# Visualized Wave Mechanics by Coupled Macroscopic Pendula: Classical Analogue
to Driven Quantum Bits

Heribert Lorenz  Center for NanoScience (CeNS) & Fakultät für Physik, Ludwig-
Maximilians-Universität (LMU), 80539 München, Germany  Sigmund Kohler
Instituto de Ciencia de Materiales de Madrid, CSIC, 28049 Madrid, Spain  Anton
Parafilo  Center for Theoretical Physics of Complex Systems (PCS), Institute
for Basic Science (IBS), Expo-ro 55, Yuseong-gu, Daejeon 34126, Korea  Mikhail
Kiselev  The Abdus Salam International Centre for Theoretical Physics (ICTP),
Strada Costiera 11, 34151 Trieste, Italy  Stefan Ludwig  Paul-Drude-Institut
für Festkörperelektronik (PDI), Leibniz-Institut im Forschungsverbund Berlin
e.V., Hausvogteiplatz 5-7, 10117 Berlin, Germany

###### Abstract

Quantum mechanics increasingly penetrates modern technologies but, due to its
non-deterministic nature seemingly contradicting our classical everyday world,
our comprehension often stays elusive. Arguing along the correspondence
principle, classical mechanics is often seen as a theory for large systems
where quantum coherence is completely averaged out. Surprisingly, it is still
possible to reconstruct the coherent dynamics of a quantum bit (qubit) by
using a classical model system. This classical-to-quantum analogue is based on
wave mechanics, which applies to both, the classical and the quantum world. In
this spirit we investigate the dynamics of macroscopic physical pendula with a
modulated coupling. As a proof of principle, we demonstrate full control of
our one-to-one analogue to a qubit by realizing Rabi oscillations, Landau-
Zener (LZ) transitions and Landau-Zener-Stückelberg-Majorana (LZSM)
interferometry. Our classical qubit demonstrator can help comprehending and
developing useful quantum technologies.

Quantum technology already has a drastic impact on society. This development
presently accelerates with our growing ability to harvest coherent quantum
dynamics for engineering game changing devices such as quantum computers or a
quantum internet. At the same time, while the mathematical framework of
quantum mechanics can be considered complete, fundamental aspects of the
underlying physics, even on the level of only few qubits are outside our
empirical world. In this situation, classical model systems capable of
enlightening the often elusive coherent dynamics of quantum systems may prove
very useful [1]. This approach might be fundamentally questioned due to a
central paradigm of quantum dynamics, which is its probabilistic nature in
contrast to the deterministic classical equation of motion (EOM).
Nevertheless, besides non-determinism and non-locality, wave properties and
the superposition principle being central elements of quantum mechanics appear
also in classical physics. For example, the quantum mechanical double split
experiment may be visualized with classical water waves. Here we visualize one
of the most basic quantum systems, a qubit, by physical macroscopic pendula.

Classical dynamics can generally be formulated in terms of second-order, non-
linear and inhomogeneous differential equations, while non relativistic
quantum mechanics is based on the first order, homogeneous and linear
Schrödinger equation. Hence, most classical systems are improper for
simulating qubit dynamics. In this article, we derive the conditions under
which classical pendula with modulated coupling nevertheless can be described
by a Schrödinger-like equation. We demonstrate this classical-to-quantum
analogue by exploring three realizations of qubit control, namely Rabi
oscillations [2], LZ transitions [3, 4] and, finally, LZSM interferometry [5,
6].

Recent developments in quantum technology have motivated theoretical [1, 7, 8,
9, 10, 11, 12] and experimental [13, 14, 15, 16] projects exploring analogues
between classical coupled oscillators and its quantum version. The most
interesting dynamics happens at avoided crossings of the eigenmodes of coupled
oscillators near resonance. Previous theoretical considerations [8, 10, 11]
and experiments [15, 16] with nanomechanical oscillators used a time-dependent
frequency difference corresponding to the detuning usually modulated in case
of qubits [17, 18, 19, 20, 21, 22, 23, 24, 25]. To experimentally establish a
classical-to-quantum analogue based on macroscopic pendula we instead modulate
the coupling, which for this system is more practical than driving the
detuning. This gimmick, for the first time allows us to continuously monitor
the coherent dynamics of a driven two-level system at ambient conditions and
to observe it with bare eyes. As we establish a one-to-one correspondence, our
coupled pendula directly visualize the coherent dynamics of a driven qubit.

##### Setup and model

Figure 1: Photograph (a) and sketch (b) of the pendula coupled via cubic
neodymium magnets (red arrows in sketch indicate magnetic moments), a lower
pair with moments $m_{\text{l}}=25.37\,\text{Am}^{2}$ and an optional upper
pair with $m_{\text{u}}=6.54\,\text{Am}^{2}$. The lower magnets are attached
at the end of the pendula while the upper ones sit inside the red cylinders.
Their respective distances from the pivots are $l_{\text{l}}=1.148\,$m versus
$l_{\text{u}}=0.635\,$m. One of the lower magnets is slowly rotated at angular
frequency $\Omega$ around the pendulum rod by a battery driven motor (inside
the transparent plastic cases). The distances $L$ between the pivots and
$L_{\text{u}}$ between the upper magnets at deflections
$\varphi^{\text{lab}}_{1}=\varphi^{\text{lab}}_{2}=0$ as well as $\Omega$ are
variable. Each pendulum weighs 4.242 kg, where a 2.1 kg brass weight (visible
in the photo) can be moved along a threaded section of each rod to vary
$\omega_{1}$ and $\omega_{2}$.

In Fig. 1 we display a photograph and a simplified sketch of the setup. It
consists of two pendula, each being described by its deflection angle
$\varphi^{\text{lab}}_{k}$ and its angular frequency $\omega_{k}$ with
$k=1,2$. The two pendula are coupled via permanent magnets and detuned by the
frequency difference $\Delta=\omega_{1}-\omega_{2}$. To probe the dynamics of
qubits, usually the energy detuning between the diabatic states is modulated.
However, modulating the coupling is mathematically equivalent after applying
the appropriate basis transformation. For our system it is more practical to
modulate the coupling. For this purpose we employ a battery driven linear
motor, which rotates one of the magnets around the axis defined by the
pendulum rod. As a result, the coupling and, at the same time, the equilibrium
deflections of the pendula, $\varphi_{k}^{\text{qs}}$, are periodically
modulated in time. The latter correspond to the quasistatic solution of the
driven system, describing the (momentary) adiabatic equilibrium position.

We consider the deviation from the adiabatic equilibrium,
$\varphi_{k}=\varphi^{\text{lab}}_{k}-\varphi_{k}^{\text{qs}}$. Aiming at a
description in the form of a Schrödinger equation, which is linear and of
first order, experiments and theory have to facilitate a linearization of the
non-linear Newton EOM. This requires small deflection angles, small frequency
differences and similar moments of inertia of the uncoupled pendula. The
linearized version of the EOM reads

$\begin{split}\ddot{\varphi}_{1}+\omega_{1}^{2}\varphi_{1}={}\omega_{0}\varepsilon(t)(\varphi_{1}-\varphi_{2}),\\\
\ddot{\varphi}_{2}+\omega_{2}^{2}\varphi_{2}={}\omega_{0}\varepsilon(t)(\varphi_{2}-\varphi_{1}),\end{split}$
(1)

where $\omega_{0}=\frac{1}{2}(\omega_{1}+\omega_{2})$ is the average pendulum
(angular) frequency and $\varepsilon(t)$ is the coupling in units of frequency
111In the interaction term, we have neglected the small difference of the
moments of interia. Moreover, the sign of $\varepsilon(t)$ is chosen such that
it matches the usual definition in the quantum mechanical two-level problem.
It is positive for attractive interaction.. The symmetry of the interaction
terms on the right-hand side of Eq. (1) is essential for resembling the
Schrödinger equation and requires neglecting the difference between the
moments of inertia. Our modulated coupling, $\varepsilon(t)$, corresponds to
the time-dependent level detuning commonly used to drive qubits, for instance
in the context of the quantum mechanical LZSM problem [27, 28]. To simplify a
comparison with typical qubit experiments we aim at a coupling of the common
form $\varepsilon(t)=\varepsilon_{0}+A\cos(\Omega t)$, which renders Eq. (1) a
Mathieu equation [29]. This modulation requires an experimental setup allowing
for the far-field approximation of the dipole-dipole interaction.

For simulating qubit experiments we would like to independently modify the
mean coupling $\varepsilon_{0}$ and the modulation amplitude $A$. To achieve
this, we use two sets of magnet pairs, see Fig. 1. The lower magnets are
attached at the distance $l_{\text{l}}$ from the pivots and the upper magnets
at $l_{\text{u}}$, where $(l_{\text{l}}-l_{\text{u}})/L$ is sufficiently large
to allow us to neglect quadrupole components of the coupling. The coupling is
then composed of the sum of the contributions of the upper versus lower
magnets, $\varepsilon=\varepsilon_{\text{u}}+\varepsilon_{\text{l}}$, where we
slowly modulate $\varepsilon_{\text{l}}$ by rotating one of the lower magnets.
The time-dependence of the reference point of the linearization,
$\varphi_{k}^{\text{qs}}(t)$, leads to harmonic mixing such that
$\varepsilon_{0}$ aquires a contribution from the rotating magnets and, vice
versa, $A$ is also affected by the static magnets.

The linearized EOM Eq. (1), which is still of second order, describes the free
oscillations of two pendula with modulated coupling. In comparison, the
Schrödinger equation of a qubit describes probability functions. These
correspond to the slowly varying occupation amplitudes of the two pendula,
given by the envelope functions, say $\Psi_{k}$, of the individual rapid
oscillations $\varphi_{k}(t)$. To separate the time scales, we therefore
employ the ansatz

$\varphi_{k}=e^{-i\omega_{0}t}\Psi_{k}+\text{c.c.}$ (2)

with a rapidly oscillating prefactor and slowly varying complex envelopes
$\Psi_{k}$. Inserting the ansatz into Eq. (1), while neglecting second order
derivatives of $\Psi_{k}$, we find

$i\frac{d}{dt}\begin{pmatrix}\Psi_{1}\\\
\Psi_{2}\end{pmatrix}=\frac{1}{2}\begin{pmatrix}\Delta-\varepsilon(t)&\varepsilon(t)\\\
\varepsilon(t)&-\Delta-\varepsilon(t)\end{pmatrix}\begin{pmatrix}\Psi_{1}\\\
\Psi_{2}\end{pmatrix},$ (3)

which for $\hbar=1$ possesses the form of a Schrödinger equation of the driven
two-level system in the representation frequently found in textbooks for the
Rabi problem (in a gauge without $\varepsilon$ in the diagonal).

For describing LZ transitions of a qubit with time-dependent detuning one
usually uses the diabatic basis, in which the constant tunnel coupling appears
in the off-diagonal matrix elements of the Hamiltonian. As we modulate the
coupling between our pendula, it is convenient to transform into the according
diabatic basis of the in-phase and out-of-phase modes,
$\varphi_{\pm}=(\varphi_{1}\pm\varphi_{2})/2$, in which the constant frequency
difference $\Delta$ appears in the off-diagonal elements of the Hamiltonian.
With $\Psi_{\pm}$ defined in accordance with Eq. (2) the presentation of the
LZ problem for our pendula then reads

$i\frac{d}{dt}\begin{pmatrix}\Psi_{+}\\\
\Psi_{-}\end{pmatrix}=\frac{1}{2}\begin{pmatrix}0&\Delta\\\
\Delta&-2\varepsilon(t)\end{pmatrix}\begin{pmatrix}\Psi_{+}\\\
\Psi_{-}\end{pmatrix}\,.$ (4)

Equations (3) and (4) provide the foundation for comparing the dynamics of
classical pendula with that of a qubit. They describe the occupation
amplitudes of our coupled pendula in the form of a Schrödinger equation in the
two alternative bases $\\{\varphi_{1},\varphi_{2}\\}$ versus
$\\{\varphi_{+},\varphi_{-}\\}$. In Appendices A and B we offer an elegant
alternative derivation based on the Lagrange formulation of classical
mechanics and starting from the non-linear Newton equation. We also
demonstrate there, how the time-dependent quasi static equilibrium,
$\varphi^{\text{qs}}_{k}(t)$, contributes to the static $\varepsilon_{0}$ and
discuss limitations of our approximations.

In a nutshell, Eq. (3) describes —for the case of a time dependent coupling
between the pendula— the dynamics of the occupation amplitudes of the
individual pendula. In this way, the eigenmodes of the uncoupled pendula
directly correspond to the wave functions of the localized sates of a qubit.
Equation (3) is the natural choice for predicting Rabi oscillations occurring
between the occupation amplitudes of the individual pendula for weak coupling.
A basis transformation yields Eq. (4), which describes the dynamics between
the occupation amplitudes of the in-phase and out-of-phase superposition modes
of the individual pendula. Without driving, they resemble the eigenmodes of
two strongly coupled pendula and correspond to the eigenfunctions of a qubit
at zero detuning. Consequently, Eq. (4) is the natural choice for predicting
the dynamics of the occupation amplitudes in the regime of LZ transitions,
where the maximum coupling exceeds the detuning. See Appendix B.3 and Table 1
therein for a one-to-one comparison between the parameters of a qubit and the
pendula.

In our experiments we control $A$ and $\varepsilon_{0}$ by varying the
distance $L$ between the lower magnets, corresponding to the distance between
the pivots, and optionally the distance $L_{\text{u}}$ between the upper
magnets [positioned inside the red horizontal cylindric housings seen in Fig.
1(a)], both defined for $\varphi_{1}=\varphi_{2}=0$. In addition, we adjust
the frequency difference, $\Delta$, by moving a heavy weight [vertical brass
cylinders partly visible in Fig. 1(a)] using a standard thread along one of
the pendulum rods. We employ a line scan camera to simultaneously image at a
rate of 10 Hz the lateral positions of both pendula within a linear pixel
array. Applying numerical filtering we then obtain the displacement angles
$\varphi_{k}(t)$ as a function of time, see Appendix E. The mean frequency of
our pendula is close to $\omega_{0}/2\pi\simeq 0.5$ Hz. To ensure the validity
of the linearized Eq. (1), we work with small deflections
$|\varphi_{k}|<1^{\circ}$, modulation frequencies $\Omega<10^{-2}\omega_{0}$
and frequency differences $|\Delta|<0.1\omega_{0}$. Thereby $\omega_{0}$ is
always the largest frequency by far supporting the separation of time scales
in Eq. (2). The quality factor of $Q>2500$ of the coupled pendula is high
enough to allow us ignoring dissipation as we did in the model above. In order
to achieve high and stable quality factors, we employ professional pendulum
clock pivots based on leaf springs provided by the company Erwin Sattler GmbH
& Co. KG. In detail our experimental results indicate, that the damping of the
coupled pendula motion is dominated by magnetic induction, i.e., eddy currents
induced inside the conducting magnets due to their relative motion. The
friction of air and the deforming pivot springs dominate the damping of the
uncoupled pendula with stable quality factors of $Q>10000$. Note, that we can
easily achieve the strong coupling regime as our maximal coupling of
$|\varepsilon|_{\text{max}}\simeq 0.2$ Hz exceeds the resonance line width of
$\omega_{0}/2\pi Q\sim 10^{-4}\,$Hz by more than three orders of magnitude.

##### Analysis and Discussion

To explore the analogy between our coupled pendula and a qubit we first
perform Rabi oscillations between the individual pendula in the limit of fast
driving with $\Omega\simeq|\Delta|\gg A$, where the driving amplitude becomes
the Rabi frequency, $\Omega_{\text{R}}\simeq A$. After that, we turn to “qubit
manipulation” using LZ transitions between the superposition modes,
$\varphi_{\pm}$, in the limit of slow driving, $\Omega\ll|\Delta|<A$.

In Figs. 2a and 2b

Figure 2: Near resonance Rabi oscillations between the two pendula with mean
frequency $\omega_{0}/2\pi\simeq 528$ mHz, frequency difference
$\Delta/2\pi=11.7\,$mHz and modulation frequency $\Omega/2\pi=11.8\,$mHz. At
$t=0$ pendulum 1 was deflected at maximally attracting lower and no upper
magnets. Individual oscillations are not visible owing to the time axis
covering 45 minutes. (a,b) Deflections $\varphi_{1}(t)$ and $\varphi_{2}(t)$
of the two pendula for the pivot distances $L=496.5\,$mm and $L=330.0\,$mm
resulting in Rabi frequencies of $\Omega_{\text{R}}/2\pi=0.47\,$mHz versus
$\Omega_{\text{R}}/2\pi=3.69\,$mHz. (c,d) Effective frequency
$\Omega_{\text{eff}}(\Delta)$ and visibility $\nu(\Delta)$ of the Rabi
oscillations for $L=454.0\,$mm. The solid lines represent model predictions.

we present Rabi oscillations for two different pivot distances $L$ but
otherwise identical conditions. We use no upper magnets and weak couplings
(large $L$), such that $\varepsilon_{0}\ll A$ and $\varepsilon(t)\simeq
A\cos(\Omega t)$. Shown are the deflections $\varphi_{k}(t)$ in respect to the
equilibrium $\varphi_{k}^{\text{qs}}$. The observed beatings between the
pendula are Rabi oscillations, where the variation between Rabi frequencies in
Fig. 2(a) versus Fig. 2(b) reflect the differences in $L$. In both cases the
energy transfer between the pendula is almost complete, as we chose a near
resonance condition $\Omega\simeq|\Delta|$. Small steps, which occur at the
repetition rate $2\Omega$ [zoom into Figs. 2(a) and 2(b) to clearly see them],
indicate side band transitions beyond the rotating wave approximation. In Fig.
2(b) these steps are bigger due to a larger modulation amplitude compared to
Fig. 2(a).

By varying $\Delta$ we next explore the Rabi dynamics near resonance. In Fig.
2(c) we present the effective Rabi frequency $\Omega_{\text{eff}}(\Delta)$
corresponding to the actual beating frequency. Likewise, in Fig. 2(d) we show
the visibility $\nu(\Delta)$ defined as the fraction of energy exchanged
between the pendula. Symbols are measured data while the lines visualize the
theory predictions,
$\Omega_{\text{eff}}=[\Omega_{\text{R}}^{2}+(\Omega-|\Delta|)^{2}]^{1/2}$ and
$\nu=\Omega_{\text{R}}^{2}/\Omega_{\text{eff}}^{2}$. The only fit parameter is
the Rabi frequency $\Omega_{\text{R}}/2\pi=0.71\,$mHz, which defines the
minimum of $\Omega_{\text{eff}}$ at resonance $\Omega=\Delta$ and which can be
used to accurately determine the magnetic moment $m_{\text{l}}$, see Appendix
C. The excellent agreement between theory and experiment underlines the high
quality of our classical mechanics experiment. Since the model curves can be
derived from the Schrödinger equation (3), the result establishes a first
analogue between classical pendula and a qubit.

An elegant method to manipulate qubits in the limit of slow modulation,
$|\Delta|\gg\Omega$, are LZ transitions [30, 31]. In Fig. 3(a)

Figure 3: LZ transitions: At $t=0$ both pendula where deflected to excite the
out-of-phase mode $\varphi_{-}$ at maximally attracting lower and no upper
magnets. (a–c) Measured deflection angles $\varphi_{1,2}(t)$ (adiabatic modes)
in a, diabatic modes $\varphi_{\pm}(t)$ in b and their occupations
$P_{\pm}(t)$ in c during the first period $T=2\pi/\Omega$ of modulation for
$\Omega/2\pi=2.27\,$mHz, $\Delta/2\pi=6.7\,$mHz, $\omega_{0}/2\pi=0.53$ Hz,
and $P_{\text{LZ}}\simeq 0.4$. Two LZ transitions occur as the pendula pass
through avoided crossings at $\varepsilon=0$ indicated by vertical dotted
lines. The occupations $P_{\pm}(t)$ as well as the beating dynamics clearly
change at each avoided crossing. (d) Sketch of the avoided crossing. Solid
lines are the eigenfrequencies
$(\mp\sqrt{\Delta^{2}+\varepsilon^{2}}-\varepsilon)/2$ and dashed lines
indicate the frequencies of the envelope functions $\Psi_{\pm}(\varepsilon)$.
(e) Mean population $\overline{P}_{+}(T/2)$ averaged over a proper time window
around $t=T/2$ in between the first two crossings. Both, $\Delta$ and the LZ
speed $v$ are varied between individual measurements. The black line follows
$1-P_{\text{LZ}}$. Our initialization at finite $\varepsilon(t=0)$ causes a
small initial population of the upper mode $\varphi_{+}$, which varies from
measurement to measurement in amplitude and phase. The gray region indicates
the corrected range of prediction accounting for the range of experimental
parameters by assuming arbitrary initial phases.

we present the deflection angles $\varphi_{k}$, while one of the magnets
completes one full rotation. Within this driving period, the pendula pass
twice through the avoided crossing at zero coupling, sketched in Fig. 3(d),
namely from positive to negative $\varepsilon$ and back. At $t=0$, we
initialized $\varphi_{1}=-\varphi_{2}$, such that $\varphi_{+}=0$. This is
evident in Fig. 3(b) plotting the in-phase and out-of-phase combinations
$\varphi_{\pm}=(\varphi_{1}\pm\varphi_{2})/2$. In Fig. 3(c) we present the
according populations $P_{\pm}$, given by the square modulus of the slowly
varying amplitude, $P_{\pm}\propto|\Psi_{\pm}|^{2}$, normalized such that
$P_{+}+P_{-}=1$. Around the two avoided crossings (indicated by vertical
dotted lines) we observe LZ transitions. The first one, which mixes
$\varphi_{+}$ and $\varphi_{-}$, is followed by beats with the time dependent
frequency $\sqrt{\Delta^{2}+\varepsilon^{2}(t)}$ clearly visible between
$\varphi_{1}$ and $\varphi_{2}$ as well as between $P_{-}$ and $P_{+}$. The
latter beats confirm theoretical predictions [17, 18, 19], namely chirped
oscillations centered around the LZ probability
$P_{\text{LZ}}=\exp(-\pi\Delta^{2}/2v)$ and $1-P_{\text{LZ}}$, respectively,
right after passing through the avoided crossing. Here,
$v=\Omega(A^{2}-\varepsilon_{0}^{2})^{1/2}$ is the sweep velocity at
$\varepsilon=0$, which depends on $L$, $L_{\text{u}}$ and $\Omega$. This
observation demonstrates an advantage of our classical two-level system,
which—in contrast to a qubit—allows us to trace the time evolution of
population probabilities in real time in a single shot measurement. Based on
the comparison with theory, we identify the long-time transition probability
of a single LZ transition by averaging out the beats of the measured
occupation $P_{+}(t)$ within an appropriate time window after half of the
modulation period $T=2\pi/\Omega$, i.e., centered between the two LZ
transitions. In Fig. 3(e) we compare the resulting $\overline{P}_{+}(T/2)$ for
a wide range of the parameters $\Delta$, $L$, $L_{\text{u}}$ and $\Omega$ with
the classic result $1-P_{\text{LZ}}$ for the limit $t\to\infty$ [3, 4, 5, 6].

The agreement between model and measurements is good albeit compared to the
Rabi experiment above the data scatter considerably around the model line.
These deviations can be explained with the initialization into $\varphi_{-}$
at finite $\varepsilon$, which for $\Delta\neq 0$ is not an eigenmode, as it
would be for an initialization at $\varepsilon\to-\infty$ [3, 4]. The weak
admixture of the second eigenmode gives rise to a weak beating between
$\varphi_{\pm}(t)$ right after initialization as visible in Fig. 3(b) for
$t\lesssim 1\,$min. Treating the relative phase between the modes
$\varphi_{\pm}(t)$ (which could be predetermined at the cost of additional
experimental effort) as an unknown, we predict the range of possible values of
$\overline{P}_{+}(\pi/\Omega)$ for arbitrary phases [gray area in Fig. 3(e)].
In the adiabatic limit, $\Delta^{2}/v\gg 1$, independent of the relative phase
the finite occupation of the upper eigenmode, $P_{+}(t=0)>0$, results in
$\overline{P}_{+}(\pi/\Omega)<1$ while $P_{\text{LZ}}=1$.

Note, that in a corresponding experiment with an actual qubit the
initialization procedure would be similar, such that the phase problem
described above occurs as well. However, only the classical qubit analog
allowed us to perform continuous measurements as those in Fig. 3(c), which
helped us to fully determine the influence of the non-zero phase at
initialization. This result is an example of the usefulness of our classical
approach for deciphering the sometimes complex dynamics of qubits.

Each LZ transition mixes the eigenmodes as demonstrated in Fig. 3(e). The
resulting superposition state then accumulates the adiabatic phase
$\int\text{d}t(\Delta^{2}+\varepsilon(t)^{2})^{1/2}$ integrating the
difference between the two eigenfrequencies with the Stokes phase added [27].
The second LZ transition is then heavily influenced by these phases [5, 6].
Indeed, multiple LZ transitions result in a complicated time evolution of
$P_{+}(t)$ as can be seen in Fig. 4(a),

Figure 4: LZSM interference: (a) Measured occupation $P_{+}(t)$ of the in-
phase mode $\varphi_{+}$, while one magnet was rotated at
$\Omega=$7.1\text{\,}\mathrm{m}\mathrm{H}\mathrm{z}$$ with (blue) and without
(red) static magnets. At $t=0$, $\varphi_{-}$ was excited as in Fig. 3. (b)
Occupation $P_{+}$ averaged over the first 5 driving periods as a function of
the average detunging $\epsilon_{0}$ and the effective driving amplitude $A$.
The data are computed using Eq. (4). (c,d) Slices through the LZSM fan along
the lines marked in panel b. Dots correspond to experimental results, while
solid black lines show theory data from panel b. The colored dots in panel b
and the arrows in panels c and d indicate the values used in panel a.

which presents two example time traces of $P_{+}(t)$. While the modulation
frequency is identical for both measurements, $\Omega/2\pi=7.1\,$mHz, we
varied $A$ and $\varepsilon_{0}$. Each trace covers five modulation periods
corresponding to ten LZ transitions. Most of them are clearly visible as more
or less pronounced steps while for some transitions $P_{+}(t)$ stays almost
unchanged beyond the perpetual beating. The steady state solution for
continuous driving averaging $P_{+}(t)$ over many periods gives rise to LZSM
interference patterns $\overline{P}_{+}(A,\varepsilon_{0})$, which can be used
for exploring qubit dynamics, decoherence or multi-color driving [23, 24]. For
a practical comparison, allowing for small deviations, we average $P_{+}(t)$
over the initial five modulation periods. In Fig. 4(b) we present the LZSM
interference pattern $\overline{P}_{+}(A,\varepsilon_{0})$ as a gray scale,
which we computed using the Schrödinger equation (4). In Figs. 4(c) and 4(d)
we finally present interference traces along the two solid lines in Fig. 4(b),
mind the color code. In Fig. 4(c) we plot $P_{+}(L^{5})$ for the case without
upper magnets, while for Fig. 4(d) we added the upper magnets and show
$P_{+}(L_{\text{u}}^{5})$ for a constant pivot distance, $L=246$ mm. Solid
lines are model curves calculated with Eq. (4) [contained in the gray scale
plot in Fig. 4(b)], while the dots present measured interference patterns.
Hereby, each point corresponds to the average of a $P_{+}(t)$ trace as those
shown in Fig. 4(a). Our measurements qualitatively reproduce the calculated
interference fringes. Quantitative deviations indicate the limitations of the
mapping of the Newton equation onto the Schrödinger equation, in particular
for large $A$ and $\epsilon_{0}$. In Appendix D we provide related background
information.

##### Summary

Wave mechanics introduced by Erwin Schrödinger in 1926 [32] provides a
mathematical description for the coherent dynamics of a qubit. However, a
continuous experimental visualization of its time evolution is hindered by the
principle of projection measurement; each measurement would destroy the
quantum coherence. In comparison, our classical qubit analog surely allows to
trace the complete time evolution in a single measurement. In order to
actually visualize Schrödingers wave mechanics using physical pendula with
modulated coupling one has to map the non-linear, second order and
inhomogeneous classical EOM to the linear, first order and homogeneous
Schrödinger equation. This mapping, which includes a linearization, a rotating
wave approximation and a time-dependent shift of the reference point, also
clarifies the experimental conditions necessary for classical qubit simulator
experiments and their physical interpretations. In this spirit, we presented
three key qubit experiments with coupled pendula, namely Rabi oscillations, LZ
transitions and LZSM interferometry. Comparing measurements with the
prediction of the Schrödinger equation we demonstrated that our classical
experiments directly visualize Schrödinger’s wave mechanics. Our classical
qubit simulator bridges the gap between the elusiveness of quantum mechanics
and the common imagination pre-shaped by classical experiences. The
experimental setup is highly versatile and might be used for exploring a
variety of phenomena beyond simulating a qubit, such as geometrical phases
[33], after adding more pendula, multi-level systems [34, 35] or coherent
transfer by adiabatic passage [36] or simulating the non-linear Schrödinger
equation [37]. Moreover, driven systems of coupled pendula may serve as
visualizer of a large variety of coupled systems in nature or even economical,
social or financial systems. However, a limit to any classical model system is
imposed by entanglement of multiple qubits, which has no classical analog.

##### Acknowledgements

The authors thank W. Kurpas and S. Manus for technical support. H.L. thanks K.
Pfeffer and R. Buchholz for frequent discussions of the ongoing experiments.
This work was financially supported by the Center for NanoScience (CeNS) at
LMU Munich, by the Spanish Ministry of Science and Innovation through Grant.
No. PID2020-117787GB-I00, the CSIC Research Platform on Quantum Technologies
PTI-001 and the Institute for Basic Science in Korea (IBS-R024-D1). The work
of M.K. is conducted within the framework of the Trieste Institute for
Theoretical Quantum Technologies (TQT).

##### Author’s contributions

S.L. and M.K. initiated the project and planned it together with H.L. The
experimental setup of the coupled pendula was designed by H.L. and S.L. H.L.
performed the experiments. The data were analyzed by all authors. M.K. and
S.K. formulated the theoretical model. S.K., M.K., and A.P. computed the
theoretical data. S.K., M.K. and S.L. wrote the article including the
appendices.

##  Appendix A Newton equation

Figure 5: Sketch (a) and photograph (b) of the pendula, which are coupled by
cubic neodymium magnets. The sketch visualizes the geometry of the pendula,
the orientations of the magnets as well as the angles and lengths used in the
derivation of the interaction potential. The two lower magnets with
magnetizations $\vec{m}_{1}$ and $\vec{m}_{2}$ (with
$|\vec{m}_{1}|=|\vec{m}_{2}|$) are attached at the end of the pendula to
motors, which are encapsulated together with batteries and a circuit board by
half transparent polethylene cylinders.The upper magnets, which have an
approximately four times smaller magnetization, can be horizontally moved
within the red ceramic cylinders in order to adjust their distance
independently from the distance $L$ between the pivots. With their
magnetizations put parallel and in line they attract each other. Brass nuts
attached to the cylinders have the purpose of balancing the weights of the
magnets such that the center of masses stay within the pendulum rods. The 2.1
kg heavy brass weights above the motors can be vertically moved to tune the
pendula’s eigenfrequencies. The line scan camera in the background
stroboscopically monitors the horizontal position of both pendula as a
function of time using narrow reflectors (not seen).

To derive the equation of motion, we employ the Lagrange formalism. For two
uncoupled physical pendula the Langrangian reads [38]

$\mathcal{L}_{k}=\frac{1}{2}J_{k}\dot{\varphi}_{k}^{2}+J_{k}\omega_{k}^{2}\cos\varphi_{k},$
(5)

where $k=1,2$ labels the two so-far uncoupled pendula and the deflection
angles $\varphi_{k}$ are zero for vertically hanging pendula described by
their moments of inertia $J_{k}$. In the second term, which is the negative
potential energy, we have already made use of the fact that the eigenfrequency
reads $\omega_{k}^{2}=gM(l_{\text{c}})_{k}/J_{k}$, with the gravity
acceleration $g$, the mass $M$, which is equal for both pendula, and the
distance $(l_{\text{c}})_{k}$ between the pivot and the center of mass of each
pendulum. Note, that up to Eq. (19), $\varphi_{k}$ denotes the deflection
angles in the lab frame with the vertical direction as reference.

###  A.1 Coupling by magnets

The two pendula are coupled by either one or two pairs of permanent neodymium
magnets as sketched in Fig. 5(a) for the case of one pair of magnets. The
cubical magnets can be seen in the photograph in Fig. 5(b). Two large magnets
with 28 mm edge lengths are fixed to the end of each rod and can be rotated by
battery driven motors. Two smaller magnets are positioned inside the red
cylinders 0.513 m above. All magnet cubes are magnetized along their
horizontal four-fold axes, which at all times remain fixed within the
oscillation plane with the excumption of one of the lower magnets optionally
rotated around the rod it is attached to.

With the origin set to the pivot of the left pendulum ($k=1$), the magnets
have positions

$\vec{r}_{1}=\begin{pmatrix}l\sin\varphi_{1}\\\ -l\cos\varphi_{1}\\\
0\end{pmatrix},\quad\vec{r}_{2}=\begin{pmatrix}L+l\sin\varphi_{2}\\\
-l\cos\varphi_{2}\\\ 0\end{pmatrix},$ (6)

where $L$ is the horizontal distance between the pivots of the pendula, and
$l$ is the distance between magnet and pivot, which is equal for both pendula.
Their magnetic dipole moments of identical magnitude $m$ are oriented as

$\vec{m}_{1}=m\begin{pmatrix}\cos\varphi_{1}\\\ \sin\varphi_{1}\\\
0\end{pmatrix},\quad\vec{m}_{2}=m\begin{pmatrix}\cos\varphi_{2}\cos(\Omega
t)\\\ \sin\varphi_{2}\cos(\Omega t)\\\ \sin(\Omega t)\end{pmatrix},$ (7)

valid for large enough $L$ as is the case in all our experiments.

The magnetic dipole-dipole coupling energy is [39]

$\mathcal{U}_{12}=\frac{\mu_{0}}{4\pi
R^{3}}[\vec{m}_{1}\cdot\vec{m}_{2}-3(\vec{m}_{1}\cdot\vec{e}_{R})(\vec{m}_{2}\cdot\vec{e}_{R})]\,,$
(8)

where $R\vec{e}_{R}=\vec{R}\equiv\vec{r}_{2}-\vec{r}_{1}$. To express this in
terms of our dynamical variables $\varphi_{k}$, we insert the vectors in Eqs.
(6) and (7) and obtain the potential energy of the coupling to read
$\mathcal{U}_{12}=\mathcal{V}_{12}\cos(\Omega t)$, where

$\mathcal{V}_{12}=\frac{\mu_{0}m^{2}}{4\pi
R^{3}}[\cos(\varphi_{1}-\varphi_{2})-3\cos(\varphi_{1}-\psi)\cos(\varphi_{2}-\psi)]\,.$
(9)

Length and orientation of $\vec{R}$ follow from straightforward geometric
considerations and read

$\displaystyle R={}$
$\displaystyle\sqrt{(L+l\sin\varphi_{2}-l\sin\varphi_{1})^{2}+(l\cos\varphi_{2}-l\cos\varphi_{1})^{2}},$
$\displaystyle\psi={}$
$\displaystyle\arctan\left(\frac{\cos\varphi_{1}-\cos\varphi_{2}}{L/l-\sin\varphi_{1}+\sin\varphi_{2}}\right)\,,$
(10)

where $\psi$ is the angle between the $x$-axis and $\vec{R}$, cf. Fig. 5. In
some of our experiments, we use two pairs of magnets, i.e., a lower pair
connected to the ends of the rods and an upper pair with a variable horizontal
distance shifted upwards along the rods. Consequently, we introduce the
respective distances $l_{\text{l}}$ (replacing $l$) and $l_{\text{u}}$ between
pivots and lower versus upper magnets. Hereby, $l_{\text{l}}-l_{\text{u}}$
exceeds the horizontal distance $L$ roughly by a factor two, such that we can
neglect the interaction between the upper and the lower magnets and consider
two separate dipole-dipole couplings. The upper magnets orientations are
fixed, such that their magnetizations are perpendicular to the respective
rods, remain in the $x$-$y$ plane and attract each other. In contrast, one of
the lower magnets is rotated such that

$\mathcal{U}_{12}=\mathcal{V}_{12}^{\text{u}}+\mathcal{V}_{12}^{\text{l}}\cos(\Omega
t).$ (11)

The equations of motion follow readily from the Lagrange equation [38]

$\frac{d}{dt}\frac{\partial\mathcal{L}}{\partial\dot{\varphi}_{k}}=\frac{\partial\mathcal{L}}{\partial\varphi_{k}}.$
(12)

For the full Lagrangian,
$\mathcal{L}=\mathcal{L}_{1}+\mathcal{L}_{2}-\mathcal{U}_{12}$, they become

$J_{k}\ddot{\varphi}_{k}+J_{k}\omega_{k}^{2}\sin\varphi_{k}=-\frac{\partial\mathcal{U}_{12}}{\partial\varphi_{k}},\quad
k=1,2.$ (13)

The derivative of the interaction potential is straightforward to calculate
but results in a bulky expression, whose explicit form is not needed for our
discussion. Equations (13) can be integrated numerically to obtain a full
solution of the pendula dynamics which considers all nonlinearities.

###  A.2 Linearization and quasistatic solution

The mapping of Newton’s equation of motion to a Schrödinger equation requires
a bilinear Lagrangian, as will become clear in Sec. B below. In our
experiment, the deflection angles are rather small, $|\varphi_{k}|\lesssim
1^{\circ}$, such that already a linearized version of the equation of motion
provides a faithful description. We will perform this linearization by an
expansion for the Lagrangian up to second order in $\varphi_{k}$. Then the
Lagrangian of the uncoupled pendula, Eq. (5), takes the bilinear form
$\mathcal{L}_{k}=J_{k}(\dot{\varphi}_{k}^{2}-\omega_{k}^{2}\varphi^{2})/2$,
where we have omitted an irrelevant constant term. The expansion of the
interaction term (9) requires more effort, because the $\varphi_{k}$ appear in
both the numerator and the denominator.

Closer inspection of the expressions, however, reveals that the
$\varphi_{k}$-dependence of the numerator is much weaker than that of the
denominator, given by $R^{-3}(\varphi_{k})$, see Eq. (10). The difference
stems from the numerical pre-factors of the second-order terms of the Taylor
expansions of $(1-x)^{-3}$ and $\cos(x)$, which are $12$ and $1/2$,
respectively. Moreover, in $R$ the $\varphi_{k}$ are weighted by a factor
$l/L$, which for our experiment is $\sim 4$. This leads to the conclusion that
the numerator is responsible for only $\sim 1\%$ of the curvature of
$\mathcal{U}_{12}$. Hence, we neglect the higher order terms of the numerator
and approximate
$\cos(\varphi_{1}-\varphi_{2})-3\cos(\varphi_{1}-\psi)\cos(\varphi_{2}-\psi)\simeq-2$.
However, we do consider the terms stemming from the expansion of the
denominator,

$\begin{split}R^{-3}={}&[L-l\varphi_{1}+l\varphi_{2}+\mathcal{O}(\varphi^{4})]^{-3}\\\
\simeq{}&L^{-3}\left(1+3\alpha+6\alpha^{2}+10\alpha^{3}\right)\end{split}$
(14)

with $\alpha=(\varphi_{1}-\varphi_{2})l/L$. Inserting the relevant terms up to
second order into Eq. (9) yields the Langrangian of the linearized problem

$\mathcal{L}=\sum_{k=1,2}\frac{J_{k}}{2}(\dot{\varphi}_{k}^{2}-\omega_{k}^{2}\varphi_{k}^{2})+\tilde{F}(\varphi_{1}-\varphi_{2})+\frac{\tilde{G}}{2}(\varphi_{1}-\varphi_{2})^{2}.$
(15)

For our two pairs of magnets, where one of the lower magnets is rotating, the
energies $\tilde{G}$ and $\tilde{F}$ take the form

$\begin{split}\tilde{G}(t)={}&G_{\text{u}}+G_{\text{l}}\cos(\Omega t),\\\
\tilde{F}(t)={}&\frac{L_{\text{u}}}{4l_{\text{u}}}G_{\text{u}}+\frac{L}{4l_{\text{l}}}G_{\text{l}}\cos(\Omega
t),\end{split}$ (16)

where the tilde indicates parameters stemming from second-order Taylor
expansion of the Lagrangian. Equation (16) neglects the time-dependent
equilibrium positions of the pendula, which we will take into account as a
correction below. The coefficients in Eq. (16), which quantify the interaction
energies are determined by $\mathcal{U}_{12}$ and read

$\begin{split}G_{\text{u}}={}&\frac{6\mu_{0}m_{\text{u}}^{2}l_{\text{u}}^{2}}{\pi
L_{\text{u}}^{5}},\\\
G_{\text{l}}={}&\frac{6\mu_{0}m_{\text{l}}^{2}l_{\text{l}}^{2}}{\pi
L^{5}},\end{split}$ (17)

where $L_{\text{u}}$ is the distance between the upper magnets at
$\varphi_{1}=\varphi_{2}=0$. $L_{\text{u}}$ is varied between experiments and
$L_{\text{u}}=\infty$ corresponds to the case without upper magnets.

To bring $\mathcal{L}$ into a bilinear form, we transform the coordinates such
that the linear term $\tilde{F}(\varphi_{1}-\varphi_{2})$ on the right-hand
side of Eq. (15) vanishes. This can be achieved by introducing coordinates
relative to the potential minimum which is located at

$\begin{split}\varphi_{1}^{\text{qs}}(t)={}&\frac{J_{2}\omega_{2}^{2}\tilde{F}(t)}{J_{1}J_{2}\omega_{1}^{2}\omega_{2}^{2}-\tilde{G}(t)(J_{1}\omega_{1}^{2}+J_{2}\omega_{2}^{2})},\\\
\varphi_{2}^{\text{qs}}(t)={}&-\frac{J_{1}\omega_{1}^{2}\tilde{F}(t)}{J_{1}J_{2}\omega_{1}^{2}\omega_{2}^{2}-\tilde{G}(t)(J_{1}\omega_{1}^{2}+J_{2}\omega_{2}^{2})},\end{split}$
(18)

where the superscript “qs” refers to the quasistatic solution. The adiabatic
equilibrium positions $\varphi_{k}^{\text{qs}}$ implicitly depend on the
center-of-mass coordinates since $J_{k}\omega_{k}^{2}=Mg(l_{\text{c}})_{k}$.

Our driving frequency is much smaller than the resonance frequencies of the
two pendula, $\Omega/\omega_{0}\lesssim 0.01$, where
$\omega_{0}=(\omega_{1}+\omega_{2})/2$. Hence, within an adiabatic
approximation, Eq. (18) describes the steady-state solution of the coupled
pendula driven by the rotation of one of the magnets. Henceforth, we use
$\varphi_{k}^{\text{qs}}(t)$ as reference point, which is achieved by the
transformation

$\varphi_{k}\to\varphi^{\text{lab}}_{k}=\varphi_{k}+\varphi_{k}^{\text{qs}}(t)\,.$
(19)

Moreover, we neglect within an adiabatic approximation the time derivatives of
$\varphi_{k}^{\text{qs}}(t)$ which removes the linear term in Eq. (15) and
results in the desired intended bilinear Lagrangian.

By applying the transformation (19) the corresponding equation of motion
becomes not only of first order and linear, but also homogeneous. Accordingly,
it assumes the form of a Schrödinger equation. However, the coordinates
$\varphi_{k}$ are no longer in the lab frame. Instead, they are the deflection
angles with respect to the time-dependent quasistatic solution in Eq. (18).

###  A.3 Dynamic potential curvature

The price for separating $\varphi_{k}$ from the time dependence of the
equilibrium positions of the pendula is that for the new coordinates the
interaction becomes time dependent even for constant $\varphi_{1}$ and
$\varphi_{2}$. In a hand-waving picture, while one magnet is rotating the
equilibrium distance between the magnets is smaller whenever the interaction
is attractive as compared to the case of repulsive interaction. As a result,
the interaction itself is accordingly modulated for any values of
$\varphi_{1}$ and $\varphi_{2}$. We will now quantify this correction.

So far, we have linearized the equation of motion by performing a Taylor
expansion of the interaction potential at the pivot distance $L$. However,
after the transformation (19), the potential curvature at
$\varphi_{1}=\varphi_{2}$ is no longer given by the energy $\tilde{G}(t)$ in
Eq. (16), but by the corresponding Taylor coefficient evaluated at
$L-l_{\text{u}}\delta\varphi^{\text{qs}}(t)$ for $G_{\text{u}}$ and
$L-l_{\text{l}}\delta\varphi^{\text{qs}}(t)$ for $G_{\text{l}}$. This can be
captured by the approximate correction

$\begin{split}\tilde{G}(t)\to
G(t)={}&G_{\text{u}}\Big{[}1-\frac{l_{\text{u}}}{L_{\text{u}}}\delta\varphi^{\text{qs}}(t)\Big{]}^{-5}\\\
&+G_{\text{l}}\cos(\Omega
t)\Big{[}1-\frac{l_{\text{l}}}{L}\delta\varphi^{\text{qs}}(t)\Big{]}^{-5}.\end{split}$
(20)

Clearly, the interaction term acquires a further time dependence via
$\delta\varphi^{\text{qs}}(t)\equiv\varphi_{1}^{\text{qs}}(t)-\varphi_{2}^{\text{qs}}(t)$
in addition to the modulation expressed in Eq. (16). The explicit dependence
of $G(t)$ on $\delta\varphi^{\text{qs}}(t)$ results in an asymmetry between
the times of attractive versus repulsive magnetic force even without upper
magnets.

To see the consequences of this correction, we restrict ourselves to weak
asymmetries such that the pendulum frequencies and moments of inertia can be
replaced by their average values $\omega_{0}$ and $J_{0}=(J_{1}+J_{2})/2$.
Then we arrive at the approximation

$\varphi_{1}^{\text{qs}}(t)=\frac{\tilde{F}(t)}{\omega_{0}^{2}J_{0}-2\tilde{G}(t)}=-\varphi_{2}^{\text{qs}}(t),$
(21)

which allows the numerical evaluation of the effective potential curvatures
using Eq. (20).

To make analytical progress, we restrict ourselves to the case in which only
the rotating lower magnets are present, while $G_{\text{u}}=0=F_{\text{u}}$.
Our aim is to show that, in consistency with the experimental observation,
nevertheless the effective potential curvature has a non-vanishing mean value.
In doing so, we keep only corrections to lowest order in $G(t)$ and, hence, in
$\delta\varphi^{\text{qs}}(t)$ such that the effective potential curvature
becomes

$\begin{split}G(t)&\simeq G_{\text{l}}\cos(\Omega
t)\Big{[}1+\frac{5}{2}\frac{G_{\text{l}}\cos(\Omega
t)}{\omega_{0}^{2}J_{0}}\Big{]}\\\ &=G_{\text{l}}\cos(\Omega
t)+\frac{5G_{\text{l}}^{2}}{4\omega_{0}^{2}J_{0}}[1+\cos(2\Omega
t)].\end{split}$ (22)

The first term describes the modulation of the interaction by the rotating
magnet in accordance to Eq. (16). The second constant term describes the
average increase of the interaction related to the fact that the dipole-dipole
interaction is more enhanced during attraction than reduced during repulsion.
The third term describes a second harmonic modulation in $\Omega$. The latter
merely distorts the shape of the driving and can be neglected on the level of
agreement we are aiming at.

Let us emphasize that the derivation of the correction (20) relies on the
assumption that the nonlinear $\varphi_{k}$ dependence of the interaction
$\mathcal{U}_{12}$ can be captured by a time-dependent quadratic term. While
this reasoning naturally has limitations, it clearly reveals that the varying
potential curvature causes an effective constant interaction term, even in the
absence of the upper magnets. It explains the observed dependences
$\varepsilon_{0}(L)$ and $A(L)$ without upper magnets, cf. the red line in
Fig. 4(b) or Fig. 11 below.

##  Appendix B Schrödinger-like Equation

Next, we bring the linearized equation of motion of our system to the form of
a Schrödinger equation that describes the time-dependent amplitude of the fast
oscillations of each pendulum with approximately the average frequency
$\omega_{0}=(\omega_{1}+\omega_{2})/2$. The pedestrians way [16, 1, 8] starts
from the linearized classical equation of motion which is of second order. To
obtain a differential equation of first order, one employs for the deflection
angles the complex ansatz $\varphi_{k}=\Re
e^{-i\omega_{0}t}\Psi_{k}+\text{c.c.}$, $k=1,2$, where $\Psi_{k}$ is a slowly
varying amplitude. Within a rotating-wave approximation, one then neglects the
second-order derivatives of $\Psi_{k}$ and all terms that oscillate with the
angular frequency $\omega_{0}$.

Here we pursue a more elegant alternative by performing these steps within the
Lagrange formalism. Accordingly, our goal is to transform Eq. (15) such that
it assumes the form of the Lagrangian of the Schrödinger equation
$i\hbar\partial_{t}\Psi=H\Psi$, which reads

$\mathcal{L}_{\text{Schr}}=i\hbar\Psi^{\dagger}\partial_{t}\Psi-\Psi^{\dagger}H\Psi,$
(23)

where $\Psi$ and $H$ denote the probability amplitudes and the Hamiltonian in
vector-matrix notation. Alternatively, one may use the symmetrized form of the
Lagrangian,
$\mathcal{L}_{\text{Schr}}=i\hbar(\Psi^{\dagger}\partial_{t}\Psi-\Psi\partial_{t}\Psi^{\dagger})/2-\Psi^{\dagger}H\Psi$,
which differs from Eq. (23) by an irrelevant total time derivative. Its
relation to the Schrödinger equation follows readily from the Langrange
equation

$\frac{d}{dt}\frac{\partial\mathcal{L}}{\partial\dot{\Psi}_{k}^{*}}=\frac{\partial\mathcal{L}}{\partial\Psi_{k}^{*}}.$
(24)

The fact that $\mathcal{L}_{\text{Schr}}$ is bilinear, makes it obvious that
it was indispensable to remove the term of Eq. (15) linear in $\varphi_{k}$
via the transformation (19) and to avoid explicit terms of higher order.

Let us stress that the resulting equation still describes classical mechanics
and, despite its form, does not constitute a quantization. In particular, the
quantum mechanical energy-frequency relation given by Planck’s constant does
not hold. Technically, this is not a problem as long as we use a Schrödinger
equation in the dimensions of frequency. If we assume $\hbar=1$, it is
identical to the usual quantum version of the Schrödinger equation in units of
frequency.

###  B.1 Hamiltonian and conservation law

Like in the standard approach, we assume that the average pendulum frequency
$\omega_{0}$ is much larger than all other frequency scales, which suggests an
ansatz with a rapid oscillation and a slowly varying amplitude. To be
specific, we define

$\varphi_{k}(t)=\Psi_{k}(t)e^{-i\omega_{0}t}+\text{c.c.},$ (25)

where $\Psi_{k}$ is generally complex. While inserting this ansatz into Eq.
(15), we keep only terms that are at least of order $\omega_{0}$. (The ansatz
is constructed such that terms $\propto\omega_{0}^{2}$ cancel each other.)
Moreover, we neglect within a rotating-wave approximation all terms that
contain the phase factor $e^{\pm i\omega_{0}t}$.

In the resulting expression, the remaining part of the first term of
$\mathcal{L}$ in Eq. (15) becomes
$2i\omega_{0}J_{k}\Psi_{k}\partial_{t}\Psi_{k}$. It is still not of the
desired form, because its pre-factor still depends via the moment of inertia
on the mode index $k$. To reach the form of a Schrödinger equation, one needs
pre-factors independent of the mode index $k$, e.g., by re-scaling the
amplitude with a factor $J_{k}^{-1/2}$. This however would no longer allow the
intuitive interpretation of the $\Psi_{\pm}$ defined below as in-phase and
out-of-phase modes. To nevertheless get rid of the $k$-dependence of the pre-
factors, we replace the $J_{k}$ by their average $J_{0}=(J_{1}+J_{2})/2$. This
is a reasonable approximation for most of our experiments, as they fulfill
$|J_{1}-J_{2}|\ll(J_{1}+J_{2})$. Finally, for convenience we divide by
$2\omega_{0}J_{0}$ (which has no consequence for the equations of motion) to
obtain a Lagrangian with dimension frequency

$\mathcal{L}=i\sum_{k}\Psi_{k}^{*}\dot{\Psi}_{k}-\frac{\Delta}{2}(|\Psi_{1}|^{2}-|\Psi_{2}|^{2})+\frac{\varepsilon}{2}|\Psi_{1}-\Psi_{2}|^{2},$
(26)

where we have introduced the frequency difference

$\Delta=\omega_{1}-\omega_{2}$ (27)

and the time dependent coupling (in units of frequency)

$\varepsilon(t)=\frac{G(t)}{\omega_{0}J_{0}}.$ (28)

Comparison with Eq. (23) demonstrates that the Lagrangian in Eq. (26)
corresponds to a Schrödinger equation of a two-level-system (TLS) with the
Hamiltonian

$H=\frac{1}{2}\begin{pmatrix}\Delta-\varepsilon(t)&\varepsilon(t)\\\
\varepsilon(t)&-\Delta-\varepsilon(t)\end{pmatrix}$ (29)

in units of frequency with $\hbar=1$.

The $U(1)$ symmetry of the Lagrangian (26) together with Noether’s theorem
immediately provides the conservation of
$N\equiv|\Psi_{+}|^{2}+|\Psi_{-}|^{2}$. Then the energy
$\mathcal{E}=\mathcal{T}+\mathcal{U}$ of the pendula obeys the proportionality

$\mathcal{E}=\omega_{0}^{2}N+2\omega_{0}\langle H\rangle.$ (30)

It is dominated by the constant term $\omega_{0}^{2}N$, while the expectation
value $\langle H\rangle$ represents a small, generally time-dependent
correction to the energy of the driven coupled pendula. Because of the
driving, normalization of the wave function corresponds to a merely
approximate energy conservation of the pendula motion.

###  B.2 Basis transformation

The preferential basis in most works on quantum dots is the one formed by
localized states. Then ac voltages applied via plunger gates appear in the
Hamiltonian as time-dependent diagonal elements. To establish a closer
connection to these systems, we introduce the in-phase and out-of-phase mode
$\varphi_{\pm}=\varphi_{1}\pm\varphi_{2}$ and the corresponding envelopes
$\Psi\pm=\Psi_{1}\pm\Psi_{2}$. This corresponds to a unitary transformation
with $S=(\sigma_{x}+\sigma_{z})/\sqrt{2}$, which formally interchanges the
Pauli matrices $\sigma_{x}$ and $\sigma_{z}$. With $H\to SHS^{\dagger}$ our
Schrödinger equation takes the form

$i\frac{d}{dt}\begin{pmatrix}\Psi_{+}\\\
\Psi_{-}\end{pmatrix}=\begin{pmatrix}0&\Delta/2\\\
\Delta/2&-\varepsilon(t)\end{pmatrix}\begin{pmatrix}\Psi_{+}\\\
\Psi_{-}\end{pmatrix},$ (31)

which represents the basis of the analogy between the pendula and the quantum
mechanical two-level system.

###  B.3 Duality and limitations

Table 1: Correspondence between a qubit described by the Schrödinger equation and the classical coupled pendula after the rotating-wave approximation, which alters the EOM independent of the carrier frequency $\omega_{0}=(\omega_{L}+\omega_{R})/2$. Each line contains two corresponding quantities. two-level system | coupled pendula
---|---
eigenstates | normal modes
tunnel oscillations | beating
tunnel coupling $\Delta$ | frequency diff. $\Delta=\omega_{1}-\omega_{2}$
energy detuning $\varepsilon(t)$ | interaction $\varepsilon(t)$
localized states | in-phase/out-of-phase mode
delocalized states | left/right pendulum mode
amplitude of wavefunctions | amplitude of pendula
occupation probability | occupation $\propto$ energy

We have chosen the diabatic basis spanned by the in-phase and out-of-phase
modes. The corresponding delocalized basis for the Hamiltonian in Eq. (31)
corresponds to a localized basis usually employed for LZSM physics with a
qubit, e.g., the one defined by the localized electron states in a double
quantum dot [23]. In Table 1 we present the cross relation between the
localized and delocalized bases corresponding to mapping our coupled pendula
to a double dot qubit.

An interesting point arises from the fact that we modulate the coupling
between the pendula while in qubits usually the detuning between localized
states is modulated. The consequence is a duality between the terms “detuning”
and “coupling” in Table 1: The coupling of the pendula by the magnetic
interaction corresponds to the detuning of the quantum dot levels, while the
detuning of the pendulum frequencies corresponds to the tunnel coupling.

The mapping to a Schrödinger equation relies on the slowly-varying envelope
approximation which requires $\Delta,\varepsilon\ll\omega_{0}$. While usually
the case, for our smallest pivot distance, $L=208\,$nm, this is fulfilled only
marginally as we find $|\varepsilon|$-values up to
$$1\text{\,}\mathrm{s}$^{-1}$ while $\omega_{0}\simeq 3.3\,\text{s}^{-1}$.
Indeed, such large $|\varepsilon|$-values already significantly reduce the
potential curvatures in some direction such that with increasing coupling, one
eigenmode eventually becomes unstable. This happens when for
$\varepsilon(t)>\omega_{0}$ the adiabatic eigenfrequencies of the linearized
Newton equation become imaginary during some time intervals, see Fig. 7 below
and its discussion in Sec. C.3.

Finally, we comment on the approximation by which we replaced the moments of
interia $J_{1}$ and $J_{2}$ by their average, i.e., we neglected terms
propotional to $\delta J=J_{1}-J_{2}$. If we had kept this term and employed
the ansatz (25), our Hamiltonian would have acquired a non Hermitian
contribution. Its size might be reduced by a more sophisticated ansatz, but
then the modes would depend on the $J_{k}$ and, thus, on the parameter sets.
For our LZSM-interferometry experiments, $\delta J\ll J_{1,2}$, such that
practical corrections are minor. In the case of our Rabi experiments $\delta
J\leq 0.2J_{1,2}$, which is sufficient for the weak couplings considered
there.

##  Appendix C Driven qubit dynamics

So far, we have shown that within the range of validity of the rotating-wave
approximation, the oscillations of our coupled pendula have an envelope which
obeys the Schrödinger equation of the TLS. Next we turn to the particular case
of a periodically time-dependent interaction which maps to a time-dependent
TLS detuning $\varepsilon(t)=\varepsilon_{0}+A\cos(\Omega t)$, while higher
harmonics do not play a relevant role. In this case the Hamiltonian in Eq.
(31) takes the form

$H(t)=\frac{\Delta}{2}\sigma_{x}+\frac{\varepsilon(t)}{2}\sigma_{z},$ (32)

where we have added an irrelevant term proportional to the unit matrix such
that $H(t)$ becomes traceless. In the following, we review the typical quantum
phenomena observed with this Hamiltonian.

###  C.1 Rabi oscillations

The most prominent textbook example is the Rabi problem found for resonant
driving with frequency $\Omega\simeq\Delta$, small amplitude
$A\ll\Omega,\Delta$ and zero detuning, $\varepsilon_{0}=0$ [40]. In this
limit, it is convenient to work in the eigenbasis of the undriven (here
uncoupled) system, henceforth denoted by a tilde, which corresponds to the
basis of individual pendulum modes. Then the Hamiltonian (32) reads

$\widetilde{H}(t)=\frac{\Delta}{2}\sigma_{z}+\frac{A}{2}\cos(\Omega
t)\sigma_{x}.$ (33)

In the quantum optical context, this model describes atomic transitions
induced by irraditation with a laser that couples to the dipole moment of the
atom.

It is now convenient to transform the Hamiltonian via the unitary
$S=e^{-i\sigma_{z}\Omega t/2}$ to a rotating frame, $\widetilde{H}\to
S^{\dagger}\widetilde{H}S-iS^{\dagger}\dot{S}$. Sufficiently close to
resonance and for small driving amplitude, $|\Delta-\Omega|\ll\Omega$ and
$A\ll\Omega$, one can neglect within a rotating-wave approximation rapidly
oscillating terms to find the time-independent two-level Hamiltonian

$\widetilde{H}_{\text{eff}}=\frac{1}{2}\begin{pmatrix}\Delta-\Omega&\Omega_{\text{R}}\\\
\Omega_{\text{R}}&-\Delta+\Omega\end{pmatrix}$ (34)

with the Rabi frequency

$\Omega_{\text{R}}=\frac{A}{2}=\frac{3\mu_{0}m_{\text{l}}^{2}l_{\text{l}}^{2}}{\pi\omega_{0}J_{0}L^{5}}\simeq\frac{3\mu_{0}m_{\text{l}}^{2}l_{\text{l}}^{2}}{\pi
Mg^{2}}\,\frac{\omega_{0}^{3}}{L^{5}}\,.$ (35)

Hence, the occupation probabilities oscillate with the frequency

$\Omega_{\text{eff}}=\sqrt{(\Delta-\Omega)^{2}+\Omega_{\text{R}}^{2}}\,.$ (36)

The latter approximation assumes $J_{0}=M\ell^{2}$ and $\omega_{0}^{2}=g/\ell$
(mathematical pendulum), such that we can eliminate $J_{0}\simeq
Mg/\omega_{0}^{4}$. For resonant driving $\Omega=\Delta$, the resulting
dynamics consists of Rabi oscillations between the ground state and the
excited state with frequency $\Omega_{\text{R}}$ with full probability
transfer, i.e., visibility $\nu=1$. Thereby, the lower frequency eigenmode of
the coupled pendula corresponds to the ground state of the quantum mechanical
TLS. Note, that in our Rabi-oscillation experiments no upper magnets are
present and, moreover, $A\ll\Delta$, such that also the effective static
interaction derived in Sec. A.3 is negligible, $\varepsilon_{0}\simeq 0$.

We used Eq. (36) for the model data shown in Fig. 2(c) plotting
$\Omega_{\text{eff}}(\Delta)$. In Fig. 6

Figure 6: Probing the magnetic dipole-dipole interaction on Rabi oscillations
(no upper magnets are used). The symbols are measured Rabi frequencies while
the straight lines are calculated with Eq. (35). The propotionality
$\Omega_{\text{R}}\propto L^{-5}$ is a consequence of the interaction energy
$G_{\text{l}}$ in Eq. (17). Model lines are calculated with material
parameters listed in Tables 4 and 4, there is no free fit parameter. (a) Rabi
frequency as a function of $L^{-5}$ for varying pivot distance $L$; (b) Rabi
frequency as a function of $L^{-5}f_{0}^{3}$ for three different $L$ (as
labeled) and varying $f_{0}=\omega_{0}/2\pi$, which is the average pendulum
eigenfrequency.

we additionally compare Eq. (35) with the measured Rabi frequency while we
varied the pivot distance $L$ and the average eigenfrequency of both pendula.
For the model curves we used the setup parameters listed in Tables 4 and 4.
The good agreement between model data and theory in all three cases confirms
the validity of our approximations within the regime of small couplings
realized in our Rabi measurements.

###  C.2 Single Landau-Zener passage

The adiabatic eigenenergies of the Hamiltonian (32) exhibit an avoided
crossing at $\varepsilon=0$. For large driving amplitudes $A$, a single
traverse of such an avoided crossing is a standard problem in time-dependent
quantum dynamics. In an idealized model, one linearizes the time-dependent
detuning to obtain $\varepsilon(t)=vt$ with the sweep velocity

$v=\pm\Omega\sqrt{A^{2}-\varepsilon_{0}^{2}}$ (37)

at the crossing. Then the Hamiltonian can be approximated by an idealized
version with linearized time-dependence,

$H_{\mathrm{LZ}}(t)=\frac{1}{2}\begin{pmatrix}vt&\Delta\\\
\Delta&-vt\end{pmatrix}=\frac{1}{2}\left(vt\sigma_{z}+\Delta\sigma_{x}\right).$
(38)

Since the dynamics takes place close to the crossing, one may extend the time
range to infinity, which allows an analytic solution. In 1932, Landau, Zener,
Stückelberg, and Majorana [3, 4, 5, 6] in four independent works found that if
the system at time $t=-\infty$ is in its adiabatic ground state, it will in
the limit $t\to\infty$ occupy the excited adiabatic state with the so-called
Landau-Zener probabilty

$P_{\rm LZ}=\exp\left(-\frac{\pi\Delta^{2}}{2|v|}\right).$ (39)

In particular for $|v|\ll\Delta^{2}$, the system adiabatically follows the
initialized ground state, while for $|v|\gg\Delta^{2}$, it non-adiabatically
switches to the excited state.

###  C.3 LZSM interference

A more recent topic in ac-driven quantum dynamics is the behavior of a quantum
system that is repeatedly swept through an avoided crossing. Then each
crossing acts as beam splitter in energy space, such that one observes
interference patterns as a function of the average detuning $\varepsilon_{0}$
and the driving amplitude $A$ [27]. Since the crossing condition
$\varepsilon(t)=0$ can only be fulfilled for sufficiently large amplitude,
$A\gtrsim|\varepsilon_{0}|$, a nontrivial interference pattern is found in the
triangle that meets this condition during the modulation, cf. Fig. 7.

Figure 7: LZSM fan diagram presenting $\overline{P}_{+}(\varepsilon_{0},A)$
computed (a) with the linearized Newton’s equation and (b) with the slowly
varying envelope approximation which has the form of a Schrödinger equation.
The dashed lines indicate $A=|\varepsilon_{0}|$, below which the avoided
crossing at $\varepsilon(t)=0$ is never reached during the modulation.

We calculated the interference pattern displayed in Fig. 7(a) based on the
linearized Newton’s equation, while for the one in Fig. 7(b) we used the
Schrödinger equation (31) which results from a slowly-varying envelope
approximation. The LZSM fan diagram computed with Newton’s equation contains a
clearly visible distortion for large values of $\varepsilon_{0}$ and $A$,
which is missing in the corresponding calculation using the Schrödinger
equation. These distortions are related with the reduced potential curvatures
discussed in Sec. B.3 above. Moreover, in the upper-right angle of Fig. 7(a),
we observe a region of saturate amplitude pointing to an instability. It
emerges when the coupling exceeds the pedulum frequencies,
$\varepsilon(t)\gtrsim\omega_{0}$ at least at some instances of time, such
that one eigenvalue of the linearized Newton equation (Eq. (1) of the main
text) becomes imaginary. Both, the distortion and the instability pose a limit
to a one-to-one comparison between coupled pendula and a quantum mechanical
two-level-system.

We demonstrate a classical analog to a qubit within this limitation
[demonstrated by the differences between Fig. 7(a) and Fig. 7(b)]. Beyond, we
provide clear evidence for LZSM interference in a macroscopic classical
system. In quantum systems, LZSM interference patterns of this kind have been
observed for the non-equilibrium population of superconducting qubit [41, 42,
43, 44, 21, 45], the current through double quantum dots [22, 46, 23, 24], and
the response of a cavity coupled to a double quantum dot [47, 34, 48].

##  Appendix D Data analysis

###  D.1 Processing of experimental data

The dynamics of the pendula covers a broad spectrum of time scales, dominated
by the fastest one, the oscillation of the pendula near their resonances with
$\omega_{0}/2\pi\sim 0.5\,$Hz. The modulation of the interaction with a
frequency $\Omega/2\pi\sim 5\,$mHz defines a much slower time scale. A central
quantity in our experiments is the modulated interaction quantified by
$(G_{\text{u}}+G_{\text{l}})/2\pi\omega_{0}J_{0}$. It defines the time scale
of an envelope of the pendula oscillations, which is slow compared to
$\omega_{0}$. In case of the Rabi experiments it is the smallest time scale,
while in our Landau-Zener experiments the modulation is the smallest time
scale. In both cases, the resulting beating dynamics can be described by a
Schrödinger-like equation and, consequently, can be compared with the dynamics
of a qubit.

Figure 8: Determining the population dynamics from raw data. (a) Measured
horizontal positions (corresponding camera pixels) as a function of time for
both pendula during a LZSM experiment (gray). Apart from an offset, they are
proportional to the pendulum angles used as labels in the figure. Individual
oscillations are not resolved. The quasistatic solutions (colored solid lines)
are determined by applying a numerical low-pass filter to the raw data. (b)
In-phase mode $\varphi_{+}(t)=[\varphi_{1}(t)+\varphi_{2}(t)]/2$ (gray) as
determined from the raw data in panel a after subtracting the
$\varphi_{k}^{\text{qs}}(t)$. The green line is the envelope $|\Psi_{+}|$. (c)
Occupation probability $P_{+}=|\Psi_{+}|^{2}$.

In Fig. 8(a) we present the horizontal positions of both pendula as a function
of time directly measured with the line scan camera visible in the background
of Fig. 5. From these, it is straightforward to calculate the deflection
angles, $\varphi^{\text{lab}}_{k}$, via a geometric relation. The
approximately 300 pendula oscillations with angular frequencies close to
$\omega_{0}$ are not resolved in Fig. 8(a), which spans a time duration of 10
minutes. We decompose these motions into the rapid oscillations $\varphi_{k}$,
centered at some unknown value $\xi_{k}$, which by and large is the adiabatic
equilibrium position $\varphi_{k}^{\text{qs}}$, i.e., we set

$\varphi^{\text{lab}}_{k}=\varphi_{k}+\xi_{k}$ (40)

with $k=1,2$. The $\varphi_{k}^{\text{qs}}(t)$ are shown as colored solid
lines in Fig. 8(a). Owing to the driving, they oscillate with the slow
modulation frequency $\Omega$.

In a first step of processing the data, we determine $\xi_{k}$ using the fact
that its dynamics is much slower than the bare pendula oscillations. On the
average, over a few periods of the pendula oscillations, the $\varphi_{k}(t)$
vanish, such that $\overline{\varphi^{\text{lab}}_{k}}=\xi_{k}(t)$. Hence,
subtraction of this time average from the measured deflection angles
$\varphi^{\text{lab}}_{k}(t)$ yields the rapid oscillations

$\varphi_{k}(t)=\varphi^{\text{lab}}_{k}(t)-\overline{\varphi^{\text{lab}}_{k}}\,.$
(41)

Provided the well separated timescales, a convenient way for performing the
time average is a convolution of $\varphi^{\text{lab}}_{k}$ with a Gaussian of
width $\sim\,10/\omega_{0}$. Note, that the precise width is practically
irrelevant.

For our Rabi experiments we then continue analyzing $\varphi_{k}$ while for
experiments in the Landau-Zener regime we consider
$\varphi_{\pm}=(\varphi_{1}\pm\varphi_{2})/2$. In Fig. 8(b) we plot
$\varphi_{+}(t)$ determined from the raw data shown in Fig. 8(a). As the fast
oscillations near $\omega_{0}$ are not resolved in this plot, the data appear
as a gray region with modulated height.

The envelope of this modulation, which is a consequence of the driving, is
$|\Psi_{+}|$. To actually determine the envelope dynamics of $\varphi_{k}(t)$,
we square both sides of Eq. (25) such that the right-hand side becomes
$2|\Psi_{k}|^{2}$ plus two terms that oscillate with angular frequency
$2\omega_{0}$. These rapidly oscillating terms can be removed by convolution
with a Gaussian as described above, which provides

$2|\Psi_{k}(t)|^{2}=\overline{|\varphi_{k}|^{2}},$ (42)

where $k=1,2,+,-$. Notice that with this procedure, we cannot obtain the
phases of $\Psi_{k}$, hence cannot determine $|\Psi_{\pm}|$ from
$|\Psi_{1,2}|$ or vice versa. Instead, both must be computed from
$\varphi_{k}$ with Eq. (42).

To emphasize the correspondence to the probability amplitude of a qubit, in
Fig. 8(c) we show the “occupation probability” $P_{+}(t)=|\Psi_{+}(t)|^{2}$
normalized, such that $P_{1}+P_{2}=P_{+}+P_{-}=1$. It visualizes the energy
transfer between the diabatic modes $\varphi_{\pm}$ for four subsequent
passages through their avoided crossing, while we modulated the magnetic
coupling between the pendula. The first pronounced step can be interpreted as
a standard Landau-Zener transition, while the three subsequent steps are
heavily influenced by the phase development between the pendula oscillations.
In between the pronounced steps occur oscillations on a faster time scale,
also visible in the individual pendula oscillations in Fig. 8(a). The time
scale of these beats corresponds to the (modulated) coupling between the
pendula. While the coupling clearly exceeds the modulation frequency in our
LZSM experiments, a look at Fig. 2 clarifies, that in the case of Rabi
experiments the modulation frequency exceeds the coupling.

###  D.2 Effective two-level-system parameters

To obtain numerical data from the Schrödinger equation (31), we need to know
the effective parameters of the driven TLS, namely $\Delta$, $\varepsilon_{0}$
and $A$. While the frequency detuning follows readily from the oscillation
frequencies of the uncoupled pendula, $\Delta=\omega_{1}-\omega_{2}$, the
interaction parameters $A$ and $\varepsilon_{0}$ require more effort. A
straightforward but tedious strategy is based on the Newton equation of the
setup with all relevant quantities such as the center of mass, the moments of
inertia, the distance between the pivots and the magnetic moments. The
effective TLS parameters can then be approximated via a Taylor expansion at
the equilibrium position.

An additional difficulty is related with our choice of using magnetic dipoles
to generate an interaction between the pendula. On the one hand, it allows us
to conveniently modulate the coupling, on the other hand the dipole-dipole
interaction gives rise to higher-order terms in the expansion of the
interaction potential, discussed above in Sec. A.3. For example, in the
absence of the upper magnets, Eq. (28) predicts for the TLS the driving
amplitude $\tilde{A}=G_{\text{l}}/\omega_{0}J_{0}$, where the tilde indicates
that we ignore the time-dependence of the potential curvature as in Eq. (16).
Moreover, together with Eq. (20), it implies that the static TLS detuning
$\varepsilon_{0}$ (i.e. the time-averaged coupling of the pendula) stemming
from the third-order term of the potential relates to the TLS driving
amplitude according to

$\varepsilon_{0}=\frac{5\tilde{A}^{2}}{4\omega_{0}}\,,\qquad A=\tilde{A}\,.$
(43)

While this expression provides the correct order of magnitude, comparison with
our experimental data reveals that it overestimates $\varepsilon_{0}$
substantially (by up to 40%).

To improve our prediction, we refine the above approach by directly evaluating
the effective interation $G$ in Eq. (20) together with the quasi static
position (21) without Taylor expansion of the denominator. For convenience, as
in the Hamiltonian (32), we express the interaction energy in terms of the
uncorrected TLS parameter
$\tilde{\varepsilon}(t)=\tilde{G}(t)/\omega_{0}J_{0}$, where we approximated
the eigenfrequencies and moments of inertia of the pendula by their averages
$\omega_{0}$ and $J_{0}$, respectively. Then the quasi static position reads

$\delta\varphi^{\text{qs}}=\frac{2\tilde{F}(t)/\omega_{0}J_{0}}{\omega_{0}-2\tilde{\varepsilon}(t)},$
(44)

while Eq. (20), which expresses the effective $G(t)$ in terms of the
uncorrected $\tilde{G}(t)$, can be replaced by an improved relation between
the effective interaction $\varepsilon(t)$ and the uncorrected interaction
$\tilde{\varepsilon}(t)$,

$\varepsilon(t)=\tilde{\varepsilon}(t)\Big{(}1-\frac{l_{\text{l}}}{L}\delta\varphi^{\text{qs}}\Big{)}^{-5}.$
(45)

If we again approximate the time dependence by its symmetrized form,
$\varepsilon(t)=\varepsilon_{0}+A\cos(\Omega t)$, we obtain the effective
parameters given by the first two Fourier coefficients of $\varepsilon(t)$,
namely

$\displaystyle\varepsilon_{0}={}$
$\displaystyle\int_{0}^{T}\frac{dt}{T}\varepsilon(t),$ (46) $\displaystyle
A={}$ $\displaystyle 2\int_{0}^{T}\frac{dt}{T}\varepsilon(t)\cos(\Omega t).$
(47)

While the prediction of $A$ and $\varepsilon_{0}$ from Eq. (45) surpasses that
of Eq. (43), it still uses an expansion which looses accuracy with increasing
coupling strength between the pendula. To circumvent this problem, below we
follow an alternative approach, where we determine $A$ and $\varepsilon_{0}$
from the measured dynamics. This experimental approach is still based on the
conjecture that the observed dynamics can be described by the Schrödinger
equation (31) with $\varepsilon(t)=\varepsilon_{0}+A\cos(\Omega t)$. In the
following, we describe three complementary methods by which this task may be
performed and explain, why the third method provides the most accurate
results.

####  D.2.1 Fourier analysis

A rather direct method to determine $A$ and $\varepsilon_{0}$ from
measurements is based on the Fourier spectra of the diabatic modes
$\varphi_{\pm}$ or, equivalently, $\Psi_{\pm}$. It works best, if
$\varepsilon(t)>\Delta$ most of the time such as in LZSM experiments. Then the
Hamiltonian is dominated by the interaction $\varepsilon(t)$, while the
detuning $\Delta$ can be neglected. Under such conditions, the mode $\Psi_{+}$
remains practically constant, such that its spectrum is dominated by a sharp
maximum at zero frequency. In contrast, the time evolution of the out-of-phase
mode is given by the phase factor $\Psi_{-}=e^{iB(t)}$, where
$\dot{B}=\varepsilon(t)$. In Fourier space, it becomes the integral

$\Psi_{-}(\omega)=\int dt\,e^{i\omega t+iB(t)}\,,$ (48)

which we evaluate within the stationary phase approximation. This means that
we replace the integral by its contributions in the vicinity of times at which
the time derivative of the exponent vanishes, i.e., when the equation
$\omega=-\dot{B}=-\varepsilon(t)$ is fulfilled. As $\varepsilon$ is bounded,
the spectrum is essentially restricted to the frequency range
$-\max(\varepsilon)\leq\omega\leq-\min(\varepsilon)$. The contribution of each
stationary point is proportional to $\sqrt{2\pi/\dot{\varepsilon}(t_{n})}$,
where $\dot{\varepsilon}$ is the second time derivative of the phase in Eq.
(48). This time derivative vanishes at the extrema of $\varepsilon(t)$, such
that the $\Psi_{-}(\omega)$ diverges at the margins of the spectrum. For the
assumed shape of the driving, we expect pronounced maxima near
$-\varepsilon_{0}\pm A$.

There is no need to remove the phase factor $e^{-i\omega_{0}t}$ as in the
ansatz (25), because it only shifts the Fourier spectrum to higher
frequencies. Thus, in Fig. 9

Figure 9: Spectrum of the in-phase and out-of-phase modes for a typical LZSM
experiment. The faint lines are the numerical Fourier transforms of the raw
data, while the thicker lines are the same data but smoothened with a Gaussian
of width $\sigma=$0.025\text{\,}\mathrm{Hz}$$. The maxima marked by arrows
allow us to roughly estimate the detuning $\epsilon_{0}$ and the driving
amplitude $A$ as discussed in the text.

we performed the analysis directly with the spectra of $\varphi_{\pm}$
measured for a typical LZSM measurement. Thin lines correspond to the
numerically Fourier transformed measured $\varphi_{\pm}$. While $\varphi_{+}$
yields a peak at the already known value $\omega_{0}$, the spectrum of
$\varphi_{-}$ is broadened due to the driving and develops a frequency comb
spectrum with two main maxima, one near $\omega_{0}-\varepsilon_{0}-A$ and a
second one near $\omega_{0}-\varepsilon_{0}+A$. To determine these maxima we
convolved the experimental spectra with a Gaussian resulting in the fat lines.
The positions of the maxima of the out-of-phase mode provide a rough estimate
of the parameters $\varepsilon_{0}$ and $A$.

Unfortunately, the positions of the maxima of the out-of-phase mode spectrum
are not exactly at $\omega_{0}-\varepsilon_{0}\pm A$. In fact, this Fourier
analysis method to determine $A$ and $\varepsilon_{0}$ systematically
underestimates $A$. In principle, this systematic error could be corrected for
in a calibration based on a comparison with alternative methods discussed
below.

####  D.2.2 Husimi analysis

The next method is capable of determining the full time dependence of
$\varepsilon(t)$ by employing a phase-space method frequently used for
visualizing semi classical structures of a quantum mechanical wave function
$\varphi(x)$. It consists of a mapping of $\varphi(x)$ to a function $Q(x,p)$,
whose structure marks the corresponding classical orbits $(x_{t},p_{t})$. As
such it provides the momentum as a function of the position, $p(x)$, with a
resolution limited by the uncertainty principle.

Replacing the phase space coordinates $(x,p)$ by time and frequency, one
obtains a mapping of a function of time, $\Psi(t)$, to $Q(t,\omega)$.
Accordingly, $Q(t,\omega)$ provides the time-resolved oscillation frequency
$\omega(t)$. For a two-level system described by the Hamiltonian (31), the
relevant frequency scale is the splitting between the eigenmodes
$\omega(t)=E(t)=\sqrt{\Delta^{2}+\varepsilon(t)^{2}}$. Thus, while the time-
frequency Husimi representation of $\Psi_{+}(t)$ is constant at zero, that of
$\Psi_{-}(t)$ traces the adiabatic splitting $E(t)$. In Fig. 10 we plot the
Husimi representation of the out-of-phase mode $\Psi_{-}(t)$ for the data
shown in Fig. 9.

Figure 10: Husimi representation $Q(t)$ (blue scale) of the out-of-phase mode
$\Psi_{-}(t)$ with its spectrum shown in Fig. 9. The average (red solid line)
provides an estimate of the time evolution of the splitting
$E(t)=\sqrt{\Delta^{2}+\varepsilon(t)^{2}}$. It’s maxima allow one to estimate
$\epsilon_{0}$ and $A$, see text.

The Husimi function can be defined as the modulus squared of the overlap of a
function with a wave packet centered at position $t$ and oscillating with a
frequency $\omega$,

$w_{t,\omega}(t^{\prime})=\exp\Big{(}{-\frac{(t-t^{\prime})^{2}}{2\sigma^{2}}-i\omega
t^{\prime}}\Big{)},$ (49)

where the width $\sigma$ shifts the uncertainty towards time (large $\sigma$)
or frequency (small $\sigma$). Thus, $Q(t,\omega)=|q(t,\omega)|^{2}$, where

$\begin{split}q(t,\omega)={}&\int
dt^{\prime}\,w_{t,\omega}^{*}(t^{\prime})\Psi(t^{\prime})\\\ ={}&e^{-i\omega
t}\int
dt^{\prime}\,w_{0,\omega}^{*}(t-t^{\prime})\Psi(t^{\prime}).\end{split}$ (50)

The convolution form obtained with the second line is convenient for the
numerical evaluation, while the phase factor does not affect $Q(t,\omega)$.

The interpretation by which we motivated the use of the Husimi function
becomes evident when one considers a function $\Psi(t)=e^{-iE(t)t}$ with some
slowly varying function $E(t)$, such that $dE/dt$ can be neglected. Evaluating
the integral in Eq. (50) within steepest descent, we have to determine the
stationary points at which the $t^{\prime}$-derivative of the exponent
vanishes. Real and imaginary part of this condition read $t^{\prime}=t$ and
$\omega=E(t)$, which means that the structure of $Q(t,\omega)$ is indeed
dominated by the momentary oscillation frequency.

The price of the Husimi analysis are the fundamental restrictions of its
resolution resulting in a broadening of $E(t)$ in time. As the Fourier
analysis above, also this method is based on the evaluation of a Fourier
integral within a stationary-phase approximation. As a consequence, it equally
suffers from an underestimation of the splitting $E(t)$ and at its turning
points and, hence, from an underestimation of $A$.

The main benefit of the Husimi analysis is its ability to directly visualize
the time evolution of the modulation of the splitting $E(t)$ and, hence, the
time-dependent coupling strength
$|\varepsilon(t)|=\sqrt{E(t)^{2}-\Delta^{2}}$, where the minimal splitting is
given by the frequency difference $\Delta$.

####  D.2.3 Analysis of the quasistatic solution

Figure 11: (a) Effective TLS modulation parameters, $A(L)$ and
$\varepsilon_{0}(L)$, reconstructed from the LZSM dynamics without upper
magnets for modulation period $T=$441\text{\,}\mathrm{s}$$. The red circles
result from the analysis of the quasistatic solution
$\varphi_{1,2}^{\text{qs}}(t)$ using Eq. (51), while the solid lines
corresponds to the numerical solution based on Eq. (45). The spectral analyses
(Fourier or Husimi analyses, not shown) yield comparable results within their
respective accuracies. (b) LZSM interference pattern (gray scale) adopted from
Fig. 7b calculated with the Schrödinger equation. The red circles depict
$A(\varepsilon_{0})$ without upper magnets determined from pairs of data
points shown in panel a with varying $L$, the solid line shows the according
prediction. Other symbols show $A(\varepsilon_{0})$ for measurements with
constant $L=217\,$mm (squares) or $L=246\,$mm (triangles) while we varied the
distance $L_{\text{u}}$ between the now present upper magnets.

To overcome such uncertainties, in our third method we determine
$\varepsilon_{0}$ and $A$ directly from the measured quasistatic solution
$\varphi_{1,2}^{\text{qs}}(t)$ discussed in Sec. A.2. It can be directly
measured by rotating a magnet without exciting the pendula otherwise, such
that $\varphi_{1,2}(t)=\varphi_{1,2}^{\text{qs}}(t)$. Alternatively, as
$\Omega\ll\omega_{0}$, one can extract $\varphi_{1,2}^{\text{qs}}(t)$ with
high accuracy from measurements with oscillating pendula by applying a digital
lowpass to separate the quasistatic dynamics from $\varphi_{1,2}(t)$, cf. Sec.
D.1. Based on our approximation $\varepsilon=\varepsilon_{0}+A\cos(\Omega t)$
we determine the extreme values $\varepsilon_{\text{min}}$ and
$\varepsilon_{\text{max}}$ and use

$\begin{split}A&=\frac{1}{2}\left(\varepsilon_{\text{max}}-\varepsilon_{\text{min}}\right)\\\
\varepsilon_{0}&=\frac{1}{2}\left(\varepsilon_{\text{max}}+\varepsilon_{\text{min}}\right)\;.\end{split}$
(51)

The symbols in Fig. 11(a) present the modulation parameters $\epsilon_{0}$ and
$A$ of the effective TLS as a function of the pivot distance $L$ determined
from a series of experiments without upper (static) magnets using Eq. (51).
The solid line corresponds to the numerical solution based on Eq. (45). The
agreement is excellent for $L\gtrsim 220\,$mm, while for smaller distances we
find noticeable deviations. They indicate, that for our stronger couplings a
quantitative derivation of the interaction parameters of the effective TLS
from the nonlinear dipole-dipole interaction has its limitations. For this
reason, we use the experimental values $\epsilon_{0}(L)$ and $A(L)$ for our
further analysis.

In Fig. 11(b) we present as a grayscale the LZSM interference pattern already
shown in Fig. 7(b), which we calculated numerically based on the Schrödinger
equation. On top we plot the values of $A(\varepsilon_{0})$ determined from
our experiments as symbols. The open circles correspond to the
$\epsilon_{0}(L)$ and $A(L)$ values shown in Fig. 11(a) for the case of no
upper magnets. The solid line behind these points is the prediction based on
Eq. (45) and corresponds to the solid lines in Fig. 11(a). Triangles and
squares show experimental values for measurements including upper magnets. In
these experiments the distance between the pivots was constant while the
distance between the upper magnets was varied. These $A(\varepsilon_{0})$
curves are the basis for the comparison of the measured and predicted LZSM
interference patterns presented in Fig. 4(c) and Fig. 4(d).

Figure 12: Avoided crossing formed by the eigenfrequencies of the coupled
pendula calculated for the detuning of $f_{1}-f_{2}=24\,$mHz. The solid lines
are computed with the linearized Newton’s equations, the dashed lines are the
eigenvalues of the Schrödinger equation. The inset displays the full spectrum
of Newton’s equations, including the negative eigenvalues.

####  D.2.4 Consistency of our approximations

Our approximations are based on the following concept: Small oscillations of a
classical many-body system can be described by a linearized equation of motion
of the form $M\ddot{\bm{x}}=V\bm{x}$, where the coordinate vector $\bm{x}$
consists of all deviations from the equilibrium position. The symmetric matrix
$V$ contains the potential curvatures, while $M$ is the diagonal matrix of the
masses of each particle. Since the masses are positive, $M^{1/2}$ is real
valued, such that the equation of motion can be written as
$\ddot{\bm{y}}=-Q\bm{y}$ with the symmetric matrix $Q=M^{-1/2}VM^{-1/2}$,
which has real and non-negative eigenvalues (see Ref. [38] or another text
book on classical mechanics). Their square roots are the eigenvalues of the
linearized equations of motion and are shown in the inset of Fig. 12.

As a consistency check for our mapping to a Schrödinger equation we verify
that its spectrum corresponds to the one of the linearized classical equation
of motion. Since we neglect the lower half of the spectrum in the rotating-
wave approximation, our results compare to the positive eigenfrequencies of
Newton’s equation. In addition, our ansatz (25) corresponds to a gauge
transformation that shifts the eigenfrequencies by $-\omega_{0}$. Hence, the
spectrum predicted by our Schrödinger equation shifted by $\omega_{0}$ finally
corresponds to the positive eigenfrequencies of Newton’s equation. In the main
panel of Fig. 12 we present both spectra in direct comparison in the absence
of the driving. For sufficiently small interaction $\varepsilon$ our mapping
is accurate. We expect, that the mapping works equally well for our driven
experiments, as we consider slow driving with $\Omega\ll\omega_{0}$.

##  Appendix E Experimental setup and methods

For visualizing the coherent wave mechanics equivalent to the dynamics of an
individual qubit, macroscopic pendula have decisive advantages. First, in
contrast to nanoscale devices, our pendula are large and have a slow clock
speed, such that their dynamics can be observed with bare eyes. Second, the
time evolution of a classical and macroscopic device can be obtained from one
single experiment, while quantum- or nanosystems require many repeated
measurements at various times. A drawback of our macroscopic pendula is, that
a continuous modulation of the eigenfrequencies, corresponding to the energy
detuning typically modulated in a qubit, is virtually impossible. It would
require moving a weight smoothly up and down, for some measurements along the
full length of a pendulum rod while it is oscillating. Therefore, in our
experiments we keep the frequency detuning fixed and instead modulate the
coupling between the pendula. Modulating the detuning or the coupling are
mathematically equivalent options, which can be demonstrated by performing a
unitary basis transformation in Sec. B.2. The in-phase and out-of-phase modes
of our coupled pendula then correspond to the diabatic (or localized) states
of a qubit, while the individual pendula eigenfrequencies correspond to its
adiabatic eigenstates, cf. Table 1.

In order to be able to modulate the coupling, we replace the usual spring
connecting both pendula by permanent magnets connected to each pendulum. We
then modulate the coupling by rotating one of the magnets with a constant
angular frequency. The setup is presented in Fig. 1 and can be experienced in
the attached movie. Our magnets are cubicles of pressed neodymium powder
coated with nickel bought from Webcraft GmbH (www.supermagnete.de).

###  E.1 Details of the setup

Each pendulum consists of a one meter long stainless steel rod with a diameter
of 12 mm, extended at the bottom end with a stainless steel thread and a
hollow polyethylen housing containing AA batteries, which can drive a linear
rotating motor via a simple circuit board. Two cubic neodymium magnets with 28
mm edge length are glued to the axes of each motor. In the experiments
discussed here, we rotate one of the two magnets with constant angular
frequency. At a distance of $0.513\text{\,}\mathrm{m}$ above them are two
smaller magnets fixed by plastic screws inside plastic cylinders [red in the
photograph in Fig. 5(b)] to the rods. These magnets can be moved horizontally
inside the cylinders. Brass nuts fixed to the opposite ends of the cylinders
function as counter weights to balance the center of masses of the pendula
within the respective rods. Heavy brass cylinders (2.14 kg) are screwed onto
the threads attached to the pendula. The vertical positions of these weights
serve for adjusting the resonance frequencies of the individual pendula. The
overall weight of each pendulum is 4.242 kg. In our experiments the air
friction can be neglected compared to the friction of the pivots and the
damping related to magnetic induction. The frame supporting the pendula is
built from hollow aluminum bars, while the pendula are fixed via their pivots
to a massive pair of stainless steel beams spanning the top of the frame. The
pivots are professional pendulum clock pivots based on leaf springs provided
by the company Erwin Sattler GmbH & Co. KG. Due to their plate geometry the
leaf springs strongly suppress unwanted pendula motions others than
oscillations in the $x$-$y$ plane. The pivots quality is essential for
providing high enough and stable quality factors. It is important to run the
experiment in a tranquil surrounding, because in particular air flows and
vibrations can cause uncontrolled phase shifts of the pendula oscillations.
Therefore we have placed the frame supporting the pendula on a massive granite
plate in a separate and quiet room in the cellar of the building. The coupling
between the pendula is provided by up to four magnets. It is modulated
whenever one of the lower magnets is rotated. At the high quality factors of
several thousands it is necessary to avoid even tiny contributions to the
coupling between the pendula mediated by the supporting frame. Initially this
was a problem in our setup, which we prevented by stiffening the frame by
adding brackets in its corners and by tightly fixing the frame at one side to
an approximately 0.5 m thick brick wall of the building.

###  E.2 Requirements, line width and strong coupling

For performing qubit simulating experiments, such as Rabi oscillations or LZSM
interferometry, the resonance frequencies of the individual pendula have to be
much higher than both, the coupling constant and the frequency difference,
while the latter two must be highly tunable. At the same time, the quality
factor must be high enough to ensure that the coupling strength exceeds the
line widths of the eigenmodes by far. Practically, it is desired that for the
duration of the experiment damping effects can be neglected, which is the case
in our experiments and greatly simplifies the data analysis. The strong
distance dependence of the magnetic dipole-dipole interaction provides the
desired tunability of the coupling via adjusting the mutual distances between
the lower and the upper magnets. Further, rotating one of the magnets allows
us to modulate the coupling of the “qubit” analogue. The price is a time
dependent momentary equilibrium deflection of the pendulum rods, which is
discussed in detail in Appendix A.

If two magnets are moved with respect to each other, their electric
conductivity gives rise to eddy currents, which result in the main damping
mechanism of our coupled pendula, similar to the functioning of an induction
break. The damping is weak, as the $Q$-factor of our coupled pendula still
ranges between $3000-6000$ depending on the average distance of the magnets.
With oscillation frequencies $\omega_{0}/2\pi\sim 0.5$ Hz, it allows us to
observe the qubit equivalent dynamics for several hours. More importantly, our
large $Q$-factors allow us to ignore damping effects within a limited time
window $\delta t\ll 2\pi Q/\omega_{0}$, which facilitates a one-to-one
comparison with a quantum mechanical two-level system.

To resolve the splitting of a two-level-system it needs to exceed the line
widths $\gamma=\omega_{0}/2Q$ of the eigenmodes,
$\sqrt{\Delta^{2}+\varepsilon^{2}}>\gamma$. For our $Q>3000$ we find
$\gamma<0.6\,\text{ms}^{-1}$ in our experiments. The splitting between the
eigenmodes can then be easilly resolved by using frequency detunings
$|\Delta|>\gamma$. Moreover, for coupled pendula with such a high $Q$-factor
it is straightforward to realize the so-called strong coupling regime defined
by $\varepsilon>\gamma$, which is reached in most of our experiments.

For achieving a meaningful comparison between our classical system and a qubit
we require a clean separation between the individual pendulum frequencies and
all other time scales $\Omega,\Delta,A\ll\omega_{0}$. We used the modulation
frequencies $\Omega/2\pi=2.3\,\text{mHz}$, $\Omega/2\pi=7.1\,\text{mHz}$ or
$\Omega/2\pi=11.7\,\text{mHz}$, modulation amplitudes of the coupling between
$0.7\,\text{mHz}\lesssim A/2\pi\lesssim 43\,\text{mHz}$, and frequency
detunings $|\Delta|/2\pi\leq 5\,\text{mHz}$. To adjust the latter, we re-
positioned 2 kg weights along the pendulum rods.

##  Appendix F Measurement Regimes

###  F.1 Rabi experiments

Rabi oscillations can be observed in the limit of small couplings and if the
much larger frequency detuning is similar to the modulation frequency,
$\Delta\sim\Omega\gg\Omega_{\text{R}}=A/2$, cf. Sec. C.1. For simplicity we
performed our Rabi experiments without upper magnets, such that (for small
couplings) $A\simeq\varepsilon_{\text{max}}$. Practically, our modulation
frequencies of a few mHz dictate a range of useful frequency differences
$\Delta$ and couplings $A<\Delta$, the latter being controlled by the distance
$L$ between the pivots.

Figure 13: Time evolution of $\varphi_{1,2}$ for beating experiments with
maximal repulsive (a) versus maximal attractive (b) couplings and the
corresponding resonant Rabi experiment with the coupling modulated at the
frequency $\Omega=\Delta$; $L=454\,$mm, $\Delta/2\pi=11.7\,$mHz. (d–f),
Fourier transforms of the time evolutions shown in panels a–c.

In Figs. 13(a) and 13(b) we present the deflections of both pendula
$\varphi_{k}(t)$ for beating experiments without driving ($A=0$,
$\varepsilon=\varepsilon_{\text{min}}$ or
$\varepsilon=\varepsilon_{\text{max}}$) for the two extreme coupling cases
with the magnets aligned either antiparallel for maximal repulsion or
collinear for maximal attraction, where the pivot distance $L=454\,$mm
corresponds to a small coupling. To initialize each measurement, we deflected
just one of the two pendula, the one corresponding to the red lines. The
energy transfer between the two pendula is clearly incomplete owing to the
finite frequency detuning, $\Delta/2\pi=11.7\,$mHz, while the beating
frequency is $\sqrt{\Delta^{2}+\varepsilon^{2}}/2\pi\simeq\Delta/2\pi$. The
latter corresponds to the difference between the respective eigenfrequencies,
directly visible in the Fourier spectra shown in Figs. 13(d) and 13(e). The
spectrum of the pendulum that was initially not deflected (blue) clearly
contains two maxima, where the frequency of the smaller peak coincides with
the main maximum of the initially deflected pendulum (red). This indicates a
finite mixing between the states represented by $\varphi_{1}$ and
$\varphi_{2}$, which are not the exact eigenmodes because of the coupling
between the pendula. Note, that the eigenfrequencies are slightly smaller for
the attractive interaction as compared to the case of repulsive interaction.

In Fig. 13(c) we present the corresponding resonant Rabi experiment with
identical parameters as above but the coupling being modulated with the
angular frequency $\Omega=\Delta$. In this case, the initial beating
experiments mark the turning points of the modulation of the coupling during
the Rabi experiment. The energy transfer between the two pendula is now
complete but happens at the Rabi frequency $\Omega_{\text{R}}/2\pi\simeq
0.7\,\text{mHz}$, hence the initially postulated condition
$\Omega_{\text{R}}\ll\Delta,\Omega$ is fulfilled.

The Fourier spectra of the Rabi experiment in Fig. 13(f) reveal two main peaks
and four side peaks for each pendulum. The main peaks are splitted by the
(effective) Rabi frequency in Eq. (36). The much smaller side peaks, each of
which is equally split by the (effective) Rabi frequency, are higher order
components split off by multiples of $\Omega$ from the main peaks. In the
resonant case $\Omega=\Delta$, the frequency values of the Fourier components
of both pendula coincide, in the non-resonant case they would be displaced by
$\Omega-\Delta$. Note that the higher order components in the Fourier spectra
are responsible for the weak stepwise modulation with frequency $\Omega$ of
the occupation of the pendula, which are weakly visible in Fig. 13(c).

###  F.2 LZSM experiments

Figure 14: Time evolution of $\varphi_{\pm}$ for beating experiments with
maximal repulsive (a) versus maximal attractive (b) couplings and the
corresponding LZSM experiment with the coupling modulated at the frequency
$\Omega/2\pi=2.27\,$mHz; $L=240\,$mm, $\Delta/2\pi=6.2\,$mHz. (d–f), Fourier
transforms of the time evolutions shown in panels a–c; for the LZSM experiment
the Fourier transform in panel f corresponds to five full periods of the
modulation, while panel c shows only one period.

For a direct comparison with the small coupling regime of Rabi experiments we
perform similar measurements within the regime of LZSM experiments at much
larger modulation of the coupling with $A>\Omega$. We consider measurements
for $L=240\,$mm, $\Delta/2\pi=6.2\,$mHz, $\Omega/2\pi=2.27\,$mHz and a sizable
$\varepsilon_{0}$ realized by including upper magnets. For such large
couplings ($A,|\varepsilon_{0}|\gg\Delta$) the diabatic modes $\varphi_{\pm}$
approximately correspond to the eigenmodes. Hence, in Fig. 14 we now plot
$\varphi_{\pm}$. The beating experiments, which we again performed for
collinear versus antiparallel magnets, summarized in the upper four panels of
Fig. 14 reveal the expected much larger range of couplings compared to the
Rabi experiment. The very different beating frequencies for attractive versus
repulsive interactions point to a sizable $\varepsilon_{0}$. Note, that the
frequency (main component of Fourier spectrum) of $\varphi_{+}$ is almost
identical for repulsive versus attractive coupling (blue in Figs. 14d and
14e), while the frequency of $\varphi_{-}$ (red) varies by roughly 20 %.

The LZSM experiment, presented by its first avoided crossing in Fig. 14(c),
shows the expected energy transfer between the in-phase and out-of-phase modes
near the avoided crossing. The additional faster beats vary in frequency
related with the time dependence of $\varepsilon$. The Fourier spectrum
plotted in Fig. 14(f) comprises five modulation periods. It reveals that the
in-phase mode stays at the frequency of the beating experiments, while the
out-of-phase mode contains frequency components spanning a slightly larger
region than that between the out-of-phase mode frequencies of the beating
experiments. Note, that the apparent splitting, say $\delta\omega$, of the in-
phase mode $\varphi_{+}$ in the Fourier spectrum of the LZSM experiment
resembles a slight difference between the frequencies that the in-phase mode
has between attractive versus repulsive beating experiments. As a result, the
the out-of phase mode spectrum is composed of two copies of a frequency comb
with the same relative shift $\delta\omega$, each one characterized by equally
spaced peaks separate by the modulation frequency $\Omega$.

Figure 15: Avoided crossings (solid lines) with Fourier components of beating
experiments (blue squares, $\Omega=0$) and main Fourier components of
modulated experiments (red circles, $\varepsilon=\varepsilon_{0}$). (a) Rabi
experiment for $f_{1}=0.53335\,$Hz, $f_{2}=0.52165\,$Hz, $L=454\,$mm and
$\Omega/2\pi=11.7\,$mHz. (b) LZSM experiment for $f_{1}=0.5290\,$Hz,
$f_{2}=0.5228\,$Hz, $L=240\,$mm and $\Omega/2\pi=2.27\,$mHz.

###  F.3 Avoided crossings for Rabi versus LZSM experiments

Figure 15 summarizes the main components of the Fourier spectra of the
experiments presented in Figs. 13 and 14 and thereby visualizes the vastly
different experimental regimes realized in a Rabi experiment, where
$A<\Delta$, versus a LZSM experiment with $A\gg\Delta$.

To highlight the differences between the regimes of the Rabi versus LZSM
experiments, in Figs. 15(a) and 15(b) we plot for the Rabi versus LZSM
experiments shown in Figs. 13 and 14 the relevant regions of the avoided
crossings predicted for the measured frequencies by the Schrödinger equation
(for the time independent quantum mechanical two-level system). The blue
squares indicate the components of the respective Fourier spectra of the
beating experiments, where we determined the values of $\varepsilon$ using the
eigenvalue equation
$\omega_{\pm}=\frac{1}{2}(-\varepsilon\pm\sqrt{\Delta^{2}+\varepsilon^{2}})$.
The red circles in Fig. 15(a) indicate the four main components of the Fourier
spectra of the Rabi experiment in Fig. 13(f), where we used
$\varepsilon=\varepsilon_{0}\simeq 0$ for simplicity. The line of red circles
in Fig. 15(b) indicates the range of the frequency comb of the Fourier
spectrum of $\varphi_{-}$ of the LZSM experiment, cf. Fig. 14(f), where we
again used $\varepsilon=\varepsilon_{0}$ for simplicity.

Clearly, for the presented Rabi experiment in Fig. 15(a),
$\varepsilon_{0}\simeq 0$ and $A\ll\Delta$, while for the LZSM experiment in
Fig. 15(b), both $\varepsilon_{0},A>\Delta$. This depicts the main difference
between the two regimes.

##  Appendix G Notations, units and magnitudes

Tables 4–4 below summarize the variables as well as their magnitudes used.

Table 2: Variables of individual pendula

variable, definition | explanation | values
---|---|---
$\varphi_{1,2}$ | deflection angles of individual pendula |  $|\varphi_{1,2}|<0.8^{\circ}\simeq 0.014\,$rad
$M$ | overall mass of each pendulum | 4.242 kg
$g$ | acceleration due to gravity in Munich (PTB table) | 9.807232 m/s2
$f_{1}$ $|$ $\omega_{1}=2\pi f_{1}$ | resonance $|$ angular frequency of pendulum 1 | $f_{1}\simeq(0.52-0.55)\,\text{Hz}$
$f_{2}$ $|$ $\omega_{2}=2\pi f_{2}$ | resonance $|$ angular frequency of pendulum 2 | $f_{2}=0.52195\,\text{Hz}$
$l_{\text{r}1}={g}/{\omega_{1}^{2}}$ | reduced length of pendulum 1 | (0.818 - 0.912) m
$l_{\text{r}2}={g}/{\omega_{2}^{2}}$ | reduced length of pendulum 2 | 0.912 m
$l_{\text{c1}}$ | center of mass of pendulum 1 (distance from pivot) |  $(0.754-0.841)\,$m
$l_{\text{c2}}$ | center of mass of pendulum 2 (distance from pivot) | 0.841 m
$J_{1}=Ml_{\text{c}1}l_{\text{r}1}$ | moment of inertia of pendulum 1 (respective pivot) | (2.619 - 3.254) kg m2
$J_{2}=Ml_{\text{c}2}l_{\text{r}2}$ | moment of inertia of pendulum 2 (respective pivot) | 3.254 kg m2
$Q_{1,2}$ | quality factor of individual uncoupled pendula | $9500-10500$
The following distances are equal for both pendula: |
$l_{\text{p}}$ | distance between pivot and point of measurement | 1.053 m
$l_{\text{l}}$ | distance between pivot and center of lower magnet | 1.148 m
$l_{\text{u}}$ | vertical distance between pivot and center of upper magnet | 0.635 m

Table 3: Variables related with the coupling between the pendula

variable, definition | explanation | values and units
---|---|---
$\varphi_{\pm}=(\varphi_{1}\pm\varphi_{2})/2$ | in-phase — out-of-phase mode | $<0.8^{\circ}$
$L$ | distance between pivots, i.e., lower magnets for $\varphi_{1}=\varphi_{2}=0$ |  $(0.205-0.454)\,$m
$m_{\text{l}}$ | magnetic moment of each lower magnet | 25.37 Am2
$L_{\text{u}}$ | distance between upper magnets for $\varphi_{1}=\varphi_{2}=0$ |  $(0.105-0.168)\,$m
$m_{\text{u}}$ | magnetic moment of each upper magnet | 6.544 Am2
$J_{0}=(J_{1}+J_{2})/2$ | mean moment of inertia |  $\simeq 3$ kg m2
$\omega_{0}=(\omega_{1}+\omega_{2})/2$ | mean angular frequency of both pendula |  $(3.28-3.37)$ s-1
$\Delta=\omega_{1}-\omega_{2}$ | angular frequency difference | $(0-29)\,\times 10^{-3}\text{s}^{-1}$
$\varepsilon={G}/{\omega_{0}J_{0}}$ | coupling constant | ($-0.4-1.3$) s-1
$G(G_{\text{l}},G_{\text{u}})$ | effective potential curvature (see Sec. A.3) |
$G_{\text{l}}={6\mu_{0}m_{\text{l}}^{2}l_{\text{l}}^{2}}/{\pi L^{5}}$ | interaction energy between lower magnets | ($0.1-5.6$) J
$G_{\text{u}}={6\mu_{0}m_{\text{u}}^{2}l_{\text{u}}^{2}}/{\pi L_{\text{u}}^{5}}$ | interaction energy between upper magnets | ($0.31-3.2$) J
$Q$ | quality factor of coupled pendula (for realized range of $L$) | $3000-6000$

Table 4: Additional variables related with the modulation of the coupling

variable, definition | explanation | values and units
---|---|---
$2\pi/\Omega$ | period of magnets’ rotation | 85.5 s, 141 s, 441 s
$\varepsilon(t)=\varepsilon_{0}+A\cos\Omega t$ | modulated coupling constant (linear approximation) |  $\left[(-0.4)\,-(+1.3)\right]$ s-1
$\Omega_{\text{R}}=G_{\text{l}}/{2\omega_{0}J_{0}}=A/2$ | rabi frequency for $\Omega_{\text{R}}\ll\omega_{0},\Delta$ and if only lower magnets are used | $(4.5-265)\,\times 10^{-3}\text{s}^{-1}$
$P_{\rm LZ}=\exp(-{\pi\Delta^{2}}/{2|v|})$ | Single passage Landau-Zener probability | $0-1$
$v=\frac{d\varepsilon(t)}{dt}\big{|}_{\varepsilon=0}=\pm\Omega\sqrt{A^{2}-\varepsilon_{0}^{2}}$ | Speed of driving at avoided crossing at $\varepsilon=0$ | ($\pm 0.0027-\pm 0.0300$) s-2
$P_{0}$ | Initial probability to occupy in-phase mode | $0-0.2$

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