# Annular confinement for electrons on liquid helium

Bartłomiej Szafran  AGH University of Science and Technology, Faculty of
Physics and Applied Computer Science,
al. Mickiewicza 30, 30-059 Kraków, Poland

###### Abstract

We discuss the annular electron confinement on the liquid helium surface
induced by a submerged tubular electrode. For a shallow liquid layer the
resulting potential has a minimum off the symmetry axis of the electrode. The
ground-state angular momentum transitions that are driven by the external
magnetic field can be resolved when the confinement radii in the first and
second Rydberg subbands of the vertical quantization are different, e.g., for
submersion depth of the tube comparable to its radius. Then, discontinuities
in the main microwave absorption line appear with the period that corresponds
to the subsequent magnetic flux quanta passing across the area within the
ground-state confinement radius.

##  I Introduction

The surface of liquid helium can serve as an ultraclean substrate for the two-
dimensional electron gas g74 . Electrons at the vacuum side are bound by the
image charges that they induce in the weakly dielectric liquid g74 ; g76 ;
co02 . The vertical quantization of electron motion in the Coulomb potential
of the image g76 ; co02  gives rise to Rydberg states of hydrogen-like
spectrum g74 ; g76 ; co02 ; kawakami21 . The pristine nature of the substrate
allowed for the first observation w792  of the Wigner crystallization at low
electron gas density w791 ; haq03 . For the study of electron states at the
surface, microwave spectroscopic techniques g76  are developed is ; yunusowa ;
zadorozhko ; kk ; lambs ; kawakami19  with recent advances on intersubband
absorption kk , excitations of high-energy Rydberg states kawakami21 ,
relaxation times yunusowa , and coupling with the superconducting resonance
cavity trap ; yang ; kool .

In addition to the vertical confinement a lateral one can be introduced by
gates defined at the sides of the container or under the surface of helium.
The submerged electrodes locally strengthen the image charge potential 1D ; Pl
; Dy ; Dyprb ; Ly ; Da ; PE ; Gl ; sab08 . The gates are considered for e.g.,
one-dimensional channels 1D  or 0D localized states Pl ; Dy ; Dyprb ; Ly ; Da
; Gl  for quantum information storage and processing Gl ; sab08 . Recently,
the coupling of Rydberg states of vertical quantization with the lateral
confinement of the Landau levels by the in-plane magnetic field has been
studied yunusowa ; zadorozhko  as a model of an atom interacting with an
oscillator potential zadorozhko .

In this paper, we study look for the annular confinement of Rydberg states by
a submerged tube-shaped electrode. In open ring-shaped solid-state devices
webb ; lb ; fuhrer  the Aharonov-Bohm ab  effect is manifested by conductance
oscillations that exhibit periodicity with the flux of the magnetic field
threading the ring. On the other hand, for closed circular quantum rings qr
the ground-state undergoes angular momentum transitions as subsequent magnetic
flux quanta fit inside the ring fuhrer ; fominprl . We discuss the possibility
of observation of these angular momentum transitions in the microwave
absorption spectra of electron states localized above the tube electrode.
Recently, the formation of quantum rings on the surface of a semiconductor by
single-atom engineering has been reported pham .

##  II Theory

The confinement of the electron gas at the liquid helium surface is due to the
dielectric constant discontinuity at the liquid/vacuum interface g74 ; g76 ;
co02 . Experiments on microwave absorption by electrons localized at the
liquid helium surface are performed in a parallel capacitor configuration
kawakami21 ; is ; yunusowa ; zadorozhko  introducing an electric field that
neutralizes the surface electron charge and tunes the transition energies
between Rydberg states yunusowa ; kawakami21 . A cross-section of the model
system is depicted in Fig. 1. The lower plate contains a circular tube
electrode protruding towards the helium surface with an inner radius $R_{1}$
and $R_{2}$. The lower plate is submerged in liquid helium of depth $H$. The
tube electrode reaches depth $d$ beneath the helium surface. For the bulk of
this work, we consider the model system with $R_{1}=400$nm, $R_{2}=500$nm and
$H=1.2\mu$m. Discussion of the geometrical parameters for absorption spectra
is provided below.

$z\uparrow$$x\rightarrow$

Figure 1: Cross section of the model system taken at $y=0$. The red areas
correspond to electrodes that are spaced by $D=2.4\mu$m. The lower electrode
contains a circular tube extension with the inner radius $R_{1}=400$nm and the
outer one $R_{2}=500$nm. The dashed line is the symmetry axis of the tube. The
lower plate electrode is submerged by a liquid helium layer of depth
$H=1.2\mu$m. The distance from the top of the tube electrode to the liquid
surface is $d$. $\varphi$ stands for the electron wave function localized
above the liquid surface, close to the tube.

Evaluation of the effective potential felt by an electron bound above the tube
gate requires solution of the generalized Poisson equation

$\nabla\cdot\left(\varepsilon({\bf r})\nabla V({\bf
r})\right)=-\frac{\rho({\bf r})}{\epsilon_{0}},$ (1)

with $\rho({\bf r})=-e\delta({\bf r}-{\bf r}_{e})$, where $\delta$ is the
Dirac delta, and ${\bf r}_{e}$ is the electron position. The dielectric
constant of liquid 3He is taken $\epsilon=1.0572$ g74 . For the electron
localized off the axis of the tube electrode, the $V$ potential does not
possess the rotational symmetry and the problem requires a solution in a 3D
computational box. The rectangular box is $4.8\mu$m wide in $x$ and $y$
coordinates with height of $D$m (see Fig. 1) in the $z$ direction (we use
$D=2.4\mu$m unless stated otherwise). At the surface of the electrodes, the
potential $V_{1}$ or $V_{2}$ is taken as the Dirichlet boundary condition for
the potential. In the experiment kawakami21  the electric field of the order
of up to $10$V/cm is applied to additionally press the electrons to the helium
surface g74 ; yunusowa ; kawakami21 . The field of this range does not
qualitatively change the results so we keep $V_{1}=V_{2}$ and use $V_{1}$ as
the reference potential. For the low-energy electron states confined above the
tube, we need to determine the potential in the region of up to $\simeq
0.75\mu$m from the axis of the tube. At the sides of the computational box
that are parallel to the $z$ axis, we apply the boundary condition of a
vanishing component of the electric field ($\nabla V|_{\bf n}=0$) normal to
the side of the computational box.

The Poisson equation is solved with the finite-element method fem  using
parabolic Lagrange-type shape function in each element (see the Appendix). The
method allows us to take a zoom of the potential within the area occupied by
electron states. In the finite element method, the potential is expressed in
the basis of shape functions $\Psi_{l}$,

$V({\bf r})=\sum_{l}c_{l}\Psi_{l}({\bf r})$ (2)

associated with a single node each, so that $\Psi_{l}({\bf
r_{k}})=\delta_{lk}$.

The Poisson equation is solved in the space spanned by the basis functions. We
substitute the expansion (2) to Eq. (1) and next project the result on a shape
function $\Psi_{k}$ with integration over the computational box $\Omega$,

$\int_{\Omega}\sum_{l}c_{l}\Psi_{k}({\bf r})\nabla\cdot\left(\varepsilon({\bf
r})\nabla\Psi_{l}({\bf r})\right)d{\bf r}=-\int_{\Omega}\Psi_{k}({\bf
r})\frac{\rho({\bf r})}{\epsilon_{0}}d{\bf r}.$ (3)

Integrating by parts we get rid of the derivative over the dielectric
constant,

$\displaystyle\sum_{l}c_{l}\int_{\Omega}\nabla\Psi_{k}({\bf
r})\cdot\left(\varepsilon({\bf r})\nabla\Psi_{l}({\bf r})\right)d{\bf r}$
$\displaystyle-\sum_{l}c_{l}\int_{S}\Psi_{k}({\bf r})\varepsilon({\bf
r})\nabla\Psi_{l}({\bf r})\cdot{\bf n}d{\bf S}$
$\displaystyle=\int_{\Omega}\Psi_{k}({\bf r})\frac{\rho({\bf
r})}{\epsilon_{0}}d{\bf r},$ (4)

where $S$ is the surface of the computational box, and ${\bf n}$ is a unit
vector normal to the surface. Equation (4) is a linear system of equations for
$c_{l}$. Implementation of the boundary conditions for the nodes at the
surface is explained in the Appendix.

(a) |  (b) |  (c)(0,0)(1,0)(2,0)(3,0)(4,0)(5,0)(0,1) |  (d)(e)
---|---|---|---
Figure 2: The results for the tube electrode at $d=200$ nm beneath the surface
of liquid helium. (a) The confinement potential $W$ for the electron in
vacuum. The color of the lines indicates the distance $z$ from the liquid
helium surface. The lowest line is taken at $z=5$ nm above the surface, and
the higher lines are plotted at spacings of 5 nm. (b) The lowest-energy part
of the spectrum in a vertical magnetic field. The color of the lines indicates
the angular momentum quantum number $m$. The ground-state at $B=0$ is $m=0$,
and changes by $-1$ with each crossing of energy levels as $B$ is increased.
(c) The energy spectrum for a wider range of energy. The energy levels for
$m=-22,-21,\dots,11$ are marked by gray lines. The lines marked by colors
indicates the energy levels that are dipole coupled to the ground state. The
lines width is proportional to the absolute value of the dipole matrix element
$d_{if}=\langle\phi_{i}|z|\phi_{f}\rangle$. $(n_{r},n_{z})$ indicates the
subbands of the energy levels, with $n_{r}$ and $n_{z}$ standing for the
number of wave function zeroes in the radial and vertical directions
respectively. The color of the lines shows the average $z$ position of the
corresponding energy level wave function. (d) The absorption spectrum
calculated as the energy difference between the excited state energy levels
and the ground-state. The size of the dots is proportional to the absolute
value of the dipole matrix element. The presented range corresponds to
$(0,0)\rightarrow(0,1)$ transition (first to second Rydberg level). (e) The
part of the energy spectrum corresponding to the (0,1) subband. The color of
the lines shows the average $z$ position.  (a)$(n_{r},n_{z})=(0,0)$
---
(b)$(1,0)$
(c)$(0,1)$
Figure 3: Probability density for $m=-1$ states at $B=0$ for $d=200$nm. (a),(b) and (c) correspond to $(n_{r},n_{z})=(0,0)$, $(1,0)$ and $(0,1)$ states.  (a) |  (b) |  (c)(1,0)(2,0)(3,0)(4,0)(0,1)(0,2) |  (d)(e)
---|---|---|---
Figure 4: Same as Fig. 2, only for $d=400$nm. (a)$(n_{r},n_{z})=(0,0)$
---
(b)$(1,0)$
(c)$(0,1)$
Figure 5: Same as Fig. 3, only for $d=400$nm.

We solve the Poisson equations for the electron position ${\bf r_{e}}$ scanned
over the finite element nodes above the liquid helium. The potential $V(\bf
r)$ calculated in this way contains both the contributions from the point
charge of the electron floating above the helium surface and the charges
induced in liquid helium and on metal electrodes. To determine the effective
potential acting on the electron near the helium surface, we need to eliminate
the electron self-interaction included in $V$ lis1 . For this purpose, we
calculate potential $V_{e}$ for the electron above the helium surface but with
all the metal electrodes removed. For $V_{e}$ we apply the Dirichlet boundary
conditions at all the sides of the computational box using the potential for
the point charge near the contact of two dielectrics i.e.
$V_{e}=\frac{1}{4\pi\epsilon_{0}}\left(\frac{e}{R_{e}}+\frac{q^{\prime}}{R_{i}}\right)$
for $z$ above the helium layer, where $R_{e}=|{\bf r}-{\bf r}_{e}|$ is the
distance from the electron position and $R_{i}=|{\bf r}-{\bf r}^{\prime}_{e}|$
the distance from its image placed at ${\bf
r}^{\prime}_{e}=(x_{e},y_{e},H-(z_{e}-H))$ with the image charge
$q^{\prime}=\frac{1-\varepsilon}{1+\varepsilon}e$. At the sides of the
computational box below the helium surface we take at the boundary the
potential
$V_{e}=\frac{1}{4\pi\varepsilon\epsilon_{0}}\frac{q^{\prime\prime}}{R_{e}}$,
where $q^{\prime\prime}=\frac{2\varepsilon}{1+\varepsilon}e$.

With $V$ and $V_{e}$ we calculate an effective potential $V_{1}=V-V_{e}$, in
which the electron self-interaction is removed. Since in the calculation for
both $V$ and $V_{e}$ we account for the presence of the helium surface, in the
evaluation of $V_{1}$ the image charge potential induced on the liquid surface
is canceled. In the Schroedinger equation that we solve for the confined
states we reintroduce the potential of the image charges induced in helium in
its analytical form g74 , $W_{s}(z_{e})=-\frac{Ze^{2}}{|z_{e}-H|}$, with
$Z=\frac{\varepsilon-1}{4(1+\varepsilon)}$. $W_{s}$ potential diverges for
$z\rightarrow H$, hence it is more readily introduced in its analytical form
to the Schroedinger equation than evaluated in the polynomial shape functions.
The confinement potential energy to be used in the Schroedinger equation is
then set as $W({\bf r_{e}})=-e\left(V{(\bf r_{e}})-V_{e}{(\bf
r_{e}})\right)+W_{s}({\bf r_{e}})$. We assume that the liquid helium is
impenetrable for the low-energy electrons of the vacuum side and take the
infinite value of $W$ below the surface.

Although for a general electron position ${\bf r_{e}}$, the potential $V({\bf
r})$ is not rotationally symmetric, the effective potential $W({\bf r_{e}})$
has this symmetry. We solve the Schroedinger equation in cylindrical
coordinates for a given angular momentum quantum number $m$,
$\varphi_{m}=\varphi(r,z)\exp(im\theta)$, with azimuthal angle $\theta$. With
the symmetric gauge ${\bf A}=(-By/2,Bx/2,0)$ for vertical magnetic field ${\bf
B}=(0,0,B)$ the Hamiltonian reads,

$\displaystyle H$ $\displaystyle=$
$\displaystyle-\frac{\hbar^{2}}{2m_{0}}\left[\frac{\partial^{2}}{\partial
z^{2}}+\frac{\partial^{2}}{\partial r^{2}}+\frac{1}{r}\frac{\partial}{\partial
r}-\frac{m^{2}}{r^{2}}\right]$
$\displaystyle+\frac{m\hbar\omega_{c}}{2}+\frac{m_{0}}{8}\omega_{c}^{2}r^{2}+W(r,z),$

with $\omega_{c}=\frac{eB}{m_{0}}$. We use the imaginary time method imaginary
to determine the eigenstates of the finite difference version of the
Hamiltonian with the mesh spacing of 2.5nm in $r$ and $z$ coordinates. The
region occupied by electrons in low-energy states is much smaller than the
computational box for the Poisson equation and a finer mesh is required to
describe the wave functions. The potential spanned by the shape functions
$W(r,z)$ can be evaluated on a finer grid.

(a) |  (b) |  (c) |  (d)
---|---|---|---
Figure 6: Same as Fig. 2 only for $d=600$ nm.

##  III Results

###  III.1 Shallow tube ($d=200$nm)

For the presentation of the results, we fix $z=0$ at the surface of the liquid
helium. The electron confinement potential depends qualitatively on the depth
at which the tube electrode is located beneath the surface. For $d=200$nm the
potential has a pronounced local maximum near $r=0$ [Fig. 2(a)] and the
minimum is located near $r=400$nm. The minimum gets shallower and shifts to
lower values of $r$ at a larger distance from the surface. The changing
profile of the lateral potential with $z$ indicates a non-separability of the
potential, in contrast to a separable potential found for a submerged rod-like
electrode Dy .

The energy spectrum near the ground state is shown in Fig. 2(b). The ground-
state quantum number $m$ changes as for the semiconductor quantum rings qr ;
fuhrer . For a strictly one-dimensional ring of radius $R_{0}$, the energy
spectrum is given by
$E_{m}=\frac{\hbar^{2}}{2m_{0}R_{0}^{2}}\left(\frac{\Phi}{\Phi_{0}}+m\right)^{2}$,
with flux quantum $\Phi_{0}=\frac{e}{h}$, and the flux threading the quantum
ring $\Phi=B\pi R_{0}^{2}$. In Fig. 2(b) we observe the changes of the ground-
state angular momentum in $B$ with the spacing of $\Delta B\simeq 8$ mT. The
radius corresponds to the magnetic field flux through a one-dimensional ring
of $R\simeq 430$nm.

The spectrum in a wider range is displayed in Fig. 2(c). The energy levels for
$m=-22,-21,\dots 11$ are marked in gray. Additionally, we marked with colors
those excited energy levels that are dipole coupled to the ground state. The
width of the color lines is proportional to the dipole matrix element
$d_{if}=|\langle\varphi_{i}|z|\varphi_{f}\rangle|$, where $\varphi_{i}$ and
$\varphi_{f}$ stand for the states participating in the transition from the
initial (ground) and final (excited) state wave functions. The color of the
lines indicates the average $z$ for the excited states. In the spectrum we see
a set of thin blue lines for states localized at $z\simeq 12$nm. The excited
states to which the transition is allowed correspond to the same $m$ as the
ground state but correspond to excitations within the plane of confinement.
For separable potential the wave function can be put in form
$\varphi_{n_{r},n_{z}}(r,z)=\varphi^{r}_{n_{r}}(r)\varphi^{z}_{n_{z}}(z)$, and
the dipole matrix element for transition from the ground state would be 0 for
the final states with $n_{r}\neq 0$ due to the in-plane orthogonality.
However, the potential $W$ is not a simple sum of the vertical and lateral
components [Fig. 2(a)] and the wave function is non-separable, which allows
for a small but nonzero dipole moment. Although the potential is non-
separable, one can attribute to states the number of wave function zeroes
along the $r$ and $z$ directions, $n_{r}$ and $n_{z}$, where $n_{z}+1$ numbers
the Rydberg subbands. At the top of Fig. 2(c) we see a thick red line that
corresponds to the second Rydberg subband with a vertical excitation (0,1).
The probability densities for $m=-1$ and $(n_{r},n_{z})$ equal to (0,0),
(1,0), and (0,1) are plotted in Fig. 3(a),(b), and (c), respectively. The
spectrum of the second Rydberg subband (0,1) [Fig. 2(d)] has a similar
quantum-ring pattern as the ground-state subband (0,0). The period is almost
the same as for the (0,0) subband of Fig. 2(b), since these states are
localized at similar $\langle r\rangle$ (cf. Figs. 3(a) and 3(c)). The
absorption spectrum $\Delta E$ for the transitions from the ground state is
depicted in Fig. 2(e). The spectrum indicates that the main transition line
from the lowest to the second Rydberg subband is nearly independent of the
external magnetic field. The $m$ changes in the ground-state (0,0) subband and
in the (0,1) subband are nearly perfectly synchronized on the $B$ scale. As a
consequence, the Aharonov-Bohm angular momentum transitions produce only weak
symptoms on the energy spectrum of the microwave absorption with fine
discontinuities of the absorption line that jump by less than $0.02$ GHz.

###  III.2 Discontinuous transition spectrum ($d=400$ nm)

The period of angular momentum transitions can be made unequal in the lowest
(0,0) and second (0,1) Rydberg subbands for the tube electrode submerged
deeper in helium. The confinement potential for the tube electrode at a depth
of $d=400$nm beneath the surface is displayed in Fig. 4(a). Compared to Fig.
2(a), the potential minimum is shifted towards the axis of the system and the
position of the minimum changes as one rises above the surface to eventually
disappear near $z=100$nm. Compared to Fig. 2 the ground-state angular momentum
transitions occur with an increased spacing on the magnetic field scale due to
reduction of the confinement ring radius [see Fig. 3(b)]. The energy level
that is coupled to the (0,0) ground-state is discontinuous as a function of
$B$ (Fig. 4(c)). In the second Rydberg (0,1) subband, the angular momentum
changes with a slower rate: the wave functions – are localized higher above
the surface and are localized closer to the axis than in the lowest (0,0)
subband [cf. Fig.5(a,c)]. As a consequence, the angular momentum transitions
in the first and second Rydberg subbands no longer occur at the same values of
$B$ and the absorption spectrum (Fig. 4(e)) contains more pronounced
discontinuities with jumps of the line of the order of 0.1 GHz.

###  III.3 Deep tube ($d=600$ nm)

For the tube $d=600$nm beneath the helium surface, the $W$ potential loses its
local maximum at the axis [Fig. 6(a)]. The energy spectrum [Fig. 6(b)] is
nearly identical with the Fock-Darwin one fd , but with lifted degeneracy of
the excited energy levels of non-equal $|m$| due to the deviation of the
confinement potential from parabolicity. The main absorption line [Fig.
6(c,d)] is nearly independent of the external magnetic field.

###  III.4 Geometry effects

The discontinuities of the absorption spectrum can be made larger for tuned
geometry. The spectrum with the inner and outer radii reduced to $R_{1}=300$
nm and $R_{2}=400$ nm is displayed in Fig. 7(a), where we use a thinner layer
of helium, $d=200$ nm instead of $d=400$ nm. For the tube of reduced radius,
the confinement potential changes faster with $z$ for these parameters which
results in the increased discontinuities in the spectrum, e.g. the ground-
state transition from $m=-2$ to $m=-3$ involves a jump of 0.3 GHz instead of
0.13 GHz as in Fig. 4(d).

For the rest of the paper, we return to $R_{1}=400$ nm, $R_{2}=500$ nm and
$d=400$ nm. The sensitivity of the spectrum with respect to the exact width of
He layer above the tube near the work point of $d=400$ nm can be studied in
Fig. 7(b). The central set of lines in the spectrum is the one of Fig. 4(d)
with $d=400$ nm. The lower set corresponds to $d=390$ nm and the upper one to
$d=410$ nm. The 10 nm change of the liquid layer depth induces a shift of the
absorption line by 1.1 GHz. For lower values of the width of the helium layer
$d$ above the tube, the effective confinement radius gets larger, hence a
slight modification of the oscillation period.

In experimental conditions the distance between the plates of the electrodes
needs to be larger than the liquid helium capillary length, i.e., not less
than 1.5 mm. Our computational model is much smaller. The length of the tube
cannot be increased much above several $\mu$m due to the limitations of the
focused ion beam methods that can define such an electrode. Therefore, the
upper plate needs to be further away from the helium surface. To study this
size effect, we kept $H$ and $d$ unchanged and increased $D$ (see Fig. 1) from
2.4 $\mu$m (as in all the results above) to 12.4 $\mu$m and $22.4$ $\mu$m. The
results are given in Fig. 8. The image charge at the upper capacitor plate
produces an electric field that pulls the electron up from the surface. This
field is reduced for larger $D$. Fig. 8 shows that as $D$ grows, the energy
position of the main absorption line is going up on the energy scale in
consistence with the study of the effect of the vertical electric field on the
transition between the ground and the excited Rydberg states g74 ; yunusowa ;
kawakami21 . At a larger distance between the top plate and the helium
surface, the discontinuities of the absorption lines are preserved. We do not
expect qualitative changes of the character of the spectrum for $D$ increasing
further.

(a)(b)

Figure 7: Absorption spectra of the transition from the lowest to the second
Rydberg level. The colors show the magnetic quantum number $m$. In (a) a thin
tube with radii $R_{1}=300$ nm and $R_{2}=400$ nm covered with helium layer of
depth $d=200$ nm is taken. In (b) $R_{1}=400$nm and $R_{2}=500$nm are applied
as elsewhere in this paper and the values of $d$ are varied.

$D=2.4\mu$m, $12.4\mu$m$22.4\mu$m

Figure 8: Absorption spectra of the transition from the lowest to the second Rydberg level for $H=1200$ nm, $d=400$ nm, $R_{1}=400$ nm and $R_{2}=500$ nm. Results for $D=2.4\mu$m, $D=12.4\mu$m and $D=22.4\mu$m are plotted with black, blue and red lines, respectively. (a)1mK |  (b) |  (c)
---|---|---
(d)10mK |  (e) |  (f)
(g)20mK |  (h) |  (i)
Figure 9: Results for the geometry of Fig. 4 in finite temperature. The first
(a-c), second (d-f) and third row (g-i) of plots correspond to $T=1$mK, 10 mK
and 20 mK, respectively. In the left column of plots (a,d,g) the dots mark the
energy levels near the ground-state with their size proportional to the
occupation of the energy level. In the central column (b,e,h) the absorption
spectrum is plotted with the size of the symbol proportional to the product of
the initial state occupation and the transition probability. The colors mark
the angular momentum for the states participating in the transition. The right
column of plots shows the intensity of the line.

###  III.5 Finite temperature

The absorption spectra presented above for zero temperature exhibit
discontinuities with the Aharonov-Bohm periodicity. The discontinuities result
from the ground-state angular momentum transitions driven by the external
magnetic field. The energy spacings between states confined in the tube
potential are small and in the finite temperature several lines will be
present at the same magnetic field. We need to look for signatures of the
absorption spectrum that will pertain in finite $T$.

The occupation probability of the energy level with energy $E_{i}$ is given by
the Fermi-Dirac distribution $p_{i}=\frac{1}{\exp(\beta(E_{i}-\mu)+1}$, where
$\beta=\frac{1}{k_{B}T}$ and $\mu$ is the chemical potential that for a
single-electron per tube solves the equation $\sum_{i}{p_{i}}=1$. The
intensity of the absorption line for initial state $i$ and final state $f$ can
be evaluated as $I_{if}=p_{i}|d_{if}|^{2}$. The low-energy spectrum is
presented in Fig. 9(a,d,g) with the dots of the size that is proportional to
$p_{i}$ for $T=1$mK, 10mK and 20 mK, respectively. The simulated transition
spectra are plotted in Fig. 9(b,e,h) with dots of size that is proportional to
$I_{if}$. Already at $1$mK the magnetic field intervals for transitions to
states of subsequent $m$ values overlap and at $10$mK the discontinuities in
the absorption spectra are no longer present. Nevertheless, the Aharonov-Bohm
periodicity can still be extracted from the absorption spectrum by observation
of the intensities of separate lines [see Fig. 9(c,f,i)]. The intensities are
maximal for the magnetic fields that appear in the middle of subsequent
ground-state angular momentum transitions for which the energy spacing between
the ground state and the excited states is the largest (see Fig. 4(b)). For
$B$ that corresponds to the maximal intensity of line $m$ the neighbor lines
with $m\pm 1$ have equal intensities, which can be useful as an additional
feature for identification of the confined Aharonov-Bohm effect.

##  IV Discussion

The results above indicate that the experimental detection of the Aharonov-
Bohm ground-state transitions by microwave absorption requires tuning of the
liquid helium level to conditions in which the radius of confinement in the
lowest and second Rydberg subbands is significantly different. Adjusting the
liquid layer above the tube electrode should be relatively straightforward.
The discontinuity of the absorption energy line due to the ground-state
transitions at $d=400$nm is of the order of 0.1 GHz [Fig. 4(d)] which is
within range of the resolution of the microwave absorption measurements
kawakami21 ; yunusowa ; lambs .

The counterparts of the discussed angular momentum transitions in a single
semiconductor quantum ring are resolved by the transport spectroscopy for
systems with attached contacts fuhrer . In the experiment on electrons on a
liquid helium surface, an array of tube electrodes could be considered. In
self-assembled semiconductor quantum rings, the ground-state transitions
induced by an external magnetic field are detected by magnetization
measurements on a system with the number of rings of the order of $10^{11}$
fominprl . For electrons on the helium surface recent microwave absorption
experiments yunusowa  are performed on $\sim 10^{7}$ electrons. For our
purpose, to eliminate the interaction between the electrons localized above
separate tube electrodes, the distances between the tubes need to be large
enough. In the present study, the rectangular shape of the computational box
does not affect the circular potential above the tube for the distance of
$1.2\mu$m between the axis of the tube and the side of the computational box.
The Neumann boundary condition at the sides is justified by the screening of
the electron charge by the image charges. Therefore, the distance between the
tubes in the array of $2.5\mu$m should be enough to switch off the interaction
between the electrons of separate tubes. The effects described in this paper
deal with a single electron per tube. The electron density in the experiments
is tuned by the electric field in the capacitor kawakami21 ; yunusowa . The
spacing between the tubes of $2.5\mu$m corresponds to $1.6\times 10^{7}$cm-2,
which is nearly equal to the surface electron density in the experimental
conditions of e.g. Ref. yunusowa .

An experiment with the array may be difficult due to the dependence of the
energy positions of the main line of the spectrum on the exact width of the
helium layer (Fig. 7), which would require a fine positioning of the
container. However, the signal from a single tube could be detected with the
recent development of the spectroscopy technique based on the capacitive
coupling of surface electrons to the capacitor plates kawakami19  defining the
external electric field that is indicated for quantum information processing
on Rydberg states as qubits. The method is based on the measurement of pA
currents induced by an electron oscillating between the lowest and excited
Rydberg states in a resonant microwave field and thus changing its distance
from both plates kawakami19 .

The present modeling neglects the coupling of the electrons to the ripplon
surface waves. The effect of the coupling to the ripplon field on the
absorption spectra has been considered in Ref.Dyprb  for a closely related
problem of 0D states induced by a rod electrode submerged 500 nm below the
surface. The Franck-Condon polaronic shift of the transition frequency between
the first and second energy level was estimated as a product of 0.022 GHz and
a factor $f_{p}$ that depends on the ratio between the effective Bohr radius
(7.6 nm) and the in-plane confinement length $a_{||}$. For the parameters of
Fig. 4, $a_{||}$ is of the order of 100 nm, for which, based on the results of
Ref. Dyprb , $f_{p}<0.01$. The ripplonic shift of the energy spectra should be
then smaller than 0.22 MHz and thus negligible. The coupling to ripplons
shifts the transition lines stronger at higher temperature, but for $T\leq 20$
meV they remain negligible lambs ; lambprl .

##  V Summary and Conclusions

We studied ring-like confinement of electrons by a tubular electrode placed
beneath the surface of liquid helium. For the tube that is submerged shallow
in helium, the vertical magnetic field induces Aharonov-Bohm angular momentum
transitions. By tuning the helium surface level one can arrange for angular
momentum transitions occurring at a period of the magnetic field that is
significantly different for the first and second Rydberg states of the
vertical quantization. Then, the absorption spectrum exhibits distinct
discontinuities at the ground-state angular momentum transitions. At the
temperatures of the order of 10 mK, the discontinuities will no longer be
present since a number of states of various $m$ will be occupied. However, the
lines can be resolved in the energy, and the observation of the intensity of
the lines as functions of the external magnetic offers a way for the
experimental detection of Aharonov-Bohm effect for bound electron states.

## Appendix: implementation of the finite element method

To determine the potential for the electron due to the image charges in helium
and metal electrodes, we solve the Poisson equation using the finite element
method. We use cubic elements of 100nm side length with spacing between the
nearest nodes of 50nm, with 27 nodes in each element. Inside the box, we have
8 nodes per element of the boundaries of the system. We work with 60025
elements and 499851 nodes.

For an element covering $x\in[x_{s},x_{e}]$ a local coordinate
$\xi_{1}\in[-1,1]$ is used with

$x(\xi_{1})=\frac{1-\xi_{1}}{2}x_{s}+\frac{1+\xi_{1}}{2}x_{e},$ (A.1)

and similar mapping for y and z directions with $\xi_{2}$ and $\xi_{3}$ local
coordinates. Within the element [see Fig. A.1] 27 nodes are defined for the
local coordinates $\xi_{1},\xi_{2}$ and $\xi_{4}$ equal $\pm 1$ or 0. We
define three parabolic shape functions

$\displaystyle\psi_{-1}(\xi)$ $\displaystyle=$ $\displaystyle\xi(\xi-1)/2$
(A.2) $\displaystyle\psi_{0}$ $\displaystyle=$ $\displaystyle(1-\xi)(1+\xi)$
(A.3) $\displaystyle\psi_{1}$ $\displaystyle=$ $\displaystyle\xi(\xi+1)/2$
(A.4)

which are Lagrange node functions with $\psi_{n}(m)=\delta_{n,m}$ for
$n,m=-1,0,1$. The 3D Lagrange functions for a node with local coordinates
$\eta_{1},\eta_{2},\eta_{3}$ is defined as
$\Psi_{\eta_{1},\eta_{2},\eta_{3}}=\psi_{\eta_{1}}(\xi_{1})\psi_{\eta_{2}}(\xi_{2})\psi_{\eta_{3}}(\xi_{3})$.
Within the cubic element the potential is spanned in the basis of 27 of these
functions

$V(x(\xi_{1}),y(\xi_{2}),z(\xi_{3}))=\sum_{l=1}^{27}c_{l}\Psi_{\eta_{l1},\eta_{l2},\eta_{l3}}(\xi_{1},\xi_{2},\xi_{3}),$
(A.5)

where $c_{l}=V({\bf r}_{l})$. The matrix elements defining the system of
equations for potential at the nodes given by Eq.(4) are divided into elements
and integrated over the local coordinates. The three-point differential
quotient allows for an exact calculation of the derivatives for parabolic
functions and the Gaussian quadrature provides exact values for integrals of
the polynomials. The large linear system of equations (about 0.5 mln) is
sparse and solved with the PARDISO library.

Figure A.1: Nodes within a single cubic element in the local system of
coordinates $\xi_{i}\in[-1,1]$ for $i=1,2,3$. The labels close to the nodes
indicate the coordinates $(\xi_{1},\xi_{2},\xi_{3})$ of the node.

The linear system of equations given by Eq. (4) is applied for all nodes
inside the computational box. For the nodes at the sides of the computational
box, equations given by boundary conditions are introduced. Due to the
properties of the Lagrange functions, the Dirichlet boundary conditions for
nodes inside the electrodes can be set as $c_{l}=V_{1}$ or $V_{2}$ for nodes
$l$ on the bottom or top electrodes, respectively. The Neumann boundary
condition for vanishing electric field component normal to the side faces of
the computational box is induced by setting equal values of the coefficients
$c_{l}=c_{m}$ where $l$ and $m$ are neighbor nodes adjacent to the boundary in
the normal direction. With the applied boundary conditions, the surface
integral in Eq. (4) is zero. At the top and bottom sides of the computational
box, i.e., in metal, the potential is constant and its gradient vanishes. At
the other side of the box, the normal component of the potential is zero
according to the Neumann boundary condition applied therein.

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