# Modelling assisted tunnelling on the Bloch sphere using the Quantum Composer

Jonas Bley1, Vieri Mattei2, Simon Goorney3, Jacob Sherson3, Stefan Heusler
4,∗,† 1 Rheinland-Pfälzische Technische Universität Kaiserslautern-Landau,
Germany  2 Purdue Department of Physics and Astronomy, West Lafayette,
Indiana, US  3 Niels Bohr Institute, University of Copenhagen, Denmark  4
Institut für Didaktik der Physik, Westfälische-Wilhems-Universität Münster,
Germany  ∗ Correspondence<EMAIL_ADDRESS>† Current address:
Institut für Didaktik der Physik, Wilhelm-Klemm-Str. 10, 48149 Münster,
Germany

###### Abstract

The Bloch sphere representation is a geometric model for all possible quantum
states of a two-level system that can be used to describe the time dynamics of
a qubit. As explicit application, we consider the time dynamics of a particle
in a double-well potential. In particular, we adopt a recent method for off-
resonant excitations, the so-called SUPER principle (Swing-UP of the quantum
emitter population) driven by periodic electromagnetic fields, to the context
of quantum tunnelling. We show that the tunnelling probability can be enhanced
significantly when an appropriate oscillation of the potential height is
introduced. Driven by a collaborative approach we call educator-developer
dialogue, an updated version of the software Quantum Composer is presented.
For educational purposes, we map the two lowest energy states of the
1D-Schrödinger equation to the Bloch sphere representation, leading to a
rather clear and intuitive physical picture for the pertinent time dynamics.

Keywords: Multiple representations, Bloch sphere, qubits, tunnelling, quantum
physics, SUPER principle, Simulation tool

##  1 Introduction

With the advent of quantum technologies, the importance of two-level systems,
also denoted as qubits, is increasing not only from a technological, but also
from a didactical perspective. In many standard textbooks [1], the topic is
addressed in a rather formal manner. Generally speaking, multiple
representations are an important tool in science education [2]. This holds
true in particular in quantum physics, as the basic concepts are rather
abstract and complex. Using the software tool Quantum Composer [3], the
mathematical formalism comes to life by interactive graphical visualisations.

In this contribution, as an illustration, we discuss tunnelling in a double
well as an explicit example. In particular, using an update of the Quantum
Composer, we apply the so-called Bloch-sphere representation to the double-
well system. The (single-particle) quantum double-well system is of great
importance to many areas of quantum physics as it enables the implementation
of a two-level quantum system, i.e. a qubit. Tunnelling in the double-well
potential is extensively explored for the symmetric case [4] and the
asymmetric (tilted) case [5]. Periodically driven potentials as examined in
this paper are usually described using the Floquet formalism [6, 7, 8, 9].

The Bloch-sphere representation is of particular importance, as it can be
applied to all two-level systems, independent of the particular physical
interpretation. A geometric model for the time dynamics of qubits on the Bloch
sphere is – for time-independent Hamiltonians – well known: Just choose a
rotation axis ${\vec{n}}$, which amounts to representing the $2\times 2$
Hamiltonian as $H=\frac{\vartheta}{2}{\vec{\sigma}}\cdot{\vec{n}}$. Then, the
time dynamics of any initial state
$|\psi\rangle=\alpha|0\rangle+\beta|1\rangle$ on the Bloch sphere translates
to a rotation of this state around the axis ${\vec{n}}$ on the Bloch sphere.
There are two eigenstates, namely those parallel/anti-parallel to ${\vec{n}}$.
This geometric picture is universal in the sense that it is independent of the
explicit physical interpretation of the degrees of freedom described by the
Hamiltonian.

A relevant concept to the time dynamics of two-level systems are so-called
Rabi oscillations [10]. These oscillations describe the behaviour of a two-
level quantum systems with an external periodic driving force. This time-
dependent driving force leads to an effective coupling between the two states.
If the driving force is in resonance with the two-level system, the Rabi
oscillations are full, i.e. the particle will oscillate between the two levels
completely. Otherwise, the particle will only partially reach the other level.
We emphasise that throughout this paper, the term ”Rabi oscillations” is used
metaphorically as we don’t explore systems with time-dependent external
driving. Rather, in the double well potential, a time-independent coupling
between the states localised in the left and the right well is present due to
the tunnelling, which can only be changed by changing the shape of the
potential. Due to the formal equivalence of the time-indepedent and the time-
dependent situation, we adopt the notion of ”Rabi oscillations” also to the
case of tunnelling.

Multiple representations introduced for Rabi oscillations can be useful for
undergraduate courses on quantum physics. We provide explicit animations and
quantum composer files for full Rabi oscillation (Level 1) and partial Rabi
oscillation (Level 2). Next, for graduate level and beyond, we consider a time
dependent double-well potential (Level 3).

This paper is organised as follows: In Sect. 2, the Quantum Composer by
Quatomic is introduced. In Sect. 3, we review the well-known general theory
for Rabi oscillations of two-level systems. We discuss the usual co-moving
frame approach and contrast it with the situation of time-independent
Hamiltonians, where this transformation is not needed. In Sect. 4, the tilted
double-well potential is introduced as an explicit model system for a time
independent Hamiltonian. We compare analytic results for the degenerate case
with numerics from the Quantum Composer (Level 1). Full Rabi oscillations,
that is, full oscillation between the left and the right minimum of the
double-well potential emerge in this case of a symmetrical double well. In the
tilted potential, we have partial oscillation between the left and the right
minimum of the double-well. In Sect. 5, we introduce the so-called SUPER
principle on the Bloch-sphere, which generalises the simple model for the time
dynamics of a qubit to a situation where the rotation axis ${\vec{n}}(t)$ on
the Bloch sphere itself is a certain function of time, allowing for the
transition from a given initial state to a given final state with almost 100%
probability by adjusting ${\vec{n}}(t)$ accordingly. In Sect. 5, we apply the
SUPER principle, which has been introduced in the context of off-resonant
excitations [11], [12] to the situation of quantum tunnelling in the tilted
double well as model system, which is the main result of the present
contribution (Level 3). Note that in contrast to [11], we do not use the co-
moving frame ansatz. In Sect. 6, we discuss further extentions and
applications of this work, and its possible didactical merits.

##  2 The Quantum Composer

In this work, we will use the digital tool Quantum Composer for numerical
simulation of the mechanics of a tilted double well potential. Composer was
developed in the group of Prof. Sherson at Aarhus University [13]. Unlike many
other fixed-interface applet approaches [14, 15, 16], which are limited to
showcasing particular sets of phenomena, Composer provides an intuitive and
user-friendly re-programmable, node-based graphical interface, as shown in
Fig. 1. These nodes can be connected to construct quantum systems which are
simulated by the underlying QEngine, a C++ library for the simulation and
control of one-dimensional quantum systems [17]. These systems under
investigation can be generated completely from scratch, or take as a basis
pre-defined scenarios loaded into the program using .flow files. This
flexibility makes the tool suitable for both education (use by teaching staff
and students), and for research purposes [18, 3, 19], with a low barrier to
entry due to the drag-and-drop nature of the programming. The Quantum Composer
can be downloaded for free for most desktop platforms [13].

Figure 1: An illustration of the drag-and-drop, node-based, graphical user
interface of Quantum Composer.

Due to its hands-on re-programmability, Quantum Composer is uniquely suited to
inquiry-based learning (IBL)[20], in which students interact with didactic
material through active discovery, questioning, evaluation, and sensemaking.
When used for teaching quantum mechanics, IBL has shown potential for
improving understanding and reducing overconfidence [21, 22]. Despite this,
there are relatively few examples of publicly available resources for inquiry-
based quantum mechanics teaching.

There are many simulation and visualisation tools [23] already available that
can be invaluable in promoting understanding, interest, and intuition for
quantum concepts [24]. However, those which are most widely used for teaching,
such as QuVis [14] from and QuILTs [25] offer a fixed set of scenarios
covering different phenomena and allow students to vary parameters. While
these are effective to enable educators to set up guided and partially
inquiry-based teaching situations, they do not allow for truly open and
exploratory inquiry, in which the system is entirely set up from scratch by
the participants. Thus, such tools may miss out on some of the benefits
associated with more open-ended IBL in understanding and promoting scientific
thinking [26]. Composer, however, being entirely reprogrammable, allows for
teaching material to be developed which can be anywhere on the spectrum of
fully guided to completely open-ended and exploratory. An example IBL approach
was previously explored using Quantum Composer to devise student exercises
based on the Predict, Explore and Revise approach [27].

For researchers, Composer may be useful as a relatively quick and simple means
to set up, visualise, and solve analytically intractable quantum systems. The
drag-and-drop interface and parameters which can be adjusted on the fly makes
it a valuable alternative to more conventional approaches taken by
researchers, such as the use of MATLAB or Python code which can be less
intuitive to use than the node-based UI of Composer [18, 3]. The tool has
previously been used to investigate the dynamics of atoms in atomic lattices
[28] and to solve quantum optimal control problems [17, 19].

In the context of this work, however, we wish to highlight another aspect of
the Composer re-programmability: the ability of teachers and instructors to
seamlessly create custom demonstrators to suit the particular student needs.
Here, instructors in advance prepare the relevant .flow files and either use
them as lecture demonstrators or distribute them to the students with suitably
highlighted experimentation parameters and visualisations preconfigured. In
addition, the Composer design team is continually available to support the
addition of new interaction and visualisation features. The current work
illustrates exactly such interactive collaboration between two instructors
(J.B. and S.H.) with the Composer design team resulting in the addition of new
features to Composer.

In addition, the Composer design team is continually available to support the
addition of new interaction and visualisation features. The current work
illustrates a form of collaboration between educators (J.B, S.H), and the
Composer developers (V.M, S.G, and J.S) which we call educator-developer
dialogue. In this interactive approach to co-design of educational material,
the digital tool is able to act as an instrument to respond to community
needs.

There are two possible avenues for this dialogue. The first consists of
educators designing material with close input from the developers of the tool.
This can be invaluable in identifying and preparing validation criteria, as in
reference [3]. The second mechanism, which is demonstrated in this article, is
where developers implement new features responding to the needs of the
educators, in this case the Bloch-sphere representation. Here we demonstrate
the value of Composer by extending the body of existing educational material,
using the example of Rabi oscillations in the double well potential. Quantum
Composer files are provided for three different levels: full Rabi oscillations
(Sect. 4.1), partial Rabi oscillations (Sect. 4.2) and Swing Up with the SUPER
principle (Sect. 5). The corresponding files can be downloaded from [29]. By
combining the Bloch sphere representation with the representation of wave
functions in the double-well, Composer helps us to develop an intuitive
physical picture of the time dynamics in the time-dependent double well using
multiple representations. This is the first application of an update of the
Quantum Composer incorporating the Bloch sphere representation, driven by an
educator-developer dialogue.

##  3 A brief review of the general theory of Rabi oscillations

In this section, we briefly review the general theory of Rabi oscillations,
which holds for many different applications. Crucially, we distinguish between
two-level systems driven by periodic fields, and those that are not driven by
periodic fields.

###  3.1 Two-level systems driven by periodic fields

We start our discussion with the usual textbook example of a two-level systems
driven by a periodic field. The unperturbed states are defined as $|g\rangle$
(ground state) and $|e\rangle$ (excited state) at energies $E_{g}$, $E_{e}$.
Let $\Delta e^{i\Omega t}$ be the interaction between the driving field and
the two-level system that introduces a coupling between the two eigenstates,
where $\Omega$ is the driving frequency of the periodic field and $A$ the
(complex) amplitude. This system is described by the following Hamiltonian

$\displaystyle H_{1}(t)=E_{g}|g\rangle\langle g|+E_{e}|e\rangle\langle
e|+\frac{1}{2}Ae^{i\Omega t}|g\rangle\langle e|+\frac{1}{2}A^{*}e^{-i\Omega
t}|e\rangle\langle g|.$ (1)

In matrix notation, we obtain (with mean energy $\bar{E}=(E_{e}+E_{g})/2$ and
energy gap $\eta\equiv E_{e}-E_{g}$ between excited and ground state)

$\displaystyle H_{1}(t)=\left[{\begin{array}[]{cc}E_{g}&\frac{1}{2}Ae^{i\Omega
t}\\\ \frac{1}{2}A^{*}e^{-i\Omega t}&E_{e}\\\ \end{array}}\right]=\bar{E}\cdot
I+\frac{1}{2}\left[{\begin{array}[]{cc}\eta&Ae^{i\Omega t}\\\ A^{*}e^{-i\Omega
t}&-\eta\\\ \end{array}}\right]$ (6)

where $I$ is the unit matrix. The Schrödinger equation describing the dynamics
of the system is given by
$i\hbar\partial_{t}|\psi\rangle=H_{1}(t)|\psi\rangle$. An elegant way to
compute the time dynamics of the state
$|\psi(t)\rangle=z_{g}(t)|g\rangle+z_{e}(t)|e\rangle$ is the transformation to
the so-called co-moving frame: Using the unitary operation
$C(t)=\exp{[-i(\Omega t/2)\sigma_{3}]}$, where $\sigma_{3}$ is the third Pauli
matrix, we define by $|\phi\rangle\equiv C(t)|\psi\rangle$ the state in the
co-moving frame. Within the co-moving frame, the time-dependence of the state
$|\phi(t)\rangle$ is then given by

$\displaystyle
i\hbar\partial_{t}|\phi\rangle=(C(t)H_{1}(t)C^{\dagger}(t)-i\hbar
C(t)\partial_{t}C^{\dagger}(t))|\phi\rangle\equiv H_{\rm CM}|\phi\rangle$ (7)

with

$\displaystyle H_{\rm
CM}=\bar{E}+\frac{1}{2}\left[{\begin{array}[]{cc}\eta-\hbar\Omega&A\\\
A^{*}&-\eta+\hbar\Omega\\\ \end{array}}\right].$ (10)

In the co-moving frame, the Hamiltonian $H_{\rm CM}$ becomes time-independent,
while the diagonal terms $\pm\eta$ are shifted to $\pm(\eta-\hbar\Omega)$.
That is, the time dependence $\Delta e^{i\Omega t}$ of the interaction is
transformed to a change in the energy shift in the co-moving frame. Resonance
emerges for $\eta\equiv E_{e}-E_{g}=\hbar\Omega$ which can then be interpreted
as absorption or emission of a photon in the transition between the states
$|e\rangle\leftrightarrow|g\rangle$.

###  3.2 Two-level systems without external driving by periodic fields

In contrast to the textbook example for Rabi oscillations induced by external
periodic fields, in this contribution, we consider two-level systems without
external driving. In other words, the Hamiltonian is time-independent from the
very beginning, and no transformation to the co-moving frame is necessary in
order to achieve this. Due to the formal equivalence between (6) and (10), all
mathematical results can be adopted, while the interpretation changes
significantly. Note that in our approach to tunnelling, we therefore do not
work in the co-moving frame.

Consider a time-independent two-level system with states $|L\rangle$ and
$|R\rangle$ at energies $E_{L}$, $E_{R}$ with mean energy $\bar{E}$ and energy
difference $\delta=E_{L}-E_{R}$. Let $\Delta$ be the (real) interaction
between the two states (which will later on be interpreted as transition
probability between the left and right potential minimum, see Eq. (37)). This
system is described by the following time-independent Hamiltonian

$\displaystyle H_{0}$ $\displaystyle=$
$\displaystyle\bar{E}+\frac{1}{2}\left[{\begin{array}[]{cc}\delta&\Delta\\\
\Delta&-\delta\\\ \end{array}}\right]$ (13) $\displaystyle=$
$\displaystyle\bar{E}+\frac{\sqrt{\delta^{2}+\Delta^{2}}}{2}\left[{\begin{array}[]{cc}\frac{\delta}{\sqrt{\delta^{2}+\Delta^{2}}}&\frac{\Delta}{\sqrt{\delta^{2}+\Delta^{2}}}\\\
\frac{\Delta}{\sqrt{\delta^{2}+\Delta^{2}}}&-\frac{\delta}{\sqrt{\delta^{2}+\Delta^{2}}}\\\
\end{array}}\right]$ (16) $\displaystyle\equiv$
$\displaystyle\bar{E}+\frac{\hbar\omega_{R}}{2}\left[{\begin{array}[]{cc}\cos(2\theta)&\sin(2\theta)\\\
\sin(2\theta)&-\cos(2\theta)\\\
\end{array}}\right]=\bar{E}+\frac{\hbar\omega_{R}}{2}\vec{\sigma}\cdot{\vec{n}}.$
(19)

First, we want to exploit the general results following from this Hamiltonian.
Later on, we use this Hamiltonian to approximate the two lowest energy states
of the double-well potential. We introduced the so-called Rabi frequency
$\hbar\omega_{R}=\sqrt{\delta^{2}+\Delta^{2}}$ and the rotation angle
$\theta$, which describes the rotation axis of the corresponding time
development in the unitary operator

$\displaystyle
U(t)=\exp[-iHt/\hbar]=\exp[-i{\bar{E}}t/\hbar-i\frac{\omega_{R}t}{2}{\vec{\sigma}}\cdot{\vec{n}}].$
(20)

$\vec{\sigma}$ are the three Pauli-matrices. Indeed, on the Bloch sphere, this
operator generates a rotation of any given initial state around the rotation
axis

$\displaystyle{\vec{n}}=(\sin(2\theta),0,\cos(2\theta))\equiv(\frac{\Delta}{\hbar\omega_{R}},0,\frac{\delta}{\hbar\omega_{R}}).$
(21)

On the Bloch sphere, the eigenstates point in the $\pm{\vec{n}}$ directions.
In Hilbert space, these eigenstates are given by

$\displaystyle|+{\vec{n}}\rangle=\left[{\begin{array}[]{c}\cos(\theta)\\\
\sin(\theta)\\\ \end{array}}\right],\ \ \ \ \
|-{\vec{n}}\rangle=\left[{\begin{array}[]{c}-\sin(\theta)\\\ \cos(\theta)\\\
\end{array}}\right].$ (26)

The eigenvalues are given by $E_{1}={\bar{E}}+\frac{1}{2}\hbar\omega_{R}$ and
$E_{2}={\bar{E}}-\frac{1}{2}\hbar\omega_{R}$, respectively. We find for
$|L\rangle$ and $|R\rangle$

$\displaystyle|L\rangle$ $\displaystyle=$
$\displaystyle\cos(\theta)|+{\vec{n}}\rangle-\sin(\theta)|-{\vec{n}}\rangle,$
(27) $\displaystyle|R\rangle$ $\displaystyle=$
$\displaystyle\sin(\theta)|+{\vec{n}}\rangle+\cos(\theta)|-{\vec{n}}\rangle.$
(28)

Figure 2: Coloured Orbits on the Bloch sphere for different rotation axis
angles $2\theta$. The rotation axis $\vec{n}$, defined by the first two energy
eigenstates of the system $E_{1}$ and $E_{2}$, is shown in red. Left: Partial
Rabi oscillations, time development of the initial state $|L\rangle$ for
$2\theta=\pi/6,\pi/4,\pi/3$ and $5\pi/12$. Right: Full Rabi oscillation,
$2\theta=\pi/2$. In this case, the transition probability to state $|R\rangle$
is $100\%$ after a $\pi-$flip.

For a given initial state $|\psi(0)\rangle$, the fidelity under time
development $|\psi(t)\rangle=U(t)|\psi(0)\rangle$ is defined as
$F(t)=|\langle\psi(0)|\psi(t)\rangle|^{2}$. For a general superposition state

$\displaystyle|\psi_{0}\rangle=\alpha|+{\vec{n}}\rangle+\beta|-{\vec{n}}\rangle$
(29)

with $|\alpha|^{2}+|b|^{2}=1$, the time dynamics is given by (up to a global
phase)

$\displaystyle|\psi(t)\rangle$
$\displaystyle=e^{-\frac{i}{2}\omega_{R}t\vec{\sigma}\cdot{\vec{n}}}(\alpha|+{\vec{n}}\rangle+\beta|-{\vec{n}}\rangle)$
(30)
$\displaystyle=e^{+\frac{i}{2}\omega_{R}t}\alpha|+{\vec{n}}\rangle+e^{-\frac{i}{2}\omega_{R}t}\beta|-{\vec{n}}\rangle$

and the fidelity, i.e. the tunnelling probability, reads

$\displaystyle F$
$\displaystyle=|\langle\psi(0)|\psi(t)\rangle|^{2}=|e^{+i\frac{1}{2}\omega_{R}t}\alpha^{2}+e^{-i\frac{1}{2}\omega_{R}t}\beta^{2}|^{2}$
(31)
$\displaystyle=1-4|\alpha|^{2}|\beta|^{2}\sin^{2}(\frac{\omega_{R}t}{2}).$
(32)

For $|\psi(0)\rangle=|L\rangle$, we obtain

$\displaystyle F=1-\sin^{2}(\frac{\omega_{R}t}{2})\sin^{2}(2\theta).$ (33)

For $2\theta=\pi/2$, the Rabi oscillation is maximal. The fidelity is then
given by

$\displaystyle F=1-\sin^{2}(\frac{\omega_{R}t}{2}).$ (34)

The state $|\psi_{0}\rangle=\alpha|+{\vec{n}}\rangle+\beta|-{\vec{n}}\rangle$
can be mapped on the Bloch sphere via Hopf mapping (the qubit vector in
$\mathbb{C}^{2}$ is mapped to the sphere $\mathcal{S}^{2}$, see e.g. [30]).
Explicitly,

$\displaystyle(\alpha,\beta)\rightarrow(\rm{Re}(2\alpha^{*}\beta),\rm{Im}(2\alpha^{*}\beta),|\alpha|^{2}-|\beta|^{2}).$
(35)

On the Bloch sphere, there is a simple geometric interpretation for the
behaviour of the fidelity. In Fig. 2, we show the rotation axis ${\vec{n}}$ of
the unitary time development, and the orbits of states at angle $\theta=0,\pi$
(the eigenstates), and $\theta_{k}=k\pi/6,\ k=1,\ldots 5$. In analogy to the
time dependent system described, for $2\theta=\pi/2$, $\left|L\right\rangle$
and $\left|R\right\rangle$ are part of the same orbit. Therefore, the
transition between the two states is certain, leading to the fidelity function
(34). We therefore call this full Rabi oscillation, in analogy to the time
dependent system described.

##  4 Tunnelling in the double well

Quantum tunnelling plays a major role in various areas of modern physics.
Early applications have been e.g. the calculation of lifetimes for alpha-decay
of nuclei, or for nuclear fusion. In recent times, from a technological
perspective, tunnelling in solid state physics, and in particular in
semiconductor physics and in spintronics [31] are of interest both for the
first and in the second quantum revolution. Furthermore, model systems such as
Bose-Einstein condensates in a double-well potential are part of actual
research [32, 33]. In the past decades, periodically driven tunnelling systems
also have been widely studied [6]. Since tunnelling is an important topic in
physics since the advent of quantum physics, it is not astonishing that also
in physics education, quantum tunnelling is part of the standard curriculum
not only at university, but also in high school [34]. Thus, it is desirable to
explore model systems for quantum tunnelling at all levels of education, and
also for research.

###  4.1 Full Rabi oscillations in the double well (Level 1)

As a model for the tunnelling system, we consider the (tilted) double-well
potential

$\displaystyle V(x)=\lambda(x^{2}-a^{2})^{2}+\mu x.$ (36)

If we consider the system to consist of two subsystems L and R, we find
respective ground states $\left|L\right\rangle$ and $\left|R\right\rangle$.
First, we want to find a relation between the parameters $(\delta,\Delta)$ of
the general theory of Sect. 3, and the parameters $(\lambda,a)$ of $V(x)$. For
this purpose, we investigate the degenerate case $\mu=\delta=0$ where the
ground state energies $E_{L}$ and $E_{R}$ are equal.

Figure 3: Tunnelling probabilities for two different potentials with $a=0.7$
and $\mu=2.8571$. Left: $V_{1}$ with $\lambda_{1}=35$,
$1-F_{1}(t)=1-\left|\left\langle L_{1}|\psi_{1}(t)\right\rangle\right|^{2}$
with
$\psi_{1}(0)=\left|L_{1}\right\rangle=0.9575\left|E_{2}\right\rangle+0.2884i\left|E_{1}\right\rangle$.
Right: $V_{2}$ with $\lambda_{2}=17$,: $1-F_{2}(t)=1-\left|\left\langle
L_{2}|\psi_{2}(t)\right\rangle\right|^{2}$ with
$\left|L_{2}\right\rangle=\psi_{1}(0)=0.9483\left|E_{2}\right\rangle+0.3172i\left|E_{1}\right\rangle$.
These two values of $\lambda$ will be used to realise assisted tunnelling with
the SUPER principle which is shown in Sect. 5. The theoretical prediction for
the fidelity $1-F(t)=4|\alpha|^{2}|\beta|^{2}\sin^{2}(\omega_{R}t/2)$
coincides with the numerical result and cannot be distinguished within the
resolution of these plots.

We will calculate $\Delta=\Delta(\lambda,a)$ within this model potential. In
the degenerate case, we obtain full Rabi oscillation between the state on the
left
$\left|L\right\rangle=1/\sqrt{2}(\left|E_{2}\right\rangle-\left|E_{1}\right\rangle)$
and on the right side of the potential
$\left|R\right\rangle=1/\sqrt{2}(\left|E_{2}\right\rangle+\left|E_{1}\right\rangle)$
(see Eq. 27 and 28 for $\theta=\pi/4$). The eigenenergies are given as
$E_{1}={\bar{E}}+\sqrt{\delta^{2}+\Delta^{2}}\rightarrow{\bar{E}}+\Delta$ and
$E_{2}={\bar{E}}-\sqrt{\delta^{2}+\Delta^{2}}\rightarrow{\bar{E}}-\Delta$. In
this case, the coupling $\Delta$ can be computed as tunnelling amplitude
either using WKB or the instanton ansatz. In both approaches, we obtain [35,
36]

$\displaystyle\frac{1}{2}\Delta(\lambda,a,\mu=0)$
$\displaystyle=\sqrt{12}\hbar\omega_{H.O.}\sqrt{\frac{S_{\rm
inst}}{2\pi\hbar}}e^{-\frac{S_{\rm inst}}{\hbar}}$ (37)
$\displaystyle=\sqrt{\frac{m^{2}\omega_{H.O.}^{5}}{2\pi\lambda}}\exp{[-\frac{1}{\hbar}(\frac{4}{3}\sqrt{2m\lambda}a^{3})]}.$

Here, $x_{\rm inst}$ is the so-called instanton solution, a classical path in
imaginary time $\tau$ connecting the left and the right potential minimum. The
instanton action is given by $S[x_{\rm inst}]=\sqrt{2m}\int\sqrt{V(x)}dx$,
leading to

$\displaystyle S_{\rm
inst.}=\sqrt{2m}\int_{-a}^{+a}\sqrt{\lambda}|x^{2}-a^{2}|dx=\frac{4}{3}\sqrt{2m\lambda}a^{3}\equiv\frac{m^{2}\omega_{H.O.}^{3}}{12\lambda}$
(38)

with $\omega_{H.O.}=\sqrt{(8a^{2}\lambda)/m}$. In such a way, we find explicit
and analytic expressions for all parameters in the general Hamiltonian (1) in
terms of the model potential (36) for the degenerate case $\mu=0$.

###  4.2 Partial Rabi oscillations in the tilted double well $\mu\neq 0$
(Level 2)

As an approximation, we can consider the two wells to be completely separate,
i.e. the system to consist of two independent subsystems L and R. Let $E_{L}$
be the energy of the ground state $\left|L\right\rangle$ of the left minimum,
and $E_{R}$ be the energy of the ground state $\left|R\right\rangle$ of the
right minimum. Concerning these two ground state energies, we may approximate
the respective potentials as harmonic oscillators. We find in the vicinity of
$x\simeq a$ with $y\equiv(x-a)$

$\displaystyle V(y)\simeq\mu(y+a)+(2a)^{2}\lambda y^{2}+\ldots$ (39)

In first order approximation, the $\mu$-dependence is irrelevant for the
ground state apart from a vertical shift by $-\mu a$ or $\mu a$, respectively,
as the term $\mu(x\pm a)$ is odd around $x=\mp a$, and the harmonic oscillator
ground state is even around its origin. Thus, we can compare the ground state
energies with the usual ground state energy of the harmonic oscillator. We
find $E_{L}=\frac{1}{2}\hbar\omega_{H.O.}-\mu a$ and
$E_{R}=\frac{1}{2}\hbar\omega_{H.O.}+\mu a$ with
$\hbar\omega_{H.O.}=\sqrt{(8a^{2}\lambda/m)}$. Thus, we find in leading order
$\delta=(E_{L}-E_{R})\simeq-2\mu a$. While it is possible to refine the
(rather bad) approximations for $E_{L/R}$ using asymptotic series [35], we
refrain from doing so here as our main concern is the introduction of assisted
tunnelling using the Quantum Composer.

We will discuss to what extent we can verify the theory of Rabi oscillations
using the Quantum Composer. This is the second level of studying Rabi
oscillations in the double-well potential with the Quantum Composer. Natural
units ($\hbar=1$ and $m=1$) are used throughout the rest of this paper. Since
the eigenvalues for the energy are
$E_{1}={\bar{E}}+\frac{1}{2}\sqrt{\Delta^{2}+\delta^{2}}$ and
$E_{2}={\bar{E}}-\frac{1}{2}\sqrt{\Delta^{2}+\delta^{2}}$, we can determine
the Rabi frequency $\omega_{R}=\sqrt{\Delta^{2}+\delta^{2}}$ from
$E_{1}-E_{2}=\omega_{R}$ and the mean energy by $E_{1}+E_{2}=2{\bar{E}}$,
obtaining $E_{1}$ and $E_{2}$ numerically using the Quantum Composer. In Fig.
3, we compare the theoretical expression for the fidelity
$1-F(t)=4|\alpha|^{2}|\beta|^{2}\sin^{2}(\omega_{R}t/2)$ (34) with the results
from the quantum composer for two different potentials
($\lambda_{1}=35,\lambda_{2}=17$). We find excellent agreement.

In the tilted potential, the instanton calculations that were shown in Sect.
4.1 for full Rabi oscillations are still possible, but much more involved
[35]. We will not embark into these troubled waters of theory not to loose
track to our main purpose, the introduction of assisted tunnelling. However,
it would be interesting for future work to link also these analytic results
with numerical results obtained with the Quantum Composer.

To conclude, Rabi oscillations can be studied in an elegant manner with the
Quantum Composer for the tilted double well potential and can give valuable
insights into some important aspects of the underlying theory.

Figure 4: The Swing Up principle illustrated in 2D. Two different potentials
$V_{1}$ and $V_{2}$ have two different rotation axis $\vec{n}_{1}$ and
$\vec{n}_{2}$ pointing towards the reader. By swapping between them, we can go
from initial state $\left|A\right\rangle$ to state $\left|B\right\rangle$
following the red path, or we can go from $\left|B\right\rangle$ to
$\left|A\right\rangle$ following the pink path. When falling out of the
swinging rhythm, there is the danger to ’loose track’. Once the state is at
$\left|B\right\rangle$, if the Swing Up process is not stopped it will follow
the pink path and start decaying back to $\left|A\right\rangle$. Note that in
a full Swing Up, a state viewed in 2D would evolve along the red path and then
”back” along the pink path to end in the desired state, orthogonal to the
initial state.

##  5 Application of the SUPER-principle to assisted tunnelling (Level 3)

The SUPER principle was originally introduced in [11], while first
experimental realisations have been reported in [12] for two level-systems
driven by periodic fields. Note that the rotating frame transformation must be
applied in this case.

In the rotating frame, the time-independent Hamiltonian (5) arises. As the
tunnelling system is of the type of equation (5) from the very beginning, no
transformation to the rotating frame is necessary. This necessity is the only
difference between the original SUPER principle and the ideas applied here.
This allows for the application to assisted tunnelling, as we will show in
what follows.

The key idea of the SUPER principle is the following: For given rotation axis
${\vec{n}}$, the orbit of all initial states under the action of (20) form a
”ring system” as shown in Fig. 4. If the rotation axis ${\vec{n}}(t)$
oscillates between two different, but near positions, the time development of
an initial state switches between these two ”ring systems”.

In [29], we show an animation of the time evolution of the state $|L\rangle$
to the state $|R\rangle$ based on the super principle, applied to the
situation shown in Fig. 5. Indeed, as we show in this section, if the
oscillation period is chosen nearby the average Rabi frequency of the two
different unitary operators (20) with rotation axis ${\vec{n}}_{1}$ and
${\vec{n}}_{2}$, a ”Swing Up” leads to an enhancement of the Rabi amplitude
even in the case of a tilted well. From our point of view, this result can be
seen as analogous to [11] in the case of Rabi oscillations with an external,
time-dependent driving field, where it was shown that the modulation frequency
has to be chosen close to the mean Rabi frequency of the two detunings.

Figure 5: Relation between potential barrier height and the corresponding time
dynamics on the Bloch sphere. By switching between these two potentials, on
the Bloch-sphere, the SWING up mechanism can be realised as shown in Fig. 4

In order to apply the SUPER principle to the particular situation of
tunnelling in a double-well, we consider a situation where the potential
height $V(0)=\lambda a^{4}$ for constant distance $a$ is periodically shifted
from a ’high’ value to a ’low’ value by changing $\lambda$. The potentials
used, and the initial and desired states are shown in Fig. 6. We consider the
situation where we start in state $|L_{1}\rangle$ with $V_{1}$ or
$|L_{2}\rangle$ with $V_{2}$. For large tilting parameter $\mu$, the maximum
fidelities $F_{1}=|\langle L_{1}|R_{1}\rangle|^{2}$ and $F_{1}=|\langle
L_{2}|R_{2}\rangle|^{2}$ are rather low (see Fig. 3 with $P_{1}=0.304$ and
$P_{2}=0.362$, respectively).

Figure 6: Two potentials $V_{1}$ with
$(\lambda_{1}=35,a_{1}=0.7,\mu_{2}=2.8571)$ and $V_{2}$ with
$(\lambda_{2}=17,a_{2}=0.7,\mu_{2}=2.8571)$ with the probability density
function of the initial state
$\left|L_{1}\right\rangle=0.9575\left|E_{2}\right\rangle_{1}+0.2884i\left|E_{1}\right\rangle_{1}$
and the desired state
$\left|R_{2}\right\rangle=0.2884i\left|E_{2}\right\rangle_{2}+0.9575\left|E_{1}\right\rangle_{2}$.
The eigenstates $\left|E_{2/1}\right\rangle_{1}$ of $V_{1}$ and
$\left|E_{2/1}\right\rangle_{2}$ of $V_{2}$ are shown in blue, with
eigenenergies in red. The corresponding Bloch-sphere representation is shown
schematically in Fig. 5

Next, we modify the system and allow for an oscillation between $V_{1}$ or
$V_{2}$, where the potential oscillation frequency is about the mean Rabi
frequency. We introduce a smoothstep function to model the transition between
the potentials. After some fine tuning that is described in 5.1, the maximum
Fidelity was increased drastically to $P=0.994$. The result is shown in Fig.
7, confirming the Swing up mechanism, leading to assisted tunnelling.

Figure 7: Assisted tunnelling by oscillation between $V_{1}$ with
$\lambda_{1}=35$ and $V_{2}$ with $\lambda_{2}=17$ at potential swap time (the
time one potential is active before swapping to the next) $T/2=1.01$. Top
Left: Potential barrier height $\lambda(t)$. At 15 time units, the potential
was held constant to trap the state at the top. Top Middle: Tunnelling
probability $F(t)=\left|\left\langle\psi(t)|R_{2}\right\rangle\right|^{2}$
with $\psi(0)=\left|L_{1}\right\rangle$. Top Right: State evolution plot.
Bottom: Look at the Bloch Sphere with initial state
$\left|L_{1}\right\rangle$, desired state $\left|R_{2}\right\rangle$ and the
eigenstates $\left|E_{2}\right\rangle$, $\left|E_{1}\right\rangle$ ”from
above” (Left), ”from the front” (Middle, comparable to Fig. 2) and ”from the
side” (Right). By periodic oscillation between $\lambda_{1}=35$ and
$\lambda_{2}=17$ and trapping the state at $t=15$ at $\lambda_{2}=17$, a
maximum fidelity of over 0.994 was achieved at $t=20.5$.

Besides the SUPER principle, other approaches to enhance the tunnelling rate
are known. Here, the notion of parametric resonance is used in a variety of
different situations. Our approach differs from parametric resonance as
introduced in [37], as we consider a tilted potential, and change the coupling
between the two wells in an appropriate way by changing the barrier height. We
do not use the rotating wave approximation. In [38], the amplitude of an
external driving field is changed to enhance the tunnelling rate. In both
cases, parametric resonance works with a weakly nonlinear, parametrically
modulated oscillator described in the rotating wave approximation with tilting
$\mu=0$.

Other approaches to enhance tunnelling – also denoted parametric resonance –
have been proposed in [39] and [40]. Here, similarly to our approach, the
potential height is varied periodically to enhance tunnelling. In this case, a
Bose-Einstein condensate is discussed, where the time dynamics is governed by
the Gross-Pitaevskii equation.

In the future, it will be interesting to study the relation of the SUPER
principle to these approaches for parametric resonance in more detail.

###  5.1 Practical hints towards realising assisted tunnelling with the
Quantum Composer

Assisted tunnelling by swapping between two double-well potentials with
different barrier heights $\lambda_{1}$ and $\lambda_{2}$ is not an easy task.
The parameters have to be chosen carefully to achieve convincing results. We
start by choosing two sets of parameters for two different potential functions
$V_{1}(\lambda_{1},a,\mu_{1})=V_{\rm high}$ and
$V_{2}(\lambda_{2},a,\mu_{2})=V_{\rm low}$, where $a$ is fixed and would
typically be given by the physical system, $\lambda_{1}\approx 2\lambda_{2}$
and $\mu_{1}=\mu_{2}$. The closer $\lambda_{1}$ and $\lambda_{2}$ are, the
closer the two corresponding rotation axes on the Bloch-sphere. Then, the
orbits (see Fig. 4) are closer together, resulting in less distance travelled
per oscillation period. This translates to more oscillations needed for a full
Swing Up. The fidelity graph becomes less steep, and in turn the Swing Up
process takes longer. Thus, we choose substantially different $\lambda$.

The Rabi frequencies $\omega_{R}^{(1)}$ and $\omega_{R}^{(2)}$ can be found
via $E_{1}^{(1)}-E_{2}^{(1)}$ and $E_{1}^{(2)}-E_{2}^{(2)}$, respectively,
that are calculated numerically in Quantum Composer. Using Eq. (21) and
double-angle formulas, we find
$\delta/\omega_{R}=\cos^{2}{(\theta)}-\sin^{2}((\theta))=2\cos^{2}{(\theta)}-1=1-2\sin^{2}(\theta)$.
With (27), we calculate the initial state $\left|L_{1}\right\rangle$ localised
in the left well as

$\displaystyle\left|L_{1}\right\rangle$
$\displaystyle=\cos{(\theta)}\left|E_{2}\right\rangle+\sin{(\theta)}\left|E_{1}\right\rangle$
(40) $\displaystyle=\sqrt{\frac{\omega_{R}^{(1)}+2\mu
a}{2\omega_{R}^{(1)}}}\left|E_{2}\right\rangle+\sqrt{\frac{\omega_{R}^{(1)}-2\mu
a}{2\omega_{R}^{(1)}}}\left|E_{1}\right\rangle$ (41)

using the approximation $\delta\approx 2\mu a$. The desired state can be
either chosen as

$\left|R_{1}\right\rangle=-\sqrt{\frac{\omega_{R}^{(1)}-2\mu
a}{2\omega_{R}^{(1)}}}\left|E_{2}\right\rangle+\sqrt{\frac{\omega_{R}^{(1)}+2\mu
a}{2\omega_{R}^{(1)}}}\left|E_{1}\right\rangle$ (42)

or

$\left|R_{2}\right\rangle=-\sqrt{\frac{\omega_{R}^{(2)}-2\mu
a}{2\omega_{R}^{(2)}}}\left|E_{2}\right\rangle+\sqrt{\frac{\omega_{R}^{(2)}+2\mu
a}{2\omega_{R}^{(2)}}}\left|E_{1}\right\rangle,$ (43)

depending on which of the two potential functions should be final. In our
example Swing Up, we chose $\left|R_{2}\right\rangle$.

Figure 8: Left: Modulation function $f(t)=\frac{1}{e^{s\cdot g(t)}+1}$ where
$s=20$ is the transition slope and
$g(t)=-\cos{\left(\frac{t-t_{0}}{T}\cdot\pi\right)}$ with potential swap time
$T/2=1.01$ and time offset $t_{0}=0.1$. Right: Potential barrier height
$\lambda(t)=\lambda_{1}\cdot f(t)+\lambda_{2}\cdot(1-f(t))$. The time of the
first swap of the potential is $T/4+t_{0}$.

Next, the potential swap time $T=\frac{2\pi}{\omega}$( where $\omega$ is the
potential oscillation frequency) should be chosen. $T/2$ is the time of one
potential being active before swapping to the other. For an optimal fidelity,
we observed that $T$ should be about 5% larger than the average Rabi period
$T_{R}$ of the two potentials,
$T_{R}=4\pi/(\omega_{R}^{(1)}+\omega_{R}^{(2)})$. Next, the time offset
$t_{0}$ at the start of the swing-up can be optimised (see Fig. 8).
$T/4+t_{0}$ is the time of the first swap of the potential.

Using these parameters, we achieve a high fidelity for assisted tunnelling in
the following manner: First, one should optimise the time offset $t_{0}$ for
the starting potential swap time $T$. Next, we optimise $T$ for this offset
and repeat. If the resulting fidelity is not satisfying, different values for
$\lambda_{1},\lambda_{2}$ or $\mu$ should be chosen. In this manner, a maximum
fidelity of over F=0.994 was achieved as is shown in more detail in Fig. 9.

Figure 9: Various maximum fidelities and the times to reach these fidelities
for different potential swap times $T/2$ and different time offsets $t_{0}$
holding the potential swap time constant ($T/4+t_{0}$ is the time of the first
potential swap, see Fig. 8). Swapping between two potentials with
$\mu=2.8571$, $a=0.7$, $V_{1}$ with $\lambda_{1}=35$ and $V_{2}$ with
$\lambda_{2}=17$. Left: The maximum fidelities reached for various potential
swap times $T/2$ at time offsets $t_{0}=0.1T/2$. Shown are the Rabi
oscillation periods of the two potentials $T_{R}^{(1)}/2=\pi/\omega_{R}^{(1)}$
and $T_{R}^{(2)}/2=\pi/\omega_{R}^{(2)}$ and the mean Rabi oscillation period
$T_{R}/2$. The maximum fidelity $F=0.987$ is achieved at
$T_{\text{max}}/2=1.052(T_{R}/2)=1.01$. This can be further improved to
$F=0.994$ by holding the potential constant at $\lambda_{2}=17$ after $t=15$
as shown in Fig. 7. Right: Maximum fidelities for various different time
offsets $2t_{0}/T_{\text{max}}$ at potential swap time $T_{\text{max}}/2$.

The process is challenging in the sense that it is very sensitive to $T$,
$t_{0}$ and the potential parameters $\lambda$ and $\mu$. This could be
because the Rabi frequency is not constant during the Swing Up process.
Rather, it increases the further $\left|\psi(t)\right\rangle$ travels towards
$\left|R\right\rangle$ as shown in Fig. 10. We have no analytical results that
predict this behaviour and can only refer to the numerical results. This can
lead to falling out of the swinging rhythm, but can be solved by careful
choice of parameters, as shown in this paper.

Figure 10: Fidelity F(t) as in Fig. 7, top middle and modulation function
$f(t)$ as in Fig. 8 in red. $f(t)=0$ corresponds to $\lambda=\lambda_{1}=17$
and $f(t)=1$ corresponds to $\lambda=\lambda_{2}=35$. The frequency of the
Swing Up increases during the process: the first 4 peaks are 6.35 time units
apart while the next 4 peaks are only 5.44 time units apart.

###  5.2 Limits of the Bloch-sphere representation for the double-well system

The $2\times 2$ Hamiltonian (13) includes only the two lowest-lying states of
an infinite number of eigenstates in the double-well. In this section, we
check the validity of the two-level system approximation (40), (42) for the
two lowest-lying states localised in the left and the right well,
respectively.

In general, these states are a superposition not only of the first two
eigenstates $\left|E_{2}\right\rangle$,$\left|E_{1}\right\rangle$, but include
more states $\left|\chi\right\rangle_{R,L}$ with higher energy. Thus, the
exact wave functions in position space reads

$\displaystyle\psi_{L}^{\rm exact}(x)$ $\displaystyle=$
$\displaystyle\left\langle x\right|L_{1}\rangle+\left\langle
x\right|\chi_{L}\rangle\equiv\psi_{L}^{2\times 2}(x)+\chi_{L}(x),$ (44)
$\displaystyle\psi_{R}^{\rm exact}(x)$ $\displaystyle=$
$\displaystyle\left\langle x\right|R_{1}\rangle+\left\langle
x\right|\chi_{R}\rangle\equiv\psi_{R}^{2\times 2}(x)+\chi_{R}(x).$ (45)

These states are fully localised either in the left or the right valley,
leading to

$\displaystyle\int_{-\infty}^{0}|\psi_{L}^{\rm exact}(x)|^{2}$
$\displaystyle=$ $\displaystyle\int_{-\infty}^{0}|\psi_{L}^{2\times
2}(x)|^{2}+\int_{-\infty}^{0}|\chi_{L}(x)|^{2}=1,$ (46)
$\displaystyle\int_{0}^{\infty}|\psi_{R}^{\rm exact}(x)|^{2}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}|\psi_{R}^{2\times
2}(x)|^{2}+\int_{0}^{\infty}|\chi_{R}(x)|^{2}=1.$ (47)

In Fig. 11, we show the expression $1-|\chi_{R}(x)|^{2}$ depending on the
values for $\lambda,\mu$. In the region with $\omega_{R}>2\mu a$, the
deviation from the two-level system is below $5\%$. Note that for
$\omega_{R}=2\mu a$, the amplitude for $\left|E_{1}\right\rangle$ in (40) (and
$\left|E_{2}\right\rangle$ in (42)) vanishes. After the transition to
$\omega_{R}<2\mu a$, the amplitude becomes imaginary, and the error increases
in an asymmetric manner. While the error stays below $15\%$ for the left
valley, it increases to up to $35\%$ in the right valley.

While the visualisation of the SUPER principle on the Bloch sphere is rather
intuitive and helps understanding the mechanism, the SUPER principle itself is
not restricted to the $2\times 2$ level system and indeed includes higher
order states during the transition shown in Fig. 7.

Figure 11: Deviations of the two-level-system approximation by including
contributions of higher order levels as defined by (44), (46). Top:
$\int_{-\infty}^{0}\textrm{d}x|\psi_{L}^{2\times 2}(x)|^{2}$ with
$\left|L\right\rangle$ from eq. (40) for different $\lambda$ and $\mu$ and
$a=0.7$ fixed. Bottom: $\int_{0}^{\infty}\textrm{d}x|\psi_{R}^{2\times
2}(x)|^{2}$ with $\left|R\right\rangle$ from eq. (42). In the region
$\omega_{R}>2\mu a$, the deviation from the two-level system is rather small
$<10\%$.

##  6 Outlook and discussion

Our conclusions can be drawn from a didactical perspective, and from the
perspective of recent research in applied physics. While periodic oscillations
of potentials have already been studied [6], to our knowledge, a systematic
enhancement of tunnelling using the SUPER-mechanism has not yet been
demonstrated.

In the context of quantum tunnelling, it seems possible to find experimental
realisations for a ”high” value and a ”low” value of the potential barrier.
Explicitly, we may think of a Bose-Einstein condensate trapped in a double-
well potential. An experimental realisation, where the atoms are magnetically
coupled to a single-mode of the microwave field inside a superconducting
resonator, has been reported in [41]. By modulating the magnetic field
strength, the trap potential could be manipulated to embark for assisted
tunnelling.

As another model system, we may think of a double-well potential realised for
an individual fluxon trapped in a Josephson tunnel junction. Again, parameters
of the potential can be manipulated by the strength of a magnetic field [42].
Since the double-well potential is the smallest subpart of the general class
of Hubbard-model systems in condensed matter physics, many other applications
of assisted tunnelling might be of interest.

Our main concern however, is physics education: Since the Quantum Composer is
available for free, it is possible to discuss Rabi oscillations in an
intuitive and direct manner using this powerful modelling tool on almost any
computer system. The result of the educator-developer dialogue is the updated
version of the Composer. With that, the explicit visualisation of the Bloch
sphere dynamics is possible. Using the Bloch-sphere representation, the SUPER
principle has a simple and general geometric interpretation as shown in Figs.
2 and 4 and even a knot theoretic interpretation [43]. Translated back to the
double-well, we may apply this mechanism to realise assisted tunnelling by
introducing an appropriate oscillation of the potential height.

Explicitly, at high school, the dependence of the Rabi frequency on the
potential parameters $(\lambda,a,\mu)$ can qualitatively be discussed using
just the graphical interface provided by the Quantum Composer. On the
undergraduate level, the quantitative relations explored in Sect. 4 can be
accessed, in particular, the dependence of $S_{\rm
inst.}\propto\sqrt{\lambda}a^{3}$ on the level spacing of the two lowest
eigenvalues, $(E_{1}-E_{2})\propto\exp{(-S_{\rm inst.})}$. On the graduate
level, assisted tunnelling and the SUPER principle can be explored numerically
as shown in Sect. 5. Here, numeric results can be used to provide a
qualitative understanding of the SUPER-mechanism as shown in Fig. 4. The
multiple representation that the Quantum Composer offers, like the state
evolution plot and the Bloch sphere, can enhance understanding of this
concept. All Quantum Composer files we used in our work can be downloaded from
[29].

Concerning research in physics, still many questions remain open. It would be
desirable to understand the relation between the numerical results from the
Quantum Composer and analytic results, such as 37, even better. Moreover,
analytic results for assisted tunnelling are not yet available, and it would
be desirable to analytically explore this approach further in accordance with
[44].

An empirical study on solving tasks with multiple representations in quantum
composer has been conducted recently in an eye-tracking study [45]. Similarly,
a possible direction of further didactical research is the exploration of the
benefits and challenges of teaching approaches using visualisation tools such
as the Quantum Composer, with various target groups and in different contexts
like the one described in this paper.

## Acknowledgements

The Aarhus team acknowledged funding from Carlsberg Foundation, Independent
Research Fund Denmark (ReGAME) and European Research Council (PQTEI).

The Münster team is grateful to D. Reiter and T. Bracht for valuable
discussions.

J.B. acknowledges support by the project QuanTUK at the RPTU in
Kaiserslautern, supported by the Federal Ministry of Education and Research
(FKZ13N15995).

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