# Machine learning in the study of phase transition of two-dimensional complex
plasmas

He Huang  College of Science, Donghua University, 201620 Shanghai, PR China
Vladimir Nosenko  Institut für Materialphysik im Weltraum, Deutsches Zentrum
für Luft- und Raumfahrt (DLR), 51147 Cologne, Germany  Han-Xiao Huang-Fu
College of Science, Donghua University, 201620 Shanghai, PR China  Hubertus M.
Thomas  Institut für Materialphysik im Weltraum, Deutsches Zentrum für Luft-
und Raumfahrt (DLR), 51147 Cologne, Germany  Cheng-Ran Du
<EMAIL_ADDRESS>College of Science, Donghua University, 201620
Shanghai, PR China  Member of Magnetic Confinement Fusion Research Centre,
Ministry of Education, PR China

###### Abstract

Machine learning is applied to investigate the phase transition of two-
dimensional complex plasmas. The Langevin dynamics simulation is employed to
prepare particle suspensions in various thermodynamic states. Based on the
resulted particle positions in two extreme conditions, bitmap images are
synthesized and imported to a convolutional neural network (ConvNet) as
training sample. As a result, a phase diagram is obtained. This trained
ConvNet model can be directly applied to the sequence of the recorded images
using video microscopy in the experiments to study the melting.

complex plasma, melting, machine learning

###### pacs:

52.27.Lw

##  I introduction

A complex plasma is a weakly ionized gas containing micron-sized dust
particles Merlino2004PT ; Morfill2009RMP ; Fortov2005PR . The particles are
negatively charged owing to the higher thermal velocities of electrons
compared to ions Goree1994PSST . The discovery of the plasma crystal in the
laboratory inspired lasting interest Thomas1994PRL ; ILin1994PRL . Using video
microscopy, one can experimentally study plenty of phenomena such as wave
propagation Menzel2010prl ; Heidemann2009prl , self-organization
Suetterlin2009prl ; Schwabe2011prl , phase separation Wysockiprl2010 ;
Killer2016prl , and slow dynamics Du2019prl  at single-particle level
Schwabe2018MST . The particles are illuminated by a laser sheet and are
recorded as bright dots by a fast video camera equipped with bandpass filter.
The structure of the complex plasmas can then be obtained by direct processes
such as Fourier transformation of the recorded images Du2019prl  or
identification of x-y positions of individual particles using particle
tracking algorithms.

Phase diagram and phase transition in complex plasmas are research topics of
great interest Melzer1996PRE ; Knapek2007PRLa ; Khrapak2011prl ; Wang2020prl .
In the laboratory, monodisperse particles are suspended in a single layer
above the lower electrode where the gravity is compensated by the
electrostatic force in the sheath Feng2010PRL ; Nunomura2005PRL . The
thermodynamic state of such two-dimensional (2D) complex plasma can be easily
controlled by the experimental conditions, such as gas pressure and discharge
power Thomas2018PPCF ; Ishihara2007JPD . Tuning experimental parameters leads
to the variation of two dimensionless parameters: the coupling parameter
$\Gamma$ and screening parameter $\kappa$. The former represents the ratio of
interaction strength to the kinetic temperature, while the latter defines the
relative scale of interparticle distance to the Debye screening length.

Melting of plasma crystal can also be induced by the localized disturbance
such as shocks or shear flows driven by external field Knapek2007PRLb ;
Harila2012POP ; Couedel2018PRE ; Couedel2010PRL ; Jaiswal2020melting ;
Melzer2000PRE ; Nosenko2002PRL ; Schwabe2017NJP . Besides, the lateral wave of
a fast moving particle above or below the plasma crystal lattice leads to heat
transport. Kinetic energy is transferred to the lattice particles via
collisions with the self-propelled extra particle, resulting in the melting of
the crystal lattice Du2014PRE ; Laut2017prl .

Recently, machine learning has become a widely used analysis technique in
addressing physical problems, such as Gardner transition Li2021PNAS , phase
transitions Nieuwenburg2017NP , crystal structures classification
Ziletti2018NC . Meanwhile, various machine learning methods have also been
applied to the analysis in the complex plasma research. For example, Bayesian
optimization framework was applied to perform a nonlinear response analysis in
a complex plasma Ding2021MachineLearning . Multilayer perceptron was used to
classify fcc, bcc, and hcp structure in the three-dimensional plasma crystal,
where features are defined based on the particle positions Dietz2017PRE .
Support vector machine was used to locate the interface in a binary complex
plasma directly based on the recorded images in the experiments performed in
the PK-3 Plus laboratory on board the International Space Station He2019JI .
Convolutional neural network was applied to reconstruct the three-dimensional
(3D) positions of particles in a dense dust cloud in a dusty plasma under
weightlessness from stereoscopic camera images Himpel2021 .

In this paper, we apply a machine learning method to investigate the phase
transition in 2D complex plasmas based on the Langevin dynamics simulations.
In Sec. II, the numerical simulation is briefly introduced. The data process
protocol and the machine learning model are described. In Sec. III, the phase
diagram is obtained using a convolutional neural network and compared with the
ones obtained using other approaches. In Sec. IV, the trained model is
directly applied to identify the change of the state of the complex plasma as
it melts. Finally, a conclusion is drawn in Sec. V.

##  II Method

Figure 1: Scheme for applying machine learning in the phase transition of 2D
complex plasmas. Training samples are prepared using Langevin dynamics
simulations and fed to the ConvNet, a convolutional neural network (a,b). The
trained model can be applied to identify the thermodynamic state of the
complex plasma in the experiment, where the raw images of the experimental
recordings are thresholded and filtered by a Gaussian kernel (c-e).

The Langevin dynamics simulation is applied to prepare complex plasmas in
various thermodynamic states. For the purpose of model training and later
application in the analysis of experiments, sequences of images are prepared
based on the simulation results. The trained model can be used to identify the
thermodynamic state of the experiments. The scheme is illustrated in Fig. 1.

###  II.1 Simulation

Figure 2: Comparison of the complex plasmas with different coupling parameters
and $\lambda_{D}=400$ $\mu$m, corresponding to $\kappa=1.18$. The particle
positions are color-coded with hexatic order parameter $|\Psi_{6}|$ in the
simulation for $\Gamma=10000$ (a) and $\Gamma=100$ (b). The pair correlation
function $g_{r}$ and the distribution of $|\Psi_{6}|$ (in percentage of
particles) are shown for plasma crystals (c,e) and liquids (d,f) with various
$\Gamma$ well above and below the melting point.  Figure 3: Architecture of
the convolutional neural network (ConvNet) used for the classification. Two
$5\times 5$ kernel 2D convolution layers are included. Each convolutional
layer is followed by a max-pooling layer. Before the fully connected layers, a
Gaussian Dropout layer is added to prevent overfit. The last two layers are
fully connected layers, to achieve binary classification.

The Langevin dynamics simulation is employed to prepare complex plasmas in
different thermodynamic states. The equation of motion including the damping
from the neutral gas and Brownian motion of microparticles is given by:

$m_{i}\ddot{\bm{r}}_{i}+m_{i}\nu_{i}\dot{\bm{r}}_{i}=-\sum_{j\neq
i}\bigtriangledown\phi_{ij}+\bm{L}_{i},$ (1)

where $\bm{r}_{i}$ is the 2D position of particle $i$, $m$ is the particle
mass, $\nu$ is the damping rate. The Langevin force $\bm{L}_{i}$ is defined by
$\left\langle\bm{L}_{i}\right\rangle=0$ and
$\left\langle\bm{L}_{i}(t)\bm{L}_{j}(t+\tau)\right\rangle=2\nu
mk_{b}T\delta_{i,j}\delta(\tau)\bm{I}$, where $k_{b}$ is the Boltzmann
constant, $T$ is the temperature of the heat bath, $\delta_{ij}$ is Kronecker
delta, $\delta(\tau)$ is the delta function, and $\bm{I}$ is the unit matrix.
In the simulation, we assume that the particles interact with each other via
the Yukawa potential,

$\phi_{ij}=\dfrac{Q_{i}Q_{j}}{4\pi\epsilon_{0}r_{ij}}exp(-\dfrac{r_{ij}}{\lambda_{D}}),$
(2)

where $\lambda_{D}$ is the Debye length,$Q_{i}$ is the charge of particle $i$
and $Q_{j}$ is the charge of a neighboring particle $j$, the interparticle
distance is $r_{ij}$. Here, we select typical experimental parameters in the
simulations. The mass is set to $3\times 10^{-13}$ kg and the particle charge
is set to $8000$ $e$. The damping rate is set to $0.9$ $s^{-1}$. The total
number of particles in the simulation is $6400$. Periodic boundary conditions
are used.

Two crucial dimensionless parameters to describe the strongly coupled systems
with Yukawa interaction are the coupling parameter $\Gamma$ and screening
parameter $\kappa$. The former is defined as

$\Gamma=\dfrac{Q^{2}}{4\pi\epsilon_{0}\Delta k_{b}T},$ (3)

where $\Delta$ is the length scale of the distance to the nearest neighbors
and can be calculated as $\Delta=(\pi n_{d})^{-1/2}$, and $n_{d}$ is the
particle number density Melzer1996PRE . The latter is defined as

$\kappa=\frac{\Delta}{\lambda_{D}}.$ (4)

In the simulations, we set the kinetic temperature $T$ to tune the coupling
parameter and the Debye length $\lambda_{D}$ to tune the screening length. In
order to cover the typical parameter range in complex plasmas, the temperature
ranges from $100$ to $70000$ K and the Debye length ranges from $150$ to
$1800$ $\mu$m for $\Delta\sim 460$ $\mu$m, corresponding to
$30\lesssim\Gamma\lesssim 22000$ and $0.25\lesssim\kappa\lesssim 3$. As a
result, complex plasmas in various thermodynamic states from liquid to crystal
are prepared.

Snapshots of particle positions with two extreme coupling strengths are
demonstrated in Fig. 2(a,b). In order to quantify the local structure , we
define the hexatic order parameter $\Psi_{6,i}$ as

$\Psi_{6,i}=\frac{1}{6}\sum_{k=1}^{6}{e^{j6\theta_{k}}},$ (5)

where we only consider six nearest neighbors and $\theta_{k}$ is the angle
between $\bm{r}_{k}-\bm{r}_{i}$ and the $x$ axis. The color coding in the
snapshots represents the order parameter $|\Psi_{6,i}|$ of particles in
simulation. For $|\Psi_{6,i}|=1$, the particle $i$ is located in the center of
a perfect hexagon cell, while for $|\Psi_{6,i}|=0$, the particle $i$ is in a
completely disordered structure.

On the one hand, for a strongly coupled 2D complex plasma, the particles self-
organize in a triangular lattice with hexagonal symmetry. A chain of
dislocations lies in the middle of the snapshot, as we see in Fig. 2(a). We
select three strong coupling situations ($\Gamma=1,5,10\times 10^{3}$) where
the temperature is well below the melting temperature $T_{m}$, and the pair
correlation functions $g_{r}$ as well as the distributions of $|\Psi_{6}|$ are
rather similar. These simulation results can be labeled as plasma crystals. On
the other hand, when the coupling parameter is small, the particle suspension
is in a liquid state and does not exhibit ordered structure, as shown in Fig.
2(b). For three selected temperatures well above the melting temperature,
$g_{r}$ and the distribution of $|\Psi_{6}|$ do not differ much. If the
temperature further increases, the first peak value of $g_{r}$ and the
averaged $|\Psi_{6}|$ may also decrease. However, the degree of the variation
is much smaller than that close to the melting temperature and thus has
marginal influence on the further analysis. The detailed structure of 2D
liquid complex plasma is rather complicated and beyond the scope of this paper
Ott2014PRE ; Ott2015CPP ; Castello2021PRE .

###  II.2 Data Preparation

The particle positions at each time step in the Langevin dynamics simulation
are transformed into a gray-scale bitmap image, resembling the images exported
from the video recording in the experiments. In order to achieve this purpose,
a few steps are necessary. First, the particle positions $\bm{r}$ in SI unit
are transformed into positions $\bm{R}_{x,y}$ in pixels by a coefficient
$\eta$, so that the interparticle distances (in pixels) appear comparable with
those in the recorded images. Second, the positions are ceiled, corresponding
to the indices of a matrix $\bm{M}$ representing a gray-scale image. This
matrix reads

$\bm{M}_{i,j}=\left\\{\begin{array}[]{rcl}255&&(i,j)\in\lceil\bm{R}_{x,y}\rceil\\\
0&&(i,j)\notin\lceil\bm{R}_{x,y}\rceil\end{array}\right..$ (6)

Finally, a Gaussian filter is applied to the matrix, convolving binary image
with a Gaussian kernel
$G_{0}\left(x,y\right)=1/{2\pi{\sigma}^{2}}\cdot\exp[-(x^{2}+y^{2})/2{\sigma}^{2}]$
and creating a gray-scale synthetic image. Here, the variance is set as
${\sigma}=9$. The synthetic images are shown in Fig. 1(b).

Similar procedures are applied to the images obtained in the experiments.
Despite the fact that the particles in the recorded bitmap images exhibit
Gaussian profiles Feng2007RSI , certain deviations still exist, especially in
the presence of overexposure. To mitigate the discrepancy while applying the
method to the experiment analysis, we apply a threshold to the experimental
images to remove noise, binarize the gray-scale bitmaps and apply the Gaussian
filter to them, as illustrated in Fig. 1(c-e). These steps may not be
necessary, but can improve the performance of the algorithm to some extent.

###  II.3 Machine Learning

ConvNet, also known as convolutional neural network, is a specific type of
deep learning network, which has been widely used in image identification in
the past years LeCun1989 ; Krizhevsky2012 ; lin2014network . In this work, we
apply ConvNet to investigate the phase behaviors of 2D complex plasmas. The
architecture of our network is shown in Fig. 3, similar to Lenet5 Lecun1998 .
It contains $2$ convolutional layers, $2$ max-pooling layers, $1$ Gaussian
Dropout layer and $2$ fully connected layers.

Two $5\times 5$ kernel 2D convolution layers are used. The convolutional
layers extract increasingly high-level feature representations of the input
and preserve their spatial relationship. The expression of the convolution can
be written as:

$\bm{h}_{k}^{l}=\varphi(\sum_{j}\bm{W}_{j,k}^{l}\bm{h}_{j}^{l-1}+\bm{b}_{k}^{l}),$
(7)

where $l$ is the layer index, $j$ is the index of input feature maps, $k$ is
the index of output feature maps. The input $\bm{h}_{j}^{l-1}$ is the $j$th
feature map at layer $l-1$, $\bm{h}_{k}^{l}$ is the $k$th feature map at layer
$l$, $\bm{W}$ is the convolutional weight tensor and the value is random
uniform. $\bm{b}$ is the bias term and the value is initialized to zero.
$\varphi(\cdot)$ is the element-wise nonlinearity function and we used the
rectified linear unit (RELU) function 2019Hinton . The number of channels is
set to $16$ for the first convolutional layer and $32$ for the second
convolutional layer.

Each convolutional layer is followed by a max-pooling layer. The max-pooling
layers only preserve the maximum value within a local receptive field and
discard all other values. By applying max-pooling layers, we reduce the number
of free parameters and introduce a small amount of translational invariance
into the network. Before the fully connected layers, we use the Gaussian
Dropout layer to prevent overfit Srivastava2014 . The last two layers are
fully connected layers. The first fully connected layer flattens all of the
feature maps after the max-pooling layer. The second fully connected layer
works like a linear classifier. We choose RELU for the fully connected hidden
layers.

The ConvNet is easy to deploy using open source platform for machine learning.
Our ConvNet is based on TensorFlow abadi2016tensorflow  and Keras
chollet2015keras . The training sample contains $2000$ labeled gray-scale
bitmap images with a size of $100\times 100$ pixel2. The training was
conducted using Adam optimization Adam2014  with batches of $100$ images for
$20$ epochs with a learning rate $10^{-3}$ and categorical crossentropy as
loss function. We train the ConvNet on $80\%$ of the dataset, and use the
remaining $20\%$ as test set. Almost $100\%$ accuracy is achieved on the
training and test datasets. This means that our model can learn the samples
and at the same time is capable of classifying the images which have not been
seen before.

##  III Phase diagram

Figure 4: Probability $P(\Gamma,N)$ of complex plasma being classified as
crystal (dark symbols) and probability $1-P(\Gamma,N)$ of being classified as
liquid (light symbols), as function of coupling parameter $\Gamma$. (a) shows
the dependence on the size of the sample image, while (b) shows the dependence
on the screening parameter $\kappa$.

Supervised machine learning method is applied to investigate the phase diagram
of 2D complex plasmas. In the conventional supervised learning, the training
samples are labeled based on some other criteria (usually not included in the
traning sample), such as Lindemann measures, for example, in the investigation
of phase behaviors of matter Chakravarty2007JCP . This requires additional
diagnostics of the particle motion while labeling the samples. In addition,
the training data usually shall cover the whole range of thermodynamic states
of interest.

However, in this work, we try a different approach Nieuwenburg2017NP ;
Carrasquilla2017NP ; Li2021PNAS . Although the training samples still need to
be labeled, as usually done in the supervised learning, they only include the
thermodynamic states in two extreme scenarios. For high coupling
($\Gamma\gtrsim 1000$) the first peak value of $g_{r}$ is higher than $7$,
while for low coupling ($\Gamma\lesssim 100$) the first peak value of $g_{r}$
is lower than $2$. The systems are in crystalline and liquid states,
respectively. We do not have to provide training sample with a thermodynamic
state near the critical value, to avoid ambiguity111We select extreme
scenarios at different coupling parameter, train the model with selected
training sample, and test its performance. It turns out that the selection of
the exact $\Gamma$ has marginal impact to the model training as long as they
are far beyond and below the critical value..

Figure 5:  Phase diagram obtained using ConvNet based on the Langevin dynamics
(red hexagons), fitted by the analytical expression Eq. 8 (red line). Melting
lines based on 2D-YOCP theory and on the numerical simulation with the
criterion $\langle|\Psi_{6}|\rangle=0.45$ are shown by blue dashed line and
blue solid hexagons, respectively. The experiments with plasma crystals and
liquids are marked by solid and empty symbols, respectively.

Once the training is completed, we apply this trained model to classify the
synthetic images of 2D complex plasma resulting from the Langevin simulation.
The simulation method has already been introduced in Sec. II.1. We prepare
particles with $T=100$ K, well below the melting temperature, in the
simulation box and have the system relax. Then we slowly heat the system until
it melts and further increase the temperature until $T=70000$ K. Synthetic
bitmap images are generated every thousand K during the temperature increase.
The simulation is repeated $20$ times with different initial conditions. The
bitmap images are fed to the ConvNet model and result in the probability
$P(\Gamma,N)$, that the image is classified as plasma crystal. Here, $N$ is
the approximate number of particles included in the bitmap image sample. The
cross of $P(\Gamma,N)$ and $1-P(\Gamma,N)$ provides the coupling strength of
the phase transition $\Gamma_{m}$.

The dependence of $P(\Gamma,N)$ on the size of bitmap images is shown in Fig.
4(a). The number of particles $N=20,80,320$ included in the bitmap image
sample corresponds to the image sizes $50\times 50,100\times 100,200\times
200$ pixel2, respectively. The results show that for images including more
than $20$ particles, the sample image size has marginal impact on the
identification of $\Gamma_{m}$. In order to study the dependence of
$\Gamma_{m}$ on the screening parameter, we also vary $\kappa$ by changing the
Debye length $\lambda_{D}$ in the simulation. As shown in Fig. 4(b), as the
screening parameter $\kappa$ increases, the cross of $P(\Gamma,N)$ and
$1-P(\Gamma,N)$ (denoting $1/\Gamma_{m}$) shifts to the left.

The phase diagram resulting from our ConvNet is shown as red hexagons in Fig.
5. The melting line can be fitted by the analytical expression Hartmann2005PRE

$\Gamma_{m}\left(\kappa\right)=\frac{\Gamma_{m}^{OCP}}{1+a\kappa^{2}+b\kappa^{3}+c\kappa^{4}},$
(8)

where $\Gamma_{m}^{OCP}=131$ is an approximation for the 2D-OCP melting point
Totsuji2004PRE ; Vaulina2002PRE . The fitting results
$a=-0.401,b=0.132,c=0.0099$ show a fairly good agreement with the 2D-YOCP
theory Melzer1996PRE  (blue dashed line) and numerical simulations, where the
melting line is found with the criterion $\langle|\Psi_{6}|\rangle=0.45$
Hartmann2005PRE  (blue solid hexagons), as well as other experiments listed in
Tab. 1.

Table 1: Dimensionless parameters $\Gamma$ and $\kappa$ and thermodynamic state of 2D complex plasmas in various experiments, corresponding to the symbols in Fig. 5. Ref. | $\Gamma$ | $\kappa$ | State
---|---|---|---
Hartmann2010PRL  | $392$ | $1.5\pm 0.3$ | crystal
Knapek2007PRLb  | $1850\pm 450$ | $0.8\pm 0.13$ | crystal
Samsonov2000PRE  | $3000,2500,2700$ | $2.3,2.4,1.8$ | crystal
$1900,4000,3000$ | $1.9,1.4,1.7$
Vorona2007  | $30,150$ | $0.375,2.4$ | liquid
Nunomura2005PRL  | $12,30$ | $1$ | liquid
Feng2010PRL  | $68$ | $0.5$ | liquid
Haralson2016pop  | $92\sim 155$ | $0.72$ | liquid

##  IV Application in Experiment

We apply our trained CovnNet to an experiment, to identify the state of the 2D
complex plasma as it melts. The experiment was carried out in Gaseous
Electronics Conference (GEC) rf reference cell Couedel2010PRL ; YanFeng2008PRL
. Argon plasma was sustained using a capacitively coupled rf discharge at
$13.56$ MHz. The input power was set at $20$ W. Melamine-formaldehyde(MF)
spherical particles with a diameter of $7.17\pm 0.07$ $\mu$m were levitated in
the plasma sheath and illuminated by a horizontal laser sheet. The particle
motion was recorded by a CMOS camera from the top. During the experiment, some
disturbances were imposed by extra particles.

Figure 6:  Evolution of local structure and kinetic temperature, as an extra
particle moves below the particle suspension. The structure is measured as the
number of neighbors $NN$ using triangulation, whose value is $5$ for a five-
fold defect and $7$ for a seven-fold defect. The extra particle is marked as
gray circle in (e) and its motion is illustrated by an arrow.

As an extra particle moved below the particle suspension with a high velocity,
it created a strong disturbance in terms of Yukawa repulsion Samsonov2000PRE ;
Du2012EPL ; Du2014PRE . The interparticle distance of the particle suspension
in the vicinity of the extra particle increases dramatically, and the ordered
structure is destroyed, illustrated by the emergence of many defects locally,
see Fig. 6(b,e). Heat is transported to the plasma crystal, local kinetic
temperature spikes, and eventually the system melts, as shown in Fig. 6(c,f).

The raw bitmap images [c.f. Fig. 7(a-c)] recorded in the experiment are
processed according to the description in Sec. II.2. The processed images of
the whole melting event are fed to the trained ConvNet and the output layer
provides the probability $P$ of the particle suspension being crystalline. The
evolution of $P$ and $\langle\Psi_{6}\rangle$ are shown in Fig. 7(d,e),
respectively. As the particle moves into the field of interest, $P$ drops
instantaneously, signifying that the plasma crystal melts. This can also be
seen as the substantial drop of $\langle\Psi_{6}\rangle$ at the same time
Hartmann2005PRE ; Schweigert1999PRL . By applying the machine learning method,
a sharper contrast on the identification of the melting transition is achieved
than that of the traditional method based on $\Psi_{6}$.

Figure 7:  Raw bitmap images as test sample for ConvNet (a-c), the evolution
of test results (d) and the hexatic order parameter
$\langle|\Psi_{6}|\rangle$. The probability $P(\Gamma)$ of being plasma
crystal is shown by the red circles, while the probability $1-P(\Gamma)$ of
being liquid is shown by the yellow circles in (d). The green strip in (d,e)
highlights the moment when the extra particle moves into the region of
interest. The scale bar corresponds to $1$ mm in (a).

##  V Conclusion

To conclude, we apply a machine learning method to study the phase transition
of 2D complex plasmas. A convolutional neural network (ConvNet) is trained
with the synthetic bitmap images based on the Langevin dynamics simulation,
where definition of feature parameters is not needed. By training the model
with the samples in two extreme scenarios in plasma crystal (very high
$\Gamma$) and in liquid (very low $\Gamma$), a phase diagram is obtained,
which agrees well with previous YOCP theory and numerical simulations. The
method can be directly applied to the analysis of experiments, as demonstrated
in this paper.

This method may be extended to investigate the phase transition in 3D complex
plasmas, where particle tracking using video microscopy is challenging. As
particle tracking is not necessary and thus the analysis is much faster than
traditional methods, it can be possibly applied to the experiments on board
the space station, where the automation of the experiment control based on the
live diagnostics is desirable. We leave this for the future work.

###### Acknowledgements.

The authors acknowledge the support from the National Natural Science
Foundation of China (NSFC), Grant No. $11975073$ and $21035003$. The students
in Class $012781$ (Summer Semester $2021$) of the lecture "Machine Learning in
Physics" have participated in developing the architecture of the machine
learning model as their semester assignment. Their reports are taken as
reference for this work. The authors acknowledge their contributions and
inspirations. We thank M. Schwabe for the helpful discussions and C. Knapek
for the valuable comments.

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# Unbounded operators and the uncertainty principle

friedrich haslinger  Fakultät für Mathematik
Universität Wien
Oskar-Morgenstern-Platz 1
A-1090 Wien, Austria<EMAIL_ADDRESS>
###### Abstract.

We study a variant of the uncertainty principle in terms of the annihilation
and creation operator on generalized Segal Bargmann spaces, which are used for
the FBI-Bargmann transform. In addtion, we compute the Berezin transform of
these operators and indicate how to use spaces of entire functions in one
variable to study the Szegő kernel for hypersurfaces in $\mathbb{C}^{2}.$

###### Key words and phrases:

Uncertainty principle, Segal-Bargmann space, FBI transform

###### 2020 Mathematics Subject Classification:

Primary 30H20, 32A36; Secondary 81S05

Supported by the Austrian Science Fund (FWF) project P36884.

##  1\. Introduction

In the Segal-Bargmann space (also Fock space)
$A^{2}(\mathbb{C}^{n},e^{-|z|^{2}})$ of entire functions
$f:\mathbb{C}^{n}\longrightarrow\mathbb{C}$ such that

$\|f\|^{2}=\int_{\mathbb{C}^{n}}|f(z)|^{2}\,e^{-|z|^{2}}\,d\lambda(z)<\infty,$

the differentiation operators $a_{j}(f)=\frac{\partial f}{\partial z_{j}}$
(annihilation) and the multiplication operators $a^{*}_{j}(f)=z_{j}f$
(creation) are unbounded, densely defined adjoint operators with

$\|f\|^{2}\leq\|a_{j}(f)\|^{2}+\|a^{*}_{j}(f)\|^{2}$

for $f\in{\text{dom}}(a_{j}),$ which corresponds to the uncertainty principle,
see [3].

The uncertainty principle in its general form states that if $A$ and $B$ are
quantum observables (i.e. self-adjoint operators), the probability
distributions of $A$ and $B$ cannot both be concentrated near single points in
any state $u$ such that

$((AB-BA)u,u)\neq 0.$

######  Theorem 1.1.

If $A$ and $B$ are self-adjoint operators on a Hilbert space $\mathcal{H},$
then

$\|(A-a)u\|\,\|(B-b)u\|\geq\frac{1}{2}\left|((AB-BA)u,u)\right|$

for all $u\in{\text{dom}}(AB)\cap{\text{dom}}(BA)$ and all $a,b\in\mathbb{R}.$

We apply this result to the position operators $X_{j}(u)=x_{j}u$ and the
momentum operators $D_{j}u=-i\frac{\partial u}{\partial x_{j}}$ on
$L^{2}(\mathbb{R}^{n})$ and get

(1) $\|(X_{j}-a)u\|\,\|(D_{j}-b)u\|\geq\frac{1}{2}\|u\|^{2}.$

With $a=b=0,$ and the inequality
$\alpha\beta\leq\frac{1}{2}(\alpha^{2}+\beta^{2})$ we get the following
variant of the uncertainty principle

(2)
$\|X_{j}(u)\|^{2}+\|D_{j}(u)\|^{2}\geq\|u\|^{2},\,u\in{\text{dom}}(X_{j})\cap{\text{dom}}(D_{j}).$

Actually, (1) and (2) are equivalent.

We will describe this inequality in the Segal-Bargmann space (Fock space)
$A^{2}(\mathbb{C}^{n},e^{-|z|^{2}}).$

Let $a_{j}(f)=\frac{\partial f}{\partial z_{j}},$ where we take

$f\in{\text{dom}}(a_{j})=\\{f\in
A^{2}(\mathbb{C}^{n},e^{-|z|^{2}}):\frac{\partial f}{\partial z_{j}}\in
A^{2}(\mathbb{C}^{n},e^{-|z|^{2}})\\}.$

$a_{j}$ (annihilation) is a densely defined, unbounded operator, its adjoint
$a_{j}^{*}$ (creation) is given by $a_{j}^{*}(f)=z_{j}f$ and we have

$\|a_{j}^{*}(f)\|^{2}=\|a_{j}(f)\|^{2}+\|f\|^{2},$

hence ${\text{dom}}(a_{j})={\text{dom}}(a_{j}^{*}),$ and

(3) $\|f\|^{2}\leq\|a_{j}(f)\|^{2}+\|a_{j}^{*}(f)\|^{2},$

for $f\in{\text{dom}}(a_{j}).$ In another similar context, this inequality is
the so-called basic estimate for the corresponding complex Laplacian operator,
see [6], [7], [8], [9], [10], [11].

Here we use the Bargmann transform
$\mathcal{B}:L^{2}(\mathbb{R}^{n})\longrightarrow
A^{2}(\mathbb{C}^{n},e^{-|z|^{2}}),$ which is given by

$\mathcal{B}(f)(z)=\pi^{-n/4}\int_{\mathbb{R}^{n}}f(x)\exp\big{[}-z^{2}/2+\sqrt{2}zx-x^{2}/2)\big{]}\,dx,\,f\in
L^{2}(\mathbb{R}^{n}),$

here $z^{2}=\sum_{j=1}^{n}z_{j}^{2}$ and $xz=\sum_{j=1}^{n}x_{j}z_{j}.$

The Bargmann transform is a unitary operator between $L^{2}(\mathbb{R}^{n})$
and $A^{2}(\mathbb{C}^{n},e^{-|z|^{2}}).$

Concerning the operators $a_{j}$ and $a_{j}^{*}$ we have

$\mathcal{B}^{*}a_{j}\mathcal{B}=\frac{1}{\sqrt{2}}(X_{j}+iD_{j})\,\
{\text{and}}\
\,\mathcal{B}^{*}a_{j}^{*}\mathcal{B}=\frac{1}{\sqrt{2}}(X_{j}-iD_{j}),$

So we get

$\|X_{j}(u)\|^{2}=\frac{1}{2}\|\mathcal{B}^{*}a_{j}\mathcal{B}(u)+\mathcal{B}^{*}a_{j}^{*}\mathcal{B}(u)\|^{2},$

and

$\|D_{j}(u)\|^{2}=\frac{1}{2}\|\mathcal{B}^{*}a_{j}\mathcal{B}(u)-\mathcal{B}^{*}a_{j}^{*}\mathcal{B}(u)\|^{2}.$

Hence

$\displaystyle\|X_{j}(u)\|^{2}+\|D_{j}(u)\|^{2}$ $\displaystyle=$
$\displaystyle\|\mathcal{B}^{*}a_{j}\mathcal{B}(u)\|^{2}+\|\mathcal{B}^{*}a^{*}_{j}\mathcal{B}(u)\|^{2}$
$\displaystyle=$
$\displaystyle\|a_{j}\mathcal{B}(u)\|^{2}+\|a^{*}_{j}\mathcal{B}(u)\|^{2}$
$\displaystyle\geq$ $\displaystyle\|\mathcal{B}(u)\|^{2}$ $\displaystyle=$
$\displaystyle\|u\|^{2},$

hence (3) and (2) are equivalent.

On the Segal-Bargmann space we have the following commutation relations:

$[a_{j},a_{j}^{*}]=a_{j}a_{j}^{*}-a_{j}^{*}a_{j}=I,$

$j=1,\dots,n.$

In this paper we study the operators $a_{j}$ and its adjoints $a_{j}^{*}$ in
more general weighted spaces of entire functions which play a crucial role in
semiclassical analysis and quantization (see [14], [16]). In sake of
simplicity we will carry out all computations for the case $n=2,$ as the
general case contains no further difficulties.

Let

$A=\begin{pmatrix}\beta_{1}&\kappa\\\ \kappa&\beta_{2}\end{pmatrix},$

with $\beta_{1},\beta_{2},\kappa\in\mathbb{R}.$ For
$z=\binom{z_{1}}{z_{2}}\in\mathbb{C}^{2}$ define

$\langle Az,z\rangle=\beta_{1}z_{1}^{2}+2\kappa
z_{1}z_{2}+\beta_{2}z_{2}^{2},$

and let

$H_{A}=\left\\{f\,{\text{entire}}:\int_{\mathbb{C}^{2}}|f(z)|^{2}\,\exp(-\Re\langle
Az,z\rangle-|z|^{2})\,d\lambda(z)<\infty\right\\}.$

Using the canonical orthonormal basis of $H_{A}$ we investigate the densely
defined operators $a_{j}(f)=\frac{\partial f}{\partial z_{j}}$ and
$b_{j}(f)=z_{j}f$ on $H_{A},$ and establish analogous estimates to the
uncertainty principle (3).

##  2\. The FBI-Bargmann transform

By means of the FBI-Bargmann transform one obtains a correspondence between
Weyl operators acting on $L^{2}(\mathbb{R}^{n})$ and Weyl operators acting on
$H_{A},$ see [14]. In [1] a similar approach is used for some other
interesting applications. We will use the canonical orthonormal basis in
$H_{A}$ to compute the adjoint operators. We apply Parseval’s identity, and it
will be important to know that convergence in the norm of $H_{A}$ implies
convergence of entire functions in the compact open topology, which follows
from

######  Lemma 2.1.

Let $\varphi:\mathbb{C}^{n}\longrightarrow\mathbb{R}$ be a real valued
continuous function and define

$A^{2}(\mathbb{C}^{n},e^{-\varphi})=\\{f\,{\text{entire}}\,:\|f\|^{2}=\int_{\mathbb{C}^{n}}|f(z)|^{2}e^{-\varphi(z)}\,d\lambda(z)<\infty\\}.$

Then, for each compact subset $K\subset\mathbb{C}^{n},$ there exists a
constant $C_{K}>0$ such that

(4) $\sup_{z\in K}|f(z)|\leq C_{K}\|f\|,$

for each $f\in A^{2}(\mathbb{C}^{n},e^{-\varphi}).$

###### Proof.

We consider only the case $n=1,$ as iterations of Cauchy’s theorem imply the
general result:

$\displaystyle|f(z)|$ $\displaystyle\leq$ $\displaystyle\frac{1}{\pi
r^{2}}\int_{D(z,r)}e^{\varphi(w)/2}|f(w)|e^{-\varphi(w)/2}\,d\lambda(w)$
$\displaystyle\leq$ $\displaystyle\frac{1}{\pi
r^{2}}\left(\int_{D(z,r)}e^{\varphi(w)}\,d\lambda(w)\right)^{1/2}\,\|f\|.$

∎

Next we indicate that the spaces $H_{A}$ and
$A^{2}(\mathbb{C}^{n},e^{-|z|^{2}})$ are isometric isomorphic. The isometry

$\Phi:A^{2}(\mathbb{C}^{n},e^{-|z|^{2}})\longrightarrow H_{A}$

is given by $\Phi(f)(z)=f(z)\exp\left(\frac{1}{2}\langle Az,z\rangle\right),$
for $f\in A^{2}(\mathbb{C}^{n},e^{-|z|^{2}}).$

The functions

$\phi_{\alpha}(z)=\frac{z_{1}^{\alpha_{1}}}{\sqrt{\pi\alpha_{1}!}}\,\frac{z_{2}^{\alpha_{2}}}{\sqrt{\pi\alpha_{2}!}},$

$\alpha_{1},\alpha_{2}=0,1,2,\dots.$ form the canonical orthonormal basis for
$A^{2}(\mathbb{C}^{2},e^{-|z|^{2}}).$ Then the functions

$\psi_{\alpha}(z)=\frac{z_{1}^{\alpha_{1}}}{\sqrt{\pi\alpha_{1}!}}\,\frac{z_{2}^{\alpha_{2}}}{\sqrt{\pi\alpha_{2}!}}\,\exp\left(\frac{1}{2}\langle
Az,z\rangle\right),$

constitute an orthonormal basis in $H_{A}:$ for the inner product in $H_{A}$
we have

$(\psi_{\alpha},\psi_{\alpha^{\prime}})=\int_{\mathbb{C}^{2}}\psi_{\alpha}(z)\left(\psi_{\alpha^{\prime}}(z)\right)^{-}\exp(-\Re\langle
Az,z\rangle-|z|^{2})\,d\lambda(z)=(\phi_{\alpha},\phi_{\alpha^{\prime}}),$

where the last inner product is the inner product of the Segal Bargmann space.

If $f\in H_{A},$ we write $f=\sum_{\alpha}f_{\alpha}\psi_{\alpha},$ where
$(f_{\alpha})_{\alpha}\in l^{2}.$ Define

${\text{dom}}(a_{j})=\\{f\in H_{A}:a_{j}(f)\in H_{A}\\},\,j=1,2.$

Similarly, let $b_{j}(f)=z_{j}f$ and

${\text{dom}}(b_{j})=\\{f\in H_{A}:b_{j}(f)\in H_{A}\\},\,j=1,2.$

The operators $a_{j},b_{j}$ are densely defined unbounded operators on $H_{A}$
with closed graph, which implies that the domains of $a_{j}^{*}$ and
$b_{j}^{*}$ are also dense in $H_{A};$ (see [6]).

We compute the images of the basis elements under $a_{j}$ and $b_{j}.$ We have

(5)
$b_{1}(\psi_{\alpha})=\sqrt{\alpha_{1}+1}\,\psi_{\alpha_{1}+1,\alpha_{2}}\,,\,b_{2}(\psi_{\alpha})=\sqrt{\alpha_{2}+1}\,\psi_{\alpha_{1},\alpha_{2}+1}.$

For $a_{j}$ we obtain

(6)
$a_{1}(\psi_{\alpha})=\sqrt{\alpha_{1}}\,\psi_{\alpha_{1}-1,\alpha_{2}}+\beta_{1}\sqrt{\alpha_{1}+1}\,\psi_{\alpha_{1}+1,\alpha_{2}}+\kappa\sqrt{\alpha_{2}+1}\,\psi_{\alpha_{1},\alpha_{2}+1}$

and

(7)
$a_{2}(\psi_{\alpha})=\sqrt{\alpha_{2}}\,\psi_{\alpha_{1},\alpha_{2}-1}+\beta_{2}\sqrt{\alpha_{2}+1}\,\psi_{\alpha_{1},\alpha_{2}+1}+\kappa\sqrt{\alpha_{1}+1}\,\psi_{\alpha_{1}+1,\alpha_{2}}.$

Now let $f\in{\text{dom}}(a_{1})$ with
$f=\sum_{\alpha}f_{\alpha}\psi_{\alpha},$ where $(f_{\alpha})_{\alpha}\in
l^{2}.$

Now we suppose that $\kappa=0.$ We use (2.1) and get from (6)

(8)
$a_{1}(f)=\sum_{\alpha}\sqrt{\alpha_{1}+1}f_{\alpha_{1}+1,\alpha_{2}}\psi_{\alpha_{1},\alpha_{2}}+\beta_{1}\sum_{\alpha_{1}\geq
1,\alpha_{2}\geq
0}\sqrt{\alpha_{1}}f_{\alpha_{1}-1,\alpha_{2}}\psi_{\alpha_{1},\alpha_{2}}.$

If $g=\sum_{\alpha}g_{\alpha}\psi_{\alpha}\in{\text{dom}}(a_{1}^{*})$ we get
$(a_{1}(f),g)=(f,a_{1}^{*}(g)).$ Comparing with the formula from above

(9) $a_{1}^{*}(g)=\sum_{\alpha_{1}\geq 1,\alpha_{2}\geq
0}\sqrt{\alpha_{1}}g_{\alpha_{1}-1,\alpha_{2}}\psi_{\alpha_{1},\alpha_{2}}+\beta_{1}\sum_{\alpha}\sqrt{\alpha_{1}+1}g_{\alpha_{1}+1,\alpha_{2}}\psi_{\alpha_{1},\alpha_{2}}.$

If $\beta=1$ it follows that $a_{1}=a_{1}^{*}.$ For general
$\beta_{1}\in\mathbb{R}$ we compute the norms using Parseval’s identity

$\displaystyle\|a_{1}^{*}(f)\|^{2}-\|a_{1}(f)\|^{2}$ $\displaystyle=$
$\displaystyle\sum_{\alpha_{2}}|f_{1,\alpha_{2}}|^{2}(\beta_{1}^{2}-1)$
$\displaystyle+$
$\displaystyle\sum_{\alpha_{2}}|f_{0,\alpha_{2}}+\beta_{1}\sqrt{2}f_{2,\alpha_{2}}|^{2}-\sum_{\alpha_{2}}|\sqrt{2}f_{2,\alpha_{2}}+\beta_{1}f_{0,\alpha_{2}}|^{2}$
$\displaystyle+$
$\displaystyle\sum_{\alpha_{2}}|\sqrt{2}f_{1,\alpha_{2}}+\beta_{1}\sqrt{3}f_{3,\alpha_{2}}|^{2}-\sum_{\alpha_{2}}|\sqrt{3}f_{3,\alpha_{2}}+\beta_{1}\sqrt{2}f_{1,\alpha_{2}}|^{2}$
$\displaystyle+$ $\displaystyle\dots$ $\displaystyle=$
$\displaystyle(1-\beta_{1}^{2})\left(\sum_{\alpha
2}|f_{0,\alpha_{2}}|^{2}+\sum_{\alpha 2}|f_{1,\alpha_{2}}|^{2}+\dots\right)$
$\displaystyle=$ $\displaystyle(1-\beta_{1}^{2})\|f\|^{2}.$

Hence ${\text{dom}}(a_{1})={\text{dom}}(a_{1}^{*}).$

Using (5) we obtain

(10) $b_{1}(f)=\sum_{\alpha_{1}\geq 1,\alpha_{2}\geq
0}\sqrt{\alpha_{1}}f_{\alpha_{1}-1,\alpha_{2}}\psi_{\alpha_{1},\alpha_{2}}\,\,{\text{and}}\,\,b_{1}^{*}(f)=\sum_{\alpha}\sqrt{\alpha_{1}+1}f_{\alpha_{1}+1,\alpha_{2}}\psi_{\alpha_{1},\alpha_{2}}.$

Analogous results can be obtained for $a_{2}^{*}$ and $b_{2}^{*}.$ Putting
together all these results we get the following

######  Theorem 2.2.

Let

$A=\begin{pmatrix}\beta_{1}&0\\\ 0&\beta_{2}\end{pmatrix},$

with $\beta_{1},\beta_{2}\in\mathbb{R}.$ Then the operators $a_{j}$ and
$b_{j}$ on $H_{A}$ have the following properties:

$\|a_{j}^{*}(f)\|^{2}-\|a_{j}(f)\|^{2}=(1-\beta_{j}^{2})\|f\|^{2},\
{\text{dom}}(a_{j})={\text{dom}}(a_{j}^{*}),$

$\|b_{j}(f)\|^{2}-\|b_{j}^{*}(f)\|^{2}=\|f\|^{2},\
{\text{dom}}(b_{j})={\text{dom}}(b_{j}^{*}),\,j=1,2.$

(11) $a_{j}^{*}(f)=\beta_{j}a_{j}(f)+(1-\beta_{j}^{2})b_{j}(f),$

for $f\in{\text{dom}}(a_{j})\cap{\text{dom}}(b_{j}).$

The commutation relations are

$[a_{j},a_{j}^{*}]=a_{j}a_{j}^{*}-a_{j}^{*}a_{j}=(1-\beta_{j}^{2})I,$

in addition, we have $[a_{j},a_{k}]=0$ and $[a_{j}^{*},a_{k}^{*}]=0,$ for
$j,k=1,2.$

If $\beta_{j}\neq 1,$ we get the basic estimate

$\|f\|^{2}\leq\frac{1}{|1-\beta_{j}^{2}|}\left(\|a_{j}^{*}(f)\|^{2}+\|a_{j}(f)\|^{2}\right).$

###### Proof.

(11) follows from (8), (9) and (10). To show the commutation relations we use
the fact that $a_{1}^{*}=\beta_{1}a_{1}+(1-\beta_{1}^{2})b_{1}$ and that
$[a_{1},b_{1}]=I$ and easily compute $[a_{1},a_{1}^{*}].$ ∎

######  Remark 2.3.

In a similar way one can handle the case where

$A=\begin{pmatrix}0&\kappa\\\ \kappa&0\end{pmatrix},$

we get

$a_{1}(f)-\kappa a_{2}^{*}(f)=(1-\kappa^{2})b_{1}^{*}(f)\,\ {\text{and}}\,\
a_{2}(f)-\kappa a_{1}^{*}(f)=(1-\kappa^{2})b_{2}^{*}(f),$

the commutation relations are

$[a_{j},a_{k}^{*}]=\delta_{jk}(1-\kappa^{2})I,\,j,k=1,2,\,$

and $[a_{j}^{*},a_{k}^{*}]=0,j,k=1,2.$

If $\kappa=1$ we get $a_{1}(f)=a_{2}^{*}(f),$ whereas, if $\kappa=-1,$ we
obtain $a_{1}(f)=-a_{2}^{*}(f).$

##  3\. The Berezin transform

The Berezin transform is effective and successful in the study of Hankel and
Toepltiz operators, see [15]. We use the methods of chapter 2 to compute the
Berezin transform of the operator $a$ on

$H_{\beta}=\\{f\
{\text{entire}}:\int_{\mathbb{C}}|f(z)|^{2}\exp{-((1+\beta)x^{2}+(1-\beta)y^{2}))}\,d\lambda(z)<\infty\\},$

where $0\leq\beta\leq 1$ and $z=x+iy.$ The functions

$\psi_{k}(z)=\frac{z^{k}}{\sqrt{\pi k!}}\,\exp(\frac{\beta}{2}z^{2}),\
k=0,1,\dots$

constitute an orthonormal basis in $H_{\beta}.$ Hence the Bergman kernel
$K_{\beta}$ of $H_{\beta}$ is

$K_{\beta}(z,w)=\sum_{k=0}^{\infty}\psi_{k}(z)(\psi_{k}(w))^{-}=\frac{1}{\pi}\exp(\frac{\beta}{2}(z^{2}+\bar{w}^{2}))\exp(z\bar{w})$

and

$k_{w}(z)=\frac{K_{\beta}(z,w)}{\sqrt{K_{\beta}(w,w)}}.$

We get

$k_{w}(z)=\frac{1}{\sqrt{\pi}}\,\exp(\frac{\beta}{2}(z^{2}+\frac{1}{2}\bar{w}^{2}-\frac{1}{2}w^{2})+z\bar{w}-\frac{|w|^{2}}{2}).$

In addition, we have

$a(k_{w})(z)=\frac{1}{\sqrt{\pi}}\,\exp(\frac{\beta}{2}(z^{2}+\frac{1}{2}\bar{w}^{2}-\frac{1}{2}w^{2})+z\bar{w}-\frac{|w|^{2}}{2})\,(\beta
z+\bar{w}).$

It is easy to see that $k_{w}\in{\text{dom}}(a).$

The Berezin transform $\tilde{a}$ of the operator $a$ on $H_{\beta}$ is given
by

$\tilde{a}(w)=(a(k_{w}),k_{w})_{\beta}$
$=\frac{1}{\pi}\int_{\mathbb{C}}\exp(\beta(x^{2}-y^{2})+2xu+2yv-u^{2}-v^{2})\,(\beta
z+\bar{w})\exp(-(1+\beta)x^{2}-(1-\beta)y^{2})\,d\lambda(z)$
$=\frac{1}{\pi}\int_{\mathbb{C}}(\beta
z+\bar{w})\,\exp(-(x-u)^{2}-(y-v)^{2})\,d\lambda(z),$

where $z=x+iy$ and $w=u+iv).$

Finally we get

$\tilde{a}(w)=\beta w+\bar{w}.$

For the adjoint operator $a^{*}$ one has

$\tilde{a^{*}}(w)=(a^{*}(k_{w}),k_{w})_{\beta}=(k_{w},a(k_{w}))_{\beta}=(\tilde{a}(w))^{-}.$

Notice that the Berezin transform of $a$ on the Segal-Bargmann space
($\beta=0$) is $\tilde{a}(w)=\bar{w}.$

As $a=a^{*}$ on $H_{1},$ the Berezin transform of $a$ must be real-valued. In
fact, on $H_{1}$ we have $\tilde{a}(w)=w+\bar{w}.$

The Berezin transform of $ab$ and $ba$ on $H_{\beta}$ is given by

$\tilde{ab}(w)=1+w^{2}+|w|^{2},\ \tilde{ba}(w)=w^{2}+|w|^{2}.$

The spaces $H_{\beta}$ can be generalized in the following way, see [12].

Let $\kappa>0$ and $l\in\mathbb{C}.$ Define

$H_{\kappa,l}=\\{f\,{\text{entire}}\,:\|f\|_{\kappa,l}^{2}=\int_{\mathbb{C}}|f(z)|^{2}\,\exp(-\kappa|z|^{2}+\Re(lz^{2}))\,d\lambda(z)<\infty\\}.$

The functions

$\psi_{j}(z)=\frac{(\sqrt{\kappa}z)^{j}}{\sqrt{\pi j!}}\,e^{-lz^{2}/2}\ \
,\,j=0,1,\dots$

constitute an orthonormal basis in $H_{\kappa,l}.$ The Bergman kernel of
$H_{\kappa,l}$ is

$K_{\kappa,l}(z,w)=\frac{1}{\pi}\,e^{-l(z^{2}+\bar{w}^{2})/2}\,e^{\kappa
z\bar{w}}.$

In order to compute the Berezin transform of the operator $a$ on
$H_{\kappa,l}$ we have to consider

$k_{\kappa,l,w}(z)=\frac{K_{\kappa,l}(z,w)}{\sqrt{K_{\kappa,l}(w,w)}}.$

We get

$k_{\kappa,l,w}(z)=\frac{1}{\sqrt{\pi}}\,\exp(-lz^{2}/2-l\bar{w}^{2}/2+\kappa
z\bar{w}+lw^{2}/4+l\bar{w}^{2}/4-\kappa|w|^{2}/2),$

as before, we have $k_{\kappa,l,w}\in{\text{dom}}(a),$ so the Berezin
transform of $a$ has the form

$\tilde{a}(w)=(a(k_{\kappa,l,w}),k_{\kappa,l,w})_{\kappa,l},$

and an easy computation shows

$\tilde{a}(w)=\int_{\mathbb{C}}(-lz+\kappa\bar{w})\,\exp(\frac{1}{2}\Re(lw^{2})-\frac{1}{2}\Re(l\bar{w}^{2}))\exp(-\kappa(x-u)^{2}-\kappa(y-v)^{2})\,d\lambda(z),$

where we set $z=x+iy$ and $w=u+iv.$ Finally we obtain

$\tilde{a}(w)=\bar{w}\exp(\frac{1}{2}\Re(lw^{2})-\frac{1}{2}\Re(l\bar{w}^{2}))-\frac{lw}{\kappa}.$

Comparing this with the Berezin transform of $a$ in $H_{\beta}$ we have to set
$\kappa=1$ and $l=-\beta$ and get

$\tilde{a}(w)=\bar{w}+\beta w.$

##  4\. Szegő and Bergman kernels

We compute the Szegő kernel of

$S_{\kappa,l}=\\{(\zeta_{1},\zeta_{2})\in\mathbb{C}^{2}:\Im\zeta_{2}=\kappa|\zeta_{1}|^{2}-\Re(l\zeta_{1}^{2})\\},$

where $\kappa>0$ and $l\in\mathbb{C}.$ Hypersurfaces of this type are used as
model domains for the Leray transform ( [2]). For this purpose we use the
Bergman kernel $K_{\tau,\kappa,l}$ of

$H_{\tau,\kappa,l}=\\{f\,{\text{entire}}\,:\|f\|_{\tau,\kappa,l}^{2}=\int_{\mathbb{C}}|f(z)|^{2}\,\exp(-4\pi\tau(\kappa|z|^{2}+\Re(lz^{2})))\,d\lambda(z)<\infty\\}.$

and a formula reducing the Szegő kernel of a hypersurface in $\mathbb{C}^{2}$
to the parameter family of Bergman kernels in spaces of entire functions in
one variable, see [4], [5], [13].

The Bergman kernel of $H_{\tau,\kappa,l}$ is

$K_{\tau,\kappa,l}(z,w)=4\tau\exp(-2\pi\tau
l(z^{2}+\bar{w}^{2})+4\pi\tau\kappa z\bar{w}).$

Let
$\mathcal{D}_{\kappa,l}=\\{(z_{1},z_{2})\in\mathbb{C}^{2}:\Im(z_{2})>\kappa|z_{1}|^{2}+\Re(lz_{1}^{2})\\}.$
Using [5], the Szegő kernel can be computed as

$S((z^{\prime},z),(w^{\prime},w))=\int_{0}^{\infty}K_{\tau,\kappa,l}(z^{\prime},w^{\prime})e^{-2\pi
i\tau(\bar{w}-z)}\,d\tau,$

which implies

$S((z^{\prime},z),(w^{\prime},w))=\frac{\kappa}{\pi^{2}}\,(i(\bar{w}-z)-2\kappa
z^{\prime}\bar{w}^{\prime}+l(z^{\prime 2}+\bar{w}^{\prime 2}))^{-2},$

where $(z^{\prime},z),(w^{\prime},w)\in\mathcal{D}_{\kappa,l}.$

## References

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# Time bound of atomic adiabatic evolution in the accelerated optical lattice

Guoling Yin  State Key Laboratory of Quantum Optics and Quantum Optics
Devices, Institute of Opto-Electronics, Shanxi University, Taiyuan 030006,
China  State Key Laboratory of Advanced Optical Communication Systems and
Networks, Department of Electronics, Peking University, Beijing 100871, China
Lingchii Kong  International Center for Quantum Materials, School of Physics,
Peking University, Beijing 100871, China  Zhongcheng Yu  State Key Laboratory
of Advanced Optical Communication Systems and Networks, Department of
Electronics, Peking University, Beijing 100871, China  Jinyuan Tian  State Key
Laboratory of Advanced Optical Communication Systems and Networks, Department
of Electronics, Peking University, Beijing 100871, China  Xuzong Chen  State
Key Laboratory of Advanced Optical Communication Systems and Networks,
Department of Electronics, Peking University, Beijing 100871, China  Xiaoji
Zhou<EMAIL_ADDRESS>State Key Laboratory of Quantum Optics and Quantum
Optics Devices, Institute of Opto-Electronics, Shanxi University, Taiyuan
030006, China  State Key Laboratory of Advanced Optical Communication Systems
and Networks, Department of Electronics, Peking University, Beijing 100871,
China

###### Abstract

The accelerated optical lattice has emerged as a valuable technique for the
investigation of quantum transport physics and has found widespread
application in quantum sensing, including atomic gravimeters and atomic
gyroscopes. In our study, we focus on the adiabatic evolution of ultra-cold
atoms within an accelerated optical lattice. Specifically, we derive a time
bound that delimits the duration of atomic adiabatic evolution in the
oscillating system under consideration. To experimentally substantiate the
theoretical predictions, precise measurements to instantaneous band
populations were conducted within a one-dimensional accelerated optical
lattice, encompassing systematic variations in both lattice’s depths and
accelerations. The obtained experimental results demonstrate a quantitatively
consistent correspondence with the anticipated theoretical expressions.
Afterwards, the atomic velocity distributions are also measured to compare
with the time bound. This research offers a quantitative framework for the
selection of parameters that ensure atom trapped throughout the acceleration
process. Moreover, it contributes an experimental criterion by which to assess
the adequacy of adiabatic conditions in an oscillating system, thereby
augmenting the current understanding of these systems from a theoretical
perspective.

##  I INTRODUCTION

Bloch electrons that are accelerated by an external static electric field have
been long discussed in solid state physics since Bloch and Zener’s formula in
the 1930s [1, 2, 3, 4]. Many striking phenomena are predicted due to the
interplay between the quantum effects and the translation symmetry [5, 6, 7,
8, 10, 11, 9]. However, observing these in experiments is very challenging
owing to the poor control of the lattice defects, impurities, interactions
between particles and etc. in solid state materials[12]. Afterwards, due to
the rise of Bose-Einstein condensate (BEC) and laser techniques, the optical
lattice became an ideal system to simulate the Bloch electrons using neutral
atoms with nearly all parameters being controlled [13]. The 1D static optical
lattice is created by two counter-propagating lasers with identical
frequencies. And, the accelerated lattice can be realized by properly scanning
the phase difference between the lattice beams [15, 14].

Nowadays, the accelerated optical lattice can be used to resemble Wannier-
Stark Hamiltonian in order to study Bloch oscillation, Landau-Zener
transition, Wannier’s ladders and etc.[20, 31, 21, 22, 24, 25, 19, 28, 23, 26,
27, 16, 17, 18, 28, 29, 30, 32, 33]. Besides, the accelerated optical lattice
can also be used in gravimeters, where the atoms can be accelerated against
gravity in order to improve sensitivity[15, 34]. During this process, many
atoms escape from the lattice and only a few are left for the following
interference if the acceleration is large or the well depth is small. It can
be caused by quantum tunneling and the number of escaped atoms is typically an
exponential function of time. The number of left atoms is usually balanced
against the time of acceleration based on researchers’ experience.

Here, we derive a time bound of atomic adiabatic evolution in an accelerated
optical lattice and observe it in the experiment. Within this time bound, the
atoms will be trapped in the initial band and cannot escape from the lattice.
The time bound is measured with different lattice depths and lattice
accelerations after using our proposed shortcut method to load atoms into the
S-band of the optical lattice, and the experimental results agree well with
the theoretical predictions. Meanwhile, this time bound is derived as a part
of adiabatic conditions in accelerated optical lattice which possesses an
oscillating Hamiltonian. The general sufficient adiabatic conditions of such
Hamiltonian are still implicit in theory. Thus, the experimental evidence of
this time bound can contribute to a judge for various adiabatic conditions
from different theories.

This paper is organized as follows. Section II formulates the atoms in an
accelerated optical lattice. In Section II.1, a time bound of adiabatic
evolution is derived, where the Hamiltonian is simplified by a two-state
approximation. In Section II.2, the time bound is explained together with the
known decay mechanisms in this system. Section III illustrates the
experimental procedures of measuring the number of atoms in the instantaneous
band. Section IV shows the experimental results. In Section IV.1, the time-
resolved distribution of atoms in the instantaneous S-band $P_{S}(t)$ with
different parameters is shown (the atoms are initially prepared in S-band). In
Section IV.2, the times when $P_{S}(t)$ decreases are found and compared with
the expression of time bound. Then we give a discussion on the time bound and
the atomic velocity in V. The conclusions are summed in VI.

##  II The time bound of adiabatic evolution

###  II.1 Theoretical Model

The non-interacting atoms in a constantly accelerated lattice can be described
by a single particle Hamiltonian, i.e.,

$\hat{H}(t)=\frac{\hat{P}^{2}}{2m}+V_{0}\sin^{2}\left[k_{L}\left(x-\frac{at^{2}}{2}\right)\right]$
(1)

where $\hat{P}$ is the momentum operator, $V_{0}$ is the well depth, $m$ is
the mass of atom, $a$ is the acceleration of optical lattice, $k_{L}=\pi/d$ is
the wave number of lasers and $d$ is the lattice constant. The atom evolves
with the time-dependent Schrödinger equation, i.e.,
$i\hbar\partial_{t}|\Psi(t)\rangle=\hat{H}(t)|\Psi(t)\rangle$ where $\Psi(t)$
is the atom’s wave function.

Suppose the initial state is the ground state. With a small acceleration, the
atom will move adiabatically, i.e., the atom will stay in the instantaneous
ground state. To give a better illustration, we derive the adiabatic
conditions. The wave function is expanded by momentum basis, i.e.,
$|\Psi(t)\rangle=\sum_{p}c_{p}(t)|p\rangle=\sum_{n,k}c_{n,k}(t)|n,k\rangle$
where the momentum $p$ is split by a reciprocal vector index $n$ and a quasi-
momentum $k$. Then the entries of Hamiltonian (1) become

$H_{n,n;k,k^{\prime}}=4n^{2}E_{r}\delta_{k,k^{\prime}},\quad H_{n,n\pm
1;k,k^{\prime}}=\frac{V_{0}}{4}e^{\pm ik_{L}at^{2}}\delta_{k,k^{\prime}}$ (2)

where $E_{r}=\hbar^{2}k_{L}^{2}/(2m)$ is the recoil energy. The quasi-momentum
$k$ which is preserved along dynamics is set to zero for simplicity. If the
atoms perform a Bloch oscillation, there will be a momenta transfer from $0$
to $2\hbar k_{L}$ within a Bloch period defined by

$T_{\mathrm{B}}\equiv\frac{2\hbar k_{L}}{ma}.$ (3)

These two momenta states outperform the others during $t\lessapprox
T_{\mathrm{B}}$. Thus, it is legal to simplify the Hamiltonian Eq.(2) using
the two-state approximation. The simplified Hamiltonian becomes

$h=\begin{pmatrix}0&{V_{0}}e^{ik_{L}at^{2}}/{4}\\\
{V_{0}}e^{-ik_{L}at^{2}}/{4}&4~{}E_{r}\end{pmatrix}.$ (4)

It is similar to the Schwinger’s Hamiltonian but with a quadratic time
dependent form. Following the results in [35], the adiabatic condition for
this oscillating Hamiltonian can be obtained by solving inequalities

$\displaystyle\left\|\Delta^{-1}\right\|\left\|\Omega\right\|$
$\displaystyle\ll 1$ (5a)
$\displaystyle\int_{0}^{t}dt^{\prime}\left(\left\|\Omega\right\|\left\|\frac{d\Delta^{-1}}{dt^{\prime}}\right\|+\left\|\Delta^{-1}\right\|\left\|\frac{d\Omega}{dt^{\prime}}\right\|\right)$
$\displaystyle\ll 1$ (5b)

where $\Delta=h_{22}-h_{11}$, $\Omega=2h_{12}$ and the sub-indices represent
the row and column of matrix $h$. Eq.(5a) is derived from the commonly used
approximate adiabatic criterion for non-oscillating Hamiltonians, which gives
$V_{0}\ll 8~{}E_{r}$. Typically, the optical lattice requires that the well
depth $V_{0}$ has the same order with $E_{r}$. Thus, one cannot expect that
the atoms move adiabatically “forever”. Eq.(5b) is used to limit the fast
oscillating phase in $\Omega$, which can be solved as

$t\ll\tau\triangleq\sqrt{\frac{8E_{r}}{V_{0}k_{L}a}}$ (6)

where $\tau$ is the time bound for adiabatic evolution with the finite
$V_{0}/E_{r}$.

###  II.2 Escape Mechanisms

Figure 1: The schematic diagram of the relation between time bound and escape
mechanisms. The blue solid lines represent the optical lattice. The gray lines
represent different bands. The red shadows represent the atoms’ distribution
in space. The atoms are initially prepared in S-band, as shown in (a). The
evolution of atoms bounded by inequality Eq.(9) is shown in (b). When the
inequality Eq.(9) is not fulfilled, the atoms can populate either in higher
bands (shown in (c1)) or in the free space (shown in (c2)).

In the above subsection, the time bound $\tau$ is derived as a part of
adiabatic conditions and can be used to predict the time when adiabatic
evolution breaks down. Suppose the atoms are initially prepared in the S-band.
The instantaneous S-band evolves with a space translation for $at^{2}/2$ from
the initial S-band. This space translation is exactly the displacement of
optical lattice. Thus, roughly speaking, atoms that stay in the instantaneous
S-band are also trapped in the lattice.

When the acceleration $a$ increases, the time bound $\tau$ exhibits a decrease
characterized by a square root dependence on $a$. As $a$ goes to infinity, the
Hamiltonian (1) can be reduced into the effective time-independent
Hamiltonian:

$H_{\text{eff}}\triangleq\frac{1}{\Delta t}\int_{\tau}^{\tau+\Delta
t}dt^{\prime}\hat{H}(t^{\prime})\xrightarrow{a\to\infty}\frac{\hat{P}^{2}}{2m}+\frac{V_{0}}{2}$
(7)

which is exactly the free particle Hamiltonian with constant potential
$V_{0}/2$. Additionally, the wave functions of instantaneous bands collapse
into the eigenstates of (7), i.e., plane waves. Literally, the atoms cannot
“feel” the lattice. Before deriving the benchmark to scale the magnitude of
acceleration $a$, we need to explain why the atoms can escape from the
lattice.

From previous knowledge[29, 28], the decay of trapped atoms originates from
two mechanisms. One is the quantum tunneling. Consider an atom is localized at
a certain well. When the optical lattice is accelerated, the barriers will
successively scatter this atom. In each scattering, the atom possesses a non-
zero probability to tunnel through. And, the tunneled components are not
accelerated by the lattice. When atomic relative velocity against the lattice
increases, the tunneling probability increases polynomially. Thus, one could
expect that, in the following scatterings, the atom becomes easier to tunnel
through. As a result, the accumulation of tunneling contributes to a decay of
trapped components.

On the other hand, the atoms may fall down freely from the lattice due to its
smooth shape when the acceleration $a$ is larger than some critical value. It
can be illustrated by a classical model. In the co-moving frame of reference,
the potential that an atom feels is $V_{0}\sin^{2}\left(k_{L}x\right)+max$.
Thus, the acceleration of atom can be obtained by Newton’s second law, namely,
$\ddot{x}=-a+a_{cl}\sin\left(2k_{L}x\right)$ where
$a_{cl}\equiv{k_{L}V_{0}}/{m}$. For a classical atom, when $a<a_{cl}$, it is
trapped in a certain well and cannot escape and is accelerated all the time;
when $a>a_{cl}$, it escapes from this well. Therefore, $a_{cl}$ is the
critical acceleration to identify the two escape mechanisms. When $a<a_{cl}$,
the decay of trapped atoms is caused by quantum tunneling; when $a>a_{cl}$,
the atoms classically fall down from the lattice.

The critical acceleration $a_{cl}$ can be used as a benchmark to scale the
magnitude of acceleration $a$. Suppose that $a\gg a_{cl}$, the atoms will
quickly become untrapped, since then, the momenta distributions of atomic wave
function become static. After that, the Hamiltonian Eq.(1) can be effectively
replaced by a free particle Hamiltonian $H_{\operatorname{eff}}$. As a result,
the adiabatic conditions in Eq.(5a) and (5b) are satisfied for
$H_{\operatorname{eff}}$. It can also be understood by using the condition
$a\gg a_{cl}$ in the expression of time bound $\tau$, i.e.,

$a\gg a_{cl}\;\Leftrightarrow\;\tau^{2}\gg T_{\mathrm{B}}^{2}$ (8)

where the r.h.s of the arrow shows a square comparison of time bound and Bloch
period. Back to our original motivation, we expect that the time bound of
adiabatic evolution faithfully reflects the time duration when atoms are
trapped and accelerated. Thus, a large acceleration, i.e., $a\gg a_{cl}$
should be forbidden; then, the Hamiltonian Eq.(1) cannot be replaced by
$H_{\operatorname{eff}}$. Therefore, the inequality Eq.(6) should be altered
by adding a condition that $a\ll a_{cl}$, which gives

$t\ll\tau,~{}\tau^{2}\ll T_{\mathrm{B}}^{2}.$ (9)

These inequalities generally limit the time bound less than one Bloch period.
Within this time bound, the atoms are totally trapped and undergo adiabatic
evolution. It should be stressed that Eq.(9) do not conflict with the fact
that the atoms can perform many Bloch oscillations with a small enough
$a/a_{cl}$ where the escape mechanisms are nearly negligible.

The above analysis can be visualized in Figure1. The atoms can be initially
prepared in the S-band as shown in Figure1(a). Within the time bound Eq.(9),
the atoms are trapped and accelerated, as shown in Figure1(b). When the
inequality Eq.(9) is not fulfilled, the atoms can populate either in higher
bands (shown in Figure1(c1) or in the free space (shown in Figure1(c2).

##  III Experimental Protocol

Figure 2: The experimental setups and sequence. (a) The experimental setups.
The 1064nm laser beam is reflected by mirror to form the optical lattice,
where the AOM is to control the laser power and a minority of light is
transmitted for feedback to stabilize the laser power. A phase modulation
between the counter-propagating lattice beams is shaped through an EOM. The
red solid lines with arrow indicate the optical path. (b) The time sequence
diagram of instantaneous band measurements. The blue line represents the
intensity of the lattice beam. It first performs the shortcut sequence from
$t_{1}$, then keep for the evolution time, and finally adiabatically turn off.
The green line shows the phase of the lattice beam, and the phase is changed
as $at^{2}/2$. The origin line expresses probe light that gives the absorption
imaging after the time of flight.

To measure the time bound, we perform the experiments with the ultracold atoms
in the 1D accelerating optical lattice. We firstly prepare a dilute BEC of
approximately $3\times 10^{5}$ ${}^{87}\text{Rb}$ atoms in a harmonic trap
[36, 18, 37]. Then we load the BEC into the S-band of the optical lattice
using our proposed shortcut method [19], where the optical lattice is created
by a laser beam and its reflected beam with wavelength $1064\,~{}\mathrm{nm}$
as shown in Figure2(a). The shortcut method requires continuously turning on
and off the lattice beams in order to modulate the initial state to the target
state, where the whole process costs hundreds of $\rm\mu s$ with the fidelity
near $1$. For example, if the well depth is $6~{}E_{r}$ and the target state
is S-band, the time sequence of operations is set as $42/6/54/24~{}\rm\mu s$
with fidelity $99.95\%$. In this process, the AOM (acousto-optic modulator) is
used to realize the sharp variation of lattice depth in tens of microseconds
with several pulses. Furthermore, we use the feedback control circuits to
ensure that the laser intensity fluctuation is below $0.2\%@1s$.

After the shortcut loading, we accelerate the optical lattice with a fixed
acceleration $a$. It is achieved by using an EOM (electro-optic modulator, in
Figure2 (a)) to modulate the phase of reflected lattice beam $\varphi$. The
mode of modulation can be carefully chosen so that $\varphi=at^{2}/2$, i.e.,
the lattice is accelerated constantly, as shown in Figure2(b). There, the
phase noise of the accelerated optical lattice is attributed to the noise of
the EOM control voltage within 50 mVpp, which gives rise to the acceleration
uncertainty below $0.02\%$. Afterwards, the band distributions are observed
using band mapping technique, where the intensity of lattice beams is
adiabatically ramped down as an exponential curve in $1~{}\rm ms$. Finally,
after the time of flight for $32~{}\rm ms$, the absorption imaging is taken to
detect the atomic imaging.

In the experiment, to observe the obvious time bound, $V_{0}/E_{r}$ ranges
from $10^{0}\sim 10^{1}$ and $a$ ranges from $10^{0}\sim
10^{4}~{}\mathrm{m/s^{2}}$, as a result, the values of time bound $\tau$ vary
from $10^{0}\sim 10^{4}~{}\rm\mu s$. Each distribution will be measured five
times to obtain the statistical means and corresponding errors.

Figure 3: The typical images of atomic instantaneous band distributions. q is
defined as quasi momentum. The indices s, p, d, f, g represent the
corresponding bands, respectively. Due to the small gap, the atoms that
occupied in P-band and F-band are collected together in experiments. The white
dashed lines represent the values of time bound $\tau$ and the white dot-
dashed lines represent the values half of the Bloch period in (a) and Bloch
period in (b). In (a), $V_{0}=6~{}E_{r}$, $a=20~{}\mathrm{m/s^{2}}$ and
$\tau/T_{\mathrm{B}}\approx 0.24$, $a/a_{cl}\approx 0.06$. In (b),
$V_{0}=2~{}E_{r}$, $a=200~{}\mathrm{m/s^{2}}$ and $\tau/T_{\mathrm{B}}\approx
1.35$, $a/a_{cl}\approx 1.82$.

##  IV Experimental results

###  IV.1 The Instantaneous Band Distributions

To observe the adiabatic evolution of atoms in the accelerated optical
lattice, the instantaneous band distributions of atoms are measured. The
typical samples with selected parameters are shown in Figure3. In Figure3 (a),
the acceleration $a$ of the optical lattice is $20~{}\mathrm{m/s^{2}}$ and the
lattice depth $V_{0}$ is $6~{}E_{r}$. Under these parameters, the time bound
$\tau$ is about $0.24T_{\mathrm{B}}$; the acceleration $a$ is about
$0.06a_{cl}$. When the time $t$ is smaller than the time bound $\tau$, the
atoms almost distribute in the instantaneous S-band, and the atoms are
adiabatically accelerated by the optical lattice. As the time $t$ increases to
about half of Bloch period $T_{\mathrm{B}}/2$, an instantaneous inter-band
transition takes place and the atoms tunnel to higher bands.

In Figure3(b), the acceleration $a$ of the optical lattice is
$200~{}\mathrm{m/s^{2}}$ and the lattice depth $V_{0}$ is $2~{}E_{r}$. Under
these parameters, the time bound $\tau$ is about $1.35T_{\mathrm{B}}$; the
acceleration $a$ is about $1.82a_{cl}$. With the evolution time increasing,
the atoms seem almost stay in the instantaneous S-band no matter if the time
$t$ is smaller than time bound $\tau$. It is because the acceleration $a$ is
so large that the instantaneous S-band function nearly collapses into plane
waves after the time bound $\tau$ as illustrated in Section II.2.

Figure 4: The instantaneous S-band distributions with different parameters.
Red stars, blue triangles and green circles represent the experimental data
with well depth $2~{}E_{r}$, $4~{}E_{r}$ and $6~{}E_{r}$, respectively. The
solid lines represent the correspond numerical results from (1). The arrows
point out the drop point of $\ln P_{S}(t)$ of experimental results. The
accelerations used in (a) and (b) are $20~{}\mathrm{m/s^{2}}$ and
$200~{}\mathrm{m/s^{2}}$, respectively.

To quantitative analyze atomic adiabatic evolution, we define the
instantaneous band distributions $P_{\text{band}}(t)$ of atoms as

$P_{\text{band}}(t)=N_{\text{band}}(t)/N,$ (10)

where $N$ is the total number of atoms, and $N_{\text{band}}(t)$ is the atoms
number in the target instantaneous band at time $t$. The instantaneous S-band
distributions $P_{S}(t)$ along time under different accelerations of lattice
are shown in Figure4 (a) ($a=20~{}\mathrm{m/s^{2}}$) and (b)
($a=200~{}\mathrm{m/s^{2}}$). The solid lines are the results of numerical
simulation on Hamiltonian Eq.(1). The symbols are the results from
experimental measurements.

In Figure4(a), the atoms are driven by the optical lattice with acceleration
$20\,\mathrm{m/s^{2}}$ and the well depths are selected as $2~{}E_{r}$ (red),
$4~{}E_{r}$ (green) and $6~{}E_{r}$ (blue). The time bound $\tau$ defined in
Eq.(6) are shown in the dashed vertical lines. When $t<\tau$, $P_{S}(t)$ are
near to value $1$. The experimental $P_{S}(t)$ around the theoretical time
bound are respectively $0.96$, $0.95$ and $0.93$ corresponding to well depth
$2~{}E_{r}$, $4~{}E_{r}$ and $6~{}E_{r}$, which means the instantaneous inter-
band transitions are depressed within $t<\tau$. Moreover, with an identical
acceleration, when the well depth is larger, the time bound is smaller, and
the adiabatic evolving time is shorter. When $t>\tau$, $P_{S}(t)$ quickly
drops to the individual converged values and oscillates around them.

In Figure4(b), the acceleration of lattice is $200~{}\mathrm{m/s^{2}}$. When
$t<\tau$, the decay of $P_{S}(t)$ is also small, and the values of
experimental data around individual $\tau$ are $0.84$, $0.80$ and $0.82$.
Comparing to the situation in Figure4(a), with the acceleration increasing,
the time bound $\tau$ becomes shorter. Further, with an identical well depth,
a larger acceleration of lattice gives a shorter adiabatic evolving time
period. When $t>\tau$, $P_{S}(t)$ drops quickly, and the increase of
acceleration also magnifies the oscillations of $P_{S}(t)$ with well depth
$4~{}E_{r}$ (green) and $6~{}E_{r}$ (blue). But, the oscillations with well
depth $2~{}E_{r}$ (red) is smaller than the situation with
$a=20~{}\mathrm{m/s^{2}}$, and the red line shows an oscillating trend around
a value that is approximate to $1$. For the green and blue lines, a finite
proportion is left in the instantaneous S-band.

###  IV.2 The Measurement of Time Bound

In order to verify the expression of time bound defined in Eq.(6), we extract
the time bound from the experimental data. We consider the time bound as the
time point where $P_{S}(t)$ rapidly decay. The insert of Figure5 shows our
method to extract time bound $\tau$. Because the $P_{S}(t)$ suddenly decays
around time bound, we use two lines to fit the $\ln(P_{S})$ before and after
the time bound respectively, as the blue and orange lines shown in the insert
window of Figure5. And, the intersection of two lines is extracted as the time
bound.

Figure 5: The relation between the time bound $\tau$ and the product $a\cdot
V_{0}$. The red line is the theoretical result from Eq.(6) and the black
circles are the experimental results. The error bar is the standard error of
five repeated measurements. The time bound is defined as the drop points of
function $\ln P_{S}(t)$. The drop points are collected using the crossovers of
two straight lines fitted by bilateral data points, as shown in the inserted
figure.

Figure5 compares the experimental time bound and the theoretical prediction
using different experimental parameters. The results show that the
experimental data align reasonably well with the theoretical curve. Increasing
the acceleration and lattice depth leads to a decrease in the time bound. If
the product of the acceleration and lattice depth remains constant, the time
bound remains the same regardless of variations in accelerations and lattice
depths. However, it is important to note that there are still disparities
between the experimental results and theoretical simulations, which can be
attributed to several factors. Firstly, in the experiment, the error in
lattice depth calibration is within 5%, while the error in acceleration
calibration is approximately 2% [18]. These calibration errors contribute to
the observed deviations. Secondly, fluctuations in optical intensity and
imperfections in the pulse waveform impact the experimental fidelity of the
shortcut method, resulting in lower fidelity compared to the theoretical
predictions. Thirdly, the condensate of ${}^{87}\mathrm{Rb}$ exhibits particle
interactions that introduce finite nonlinear effects to the system. To
mitigate these effects, we select sufficiently large lattice potentials as an
experimental parameter.

##  V The measurements of atomic velocity distributions

We also perform the measurements of atomic velocity distributions at different
time in order to study its relation with time bound $\tau$. The typical images
are shown in Figure6. The atoms are initially prepared in the S-band, where
most of atoms distribute at a zero-velocity state, and a small part of atoms
distribute at the $\pm 2v_{L}$ states. In Figure6(a), where $a\ll a_{cl}$, the
atoms are accelerated, and a velocity eigenvalue transfer (with amplitude
$2v_{L}$) can be observed around $T_{\mathrm{B}}/2$ which is regarded as one
character of Bloch oscillation.

Figure 6: The typical images of the atomic velocity distributions. $v_{L}$ is
defined as $\hbar k_{L}/m$. The white dashed lines represent the values of
time bound $\tau$ and the white dot-dashed lines represent the values half of
Bloch period in (a) and the Bloch period in (b). In (a), $V_{0}=6~{}E_{r}$,
$a=20~{}\rm m/s^{2}$ and $\tau/T_{\mathrm{B}}\approx 0.24$, $a/a_{cl}\approx
0.06$. In (b), $V_{0}=2~{}E_{r}$, $a=200~{}\rm m/s^{2}$ and
$\tau/T_{\mathrm{B}}\approx 1.35$, $a/a_{cl}\approx 1.82$.

In Figure6(b), where $a>a_{cl}$, the velocity distribution of atoms preserves
the same for more than one Bloch period. It shows that the atoms are not
accelerated by the lattice because they have almost escaped from the lattice,
which is illustrated in Section II.2.

Figure 7: The atomic relative velocity against the lattice’s velocity. The
blue lines and the red lines are respectively the experimental data for well
depth $2~{}E_{r}$ and $6~{}E_{r}$. The gray short dash lines represent the
velocity reference of the accelerated lattice. Figure(a) to (d) are set with
different accelerations, i.e., $a=20$, $50$, $100$, $200~{}\mathrm{m/s^{2}}$.
Inset in (a): the numerical simulation of atomic relative velocity for $10$
Bloch periods with $a/a_{cl}=0.061$.

The atomic relative velocity against the lattice’s velocity with different
parameters are shown in Figure7. In Figure7(a), the blue circles are the
relative velocity with $a=0.061a_{cl}$, and it oscillates up and down around
zero (denoted by the gray short dash line) for more than one Bloch period.
Besides, its crossover with zero is at the half of Bloch period. It is a sign
that the atoms are performing Bloch oscillations. In the inset of Figure7(a),
we calculate numerically further for longer time, which shows the Bloch
oscillations last over $10$ Bloch periods with less distortion. It means that
the time bound does not prevent the Bloch oscillation over many periods under
such small $a/a_{cl}$.In the experiment, due to limited by the maximum control
voltage of EOM, we can’t measure any more Bloch periods. The red circles are
the relative velocity with $a=0.18a_{cl}$, and they also stay very close to
zero. However, because the crossover is not at the half of Bloch period, it is
not a standard Bloch oscillation and the atoms are slightly left behind the
lattice. In Figure7.(b) and (c), the acceleration $a$ is near $a_{cl}$. The
blue circles show deformed oscillations with the crossovers drifting because
some of the components have escaped from the lattice. The red circles fall far
behind the lattice and show a trend to keep a finite velocity. It is a signal
that a large component of atoms is not accelerated and they move like free
particles with a static velocity distribution. In this way, the expectation of
atoms’ velocity becomes static since the escape. In Figure7.(d), $a$ is
increased to $200~{}\mathrm{m/s^{2}}$, which is larger than the classical
critical velocity $a_{cl}$ of the red circles. The clustering of red circles
follows the velocity of lattice without oscillations, which illustrates the
absence of atom acceleration.

##  VI conclusions

The present study shows that the atomic adiabatic evolution in the accelerated
optical lattice is limited by a time bound $\tau$. This time bound is obtained
through a two-momenta state approximation and is expressed in Eq. (6). The
theoretical framework is verified through measurements of the instantaneous
band distributions at different times $P_{S}(t)$ with varying lattice’s depths
and accelerations, as shown in Figure4. The drop points of $P_{S}(t)$ are
fitted and collected to compare with the theoretical prediction. And it shows
that they agree well in Figure5. To further ensure that the atoms keep trapped
within the lattice during the adiabatic evolution, a limit on the acceleration
of the optical lattice is necessary. Specifically, it is recommended that the
acceleration be much smaller than a characteristic acceleration $a_{cl}$, as
indicated in Eq. (8) and (9). The verification of this condition is
demonstrated through the observation of atomic velocity distributions at
various times.

In many experimental cases, the accelerated optical lattice is used to
accelerate atoms. However, during the acceleration process, most atoms become
untrapped and fall freely due to escape mechanisms. Additionally, the
proportion of untrapped atoms increases as the final atomic velocity becomes
larger. This highlights the challenge of maintaining a high population of
trapped atoms during the acceleration process. The derived time bound helps to
optimize the experimental parameters to strike a balance between achieving a
desirable final atomic velocity and preserving a sufficient population of
trapped atoms for subsequent experiments.

It should be noted that the characteristic acceleration $a_{cl}$ is derived
from our understanding of escape mechanisms rather than detailed calculations.
So, its validity and limitations need further discussion in the future
studies.

##  VII acknowledgments

The authors thank professor Biao Wu for helpful comments. This work was
supported by the National Key Research and Development Program of China
(Grants No. 2021YFA0718300 and No. 2021YFA1400900), the National Natural
Science Foundation of China (Grants No. 11934002 and No. 11920101004), the
Science and Technology Major Project of Shanxi (Grant No. 202101030201022),
the National Key R&D Program of China (Grant No. 2018YFA0305602), the Space
Application System of China Manned Space Progra.

Guoling Yin and Lingchii Kong contributed equally to this work.

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