\subsection*{Introduction}
 Great interest has been devoted to  theoretical and experimental studies of  various low-dimensional
 systems such as tunnel junctions, quantum point contacts (QPC), quantum nanowires, and  2D
  electron gas based nanostructures,  just to name a few. These systems show a large variety
  of fascinating quantum-mechanical effects  on the macroscopic scale, e.g. the weak localization
   \cite{Heinzel,Dittrich}, the quantum Hall effect \cite{QH}, the macroscopic quantum tunnelling
   \cite{Devoret}, the conductance quantization in QPCs \cite{Heinzel,QH,Beenakker} etc.

The quantum-mechanical dynamics of quasiparticles and the
electronic transport become even
  more complex and intriguing when
 \emph{bound states} are present in such systems.
  Indeed, it was shown in Ref. \cite{ARinQPC1,ARinQPC2} that the bound states manifest
  themselves by narrow drops (the anti-resonance) in the gate voltage dependent conductance
  of QPCs. It is well known also that the bound states naturally arising on the boundaries
  of various systems such as graphene nanoribbons \cite{Castro},  nanowires \cite{Charlier,Mourik},
  2D electron gas under magnetic field, greatly influence the transport properties.

Specific magnetic field dependent bound states can be artificially
created if a lateral junction (barrier) is fabricated inside of a
2D electron gas subject to an externally applied magnetic field
(see Fig. 1) \cite{exp1,kang,twin,barrier,grapheneLat}.  These
highly localized
 states are formed due to the quantum-mechanical  interference of magnetic edge states occurring
 on the both sides of the barrier.   The localization length
 in the transverse direction $x$ is
of the order of the magnetic
length $\ell_c~\simeq~\sqrt{\hbar c/(eH)}$ in the regime of a strong externally applied
magnetic field, $H$.

However, in the longitudinal direction $y$ these states show
delocalized behavior, and the dynamics of electrons is
characterized by the component of electron momentum $p_y$. Thus,
in such a 1D channel the  energy  spectrum of electrons,
$\epsilon_n(p_y)$, contains many bands, and it is shown for two
lowest bands in Fig. 2A. In the absence of quantum tunnelling
across the junction the left-right symmetry of the electron states
results in a large number of degenerate states in the electronic
spectrum (see Fig. 2A, dashed line). The quantum tunnelling
provides the lifting of the degeneracy, and the energy spectrum
shows a complex structure  with bands and gaps (see Fig. 2A, solid
line). Notice here that in the region of a rather small magnetic
field such a spectrum was calculated in Ref. \cite{twin,barrier},
and in the region of a strong magnetic field, i.e in the quantum
Hall regime, it was obtained in Ref. \cite{kang}. Since this
spectrum having an origin in the coherent quantum-interference
phenomenon,  is extremely sensitive to various interacting
effects, the renormalization of the above-mentioned spectrum  due
to the Coulomb interaction and /or the impurities has been
theoretically studied in papers
\cite{Mitra,Kollar,Nonoyama,Kim,Aranzana}.



  \begin{figure}
 
 \centerline{\includegraphics[width=0.7\columnwidth]{kadigrobov_fig1.eps}}
 
 \caption{\textbf{The  point contact with a lateral junction in the quantum Hall regime.}
 A scattering process (propagation and reflection) of magnetic edge states on the electron
  states localized on the lateral junction, is schematically shown.}

 \label{edgestates}
 \end{figure}
 


The bands and gaps in the energy spectrum of electronic states
bounded to the junction, directly manifest themselves  in  a large
amount of interesting effects as the transport in the longitudinal
direction is studied. Indeed, the giant oscillations of the
longitudinal conductance with the gate voltage, strongly nonlinear
current-voltage
 characteristics and coherent Bloch oscillations under a weak
 electric fields, have been
 predicted and theoretically studied \cite{twin,barrier}.



It is  naturally to suppose that the above-mentioned  unique
electronic spectrum can be probed by the electric current flowing
\emph{across } a lateral junction. In this article we show that the
bands and gaps in the quasiparticle spectrum of the states bounded
to a lateral junction,
 manifest themselves in  giant oscillations of the conductance
 $G_x$ as the applied magnetic
  field or dc voltage are varied. The origin of these oscillations is
  in the magnetic edge
  states transmission coefficient $D(E)$ which shows  peaks and drops as
  the electron energy is tuned.

Qualitatively, the quantum-mechanical dynamics of electrons in a
strong magnetic field and in the presence of a lateral junction
can be considered as following: the lateral junction serves as a
tunable quantum-mechanical scatterer for  propagating edge states
(see Fig. 1). E.g., the edge state $1$ propagating from the left
lead along the lower boundary is scattered by the barrier into the
edge state $2$ going to the right lead,  and into the edge state
$3$ which reflects from the barrier to the left lead along the
upper boundary. The highest probability of such reflection
 occurs if the propagation of electrons \emph{along} the junction is allowed. It takes place if
 the electron energy $ E$ is inside of the energy bands of bound states. In this case the
 transmission coefficient $D(E)$ shows minimal values. As the energy of electrons is tuned to
 gaps in the spectrum of the bound states, the edge states propagation along the junction  is
 forbidden, and a great enhancement of $D(E)$ is obtained. These oscillations of $D(E)$ transform
 in  giant oscillations of $G_x$ under variation of
 the  magnetic field or the dc voltage. Such oscillations of
 $G_x$  have been verified experimentally in  the Ref. \cite{kang} but
 the quantitative analysis has  not been  done. Notice here,
 that in the Ref.\cite{Takagaki}, the electron current flowing perpendicular to
the lateral junction in a 2D electron gas under a strong magnetic
filed was numerically calculated, and similar oscillations of the
conductance were obtained.

Below we present a complete analytical solution of the transverse
transport of electrons in such a system that allows to clarify
both qualitatively and quantitatively experimental features
\cite{kang} such as  the small value of the observed conductance
in comparison to the expected Landauer conductance, the
oscillations of the conductance as a function of the magnetic
field or voltage, the dependence of the conductance on the lateral
size of the system, etc. Our analysis based on the
quantum-mechanical scattering of propagating magnetic edge states
on the bound states is in a good accord with both the qualitative
scenario and experimental observations.

Let us consider a QPC fabricated in a two-dimensional electron gas
in the presence of a lateral junction and subject to an externally
applied magnetic field. The magnetic field $H$ is perpendicular to the QPC plane.
The QPC is characterized by the coordinate-dependent electrostatic potential
$\widetilde V (x, y)=\frac{m \omega_1^2}{2}y^2 + V (x)$,
where  the last term describes the potential barrier between two
parts of the electron gas. The parameter $\omega_1$ characterizes
the curvature of the confinement potential of the QPC, $m$ is the
electron effective mass.

Quantum dynamics of electrons in the QPC with the lateral junction is
described by the wave function $\Psi(x,y)$ satisfying
the two-dimensional Schr\"odinger equation:


\begin{eqnarray}
 &-&\frac{\hbar^2}{2 m} \frac{\partial^2 \Psi}{\partial x^2}+
 \Big[\frac{1}{2 m}\big(-i \hbar \frac{\partial }{\partial y}-
 \frac{e H x}{c}\big)^2 \nonumber \\
 &+&\frac{m \omega_1^2}{2}y^2 + V (x)-E\Big]\Psi =0,
\label{Shcroedinger}
\end{eqnarray}
where the Landau  gauge, i.e. the vector-potential  ${\bf A} =
(0,Hx,0)$, is used. Here, the axis $y$ is parallel to the barrier
and the $x$-axis is directed along the QPC (see Fig. 1).
Introducing the dimensionless variables, $\xi=x/\ell_c$ and
$\varepsilon = E/(\hbar \omega_c)$ (the $\omega_c$ is the
cyclotron frequency), we write the total wave function $\Psi$ as
\begin{eqnarray}
 \Psi(\xi,y)= \int_{-\infty}^\infty Q(p_y,\xi)\exp\{i\frac{ p_y y }{\hbar}\}d p_y, \nonumber \\
 Q(p_y,\xi)=\sum_{n=0}^{\infty} R_n(p_y)\varphi_{n,p_y}(\xi)
\label{Fourier}
\end{eqnarray}
Here, the partial wave functions $\varphi_{n,p_y}(\xi)$ describe  the magnetic field dependent electron states bounded to the lateral junction and satisfy to the equation
\begin{eqnarray}
  \frac{\partial^2 \varphi_{n,p_y}}{\partial \xi^2}-
  \Big[\Big(\frac{p_y \ell_c}{\hbar} -\xi \Big)^2+
 \nu (\xi)-2 \varepsilon_{n}(p_y) \Big]\varphi_{n,p_y}(\xi) =0
\label{Shcroedinger2}
\end{eqnarray}
with the boundary conditions  being $\varphi_{n,p_y}(\xi)
\rightarrow 0$ at $\xi \rightarrow \pm \infty$. Here, the
dimensionless potential of the junction is $\nu (\xi) = 2m V(\xi
l_c) \ell_c^2/\hbar^2$. These bound states are formed from the
edge states propagating along the barrier
 $\phi_{n,p_y}^{(l)}(\xi)$ ($\phi_{n,p_y}^{(r)}(\xi)$) on the left (right) parts of
the lateral junction. In the quantum Hall regime as $E~\simeq~\hbar \omega_c$ these bound
 states decay in the transverse direction  on the distance $\ell_c$. In the absence of
 quantum tunneling the electronic spectrum contains a large amount of degenerate states. The quantum-mechanical interference between  left and right edge states results in a lifting of this degeneracy, and
a peculiar one-dimensional spectrum $\varepsilon_n(p_y)$ with an
alternating  sequence of narrow energy bands $\sim
\sqrt{1-|t|^2}\; \hbar \omega_c$ and energy gaps,
\cite{twin,kang,barrier} $\Delta_n \sim |t|\;\hbar \omega_c$,
where $|t|^2$ is the barrier transparency (see Fig. 2A).


The transverse electronic transport is determined by the  partial wave
functions $R_n (p_y)$ which, in turn, are entangled with the bound
states. Next, we explicitly consider this quantum entanglement in
the case that the energy of electrons is in the vicinity of the
first "crossing point". This crossing point occurs at  $p_y = 0$,
and the energy $\epsilon_0=3/2$ (in dimensionless units).

The generic total quantum-mechanical state is presented as a
superposition of basis functions, i.e.
\begin{eqnarray}
Q_0(p_y,\xi)=C_1(\varepsilon)R^{(l)}_0 (p_y)\phi^{(l)}_{0, p_y}(\xi)
 +C_2(\varepsilon)R^{(r)}_0(p_y)\phi^{(r)}_{0, p_y}(\xi)
\label{Superposition}
\end{eqnarray}
Here, $R^{(l,r)}_0 = R_0 \pm R_1$ are the functions normalized to
the unit flux density, the energy dependent coefficients
$C_1(\varepsilon)$ and $C_2(\varepsilon)$ determine the
transmission coefficient $D(E)$ and, therefore, the electronic
transport.

The partial wave functions $R_0^{(l,r)}(p_y)$ are satisfied to the
following set of coupled  equations (its derivation is given in
Supplementary Materials):
\begin{eqnarray}
\left(\frac{\alpha^2}{2} \frac{\hbar^2}{\ell_c^2} \frac{d^2 }{d
p_{y}^2} + \varepsilon_0^{(l)}(p_y) -\varepsilon \right)R_0^{(l)}
+
\tilde{\Delta}R_0^{(r)}  =0;\nonumber \\
\left(\frac{\alpha^2}{2} \frac{\hbar^2}{\ell_c^2} \frac{d^2 }{d
p_y^2} + \varepsilon_0^{(r)}(p_y) -\varepsilon \right)
  R_0^{(r)}  +\tilde{\Delta} R_0^{(l)} =0,
\label{BasicEquation}
\end{eqnarray}
where $\alpha =\omega_1 /\omega_c $, and
$\varepsilon_0^{(l,r)}(p_y)$ are the energy spectrum of the left(
right) edge states in the absence of tunneling (dashed line in
Fig. 2A), the dimensionless energy gap
$\tilde{\Delta}=\Delta/(\hbar \omega_c)$ is determined by the
quantum tunneling across the barrier.





As  $\alpha \ll 1$  one may solve  this set of equations in the quasiclassical approximation.
 Indeed, substituting  $\Psi_{1,2} =A_{1,2}(p_y)\exp\{i S(p_y)\ell_c/(\hbar\alpha)\}$ in
 Eq.(\ref{BasicEquation}) and introducing the classical momentum as $P =d S/dp_y$ one
 finds that the electron dynamics is determined by the quasiclassical phase trajectories
 $P(p_y)$ as
\begin{eqnarray}
P^2=\varepsilon - \frac{\varepsilon_0^{(l)}(p_y)+\varepsilon_0^{(r)}(p_y)}{2} \pm \nonumber \\
\pm \sqrt{\Big(\frac{
\varepsilon_0^{(l)}(p_y)-\varepsilon_0^{(r)}(p_y)}{2}\Big)^2+\tilde{\Delta}^2}.
\label{momentum}
\end{eqnarray}



Notice here that the quasiclassical approximation is valid as
$|\varepsilon-\varepsilon_0^{(l,r)}(p_y)| \gg \alpha^{2/3}$.

The quasiclassical phase trajectories determined by
Eq.(\ref{momentum}) for different values of the electron energy
$\varepsilon$ are shown in Fig. 2B. As the well separated phase
trajectories approach to each other the quantum tunneling  occurs
between them. However, in  contrast to the standard interband
transitions \cite{LandauZienerBlount}, in the case under
consideration the tunnelling takes place in the vicinity of
Lifshitz's phase transition \cite{Lifshitz}. Indeed, as it follows
from Eq.(\ref{momentum}) there are two critical energies at which
the topology of trajectories changes: at first, in the vicinity of
the energy $\varepsilon_{cr}^{(1)} = \varepsilon_0
-\tilde{\Delta}$ the mutual directions of the motion on two open
trajectories vary (see Fig. 2Ba and 2Bb); secondly, a new closed
orbit arises at  the critical energy $\varepsilon_{cr}^{(2)} =
\epsilon_0 +\tilde{\Delta}$ (see Fig. 2Bc and 2Bd).
The quasiclassical approximation breaks down not only at the turning points but also
in the vicinity of the point $p_y=0$ where the quantum tunneling takes place.

  \begin{figure}
 
 \centerline{\includegraphics[width=0.7\columnwidth]{kadigrobov_fig2.eps}}
 
 \caption{\textbf{The energy spectrum of electron states bounded to the lateral
 junction and corresponding quasiclassical trajectories.}
A) The lowest two bands  of the  spectrum $\epsilon_n(p_y)$  are
shown by dashed
  (in the absence of tunneling) and solid (in the
  presence of  tunneling) lines.
 B) The phase trajectories $P(p_y)$  for different values of the electron energy:
 a)$\varepsilon<\varepsilon_{cr}^{(1)}=\varepsilon_0-\tilde{\Delta}$;
 b)$\varepsilon=\varepsilon_{cr}^{(1)}$;
  c) $\varepsilon=\varepsilon_{cr}^{(2)}=\varepsilon_0+\tilde{\Delta}$;
   d)$\varepsilon>\varepsilon_{cr}^{(2)}$ (here $2 \tilde{\Delta}$ is the energy gap).}

 \label{trajectories}
 \end{figure}
 

In order to properly elaborate quantum tunnelling between various
quasiclassical phase trajectories we apply  the perturbation
analysis which is valid in a whole ranges of energies but the
energy gap $\tilde{\Delta}$ is assumed to be small. Introducing
the Fourier transformation as
\begin{eqnarray}\label{Fourier2}
\Phi(p_y)_{1,2}=\int_{-\infty}^{\infty} g_{1,2}(Y)
 \exp \{i\frac{Y p_y \ell_c }{\hbar\alpha}\} dY
\end{eqnarray}
and presenting  the Fourier transform $g_{1,2}(Y)$ in the form
\begin{eqnarray}\label{u}
g_{1,2}=u_{1,2}(\zeta)e^{\{\pm i(\frac{2}{3}\zeta^3+2 \eta \zeta)\}},
\end{eqnarray}
one gets the following set of equations \cite{PRL} for the new
variable $\zeta=Y[\frac{\ell_c \omega_c}{\alpha v}]^{1/3}$ (here,
$v=|d\varepsilon_0^{(l)}(p_y)/dp_y|$ is the electron velocity):
\begin{eqnarray}\label{SystemH3}
&&\hspace{5mm} i\frac{du_1(\zeta)}{d\zeta}=-
\gamma e^{-i\left(\frac{2}{3}\zeta^3 +
2\eta \zeta\right)} u_2(\zeta) \nonumber \\
&&\hspace{5mm} i\frac{du_2(\zeta)}{d\zeta}=+
\gamma e^{i\left(\frac{2}{3}\zeta^3 + 2\eta\zeta\right)} u_1(\zeta),
\end{eqnarray}
where the parameters
\begin{eqnarray}
\gamma = 2^{1/3} \tilde{\Delta} \Big[\frac{\ell_c \omega_c}{\alpha
v}\Big]^{2/3}, \hspace{3mm} \eta = 2^{1/3}(\varepsilon_0
-\varepsilon)\Big[\frac{\ell_c \omega_c}{\alpha v} \Big]^{2/3}.
\label{Parameters}
\end{eqnarray}
Here, the parameter $\eta$ controls  the topology of the phase
trajectories (see Fig.\ref{trajectories}B) and the parameter
$\gamma$ determines the probability of quantum tunnelling between
trajectories.


The perturbation analysis of Eq.(\ref{SystemH3}) in the limit of
$\gamma \ll 1$ allows one to obtain the probability $D(E)$ for a
particle in the momentum space  to be transmitted from $p_y
\rightarrow + \infty $  to $p_y \rightarrow - \infty $ as
\begin{eqnarray}\label{probability2}
D(\epsilon) =  \gamma^2 \Big[\pi 2^{2/3} \textrm{Ai} \left( 2^{2/3}
\eta \right) \Big]^2,
\end{eqnarray}
where $\textrm{Ai}(x)$ is the Airy function \cite{Stegan} (see
Supplementary Material, Section II). The typical dependence of
$D(\epsilon)$ is shown in Fig. 3.



  \begin{figure}
 
 \centerline{\includegraphics[width=0.7\columnwidth]{kadigrobov_fig3.eps}}
 
 \caption{\textbf{Energy dependent transmission probability.}
 Dependence of the transmission probability $D$ on the normalized
 energy $\varepsilon$ is   shown. The transparency of  the lateral barrier (which acts as a
 tunable quantum scatterer) is zero as long as the electron energy is below the lowest energy
 gap at $E_0/\hbar \omega_c = 1.5$ (see Fig. \ref{trajectories}A). Further increase
 of $E$ results in a narrow sharp peak accompanied by fast
 transparency oscillations (thin solid line) caused by the quantum
 interference inside the effective scatter.  The fast oscillations are  partially washed out at finite
 temperature $T$ (thick solid line).   The parameters $\alpha=10^{-2}$, $v/(l_c \omega_c)= 2$,
 and $k_B T/\hbar \omega_c=10^{-2}$  have been used. }

 \label{edgestates}
 \end{figure}
 
One can see that  $D(E)$ is an extremely small as $E<(3/2)\hbar
\omega_c$. In the opposite
 regime
at $E>(3/2)\hbar \omega_c$,  $D(E)$ displays both a great
enhancement and fast oscillations
 with
a small period of $\delta E_1~\simeq~\hbar \omega_c \alpha^{2/3}$.
The origin of these oscillations is the quantization of a
quantum-mechanical phase of the wave function of electrons moving
along the closed orbit  (see, Fig. 2Bd). At finite temperatures
these fast oscillations  are partially washed out, and it results
in a strong decay of the function $D(E)$ as  $E>(3/2)\hbar
\omega_c$ (see Fig. 3, thick solid line).


As we turn to the electronic transport in the quantum Hall regime
it can be shown (see Supplementary Material,  for details) that
$D(E)$ determines also the energy  dependent transmission
coefficient of electrons propagating across the lateral junction.
Thus,  $D(\mu)$ determines the linear conductance $G_x$. Here,
$\mu$ is the chemical potential of the system.

The current-voltage characteristic $I(V)$ of the QPC in a broad voltage region is obtained by
making
use of the Landauer-B\"{u}ttiker approach \cite{QH} as:

\begin{eqnarray}\label{current}
I(V) = \frac{2 e}{h} \sum_{n}\int d E D_n(E) \\\nonumber
[f(E+eV/2)- f(E-e V/2)]~,
\end{eqnarray}
where $f(E)$ is the Fermi-Dirac distribution function. Notice
here, that the transmission coefficients for upper bands, $D_n$,
are also determined by the energy differences to the crossing
points $\varepsilon_n$ as
$$
D_n=\gamma^2_n \Big[\pi 2^{2/3} \textrm{Ai} \left( 2^{2/3} \eta_n
\right) \Big]^2
$$
where $$\eta_n \approx 2^{1/3}(\varepsilon_n
-\varepsilon)[\frac{\ell_c \omega_c}{\alpha v_n}]^{2/3}$$ and
$$\gamma_n \approx 2^{1/3}\Delta_n \Big[\frac{\ell_c
\omega_c}{\alpha v_n}\Big]^{2/3}$$ (see Supplementary Materials,
Section III, for details) . The differential conductance $G_x$ is
determined as $G_x=dI/dV$. The typical dependence of $G_x(V)$
determined by Eqs. (\ref{probability2}) and (\ref{current}) is
shown in Fig. 4.

The dependence of $G_x(V)$ displays giant oscillations with the period
$\delta E_2~\simeq~\hbar \omega_c$. The two reasons determine the appearance of
peaks in the $G_x(V)$ dependence: at first, the applied voltage distorts the spectrum of bound
states to $\epsilon_n(p_y, eV)$, and  secondly, the voltage tunable energy of electrons $E=\mu+eV/2$
traces the energy gaps $\Delta_n$ in the spectrum of bound states.

The consistent experimental study of the conductance of a 2D
electron gas biased in the quantum Hall regime and in the presence
of a lateral junction has been carried out in Ref. \cite{kang}.
Our quantitative analysis presented in Fig. 3,4 shows all
important features observed in \cite{kang}, namely a great
enhancement and fast oscillations of the linear conductance $G_x
(0)$ as the magnetic field was decreased (a decrease of the
magnetic field results in an effective increase of the chemical
 potential $\mu$), and unique giant oscillations of $G_x(V)$.

  \begin{figure}
 
 \centerline{\includegraphics[width=1\columnwidth]{kadigrobov_fig4.eps}}

 
 \caption{\textbf{Dependence of the normalized conductance
on the applied dc voltage.}
 The parameters   $\alpha=10^{-2}$,  $\mu_0/\hbar \omega_c=1$, $v/ l_c \omega_c=2$
 and $k_B T/\hbar \omega_c=10^{-2}$ have been used
 (here $G_0= (4 e/h)\gamma^2$ and $\mu_0$ is
 the chemical
 potential in the absence of the applied voltage).}
 \label{conductance}
\end{figure}
 
In conclusion, we have shown that in the quantum Hall regime a lateral junction formed  in the QPC serves as a unique
quantum-mechanical scatterer for propagating magnetic edge states. Such a lateral junction hosts the electrons bound states having a
peculiar magnetic field dependent 1D spectrum
$\varepsilon_n(p_y)$. This spectrum contains an alternating
sequence of narrow energy bands and gaps. These band and gaps
manifests themselves by  giant oscillations of the transmission
coefficient $T(E)$ on the electron energy $E$ and the voltage
dependent conductance $G_x(V)$. Our theoretical analysis based on the coherent quantum-mechanical superposition of localized and delocalized magnetic edge states  is in a good accord with the experemental observations \cite{kang}. Such a generic  approach [see the Eqs. (\ref{BasicEquation}) and (\ref{probability2})] can be applied to the variety of solid state systems where the physical properties are determined by the coherent quantum dynamics of interacting quantum-mechanical objects, e.g. the magnetic edge states propagation in graphene based nanostructures \cite{grapheneLat} or the superconducting quantum metamaterials \cite{SM}.



\textbf{Acknowledgements:}The financial support of the Ministry of Education and Science of
the Russian Federation  in the framework of
Increase Competitiveness Program of NUST "MISiS"($K2-2014-015$) and
the SPP programme "Graphene" of the DFG are acknowledged.

\vspace{3cm}
