 0.6, and 1.7 mm, respectively.
The decomposition into segments is defined by the structure of the bottom
layer, which is divided into three electrodes separated by 2-$\mu$m wide gaps
to form bottom electrodes for each of the three segments. The corresponding dc
bias voltages applied to these electrodes are denoted as $V_{\rm LR}$, $V_{\rm
RR}$, and $V_{\rm ch}$, see figure 2(a). The two 1.7-mm long segments serve as
left and right electron reservoirs (LR and RR, respectively) which are
connected by the third 0.6-mm long middle segment. The top gold layer consists
of a single electrode, the guard electrode, which lies in the plane of the
liquid helium surface. The guard electrode defines the channel boundary and,
together with the bottom electrodes, serves to provide the confinement of SE
inside the channels and their tuning to the Rydberg resonance via the Stark
shift, as will be discussed in section 3. The corresponding dc bias voltage
applied to this electrode is denoted as $V_{\rm gu}$.

###  2.4 Device operation

The microchannel device is mounted on a printed circuit board (PCB) and bonded
with the conducting pads of PCB using a 50-$\mu$m diameter gold wire, see
figure 2(b). PCB is placed inside a leak-tight copper cell and is connected to
the control and detection circuits via hermetic SMP connectors mounted at the
top flange of the cell. The cell is cooled down below 1 K at the mixing
chamber of a dilution refrigerator. Helium-4 gas is condensed into the cell,
such that the liquid level is placed slightly below PCB, and superfluid helium
fills the channels of the device by the capillary action. The surface of
liquid in the channels is charged with electrons produced by the thermionic
emission from a tungsten filament placed about 1 mm above the device, with the
applied bias voltages $V_{\rm LR}=V_{\rm RR}=V_{\rm ch}=1$ V and $V_{\rm
gu}=0$ V.

The Rydberg transition of SE in the microchannel device is excited by the
pulse-modulated MW radiation, which is supplied from a room-temperature
source, guided into the cryostat through an overmoded (WR-28) stainless steel
waveguide and sent into the experimental cell through a tapper waveguide
transition. The MW power measured at the output of the source was about 1 mW
in the frequency range 350-500 GHz, while the total transmission loss of the
waveguide system was measured to be about -10 dB. The MW power at the location
of the microchannel device inside the cell could not be directly measured but
is estimated to be in the range $10\textrm{--}100$ $\mu$W.

In the experiment reported here, SE are excited by the MW radiation at a fixed
frequency $\omega$, while the resonant frequency $\omega_{21}$ corresponding
to the transition of SE from the ground state to the first excited Rydberg
state is tuned in resonance with $\omega$ via the Stark shift by varying the
pressing electric field $E_{\perp}$ exerted on the electrons perpendicular to
the surface of liquid [26]. This field is determined by the dc bias voltages
applied to the device electrodes, as well as by the image charges induced in
the electrodes by SE. The latter depend on the density of SE in the channel
and the channel depth. The image current due to the MW-excited electrons is
measured at the bottom electrode of the middle segment, while the bias
voltages applied to the device electrodes are varied to change $E_{\perp}$.
Note that a relatively large depth of 4 $\mu$m allows us to apply relatively
large electrode voltages, thus to preserve the resolution of the transition
line while voltages are swept through the resonance. For typical density of SE
in our device [24] and the total middle-segment channel area of about $6\times
10^{-3}$ cm-2, we expect to have on the order $10^{5}$ electrons in the middle
segment.

##  3 Results

Figure 3: In-phase voltage signals recorded by the lock-in amplifier
referenced at the modulation frequency $\omega_{\rm m}/2\pi=100$ kHz versus
the bottom electrode voltage $V_{\rm bot}=V_{\rm LR}=V_{\rm RR}=V_{\rm ch}$.
Each trace is recorded with a fixed value of the MW frequency $\omega/2\pi$
given in the plot legend.

Figure 3 shows some exemplary image-charge signals from the excited SE for
several values of the MW frequency $\omega$ while the Rydberg transition
frequency of SE is tuned in resonance by varying the bias voltage $V_{\rm
bot}$ applied to the bottom electrodes, with $V_{\rm bot}=V_{\rm LR}=V_{\rm
RR}=V_{\rm ch}$ and $V_{\rm gu}=0$. Note that by keeping the voltages applied
to all three bottom electrodes equal to each other, we ensure that the density
of electrons across all three segments is uniform and constant, by the
conservation of the total number of SE in the device. The data reported here
were taken at $T=0.2$ K, which was measured by a ruthenium oxide sensor
attached to the experimental cell. At such temperature, the thermal population
of the excited Rydberg states is negligible. The in-phase voltage signal
$V_{\rm s}$ is recorded by a lock-in amplifier referenced at the MW modulation
frequency $\omega_{\rm m}/2\pi=100$ kHz and with the time constant of 1 s. The
resonant peak for each trace corresponds to the transition of SE from the
ground state to excited Rydberg states. The position of resonance shifts
towards higher values of $V_{\rm bot}$ with increasing MW frequency,
consequence of the linear Stark shift of the Rydberg energy levels of SE [26].
Figure 4(a) shows a summary plot (symbols) of the position of the observed
resonance peaks versus $\omega/2\pi$, which demonstrates the linear dependence
with the slope $k_{1}=0.0077$ V/GHz (red line). Using this result, we can
estimate the average half-width at half-maximum (HWHM) of the resonance peaks
of about 10 GHz. At $T=0.2$ K, the intrinsic linewidth of the Rydberg
resonance due to the scattering of electrons from ripplons is expected to be
of the order of 10 MHz [27]. A thousandfold larger broadening observed in our
experiment is due to the inhomogeneous broadening of the Rydberg transition of
SE caused by the variation of the pressing electric field $E_{\perp}$ across
the channel. This result is consistent with the numerically calculated
distribution of $E_{\perp}$ for our channel geometry [24]. Note that,
comparing to the resonance signal measured by the indirect resistive method
[24], a typical resonance peak reported here is much more symmetric, as should
be expected from the electric field distribution in the given channel
geometry. This is because the image-charge detection method employed here
directly probes the excited-state population $\rho_{22}$ of SE, as follows
from (2). Also, note that the observed magnitude of the image-charge resonant
signal is in a good agreement with our theoretical estimation given in section
2.1.

Figure 4: (a) Position of the maxima of the resonance signals observed in the
experiment versus the MW frequency (symbols). The red line is a linear fit to
the experimental data with the slope $k_{1}=0.0077$ V/GHz. (b) Numerically
calculated dependence of the transition frequency $\omega_{21}/2\pi$ of SE on
liquid helium-4 for different values of the pressing electric field
$E_{\perp}$.

In order to compare the position of the resonance signal with theory, we
numerically solved the eigenvalue problem for the quantized motion of an
electron perpendicular to the liquid helium surface to find the dependence of
its transition frequency $\omega_{21}$ on the electric field $E_{\perp}$. The
result is shown in figure 4(b). In the given range of $E_{\perp}$, this
dependence is almost linear with the slope $k_{2}=0.23$ GHz/(V/cm). Assuming
that the resonance signal observed in the experiment corresponds to the
resonant condition $\omega=\omega_{21}$, we obtain the linear relation between
the pressing electric field $E_{\perp}$ acting on SE and the applied bias
voltage $V_{\rm bot}$, with the slope equal to $k=(k_{1}k_{2})^{-1}=565$ cm-1.
In [24], an approximate theoretical relation between $E_{\perp}$ and the
voltages applied to the device electrodes was established, which for our
device reads

$E_{\perp}=\frac{V_{\rm bot}}{2d}-\frac{\beta V_{\rm gu}}{2\alpha
d}-\frac{V_{\rm e}}{d}\left(1-\frac{1}{2\alpha}\right).$ (3)

Here, $\alpha$ and $\beta$ are weighted contributions ($\alpha+\beta=1$) to
the total capacitance between the charged surface of liquid and device
electrodes due to the bottom channel electrode and the top guard electrode,
respectively, while $V_{\rm e}$ is the electric potential of the charged
surface. For our channel geometry, we have $\alpha=0.8$ and $\beta=0.2$ [24].
To account for the experimental conditions in figure 4(a), we set $V_{\rm
gu}=0$. Moreover, unlike for the device with a single-channel middle segment
used in [24], for the current device with a multi-channel middle segment the
potential $V_{\rm e}$ depends on the value of the applied bias voltage $V_{\rm
bot}$. In order to find the relationship between $V_{\rm e}$ and $V_{\rm
bot}$, we notice that the electric potential of the uncharged surface of
liquid in the channel is given by $V_{\rm u}=\alpha V_{\rm bot}+\beta V_{\rm
gu}=\alpha V_{\rm bot}$ (with $V_{\rm gu}=0$). The difference between the
electric potentials of the uncharged and charged surface of liquid $\Delta
V=V_{\rm u}-V_{\rm e}$ depends only on the number of SE and the channel
geometry, therefore is fixed during the variation of $V_{\rm bot}$. Plugging
the relation $V_{\rm e}=\alpha V_{\rm bot}-\Delta V$ into (3), we obtain

$E_{\perp}=\frac{V_{\rm bot}}{d}(1-\alpha)+\frac{\Delta
V}{d}\left(1-\frac{1}{2\alpha}\right),$ (4)

which predicts the linear dependence between $E_{\perp}$ and $V_{\rm bot}$
with the slope $(1-\alpha)/d=500$ cm-1. This is fairly close the the
experimentally observed slope of $565$ cm-1. Note that in an actual device the
charged surface of liquid helium in the channel is somewhat depressed due to
the force exerted on SE by the electric field $E_{\perp}$. The smaller value
of $d$ results in the higher value of the slope, in agreement with our result.

Figure 5: In-phase voltage signal recorded by the lock-in amplifier referenced
at the modulation frequency $\omega_{\rm m}/2\pi=200$ kHz versus the bottom
electrode voltage $V_{\rm bot}=V_{\rm LR}=V_{\rm RR}=V_{\rm ch}$. Each trace
is recorded with for the same MW frequency $\omega/2\pi=480$ GHz and different
values of the negative dc bias voltage $V_{\rm gu}$ applied to the guard
electrode, which is indicated (in Volts) for each trace.

The above relation (3) also suggests that the resonance position in $V_{\rm
bot}$ for a fixed MW frequency $\omega$ can be adjusted by varying the guard
voltage $V_{\rm gu}$. Fig. 5 shows the resonant image-charge signals for MW
radiation with a fixed frequency 480 GHz and for several fixed negative biases
$V_{\rm gu}$. The position of the resonance shifts towards smaller values of
$V_{\rm bot}$ for more negative values of $V_{\rm gu}$. The plotted dependence
of $V_{\rm bot}$ at resonance versus $V_{\rm gu}$ is linear with the slope
very close to 1. Using the above relationship between $V_{e}$ and the applied
electrode voltages, with $V_{e}=\alpha V_{\rm bot}+\beta V_{\rm gu}-\Delta V$,
we obtain from (3)

$E_{\perp}=\frac{V_{\rm bot}}{d}(1-\alpha)-\frac{V_{\rm
gu}}{d}\beta-\frac{\Delta V}{d}\left(1-\frac{1}{2\alpha}\right).$ (5)

This equation predicts the linear relationship between the pressing field
$E_{\perp}$ and the voltage difference $V_{\rm bot}-V_{\rm gu}$ with slope
equal to $\beta/d$. This is in an excellent agreement with our experimental
result shown in figure 5, where SE are excited by the MW radiation at a fixed
frequency $\omega$ corresponding to a certain value of $E_{\perp}$, while the
values of $V_{\rm bot}$ and $V_{\rm gu}$ satisfy the relation $V_{\rm
bot}-V_{\rm gu}\simeq 1.88$ V, with the slope of the linear relationship
between $V_{\rm bot}$ and $V_{\rm gu}$ equal to 1.

Figure 6: (a) In-phase voltage signal recorded by the lock-in amplifier
referenced at the modulation frequency $\omega_{\rm m}/2\pi=200$ kHz versus
the channel electrode voltage $V_{\rm ch}$. Each trace is recorded for the
same MW frequency $\omega/2\pi=480$ GHz and different values of the positive
dc bias voltage $V_{\rm R}=V_{\rm LR}=V_{\rm RR}$ applied to the reservoir
electrodes, as indicated (in Volts) for each trace. (b) Position of the maxima
of the resonance signals shown in (a) versus $V_{\rm R}$ (symbols). The red
line is a linear fit to the experimental data with the slope equal to
approximately $0.48$.

As mentioned earlier, in the experiments described above the bias voltages
$V_{\rm ch}$, $V_{\rm LR}$ and $V_{\rm RR}$ applied to all three bottom
electrodes of the device were varied simultaneously to ensure that the density
of SE in the middle segment is constant during the voltage sweep.
Alternatively, we can fix the dc bias $V_{\rm R}$ applied to the reservoir
electrodes and tune the transition frequency of SE in the middle segment into
resonance with MWs by varying $V_{\rm ch}$. In this case, the density of SE in
the middle segment is expected to vary as electrons redistribute between the
middle segment and reservoirs, while keeping the total number of SE fixed.
Figure 6(a) shows the resonant image-charge signals for the MW radiation with
a fixed frequency 480 GHz and for several fixed positive biases $V_{\rm R}$,
with $V_{\rm R}=V_{\rm LR}=V_{\rm RR}$ and $V_{\rm gu}=0$. The resonance
shifts towards higher values of $V_{\rm ch}$ with increasing $V_{\rm R}$,
while the signal amplitude decreases. Qualitatively, both effects appear
because the density of electrons in the middle segment decreases with
increasing $V_{\rm R}$. In figure 6(b), we plot the position of the resonance
peaks shown in figure 6(a) versus the reservoir voltage $V_{\rm R}$. The
dependence is linear (red line) with the slop equal to approximately 0.48.

In order to quantitatively account for the observed resonance shift, we note
that the density of SE in the middle segment varies with the bias voltages
applied to the device electrodes according to [24]

$n_{s}=\frac{\varepsilon\varepsilon_{0}}{\alpha ed}\left(\alpha V_{\rm
ch}+\beta V_{\rm gu}-V_{\rm e}\right),$ (6)

where $\varepsilon$ is the dielectric constant of liquid helium and
$\varepsilon_{0}$ is the vacuum permittivity. A similar equation is valid for
the density of SE in the reservoirs, with $V_{\rm ch}$ replaced by $V_{\rm
R}$, owing to the fact that the electric potential of the charged surface of
liquid $V_{\rm e}$ is uniform across the whole device. Thus, we can establish
the following relationship between $V_{\rm e}$, $V_{\rm ch}$ and $V_{\rm R}$
(with $V_{\rm gu}=0$)

$N_{e}=\frac{\varepsilon\varepsilon_{0}}{ed}\left(S_{\rm ch}V_{\rm ch}+S_{\rm
R}V_{\rm R}\right)-\frac{\varepsilon\varepsilon_{0}}{\alpha ed}SV_{\rm e},$
(7)

where $N_{\rm e}$ is the total number of SE in the device, $S_{\rm ch}$ and
$S_{\rm R}$ is the area of the charged surface of liquid in the middle segment
and reservoirs, respectively, and $S=S_{\rm ch}+S_{\rm R}$. Plugging this
relation into (3) with $V_{\rm bot}=V_{\rm ch}$ and $V_{\rm gu}=0$, we obtain
a linear dependence between $V_{\rm R}$ and $V_{\rm ch}$ with the slope equal
to $(\alpha-1/2)a/[1/2-(\alpha-1/2)b]$, where $a=S_{\rm R}/S$ and $b=S_{\rm
ch}/S$. For our device, we have $a=34/40$ and $b=6/40$ (see figure 2(a)),
which predicts the slope value equal to 0.56. This is in a satisfactory
agreement with our experimental result given in figure 6(b).

##  4 Discussion

The experimental results presented in the previous section demonstrate high-
level control and tunability of SE in a device comprised of an array of
microchanels. The Rydberg energy spectrum of SE in such a device is well
described by simple electrostatics, giving a very good agreement with the
measured image-charge spectra. This presents SE confined in microchannels as a
promising platform for further work towards quantum-state detection and qubit
implementation with electrons on helium. In our experiment, an image-charge
signal from about $10^{5}$ electrons excited by the MW radiation is readily
observed with the signal-to-noise (SNR) ratio approaching 100, thus showing
feasibility to scale down the Rydberg-transition detection to a much smaller
number of electrons. The most straightforward approach towards increasing the
sensitivity of the image-charge detection is to further increase the change in
the image charge $|\delta q|$ for an excited electron, see the discussion in
section 1, by employing a microchannel device with a smaller channel depth
$d$. However, this would bring electrons closer to the metal electrodes, thus
shifting their Rydberg transition frequency into the THz range due to the
Stark shift by the electric field of image charges. A more suitable approach
is to increase the sensitivity of the image-current detection method employed
together with the microchannel setup. Below, we discuss two methods towards
such an improvement.

Figure 7:  Thevenin equivalent circuit of the image-charge detection setup
with an arbitrary load impedance $Z$ (a) and with a parallel $LCR$ circuit
(b). The lock-in amplifier is referenced at the MW modulation frequency
$\omega_{\rm m}$ that matches the resonance frequency $\omega_{0}$ of the tank
circuit. (c) RF reflection method for a dispersive readout of the Rydberg
excitation of SE in a microchannel. Electrons are excited by continuous wave
(cw) microwaves, while their transition probability is modulated at the
frequency of the RF driving.

From the equivalent Thevenin circuit of the setup shown in figure 1(c), it is
clear that the output voltage signal $u_{\rm out}$ is limited by the value of
the parasitic capacitance $C_{\rm p}$. In the experiment reported here, this
capacitance is about 20 pF and is determined by the capacitance of the
connecting cryogenic cable. There are serious technical limitations due to the
dilution refrigerator operation which prevent us from shortening this cable,
thus decreasing the value of $C_{\rm p}$. A better solution is to employ a
resonant (tank) circuit to eliminate the parasitic capacitance and to increase
the output voltage signal. First, let us consider the equivalent Thevenin
circuit shown in figure 6(a), with a load impedance $Z$. As before, we assume
an infinite input impedance of the cryogenic amplifier. Then, the output
voltage is given by

$u_{\rm out}=\frac{Zu}{\left[Z+(j\omega_{\rm m}C_{0})^{-1}\right]}.$ (8)

Next, let us consider the load impedance in the form of a parallel $LCR$
circuit, with $Z^{-1}=j\omega_{\rm m}C+R^{-1}+(j\omega_{\rm m}L)^{-1}$.
Plugging this into (8) and using $u=-\Delta q/C_{0}$, we obtain

$u_{\rm out}=\frac{-j\Delta q\omega_{\rm m}}{\left[j(C_{0}+C)\omega_{\rm
m}^{-1}(\omega_{\rm m}^{2}-\omega_{0}^{2})+R^{-1}\right]},$ (9)

where $\omega_{0}=[L(C_{0}+C)]^{-1/2}$. Note that at the resonance condition
corresponding to $\omega_{\rm m}=\omega_{0}$ we obtain $u_{\rm out}=-jR\Delta
q\omega_{\rm m}$. In other words, the output voltage signal amplitude is given
by $iR$, where $i$ is the equivalent Norton current (see figure 1(b)) and $R$
is the impedance of the parallel $LCR$ circuit at resonance. Note that such a
circuit eliminates any parasitic capacitance $C_{\rm p}$ (see figure 6(b))
which can be simply added to the capacitance of the tank circuit causing a
shift in its resonance frequency $\omega_{0}$.

For the parallel $LCR$ circuit, a promising approach is to use a lumped-
element superconducting resonator [28]. Such resonators can be placed at
cryogenic temperatures in proximity to the experimental cell. When cooled
below the superconducting transition temperature, they achieve a high quality
factor $Q\sim 10^{4}\textrm{--}10^{5}$ at the resonance frequency
$\omega_{0}/2\pi\sim 1$ MHz, even in the presence of a strong magnetic field
[29, 30]. This results in a high value of the effective parallel resistance
$R=Q\omega_{0}L\sim 10^{2}\textrm{--}10^{3}$ M$\Omega$. Even with limitations
associated with a finite impedance of the cryogenic amplifier and added losses
in the circuit, this method allows detection of the image current on the order
of 10 fA [31]. In the experimental setup reported here, excitation of a single
electron will produce an image current $i=\kappa e\omega_{\rm m}(\Delta
z/d)\sim 10^{-14}$ A for the MW modulation frequency $\omega_{\rm m}/(2\pi)=1$
MHz. This approach shows the feasibility of detecting a single electron in our
setup.

Another approach towards improving the image-charge detection method is based
on a dispersive readout of the Rydberg transition, as was outlined previously
[11]. In this method, we measure a change in the capacitance due to variation
of the image charge induced in the capacitor plates by SE excited by the
continuous-wave (cw) MW radiation. In order to realize this method, we connect
the capacitor $C_{0}$ formed by the device electrodes in series with an
inductor $L$ and drive this tank circuit with an RF signal at the frequency
$\omega_{\rm RF}$ close to the resonance frequency of the circuit, see figure
7(c). The ac component of the voltage $u_{C}$ across the capacitor causes a
periodic modulation (with the frequency $\omega_{\rm RF}$) of the excited-
state population $\rho_{22}$ of SE via the Landau-Zener-Stückelberg (LZS)
transitions [32], providing that the Rydberg transition frequency of SE is
sufficiently close to the frequency of the applied MW radiation. This causes a
change in the capacitance value given by

$\Delta C=\frac{q(t)}{u_{C}(t)}=\frac{\bar{\rho}_{22}N_{e}\kappa e\Delta
z}{\bar{u}_{C}d},$ (10)

where $\bar{u}_{C}$ and $\bar{\rho}_{22}$ is the amplitude of the harmonic
variation of $u_{C}$ and $\rho_{22}$, respectively. In order to obtain the
maximum $\bar{\rho}_{22}$ by the constructive LZS interference [32], SE must
be driven through the Rydberg resonance with an amplitude $\bar{u}_{C}\approx
k_{1}\Delta\omega_{21}/(2\pi)$, where $k_{1}$ is the slope of the linear plot
in Fig. 4(a) and $\Delta\omega_{21}$ is the broadening of the transition line.
Using results of the experiment reported here, we estimate the change in
capacitance $\Delta C\sim 10^{-16}$ F for $N_{e}=10^{5}$,
$\Delta\omega_{21}=10$ GHz, and by assuming $\bar{\rho}_{22}=0.1$ [33]. Note
that for a single electron the inhomogeneous broadening of the transition line
must be replaced with the intrinsic linewidth due to the ripplon scattering.
For sufficiently low temperatures below 100 mK, the intrinsic linewidth is
expected to be on the order of 1 MHz [26]. This corresponds to the capacitance
change of $10$ aF for a single-electron excitation, which can be detected by
observing the spectral shift in the RF signal reflected from the tank circuit
[34], see figure 7(c). This makes such a dispersive readout of the single-
electron Rydberg transition to be very promising. Note that in this method any
parasitic capacitance $C_{\rm p}$ also contributes to the tank circuit (see
figure 7(c)), therefore to the dispersive changes in the circuit. Thus, the
value of this capacitance has to be minimized. In practice, this value is
limited to about 1 pF [34].

##  5 Conclusion

We reported on the first observation of the Rydberg transition of SE on
superfluid helium in a microchanel device by the image-charge detection
method. This work is largely motivated by the possibility for the quantum-
state detection of a single electron in a nano-structured device for qubit
implementation. It is shown that the sensitivity of the image-charge detection
method enhances thousandfold comparing with the previous work [11] due to
close proximity of SE from the electrodes in the microchannel structure. The
observed Rydberg energy spectrum can be controlled with high accuracy by the
dc bias voltages applied to the device and is in a very good agreement with
our theoretical calculations. It is shown that, by improving the image-current
measurement circuit, this detection method can be scaled down to the detection
of the Rydberg transition of a single confined electron. This would open a
pathway towards the spin-state detection, which has not been demonstrated in
this system yet. Together with the high-level control of the electron
transport, which was demonstrated in the microchannel devices previously [13,
14, 15, 16, 20, 24], our results show that SE confined in such devices present
a suitable platform for electron manipulation and their quantum-state
detection, with further prospects for qubit implementation and building a
scalable quantum computer [21, 23].

This work is supported by the internal grant ”KICKS” from Okinawa Institute of
Science and Technology (OIST) Graduate University. We thank Erika Kawakami for
helpful discussions and Research Support Division of OIST Graduate University
for technical support.

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