#  Topological power pumping in quantum circuits

J. Luneau<EMAIL_ADDRESS>ENS de Lyon, CNRS, Laboratoire de
Physique, F-69342 Lyon, France  C. Dutreix  Univ. Bordeaux, CNRS, LOMA, UMR
5798, F-33405 Talence, France  Q. Ficheux  Department of Physics, ETH Zürich,
CH-8093 Zürich, Switzerland  P. Delplace  ENS de Lyon, CNRS, Laboratoire de
Physique, F-69342 Lyon, France  B. Douçot  Laboratoire de Physique Théorique
et Hautes Energies, Sorbonne Université and CNRS UMR 7589, 4 place Jussieu,
75252 Paris Cedex 05, France  B. Huard  ENS de Lyon, CNRS, Laboratoire de
Physique, F-69342 Lyon, France  D. Carpentier  ENS de Lyon, CNRS, Laboratoire
de Physique, F-69342 Lyon, France

###### Abstract

In this article, we develop a description of topological pumps as slow
classical dynamical variables coupled by a quantum system. We discuss the
cases of quantum Hall pumps, Thouless pumps, and the more recent Floquet pumps
based frequency converters. This last case corresponds to a quantum
topological coupling between classical modes described by action-angle
variables on which we focus. We propose a realization of such a topological
coupler based on a superconducting qutrit suitably driven by three modulated
drives. A detailed experimental protocol allowing to measure the quantized
topological power transfer between the different modes of a superconducting
circuit is discussed.

##  I Introduction

Topological properties of matter have always been discussed in relation with
topological pumping. In an enlighting gedankenexperiment, B. Laughlin related
the topological nature of the transverse conductivity of the dimension $D=2$
quantum Hall state to a transfer of charge between two edges in a Corbino
geometry [1]. A modern interpretation views the Hall sample as effectively
wrapped on a cylinder, realizing a $D=1$ quantum hall charge pump as the
enclosed magnetic flux is smoothly increased [2, 3]. This quantized adiabatic
pumping of a time-dependent quantum system was soon generalized beyond the
Quantum Hall Effect to a driven one dimensional crystal by D. Thouless [4]. In
a Thouless pump, exemplified by the Rice-Mele model [5], the time-varying flux
of the quantum Hall effect is replaced by a time-periodic phase
$\phi(t)=\omega t$ which accounts generically for the external drive. The
corresponding dynamics is periodic both in the linear spatial dimension of the
crystal, but also in time. In contrast with previous implementations of
geometrical pumps, such topological pumps are characterized by a Chern number
and were only recently realized using cold atoms lattices [6, 7, 8], optical
waveguides [9, 10, 11], magnetically coupled mechanical resonators [12] or
stiffness-modulated elastic plates [13].

An extension of this pumping was unveiled recently in quantum systems of
effective spatial dimension $D=0$, but with two temporal dimensions
corresponding to a drive at two different frequencies $\omega_{1}$ and
$\omega_{2}$. The initial theoretical proposal arose in Cooper pair pumps
realized in Josephson junction driven by two independent superconductors’
phase differences [14, 15, 16, 17, 18, 19] and later on as a frequency
converter in a driven two level system [20]. This led to the realization of
such a frequency converter using a single nitrogen-vacancy center in diamond
[21] or with a superconducting qubit [22], although in both cases measuring
the topological transfer was beyond experimental reach.

In all existing realizations, the topological pumping is probed indirectly, as
a property of the time-dependent quantum system. The dynamics of these quantum
systems is assumed to be adiabatic, i.e. restricted to an eigensubspace well-
separated in energy from other quantum eigenstates. Pumping manifests itself
as an anomalous velocity in real space for the Quantum Hall and Thouless
pumps, or an anomalous velocity in the harmonic Floquet space for the
frequency converter. This anomalous velocity was initially identified in the
case of the quantum Hall effect in a crystal [23, 24, 25], and originates from
a Berry curvature whose average value defines a Chern number characterizing
the topological nature of the pump. It was recently measured in the cold atom
[6, 7, 8] and optical waveguides [10] realizations of a Thouless pump.
Alternatively to these bulk measurements, the topology of the $D=1$ pumps can
be inferred from the occurrence of states at the boundary of the chain during
a period of the drive. A direct probe of evolution of these edge states allows
to characterize the pumping [12]. Indeed, these topological edge states
manifest themselves in the scattering description of the quantum pump [26, 27,
28, 29, 30]: they lead to resonances of the reflection matrix $R$ during one
cycle of the driven quantum chain, whose associated $\pi$ phase shift lead to
the winding of the phase of $\mathrm{det}R$ [3]. Such a theoretical approach
was recently extended to generalized pumps obtained by wrapping $D=2$
topological insulators around a cylinder [31, 32].

Although previous descriptions of pumping focused on the driven quantum
system, they call for a key generalization that embeds the coupled degrees of
freedom of the environment if one wants to model the observable topological
transfer itself. In this article, we present such a global framework that
allows to characterize the actual pumping and propose a realistic experiment
that directly measures this topological pumping. In practice, a classical
treatment of the degrees of freedom of the environment is sufficient to
observe a topological transfer and we restrict our attention to that limit.
While a similar treatment dates back to the Born-Oppenheimer approximation
[33, 34, 35], here we focus on classical modes or action-angle variables which
describe the periodic drive of a quantum system. The interaction between these
slow classical degrees of freedom and the gapped quantum system, when treated
within the adiabatic approximation, effectively reduces on average to a
topological coupling leading to a quantized pumping between the classical
environments.

To illustrate the virtues of this formalism, we propose a realization of a
Chern mode coupler enabling to measure the topologically protected power
transferred between electromagnetic modes. The proposed setup consists in
using a qutrit, realized from the first three levels of a highly anharmonic
superconducting circuit, a fluxonium, driven at multiple microwave
frequencies. The driven fluxonium realizes a time-dependent version of a
Haldane model on a Lieb lattice. The corresponding phase diagram can directly
be revealed by measuring the power transfer between the microwave modes. The
topological pumping leads to a topological redistribution of energy between
the three microwave modes.

##  II Slow classical modes coupled to a fast quantum state

###  II.1 Dynamics of classical degrees of freedom adiabatically coupled to a
quantum system

Figure 1:  We consider a general gapped and fast system (in blue) coupled to
slow variables of the environment (in red). The energy separation of the fast
quantum system is assumed to be much larger than that of the slow degrees of
freedom, allowing a description of the latter by classical pairs of conjugate
variables. While of different nature, we denote generically these variables by
$q_{\alpha},p_{\alpha}$. The dynamics of each pair follows from a classical
Hamiltonian $\mathcal{H}_{\alpha}$. The quantum system couples instantaneously
to a single variable $q_{\alpha}$ of each degree of freedom of the classical
environment.

We provide a general model of coupled fast and slow degrees of freedom by
resorting to a mixed classical - quantum description, see Fig. 1, similar in
spirit to that used by M.V. Berry and J. Robbins [35] and by Q. Zhang et al.
[36]. The fast degrees of freedom are generically described by a quantum
Hamiltonian $H$. The slow degrees of freedom are associated to an energy
separation (much) smaller than that of the fast system, allowing for a
classical description by pairs of conjugated variables
$q_{\alpha},p_{\alpha}$, $\alpha=1,...N$ satisfying the Poisson bracket
relations $\\{q_{\alpha},p_{\beta}\\}=\delta_{\alpha\beta}$ [37]. We consider
the common case where the quantum system couples to only one of the variable,
$q_{\alpha}$, of each pair of conjugate variables, which includes in
particular the case of a driven quantum system.

####  II.1.1 Hamilton equations of motion.

The dynamics of each pair of classical variables, prior to the coupling to the
quantum system, is assumed to be slow and described by classical dynamics
deduced from a classical Hamiltonian
${\mathcal{H}}_{\alpha}(q_{\alpha},p_{\alpha})$ following

$\displaystyle\dot{q}_{\alpha}^{(0)}$
$\displaystyle=\frac{\partial{\mathcal{H}}_{\alpha}}{\partial
p_{\alpha}}~{}~{}~{}~{}~{}\text{and}~{}~{}~{}~{}~{}\dot{p}_{\alpha}^{(0)}=-\frac{\partial{\mathcal{H}}_{\alpha}}{\partial
q_{\alpha}}.$ (1)

The quantum system is described by a Hamiltonian $H(\\{q_{\alpha}\\})$
parametrized by the states of the classical variables $q_{\alpha}$. We focus
on quantum systems that remain gaped during the evolution of the classical
modes. This allows to approximate at shorter times the dynamics of the quantum
system by an adiabatic evolution, driven by the slow dynamics of the classical
modes. More precisely, we consider a quantum system prepared at time $t=0$
into one of the eigenstates denoted $\ket{\psi_{\nu}(t=0)}$ of the $t=0$
Hamiltonian $H(\\{q_{\alpha}(t=0)\\}$. This amounts to assume an initial
correlation between the state $\\{q_{\alpha}(t=0)\\}$ of the classical
environment and the quantum system. Through the coupling to $q_{\alpha}(t)$,
the quantum system will slowly evolve in time on a timescale dictated by the
slow environment. We denote its instantaneous state $\ket{\Psi(t)}$ and assume
that it remains approximately withing the same eigensubspace of the
Hamiltonian, which is the essence of the _adiabatic approximation_. In return,
the coupling of the classical degrees of freedom to the quantum system also
perturbs their dynamics and results in an effective coupling between different
pairs of slow degrees of freedom which is the focus of this paper. The
modified equations of motion for the slow variables are

$\displaystyle\dot{q}_{\alpha}$ $\displaystyle=\dot{q}_{\alpha}^{(0)}$ (2)
$\displaystyle\dot{p}_{\alpha}$
$\displaystyle=\dot{p}_{\alpha}^{(0)}-\bra{\Psi(t)}\frac{\partial H}{\partial
q_{\alpha}}\ket{\Psi(t)}.$ (3)

The second term in (3) was discussed as a geometric force when $q_{\alpha}$ is
a position in [35]. This decomposition of dynamics into slow and fast degrees
of freedom, in the spirit of the Born-Oppenheimer decomposition [38, 33, 39],
reduces to the standard semi-classical analysis for the specific case of the
motion of electrons in insulators [40].

Let us stress that the above description ensures the conservation of the total
energy of both quantum and classical degrees of freedom. Indeed, the
Schrödinger equation in finite dimension corresponds to classical Hamiltonian
equations associated to an Hamiltonian function
$E(\Psi)=\bra{\Psi}H\ket{\Psi}$ and to a natural Poisson bracket structure on
the Hilbert space [41]. Therefore the total phase space also inherits a
Poisson bracket structure and the above equations of motion from the
Hamiltonian function
$\mathcal{H}(\\{q_{\alpha}\\},\\{p_{\alpha}\\},\Psi)=\sum_{\alpha}\mathcal{H}_{\alpha}(q_{\alpha},p_{\alpha})+\bra{\Psi}H(\\{q_{\alpha}\\})\ket{\Psi}$,
thus abiding by the conservation of the total energy.

To evaluate the matrix elements in (3), we now describe the adiabatic
evolution of the quantum state $\ket{\Psi(t)}$. Due to the slow evolution of
the Hamiltonian $H(\\{q_{\alpha}(t)\\})$, the state $\ket{\Psi(t)}$ of the
quantum system does not identify with the instantaneous eigenstate
$\ket{\psi_{\nu}(t)}$ defined by
$H(\\{q_{\alpha}(t)\\})\ket{\psi_{\nu}(t)}=E_{\nu}(\\{q_{\alpha}(t)\\})\ket{\psi_{\nu}(t)}$.
The corresponding correction to the dynamics of the slow classical variables
$p_{\alpha}$ in Eq. (3) is now expressed as

$\displaystyle\dot{p}_{\alpha}$
$\displaystyle=\dot{p}_{\alpha}^{(0)}-\frac{\partial E_{\nu}}{\partial
q_{\alpha}}+\hbar\sum_{\beta\neq\alpha}\dot{q}^{(0)}_{\beta}{F}^{(\nu)}_{q_{\alpha}q_{\beta}},$
(4)

see Appendix A for the detailed derivation. The first correction relates to
the energy variation of the quantum state and follows from the standard
classical - quantum coupling. The last correction is more exotic: it manifests
an effective coupling between the different slow variables
$q_{\alpha},p_{\alpha}$. The strength of this transverse coupling depends on
the geometry of the eigenstates $\ket{\psi_{\nu}}$ of the quantum system
through the components ${F}_{q_{\alpha}q_{\beta}}^{(\nu)}$ of the two-form
Berry curvature which are defined as

${F}_{q_{\alpha}q_{\beta}}^{(\nu)}=\\\
i\sum_{\mu\neq\nu}\frac{\bra{\psi_{\nu}}\partial_{q_{\alpha}}H\ket{\psi_{\mu}}\bra{\psi_{\mu}}\partial_{q_{\beta}}H\ket{\psi_{\nu}}}{(E_{\nu}-E_{\mu})^{2}}-(\alpha\leftrightarrow\beta).$
(5)

####  II.1.2 Condition of adiabaticity

The above adiabatic approximation in one eigensubspace $\ket{\psi_{\nu}}$
holds as long as the transitions to a different eigensubspace
$\ket{\psi_{\mu}}$ can be neglected. These non-adiabatic transitions can be
described as Landau-Zener transitions [42, 43], with a probability of
transition from one eigenstate to another behaving as
$\exp(-\pi/(4\max_{t}\epsilon_{\mu\nu}))$ where the time dependent parameter
$\epsilon_{\mu\nu}$ reads

$\epsilon_{\mu\nu}=\hbar\frac{\left|\bra{\psi_{\mu}}\frac{\differential
H}{\differential t}\ket{\psi_{\nu}}\right|}{(E_{\mu}-E_{\nu})^{2}}.$ (6)

In this formula,
$\tau_{\mu\nu}^{\mathrm{col}}=|E_{\mu}-E_{\nu}|/\left|\bra{\psi_{\mu}}\frac{\differential
H}{\differential t}\ket{\psi_{\nu}}\right|$ corresponds to the characteristic
time of the Landau-Zener collision, or equivalently to an energy spread
$\delta E=\hbar/\tau_{\mu\nu}^{\mathrm{col}}$. Transitions occur when this
spread is comparable to the gap $E_{\mu}-E_{\nu}$, and the parameter
$\epsilon_{\mu\nu}$, which controls the validity of the adiabatic
approximation, identifies with this ratio $\epsilon_{\mu\nu}=\delta
E/|E_{\mu}-E_{\nu}|$. For an alternative derivation of this small parameter
$\epsilon_{\mu\nu}$ within the adiabatic expansion [44], see Appendix A.

Along a trajectory in phase space, the gap of the quantum system varies. The
transition dynamics at each minimum can be described following the above
Landau-Zener approach, leading to a series of transitions. It is then possible
to determine the characteristic time of validity of the adiabatic
approximation $\tau_{\mathrm{adiab}}$ by constraining the cumulative
transition probability to be e.g. of order $0.1$. Introducing the mean free
time $\tau_{\mathrm{mft}}$ separating the evolution in phase space between two
minima of the gap, we get

$\tau_{\mathrm{adiab}}\approx
0.1~{}\tau_{\mathrm{mft}}~{}\exp\left(\frac{\pi}{4\max_{t}\epsilon_{\mu\nu}}\right).$
(7)

where the maximum of the parameter $\epsilon_{\mu\nu}$ is evaluated along the
phase space trajectory during $\tau_{\mathrm{mft}}$. Note that for the above
analysis to be consistent, the collision time $\tau_{\mu\nu}^{\mathrm{col}}$
must be smaller than the mean free time $\tau_{\mathrm{mft}}$. Since we focus
on aperiodic evolution, we also neglected the effect of relative phases
accumulated between the transitions [45].

###  II.2 Geometrical power transfer

The geometrical coupling in (4) between the different subset of slow variables
$q_{\alpha}(t),p_{\alpha}(t)$ is associated with an energy transfer between
them. The change of energy of each classical degree of freedom is:

$\displaystyle\frac{\differential\mathcal{E}_{\alpha}}{\differential t}$
$\displaystyle=\dot{q}_{\alpha}\frac{\partial\mathcal{H}_{\alpha}}{\partial
q_{\alpha}}+\dot{p}_{\alpha}\frac{\partial\mathcal{H}_{\alpha}}{\partial
p_{\alpha}}$ $\displaystyle=-\dot{q}_{\alpha}^{(0)}\frac{\partial
E_{\nu}}{\partial{q_{\alpha}}}+\hbar\sum_{\beta\neq\alpha}\dot{q}^{(0)}_{\alpha}\dot{q}_{\beta}^{(0)}{F}^{(\nu)}_{q_{\alpha}q_{\beta}}.$
(8)

The antisymmetry of the Berry curvature implies the conservation of the total
energy.

###  II.3 Nature of classical degrees of freedom

Let us discuss briefly the implications of the previous modification of
Hamilton equations for different types of classical slow degrees of freedom
coupled to a quantum system.

####  II.3.1 Massive classical particles

The initial context of the Born-Oppenheimer approximation, at the origin of
the adiabatic approximation, was the description of light particles, the
electrons, coupled to heavy particles, the nucleus. In this situation, the
slow degrees of freedom described classically are those of the massive
particle: its position $q_{\alpha}$ and conjugated momentum $p_{\alpha}$. The
corresponding Hamiltonian is
$\mathcal{H}_{\alpha}=p_{\alpha}^{2}/(2M)+V(q_{\alpha})$, parametrized by the
mass $M$ and potential $V(q)$. The equations of motion in this case take the
form

$\displaystyle\dot{q}_{\alpha}$ $\displaystyle=\frac{p_{\alpha}}{M},$ (9)
$\displaystyle\dot{p}_{\alpha}$
$\displaystyle=-V^{\prime}(q_{\alpha})-\partial_{q_{\alpha}}E_{\nu}+\hbar\sum_{\beta\neq\alpha}\frac{p_{\beta}}{M}{F}^{(\nu)}_{q_{\alpha}q_{\beta}}.$
(10)

Equation (10) describes the associated anomalous geometrical force [34, 35].

####  II.3.2 Classical modes

While the previous adiabatic formalism was initially designed with classical
massive degree of freedom in mind, it also applies to the case of slow action
angle $\phi_{\alpha},n_{\alpha}$ variables, which we will call a _classical
mode_. In more details, we consider a variable $p_{\alpha}=\hbar n_{\alpha}$
where $n_{\alpha}$ takes only integer values and its canonical phase
$q_{\alpha}=\phi_{\alpha}$, a $2\pi$ periodic phase. Of particular interest is
the situation of a monochromatic mode, corresponding to the Hamiltonian
$\mathcal{H}_{\alpha}=\hbar\omega_{\alpha}n_{\alpha}=\omega_{\alpha}p_{\alpha}$
whose linearity in $n_{\alpha}$ is the distinctive feature compared with the
massive case. In this situation, the modified Hamilton equations of motion
read

$\displaystyle\dot{\phi}_{\alpha}=\dot{q}_{\alpha}$
$\displaystyle=\omega_{\alpha},$ (11)
$\displaystyle\hbar\dot{n}_{\alpha}=\dot{p}_{\alpha}$
$\displaystyle=-\frac{\partial
E_{\nu}}{\partial\phi_{\alpha}}+\hbar\sum_{\beta\neq\alpha}\omega_{\beta}{F}^{(\nu)}_{\phi_{\alpha}\phi_{\beta}}.$
(12)

In the case of classical modes, the Eq. (12) describes the filling rate of
mode $\alpha$. The energy transferred between the different modes corresponds
to

$\displaystyle\frac{\differential\mathcal{E}_{\alpha}}{\differential t}=$
$\displaystyle-\omega_{\alpha}\frac{\partial
E_{\nu}}{\partial\phi_{\alpha}}+\hbar\sum_{\beta\neq\alpha}\omega_{\alpha}\omega_{\beta}{F}^{(\nu)}_{\phi_{\alpha}\phi_{\beta}}.$
(13)

##  III Topological pumping

###  III.1 Topological versus geometrical couplings

In the above section, we have shown how a geometrical quantity, the Berry
curvature, encodes the strength of the effective coupling mediated by a
quantum system between classical variables. Of particular interest is the case
of two classical variables, e.g. $q_{1}$ and $q_{2}$ with a compact
configuration denoted $[0,2\pi]^{2}$. In this situation, introducing the
integer ${\mathcal{C}}_{12}^{(\nu)}$ Chern number, the quantity

$\int_{[0,2\pi]^{2}}\differential q_{1}\differential
q_{2}~{}{F}^{(\nu)}_{q_{1}q_{2}}=2\pi~{}{\mathcal{C}}_{12}^{(\nu)}$ (14)

is a topological quantity quantized in units of $2\pi$, i.e. it is insensitive
of perturbations of the quantum Hamiltonian $H$ provided the gap between
$E_{\nu}$ and other states does not close. Noting that (14) is nothing but the
averaged Berry curvature over the configuration space, the case of a
Hamiltonian $H$ characterized by a topological Chern number corresponds to a
quantized averaged coupling in equation (4). When the Berry curvature
fluctuates in the configuration space around its topological average, a
quantized coupling between the classical variables is recovered when averaging
over initial position in the configuration space, which is hardly practical.
Instead, we can resort to a time average: if the evolution of classical
variables is ergodic, an average over the configuration space can be replaced
by an average over a sufficiently long time.

The above situation of two classical environments topologically coupled
through a quantum system correspond to a topological pump. Historically, the
relation between a topological Chern invariant and a pumping process
originates from the Laughlin’s description of a charge transfer between inner
and outer edges of a quantum Hall sample in a Corbino geometry [1]. Later on,
D. Thouless proposed another topological pumping mechanism by periodically
modulating a $D=1$ crystal. Much more recently, realizations of a topological
pump were proposed as a junction between superconductors [14, 15, 17] or by
driving a two level system at two different frequencies [46]. The topological
nature of these processes can be inferred from the dynamics of the quantum
degrees of freedom : in all cases the corresponding band structure is gapped,
and the eigenmanifold in which the adiabatic dynamics takes place is
characterized by a Chern number (14). Alternatively, such topological pumps
can be characterized from a scattering point of view by probing the appearance
of topological edges during the evolution of the quantum system through phase
shifts of the reflection matrix [3, 31, 32]. In the following, we develop a
’Born-Oppeiheimer description’ of this topological pumping by showing how they
can naturally be described with the formalism of section II of adiabatic
topological coupling between classical slow variables of different nature.

###  III.2 $D=2$ Quantum Hall pump

Figure 2: (a) Schematic Quantum Hall circuit where two LC branches, described
by classical conjugate variables $\Phi$ and $Q$, are connected through a
quantum Hall sample. (b) Schematic Thouless pump driven by a phase
$\phi_{1}=\omega_{1}t$ conjugate to a variable $n_{1}$, and coupled to an LC
branch. Topological pumping gives rise to a current in the $LC$ circuit. (c)
Topological mode coupler, or frequency converter, in which two classical modes
described by $\phi_{1},n_{1}$ and $\phi_{2},n_{2}$ variables are coupled
topologically through a quantum to system.

In an enlighting argument, B. Laughlin related the quantization of the
transverse conductivity of the quantum Hall state to a transfer of charge
between edges in a Corbino geometry as the flux threading the disk is
increased by one quantum [1]. Later on, Niu, Thouless and Wu introduced the
notion of generalized boundary conditions for quantum Hall states [47]. The
Quantum Hall topological properties are expressed as the Chern number of the
ensemble of many-body groundstates over the closed manifold of phase boundary
conditions. These boundary conditions parameters were later related to
electromotive forces through loops connecting opposite edges of the sample
[48, 49] effectively generalizing the topology of Laughlin’s gedanken
experiment to that of a torus and allowing for a dynamical description of the
quantum Hall effect over a classical parameter space [50].

Here we consider the classical phases entering the generalized boundary
conditions as dynamical variables. This effectively amounts to realize a
quantum Hall topological pump between two $LC$ harmonic circuits. Let us
consider a quantum Hall sample coupled to two independent electrical circuits
in the $x$ and $y$ direction, see Fig. 2(a). The coupling between each circuit
and the quantum Hall sample follows from the boundary conditions of Niu et al.
[47] on the many-body ground state wave function $\Psi(x_{i},y_{i})$:

$\displaystyle\Psi(x_{i}+L_{x},y_{i})$
$\displaystyle=e^{i\Phi_{1}}~{}\Psi(x_{i},y_{i}),$ (15a)
$\displaystyle\Psi(x_{i},y_{i}+L_{y})$
$\displaystyle=e^{i\Phi_{2}}~{}\Psi(x_{i},y_{i}).$ (15b)

The two phases $\Phi_{1},\Phi_{2}$ are related to the voltage drop
$V_{\alpha}$ in each directions as
$\Phi_{\alpha}(t)=(e/\hbar)\int^{t}V_{\alpha}(t^{\prime})dt^{\prime}$ . If we
model each electric branch associated to this voltage drop as an LC circuit
[51], these phases are dynamical classical variables, whose canonically
conjugate momenta are the (rescaled) accumulated charge in each circuit
$Q_{\alpha}=(\hbar/e)\int^{t}I_{\alpha}(t^{\prime})dt^{\prime}$, $I_{\alpha}$
being the current in each circuits. The classical Hamiltonian describing each
LC circuit is

$\mathcal{H}_{\alpha}=\frac{e^{2}}{2\hbar^{2}C_{\alpha}}Q_{\alpha}^{2}+\frac{\hbar^{2}}{2e^{2}L_{\alpha}}\Phi_{\alpha}^{2}~{},$
(16)

where $L_{\alpha}$ is the inductance and $C_{\alpha}$ the capacitance of the
corresponding circuit [51]. Hence the dynamics of an LC circuit identifies to
that of a massive particle of position $q_{\alpha}=\Phi_{\alpha}$ and momentum
$p_{\alpha}=Q_{\alpha}$, in a harmonic potential. The Hamilton’s equations
(9,10) include a correction to the usual relations between flux and current

$\displaystyle\dot{Q}_{1}$
$\displaystyle=\frac{\hbar}{e}I_{1}=-\left(\frac{\hbar}{e}\right)^{2}\frac{\Phi_{1}}{L_{1}}+\hbar
F_{\Phi_{1}\Phi_{2}}\frac{e}{\hbar}V_{2}$ (17)

via the Berry curvature $F_{\Phi_{1}\Phi_{2}}$ of the quantum Hall effect
ground states derived by Niu et al. [23]. This Berry curvature being
independent of the external fluxes and thus constant over the parameter space,
it is related to the quantum Hall Chern number ${\mathcal{C}}_{12}$ via
$F_{\Phi_{1}\Phi_{2}}={\mathcal{C}}_{12}/(2\pi)$. Thus the corrected classical
equation of motion of the $LC$ circuits reads

$I_{1}=-\frac{\hbar}{e}\frac{\Phi_{1}}{L_{1}}+\frac{e^{2}}{h}{\mathcal{C}}_{12}V_{2}.$
(18)

The usual case of an ideal amp-meter is recovered in the limit
$L_{1}\to\infty,C_{1}\to 0$.

Note that the energy transferred from one LC circuit to the other, following
(8), is

$\frac{\differential E_{1}}{\differential t}=\hbar
F_{\Phi_{1}\Phi_{2}}\dot{\Phi}_{1}\dot{\Phi}_{2}={\mathcal{C}}_{12}\left(\frac{e^{2}}{h}\right)V_{1}V_{2}=\delta
I_{1}V_{1}.$ (19)

In the limit of an ideal Hall measurement, where the $L_{1}C_{1}$ circuit
corresponds to an amp-meter, we get $V_{1}=0$, and no energy is transferred
between the two circuits.

###  III.3 $D=1$ Thouless pump

Let us now turn to the canonical example of a topological pump, proposed by D.
Thouless [4], which consists in a $D=1$ crystal suitably periodically driven
in time, such as the Rice-Mele model [5]. The single electron dynamics is thus
described by a time-dependent Bloch Hamiltonian $H(k,\phi(t))$ periodic both
in momentum over the Brillouin zone, and in $2\pi$ periodic phase
$\phi(t)=\omega t$. This Hamiltonian is assumed to be gapped at all time, and
with energy band $\nu$ eigenstates $\ket{\psi_{\nu}(k,\phi)}$ possessing a
finite Chern number ${\mathcal{C}}_{k\phi}^{(\nu)}$ over the $2$-torus
constituted of the $D=1$ Brillouin zone and periodic phase configuration space
of phase $\phi$. In such a system, topological pumping is usually described as
the appearance of a steady current in the bulk of a closed ring, corresponding
to an anomalous geometric velocity for semi-classical states in band $\nu$
$\left<\dot{x}_{\nu}\right>=\left<F_{k\phi}\right>\partial_{t}\phi=2\pi~{}{\mathcal{C}}_{k\phi}^{(\nu)}~{}\omega$
[52, 40].

We propose an alternative description of such a Thouless pump, by considering
an open $D=1$ crystal of size $L$ connected on both ends to an $LC$ circuit,
analogous to the Quantum Hall pump, see Fig. 2(b). The coupling between the
charged particles in the crystal and the $LC$ circuit follows from the
boundary conditions (15) on the many-body groundstate wavefunction
$\Psi(x=0,\phi(t))=e^{i\Phi_{1}}\Psi(x=L,\phi(t))$. This amounts to couple the
crystal to a pair of classical conjugated variables $Q_{1},\Phi_{1}$ identical
to those in the Quantum Hall pump, with a classical Hamiltonian (16)
describing their dynamics. The periodic driving of the Hamiltonian is now
interpreted as the coupling between the charges of the crystal and a dynamical
classical variable $\phi_{2}=\omega t$, conjugated to a variable $n_{2}$. The
dynamics $\dot{\phi}_{2}=\omega,\dot{n}_{2}=0$, correspond to that of a
classical mode introduced in section II.3.2. In this representation, the
topological coupling between the $LC$ circuit and the classical mode leads to
modified equation of motion in the $LC$ circuit, manifesting the appearance of
a charged current. The modified Hamilton equation (9) reads

$\dot{Q}_{1}=\frac{\hbar}{e}I_{1}=-\left(\frac{\hbar}{e}\right)^{2}\frac{\Phi_{1}}{L}+\hbar
F_{\Phi_{1}\phi_{2}}\omega,$ (20)

which leads to a steady charge current $I_{1}=e2\pi F_{\Phi_{1}\phi_{2}}/T$
through the driven crystal, corresponding to an average number
${\mathcal{C}}_{\Phi_{1}\phi_{2}}^{(\nu)}={\mathcal{C}}_{k\phi}^{(\nu)}$ of
charged transferred across the chain per period $T$ of the drive. This results
identifies with the standard steady anomalous velocity in the bulk of the
Thouless pump.

###  III.4 $D=0$ Power pump

In the previous section, we interpreted a time-periodic quantum Hamiltonian as
a coupling between a fast quantum system and a slow classical mode. A natural
extension consists in considering a single quantum system coupled to the
phases $\phi_{\alpha}$ of an arbitrary number $N$ of classical modes, see Fig.
2(c). The quantum dynamics of such a system can be described within Floquet
theory [20]. The topological Chern numbers of such a quantum system are
defined in Floquet space, and, when non zero, leads to a frequency conversion
mechanism. The description of such a _quantum mode coupler_ is natural in
terms of an effective classical dynamics of the modes. We assume that the
quantum system couples only to the phases of the modes, corresponding to a
Hamiltonian $H(\phi_{1}(t),\dots,\phi_{N}(t))$. In such a case, the equations
of motions are given by (11, 12) with a power leaving each mode given by Eq.
(13). In the particular case of two modes of frequency
$\omega_{1},\omega_{2}$, we recover the result of Martin et al. for the
averaged power between two modes given by
$\differential\mathcal{E}_{1}/\differential
t=\hbar\omega_{1}\omega_{2}~{}{\mathcal{C}}_{\phi_{1}\phi_{2}}^{(\nu)}/(2\pi)$
[20]. In the following section, building on this general description of
topological pumping we propose a realization of such a quantum topological
coupler between microwaves modes using an artificial $3$ level atom, a qutrit.

##  IV Topological pump in quantum circuits : the topological qutrit

###  IV.1 Case of a qubit

It is possible to apply the concept of geometrical and topological response of
gapped states to individual quantum systems. The seminal work of Martin,
Refael, and Halperin [46] proposed to use a spin-1/2 under two frequency
drives to observe the quantized pumping of energy from one drive to the other.
Measuring such a power transfer presents a substantial experimental challenge.
Recently, Malz and Smith [22] used the IBM Quantum Experience to observe the
inner dynamics of a superconducting qubit state that would correspond to a
topological quantum transition. However, their control scheme was mixing the
flows of power between the drives – hindering a direct measurement of the
power transfer. Similarly, other experiments on NV centers demonstrate a
topological transition in the qubit dynamics but could not explore the
quantized pumping of power [21].

Essentially, the model proposed in Ref. [46] consists in engineering the
Hamiltonian

$H(\phi_{1},\phi_{2})\propto\sin(\phi_{1})\sigma_{X}+\sin(\phi_{2})\sigma_{Y}\\\
+\left(M-\cos(\phi_{1})-\cos(\phi_{2})\right)\sigma_{Z},$ (21)

where $\sigma_{i}$ are the Pauli operators of the qubit, $M$ is a parameter
that drives the topological transition and the phases $\phi_{1}=\omega_{1}t$
and $\phi_{2}=\omega_{2}t$ are driven at two incommensurate frequencies. We
envision three ways to realize this Hamiltonian.

  * •

As in Ref. [22], it is possible to drive a single superconducting qubit, i.e.
a transmon, with a complex amplitude $\left[X(t)+iY(t)\right]e^{-2i\pi
f_{q}t-2i\int_{0}^{t}Z(\tau)\mathrm{d}\tau}$, where $f_{q}$ is the qubit
frequency in order to implement any driving term of the sort
$H(t)=X(t)\sigma_{X}+Y(t)\sigma_{Y}+Z(t)\sigma_{Z}$. However, while it is
possible to infer what is the transferred power between frequency components
at $\omega_{1}$ and $\omega_{2}$ from the measured qubit dynamics, this power
flow lacks physical embodiment and cannot be measured using any known
apparatus.

  * •

Alternatively, the $\sigma_{Z}$ term in the Hamiltonian (21) can be achieved
by controlling the frequency of the qubit directly, hence physically
separating the source of power from the three terms corresponding to each
Pauli operator in Eq. (21). The frequency of flux tunable qubits can be tuned
rapidly using a on-chip flux control. However, measuring the power of the
drive used for such a flux-tunable bias has never been achieved to our
knowledge and requires involved technical development.

  * •

Lastly, the frequency of the qubit can be controlled by exploiting the ac-
Stark shift created by a drive far detuned from the qubit transition
frequency. The ac-Stark shift is a commonly used method to engineer the
spectrum of artificial atoms but the relatively low anharmonicity of transmons
imposes a finite bandwidth on the control parameter $Z(t)$.

All these solutions present serious practical limitations either on the
achievable Z-control or – more importantly – on the ability to measure the
quantized power transfer. We propose to circumvent this difficulty by
extending the size of the Hilbert space. This new pumping schemes uses a
qutrit to create gapped states.

###  IV.2 Implementation with a superconducting qutrit

####  IV.2.1 Principle of the experiment

Figure 3:  (a) Schematic of the experimental setup. A fluxonium circuit is
embedded in a host cavity. The transitions between the first three levels of
the fluxonium are driven with a detuning $\delta_{i}$, an amplitude $\Omega$,
and a modulation frequency $\dot{\phi}_{i}$ (blue, red and green). The power
of the outgoing signals are recorded with a power spectrum analyzer that
provides the instantaneous photon flux of each frequency mode. (b) Spectrum of
the driving tones. Each fluxonium transition is driven with two side-bands
used to implement a topologically protected power transfer. (c) For
decoherence rates smaller than the modulation frequencies $\dot{\phi}$, the
power of each sideband can be resolved. The quantized power transferred is
expressed as a function of the difference of the spectral power $\Delta S_{i}$
in the sidebands of each reflected driving tone according to Eq. (32).

The experiment we propose consists in driving a superconducting qutrit at
several frequencies in order to establish a topologically given power flow
between microwave modes at various frequencies. We propose to use a
superconducting circuit behaving as a qutrit where every transitions can be
addressed individually with a well defined phase. In the following, we denote
the transitions $|0\rangle-|1\rangle$ as $1$, $|1\rangle-|2\rangle$ as $2$,
and $|0\rangle-|2\rangle$ as $3$ for the sake of simplicity. By modulating the
drive amplitude $\Omega_{i}$ of each transition $i$ at a frequency
$\dot{\phi}_{i}=\omega_{i}$, it is possible to engineer an effective
Hamiltonian in a phase space defined by the phases
$\phi_{1},\phi_{2},\phi_{3}$. By enforcing $\phi_{3}=\phi_{1}-\phi_{2}$, the
dimension is reduced while still enabling the observation of a topological
transition in the power transferred between the various driving tones. The
difference between any two transition frequencies of the qutrit shall be much
greater than any drive amplitude $\Omega_{i}$ and modulation frequency
$\omega_{i}$ in order to enable the direct measurement of the power transfer
between any driving modes. Finally, all the above-mentioned timescales should
be much smaller than the coherence time of the qutrit transitions.

An example of a superconducting circuit satisfying theses requirements is the
fluxonium artificial atom. The fluxonium is a highly anharmonic
superconducting circuit whose first three energy levels can be used as a
qutrit. The circuit is a loop composed of a Josephson junction shunted by a
large inductance, see Fig. 3(a). When the loop is threaded by an external
magnetic flux corresponding to almost (but not exactly) half a flux quantum,
no selection rule prevents the direct driving of all three transitions while
the circuit transitions have been shown to display record long coherence times
[53, 54, 55]. The circuit is embedded in a cavity with a single port connected
to a transmission line. Incoming microwave modes carry the modulated drives
(see Fig. 3(b)) to the qutrit and outgoing modes carry the reflected signal
before being measured by a power spectrum analyzer. At the input, each drive
at frequency $f_{i}$ is modulated in amplitude at a frequency $\omega_{i}$,
resulting in the pairs of sidebands in Fig. 3(b). The geometrical and
topological signatures can be observed in the power transfer between these
various frequency modes. We model the propagating mode in the transmission
lines at frequency $f$ as a classical mode of energy $hfn_{f}$ such that the
net photon flux is given by the difference between the outgoing and incoming
signals at this frequency $(S_{\text{out}}[f]-S_{\text{in}}[f])/hf$.
Precisely, one first needs to probe the difference $\Delta S_{i}$ in power
spectral density between two sidebands (see Fig. 3(c)) and convert it in
photon flux $\dot{n}_{i}=\Delta S_{i}/hf_{i}$ (see Appendix C for a refined
expression).

####  IV.2.2 Hamiltonian in the rotating frame

The fluxonium Hamiltonian reads [56]

$H_{\mathrm{fluxonium}}=4E_{C}\hat{N}^{2}+\frac{E_{L}}{2}\hat{\varphi}^{2}-E_{J}\cos(\hat{\varphi}-\varphi_{\text{ext}}),$
(22)

where $\hat{N}$ is the charge on the capacitor of the circuit, $\hat{\varphi}$
is the phase twist across the inductance, $\varphi_{\mathrm{ext}}$ is the
external magnetic flux threading the loop, and $E_{C},~{}E_{L},~{}E_{J}$ are
respectively the charging energy, inductive energy and Josephson energy of the
circuit. The fluxonium is addressed by a microwave drive applied on a
capacitance, so that each pump induces a term proportional to
$\cos(\phi_{i})\cos(\theta_{i})\hat{N}$ in the Hamiltonian, where
$\theta_{i}(t)=2\pi f_{i}t+\theta_{i}^{0}$ is the phase of each tone and
$\phi_{i}(t)=\omega_{i}t$ is the phase of their amplitude modulation. We
denote as $\ket{0},\ket{1},\ket{2}$ the first three energy levels of the
fluxonium (22). The charge operator $\hat{N}$ is off-diagonal in this basis
$(\ket{0},\ket{1},\ket{2})$. Hence the effect of the drives is purely off-
diagonal. The frequencies of the three drives $f_{1}$, $f_{2}$ and $f_{3}$ are
constrained to satisfy $f_{3}=f_{1}+f_{2}$ so that each tone drives a single
transition of the fluxonium. This constraint can be enforced in the microwave
domain by using mixers to generate a tone at $f_{3}$ using tones at $f_{1}$
and $f_{2}$ or by direct numerical synthesis.

We move to a rotating frame by applying the unitary diagonal transformation
$U(t)=\textrm{diag}(1,\exp(-2i\pi f_{1}t),\exp(-2i\pi f_{3}t))$. One can
choose the initial phases $\theta_{i}^{0}$ of the tones so that in the
rotating frame and using the rotating wave approximation, the dynamics of the
qutrit is governed by the Hamiltonian

$\tilde{H}(\phi_{1},\phi_{2},\phi_{3})=\\\
\hbar\left(\begin{array}[]{ccc}0&\Omega_{1}\cos(\phi_{1})&-i\Omega_{3}\cos(\phi_{3})\\\
\Omega_{1}\cos(\phi_{1})&\delta_{1}&\Omega_{2}\cos(\phi_{2})\\\
i\Omega_{3}\cos(\phi_{3})&\Omega_{2}\cos(\phi_{2})&\delta_{3}\end{array}\right)$
(23)

which depends on the phases $\phi_{i}(t)=\omega_{i}t$ of the drive amplitude
modulations, where $\delta_{1}=2\pi f_{01}-2\pi f_{1}$ and $\delta_{3}=2\pi
f_{02}-2\pi f_{3}$ are the frequency detuning between the fluxonium transition
frequencies and the drive frequencies (see Appendix B for details). Note that
the above constraint on drive frequencies sets
$\delta_{2}=\delta_{3}-\delta_{1}$. Besides, in order to simplify the
dynamics, we impose an additional constraint on the modulation frequencies,
namely $\phi_{3}=\phi_{1}-\phi_{2}$. Therefore the effective Hamiltonian can
be described by an Hamiltonian evolution controlled by two phases only
$H(\phi_{1},\phi_{2})\equiv\tilde{H}(\phi_{1},\phi_{2},\phi_{1}-\phi_{2})$.
The rotating wave approximation is valid if the detunings $\delta_{i}$ and the
drive amplitudes $\Omega_{i}$ are much lower than the fluxonium transitions
frequencies $f_{ij}$ and the difference between any two transition
frequencies, which is one of the requirements detailed above.

####  IV.2.3 Chern insulator on the Lieb lattice

Figure 4: (a) Schematic representations of a Haldane model on the Lieb lattice
and its square Brillouin zone with the four inversion-invariant momenta. The
sublattices 0, 1, and 2 respectively have on-site potential energies 0,
$\delta_{1}$, and $\delta_{3}$. The regular and bold black lines represent the
nearest-neighbor tight-binding amplitudes $\Omega_{1}$ and $\Omega_{2}$. The
dashed black lines depict the next-nearest-neighbor coupling $\Omega_{3}$. (b)
Two allowed configurations of the band parities $p_{\nu}$ at the inversion-
invariant momenta. A trivial insulator can be adiabatically connected to an
atomic limit, which is necessarily characterized here by positive parity
products $\pi_{\nu}=+1$. A band insulator exhibiting negative parity products
$\pi_{\nu}=-1$ is therefore topological. Enumerating all the configurations of
the parity eigenvalues allowed by the model parameters leads to the
topological phase diagram in panel (c) (see Appendix D). (c) Phase diagram of
the qutrit model. Each colored region corresponds to a set of band Chern
numbers. On the boundaries the gap closes for at least one combination of
phases $\phi_{1}$ and $\phi_{2}$.

In this section, we show that the fluxonium qutrit emulates in time the
physics in momentum space of a Chern insulator on a Lieb lattice, which is
illustrated in Fig. 4(a). This proposal thus provides a missing implementation
of a new three level topological model, which in particular supersedes a
recent proposal based on molecular enantiomers [57]. In this quantum
simulation correspondence, the drive detunings $\delta_{1}$ and $\delta_{3}$
correspond to onsite potential energies on the Lieb lattice, while the drive
amplitudes $\Omega_{1}$ and $\Omega_{2}$ mimic nearest-neighbour tight-binding
amplitudes, and $\Omega_{3}$ simulates a next-nearest-neighbour coupling along
one diagonal direction. The relative $\pi$ phase between the
$\Omega_{1},\Omega_{2}$ and $\Omega_{3}$ terms in Eq. (23) originates from a
periodic pattern of staggered magnetic fluxes, as shown in Fig. 4(a). These
fluxes break the time-reversal symmetry of the tight-binding model, while
preserving the translation invariance of the Bravais lattice. We thus end up
with a generalization of the celebrated Haldane’s model [58] but on the Lieb
instead of the honeycomb lattice. In the sublattice basis $(0,1,2)$ (see Fig.
4(a)), the corresponding Bloch Hamiltonian is

$\displaystyle H(\bm{k})=$ (24)
$\displaystyle\hbar\left(\begin{array}[]{ccc}0&\Omega_{1}\cos\left(\frac{k_{x}}{2}\right)&-i\Omega_{3}\cos\left(\frac{k_{x}-k_{y}}{2}\right)\\\
\Omega_{1}\cos\left(\frac{k_{x}}{2}\right)&\delta_{1}&\Omega_{2}\cos\left(\frac{k_{y}}{2}\right)\\\
i\Omega_{3}\cos\left(\frac{k_{x}-k_{y}}{2}\right)&\Omega_{2}\cos\left(\frac{k_{y}}{2}\right)&\delta_{3}\\\
\end{array}\right).$ (28)

The pseudo-momentum $\bm{k}$ is dimensionless, corresponding to a lattice
constant chosen as a length unit. The qutrit Hamiltonian (23) is then
recovered through the substitutions $k_{x}\rightarrow 2\phi_{1}$ and
$k_{y}\rightarrow 2\phi_{2}$.

We now aim at determining the ranges of drive parameters
$\Omega_{i},\delta_{i}$ that lead to topologically nontrivial band structures
for the model of Eq. (24). By definition, a topological band structures cannot
be smoothly deformed to that of an atomic limit [59]. In contrast, a trivial
band structure admit some band representations of the crystal space group on a
basis of symmetric localized orbitals [60, 61, 62, 63]. An efficient strategy
to detect topological band structures then consists of enumerating all
possible band representations of a space group and identifying band structures
that do not support such representations. This strategy lies at the heart of
the recent paradigm of Topological Quantum Chemistry and led to the
predictions of exhaustive catalogues of topological materials [64, 65, 66, 67,
68]. We can use this methodology to efficiently determine the phase diagram of
model (24). We first determine the band representations of the Lieb lattice in
Fig. 4(a) for the atomic limit $\Omega_{1,2,3}=0$. For non-degenerate onsite
energies $\delta_{1,3}\neq 0$ and $\delta_{1}\neq\delta_{3}$, and in the
presence of staggered magnetic fluxes, the lattice only has inversion symmetry
and belongs to the wallpaper space group $p2$. The three orbitals occupy the
maximal Wyckoff positions $q_{0}=(1/2,0)$, $q_{1}=(0,0)$, and $q_{2}=(0,1/2)$
in the primitive unit cell. Their elementary band representations are
determined from the band parities $p_{\nu}=\pm 1$ — i.e. eigenvalues of the
parity operator — at the inversion-invariant momenta in the Brillouin zone
depicted in Fig. 4(a) [69]. It leads to the band representations summarised in
the top panel in Fig. 4(b). The parity product

$\displaystyle\pi_{\nu}=p_{\nu}(\bm{\Gamma})p_{\nu}(\bm{X})p_{\nu}(\bm{Y})p_{\nu}(\bm{M})$
(29)

of each band $\nu$ is always positive in the atomic limit. Therefore, band
structures with negative parity products fall outside these band
representations and are topological.

Away from the atomic limit, i.e. for $\Omega_{1,2,3}\neq 0$, we enumerate all
the possible parity configurations of the band structure $H(\bm{k})$ at the
inversion-invariant momenta (see Appendix D and Fig. 8). Each configuration
leads to a colored region in the $\delta_{1}\delta_{3}$-plane in Fig. 4(c).
The ivory colored regions correspond to the parity configurations of the
atomic limit in Fig. 4(b). Thus, they describe trivial band insulators. In
contrast, we find that the red, green, and yellow regions exhibit negative
parity products, thus characterizing topological insulators. As an
illustration, the bottom panel in Fig. 4(b) specifies the parity configuration
of the yellow region, where $\pi_{\nu}=-1$ for $\nu=0$ and $\nu=2$. It shows
that the change of parity products between the trivial atomic limit and the
topological insulator requires the bands $\nu=1$ and $\nu=0$ ($\nu=2$) to
switch parities at point $\bm{X}$ ($\bm{Y}$) of the Brillouin zone. More
generally, a parity switch cannot occur continuously and requires the band gap
to close at one of the inversion-invariant momentum (see Appendix D). This
provides a very efficient determination of the phase diagram of Fig. 4(c) by
considering the gap closing at these inversion-invariant momenta as a function
of the $\Omega_{i}$ and $\delta_{i}$. Such band crossings mark the borders
between topologically distinct regions in the phase diagram represented in
Fig. 4(c), where from now on we use the shorthand notation for the Chern
number ${\mathcal{C}}^{(\nu)}={\mathcal{C}}^{(\nu)}_{12}$. One can further
show that this Chern number is non zero when the parity product is negative
[70, 71]. Since the qutrit Hamiltonian in Eq. (23) is recovered via the
substitutions $k_{x,y}\rightarrow 2\phi_{1,2}$, the Chern number for the
fluxonium qutrit is four times larger than for the Lieb insulator, hence the
values summarized in the table in Fig. 4(c).

Figure 5:  Pumping by a qutrit initialized in its ground state. (a) Filling of
the different modes $n_{i}$ as a function of time, for three different set of
pump parameters corresponding to points A, B and C in Fig. 6(a). The driving
amplitudes are chosen all equals $\Omega_{1,2,3}=\Omega$. The filling rate are
found to depend on the parameters of the pump. (b) The two different
combinations of fillings display the topological power
$\hbar\omega_{1}\omega_{2}{\mathcal{C}}^{(0)}/2\pi$ (grey dashed lines),
invariant on the region of stability of the phase diagram. where
${\mathcal{C}}^{(0)}=4$ is the Chern number of the ground state for all
parameter sets A,B,C.

####  IV.2.4 Topological power transfer between three modes

We now discuss how the topological nature of the qutrit pump manifests itself
in filling rate of the three modes. The dynamical system consist of three
classical modes described by the classical phases $\phi_{i}$ conjugated to
$n_{i}$ coupled to the qutrit through the Hamiltonian (23) in the rotating
frame. The equations of dynamics have the same form in the rotating frame
$\dot{n}_{i}=-\frac{1}{\hbar}\bra{\Psi}\partial_{\phi_{i}}\tilde{H}\ket{\Psi},i=1,2,3$
where the dynamics of the qutrit state $\ket{\Psi(t)}$ in the rotating frame
is governed by the Hamiltonian (23), see Appendix E for details. As said
above, the frequencies of amplitude modulation satisfies
$\omega_{3}=\omega_{1}-\omega_{2}$ such that
$\phi_{III}=\phi_{1}-\phi_{2}-\phi_{3}$ is a constant of motion, so we can
keep $\phi_{1}-\phi_{2}-\phi_{3}=0$ at all time. We consider the following
canonical transformation
$n_{I}=n_{1}+n_{3},n_{II}=n_{2}-n_{3},\phi_{I}=\phi_{1},\phi_{II}=\phi_{2}$.
The dynamics of $n_{I}$ and $n_{II}$ is given by

$\displaystyle\dot{n}_{1}+\dot{n}_{3}=-\frac{1}{\hbar}\frac{\partial
E_{\nu}}{\partial\phi_{1}}+{F}^{(\nu)}_{\phi_{1}\phi_{2}}\omega_{2}$ (30a)
$\displaystyle\dot{n}_{2}-\dot{n}_{3}=-\frac{1}{\hbar}\frac{\partial
E_{\nu}}{\partial\phi_{2}}-{F}^{(\nu)}_{\phi_{1}\phi_{2}}\omega_{1}$ (30b)

with $E_{\nu}$ the energy and ${F}^{(\nu)}$ the Berry curvature of the band
$\nu$ of the Hamiltonian
$H(\phi_{1},\phi_{2})\equiv\tilde{H}(\phi_{1},\phi_{2},\phi_{1}-\phi_{2})$ in
which the qutrit is initially prepared (see Appendix E). Then, the topological
power transfer between the modes $1,2,3$ is

$\hbar\omega_{1}\langle\dot{n}_{1}+\dot{n}_{3}\rangle_{t}=\hbar\frac{{\mathcal{C}}^{(\nu)}}{2\pi}\omega_{1}\omega_{2}=-\hbar\omega_{2}\langle\dot{n}_{2}-\dot{n}_{3}\rangle_{t},$
(31)

with ${\mathcal{C}}^{(\nu)}$ the Chern number of the band $\nu$ of the
Hamiltonian. From a more experimental point of view, the power transfer to
demonstrate is

$\frac{\Delta S_{1}}{hf_{1}}+\frac{\Delta
S_{3}}{hf_{3}}=\frac{{\mathcal{C}}^{(\nu)}}{2\pi}\omega_{2}\textrm{ and
}\frac{\Delta S_{2}}{hf_{2}}-\frac{\Delta
S_{3}}{hf_{3}}=-\frac{{\mathcal{C}}^{(\nu)}}{2\pi}\omega_{1}.$ (32)

In order to get a sense of how feasible this measurement is, let us set some
possible figures for the experiment that fulfill the criterion discussed
earlier. The fluxonium frequencies could be set to $f_{01}=4~{}\mathrm{GHz}$,
$f_{12}=6~{}\mathrm{GHz}$ and $f_{02}=10~{}\mathrm{GHz}$. It is then possible
to drive the transitions with $\Omega_{1,2,3}/2\pi=100~{}\mathrm{MHz}$ and a
similar range of variation for the detunings. The modulation frequencies could
then be $\omega_{1}/2\pi=5~{}\mathrm{MHz}$, $\omega_{2}/2\pi\simeq
3~{}\mathrm{MHz}$. From the simulations below, we see that the topological
power transfer can be resolved in about 30 periods $2\pi/\omega_{i}$, which is
a few $\mu\mathrm{s}$. This is well below the typical decoherence times of
fluxonium qubits, which will thus not limit the dynamics of the system during
the measurement. Verifying Eq. (32) thus requires to measure instantaneous
powers in the range of $hf_{i}\omega_{i}$. This corresponds to a power of
several dozens of aW, which is a level of precision that is now routinely
reached experimentally [72, 73, 74].

Figure 6:  Topological transitions for bands 0 and 1 in the case of same drive
amplitudes $\Omega_{1,2,3}=\Omega$. For each working point of the phase
diagram, we fit the time evolution of the energy
$E_{1}=\hbar\omega_{1}(n_{1}+n_{3})$ by a line on a time $\Delta
t=8\mu\mathrm{s}$ to construct the reduced power rate
$2\pi\dot{E}_{1}/\hbar\omega_{1}\omega_{2}$ (red dots) when the qutrit is
initialized in band 0, panels (b,e,h), or in band 1, panels (c,f,i). The
average populations of the qutrit $\bar{p}_{\mu}=\frac{1}{\Delta
t}\int_{0}^{\Delta t}|\innerproduct{\psi_{\mu}(t)}{\Psi(t)}|^{2}\differential
t$ are displayed to evaluate adiabaticity. (a-c) Transition line DE, with a
transition between bands 1 and 2 at $\delta_{1}/\Omega=-1$, and between bands
0 and 1 at $\delta_{1}/\Omega=1$. (d-f) Transition line HI, with a transition
between bands 0 and 1 at $\delta_{1}/\Omega=-1$, and between bands 1 and 2 at
$\delta_{1}/\Omega=1$. (g-i) Transition line FG, with a transition between the
three bands at $\delta_{1}/\Omega=\pm\sqrt{2}/2$.

###  IV.3 Numerical analysis of topological pumping

####  IV.3.1 Topological power transfer

In order to perform numerical simulations of the proposed experiment, we solve
the time-dependent Schrödinger equation of the qutrit under the rotating wave
approximation (23). We first determine the optimal parameters for the pump:
the stability of the adiabaticity evolution requires the largest gap. This is
reached at the resonant drive $\delta_{1}=\delta_{3}=0$, point A in the phase
diagram in Fig. 6(a), and with equal driving amplitudes
$\Omega_{1,2,3}=\Omega$, resulting in a gap $0.87~{}\hbar\Omega$. In these
conditions, from the analysis of Fig. 4(c) both bands 0 and 2 have non-zero
Chern number ${\mathcal{C}}=\pm 4$, whereas the band 1 is topologically
trivial with ${\mathcal{C}}=0$. Therefore the two lowest energy states do not
constitute an effective topological qubit, leading to different dynamics than
for a conventional $2$ level pump as we will see below. To ensure an ergodic
exploration of the classical phase space of the pump, we choose the ratio
between phases frequencies $\omega_{1}/\omega_{2}=(1+\sqrt{5})/2$.

Keeping parameters of the pump to point A in the phase diagram in Fig. 6(a),
we initialize the qutrit at $t=0$ in its ground state. The evolution of the
filling $n_{i}$ of each individual modes is represented in Fig. 5(a), and
varies linearly in time. However the filling rate $\dot{n}_{i}$ are not set by
the topological nature of the pump, and depend on the precise values of the
pump parameters: changing slightly these from point A to point B in the same
topological region of Fig. 6(a) leads to different filling rates, as seen in
Fig. 5(a). On the other hand, the topological power transfer defined in Eq.
(31) is insensitive of the precise values of the pump parameters, as shown in
Fig. 5(b). Besides the linear topological evolution in time, this power
transfer displays temporal fluctuations which have two different origins as
deduced from Eq. (30): a dominant spectral term, corresponding to variation of
the energy $E_{0}$ of the qutrit, and a geometrical contribution originating
from fluctuations of the Berry curvature around its topologically quantized
average value (see Appendix F). Thus the order of magnitude of the correlation
timescale of these fluctuations correspond to the period of the drive. A
reasonable requirement to detect the topological power transfer is to average
it over $30$ such independent fluctuations, leading to a measurement time of
8$\mu$s, as announced in the previous section.

####  IV.3.2 Numerical detection of topological transitions

Having established that the average topological power transfer gives access to
the Chern number of the band in which the qutrit was initialized, we now
address the detection of the topological phase transitions of Fig. 4(c) when
the detuning parameters $\delta_{1},\delta_{2}$ are varied. The richness of
the adiabatic dynamics of the qutrit pumps require different experimental
protocols adjusted to each phase transition. For example, sets A and C of
parameters in Fig. 6(a) lead to exactly the same topological power rate for a
qutrit initialized in the ground state, as shown in Fig. 5(b), but corresponds
to a different topological qutrit phase. Indeed, the corresponding qutrit
phases differ by the topological nature of the excited bands 1 and 2, while
the nature of the ground state is unchanged. Thus detecting this particular
transition requires an initialization of the qutrit in the first excited state
1. To detect all transitions, we monitor the evolution of the pumps with both
initialization in the 0 and 1 states. Figure 6 displays the resulting power
rate, determined by a linear fit using Eq. (31), as well as the average
populations $\bar{p}_{\mu}=\frac{1}{\Delta t}\int_{0}^{\Delta
t}|\innerproduct{\psi_{\mu}(t)}{\Psi(t)}|^{2}\differential t$ of the qutrit
state $\ket{\Psi(t)}$ on the three instantaneous eigenstates
$\ket{\psi_{\mu}(t)}$. Figures 6(b,e,h) correspond to a pump with qutrit
initialized in band 0, while Fig. 6(c,f,i) correspond to an initial
preparation in band 1.

Along the lines DE and HI in Fig. 6(a,d), the topological transitions occur
between two bands only, whereas along the line FG in Fig. 6(g) the gaps
between all three bands close at the transitions. Along any line, the
transition is detected by the evolution of the populations. Moreover, a
quantized power transfer in a given state appears as a direct test of the
adiabatic nature of the evolution, related to the distance to the transitions.
In that respect, optimal choice of parameters for the pump correspond to point
A in the yellow topological phase of the phase diagram, in which the bands 0
and 2 are non-trivial and spectrally separated by a trivial band 1 and thus
generically separated from a trivial phase by two transitions. Far from any
transition near point A, the instantaneous energy separation with different
eigenstates is large, resulting in an adiabatic evolution : the average
population of the qutrit in the initial band remains close to $1$ and the
power rate is quantized and set by the Chern number of the band.

In the other topological states of the qutrit, the effects of non-adiabaticity
are manifest, resulting from shorter distances to phase transitions and thus
small gaps. For example along line DE for a qutrit prepared in band 1, while
the Chern number ${\mathcal{C}}^{(1)}$ takes values $-4$ and +4 for
respectively $\delta_{1}/\Omega<-1$ and $\delta_{1}/\Omega>1$, the qutrit does
not evolve adiabatically and the dynamics of the classical variables (30) must
be corrected, leading to an unquantized power transfer shown in Fig. 6(c).
Similarly in Fig. 6(b) the Chern number $+4$ of band 0 for
$\delta_{1}/\Omega<1$ manifests itself as a pleateau of power rate for a
reduced set of parameters for $-1.5<\delta_{1}/\Omega<0.5$ (between points C
and B).

Figure 7: Non-adiabatic effect at longer time. The time evolution for the
qutrit prepared in band 0 of the topological energy combination
$\hbar\omega_{1}(n_{1}+n_{3})$ and qutrit populations $p_{\mu}(t)$ in the
instantaneous eigenstates is displayed for three working point A, B and C in
the phase diagram of Fig. 6(a). a) Point A, case of resonance
$\delta_{1}=\delta_{3}=0$ where the evolution of the qutrit remains adiabatic
during 200 period so the pumping rate is stable. b) Point B, limit case beyond
which pumping is no longer quantized on Fig. 6(b), where the evolution of the
qutrit is no longer adiabatic after approximately $8\mu\mathrm{s}$ so the
pumping rate is no longer quantized. c) Point C, other limit case beyond which
pumping is no longer quantized on Fig. 6(b), where in this phase the Chern
number of band 1 is opposite to band 0 so we see pumping in the other
direction when band 1 is mainly populated. Downsampling of data has been
applied for clarity of presentation.

####  IV.3.3 Non-adiabatic pumping

The importance of the non-adiabatic effects can be anticipated from the
estimation of the time of validity of the adiabatic approximation
$\tau_{\mathrm{adiab}}$ introduced in Eq. (7). In this expression, the maximum
of the adiabatic parameter $\epsilon_{\mu\nu}$ is taken on all the values of
phases $\phi_{1}$ and $\phi_{2}$, and the mean free time $\tau_{\mathrm{mft}}$
is the order of magnitude of the phases periodicity, we take
$\tau_{\mathrm{mft}}=2\pi/\omega_{2}\sim 0.3\mu\mathrm{s}$. For the point A, B
and C of the phase diagram, the Landau-Zener collision time
$\tau_{01}^{\mathrm{col}}$ associated to the transition between states 0 and 1
is one order of magnitude below this mean free time, which is consistent with
the derivation of the adiabatic time $\tau_{\mathrm{adiab}}$ discussed in
section II.1.2. For the point A of maximal stability with a preparation in
band 0, we get $\tau_{\mathrm{adiab}}^{\mathrm{A}}\sim 150\mathrm{ms}$ so the
non-adiabatic effect will not be limiting for experiments with these
parameters values. This is illustrated in Fig. 7(a), where the evolution for
100$\mu\mathrm{s}$ of the energy $\hbar\omega_{1}(n_{1}+n_{3})$ and the
populations $p_{\mu}(t)=|\innerproduct{\psi_{\mu}(t)}{\Psi(t)}|^{2}$ of the
qutrit state $\ket{\Psi(t)}$ on the three instantaneous eigenstates
$\ket{\psi_{\mu}(t)}$ is displayed. The qutrit stays in the ground state with
$p_{0}(t)>0.997$, and the energy is transferred at the topologically quantized
rate. Figure 7(b) corresponds to point B closer to the topological transition
line towards the phase where ${\mathcal{C}}^{(0)}=0$. The estimated time of
adiabaticity for band 0 is $\tau_{\mathrm{adiab}}^{\mathrm{B}}\sim
6\mu\mathrm{s}$. After this typical time, the population on state 1 exceeds
0.1, in agreement with the definition of $\tau_{\mathrm{adiab}}$, and the
pumping rate deviate from its topologically quantized value. Figure 7(c)
corresponds to point C, where we crossed a different topological transition
line, where the ground state remains topological but the first excited state
switches from trivial to non-trivial with Chern number
${\mathcal{C}}^{(1)}=-4$. We compute here
$\tau_{\mathrm{adiab}}^{\mathrm{C}}\sim 8\mu\mathrm{s}$ in agreement with the
observed deviations from the adiabatic evolution. At longer times, about
70$\mu$s, the first excited state is mostly populated and the energy pumping
is reversed, manifesting the associated change of Chern number.

##  V Discussion

We have proposed an experiment that is able to observe a topologically
protected power exchange. The topological properties appear in the measured
incoming and outgoing energy flows that drive a quantum system. Our general
description of topological quantum pumps includes the classical degrees of
freedom that carry these flows. Considering a qutrit instead of a qubit is key
for two aspects. First, it solves the tremendous challenge to probe the power
flows that drive a two level topological pump. Second, owing to its richer
dynamics, which simulates the topological band properties of a $3$ band Chern
model, it gives access to various protocols of pumping which can be freely
chosen by setting the initial state of the qutrit. Besides the fascinating
perspective to actually measure this topological power transfer, such a system
also opens the paths to the study of the interplay between decoherence and the
topological adiabatic evolution of the quantum system. Our framework raises
two natural questions. Topological pumping requires an initial correlation
between the qutrit and the classical modes. Then how long does the pumping
survive in presence of qutrit decoherence? Besides, could we revert the
perspective and use the developed framework and measurable pumping rate as a
tool to characterize the correlations between the quantum system and the
classical modes. Finally, let us stress that while we have proposed an
experiment demonstrating the topologically protected transfer of microwave
power using a superconducting circuit, our general framework can be applied to
any quantum system and its driving environment such as cold atoms, mechanical
oscillators or polaritons. Symmetries are essential to classify topological
matter and in particular topological pumps [31]. The enforcement of these
symmetries to protected topological pumping in these different systems is a
stimulating perspective.

###### Acknowledgements.

This research was supported by IDEX Lyon project ToRe (Contract No.
ANR-16-IDEX-0005). C.D. also acknowledges the support of Idex Bordeaux (Maesim
Risky project of the LAPHIA Programme) and Quantum Matter Bordeaux. We are
grateful to Anton Akhmerov, Antoine Essig, Leonid Glazman, Sébastien Jezouin
and Christophe Mora for insightful discussions.

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##  Appendix A First-order adiabatic evolution

For the sake of completeness we detail here the derivation of the first-order
adiabatic evolution usually derived in different manner in the literature, see
for example [33, 4, 75, 76, 50]. The quantum Hamiltonian $H(t)$ depends
continuously on time through the set of variables $\\{q_{\alpha}(t)\\}$. The
time-dependent Schrödinger equation is

$\displaystyle i\hbar\frac{\differential}{\differential
t}|\Psi(t)\rangle=H(t)|\Psi(t)\rangle~{}.$ (33)

We look for a quantum state of the form

$\displaystyle|\Psi(t)\rangle=\sum_{\nu}a_{\nu}(t)|\psi_{\nu}(t)\rangle~{},$
(34)

where $|\psi_{\nu}(t)\rangle$ are the instantaneous eigenstates of $H$ and
satisfy

$\displaystyle H(t)|\psi_{\nu}(t)\rangle=E_{\nu}(t)|\psi_{\nu}(t)\rangle~{}.$
(35)

This leads to the equation of motion

$\dot{a}_{\nu}(t)=i\left[i\langle\psi_{\nu}(t)|\frac{\differential}{\differential
t}|\psi_{\nu}(t)\rangle-\frac{1}{\hbar}E_{\nu}(t)\right]a_{\nu}(t)\\\
-\sum_{\mu\neq\nu}\langle\psi_{\nu}(t)|\frac{\differential}{\differential
t}|\psi_{\mu}(t)\rangle a_{\mu}(t)~{}.$ (36)

We can get rid of the first term in the right-hand side by introducing the
phases

$\displaystyle\Delta_{\nu}(t)=-\frac{1}{\hbar}\int_{0}^{t}\differential
t^{\prime}~{}E_{\nu}(t^{\prime})$ (37)

and

$\displaystyle\Gamma_{\nu}(t)=\int_{0}^{t}\differential
t^{\prime}~{}\bra{\psi_{\nu}(t^{\prime})}i\frac{\differential}{\differential
t^{\prime}}\ket{\psi_{\nu}(t^{\prime})},$ (38)

and making the substitution
$a_{\nu}(t)=e^{i\Delta_{\nu}(t)+i\Gamma_{\nu}(t)}\tilde{a}_{\nu}(t)$. The time
integration of Eq. (36) then results in

$\tilde{a}_{\nu}(\tau)-\tilde{a}_{\nu}(0)=\\\
-\sum_{\mu\neq\nu}\int_{0}^{\tau}d\tau^{\prime}A_{\mu\nu}(\tau^{\prime})e^{i\Gamma_{\mu\nu}(\tau^{\prime})}e^{\frac{1}{\eta}\int_{0}^{\tau^{\prime}}d\tau^{\prime\prime}\Delta_{\mu\nu}(\tau^{\prime\prime})}$
(39)

where
$A_{\mu\nu}(\tau)=\langle\psi_{\nu}(\tau)|\partial_{\tau}\psi_{\mu}(\tau)\rangle\tilde{a}_{\mu}(\tau)$,
$\Gamma_{\mu\nu}(\tau)=\Gamma_{\mu}(\tau)-\Gamma_{\nu}(\tau)$,
$\Delta_{\mu\nu}(\tau)=-i(E_{\mu}(\tau)-E_{\nu}(\tau))/(\hbar\Delta)$, and
$\Delta=\min_{\mu,t}|E_{\mu}(t)-E_{\nu}(t)|/\hbar$ is assumed to be non zero.
Here, we have also introduced a dimensionless “time” $\tau=\eta\Delta t$,
where $\eta$ is a small parameter. We can evaluate the time integral in Eq.
(39) by integrating by part on the exponential term

$e^{\frac{1}{\eta}\int_{0}^{\tau^{\prime}}d\tau^{\prime\prime}\Delta_{\mu\nu}(\tau^{\prime\prime})}=\frac{\eta}{\Delta_{\mu\nu}(\tau^{\prime})}\frac{\differential}{\differential\tau^{\prime}}\left[e^{\frac{1}{\eta}\int_{0}^{\tau^{\prime}}d\tau^{\prime\prime}\Delta_{\mu\nu}(\tau^{\prime\prime})}\right].$
(40)

The first-order correction in the adiabatic limit results in

$\tilde{a}_{\nu}(\tau)-\tilde{a}_{\nu}(0)=\\\
-i\hbar\eta\Delta\sum_{\mu\neq\nu}\frac{\langle\psi_{\nu}(\tau)|\partial_{\tau}\psi_{\mu}(\tau)\rangle}{E_{\mu}(\tau)-E_{\nu}(\tau)}\tilde{a}_{\mu}(\tau)e^{i\Gamma_{\mu\nu}(\tau)}e^{\frac{1}{\eta}\int_{0}^{\tau}d\tau^{\prime}\Delta_{\mu\nu}(\tau^{\prime})}\\\
+i\hbar\eta\Delta\sum_{\mu\neq\nu}\frac{\langle\psi_{\nu}(0)|\partial_{\tau}\psi_{\mu}(0)\rangle}{E_{\mu}(0)-E_{\nu}(0)}\tilde{a}_{\mu}(0)+\mathcal{O}(\eta^{2}).$
(41)

If the quantum system is initially prepared in the instantaneous eigenstate
$|\psi_{\nu}\rangle$ — i.e. $a_{\nu}(0)=1$ and $a_{\mu\neq\nu}(0)=0$ — we find
the time dependent state

$\ket{\Psi(t)}=e^{i\Gamma_{\nu}(t)+i\Delta_{\nu}(t)}\left(\ket{\psi_{\nu}(t)}-i\hbar\sum_{\mu\neq\nu}\frac{\bra{\psi_{\mu}(t)}\frac{\differential}{\differential
t}\ket{\psi_{\nu}(t)}}{E_{\nu}(t)-E_{\mu}(t)}\ket{\psi_{\mu}(t)}\right)\\\
+i\hbar\sum_{\mu\neq\nu}e^{i\Gamma_{\mu}(t)+i\Delta_{\mu}(t)}\left[\frac{\bra{\psi_{\mu}(t)}\frac{\differential}{\differential
t}\ket{\psi_{\nu}(t)}}{E_{\nu}(t)-E_{\mu}(t)}\right]_{t=0}\ket{\psi_{\mu}(t)}+\mathcal{O}(\eta^{2})$
(42)

where the components on $\ket{\psi_{\mu\neq\nu}}$ are first order terms in
$\eta$. Using the identity
$\bra{\psi_{\mu}(t)}\frac{\differential}{\differential
t}\ket{\psi_{\nu}(t)}=\bra{\psi_{\mu}}\frac{\differential H}{\differential
t}\ket{\psi_{\nu}}/(E_{\nu}-E_{\mu})$, the correction at first order in $\eta$
to the equation of motion of the variable $p_{\alpha}$ is

$-\bra{\Psi}\frac{\partial H}{\partial
q_{\alpha}}\ket{\Psi}=-\bra{\psi_{\nu}}\frac{\partial H}{\partial
p_{\alpha}}\ket{\psi_{\nu}}\\\
+i\hbar\sum_{\mu\neq\nu}\frac{\bra{\psi_{\nu}}\frac{\partial H}{\partial
q_{\alpha}}\ket{\psi_{\mu}}\bra{\psi_{\mu}}\frac{\differential
H}{\differential t}\ket{\psi_{\nu}}}{\left(E_{\nu}-E_{\mu}\right)^{2}}\\\
-i\hbar\sum_{\mu\neq\nu}\frac{\bra{\psi_{\nu}}\frac{\differential
H}{\differential t}\ket{\psi_{\mu}}\bra{\psi_{\mu}}\frac{\partial H}{\partial
q_{\alpha}}\ket{\psi_{\nu}}}{\left(E_{\nu}-E_{\mu}\right)^{2}}$ (43)

where the last term of (42) induces only first order terms rapidly oscillating
at the Bohr frequencies $(E_{\nu}-E_{\mu})/\hbar$ which has no effect on the
dynamics at the time-scale of the slow variables. We note that this terms are
due to the choice of initial condition and could be eliminated if we choose
$a_{\nu}(0)=1$ and
$a_{\mu\neq\nu}(0)=-i\hbar\left[\bra{\psi_{\mu}(t)}\frac{\differential}{\differential
t}\ket{\psi_{\nu}(t)}/(E_{\nu}(t)-E_{\mu}(t))\right]_{t=0}$. For the
implementation with a superconducting qutrit, a qutrit is fully controllable
and any quantum state can be prepared [77, 78, 79]. Using this initial
condition, we recover the adiabatic parameter $\epsilon_{\mu\nu}$ introduce in
section II.1.2 in the population on an excited state
$|\innerproduct{\psi_{\nu}(t)}{\Psi(t)}|^{2}=\epsilon_{\mu\nu}(t)^{2}$.
Returning to equation (43), the time dependence of the Hamiltonian
$H(t)=H(\\{q_{\beta}(t)\\})$ is due to the coupling to the classical variables
$q_{\beta}$, whose classical dynamics is not modified by the coupling to the
quantum system,
$\dot{q}_{\beta}=\dot{q}_{\beta}^{(0)}=\frac{\partial\mathcal{H}_{\beta}}{\partial
p_{\beta}}$. Thus, using the relation $\bra{\psi_{\nu}}\frac{\partial
H}{\partial q_{\alpha}}\ket{\psi_{\nu}}=\frac{\partial E_{\nu}}{\partial
p_{\alpha}}$ and the expression of the components of the Berry curvature two
form given in (5), we obtain for the correction of $\dot{p}_{\alpha}$

$-\bra{\Psi}\frac{\partial H}{\partial q_{\alpha}}\ket{\Psi}=-\frac{\partial
E_{\nu}}{\partial
q_{\alpha}}+\hbar\sum_{\beta\neq\alpha}\dot{q}_{\beta}^{(0)}F_{q_{\alpha}q_{\beta}}.$
(44)

##  Appendix B Details on the derivation of the Hamiltonian

We detail here the derivation of the Hamiltonian of the qutrit in the rotating
frame explained in section IV.2.2. The circuit is driven by three pumps whose
amplitudes is modulated in time according to the drive Hamiltonian

$H_{\mathrm{drive}}=\hbar\sum_{i=1}^{3}g_{i}\cos(\phi_{i})\cos(\theta_{i})\hat{N}.$
(45)

with $\theta_{i}(t)=2\pi f_{i}t+\theta_{i}^{0}$ the phase of the the
electromagnetic field of frequency $f_{i}$, $\phi_{i}(t)=\omega_{i}t$ the
phase of the time modulation of the amplitude, and $g_{i}$ the coupling rates.
In this basis $(\ket{0},\ket{1},\ket{2})$ of the three eigenstates of
$H_{\text{flux}}$ (22) of lowest energy, the diagonal elements of $\hat{N}$
are null so the Hamiltonian in the laboratory frame
$H_{\text{labo}}=H_{\text{flux}}+H_{\text{drive}}$ has the form:

$H_{\text{labo}}=\\\
\hbar\left(\begin{array}[]{ccc}0&\displaystyle\sum_{i=1}^{3}\Omega_{i,01}\cos(\phi_{i})\cos(\theta_{i})&\displaystyle\sum_{i=1}^{3}\Omega_{i,02}\cos(\phi_{i})\cos(\theta_{i})\\\
c.c.&2\pi
f_{01}&\displaystyle\sum_{i=1}^{3}\Omega_{i,12}\cos(\phi_{i})\cos(\theta_{i})\\\
c.c.&c.c.&2\pi f_{02}\end{array}\right)$ (46)

where $f_{01}$ and $f_{02}$ are the transition frequencies of
$H_{\text{flux}}$, and $\Omega_{i,ab}=g_{i}\bra{a}\hat{N}\ket{b}$. We change
of reference frame with the unitary transformation
$U(t)=\textrm{diag}(1,\exp(-i2\pi f_{1}t),\exp(-i2\pi f_{3}t))$. The
frequencies of the three pumps satisfy the constraint $f_{3}=f_{1}+f_{2}$, so
the Hamiltonian in the rotating frame is

$\displaystyle H_{\text{rot}}$
$\displaystyle=U^{\dagger}H_{\text{labo}}U-i\hbar
U^{\dagger}\frac{\differential U}{\differential
t}=\hbar\left(\begin{array}[]{ccc}0&e^{-i2\pi
f_{1}t}\displaystyle\sum_{i=1}^{3}\Omega_{i,01}\cos(\phi_{i})\cos(\theta_{i})&e^{-i2\pi
f_{3}t}\displaystyle\sum_{i=1}^{3}\Omega_{i,02}\cos(\phi_{i})\cos(\theta_{i})\\\
c.c.&\delta_{1}&e^{-i2\pi
f_{2}t}\displaystyle\sum_{i=1}^{3}\Omega_{i,12}\cos(\phi_{i})\cos(\theta_{i})\\\
c.c.&c.c.&\delta_{3}\end{array}\right)$ (50)

with the detuning $\delta_{i}$ given by

$\displaystyle\delta_{1}$ $\displaystyle=2\pi f_{01}-2\pi f_{1}$ (51)
$\displaystyle\delta_{2}$ $\displaystyle=2\pi f_{12}-2\pi f_{2}$ (52)
$\displaystyle\delta_{3}$ $\displaystyle=2\pi f_{02}-2\pi
f_{3}=\delta_{1}+\delta_{2}.$ (53)

The phases of the drives are given by $\theta_{i}(t)=2\pi
f_{i}t+\theta_{i}^{0}$. In the rotating wave approximation, we ignore the
terms of the Hamiltonian in the rotating frame which oscillate at the
frequency of the drives, so we approximate the Hamiltonian by

$H_{\text{rot}}(t)\simeq H(\phi_{1},\phi_{2},\phi_{3})=\\\
\hbar\begin{pmatrix}0&\frac{1}{2}\Omega_{1,01}\cos(\phi_{1})e^{i\theta_{1}^{0}}&\frac{1}{2}\Omega_{3,02}\cos(\phi_{3})e^{i\theta_{3}^{0}}\\\
c.c.&\delta_{01}&\frac{1}{2}\Omega_{2,12}\cos(\phi_{2})e^{i\theta_{2}^{0}}\\\
c.c.&c.c.&\delta_{02}\end{pmatrix}$ (54)

where the other terms oscillates at the frequency $f_{i}\pm f_{j}$. We recover
the Hamiltonian (23) by noting the drive amplitudes
$\Omega_{1}=\frac{1}{2}|\Omega_{1,01}|=\frac{1}{2}g_{1}|\bra{0}\hat{N}\ket{1}|$,
$\Omega_{2}=\frac{1}{2}|\Omega_{2,12}|=\frac{1}{2}g_{2}|\bra{1}\hat{N}\ket{2}|$
and
$\Omega_{3}=\frac{1}{2}|\Omega_{3,02}|=\frac{1}{2}g_{3}|\bra{0}\hat{N}\ket{2}|$,
and by choosing the initial pump phases $\theta_{i}^{0}$ to set the complex
phase of the couplings at the desired value. The rotating wave approximation
is valid if the drive amplitudes and detunings are much lower than the
frequencies $f_{i}\pm f_{j}$ of the oscillating terms, which means that they
must be much lower than the difference between any two transition frequencies
of the fluxonium as said in the section IV.2.1.

##  Appendix C Classical variables coupled to the qutrit

The fluxonium is coupled to three pumps whose amplitude are modulated in time.
The coupling Hamiltonian in the laboratory frame is

$H_{\text{drive}}=\hbar\left(\sum_{i=1}^{3}g_{i}\cos(\phi_{i})\cos(\theta_{i})\right)\hat{N}$
(55)

where $\theta_{i}(t)=2\pi f_{i}t$ is the phase of the pump and
$\phi_{i}(t)=\omega_{i}t$ is the phase of time-modulation of the amplitude of
the pump. The term in parenthesis is proportional to the amplitude of the
propagating wave on the line

$\displaystyle\mathcal{A}(t)$
$\displaystyle=\sum_{i=1}^{3}2\mathcal{A}_{i}\cos(\phi_{i}(t))\cos(\theta_{i}(t))$
(56)
$\displaystyle=\sum_{i=1}^{3}\mathcal{A}_{i}\left(\cos(\theta_{i}^{+}(t))+\cos(\theta_{i}^{-}(t))\right)$
(57)

with $\theta_{i}^{\pm}=\theta_{i}\pm\phi_{i}$. Thus, the transmission line
contains six modes at frequencies $f_{i}^{\pm}=f_{i}\pm\omega_{i}/2\pi$, for
$i=1,2,3$ (see figure 3). As explained in the section IV.2.1, we model the
propagating mode at frequency $f_{i}^{\pm}$ as a classical mode of energy
$hfn_{i}^{\pm}$, where the phase $\theta_{i}^{\pm}$ of each mode is conjugated
to $\hbar n_{i}^{\pm}$, such that the net photon flux is given by the
difference between the outgoing and incoming signals at this frequency
$hf_{i}^{\pm}\dot{n}_{i}^{\pm}=S_{\text{out}}[f_{i}^{\pm}]-S_{\text{in}}[f_{i}^{\pm}]$.
The topological pumping describes the dynamics of the variables conjugated to
$\phi_{i}$. The change of variables

$\displaystyle\theta_{i}$
$\displaystyle=\frac{1}{2}\left(\theta_{i}^{+}+\theta_{i}^{-}\right)$ (58)
$\displaystyle\phi_{i}$
$\displaystyle=\frac{1}{2}\left(\theta_{i}^{+}-\theta_{i}^{-}\right)$ (59)
$\displaystyle m_{i}$ $\displaystyle=n_{i}^{+}+n_{i}^{-}$ (60) $\displaystyle
n_{i}$ $\displaystyle=n_{i}^{+}-n_{i}^{-}$ (61)

is a canonical change of variables, which means that it preserves the Poisson
brackets, so the variable $\hbar n_{i}=\hbar(n_{i}^{+}-n_{i}^{-})$ is
conjugated to the phase $\phi_{i}$. The topological pumping relates the rates
$\dot{n}_{i}$, so we want to measure

$\displaystyle\dot{n}_{i}$
$\displaystyle=\frac{S_{\text{out}}[f_{i}^{+}]-S_{\text{in}}[f_{i}^{+}]}{hf_{i}^{+}}-\frac{S_{\text{out}}[f_{i}^{-}]-S_{\text{in}}[f_{i}^{-}]}{hf_{i}^{-}}$
(62) $\displaystyle\simeq\frac{\Delta S_{i}}{hf_{i}}$ (63)

with $\Delta S_{i}=S_{\text{out}}[f_{i}^{+}]-S_{\text{out}}[f_{i}^{-}]$ if we
consider $f_{i}^{\pm}\simeq f_{i}$ and
$S_{\text{in}}[f_{i}^{+}]=S_{\text{in}}[f_{i}^{-}]$.

##  Appendix D Chern insulator on the Lieb lattice

The three-band insulator on the Lieb lattice satisfies inversion symmetry. We
write the inversion operator as $P(\bm{k})$=Diag($e^{ik_{x}}$,1,$e^{ik_{y}}$).
It leads to four inversion-invariant momenta in the 2D Brillouin zone:
$\bm{\Gamma}=(0,0)$, $\bm{X}=(\pi,0)$, $\bm{Y}=(0,\pi)$, and
$\bm{M}=(\pi,\pi)$. At these high-symmetry points, the Bloch Hamiltonian
commutes with the inversion operator. Thus, the parities $p_{\nu}$ –
eigenvalues of the inversion operator – are good quantum numbers to label each
energy band $\nu$ at the inversion-invariant momenta. We sort the energy bands
as $E_{0}<E_{1}<E_{2}$ and introduce the triplet
$\bm{\pi}=\left(\pi_{0},\pi_{1},\pi_{2}\right)$, where $\pi_{\nu}$ is the band
parity product

$\displaystyle\pi_{\nu}=p_{\nu}(\bm{\Gamma})p_{\nu}(\bm{X})p_{\nu}(\bm{Y})p_{\nu}(\bm{M})=\pm
1~{}.$ (64)

We now determine the different configurations of parity products allowed for
the three bands emulated by the fluxonium.

At momentum $\bm{\Gamma}$, the parity operator reduces to the identity matrix.
All energy bands have the same parity, regardless of the Hamiltonian
parameters. This leads to the parity triplet
$\bm{p}(\bm{\Gamma})=(p_{0}(\bm{\Gamma}),p_{1}(\bm{\Gamma}),p_{2}(\bm{\Gamma}))=(+,+,+)$.

Figure 8: Parity product of the bands: The central map represents the minimum
value of the lower gap ($\Delta_{01}$) or upper gap ($\Delta_{12}$) over the
Brillouin zone, as a function of the detunings $\delta_{1}$ and $\delta_{3}$.
It is obtained from numerical diagonalization of the Bloch Hamiltonian (24).
Energy is in units of $\Omega_{1}=\Omega_{2}=\Omega_{3}=\Omega$. The white
areas correspond to values of the detuning for which a band gap closes. They
are well described by the dashed (dotted) lines obtained analytically from the
closing conditions of gap $\Delta_{01}$ ($\Delta_{12}$) at the inversion-
invariant momenta $\bm{\Gamma}$, $\bm{X}$, $\bm{Y}$, and $\bm{M}$ in the
Brillouin zone. These degeneracy lines mark the transitions between
topologically nonequivalent band insulators, where the parity product
$\pi_{\nu}$ of certain bands changes signs. The inversion eigenvalues labeling
the bands at $\bm{\Gamma}$, $\bm{X}$, $\bm{Y}$, and $\bm{M}$ are shown in the
green boxed insets, as a parity triplet $\bm{p}=(p_{0},p_{1},p_{2})$
associated with the band energies $E_{0}<E_{1}<E_{2}$. It is only shown for
insulating regions that exhibit negative band parity products. In such
regions, the band structures do not support any band representation and cannot
be adiabatically connected to an atomic limit. The fluxonium qutrit then
simulates nontrivial Chern insulators.

At momentum $\bm{X}$, the parity operator and the Bloch Hamiltonian read
$P(\bm{X})=\text{Diag}(-1,1,1)$ and

$\displaystyle H(\bm{X})=\left(\begin{array}[]{ccc}0&0&0\\\
0&\delta_{1}&\Omega_{2}\\\ 0&\Omega_{2}&\delta_{3}\\\ \end{array}\right).$
(68)

The eigenspace of parity $-1$ is associated with the energy level
$E_{*}(\bm{X})=0$. This is fixed regardless of the Hamiltonian parameters. In
contrast, the eigenspace of parity $+1$ refers to the energy levels

$\displaystyle
E_{\pm}(\bm{X})=\frac{\delta_{1}+\delta_{3}}{2}\pm\frac{1}{2}\sqrt{4\Omega_{2}^{2}+(\delta_{1}-\delta_{3})^{2}}~{},$
(69)

which depends on the fluxonium drives. This allows three different
configurations of parities:

  * •

$E_{*}(\bm{X})<E_{-}(\bm{X})<E_{+}(\bm{X})$ with parities
$\bm{p}(\bm{X})=(-,+,+)$. It occurs when $\delta_{1}>0$, $\delta_{3}>0$, and
$\delta_{3}>\Omega_{2}^{2}/\delta_{1}$.

  * •

$E_{-}(\bm{X})<E_{+}(\bm{X})<E_{*}(\bm{X})$ with parities
$\bm{p}(\bm{X})=(+,+,-)$. It occurs when $\delta_{1}<0$, $\delta_{3}<0$, and
$\delta_{3}<\Omega_{2}^{2}/\delta_{1}$.

  * •

$E_{-}(\bm{X})<E_{*}(\bm{X})<E_{+}(\bm{X})$ with parities
$\bm{p}(\bm{X})=(+,-,+)$ otherwise.

At momentum $\bm{Y}$, the parity operator and the Bloch Hamiltonian read
$P(\bm{Y})=\textrm{Diag}(1,1,-1)$ and

$\displaystyle H(\bm{Y})=\left(\begin{array}[]{ccc}0&\Omega_{1}&0\\\
\Omega_{1}&\delta_{1}&0\\\ 0&0&\delta_{3}\\\ \end{array}\right).$ (73)

The eigenspace of parity $-1$ is associated with the energy level
$E_{*}(\bm{Y})=\delta_{3}$. The eigenspace of parity $+1$ refers to the energy
levels

$\displaystyle
E_{\pm}(\bm{Y})=\frac{\delta_{1}}{2}\pm\frac{1}{2}\sqrt{4\Omega_{1}^{2}+\delta_{1}^{2}}~{}.$
(74)

This allows three different configurations of parities:

  * •

$E_{*}(\bm{Y})<E_{-}(\bm{Y})<E_{+}(\bm{Y})$ with parities
$\bm{p}(\bm{Y})=(-,+,+)$. It occurs when
$2\delta_{3}<\delta_{1}-\sqrt{4\Omega_{1}^{2}+\delta_{1}^{2}}$.

  * •

$E_{-}(\bm{Y})<E_{+}(\bm{Y})<E_{*}(\bm{Y})$ with parities
$\bm{p}(\bm{Y})=(+,+,-)$. It occurs when
$2\delta_{3}>\delta_{1}+\sqrt{4\Omega_{1}^{2}+\delta_{1}^{2}}$.

  * •

$E_{-}(\bm{Y})<E_{*}(\bm{Y})<E_{+}(\bm{Y})$ with parities
$\bm{p}(\bm{Y})=(+,-,+)$ otherwise.

At momentum $\bm{M}$, the parity operator and the Hamiltonian read
$P(\bm{M})=\textrm{Diag}(-1,1,-1)$ and

$\displaystyle H(\bm{M})=\left(\begin{array}[]{ccc}0&0&-i\Omega_{3}\\\
0&\delta_{1}&0\\\ i\Omega_{3}&0&\delta_{3}\\\ \end{array}\right).$ (78)

The eigenspace of parity $+1$ is associated with the energy level
$E_{*}(\bm{M})=\delta_{1}$. The eigenspace of parity $-1$ refers to the energy
levels

$\displaystyle
E_{\pm}(\bm{M})=\frac{\delta_{3}}{2}\pm\frac{1}{2}\sqrt{4\Omega_{3}^{2}+\delta_{3}^{2}}~{}.$
(79)

This allows three different energy configurations:

  * •

$E_{*}(\bm{M})<E_{-}(\bm{M})<E_{+}(\bm{M})$ with parities
$\bm{p}(\bm{M})=(+,-,-)$. It occurs when $\delta_{1}<0$ and
$\delta_{3}>\delta_{1}-\Omega_{3}^{2}/\delta_{1}$.

  * •

$E_{-}(\bm{M})<E_{+}(\bm{M})<E_{*}(\bm{M})$ with parities
$\bm{p}(\bm{M})=(-,-,+)$. It occurs when $\delta_{1}>0$ and
$\delta_{3}<\delta_{1}-\Omega_{3}^{2}/\delta_{1}$.

  * •

$E_{-}(\bm{M})<E_{*}(\bm{M})<E_{+}(\bm{M})$ with parities
$\bm{p}(\bm{M})=(-,+,-)$ otherwise.

This shows that any change of band parity requires the band gap to close at
the inversion-invariant momenta. All the band-parity configurations determined
above, as well as their parity products, are summarized in the insets of Fig.
8 as a function of $\delta_{1}$ and $\delta_{3}$.

##  Appendix E Dynamics in the rotating frame

The equations of dynamics of the classical variables $n_{i}$ conjugated to the
phases $\phi_{i}$ are first written in the laboratory frame

$\dot{n}_{i}=-\frac{1}{\hbar}\bra{\Psi_{\text{labo}}(t)}\frac{\partial
H_{\text{labo}}}{\partial\phi_{i}}\ket{\Psi_{\text{labo}}(t)},$ (80)

where the dynamics of state $\ket{\Psi_{\text{labo}}(t)}$ of the qutrit in the
laboratory frame is governed by the Hamiltonian
$H_{\text{labo}}(\\{\phi_{j},\theta_{j}\\})$, introduce before (46). In the
rotating frame, the dynamics of
$\ket{\Psi_{\text{rot}}(t)}=U^{\dagger}(t)\ket{\Psi_{\text{labo}}(t)}$ is
governed by $H_{\text{rot}}=U^{\dagger}H_{\text{labo}}U-i\hbar
U^{\dagger}\frac{\differential U}{\differential t}$ which satisfies
$\frac{\partial H_{\text{rot}}}{\partial\phi_{i}}=U^{\dagger}\frac{\partial
H_{\text{labo}}}{\partial\phi_{i}}U$ since the unitary transformation $U(t)$
does not depend on the phase $\phi_{i}$. Thus, the equation of the dynamics of
$n_{i}$ has the same form in the rotating frame

$\dot{n}_{i}=-\frac{1}{\hbar}\bra{\Psi_{\text{rot}}(t)}\frac{\partial
H_{\text{rot}}}{\partial\phi_{i}}\ket{\Psi_{\text{rot}}(t)}$ (81)

where in the rotating wave approximation, we consider
$H_{\text{rot}}(\\{\phi_{j},\theta_{j}\\},t)\simeq\tilde{H}(\phi_{1},\phi_{2},\phi_{3})$,
with the 3 phases Hamiltonian given by (23). We consider the following
canonical change of variable

$\displaystyle n_{I}$ $\displaystyle=n_{1}+n_{3}$
$\displaystyle\quad;\quad\phi_{I}$ $\displaystyle=\phi_{1}$ (82a)
$\displaystyle n_{II}$ $\displaystyle=n_{2}-n_{3}$
$\displaystyle\quad;\quad\phi_{II}$ $\displaystyle=\phi_{2}$ (82b)
$\displaystyle n_{III}$ $\displaystyle=-n_{3}$
$\displaystyle\quad;\quad\phi_{III}$
$\displaystyle=\phi_{1}-\phi_{2}-\phi_{3}$ (82c)

satisfying $\\{\phi_{A},n_{B}\\}=1/\hbar$, $A,B=I,II,III$. Since these new
variables are conjugated, the equations of motion are

$\dot{n}_{A}=-\frac{1}{\hbar}\frac{\partial
E_{\nu}}{\partial\phi_{A}}+\sum_{B\neq
A}\dot{\phi}_{B}{F}^{(\nu)}_{\phi_{A}\phi_{B}}$ (83)

with $E_{\nu}$ the energy and ${F}^{(\nu)}$ the Berry curvature of the band
$\nu$ of
$H(\phi_{I},\phi_{II},\phi_{III})=\tilde{H}(\phi_{I},\phi_{II},\phi_{I}-\phi_{II}-\phi_{III})$.
The frequencies of the phases $\phi_{1},\phi_{2},\phi_{3}$ are chosen such
that $\omega_{3}=\omega_{1}-\omega_{2}$, thus $\dot{\phi}_{III}=0$ and we keep
$\phi_{III}=0$ at all time with the initial condition. Thus, the equations of
motion reduce to (30).

##  Appendix F Temporal fluctuations

During the adiabatic evolution, the time derivative of the energy of a mode
can be decomposed in a sum of three terms

$\hbar\omega_{1}(\dot{n}_{1}+\dot{n}_{3})=-\omega_{1}\frac{\partial
E_{\nu}}{\partial\phi_{1}}+\hbar\omega_{1}\omega_{2}(F_{\phi_{1}\phi_{2}}^{(\nu)}-\frac{{\mathcal{C}}^{(\nu)}}{2\pi})+\hbar\frac{\omega_{1}\omega_{2}}{2\pi}{\mathcal{C}}^{(\nu)}.$
(84)

The first term is the variation of the energy $E_{\nu}$ of the band $\nu$ of
the qutrit, corresponding to an energy exchange between the qutrit and the
mode. The second term correspond to the fluctuation of the Berry curvature
${F}^{(\nu)}_{\phi_{1}\phi_{2}}$ around its topologically quantized average
value $\frac{{\mathcal{C}}^{(\nu)}}{2\pi}$, with ${\mathcal{C}}^{(\nu)}$ the
Chern number. This corresponds to the fluctuation of the geometrical transfer
of energy between the two modes. The last term is the topological power rate,
the only non-zero term in time-average. The two first terms are responsible
for the time fluctuation of the energy of the mode. In figure 9 is represented
the time-integration of each term in the case of resonance
$\delta_{1}=\delta_{3}=0$, point A in Fig. 6(a). We see that the temporal
fluctuation of the energy is mainly due to the energy exchange between the
qutrit and the mode, and the fluctuation of the Berry curvature is much lower.
This is the case for every value of parameters in the region of interest of
the phase diagram.

Figure 9: Different terms in the variation of the energy
$\hbar\omega_{1}(n_{1}+n_{3})$, in the case of resonance
$\delta_{1}=\delta_{3}=0$, point A in Fig. 6(a). In blue: time-integration of
the term of variation of the energy of the qutrit $-\omega_{1}\frac{\partial
E_{\nu}}{\partial\phi_{1}}$. In orange: time-integration of the term of
fluctuation of the geometrical coupling
$\hbar\omega_{1}\omega_{2}(F_{\phi_{1}\phi_{2}}^{(\nu)}-\frac{{\mathcal{C}}^{(\nu)}}{2\pi})$.
In green: topological energy transfer at constant rate
$\hbar\frac{\omega_{1}\omega_{2}}{2\pi}{\mathcal{C}}^{(\nu)}$. The fluctuation
of the energy of the qutrit is the predominant source of temporal fluctuation
of the energy.