|  Non-Uniform Magnetic Fields for Single-Electron Control†
---|---
|  Mauro Ballicchia,∗a Clemens Etl,a Mihail Nedjalkov,a Josef Weinbuba
|  Controlling single-electron states becomes increasingly important due to
the wide-ranging advances in electron quantum optics. Single-electron control
enables coherent manipulation of individual electrons and the ability to
exploit the wave nature of electrons, which offers various opportunities for
quantum information processing, sensing, and metrology. A unique opportunity
offering new degrees of freedom for single-electron control is provided when
considering non-uniform magnetic fields. Considering the modeling perspective,
conventional electron quantum transport theories are commonly based on gauge-
dependent electromagnetic potentials. A direct formulation in terms of
intuitive electromagnetic fields is thus not possible. In an effort to rectify
this, a gauge-invariant formulation of the Wigner equation for general
electromagnetic fields has been proposed in [Nedjalkov et al., Phys. Rev. B.,
2019, 99, 014423]. However, the complexity of this equation requires to derive
a more convenient formulation for linear electromagnetic fields [Nedjalkov et
al., Phys. Rev. A., 2022, 106, 052213]. This formulation directly includes the
classical formulation of the Lorentz force and higher-order terms depending on
the magnetic field gradient, that are negligible for small variations of the
magnetic field. In this work, we generalize this equation in order to include
a general, non-uniform electric field and a linear, non-uniform magnetic
field. The thus obtained formulation has been applied to investigate the
capabilities of a linear, non-uniform magnetic field to control single-
electron states in terms of trajectory, interference patterns, and dispersion.
This has led to explore a new type of transport inside electronic waveguides
based on snake trajectories and also to explore the possibility to split
wavepackets to realize edge states.

##

††footnotetext: a Institute for Microelectronics, TU Wien, Gusshausstrasse
27-29, 1040 Wien, Austria. E-Mail<EMAIL_ADDRESS>
##  1 Introduction

The field of electron quantum optics studies and applies electromagnetic (EM)
field-controlled phenomena for manipulating electron states in solid-state
quantum systems 1, 2, 3, 4, 5. Ideally, transport theories should describe
both, processes governing the electron trajectories as well as wave phenomena
such as interference and diffraction in multiple dimensions. Theories based on
EM potentials rely on the formal mathematical apparatus related to the choice
of the scalar and vector potentials, which, however, obscures the physical
aspects and thus the heuristic understanding of the electron evolution. This
is also true for the Wigner theory which among the alternative formulations of
quantum mechanics utilizes classical concepts of a phase space and a quasi-
distribution function. Indeed, the underlying Wigner quantum mechanics is
formulated in electrostatic terms under the choice of zero vector potential 6.
Central quantities in the derived evolution (transport) equation are the
Wigner function $f_{w}$ and Wigner potential $V_{w}$, defined by the Weyl
transform of the density matrix and the scalar potential, respectively. The
inclusion of the magnetic field, however, introduces additional terms, which
depend on the choice of the gauge 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17. Six
decades ago Stratonovich 18 generalized the Weyl transform to replace the
canonical momentum as a phase space coordinate with the kinetic momentum. The
latter, being a physical quantity, is gauge invariant, and so is the transport
equation for $f_{w}$ derived by the Weyl-Stratonovich transform. Details can
be found, e.g., in 19, 20, 21 and the references therein. The potentials are
completely removed from the theory, which offers the advantage of depending on
physical factors only, such as the EM fields ${\bf E},{\bf B}$. The equation
is mathematically challenging as it depends on multi-dimensional integrals of
$f_{w}$ with the terms $D^{F}({\bf E})$, $H^{F}({\bf B})$, and $I^{F}({\bf
B})$, which include the Fourier transform $FT=\int d{\bf
s}e^{-\frac{i}{\hbar}{\bf p}{\bf s}}$ of the quantities
$\int_{-1}^{1}d\tau\left({\bf s}\cdot{\bf E}({\bf r}+\frac{{\bf
s}\tau}{2})\right)$ and $\int_{-1}^{1}d\tau\left({\bf s}\times{\bf B}({\bf
r}+\frac{{\bf s}\tau}{2})\right)$, respectively. Moreover, contrary to the
electrostatic counterpart, which has been analyzed and applied for more than
three decades to a plethora of quantum transport processes 22, 23, there is
limited experience with the properties of the gauge-invariant equation. It is
thus desirable to gain first experience by reducing the complexity to simplify
the equation. A first step in this direction is based on the fact that for
stationary EM conditions and a homogeneous ${\bf B}$ the $I^{F}$-term can be
neglected. Also, $H^{F}$ introduces the magnetic Lorentz force, while $D^{F}$
can be expressed via $V_{w}$ for a stationary ${\bf E}$ 21. As a homogeneous
magnetic field can be associated to the zeroth-order term in the Taylor
expansion of ${\bf B}({\bf r})$, as a next step, it is reasonable to take into
account the next (linear) term in the expansion. Therefore, in what follows,
we focus on the effects introduced by a magnetic field with a spatial
linearity. Therefore, we consider the following physical settings: The
dimensionality of the problem is reduced to a two-dimensional electron
evolution in the ${\bf r}=(x,y)$-plane. The inhomogeneous magnetic field is
normal to the plane in $z$-direction: ${\bf B}={(0,0,B(y))}$ . In this way,
the Lorentz magnetic force is in the plane. The $y$-coordinate is chosen along
the linearity of ${\bf B}$, so that $B(y)=B_{0}+B_{1}y$. The electric field
${\bf E}({\bf r})$ has a general spatial dependence. Being stationary, the
electric field allows to reintroduce the Wigner potential $V_{w}({\bf p},{\bf
r})$, a quantity which has been physically well analyzed over the last
decades. In Section 2, we formulate the corresponding magnetic field aware
evolution equation for the Wigner function. An analysis of the operators,
which compose the equation, is presented.

In section 3, certain physical effects incorporated in the solution are
identified in the case of a weak non-linearity. Interesting spatial
correlations between the electric and magnetic fields are observed, which
affect the process of magnetotunneling 24, 25. They are observed in both,
density and negativity distributions obtained from the Wigner solution,
suggesting the existence of both local and non-local interplays of these
fields, and indicating effects, which can be used for controlling the electron
evolution. In particular, different settings of the non-linear magnetic field
can be used to guide electron trajectories to a desired region of the space.
Furthermore, it is shown that certain evolution patterns such as snake
trajectories and edge states 26, 27, 2, which are expected from classical
considerations, are maintained by the quantum evolution. They can be used to
guide and manipulate an electron state by restraining it in a desired region,
splitting the density distribution, or by affecting its spreading.

##  2 Wigner Equation for Linear Magnetic Fields

The equation has been initially formulated for the case of both linear
electric and magnetic fields (see equation (25) in 21). A linear electric
field can be accounted for either by an accelerating (Newtonian) force or,
equivalently, by the corresponding Wigner potential term. This duality has
been used to verify certain quantum particle concepts used in the
electrostatic Wigner theory 28. For an electric field with a general spatial
shape it is straightforward to reintroduce the Wigner potential in the
evolution equation (using equations (44) and (45) from 21):

$\displaystyle\left(\frac{\partial}{\partial t}+\frac{{\bf
p}}{m}\cdot\frac{\partial}{\partial{\bf r}}+{\bf
F}(B(y))\cdot\frac{\partial}{\partial{\bf p}}\right)f_{w}\bigl{(}{\bf p},{\bf
r}\bigr{)}=$ $\displaystyle\int d{\bf p}^{\prime}V_{w}({\bf p}-{\bf
p}^{\prime},{\bf r})f_{w}({\bf p}^{\prime},{\bf r})+$ (1)
$\displaystyle\frac{B_{1}\hbar^{2}}{m}\frac{e}{12}\left(\frac{\partial^{2}}{\partial
p_{y}^{2}}\frac{\partial}{\partial x}-\frac{\partial}{\partial
p_{x}}\frac{\partial}{\partial p_{y}}\frac{\partial}{\partial
y}\right)f_{w}\bigl{(}{\bf p},{\bf r}\bigr{)}$

The left-hand side contains the Liouville operator ($Lo(B(y))$), where ${\bf
F}$ is the magnetic Lorentz force ${\bf F}(B(y))=\frac{e}{m}{{\bf p}\times{\bf
B}({\bf r})}=\frac{e}{m}(p_{y}B(y),-p_{x}B(y),0)$ ***In the Lorentz force, in
equation 2, and in the following, ${\bf p}$ refers to the kinetic momentum
instead of the canonical momentum. . On the right-hand side is the Wigner
potential term, followed by a term, which involves higher-order mixed
derivatives. Without EM fields the equation reduces to $Lo(0)=0$, which
resembles the force-less Vlasov equation. The equation involves Newtonian
trajectories, however, the quantum character of the evolution depends on the
initial condition 29: The latter can contain, e.g., negative values in
contrast to the classical distribution function. An electric field is
accounted for by the Wigner potential term ${\cal V}_{w}$, so that the
equation takes the well-known form $Lo(0)={\cal V}_{w}$. $Lo(0)$ is associated
with force-less Newtonian trajectories, however, this does not challenge the
quantum character of the theory: $Lo(0)$ together with ${\cal V}_{w}$ gives
rise to interference, non-locality, tunneling, negativity, and oscillatory
behavior of $f_{w}$. The evolution is fully coherent as the theory is fully
equivalent to wave mechanics 29, 30. Next, if $B_{0}\neq 0$, (1) becomes the
homogeneous magnetic field equation discussed in 20 (equation (49)). The
Liouville operator in equation (1) involves Newtonian trajectories driven by
the inhomogeneous magnetic field $B(y)=B_{0}+B_{1}y$, which affects the
interplay with ${\cal V}_{w}$. The last operator with the higher-order
derivatives is proportional to $B_{1}$. This fact provides the opportunity to
discriminate the two operators by considering a small $B_{1}$. For large $y$,
the magnitude of the linear component $B_{1}y$ can become larger than $B_{0}$
so that if both have opposite signs, the sign of the magnetic field $B(y)$ is
changed. Furthermore, an analysis of physically relevant settings shows that
the higher-order derivatives operator can be a few orders of magnitude smaller
than the Wigner potential. The latter is characterized by the quantity
$\gamma\simeq 10^{14-15}/sec$ 6, characterizing the electric conditions in
nanostructures. This is a quantity equivalent to the total out-scattering rate
in Boltzmann transport models, which, for instance for phonons is
$10^{12-13}/sec$. These considerations suggest that the interplay of
$Lo(B(y))$ and $V_{w}$ can give rise to important physical effects which
dominate the transport for small $B_{1}$ values. Accordingly, we neglect the
last row in (1) and consider the equation $Lo(B(y))={\cal V}_{w}$. It
resembles the standard Wigner equation, which is associated with a signed-
particle model 6†††The inclusion of the higher-order derivative terms demands
an advanced quantum particle model, which is currently in development 31.. The
signed-particle model represents an electron state by an ensemble of numerical
particles. The particles have special features that carry the quantum
information, however, they evolve in the phase space over Newtonian
trajectories dictated by $Lo$. Thus the left-hand side of (1) determines the
trajectories of the particles in the evolving ensemble so that the effects of
the classical Lorentz force (due to the linear magnetic field) are
incorporated also in the quantum evolution. This is demonstrated by the
existence of snake and edge modes in the quantum evolution considered in the
next section: There, we consider the essential cases of magnetotunneling and
ways to manipulate the state evolution by choosing different settings for
$B_{0}$ and $B_{1}$. Finally, it is important to highlight that such level of
physical insight is a unique feature of the Wigner formalism.

##  3 Magnetic Field Effects in the Quantum Evolution

In general, a magnetic field is able to control the trajectory of a classical
charged particle. However, the quantum case has a wave-like nature which needs
to be incorporated. This is possible by using a signed-particle model which is
a numerical model to describe the evolution of a quantum electron state and,
moreover, the magnetic field effect in this evolution. In the following sub-
sections, we show and study the effect of applying a non-uniform linear
magnetic field to a selection of specific quantum electron state evolution
scenarios, and we highlight the use of magnetic fields for controlling the
electron state. Where possible, we compare to classical transport results, as
the Wigner signed-particle model we employ provides a seamless transition to
the classical transport picture. In Section 3.1, we analyze a magnetotunneling
structure where the electron state interacts with an "electric" potential
barrier and an external magnetic field orthogonal to the trajectory. In
Section 3.2, we analyze the evolution of a quantum electron state in an
electronic waveguide with two opposite external magnetic field configurations,
resulting in a so-called snake trajectory and the formation of an edge state,
respectively.

###  3.1 Magnetotunneling

Magnetotunneling refers to the effects of a magnetic field on tunneling
processes of an electron state through a single-barrier potential. It has been
analyzed in the framework of the Wigner formalism in 32, showing how a
coherence pattern remains after a barrier, revealing a sort of "two-peak"
waveform. In other work, magnetotunneling has been investigated in 20 with a
comparison between the classical and the quantum behaviour in presence of a
uniform magnetic field $B(y)=B_{0}$. The "classical" behaviour corresponds to
treating the potential barrier as classical electric force. In this case, the
particle, having a kinetic energy less than the barrier, is completely
reflected. In the quantum case it was demonstrated that there is tunneling, as
was expected, e.g., from 32, but in addition it was shown that the process of
transmission (including the interference effects) is clearly affected by the
magnetic field. In the following, we are going to deepen this analysis, in
particular, studying the case where a non-uniform magnetic field is included
in the Liouville operator, i.e., where $Lo(B(y))$ includes a non-uniform
magnetic field $B(y)=B_{0}+B_{1}y$, and the barrier is quantum mechanically
represented by the Wigner potential ${\cal V}_{w}$.

0T –0T –0T –0T –B(y)

Fig. 1: Case 1: The steady-state electron density, $n(x,y)$, of a Wigner state
(electron) injected at the bottom, evolving towards $+y$-direction. No
magnetic field is applied (see $B(y)$ indicators on the left). The dashed line
indicates the mean path of the state’s evolution. Green lines indicate the
$1\,\rm nm$ thick barrier of $0.3\,\rm eV$. The density shows a fine
oscillatory structure above the barrier.

-6T –-6T –-6T –-6T –B(y)

Fig. 2: Case 2: The steady-state electron density, $n(x,y)$, of a Wigner state
(electron) injected at the bottom under a constant magnetic field (see $B(y)$
indicators on the left) which bends the density and thus the mean path,
indicated by the dashed line. The oscillatory structure above the barrier is
strongly reduced by the magnetic field.

####  3.1.1 Simulation Setup

The simulation domain is $(x,y)=(40\,\rm nm,60\,\rm nm)$ as shown in Figs.
2-4. A $0.3\,\rm eV$ potential barrier is placed at $y=30\,\rm nm$. The
barrier thickness is $1\,\rm nm$ and is physically modeled by the Wigner
potential $V_{w}$ in all shown experiments. The initial condition corresponds
to a minimum uncertainty Wigner state $\sigma_{x}=\sigma_{y}=3\,\rm nm$, is
periodically injected into the simulation domain from the bottom, and evolves
towards the barrier with a kinetic energy of $0.1\,\rm eV$. The initial mean
velocity is $0$ in $x$-direction so almost all the energy at $t=0\,\rm fs$ is
directed towards the potential barrier. We consider four cases of
$(B_{0},B_{1})$:

  * Case 1 is $(B_{0},B_{1})=(0,0)$ and represents a reference case for the entire experiment since it allows to analyse quantum tunneling through the barrier without the influence of an external magnetic field.

  * Case 2 is $(B_{0},B_{1})=(-6\,\textrm{T},0)$ and represents the effect of a uniform magnetic field directed orthogonal to the simulation domain.

  * Case 3 is $(B_{0},B_{1})=(-6\,\textrm{T},0.2\,\textrm{T}\textrm{nm}^{-1})$. The magnetic field changes its sign around the barrier; the magnetic field is thus zero at the barrier.

  * Case 4 is $(B_{0},B_{1})=(-2\,\textrm{T},-0.2\,\textrm{T}\textrm{nm}^{-1})$. The linear component $B_{1}y$ increases the effect of the magnetic field ${\bf B}$ along $y$ so that the magnitude of the magnetic field is non-zero in the region of the barrier.

-6T –-3T –0T –3T –B(y)

Fig. 3: Case 3: The steady-state electron density, $n(x,y)$, of a Wigner state
(electron) injected at the bottom, under a linear magnetic that goes from
$-6\,\rm T$ to $6\,\rm T$ (see $B(y)$ indicators on the left) and thus the
magnetic field becomes zero at the barrier and switches the sign, giving rise
to a snake type of evolution. The density in the upper half of the domain
shows again a fine oscillatory structure.

-2 T –-5 T –-8 T –-11 T –B(y)

Fig. 4: Case 4: The steady-state electron density $n(x,y)$, of a Wigner state
(electron) is injected at the bottom. The magnetic field is gradually
increased towards $+y$-direction and is particularly large at and above the
barrier (see $B(y)$ indicators on the left). The magnetic field suppresses the
oscillations of the density, similar to case 2. The mean path (white dashed
line) is compared to the mean path of case 2 (orange dot-dashed line):
Although they differ, they both guide the state to the same final position.

Fig. 5: Density distribution along $y$-direction $n(y)$: Cases 1&3 and 2&4
clearly group together, suggesting that the oscillations are suppressed in the
presence of a magnetic field in the region of the barrier, further indicating
that the EM fields interact locally.

Fig. 6: Negativity along $y$-direction $Neg_{w}(y)$. Negative values after the
injection ($y<25\,\rm nm$) demonstrates the non-local action of the barrier
already without magnetic field. The negativity increases with the increase of
$B(y)$ in this region, which suggests again a non-local interplay of the EM
fields. Oscillatory behavior is slightly visible for the first two peaks in
presence of strong magnetic field: case 2&4\.

####  3.1.2 Density and Negativity Analysis

Figs. 2-4 shows the electron density for all the four cases. The steady-state
electron density, $n(x,y)$, is obtained by the integration over ${\bf p}$ of
the steady-state Wigner function $f_{w}({\bf x},{\bf p})$. In accordance with
the Ehrenfest theorem, in all the four cases the mean densities follow the
classical paths, that are indicated by the dotted lines.

In Fig. 2, that is related to $B(y)=0$, the classical path is the central line
$x=20\,\rm nm$, since no magnetic field bends the trajectory. The density
perfectly reflects the symmetry with respect to the central dashed line and
shows a fine oscillatory structure above the barrier, after the tunneling. The
latter is almost completely destroyed by the constant magnetic field, as shown
in Fig. 2. There, the magnetic field bends the path, guiding the electron
towards a specific position, where, considering possible applications, an
additional channel could be envisioned, outlining a possible use-case for
single electron control.

In case 3 the magnetic field changes its sign around the barrier, giving rise
to a snake type of evolution 1, as shown in Fig. 4. Besides, the fine
structure of the density above $y=30nm$ is recovered, similar to the case
shown in Fig. 2. Observing that the magnetic field is zero at the barrier as
in Fig. 2, we associate this effect with the existence of a local interplay
with the EM fields. Indeed, in Fig. 4, when $B(y)$ around the barrier is
particularly large (similar to case 2), the oscillations are again suppressed.

In case 4 the electron density, shown in Fig. 4, is very similar to the
electron density of case 2, since the magnetic field is negative in the entire
simulation domain, but gradually increases towards $+y$-direction starting
from $-2\,\rm T$ until $-15\,\rm T$. The mean path (white dashed line) is
compared to the mean path of case 2 (orange dot-dashed line): Although they
differ, they both guide the state to the same final position. An important
observation is that the magnetic field is particularly large at and above the
barrier in this case, and indeed it suppresses the oscillations of the
density, similar to case 2 (Fig. 2).

Since the linear change of the magnetic field is along the $y$ axis and no
significant asymmetries with respect to the classical trajectory are observed
along $x$, to better analyze the four cases, the steady-state electron density
$n(x,y)$ has been integrated over $x$, obtaining the density along $y$, i.e.,
$n(y)$. One of the advantages of the Wigner formalism and, in particular, of
the signed-particle model is to have access to Wigner function and its
negativity, obtained by $Neg_{w}({\bf r},{\bf p})=f_{w}\theta(-f_{w})$ where
$\theta$ is the Heaviside function. Wigner function negativity indicates
quantum behavior of a state 33. We thus also investigate the negativity along
$y$, $Neg_{w}(y)$, integrating over the momentum ${\bf p}$ and over $x$.

From the comparison of $n(y)$ shown in Fig. 6 it is possible to notice that
the oscillatory behaviour in case 1 is smaller than in case 2 before the
barrier. It is almost completely destroyed by the magnetic field after the
barrier in case 2, only a small variation related to the first two peaks is
slightly visible, while in case 1 it is well visible. A further confirmation
of the action of the magnetic field on the oscillatory behaviour is given by
the fact that case 3, where the magnetic field is almost zero around the
potential barrier, behaves similar to case 1, while in case 4, where the
magnetic field is even stronger than $-6\,\rm T$, the behaviour is very
similar to case 2. This observation strongly suggests the existence of a local
effect of the magnetic field on the process of tunneling.

The negativity, shown in Fig. 6, demonstrates an oscillatory behaviour around
the barrier, which is a manifestation of quantum effects. The appearance of
negative values after the injection of the entirely positive initial state
below the barrier ($y<25\,\rm nm$) demonstrates the non-local action of the
barrier already without a magnetic field. The oscillations of the negativity
are well visible after the barrier for case 1 and 3, while they are
drastically reduced for case 2 and 4: Only the first two peaks are slightly
visible as a change of the slope (gradient). In any case the negativity shows
this behaviour more evidently than the density. This observation provides a
confirmation that the magnetic field tends to destroy the oscillatory
behaviour of tunneled electrons and also confirms the results presented in 32.
It’s also possible to observe that the negativity increases with the magnitude
of the magnetic field already far before the barrier, which can be seen in all
the cases where a magnetic field is different from zero, i.e., cases 2,3, and
4. This suggests another, non-local effect of the interplay of the EM fields.

Fig. 7: Snake trajectory: Evolution of a minimum uncertainty Wigner state
along a waveguide under an orthogonal, non-uniform linear magnetic field
$B(x)=B_{0}+B_{1}x$ with
$(B_{0},B_{1})=(-1\,\textrm{T},0.0033\,\textrm{T}\textrm{nm}^{-1})$. The right
part shows the magnetic field $B(x)$ in Tesla (T). The left part shows the
electron density at
$t=\\{0,1.6\textrm{ps},3.2\textrm{ps},5\textrm{ps},6.4\textrm{ps}\\}.$ The
electron state follows a snake trajectory indicated by the white dashed line.
Fig. 8: Edge state: Evolution of a minimum uncertainty Wigner package along a
waveguide under an orthogonal non-uniform linear magnetic field
$B(x)=B_{0}+B_{1}x$ with
$(B_{0},B_{1})=(1\,\textrm{T},-0.0033\,\textrm{T}\textrm{nm}^{-1})$. The right
part shows the magnetic field $B(x)$ in Tesla (T). The left part shows the
electron density at $t=\\{0,2.5\textrm{ps},5\textrm{ps},8\textrm{ps}\\}.$ At
the end, the wavepacket becomes a split edge state.

###  3.2 Snake Trajectory and Edge State

In this section, we are going to explore the ability of a non-uniform magnetic
field to control the evolution of an electron-state, not only in terms of
trajectory and interference patterns, as in the magnetotunneling application,
but also in terms of shape and dispersion of the wave-packet. In particular,
we are identifying two opposite configurations of the linear magnetic field
where the first contributes to maintain the localization of the wave-packet,
as it happens in the case of a snake trajectory, and the second to de-
localize, which gives rise to the creation of an edge state.

####  3.2.1 Simulation Setup

In the two following experiments, we consider a portion of an electronic
waveguide in the plane $x-y$, with a magnetic field, ${\bf B}({\bf r})$, that
is orthogonal to the plane $x-y$ and presents a linear dependence in direction
$x$, which is orthogonal to the electron evolution (electron state evolves
along $y$). The waveguide width is $600\,\rm nm$ along $x$-direction. The
magnetic field is defined by ${\bf B}({\bf r})={(0,0,B(x))}$ with
$B(x)=B_{0}+B_{1}x$. Equation (2), that describes the electron state
evolution, keeps the same form except that $y$ and $x$ are exchanged. The
waveguide length for the snake trajectory and edge state experiment is
$2500\,\rm nm$ and $3000\,\rm nm$, respectively. We evolve a minimum
uncertainty Wigner state with $\sigma_{x}=\sigma_{y}=35\,\rm nm$,
$m_{\textrm{eff}}=0.19m_{e}$ and initial energy of $0.045\,\rm eV$ along $y$.
The electron state is placed at $y_{0}=300\rm nm$ and we use a Gaussian state
that is fully included in the simulation domain at $t=0$.

###  3.3 Snake Trajectory

In the snake trajectory experiment the value of the magnetic field is defined
by $B_{0}=-1\,\rm T$ and $B_{1}=0.0033\,\rm Tnm^{-1}$. This means that the
magnetic field starts at $-1\,\rm T$ at $x=0\,\rm nm$ and increases to $1\,\rm
T$ at $x=600\,\rm nm$ and is constant along the $y$-direction, as shown in the
right part of Fig. 7. The initial position of the Wigner state is
$(x_{0},y_{0})=(370\,\textrm{nm},300\,\textrm{nm})$ and is placed $70\,\rm nm$
to the right of the waveguide center which is at $x=300\,\rm nm$, where
$B=0\,\rm T$), see Fig. 7.

The left part of Fig. 7 shows the evolution of the electron state along the
waveguide. Fig. 7 reports the density of the quantum states in specific time
instants
$t=\\{0\,\textrm{ps},1.6\,\textrm{ps},3.2\,\textrm{ps},5.0\,\textrm{ps},6.4\,\textrm{ps}\\}$.
The dashed white line represents the trajectory of a classical "point-like"
particle that evolves along the wire starting from the middle of the quantum
state with the same mean velocity. Furthermore, the chosen magnetic field
affects the shape of the electron state, as shown by the distribution of the
electron density. At $t=0$, the state is perfectly circular. During the
evolution the "width" of the wavepacket tends to decrease approaching the
center ($t=1.6\rm ps$), since the magnetic field decreases and reaches $0\rm
T$, followed by an increase again as it oscillates to the other side
($t=3.2\rm ps$). On the contrary, the natural spreading (i.e., increase of the
dispersion) characterising a freely evolving Gaussian state is retained in the
direction of the evolution. This particular choice of the initial condition
and of the external linear magnetic field allows to limit the dispersion of
the electron state during the evolution and shows how a linear magnetic field
can be used to control the shape of the density of an electron state during
its evolution.

###  3.4 Edge State

In the edge state experiment, the magnetic force is opposite to the one used
in Section 3.3, as can be seen from the right part of Fig. 8. Thus
$B_{0}=1\,\rm T$ and $B_{1}=-0.0033\,\rm Tnm^{-1}$, so the magnetic field
ranges from $1\,\rm T$ to $-1\,\rm T$ along $x$, and is constant along the
$y$-direction. The initial position of the Wigner state ($t=0\,\rm ps$) is
perfectly centered, i.e., $(x_{0},y_{0})=(300\,\textrm{nm},300\,\textrm{nm})$,
where $B=0\,\rm T$, as can be seen in the left part of Fig. 8. In the
classical case, a particle evolving in the center and along the line
$x=300\,\rm nm$ (where the magnetic force is zero) will not be affected by the
action of the magnetic field. In the quantum case, the electron state density
is distributed in space, according to Heisenberg’s uncertainty principle, and
represented by an ensemble of numerical particles, so that the evolution is
always affected by the action of the magnetic field. It is thus clear that in
the quantum case the signed-particle model is useful as it allows to describe
and heuristically understand the electron evolution. Parts of the electron
density on the left of the central line are pushed towards the left border of
the waveguide, while other parts on the right are pushed towards the right
border. The boundary conditions reflect the parts of the split state back
towards the center. An analysis of the left part of Fig. 8 shows the density
of the quantum states at specific times
$t=\\{0\,\textrm{ps},2.5\,\textrm{ps},5.0\,\textrm{ps},8.0\,\textrm{ps}\\}$.
As we can see at $t=2.5\,\rm ps$ the electron density width increases along
the $x$-direction during the evolution. This dispersion is driven by the
Lorenz force and provides a curvilinear shape along $x$ to the wavepacket,
similar to an "arc". When the electron state interacts with the lateral
borders of the waveguide it is reflected towards the center, this can be seen
by the fact that the bending of the curvilinear shape disappears near the
borders. At $t=5.0\,\rm ps$ the electron state is completely reflected and the
density takes an almost "bimodal distribution" form along $x$. It is almost
separated in two parts, one on the left and one on the right of the line
$x=300\,\rm nm$, where the magnetic field is zero. Only a small portion of the
density is still in the center of the waveguide. The electron density at
$t=8.0\,\rm ps$ keeps the bimodal distribution which is more defined, and
evolves along the waveguide as two parallel parts of an edge state that travel
in the same direction, due to the opposite sign of the magnetic field. This
behaviour is further confirmed by the dispersion of the two parts along the
direction of motion, that is clearly visible comparing to the density at
$t=5.0\,\rm ps$ with the density at $t=8.0\,\rm ps$. This configuration of the
magnetic field splits the state and pushes its parts towards the boundaries
which reflect them back. This mechanism of electron splitting can be an
attractive option for research in electron quantum optics, where "beam
splitter" concepts are a vital building block.

##  4 Summary

Non-uniform magnetic fields offer unique capabilities to control single
electron states. Here, we investigate a generalization of equation (25)
proposed in 21 for non-linear electric fields. The obtained equation involves
the conventional Wigner potential, the magnetic Lorentz force, that includes
also linear magnetic fields, and higher-order terms that depend on the linear
coefficient of ${\bf B}$. The higher-order terms can be neglected for small
magnitudes which allows to perform numerical calculations in terms of the
well-developed signed-particle model. Experiments on magnetotunneling are
shown and, in particular, the capability of linear, non-uniform magnetic
fields to control an electron state trajectory. Moreover, the possibility to
change the value of the magnetic field allows to specifically influence an
interference pattern or an oscillatory behaviour. We also show the capability
of linear, non-uniform magnetic fields to control the spatial dispersion of an
electron state, or to split the electron state. It should be stressed that
both cases of evolution have a classical analog of evolving initial Gaussian
distributions of an ensemble of non-interacting electrons. Indeed, for the
considered physical setup classical and quantum evolution rules are the same.
However, in the former, the initial condition can be only non-negative, while
in the latter also negative values are possible, characterising the Wigner
quasi-distribution function.

## Conflicts of interest

There are no conflicts to declare.

## Acknowledgements

The financial support by the Austrian Science Fund (FWF): P33609 and P37080 is
gratefully acknowledged. The computational results have been achieved using
the Vienna Scientific Cluster (VSC).

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