# Non-adiabatic quantum control of valley states in silicon

Alan Gardin1<EMAIL_ADDRESS>Ross D. Monaghan1 Tyler Whittaker1
Rajib Rahman2 Giuseppe C. Tettamanzi1<EMAIL_ADDRESS>1School of Physics & Institute of Photonics and Advanced Sensing, The
University of Adelaide, Adelaide SA 5005, Australia.  2School of Physics, The
University of New South Wales, Sydney, NSW 2052, Australia.

###### Abstract

Non-adiabatic quantum effects, often experimentally observed in semiconductors
nano-devices such as single-electron pumps operating at high frequencies, can
result in undesirable and uncontrollable behaviour. However, when combined
with the valley degree of freedom inherent to silicon, these unfavourable
effects may be leveraged for quantum information processing schemes. By using
an explicit time evolution of the Schrodinger equation, we study numerically
non-adiabatic transitions between the two lowest valley states of an electron
in a quantum dot formed in a SiGe/Si heterostructure. The presence of a single
atomic layer step at the top SiGe/Si interface opens an anti-crossing in the
electronic spectrum as the centre of the quantum dot is varied. We show that
an electric field applied perpendicularly to the interface allows tuning of
the anti-crossing energy gap. As a result, by moving the electron through this
anti-crossing, and by electrically varying the energy gap, it is possible to
electrically control the probabilities of the two lowest valley states.

##  I Introduction

Adiabatic theorem and manifestations in single-electron pumps. According to
the adiabatic theorem, a system initially in its ground state, and whose
Hamiltonian evolves slowly in time, is expected to remain in the ground state
of the instantaneous Hamiltonian, provided the ground state is non-degenerate.
However, if the Hamiltonian varies quickly, then this approximation is no
longer valid and the state is best described by a superposition of eigenstates
of the instantaneous Hamiltonian. This superposition of states induces spatial
oscillations of the wave-function, which are an observable manifestation of
the violation of the adiabatic theorem. This is encountered in single electron
pumps operating at fast pumping frequencies, where being in the non-adiabatic
regime leads to a decrease in the accuracy and precision of the pump [1]. For
instance, in the work of Yamahata  _et al._ [2] non-adiabatic oscillations
were experimentally measured in single-electron pumps formed by a silicon
nanowire field effect transistor. They also presented a one-dimensional
simulation of non-adiabatic oscillations between electron orbitals for a
moving quantum dot and demonstrated the resulting spatial oscillations of the
wave function.

Valleys in silicon and SiGe structures.  In a [001] grown SixGe1-x
heterostructure, the tensile strain between the Si and SiGe layers breaks the
six-fold valley degeneracy of bulk silicon, leading to a four-fold degeneracy
($k_{x}$ and $k_{y}$ valleys) raised in energy and a two-fold degeneracy
($k_{z}$ valleys) lowered in energy. The latter can be further broken by
interface roughness, impurities, or electric fields [3, 4]. It is sufficient
to focus on the two lowest $k_{z}$ valley states because of the high energy
gap (typically around 200 meV [3]) between them and the other $k_{x}$ and
$k_{y}$ valley states.

Non-adiabatic transitions with valleys is unexplored. Whilst Yamahata  _et
al._ considered electron orbitals, they do not take into account the valley
degrees of freedom inherent to silicon, which are expected to couple with the
orbital states in any realistic device [5, 3, 6]. Hence it may be possible,
due to valley-orbit coupling, to observe non-adiabatic transitions between
valley states. Note that coherent oscillations between valley states has been
previously studied in pure valley qubit schemes based on double quantum dots
[7, 8], while we instead focus on a single quantum dot. Similarly, Boross  _et
al._ [9] considered coherent oscillations between valley states in a single
dot, but their driving scheme is limited to a weak electric field, contrary to
our scheme.

Model and contribution of the paper. In this paper, we study the dynamics of a
single electron trapped in a quantum dot formed by a [001] grown Si0.8Ge0.2/Si
heterostructure. The appearance of valley physics stems from our modelling of
the SiGe/Si heterostructure by use of a two-band model. Non-adiabatic effects
are then introduced by quickly varying the centre of the quantum dot
potential. We then successively consider two two-dimensional models of a
simplified SiGe/Si heterostructure, both introduced in section II.

In the first model, we present an ideally grown heterostructure, where the
orbital degrees of freedom originating from the quantum dot are decoupled from
the valley degrees of freedom, and thusly observe only non-adiabatic
transitions between orbital states. An analytic study of such transitions is
tackled in section III.

In the second model, we model a miscut SiGe/Si heterostructure by introducing
a single atomic step at the interface, akin to that of Boross  _et al._. In
section IV, we explain why the introduction of this step alters the spectrum
of the system as the quantum dot position is varied. The key physics is that
an anti-crossing between the two lowest valley states opens, with an energy
gap tuneable by an electric field applied along the $z$ direction. This
behaviour is verified by a realistic tight-binding calculation using NEMO 3D’s
20 band $sp^{3}d^{5}s^{*}$ model [10]. We detail this in appendix A. As a
result of the anti-crossing, the non-adiabatic pulsing of the electron through
this step allows one to induce transitions from one valley state to another.
Section V discusses the main result of the paper, namely that the final state
can be electrically controlled by varying the anti-crossing energy gap. Due to
the appearance of the avoided crossing in the spectrum, we model the
transition using the Landau-Zenner approximation [11] and achieve good
concordance. From studying the results there exists a range of driving
parameters in which the final state forms a good two-level system, opening the
door for qubit applications. We discuss the viability of such non-adiabatic
valley qubit in section VI.

##  II Models

###  II.1 Ideal model

Quantum dot potential. We now present the two-dimensional model, in the
$(x,z)$ plane, used throughout this paper. We assume the growth axis $z$ of
our heterostructure to be orthogonal to the $x$ axis, as shown in figure 1. In
the $x$ direction, the electron has a transverse effective mass
$m_{x}=0.19\,m_{e}$. The quantum dot potential $V_{QD}$ is a harmonic
potential of angular frequency $\omega$ whose minimum follows a trajectory
$\xi$, hence $V_{QD}(x,t)=\frac{1}{2}m_{x}\omega^{2}(x-\xi(t))^{2}$. We choose
$\omega$ such that we obtain an orbital level spacing
$\Delta\epsilon=\hbar\omega=$ 2 meV for the orbital states. This value is
chosen to isolate the higher orbital states from the two lowest valley states.
Note that the $y$ axis is neglected in our work since it does not affect the
physics discussed.

Quantum dot trajectory. The evolution in time of the quantum dot position is
the only source of time-dependence in Hamiltonian. The quantum dot is
initially at position $x_{0}=0$ nm at $t_{0}=0$ ps, and moves with constant
velocity $v_{QD}$ until it reaches $x_{1}$ at time $t_{1}$. The parameters
$x_{1}$ and $t_{1}$ are varied to obtain different values for $v_{QD}$. The
trajectory $\xi$ of the quantum dot can be written:

$\xi(t)=\begin{cases*}x_{0}\text{ for }t\leqslant t_{0}\\\
x_{0}+v_{QD}(t-t_{0})\text{ for }t_{0}\leqslant t\leqslant t_{1}\\\
x_{1}\quad\quad\;\,\text{ for }t_{1}\leqslant t\end{cases*}$ (1)

The Hamiltonian describing the dynamics along the $x$ axis is

$H_{x}(x,t)=\frac{p_{x}^{2}}{2m_{x}}+V_{QD}(x,t)$ (2)

and is identical for all $z$ in the ideal model.

Figure 1: Geometry of the two-dimensional SiGe heterostructure model with the
$y$ axis neglected. We apply a time-dependent harmonic quantum dot potential
$V_{QD}$ in $x$, and an electric field in $z$. The quantum well in the $z$
axis, originating from the SiGe/Si interfaces, is composed of $N_{z}$ atoms
and decreases by one atomic layer at the step (green circle). The dashed
parabola represents the potential profile at $t^{\prime}>t$, reflecting the
spatial movement of the harmonic potential.

Description of the $z$ confinement.  The $z$ confinement and the valley
effects are modelled using the tight-binding model for strained silicon
quantum wells of Boykin  _et al._ [13, 12]. The resulting Hamiltonian for the
$z$ slices depends on three parameters: the onsite energy $\eta$, the nearest
neighbor hopping $v$ and the second-neighbor hopping $u$ giving the
Hamiltonian in equation 3.

$H_{z}(\epsilon,u,v)=\begin{pmatrix}\eta&v&u&0&\dots&\dots&0\\\
v&\eta&v&u&0&\ddots&\vdots\\\ u&v&\eta&v&u&\ddots&\vdots\\\
0&\ddots&\ddots&\ddots&\ddots&\ddots&0\\\ \vdots&\ddots&u&v&\eta&v&u\\\
\vdots&\ddots&0&u&v&\eta&v\\\ 0&\dots&\dots&0&u&v&\eta\\\ \end{pmatrix}$ (3)

In this Hamiltonian, each site represents an atom of the quantum well in the
$z$ axis. We will define $N_{z}=40$ to be the total number of atoms in the $z$
axis, where each atom is separated by $1.366$Å. We will also refer to $N_{z}$
as the quantum well width, expressed in number of atoms. A monolayer
corresponds to two atoms, so that our choice of quantum well width corresponds
to 20 monolayers. Note that varying the width of the Si/SiGe quantum well, as
we could do in the simulations, is experimentally feasible.

Expected eigenstates and spectrum.  In our model, the quantum well width is
$L_{z}=N_{z}\times 0.1366=5.464$ nm. The energies of this infinite quantum
well barrier are given by
$E_{m}=m^{2}\frac{\hbar^{2}\pi^{2}}{2m_{l}L_{z}^{2}}$ with $m_{l}=0.92m_{e}$
the electron’s longitudinal effective mass and $m\in\mathbb{N}^{*}$ the
orbital number. The energy gap between the first and second quantum well
eigenstates is then $3\frac{\hbar^{2}\pi^{2}}{2m_{l}L_{z}^{2}}=13$ meV.
Because of this high energy gap, it is reasonable to only consider the first
eigenstate of the quantum well. The eigenstates of $H_{x}$, which are
eigenstates of the harmonic oscillator, are denoted by $\varphi_{n}$ with
$n\in\mathbb{N}$ describing the orbital number.

The Hamiltonian describing the $z$ confinement gives a valley splitting less
than the orbital spacing $\Delta\epsilon=\hbar\omega$ induced by to the
harmonic oscillator. As a consequence, we expect to observe two well defined
valley states for each orbital. We will denote the corresponding eigenstates
$\psi_{n,v}$ and energies $\epsilon_{n,v}$, with $(n,v)$ where
$n\in\mathbb{N}$ is the orbital index and $v\in\quantity{-,+}$ the valley
index. It will be convenient later on to index the states and the energies
with a single index for readability. Since the $v=-$ valley states always have
lower energies than the $v=+$ states for a given $n$, we will use the natural
mapping $(0,-)\to 0$, $(0,+)\to 1$, $(1,-)\to 2$ and so on to label the
energies and eigenstates by increasing energies. The meaning of the indices
will be understood from context depending on whether there is one or two
indices.

As a final note, the full Hamiltonian $H$ for this idealised model is
separable in $x$ and $z$, and hence the valley degrees of freedom do not
couple to the orbital degree of freedom. The eigenstates are the product of
the eigenstates of the Hamiltonians $H_{x}$ and $H_{z}$, and indeed the
indices $(n,v)$ are good quantum numbers.

###  II.2 Single step model

Description of the interface step, and experimental justification. As in the
work of Boross  _et al._ , we introduce an interface step consisting of a
single atomic layer at $x_{s}=30$ nm, modelled by decreasing the width of the
quantum well for all $z$ slices where $x\geqslant x_{s}$ (see figure 1). This
interface step is motivated by the single atomic layer steps observed in
SiGe/Si heterostructures grown on slightly vicinal [0 0 1] Si substrates [14,
15, 16, 17]. In particular, these steps occur at polar miscut angles
$\theta\leq 2^{\circ}$, with terrace width on the order of 10nm [15, 16, 17].
Given $\theta$ can be controlled by manufacturers to within $0.1^{\circ}$ [14]
and uniform terrace width can be achieved as in [15], one could manufacture a
structure with an interface step similar to that modelled in this paper.

Differences with Boross  _et al._. Unlike Boross  _et al._ , we do not allow
the wave function to penetrate inside the SiGe layer, instead enforcing hard-
wall boundary conditions, however, the physics observed is expected to be
similar. And indeed as shown in figure 5, the evolution of the spectrum is
similar to that obtained in Boross  _et al._. Another difference in our work
is that we do not limit ourselves to weak perturbations along the $x$ axis: we
study the opposite limit of strong non-adiabatic perturbations. Furthermore,
our modelling has been validated with NEMO 3D’s 20 band $sp^{3}d^{5}s^{*}$
model, see appendix A.

###  II.3 Methods

We numerically solve the ground state of the total Hamiltonian $H$ at the
initial time $t_{0}=0$ ps and use this result as the initial wave function for
solving the time-dependent Schrödinger equation. The time-dependent
Schrödinger equation is solved using the Crank-Nicolson scheme [18]. The
onsite energy parameter $\eta$ of $H_{z}$ (equation 3) is set to 1.395 eV such
that the spectrum has negative values close to zero, allowing for easier
differentiation between “real” eigenvectors and “artificial” eigenvectors
(eigenvalues of 0). Indeed, this is necessitated by the setting the column and
row of the relevant lattice sites to zero in the Hamiltonian to produce the
step in the interface, described above, as this introduces null eigenvalues.
The difference between our chosen value of $\eta$ and the choice made in
Boykin  _et al._ is then restored after diagonalisation. The values used for
the off-diagonal elements can be found in figure 2 of Boykin  _et al._ [12].

In all simulations, we fix $x_{0}=0$ nm and $t_{0}=0$ ps. For the simulations
involving the ideal model, we fix $t_{1}-t_{0}$ to 5 ps and vary $x_{1}$ to
change $v_{QD}$. This choice ensures that the perturbation is applied for the
same amount of time in all simulations. For the step model however, we fix
$x_{1}=55$ nm and vary $t_{1}$ to change $v_{QD}$. As discussed later, the
spatial evolution of the dot relative to the step is critical for analysis of
the simulations, such that fixing the initial and final positions and changing
the time is the more appropriate choice of altering the velocity.

##  III Non adiabatic effects in the ideal model

###  III.1 Preliminary remarks

Introduction. In this section, we comment on the numerical results obtained
with the ideal model (figure 2) of section II.1, i.e when no interface step is
present. As explained in the previous section, the Hamiltonian is separable in
the $x$ and $z$ coordinates. Since the time-dependence only originates from
the $x$ axis, the dynamics of the system reduce to that of a driven quantum
harmonic oscillator described by the Hamiltonian $H_{x}$ of equation 2. As a
consequence, the only parameter we vary in our simulations is the quantum dot
speed $v_{QD}$ (in particular, the applied electric field $E_{z}$ does not
influence the numerical results). This model is essentially the same as the
one from Yamahata  _et al._ [2], with the difference being the trajectory
$\xi$ of the quantum dot. Our choice for the trajectory $\xi$ (see equation 1)
will simplify the interpretation of the numerical results.

Figure 2:  Time evolution of the probabilities for the ideal model (no step
and hence no valley physics). For small speeds, the system mostly occupies the
two lowest energy orbitals, forming a two-dimensional Hilbert space, as needed
for a qubit. However, a higher speed introduces higher excited states and does
not behave as a two-level system anymore. Note that the period of oscillations
is constant and the higher the speed the bigger the amplitudes of the
probability oscillations.

Oscillatory features of the probabilities: amplitude and period.  As a
preliminary remark, one can notice that the states of energy $\epsilon_{n,+}$
have a zero probability of occupation for all time. Indeed, since the dynamics
only originate from the orbital degree of freedom, and no valley orbit
coupling exist in this ideal model, no transition to a state of valley index
$v=+$ can happen because we started in a state with $v=-$.

A second striking feature is the oscillatory behaviour of the probabilities.
Noticeably, the period of the oscillations is constant (it depends only on the
potential of the quantum dot), but the amplitude can be increased through
$v_{QD}$. An analytical derivation of this fact is performed in the next
subsection under the adiabatic approximation. Note, however, that higher
values of $v_{QD}$ slowly increase the probabilities of higher orbital states
(i.e states with $n\geqslant 1$), as can be seen in the last row. For low
values of $v_{QD}$, the coherent oscillations between the two states are
effectively Rabi oscillations encountered in qubits, but this does not hold at
higher quantum dot speeds.

Figure 3:  Adiabatic and non-adiabatic driving of a classical harmonic
oscillator. a) In the adiabatic case, the mass follows the position of the
spring. b) In the non-adiabatic case, the spring is moved so quickly that the
mass initially stays at the initial position - it does not have enough time to
react to the external perturbation.

Classical analogy and spatial oscillations of the probability density.  A
consequence of the presence of states other than the ground state manifests
itself in the spatial oscillations of the probability density. The spatial
oscillations of the electron can easily be understood classically as follows.
Applying the Heisenberg equation of motion to the operators $\hat{x}$ and
$\hat{p}_{x}$, we obtain:

$\displaystyle\derivative{\expectationvalue{\hat{p}_{x}}}{t}=$
$\displaystyle\expectationvalue{\frac{1}{i\hbar}\left[\hat{p}_{x},\hat{H}_{x}\right]}$
$\displaystyle=$ $\displaystyle
m\omega^{2}\left(\xi(t)-\expectationvalue{\hat{x}}(t)\right)$
$\displaystyle\derivative{\expectationvalue{\hat{x}}}{t}=$
$\displaystyle\frac{\expectationvalue{\hat{p}_{x}}}{m}$

From which we can derive the following equation of motion for
$\expectationvalue{\hat{x}}$:

$\derivative[2]{\expectationvalue{\hat{x}}}{t}+\omega^{2}\expectationvalue{\hat{x}}=\omega^{2}\xi(t)$
(4)

This is exactly the equation of motion of a classical harmonic oscillator
driven by some force $\xi$. Hence we see that the average position behaves
exactly as in the classical case of a driven harmonic oscillator. The period
of the spatial oscillations depends only on $\omega$ and not on the initial
conditions, whilst the amplitude increases with an increase in the driving
force.

Classically, the movement of the quantum dot can be understood by pushing a
spring to which a mass is attached, as shown in figure 3. Pushing the spring
adiabatically, the mass is expected to follow the movement of the spring, i.e
adapt to the perturbation at each time. However, pushing the spring over the
same distance during a shorter period of time, one expects the mass to
oscillate, as it does not have time to adjust to the perturbation.

Spatial oscillations from a quantum superposition.  Returning to the quantum
case, one can assume that the effect of a non-adiabatic perturbation is to put
the wave function into a superposition of states. For instance, assuming that
at the end of the quantum dot’s trajectory, i.e for $t=t_{1}$, and for
relatively low values of $v_{QD}$, the state is of the form:

$\ket{\psi(t=t_{1})}=\sqrt{1-p}\ket{\varphi_{0}}+\sqrt{p}\ket{\varphi_{1}}$

where $\varphi_{0}$ and $\varphi_{1}$ are respectively the ground and first
excited state of the harmonic oscillator at $t_{1}$, and $p$ if the state
probability. Since for $t>t_{1}$ the Hamiltonian becomes time-independent (it
corresponds to a harmonic oscillator centred at $\xi(t_{1})=x_{1}$), the time
evolution of the state is of the form:

$\ket{\psi(t)}=\sqrt{1-p}\ket{\varphi_{0}}+e^{-i2\pi\frac{t}{\tau}}\sqrt{p}\ket{\varphi_{1}}$

with $\tau=\frac{2\pi\hbar}{\Delta\epsilon}$ and period corresponding to a
pulsation $\omega=\frac{\Delta\epsilon}{\hbar}$. Note that here the
superposition is time-dependent but the state probability is time-independent,
since the probability $p$ does not depend on time.

The average position for $t>t_{1}$ can then be expressed as:

$\displaystyle\expectationvalue{x}(t)$
$\displaystyle=\matrixelement{\psi(t)}{\hat{x}}{\psi(t)}$
$\displaystyle=(1-p)\matrixelement{\varphi_{0}}{\hat{x}}{\varphi_{0}}+p\matrixelement{\varphi_{1}}{\hat{x}}{\varphi_{1}}$
$\displaystyle+2\sqrt{p(1-p)}\cos\left(2\pi\frac{t}{\tau}\right)\Re{\matrixelement{\varphi_{0}}{\hat{x}}{\varphi_{1}}}$
$\displaystyle=x_{1}+\sqrt{\frac{2\hbar
p(1-p)}{m\omega}}\cos\left(2\pi\frac{t}{\tau}\right)$

where we have used equation (4.174) of reference [19] to write
$\matrixelement{\varphi_{0}}{\hat{x}}{\varphi_{1}}=\sqrt{\frac{\hbar}{2m\omega}}$.
Hence, we recover the fact that the probability density oscillates in space,
due to a quantum superposition of states. These spatial oscillations of the
probability density are at the root of the oscillations experimentally
observed by Yamahata  _et al._.

###  III.2 Transition probabilities in the weak non-adiabatic regime.

In this section, we follow a method by B. H. Bransden [19] to derive an
analytical expression for the probability of the first orbital excited state,
i.e the state $\psi_{1,-}$. To simplify notations, we will drop the time-
dependence of $\xi$ and denote by $m=m_{x}$ the electron’s transverse
effective mass and $\hat{H}(t)=H_{x}$ the Hamiltonian. We will also make use
of the single index notation defined in section II, such that we study the
transition $\psi_{0,-}\equiv\psi_{0}\to\psi_{2}\equiv\psi_{1,-}$. One can
expand the Hamiltonian of the 1D model as follows:

$\hat{H}(t)=\frac{\hat{p_{x}}^{2}}{2m}+\frac{1}{2}m\omega^{2}\hat{x}^{2}+\frac{1}{2}m\omega^{2}\xi^{2}-m\omega^{2}\xi\hat{x}$

The derivative with respect to time comes out as:

$\displaystyle\partialderivative{\hat{H}}{t}=m\omega^{2}\dot{\xi}(\xi-\hat{x})^{2}=-m\omega^{2}\dot{\xi}(\hat{x}-\xi)^{2}$
(5)

Following B. H. Bransden, the probability amplitude of the first orbital
excited state $\psi_{2}$ under the adiabatic approximation is (equation
9.100):

$c_{2}(t)=\hbar^{-1}\int_{t_{0}}^{t}\differential{t^{\prime}}\frac{\matrixelement{\varphi_{1}(t^{\prime})}{\partialderivative{\hat{H}}{t^{\prime}}}{\varphi_{0}(t^{\prime})}}{\omega}\exp\left(i\int_{t_{0}}^{t^{\prime}}\omega\differential{u}\right)$
(6)

with $\\{\varphi_{n}(t)\\}_{n\in\mathbb{N}}$ denoting the eigenstates of the
harmonic oscillator centred at $\xi(t)$. We can simplify the numerator using
equation 5:

$\displaystyle\matrixelement{\varphi_{1}(t^{\prime})}{\partialderivative{\hat{H}}{t^{\prime}}}{\varphi_{0}(t^{\prime})}$
$\displaystyle=-m\omega^{2}\dot{\xi}\matrixelement{\varphi_{1}(t^{\prime})}{\hat{x}-\xi(t^{\prime})}{\varphi_{0}(^{\prime}t)}$
$\displaystyle=-m\omega^{2}\dot{\xi}\matrixelement{\varphi_{1}(0)}{\hat{x}-\xi(0)}{\varphi_{0}(0)}$

Using the generating function for the Hermite polynomials (see equation
(4.174) of reference [19]), one has the following identity:

$\matrixelement{\varphi_{n}}{\hat{x}}{\varphi_{m}}=\begin{cases*}\sqrt{\frac{\hbar}{m\omega}}\sqrt{\frac{n}{2}}\text{
if }m=n-1\\\ \sqrt{\frac{\hbar}{m\omega}}\sqrt{\frac{n+1}{2}}\text{ if
}m=n+1\\\ 0\text{ else }\end{cases*}$ (7)

Hence:

$\displaystyle\matrixelement{\varphi_{1}(t^{\prime})}{\partialderivative{\hat{H}}{t^{\prime}}}{\varphi_{0}(t^{\prime})}$
$\displaystyle=-m\omega^{2}\dot{\xi}\sqrt{\frac{\hbar}{2m\omega}}$

We can then obtain a more explicit expression for the probability amplitude of
the first orbital excited state:

$\displaystyle c_{2}(t)$
$\displaystyle=\frac{-m\omega^{2}}{\hbar\omega}\sqrt{\frac{\hbar}{2m\omega}}\int_{t_{0}}^{t}\differential{t^{\prime}}\;\dot{\xi}(t^{\prime})\exp\quantity(i\omega(t^{\prime}-t_{0}))$
$\displaystyle=-\omega\sqrt{\frac{m}{2\hbar\omega}}\int_{t_{0}}^{t}\differential{t^{\prime}}\;\dot{\xi}(t^{\prime})\exp\quantity(i\omega(t^{\prime}-t_{0}))$

At this stage, it is useful to introduce the maximum speed of a quantum
harmonic oscillator in the ground state. Assuming that the total energy
corresponds to kinetic energy, we have
$\epsilon_{0}=\frac{\hbar\omega}{2}=\frac{1}{2}mv_{max}^{2}$, where
$v_{\text{max}}\equiv\sqrt{\frac{\hbar\omega}{m}}=\sqrt{\frac{\Delta\epsilon}{m}}$
the maximal speed. We can then rewrite the expression for $c_{2}(t)$ as:

$\displaystyle
c_{2}(t)=\frac{-\omega}{\sqrt{2}}\int_{t_{0}}^{t}\differential{t^{\prime}}\frac{\dot{\xi}(t^{\prime})}{v_{\text{max}}}\exp\quantity(i\omega(t^{\prime}-t_{0}))$
(8)

For $t_{0}\leqslant t\leqslant t_{1}$, $\dot{\xi}(t)=v_{QD}$ we can obtain the
transition probability from the modulus squared of $c_{2}(t)$:

$P_{2}(t)=2\left(\frac{v_{QD}}{v_{\text{max}}}\right)^{2}\left|\sin\left(\frac{\omega}{2}(t-t_{0})\right)\right|^{2}$
(9)

Hence, the state probabilities oscillate with an amplitude proportional to
$v_{QD}^{2}$ as observed in figure 2. The period of oscillations can be made
more apparent by rearranging the equation above:

$\displaystyle P_{2}(t)$
$\displaystyle=2\left(\frac{v_{QD}}{v_{\text{max}}}\right)^{2}\left|\sin\left(\frac{\omega}{2}(t-t_{0})\right)^{2}\right|$
$\displaystyle=\left(\frac{v_{QD}}{v_{\text{max}}}\right)^{2}\left|1-\cos\left(2\pi\frac{(t-t_{0})}{\frac{2\pi}{\omega}}\right)\right|$
$\displaystyle=\left(\frac{v_{QD}}{v_{\text{max}}}\right)^{2}\left|1-\cos\left(2\pi\frac{(t-t_{0})}{\tau}\right)\right|$

with $\tau=\frac{2\pi\hbar}{\Delta\epsilon}$ as previously defined. In the
context of our simulations, the orbital spacing $\Delta\epsilon=2$ meV gives
$\tau=2.1$ ps, which matches the period observed in figure 2.

In appendix B, a similar calculation is performed to estimate the probability
of the second excited orbital state $\psi_{2,-}=\psi_{4}$.

###  III.3 Criterion for the transition to the non-adiabatic regime

Strictly speaking, an adiabatic evolution implies that the probability of
being in eigenstate remains the same throughout the infinitely slow
perturbation. For instance, if the initial probability of the state $\psi_{2}$
is initially zero, then it should continue being zero at all future times. In
practice, it is convenient to adopt a less restrictive definition by comparing
how well an adiabatic approximation matches with the actual results.

Using equation 9 we can find an upper bound for the probability of the first
orbital in the adiabatic approximation, namely the maximum value of $P_{0\to
2}$ is given by

$\max_{t}P_{0\to 2}(t)=2m\frac{v_{QD}^{2}}{\Delta\epsilon}\quad.$ (10)

As the bound is directly proprtional to the square of $v_{QD}$ (the orbital
spacing $\Delta\epsilon$ is fixed to 2 meV), one can appreciate that the
physics is particularly sensitive to the quantum dot speed. This upper bound
is plotted in figure 4 (blue solid line).

As discussed in the previous subsection, and considering the results of figure
2, the value of the final probabilities depend on the time at which we look
for a fixed value of $v_{QD}$. As a consequence, we plot in figure 4 the the
maximum value of the probability of $P_{0\to 2}$ depending on $v_{QD}$, as
obtained from the simulations (red dashed line).

Figure 4:  Maximum value of the probability of the first excited orbital state
$\psi_{2}$ for different values of $v_{QD}$. The blue solid line is the upper
bound of equation 10 derived in the adiabatic approximation. The red dashed
line correspond to the maximum over time of the probability of observing the
state $\psi_{2}$ in our numerical simulations. The dotted yellow line gives an
indication of how well the adiabatic approximation agrees with the numerical
results. For small values of $v_{QD}$, the upper bound is a very good
approximation, but the error widens at $v_{QD}$ increases, reflecting the
smooth transition to the non-adiabatic regime.

We remark that the numerical data matches the adiabatic approximation for low
$v_{QD}$, i.e when the adiabatic approximation is valid. However, as $v_{QD}$
increases, the adiabatic approximation, utilised to derive the analytic
expression, becomes invalid and the difference between the upper bound and the
numerical results widens as expected. Indeed, this can easily be interpreted
by looking at the results of figure 2 where we see higher orbital states
appearing as $v_{QD}$ increases. As a consequence, some of the probability
leaks to these higher orbital states, explaining why the results diverge. The
yellow dotted line gives an estimation of how “non-adiabatic” the system is.
Low values testify to an adiabatic system for the orbitals, while values above
a certain threshold (arbitraly chosen) indicate a non-adiabatic regime.

To conclude this section, we would like to emphasise that the plot of figure 4
only considers the orbital degrees of freedom, as the valley degrees of
freedom do not play a role in the ideal model. Still, figure 4 will prove
useful for analysing the results involving the valley degrees of freedom of
the step model, discussed in section V.

##  IV Spectrum in the step model

We now perform the same analysis but with the step model. Critically, the
perturbation now couples the orbitals with the valley states of the electron.

General remarks on the spectrum.  We now discuss the salient features of the
spectrum of the step model as the minimum of the quantum dot potential is
varied. The results are plotted in figure 5 for different electric fields
$E_{z}$. To simplify notations, we continue to label the energies with the
indices $(n,v)\in\mathbb{N}\times\quantity{-,+}$, but one should be reminded
that near the step, the strong valley-orbit coupling prevents $(n,v)$ from
being good quantum numbers. Note that far away from the steps, the wave
functions are well defined valley states, and the indices $(n,v)$ are still
good quantum numbers.

Figure 5:  Spatial evolution of the spectra of the single step model for
different electric fields $E_{z}$. In this figure, the reference energy has
been set to 0 meV to increase readability. An anti-crossing opens after the
step located at $x_{s}=30$ nm, and can be understood by the probability
densities of the two lowest states. The numerical results show that the energy
gap at the anti-crossing can be tuned by the electric field $E_{z}$ applied
perpendicular to the SiGe interfaces (see figure 1). Note also the global
increase of the energies as the electron is confined to a tighter region, as
we would expect.

Origin of the anti-crossing.  A critical observation is the existence of an
anti-crossing near the step location (at $x_{s}=30$ nm). As mentioned in the
previous paragraph, the step causes the coupling of the valley and orbital
degrees of freedom. In the absence of disorder at the well interface there
exists globally defined valley states, where the valley states are
translationally invariant, however, the step potential explicitly breaks this
symmetry resulting in the observed anti-crossing.

The physics behind this avoided crossing can be understood by examining the
probability density of the two lowest energetic states, and is well described
in Boross  _et al._. Indeed, as shown in both atomistic simulations [13, 12]
and effective mass theories [20, 5], the wave functions exhibit modulations on
the atomic scale of the envelope function. The two lowest $k_{z}$ valley
states are symmetric and anti-symmetric superpositions of $\pm k_{0}$ with
$k_{0}\simeq 0.82\frac{2\pi}{a}$, where $a=5.41$ Å is the lattice spacing of
the silicon crystal. The atomic scale oscillations of the two valley states
are similar except for a spatial de-phasing. A consequence of this spatial de-
phasing is that one of the state will have a maxima at the step location,
while the other will have a minima. Hence, as the quantum dot goes through the
step, the lowest energetic state will feel the step, increasing its energy,
while the other will not.

Control of the anti-crossing gap with electric field.  As shown in figure 5,
the energy gap at the anti-crossing of the two lowest valley states can be
electrically controlled through the applied electric field $E_{z}$. The
dependence of the gap with the electric field is non-trivial and also depends
on the quantum well width, which we have fixed to $N_{z}=40$ atoms in this
paper. As discussed in later sections, the values of $E_{z}$ for which the
energy gap of the anti-crossing is small will prove particularly useful for
electrical control of the final state. One could question whether this
behaviour is expected in real devices, or if it is an artifact of the two-band
model we are using to model the valleys. In appendix A, NEMO 3D calculations
using a 20 band $sp^{3}d^{5}s^{*}$ model show that the gap at the anti-
crossing is indeed controllable with the electric field $E_{z}$. Akin to the
results of the two-band model, the anti-crossing gap varies in the same non-
trivial way, but it seems the trend can be reproduced only for certain quantum
well widths. As a result, we expect the control of the anti-crossing gap to be
feasible in real devices.

Higher orbital anti-crossing. We notice that a similar anti-crossing happens
between valley states of higher orbitals too (states of energies
$\epsilon_{1,-}$ and $\epsilon_{1,+}$). However, this anti-crossing happens
before that of the ground state orbital. This makes sense since the higher
orbital states of the harmonic oscillators have a wider spatial extent over
the $x$ position as the orbital number $n$ increases. In turn, the step is
felt earlier, explaining the earlier appearance of the anti-crossing.

Figure 6:  Evolution of the probabilities at $v_{QD}=1$ nm/ps for different
electric fields $E_{z}$. The probabilities of higher energy states were at
most 0.25 %, so they have been omitted here for readability. To a good
approximation $v_{QD}=1$ nm/ps results in a two-level system (see figure 4)
and can form the basis states of a qubit. Through the applied electric field
$E_{z}$, the energy gap of the anti-crossing can be tuned, which results in
the control of the valley states.

##  V Non-adiabatic effects in the step model

###  V.1 General comments

Results at a fixed low quantum dot speed. The movement of the quantum dot
potential has the effect of bringing energy into the system. This additional
energy allows the ground state to overcome the anti-crossing gap observed in
the spectrum. As intuitively expected, the smaller the gap between the two
lowest eigenstates, for the same perturbation the stronger the probability of
transitioning to other valley states. This is illustrated well in figure 6 for
$v_{QD}=1$ nm/ps where the value of $v_{QD}$ is small enough that the system
is mostly confined to the two valleys of the first orbital state (this can be
verified by examining the yellow dashed line of figure 4, showing that the
perturbation is adiabatic for the orbitals). For this fixed quantum dot speed,
the variation of the applied electric field is the only parameter used to tune
the final probabilities of the valley states. Comparing the gaps in the
spectra of figure 5 and the final probabilities of figure 6, one can relate
the final probabilities to the tuning of the anti-crossing gap.

Figure 7:  Probability of the state $\psi_{0,+}$. This allows to find the
combination of parameters $(v_{QD},E_{z})$ to reach a specific final
probability for $\psi_{0,+}$. For the quantum well width we chose, the anti-
crossing gap is the smallest at values of $E_{z}\simeq 6$ mV/nm. This explains
the high transition probabilities observed for this range of electric fields.
For high values of $v_{QD}$, transition to higher orbital states take place,
which explains the reduction of the probability occurring around $v_{QD}=9$
nm/ps.

Results for different combinations of quantum dot speeds and electric fields.
In the ideal model, the final probabilities were tuned through $v_{QD}$ (see
figure 4). In the step model too, the quantum dot speed $v_{QD}$ can be used
to tune the final probabilities. In figure 7, we plot the probability of the
state $\psi_{0,+}$ for a combination of electric fields $E_{z}$ and quantum
dot speeds $v_{QD}$. As expected from the spectra of figure 5, the “sweet
spot” for transitioning to the state $\psi_{0,+}$ corresponds to small anti-
crossing gaps, which happens in a range of electric fields
$E_{z}\in\quantity[4,6]$ mV/nm. One can also remark that increasing the
quantum dot speed increases the transition probability for relatively small
values of $v_{QD}$. For higher values of $v_{QD}$ however, the transition
probability actually decreases with increasing $v_{QD}$, since transition to
higher orbital states starts to take place instead. This is easily interpreted
from figure 4, which testifies to the presence of higher orbital states.

Higher orbital valley flipping. The spectra of figure 5 show that an anti-
crossing exists between the states of the first excited orbital. Since the
transition between valley states is due to the anti-crossing, it is legitimate
to expect valley flipping behaviour for the first excited orbital. Still,
according to the spectrum, this valley exchange should happen before that of
the ground orbital states, as the anti-crossing happens at an earlier position
as explained in section IV. This behaviour is indeed verified at higher
quantum dot speeds as shown in figure 8.

Figure 8:  In this figure, $v_{QD}=9$ nm/ps and $E_{z}=$ 6 mV/nm. We can
observe an exchange of valley states for the orbital ($n=1$), due to an anti-
crossing in the spectrum happening earlier than that of the ground orbital
($n=0$). Another interesting feature is the change in both the amplitude and
the period of the oscillations after the valley flipping occurs.

###  V.2 Landau-Zener approximation

Standard Landau-Zener formula and anti-crossings. The probability of non-
adiabatic transitions in a two-level system with an avoided crossing has been
heavily studied, see reference [11] for instance. In particular, the Landau-
Zener formula can be used to estimate the transition probabilities due to non-
adiabatic effects. The benefit of such an analysis is that one can simply
compute the final state probabilities without solving the full time-dependent
Schrödinger equation.

Under the Landau-Zener approximation, the energy separation between the
uncoupled states is a linear function of time with gradient $\alpha$, which is
assumed to extend over all time. Furthermore, we denote the coupling induced
by the step potential as $W$ and also assume it is constant in time. Knowing
this, we can define

$\Gamma=\frac{W^{2}}{\hbar|\alpha|}$ (11)

where the probability of transitioning to the excited state is given by the
formula

$P_{LZ}=e^{-2\pi\Gamma}$ (12)

Figure 9:  Result of the Landau-Zener approximation. (a) The simulated
transition probabilities in the step model. (b) Transition probabilities under
the Landau-Zener approximation in the step model. We can observe a remarkable
agreement between our numerical results and those obtained from the Landau-
Zener approximation.

Fit to Landau-Zener. In a realistic model the perturbation is only linear over
a finite range of time, as our numerical results for the spectra show in
figure 5. An analytical modelling of our spectra is complicated, as the
expression of the wave functions is not analytical in a quantum well with an
applied electric field. As a consequence, we take the approach of fitting our
numerical results to a simpler model. Luckily, such a simpler model is
provided by Rubbmark  _et al._ [11]. Hence we fit the energy level separation
between the two lowest energy states ($\epsilon_{0,+}-\epsilon_{0,-}$) to
equation (18) of reference [11] reproduced below ($\alpha$ and $\tau$ are the
fitting parameters). The resulting transition probabilities are plotted in
figure 9.b for different values of $v_{QD}$ and $E_{z}$.

$\epsilon_{0,+}(t)-\epsilon_{0,-}(t)=\alpha\tau\quantity(\frac{1}{1+\exp\quantity(-4t/\tau)}-\frac{1}{2})$
(13)

A surprising result is that the evolution of the transition probabilities is
well captured by the Landau-Zener fit for almost all electric field values and
the quantum dot speeds. In figure 9.a, the simulated transition probabilities
$1-P_{0,-}$ are plotted. The Landau-Zener approximation in figure 9.b shows
the same evolution as that of our numerical results. In figure 10, we plot the
difference between the simulated transition probabilities and the ones
obtained from the Landau-Zener approximation $P_{\text{LZ}}$. This illustrates
that the Landau-Zener approximation deviates only slightly from our numerical
results. However, one should note that for large $v_{QD}$ one would certainly
observe transitions to higher orbitals and valley states whilst Landau-Zener
only considers the transition to the other state in the two-level system.
Regardless, the maximum value of the relative error across all values of
electric field and quantum dot speed was only 1.0% (not shown in figure 10),
and hence it provides a good measure of weather the electron is in the ground
state or not - with perhaps no qualification on which state is has
transitioned to.

##  VI Qubit application

Validity of the two-level system and qubit application. The choice of the wide
energy level spacing $\Delta\epsilon=\hbar\omega=2$ meV for the orbital states
allows one to prevent transitions to higher orbital states, thus allowing for
a two level valley system to form. However large values of $v_{QD}$ can still
lead to leakage to higher orbitals, and the electric field $E_{z}$ also plays
a complex role. Regardless, there does exist a range of parameters for which
the probability of higher orbital states is negligible and a set of qubit
basis states is realised. We can estimate the range of parameters for which
the final state is a two-level system numerically by summing the probabilities
of the two states $\psi_{0,-}$ and $\psi_{0,+}$ forming our preferred two-
level system. Using the two lowest energy states as a computational basis, our
model effectively describes the time-evolution of a pure valley qubit with
all-electrical control.

Figure 10:  Difference between the simulated transition probabilities and
those obtained under the Landau-Zener approximation. We can see that the
Landau-Zener approximation gives good agreement with our numerical results.
The maximum value of the relative error was 1.0 %, not shown in this figure.

Coherence time and protection from charge noise. The use of the valley degrees
of freedom to encode the quantum information is interesting in that it
provides certain immunity to charge noise [21]. Charge noise is one of the
leading sources of decoherence of silicon qubits [22, 21, 23, 24]. The
resistance to charge noise has been experimentally verified through Laudau-
Zener interferometry in a Si/SiGe double quantum dot by Mi  _et al._ [25].
Previous implementations of qubits leveraging valley states in double quantum
dots have shown a coherence time on the scale of nanoseconds [26, 27, 7, 8].
Although those qubits were implemented in double quantum dots, whilst our
system is a single quantum dot, we expect the coherence time of our
architecture to have a similar order of magnitude.

Experimental advantage.  The all-electrical control of the final state
probabilities is critical for scalability, as one would not need magnetic
fields for qubit manipulation as in most spin qubit implementations [23, 24].
The absence of applied magnetic field, and hence microwave lines, makes the
integration of such a system significantly more simple. A second advantage is
that we are not constrained by any adiabatic condition, as a consequence,
provided the decoherence times are similar to that of other hybrid valley
qubits, we can perform orders of magnitude more gate operations.

Coupling, readout and other issues.  Serious difficulties are to be overcome
for this scheme to be usable in practice. First, engineering the electrical
pulses used in our simulation may be challenging to implement. Indeed, the
engineering required to control the spatial evolution of the quantum dot
potential, i.e the trajectory $\xi$, may be complicated due to the time scales
involved. As argued by Boross  _et al._ , it may be somewhat achievable using
a double quantum dot architecture and varying the barrier and detuning to move
the electron through the step [28, 2]. This will inevitably be complicated by
the additional gates needed to apply the electric field $E_{z}$, due to the
existence of cross couplings. On the other hand, we have been limiting
ourselves to constant speeds for $\xi$, and hence more complicated
trajectories are unexplored and may be fruitful.

Finally, both readout and coupling to other such qubits has not been explored
and may pose a significant technical challenge for the scalability of the
proposed scheme. A potential avenue towards readout could be implemented with
a form of charge sensing, as is done in most silicon qubits [24]. This would
require the incorporation of another dot after the step, and adjusting the
energy levels so that only the highest energy valley states tunnels to the
second dot.

Figure 11:  Evolution of the probability of being in a two-level system based
on the two lowest valley states $\psi_{0,-}$ and $\psi_{0,+}$. The bright
yellow/orange regions give a range of parameters for which the final state is
a good two-level system suitable for qubit applications.

##  VII Conclusion

To summarise our work, we have shown that displacing a quantum dot potential
through an interface step leads to an anti-crossing in the spectrum of the two
lowest energetic states. A critical observation, which was verified with
tight-binding calculations of NEMO 3D, is that the energy gap at the anti-
crossing can be controlled by applying an electric field perpendicular to the
interface. Since the transition probabilities depend on the energy gap, and
the latter is electrically controlled, we can achieve all-electrical control
of the final state probabilities. There is a range of parameters for which the
final state behaves as a charge qubit encoded on valley states, though readout
and coupling have not been explored.

There are two main limitations of our results. The first is the qualitative
nature due to our highly idealized modelling. Indeed, we enforced hardwall
boundary conditions for the $z$ confinement, so that the wavefunction does not
penetrate through the SiGe layers, while a more realistic choice would be
adding a finite height barrier of around 150 meV. We also neglected both the
strain and alloy disorder of the Ge atoms. We also note that our work is only
valid in the low density limit hence ignoring both Coulomb and exchange
effects. Despite these remarks, the results obtained from NEMO 3D indicate
that the physics discussed should still be applicable to real devices, with
only qualitative modifications. Finally, the precise engineering of both the
device geometry and the applied voltages for adequate control poses additional
experimental challenges.

The second limitation is due to the experimental feasibility, namely the
precise control of the fast electrical pulses to move the quantum dot and the
existence of a single atomic step. Indeed, we expect multiple steps to form
over the spatial range chosen for our simulations. We have also neglected
relaxation mechanism, as these are supposed to occur on a timescale greater
than the operation times.

Nevertheless, non-adiabatic control of valley states may open novel quantum
information processing schemes. In this work, we have limited ourselves to the
study of a single quantum dot, but we suspect that the results could be
adaptable to more complicated double dot systems making the possible
implementation of these ideas more realistic. Additionally, in some emerging
2D and topological materials, spin and valleys are strongly momentum locked,
thus fast manipulation of valleys could lead to non-adiabatic control of
spins.

###### Acknowledgements.

We acknowledge useful discussions with Edyta Osika concerning alternative
simulation methods, and Vyacheslavs Kashcheyevs concerning the anti-crossing.
This work is was partly funded under the Nicolas Baudin initiative and Thales
Australia. Associate Professor Rajib Rahman acknowledges support from the US
Army Research Office under grant number W911NF-17-1-0202. The research was
undertaken with the assistance of re-sources and services from the National
Computational Infrastructure (NCI) under NCMAS 2020 and 2021 allo-cation,
supported by the Australian Government. This work was also supported with
supercomputing resources provided by the Phoenix HPC service at the University
of Adelaide.

##  Appendix A  Validation of the model

Objective.  The Boykin  _et al._ two-band tight-binding model has been
verified in a Si/SiGe quantum well with flat hardwall boundary conditions [12]
but it has not yet been used to model a quantum well with a monatomic step
interface. Most results discussed in the main text depend on two critical
facts: the existence of an anti-crossing between the two lowest valley states,
and the relative control of the anti-crossing gap with the electric field
$E_{z}$. In this appendix, we compare the results obtained from the step model
with those obtained with NEMO 3D’s 20 band $sp^{3}d^{5}s^{*}$ model [10].

NEMO model.  In our step model of subsection II.2, we used hardwall boundary
conditions, which corresponds to an infinite barrier height. To model this in
NEMO 3D, we utilised a SiO2 interface which has a large barrier height of $3$
eV. To reduce simulation time, we also adopted a tighter confinement for the
quantum dot. The curvature of the quantum dot was set to $k=10^{-1}$ meV/nm2
in all results presented in this appendix.

Figure 12:  Evolution of the spectrum with the location of the quantum dot
relative to the step at $0$ nm. The anti-crossing can happen either before (a)
or after (b) the step, a fact captured in both our step model (solid lines)
and NEMO 3D’s 20 band $sp^{3}d^{5}s^{*}$ model (dotted lines). In (a), the
quantum well width is 16.5 monolayers for the solid lines and 18 monolayers
for NEMO 3D. In (b), the quantum well width is 21 monolayers for the solid
lines, and 20 monolayers for NEMO 3D. The applied electric field is $E_{z}=10$
mV/nm for all the curves in this figure.

Limitations.  In their paper, Boykin  _et al._ showed that the two-band model
they proposed correctly reproduced the trends expected for the valley
splitting evolution with the width of the quantum well. In particular, the
valley splittings have the correct order of magnitude but differ from NEMO 3D
calculations. As such, their model is qualitative and not quantitative, and we
expect to observe the same differences with our model. Similarly, we shall
focus on only verifying that the critical trends necessary for our
conclusions, namely the anti-crossing and the control of the energy gap, are
mirrored by NEMO 3D.

Results.  In figure 12, we compare spectra obtained using our step model
(solid lines) and those obtained from NEMO 3D calculations (dotted lines). The
reference energy was set to 0 to facilitate the comparison of both spectra. We
remark that in both simulations, the location of the anti-crossing can happen
before or after the step depending on the quantum well width. In both cases,
the existence of the anti-crossing is verified by NEMO 3D.

Similarly, the tuning of the anti-crossing gap with the electric field is
observed in both simulations, as shown in figure 13. As mentioned in the main
text, the tuning of the gap with the electric field depends on the quantum
well width. Some key features are exhibited by both our step model and the
NEMO 3D calculations. Namely, the range of the tuning of the gap, the
existence of a minimum, and the location of this minimum all depend on the
quantum well width.

##  Appendix B Estimation of the probability of the second excited orbital
state in the ideal model

In this appendix we estimate the form of the probability $P_{4}$ of the state
$\psi_{4}=\psi_{2,-}$ in the ideal model. We adopt the notations of section
III.2. From equation (4.174) and (9.97) of reference [19], we can estimate
that the probability amplitude of the second excited state $c_{4}(t)$ is
related to the amplitude of the first excited state $c_{2}(t)$ by:

$\dot{c_{4}}(t)=\frac{c_{2}(t)}{\hbar\omega}\matrixelement{\varphi_{2}(t)}{\partialderivative{\hat{H}}{t}}{\varphi_{1}(t)}e^{i\omega(t-t_{0})}$

Following equation 7, we find that
$\matrixelement{\varphi_{2}(t)}{\partialderivative{\hat{H}}{t}}{\varphi_{1}(t)}=-m\omega^{2}\dot{\xi}\sqrt{\frac{\hbar}{m\omega}}$.
Using equation 8 we find:

$\dot{c_{2}}(t)=-i\omega\left(\frac{v_{QD}}{v_{\text{max}}}\right)^{2}\left(e^{2i\omega(t-t_{0})}-e^{i\omega(t-t_{0})}\right)$

The integration is straightforward and gives:

$c_{2}(t)=2\left(\frac{v_{QD}}{v_{\text{max}}}\right)^{2}\sin\left(\frac{\omega}{2}(t-t_{0})\right)^{2}e^{i\omega(t-t_{0})}$

Finally, the transition probability to the second excited state can be
computed as:

$\displaystyle P_{0\to 4}(t)$
$\displaystyle=4\left(\frac{v_{QD}}{v_{\text{max}}}\right)^{4}\left|\sin\left(\frac{\omega}{2}(t-t_{0})\right)\right|^{4}$
(14) $\displaystyle=P_{0\to 2}(t)^{2}$

Comparing equation 9 and equation 14 we see that the probability of the
electron being in the state $\psi_{4}$ at time $t$ is the square of it being
in the state $\psi_{2}$.

Figure 13:  Evolution of the anti-crossing energy gap with the electric field
in the step model (solid dotted blue) and NEMO simulations (dashed circled
red). Depending on the quantum well width, the magnitude and location of the
minimum of the gap can change. (a) In the step model, we used 16.5 monolayers,
while 19 monolayers were used for NEMO 3D. (b) In the step model, we used 19
monolayers, while 20 monolayers were used for NEMO 3D.

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