#  Topological dynamics of adiabatic cat states

J. Luneau<EMAIL_ADDRESS>ENSL, 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  D. Carpentier  ENSL, CNRS, Laboratoire de Physique, F-69342
Lyon, France

###### Abstract

We consider a qubit topologically coupled to two quantum modes. We show that
any initial separable state of this system generically evolves into an
adiabatic cat state. Such a state is a superposition of two adiabatic states
in which the qubit is entangled between the modes. The topological coupling
between the qubit and the modes gives rise to the separation in energy between
these two components which evolve into states with distinguishable energy
content.

Traditionally, a pump is a device that transfers energy from a source - an
engine - to a fluid. The transfer is achieved through a suitably designed
mechanical coupling. Topological pumping is a modern extension of such a
device. The first historical example of such a topological pump was provided
by Thouless who considered a periodically modulated in time one dimensional
crystal [1], realized experimentally in various forms [2, 3, 4, 5, 6, 7, 8, 9,
10, 11, 12, 13]. A simpler realization was provided recently in the form of a
qubit driven at two different frequencies [14], later extended to more complex
driven zero dimensional devices [15, 16, 17, 18, 19, 20, 21]. These studies
discussed pumping through the dynamics of the pump described as a driven
quantum system. Indeed the slow and periodically modulated parameters
generates a quantized current either through the quantum system, or,in the
last case, in an abstract space of harmonics of the drives. This current
thereby effectively describes a transfer between the drives.

In this paper, we reconcile the description of such a topological device with
that of traditional pumps by considering on equal footing the qubit and the
drives. This amounts to describe these drives quantum mechanically instead of
classical parameters. We model them as quantum modes, characterized by a pair
of conjugated operators accounting for their phase and number of quanta. This
extends previous mixed quantum-classical descriptions of the drives [22, 23,
24]. We focus on the regime of adiabatic dynamics in which these quantum modes
are slow compared to the qubit, neglecting effects induced by a faster drive,
recently discussed in the context of topological Floquet systems [25, 26, 27,
28, 27, 29, 30, 31].

The dynamics of a system with slow and fast degrees of freedom has been
largely studied in terms of an effective Hamiltonian [32, 33, 34]. Here we
focus on the adiabatically evolved quantum states. Our quantum mechanical
description shows that an initial separable state generically evolves into a
superposition of two components. Each component is an adiabatic state of the
driven qubit. The topological nature of the coupling between the qubit and the
modes splits these two components apart in energy: for each component, an
energy transfer occurs between the two quantum modes, in opposite directions
for the two components. This topological dynamics effectively creates an
adiabatic cat states: a superposition of two quantum adiabatic states
distinguishable through measures of the modes’ energy. Besides, we show that
in each of the component of this cat state, the qubit is entangled with the
two modes. The origin of this entanglement lies in the geometrical properties
of the coupling at the origin of the topological pumping. Hence a topological
coupling necessarily entangles a pump with its driving modes.

Interestingly, similar qubit-modes quantum systems were recently proposed [35]
and experimentally realized in quantum optical devices [36] to simulate
topological lattice models. In this context, the Hamiltonians of a qubit
coupled to cavities was expressed in terms of Fock-state lattices, and shown,
with two cavities, to realize a chiral topological phase [37, 38], and , with
three cavities, the quantum or valley Hall effect [35, 39]. Indeed the focus
of these realizations was on synthetic topological models and their associated
zero-energy states. Our approach bridges the gap between the study of
topological pumping of driven systems and these studies of quantum optical
devices.

Our paper is organized as follows. In Sec. I we introduce the model of a qubit
driven by two quantum modes (Sec. I.1) and discuss qualitatively the typical
dynamics of adiabatic cat states (Sec. I.2). In Sec. II we characterize the
two components of the cat states as adiabatic states and identify their
effective dynamics which splits them apart in energy. We characterize the
weight of each cat component for a separable initial state. In Sec. III we
study each cat component, relating the entanglement between the qubit and the
modes to the quantum geometry of the adiabatic states (Sec. III.1), and
discussing the evolution of the number of quanta of each mode in relation with
Bloch oscillations and Bloch breathing (Sec. III.2).

##  I A qubit driven by two quantum modes

Figure 1:  Phase and number representations of a quantum qubit-2 modes model.
(a) Phase representation, convenient to represent the dynamics of the qubit.
At each value of the phases $\bm{\Phi}$ are associated qubit eigenstates
$\ket{\psi_{\pm}^{0}(\bm{\Phi})}$ represented by a vector
$\bm{b}_{\pm}^{0}(\bm{\Phi})=\pm\bm{h}(\bm{\Phi})/|\bm{h}(\bm{\Phi})|$ on the
Bloch sphere (in grey). The adiabatic states $\ket{\psi_{\pm}(\bm{\Phi})}$,
represented by a vector $\bm{b}_{\pm}(\bm{\Phi})$, are a perturbative
deformation of the eigenstates. (b) Number representation, convenient to
represent the dynamics of the modes. In this viewpoint the model can be
interpreted as an unusual model of spin-half particle on a discrete lattice
where $\bm{N}=(n_{1},n_{2})\in\mathbb{Z}^{2}$ represents its position on the
lattice and $\bm{\Phi}\in[0,2\pi]^{2}$ the Bloch momenta. This particle is
submitted to an electric field $\hbar\bm{\omega}\cdot\hat{\bm{N}}$ and a
strong spin-orbit coupling $\bm{h}(\hat{\bm{\Phi}})\cdot\bm{\sigma}$. As a
consequence, the adiabatic states are associated to energy bands $E_{\pm}$
tilted in the direction $\bm{\omega}$ of the electric field and separated by
the gap $\Delta$ due to the spin-orbit coupling.

###  I.1 Model

We consider the dynamics of two quantum modes coupled to a fast quantum degree
of freedom, chosen for clarity as a two level system, a qubit. Each slow mode
is described by a phase operator $\hat{\phi}_{i}$ of continuum spectrum
$[0,2\pi]$, conjugated to a number operator $\hat{n}_{i}$ of discrete spectrum
$\mathbb{Z}$, such that $[\hat{n}_{i},\hat{\phi}_{j}]={\rm i}\delta_{ij}$
[40]. We assume that the fast degree of freedom couples only to the phases of
the modes, through a Hamiltonian $H(\hat{\bm{\Phi}})$ with
$\hat{\bm{\Phi}}=(\hat{\phi}_{1},\hat{\phi}_{2})$. Noting
$\bm{\omega}=(\omega_{1},\omega_{2})$ the frequencies of the modes, and
$\hat{\bm{N}}=(\hat{n}_{1},\hat{n}_{2})$ their respective number operators,
the dynamics of the full quantum system is governed by the Hamiltonian

$\hat{H}_{\rm
tot}=\hbar\bm{\omega}\cdot\hat{\bm{N}}\otimes\mathbb{I}+H(\hat{\bm{\Phi}})\,,\
H(\hat{\bm{\Phi}})=\sum_{\alpha=1}^{3}h_{\alpha}(\hat{\bm{\Phi}})\otimes\sigma_{\alpha}\,.$
(1)

Such a model appears as the natural quantization of a $2$-tone Floquet
systems, where the time-dependant parameters $\phi_{i}(t)=\omega_{i}t$ of the
qubit Hamiltonian $\bm{h}(\phi_{1}(t),\phi_{2}(t))\cdot\bm{\sigma}$ are here
considered as true quantum degrees of freedom. Note that we will focus on the
adiabatic dynamics of such a Floquet model, valid for slow frequencies
$\hbar\omega_{i}$ compared with the qubit’s spectral gap $2|\bm{h}|$.
Physically, the two modes of model (1) often results from the coupling to a
harmonic oscillator [35, 22, 37, 24]. Describing this oscillator by a pair of
conjugated operators $\hat{n}_{i},\hat{\phi}_{i}$ is valid for states with a
number of quanta $\bar{n}$ large compared to any dynamical variation of and
spreading of $n$, $\Delta n\ll\bar{n}$ [22].

The modes’ intrinsic energies depend on the number of quanta $\bm{N}$ while
the qubit’s energy depend on their phases: hence we will use two dual
representations of the dynamics of the system through this paper. When
focusing on the qubit’s evolution, the phase representation is natural,
represented in Fig. 1(a): at each value of the phases $\bm{\Phi}$ are
associated qubit’s eigenstates $\ket{\psi_{\pm}^{0}(\bm{\Phi})}$. The modes’
dynamics translate into an evolution with time of the phase, and thus an
evolution of the associated qubit’s states $\ket{\psi_{\pm}(\bm{\Phi})}$ which
slightly differ from the eigenstates and will be discussed in section II.1.

Focusing on the quantum modes, their dynamics is conveniently represented in
number representation, Fig. 1(b). In this viewpoint, we can interpret the
model as that of a particle on a 2D lattice of sites $\bm{N}=(n_{1},n_{2})$,
$\bm{\Phi}=(\phi_{1},\phi_{2})$ being the associated Bloch momenta in the
first Brillouin zone. The Hamiltonian (1) describes its motion, submitted to
both a spin-orbit coupling $H(\hat{\bm{\Phi}})$ and an electric field
$\hbar\bm{\omega}$. We will use this analogy to relate the geometrical and
topological properties of gapped phases on a lattice to those of the above
quantum model. Note that in this case, there is no embedding of this lattice
in $\mathbb{R}^{2}$ as opposed to the Bloch theory of crystals. The position
operator identifies with coordinate operator on the lattice. As a consequence,
there is no ambiguity in a choice of Bloch convention and definition of the
Berry curvature [41, 42].

In the following, we consider a topological coupling between the qubit and the
two quantum modes. This corresponds to the situation where the qubit remains
gapped irrespective of the phase $\bm{\Phi}$ of the quantum modes, i.e.
$|\bm{h(\bm{\Phi})}|\geq\Delta>0$ for all values of $\bm{\Phi}$. Besides, the
topological nature of the drive originates from the condition that the map
$\bm{\Phi}\in[0,2\pi]^{2}\to\bm{h(\bm{\Phi})}$ wraps around the origin in
$\mathbb{R}^{3}$. This is a condition of strong coupling between the qubit and
the drive. Indeed, if we represent the qubit’s eigenstates by a vector
$\bm{b}_{\pm}^{0}(\bm{\Phi})=\pm\bm{h}(\bm{\Phi})/|\bm{h}(\bm{\Phi})|$ on the
Bloch sphere, then any point of the sphere correspond to a ground state of the
qubit for a particular phase state of the drive. This is in contrast with the
familiar weak coupling limit where states around the south pole,
$\ket{\downarrow}$, are associated to the ground states and those close to the
north pole, $\ket{\uparrow}$, to excited states. In the present case,
knowledge of the state of the qubit is not sufficient to determine whether it
is in the excited or ground state: information on the state of the driving
modes is necessary.

Throughout this paper, the numerical results are obtained by considering an
example of such a topological coupling provided by the quantum version of the
Bloch Hamiltonian of the half Bernevig-Hughes-Zhang (BHZ) model [43]:

$\displaystyle h_{x}(\hat{\bm{\Phi}})$
$\displaystyle=\frac{\Delta}{2}\sin(\hat{\phi}_{1}),$ (2a) $\displaystyle
h_{y}(\hat{\bm{\Phi}})$ $\displaystyle=-\frac{\Delta}{2}\sin(\hat{\phi}_{2}),$
(2b) $\displaystyle h_{z}(\hat{\bm{\Phi}})$
$\displaystyle=\frac{\Delta}{2}\left(1-\cos(\hat{\phi}_{1})-\cos(\hat{\phi}_{2})\right),$
(2c)

where the parameter $\Delta>0$ is the gap of the qubit. See appendix F for
details on the numerical method.

###  I.2 Topological dynamics of adiabatic of cat states

Figure 2:  Typical dynamics of adiabatic cat states. Distribution of number of
quanta of the two modes
$P_{n_{1}n_{2}}=\bra{n_{1},n_{2}}\hat{\rho}_{12}\ket{n_{1},n_{2}}$ at
different times for three initial states. The modes are prepared in a Gaussian
state with an average value of phase $\phi_{1}^{0}=\phi_{2}^{0}=0$ and an
equal width in number of quanta $\Delta n_{1}=\Delta n_{2}=\Delta n$,
corresponding to a width $\Delta\phi=1/(2\Delta n)$ in phase. The qubit is
prepared in $(\ket{\uparrow}+\ket{\downarrow})/\sqrt{2}$. The evolution of
states with different initial width $\Delta n$ is represented: line (a),
$\Delta n=5$, $\Delta\phi\simeq 0.03\pi$; line (b), $\Delta n=0.7$,
$\Delta\phi\simeq 0.23\pi$, and line (c), Quasi-Fock state $\Delta
n=1/(2\pi)$, delocalized in phase $\Delta\phi=\pi$. The columns (2) to (5)
represent the time evolved state at respectively $t=8/3$, $16/3$, $8$ and
$t=32/3$ in units of the period of the first mode $T_{1}=2\pi/\omega_{1}$. The
dynamics splits the initial state in a cat state in the sense of a
superposition of two states with distinguishable energy content.

In this section, we illustrate the topological dynamics of the system starting
from a typical state. This dynamics is analyzed quantitatively in the
remaining of this paper. We focus on separable initial state, easier to
prepare experimentally:

$\ket{\Psi(t=0)}=\ket{\chi_{1}}\otimes\ket{\chi_{2}}\otimes\ket{\psi_{q}}.$
(3)

Each quantum mode is prepared in a Gaussian state $\ket{\chi_{i}}$,
characterized by an average number of quanta $n_{i}^{0}$ and a phase
$\phi_{i}^{0}=0$, with widths $\Delta n_{i},\Delta\phi_{i}$ satisfying
$\Delta\phi_{i}\Delta n_{i}=\frac{1}{2}$. The qubit is prepared in a
superposition
$\ket{\psi_{q}}=(\ket{\uparrow_{z}}+\ket{\downarrow_{z}})/\sqrt{2}$.

We consider modes with frequency of the same order of magnitude,
$\hbar\omega_{1}=0.075\Delta$ and $\omega_{2}/\omega_{1}=(1+\sqrt{5})/2\simeq
1.618$, such that in the following, time is arbitrarily expressed in unit of
period of the first mode $T_{1}=2\pi/\omega_{1}$. In Fig. 2, we represent the
dynamics of this state $\ket{\Psi(t)}$ by displaying the associated number
distribution of the modes
$P_{n_{1}n_{2}}(t)=\bra{n_{1},n_{2}}\hat{\rho}_{12}(t)\ket{n_{1},n_{2}}$, with
$\hat{\rho}_{12}(t)$ the corresponding reduced density matrix of the modes.
Three initial states with respective initial number width $\Delta n=\Delta
n_{1}=\Delta n_{2}=5$, $\Delta n=0.7$ and $\Delta n=1/(2\pi)$ (quasi-Fock
state delocalized in phase $\Delta\phi=\pi$) are shown respectively in Fig. 2
(a1), (b1) and (c1). The time evolved states at respectively $t=0,8/3,16/3,8$
and $t=32/3$ are represented respectively on columns 1 to 5 of Fig. 2. We
observe a splitting of the initial state into a superposition of two states

$\ket{\Psi(t)}=\ket{\Psi_{-}(t)}+\ket{\Psi_{+}(t)}.$ (4)

The photon number distribution of $\ket{\Psi_{-}(t)}$ and $\ket{\Psi_{+}(t)}$
drift in opposite directions, corresponding to energy transfers between modes
$1$ and $2$ in opposite directions. This drift is a manifestation of the
topological pumping discussed in classical-quantum models [14, 23]. This
pumping is conveniently represented by introducing rotated number coordinates

$\displaystyle n_{\scriptscriptstyle E}$
$\displaystyle=\frac{1}{|\bm{\omega}|}(\omega_{1}n_{1}+\omega_{2}n_{2}),$ (5)
$\displaystyle n_{\perp}$
$\displaystyle=\frac{1}{|\bm{\omega}|}(-\omega_{2}n_{1}+\omega_{1}n_{2}),$ (6)

with $|\bm{\omega}|=\sqrt{\omega_{1}^{2}+\omega_{2}^{2}}$.
$n_{\scriptscriptstyle E}|\bm{\omega}|$ corresponds to the total energy of the
modes and is constant up to the instantaneous energy exchange with the qubit.
$n_{\perp}$ is the coordinate in the direction perpendicular to $\bm{\omega}$.
A transfer of energy between mode 1 and mode 2 naturally translates into a
drift in the $n_{\perp}$ direction, at fixed average $n_{\scriptscriptstyle
E}$. We also observe that for small initial $\Delta n$, corresponding to lines
(b) and (c), each component $\ket{\Psi_{-}(t)}$ and $\ket{\Psi_{+}(t)}$
undergoes a complex breathing dynamics around the drift. This oscillatory
behavior is reminiscent of Bloch oscillations of the associated particle
submitted to an electric field, superposed with a topological drift
originating from the anomalous transverse velocity. After an initial time of
separation, the number distributions for $\ket{\Psi_{-}(t)}$ and
$\ket{\Psi_{+}(t)}$ no longer overlap (Fig. 2 columns 4 and 5). The system is
then in a cat state: a superposition of two states with well distinguishable
energy content.

We will now study quantitatively these cat states and their dynamics. In
section II, we identify the two components $\ket{\Psi_{\pm}(t)}$ as adiabatic
states (Sec. II.1). We study their topological dynamical separation into a cat
state (Sec. II.2). We characterize the weight of each component of the cat
(Sec. II.3) and identify a family of cat states with equal weight on each
component. In section III.1 we analyze the entanglement between the qubit and
the modes for each cat component, and relate it to the quantum geometry of the
adiabatic states. Finally we discuss the dynamics of each cat component around
the average drift, in relation with Bloch oscillations and Bloch breathing on
the associated lattice (Sec. III.2).

##  II Adiabatic decomposition

###  II.1 Adiabatic projector

When the driving frequencies remain small compared to the qubit’s gap,
$\hbar\omega_{i}\ll\Delta$, we naturally describe the effective dynamics of
the coupled qubit and drives in terms of fast and slow quantum degrees of
freedom. This is traditionally the realm of the Born-Oppenheimer
approximation. Historically both degrees of freedom were those of massive
particles, the slow modes being associated with the heavy nucleus of a
molecule and the fast ones with the light electrons [44, 45, 32]. In this
context, the Born-Oppenheimer approximation assumes that the time evolved
state stays closed to the instantaneous eigenstates of the fast degrees of
freedom and describes the resulting effective dynamics of the slow degree of
freedom. The distinctive characteristic of the present quantum modes - qubit
model from the usual Born-Oppenheimer setting is the linearity of the
Hamiltonian (1) in the variable $\hat{\bm{N}}$. This allows to express in a
simple form the corrections to the Born-Oppenheimer approximation for the
adiabatic states. Let us explain the procedure to construct such states, while
referring to appendix B for technical details.

We note $\ket{\bm{\Phi}}$ the eigenstates of the phase operator of the modes
$\hat{\bm{\Phi}}$. Due to the linearity in $\hat{\bm{N}}$ of the Hamiltonian
(1), the time evolution of a phase eigenstate
$\ket{\bm{\Phi}}\otimes\ket{\psi}$, where $\ket{\psi}$ is a state of the
qubit, is $\ket{\bm{\Phi}-\bm{\omega}t}\otimes U(t;\bm{\Phi})\ket{\psi}$. The
time evolution operator $U$ is deduced from that for the Floquet model
parametrized by classical phases $\bm{\omega}t$:

$U(t;\bm{\Phi})=\mathcal{T}\exp\left[-\frac{{\rm
i}}{\hbar}\int_{0}^{t}\differential\tau~{}H(\bm{\Phi}-\bm{\omega}\tau)\right]\
,$ (7)

where $\mathcal{T}$ denotes time ordering.

##### Adiabatic projector of the qubit.

We denote by $\ket{\psi^{0}_{\nu}(\bm{\Phi})}$, $\nu=\pm$, the normalized
eigenstates of the two-level driven Hamiltonian
$H(\bm{\Phi})\ket{\psi_{\nu}^{0}(\bm{\Phi})}=\nu|\bm{h}(\bm{\Phi})|\ket{\psi_{\nu}^{0}(\bm{\Phi})}$
for each $\bm{\Phi}$ in $[0,2\pi]^{2}$. For small frequencies $\omega_{i}$,
given the time evolution of phase eigenstates, we can reasonably expect that
the coupled qubit-modes system prepared in an eigenstate
$\ket{\bm{\Phi}}\otimes\ket{\psi_{\nu}^{0}(\bm{\Phi})}$ will remain in a
translated eigenstate. However this simple picture is only qualitatively
valid: eigenstates get hybridized by adiabatic dynamics, even at arbitrarily
small driving frequencies. As a consequence, we identify the family of
adiabatic states $\ket{\psi_{\nu}(\bm{\Phi})}$ such that the dynamics occurs
within each family of states
$\ket{\bm{\Phi}}\otimes\ket{\psi_{\nu}(\bm{\Phi})}$. Adiabatic evolution is
then represented as a transport from
$\ket{\bm{\Phi}}\otimes\ket{\psi_{\nu}(\bm{\Phi})}$ to
$\ket{\bm{\Phi}^{\prime}}\otimes\ket{\psi_{\nu}(\bm{\Phi}^{\prime})}$ within
this family, indexed by translations of
$\bm{\Phi}\to\bm{\Phi}^{\prime}=\bm{\Phi}-\bm{\omega}t$.

In practice, the adiabatic states $\ket{\psi_{\nu}(\bm{\Phi})}$ are
conveniently determined from the adiabatic projector
$\pi_{\nu}(\bm{\Phi})=\ket{\psi_{\nu}(\bm{\Phi}}\bra{\psi_{\nu}(\bm{\Phi})}$
[46, 47]. Introducing the dimensionless perturbative parameter $\epsilon$ by
$\bm{\omega}=\epsilon\bm{\Omega}$, $\pi_{\nu}(\bm{\Phi})$ is defined as a
series in $\epsilon$:
$\pi_{\nu}(\bm{\Phi})=\sum_{k}\epsilon^{k}\pi_{\nu,k}(\bm{\Phi})$. Stability
of each family of adiabatic states under the dynamics amounts to impose the
condition

$U(t;\bm{\Phi})\pi_{\nu}(\bm{\Phi})U(t;\bm{\Phi})^{\dagger}=\pi_{\nu}(\bm{\Phi}-\bm{\omega}t)\,,$
(8)

where the evolution operator is defined in (7). Solving order by order in
$\epsilon$ for this equation, together with the constituting property of a
projector $\pi^{2}_{\nu}(\bm{\Phi})=\pi_{\nu}(\bm{\Phi})$, leads to the
solution at order $0$:
$\pi_{\nu,0}(\bm{\Phi})=\ket{\psi_{\nu}^{0}(\bm{\Phi})}\bra{\psi_{\nu}^{0}(\bm{\Phi})}$
and to first order

$\pi_{\nu,1}(\bm{\Phi})=\\\
\sum_{\mu\neq\nu}\ket{\psi_{\mu}^{0}(\bm{\Phi})}\frac{\sum_{i}\hbar\omega_{i}A_{\mu\nu,i}^{0}(\bm{\Phi})}{E_{\mu}^{0}(\bm{\Phi})-E_{\nu}^{0}(\bm{\Phi})}\bra{\psi_{\nu}^{0}(\bm{\Phi})}+\mathrm{h.c.}$
(9)

with
$A_{\mu\nu,i}^{0}(\bm{\Phi})=i\innerproduct{\psi_{\mu}^{0}(\bm{\Phi})}{\partial_{\phi_{i}}\psi_{\nu}^{0}(\bm{\Phi})}$
the components of the non-abelian Berry connection of the eigenstates, and
$E_{\pm}^{0}(\bm{\Phi})=\pm|\bm{h}(\bm{\Phi})|$ the eigenenergies.

##### Adiabatic grading of the Hilbert space.

The above adiabatic decomposition of the qubit states allows for a natural
decomposition of all states of the qubit-modes system. We proceed by extending
the adiabatic projector of the qubit $\pi_{\nu}(\bm{\Phi})$ to a the projector
acting on the Hilbert space of the whole system

$\hat{P}_{\nu}=\int\differential\bm{\Phi}~{}\ket{\bm{\Phi}}\bra{\bm{\Phi}}\otimes\pi_{\nu}(\bm{\Phi})\
,\ \nu=\pm\ .$ (10)

This projector provides a decomposition of any qubit-mode state $\ket{\Psi}$
into two adiabatic states

$\ket{\Psi}=\ket{\Psi_{-}}+\ket{\Psi_{+}}\ ,\
\ket{\Psi_{\nu}}=\hat{P}_{\nu}\ket{\Psi}\,,\ \nu=\pm\ .$ (11)

For a separable initial state (3), each adiabatic component $\ket{\Psi_{\nu}}$
is characterized by a wave amplitude $\chi_{\nu}(\bm{\Phi})$ according to

$\displaystyle\ket{\Psi_{\nu}}$
$\displaystyle=\int\differential^{2}\bm{\Phi}~{}\chi_{\nu}(\bm{\Phi})\ket{\bm{\Phi}}\otimes\ket{\psi_{\nu}(\bm{\Phi})}\
,$ (12) $\displaystyle\chi_{\nu}(\bm{\Phi})$
$\displaystyle=\chi(\bm{\Phi})\innerproduct{\psi_{\nu}(\bm{\Phi})}{\psi_{q}}\
.$ (13)

with
$\chi(\bm{\Phi})=\innerproduct{\phi_{1}}{\chi_{1}}\innerproduct{\phi_{2}}{\chi_{2}}$
the wavefunction of the modes in the initial state (3). This splits the total
Hilbert space $\mathcal{H}_{\rm tot}$ in two adiabatic subspaces
$\mathcal{H}_{\rm tot}=\mathcal{H}_{-}\oplus\mathcal{H}_{+}$, where
$\mathcal{H}_{-}$ and $\mathcal{H}_{+}$ are respectively the images of the
projectors $\hat{P}_{-}$ and $\hat{P}_{+}$.

##### Topology of the family of adiabatic states.

The ensemble of adiabatic states $\ket{\psi_{\nu}(\bm{\Phi})}$ parametrized by
the classical configuration space $\bm{\Phi}\in[0,2\pi]^{2}$ defines a vector
bundle. This vector bundle is a smooth deformation of the eigenstates bundles
$\ket{\psi^{0}_{\nu}(\bm{\Phi})}$ associated to a spectral projector (Fig.
1(a)). As a consequence, the local curvature associated with the adiabatic
bundle

$F_{\nu}(\bm{\Phi})=i\innerproduct{\partial_{\phi_{1}}\psi_{\nu}(\bm{\Phi})}{\partial_{\phi_{2}}\psi_{\nu}(\bm{\Phi})}-(1\leftrightarrow
2),$ (14)

differs from the canonical Berry curvature associated with the eigenstates
bundle [48]. The curvature (14) generalizes the Berry curvature to all orders
in the adiabatic parameter $\epsilon$, in a similar way that the Aharonov-
Anandan phase [49] generalizes the Berry phase. On the other hand, the Chern
number ${\mathcal{C}}_{\nu}$ of both bundles are identical. Indeed, the
switching on of finite but small frequencies $\omega_{1}$ and $\omega_{2}$ is
a smooth transformation of the fiber bundle of the eigenstates
$\ket{\psi_{\nu}^{0}(\bm{\Phi})}$ to that of the adiabatic states
$\ket{\psi_{\nu}(\bm{\Phi})}$. Such a smooth transformation does not change
the bundle topology. This fails for larger frequencies comparable with the
spectral gap.

The decomposition of the total Hilbert space is not a spectral decomposition
of the total Hamiltonian. It cannot be deduced from a measure on the two-level
system alone, since for a topologically non-trivial decomposition of states
$\ket{\psi_{\pm}(\bm{\Phi})}$, any qubit state corresponds either to a state
$\ket{\psi_{-}(\bm{\Phi})}$ of the ground bundle or to a state
$\ket{\psi_{+}(\bm{\Phi}^{\prime})}$ in its complementary bundle depending on
the states $\bm{\Phi}$ and $\bm{\Phi}^{\prime}$ of the modes.

##### Non-adiabatic Landau-Zener transitions.

The adiabatic splitting of the Hilbert space is defined perturbatively in the
adiabatic parameter $\epsilon$ by the stability condition (8). This stability
is valid up to non-perturbative effects with typical exponential dependance of
the form $\exp(-\alpha/\epsilon)$ [46, 47]. The adiabatic dynamics is the
effective dynamics on each subspace, and the non-perturbative transition
between the two subspaces are Landau-Zener transitions. The amplitude of the
Landau-Zener transitions can be estimated to obtain the time of validity of
the adiabatic approximation [23], $\tau_{\rm adiab}\approx
0.1\exp(\pi/(4\varepsilon_{\rm adiab}))T_{1}$, with
$\varepsilon_{\text{adiab}}=\max_{\bm{\Phi}}\hbar|\bra{\psi_{+}^{0}}\frac{\differential
H}{\differential t}\ket{\psi_{-}^{0}}|/(E_{+}^{0}-E_{-}^{0})^{2}$. In this
work, we choose the coupling and the frequencies of the modes such that
$\tau_{\rm adiab}\approx 3100T_{1}$, allowing to neglect such Landau-Zener
transitions. Within this approximation the weight on each adiabatic subspace

$W_{\nu}(\Psi)=||\hat{P}_{\nu}\ket{\Psi}||^{2}=\innerproduct{\Psi_{\nu}}{\Psi_{\nu}}$
(15)

is a conserved quantity.

###  II.2 Topological splitting of adiabatic components

In this section, we show that the topological dynamics splits in energy the
two adiabatic components, thereby creating an adiabatic cat state.

The energy function
$E_{\nu}(\bm{\Phi})=\bra{\psi_{\nu}(\bm{\Phi})}H(\bm{\Phi})\ket{\psi_{\nu}(\bm{\Phi})}$
as well as the curvature (14) govern the adiabatic dynamics of the slow modes.
Using the Ehrenfest theorem, for each adiabatic component of the initial
state, we get

$\displaystyle\frac{\differential}{\differential
t}\langle\hat{n}_{1}\rangle_{\Psi_{\nu}(t)}$
$\displaystyle=\int\differential^{2}\bm{\Phi}~{}\frac{|\chi_{\nu}(\bm{\Phi})|^{2}}{W_{\nu}(\Psi)}\bra{\psi_{\nu}(\bm{\Phi}-\bm{\omega}t)}\frac{1}{\hbar}\frac{\partial
H}{\partial\phi_{1}}\ket{\psi_{\nu}(\bm{\Phi}-\bm{\omega}t)}$
$\displaystyle=\int\differential^{2}\bm{\Phi}~{}\frac{|\chi_{\nu}(\bm{\Phi})|^{2}}{W_{\nu}(\Psi)}\left(\frac{1}{\hbar}\frac{\partial
E_{\nu}}{\partial\phi_{1}}(\bm{\Phi}-\bm{\omega}t)+\omega_{2}F_{\nu}(\bm{\Phi}-\bm{\omega}t)\right).$
(16)

This expression identifies with the average power transfer obtained within a
classical-quantum description [14, 23] weighted by the normalized phase
wavepacket density $|\chi_{\nu}(\bm{\Phi})|^{2}/W_{\nu}(\Psi)$ of the
adiabatic component. A similar relation obtained by the exchange
$1\leftrightarrow 2$ holds for the second mode. See appendix D for details.

Similarly than within a hybrid classical-quantum description of a topological
pump [14, 23], we assume an incommensurate ratio between the frequencies
$\omega_{1}$ and $\omega_{2}$ such that a time average of the rate of change
(16) reduces by ergodicity to an average over the phases
$\bm{\Phi}\in[0,2\pi]^{2}$. The average of the derivative of the energy in Eq.
(16) vanishes by periodicity, while the average of the adiabatic curvature
$F_{\nu}(\bm{\Phi})$ is quantized by the first Chern number
${\mathcal{C}}_{\nu}$ of the vector bundle of adiabatic states
$\ket{\psi_{\nu}(\bm{\Phi})}$. The topological coupling between the modes and
the qubit corresponds to two non-vanishing Chern numbers
${\mathcal{C}}_{+}=-{\mathcal{C}}_{-}$. In this situation, we recover a
topological pumping or topological frequency conversion between the two modes.
In terms of the rotated coordinates of Eq. (5,6), the topological pumping
corresponds to opposite evolution of $n_{\perp}$ for the two adiabatic
components $\ket{\Psi_{\pm}}$:

$\langle\hat{n}_{\perp}\rangle_{\Psi_{\pm}(t)}=\langle\hat{n}_{\perp}\rangle_{\Psi_{\pm}(0)}\mp\frac{|\bm{\omega}|t}{2\pi}{\mathcal{C}}_{-}+\delta
n_{\perp}(t),$ (17)

where $\delta n_{\perp}(t)$ denotes bounded oscillations, the temporal
fluctuations of pumping, discussed in Sec. III.2.

As a result, the topological dynamics splits in energy the two adiabatic
components of the initial state from each other. A cat state is created when
the two adiabatic components no longer overlap. We note $\Delta
n_{\perp}^{\max}$ the maximal spread in $n_{\perp}$ developed during the
dynamics, which is discussed in section III.2. The time of separation of the
two cat components reads

$t_{\rm sep}\approx\frac{2\pi}{|\bm{\omega}|{\mathcal{C}}}\Delta
n_{\perp}^{\max}$ (18)

with ${\mathcal{C}}=|{\mathcal{C}}_{\pm}|$. After $t_{\rm sep}$, the weight of
the state on the region $n_{\perp}<n_{\perp}^{0}$ identifies with the
adiabatic weight $W_{-}(\Psi)$.

Figure 3:  Weight $W_{-}$ of the adiabatic states superposition. The modes are
in a Gaussian state centered on $\phi_{1}^{0}=\phi_{2}^{0}=0$. We vary the
width $\Delta\phi$ of the Gaussian state and the qubit initial state
$\ket{\psi_{q}}$ represented by its polarization $\bm{Q}$ on the Bloch sphere.
(a) Averaged ground state $\bar{\bm{b}}_{-}^{0}$, lying on the $z$-axis. The
average adiabatic state $\bar{\bm{b}}_{-}$ is pertubatively close to
$\bar{\bm{b}}_{-}^{0}$. (b) The phase density $|\chi(\bm{\Phi})|^{2}$ on the
torus translates via the map $\bm{\Phi}\mapsto\bm{b}_{-}(\bm{\Phi})$ to a
density of adiabatic states on the Bloch sphere represented in color for
different $\Delta\phi$. $\overline{\bm{b}}_{-}$ is the average of this
density. (c) Weight of the cat depending on the width in phase $\Delta\phi$.
The average Bloch vector remains approximately on the z-axis, such that a cat
with $W_{-}=\frac{1}{2}$ is obtained for the qubit on the equator
$\theta_{q}=\pi/2$. The deviation from $W_{-}=1/2$ at $\theta_{q}=\pi/2$ is
due to the difference between the eigenstates and the adiabatic states. In the
limit of small $\Delta\phi$ for $\ket{\psi_{q}}=\ket{\psi_{-}(\bm{\Phi}^{0})}$
($\theta_{q}=0$) and $\ket{\psi_{q}}=\ket{\psi_{+}(\bm{\Phi}^{0})}$
($\theta_{q}=\pi$) the weight of adiabaticity is controlled by the quantum
metric $g_{-,ij}(\bm{\Phi}^{0})$ of the adiabatic states. (d) Weight of the
cat depending on the initial state of the qubit, varying $\theta_{q}$ at
$\varphi_{q}=0$. The norm of the average adiabatic state defines the bounds
$(1\pm|\bar{\bm{b}}_{-}|)/2$ of the weight. For $\Delta\phi=0.38\pi$,
$|\bar{\bm{b}}_{\pm}|\simeq 0$ such that a cat with almost equal weight is
created independently of the initial state of the qubit.

###  II.3 Weight of the adiabatic cat state

In this section, we focus on the weight $W_{\nu}(\Psi)$ (15) of each component
of the cat state. This will allow us to identify the conditions on the initial
states to realize ideal adiabatic cat states with equal weights
$W_{+}(\Psi)=W_{-}(\Psi)$.

We can represent the states $\ket{\psi_{\nu}(\bm{\Phi})}$ and $\ket{\psi_{q}}$
by vectors on the Bloch sphere, respectively $\bm{b}_{\nu}(\bm{\Phi})$ and
$\bm{Q}$, see Fig. 3(a). The qubit’s state representation $\bm{Q}$ is
parametrized by the angle $\theta_{q}$ with the $z$-axis and its azimuthal
angle $\varphi_{q}$. In this section, we focus without loss of generality on
$\varphi_{q}=0$, i.e. on $\bm{Q}$ lying in the $xz$ plane, and on the mode
prepared in a Gaussian state centered on $\phi_{1}^{0}=\phi_{2}^{0}=0$. The
adiabatic states are obtained in a perturbative expansion around the
eigenstates $\ket{\psi^{0}_{\nu}(\bm{\Phi})}$. Hence they are represented by
$\bm{b}_{\nu}(\bm{\Phi})$ which is perturbatively close to
$\bm{b}_{\nu}^{0}(\bm{\Phi})$.

####  II.3.1 General expression of the weights

The phase states in the decomposition (12) being orthogonal, the weight
$W_{\nu}(\Psi)$, $\nu=\pm$ is the weight of the wave amplitude
$\chi_{\nu}(\bm{\Phi})=\chi(\bm{\Phi})\innerproduct{\psi_{\nu}(\bm{\Phi})}{\psi_{q}}$:

$W_{\nu}(\Psi)=\int\differential^{2}\bm{\Phi}~{}|\chi_{\nu}(\bm{\Phi})|^{2}.$
(19)

The overlap between $\ket{\psi_{\nu}(\bm{\Phi})}$ and $\ket{\psi_{q}}$ is
$|\innerproduct{\psi_{\nu}(\bm{\Phi})}{\psi_{q}}|^{2}=(1+\bm{b}_{\nu}(\bm{\Phi})\cdot\bm{Q})/2$
such that the weight (19) now reads

$W_{\nu}(\Psi)=\frac{1}{2}\left(1+\overline{\bm{b}}_{\nu}\cdot\bm{Q}\right)\
,\ \nu=\pm,$ (20)

with

$\overline{\bm{b}}_{\nu}=\int\differential^{2}\bm{\Phi}~{}|\chi(\bm{\Phi})|^{2}~{}\bm{b}_{\nu}(\bm{\Phi})$
(21)

the statistical average of the adiabatic states with respect to the initial
phase distribution $|\chi(\bm{\Phi})|^{2}$. The phase density
$|\chi(\bm{\Phi})|^{2}$ on the torus translates via the map
$\bm{\Phi}\mapsto\bm{b}_{\nu}(\bm{\Phi})$ to a density of adiabatic states on
the Bloch sphere. $\overline{\bm{b}}_{\nu}$ is the average of this density. We
represent the density associated to the ground state
$\bm{b}_{-}^{0}(\bm{\Phi})$ in colored density plot on Fig. 3(b) as well as
the average ground state $\bar{\bm{b}}_{-}^{0}$ for four widths
$\Delta\phi=\Delta\phi_{1}=\Delta\phi_{2}$. $\bar{\bm{b}}_{-}$ is
perturbatively close to $\bar{\bm{b}}_{-}^{0}$.

For small $\Delta\phi$, the phase density is localized around $\bm{\Phi}^{0}$
and $\bar{\bm{b}}_{\nu}\approx\bm{b}_{\nu}(\bm{\Phi}^{0})$ close to the
surface of the Bloch sphere. When increasing the width $\Delta\phi$, we
average vectors over an increasing support on the Bloch sphere, reducing the
norm $|\bar{\bm{b}}_{\pm}|$ which controls the minimum weight of the cat (Fig.
3(d)). For the chosen value $\phi_{1}^{0}=\phi_{2}^{0}=0$,
$\bar{\bm{b}}_{-}^{0}$ lies on the $z$-axis as inferred from (2). In the
extreme case of $\Delta\phi=\pi$ (Fock state), the density is homogeneous on
the torus. However, while $\bm{b}_{\pm}(\bm{\Phi})$ covers the whole sphere,
the associated density is not homogeneous due to the anisotropy of the
couplings (2). This leads to a non vanishing $\bar{\bm{b}}_{-}$.

####  II.3.2 Symmetric cat states

Following the analysis above, we can identify two types of initial states that
give rise to symmetric cat states with $W_{+}=W_{-}=\frac{1}{2}$. The first
class is obtained by preparing the qubit orthogonally to this average
adiabatic state $\bar{\bm{b}}_{\pm}$. For our initial phase $\bm{\Phi}^{0}$,
this average adiabatic state is perturbatively closed to the $z$-axis as
discussed above. Hence a qubit prepared on the equator of the Bloch sphere
corresponds to two almost equal weights for all $\Delta\phi$. This corresponds
to the choice made for Fig. 2. This case is represented by the orange curve on
Fig. 3(c). The adiabatic weight is computed numerically from the topological
splitting of the adiabatic components discussed in the previous section. The
deviation to $W_{-}=1/2$ originates from the difference between the
eigenstates and the adiabatic states, see appendix G for details. The second
class of symmetric cat states is obtained for well chosen gaussian states of
the modes: the weight (20) of the cat is bounded by
$(1+|\bar{\bm{b}}_{\pm}|)/2$. For $\Delta\phi=0.38\pi$,
$\bar{\bm{b}}_{\pm}\simeq 0$ such that the cat has (almost equal) weights
$W_{+}=W_{-}$ independently of the initial state of the qubit $\theta_{q}$
(Fig. 3(d) green curve).

####  II.3.3 Quasi phase states

The only separable states lying in an adiabatic subspace, corresponding to
$W_{+}=1$ or $W_{-}=1$, are pure phase states
$\ket{\bm{\Phi}^{0}}\otimes\ket{\psi_{\pm}(\bm{\Phi}^{0})}$ for which
$|\overline{\bm{b}}_{\pm}|=1$, shown as the blue and green curves of Fig. 3(c)
in limit $\Delta\phi=0$. Given that these states are fully delocalized in
quanta number $\bm{N}$, and thus in energy according to (1), we expect them to
be hard to realize. Any other separable state lies at a finite distance from
each adiabatic subspace and will be split into a cat state under time
evolution. Let us comment on this adiabatic decomposition for almost pure
phase states with small $\Delta\phi$. In this case, the correction to
adiabaticity is controlled by the quantum metric $g_{\pm,ij}$ of the adiabatic
states [50, 33]:

$g_{\pm,ij}=\operatorname{Re}\bra{\partial_{\phi_{i}}\psi_{\pm}}\left(1-\ket{\psi_{\pm}}\bra{\psi_{\pm}}\right)\ket{\partial_{\phi_{j}}\psi_{\pm}}.$
(22)

Indeed, the weight (19) is dominated by the local variations of the adiabatic
states $\ket{\psi_{\pm}(\bm{\Phi})}$ over the narrow phase support
$|\chi(\bm{\Phi})|^{2}$. These variations are encoded by the quantum metric:
$|\innerproduct{\psi_{\pm}(\bm{\Phi}^{0}+\delta\bm{\Phi})}{\psi_{\pm}(\bm{\Phi}^{0})}|^{2}=1-\sum_{i,j}g_{\pm,ij}(\bm{\Phi}^{0})\delta\phi_{i}\delta\phi_{j}+\mathcal{O}(\delta\phi^{3})$.
In the limit of a small width $(\Delta\phi_{1},\Delta\phi_{2})$ the weight
(19) for $\ket{\psi_{q}}=\ket{\psi_{\pm}(\bm{\Phi}^{0})}$ reduces to

$W_{\pm}(\Psi)=1-(\Delta\phi_{1})^{2}g_{\pm,11}(\bm{\Phi}^{0})\\\
-(\Delta\phi_{2})^{2}g_{\pm,22}(\bm{\Phi}^{0})+\mathcal{O}(\Delta\phi^{4}).$
(23)

Hence for a state close to a phase state, the first correction to $W_{\nu}$ is
quadratic in $\Delta\phi=1/(2\Delta n)$ with a factor set by the quantum
metric of the adiabatic states, as shown in black dashed line on Fig. 3(c).

##  III Characterization of cat components

Having characterized the balance between the two components of an adiabatic
cat state, we now study the dynamics of each component. We will focus first on
the entanglement between the qubit and the two modes, before focusing on their
Bloch oscillatory dynamics in the number of quanta representation.

###  III.1 Entanglement

Adiabatic states naturally entangle the fast qubit with the slow driving
modes, a phenomenon out-of-reach of previous Floquet or classical descriptions
of the drives [14, 15, 17, 18, 19, 20, 21, 23]. We focus on cat states with
almost equal weight $W_{\nu}\simeq 1/2$, obtained with $\bm{\Phi}^{0}=0$ and
$\theta_{q}=\pi/2$ following the analysis of section II.3. In the following,
we study the entanglement of the qubit with the modes for the different types
of cat states, varying the initial spread $\Delta\phi$ of the modes and the
initial azimuthal angle $\varphi_{q}$ of the qubit.

The entanglement between the qubit and the two modes in an adiabatic component
$\ket{\Psi_{\nu}(t)}$, $\nu=\pm$, is captured by the purity
$\gamma_{\nu}(t)=\Tr(\rho_{q,\nu}^{2}(t))=(1+|\bm{Q}_{\nu}(t)|^{2})/2$ of the
qubit, where $\rho_{q,\nu}(t)$ is the reduced density matrix of the qubit and
$\bm{Q}_{\nu}(t)$ its polarization. From the adiabatic time evolution (8), we
deduce the reduced density matrix $\rho_{q,\nu}(t)$ of the qubit in the
adiabatic state $\ket{\Psi_{\nu}(t)}$ as

$\displaystyle\rho_{q,\nu}(t)$
$\displaystyle=\int\differential^{2}\bm{\Phi}~{}\frac{|\chi_{\nu}(\bm{\Phi}+\bm{\omega}t)|^{2}}{W_{\nu}}\ket{\psi_{\nu}(\bm{\Phi})}\bra{\psi_{\nu}(\bm{\Phi})}$
(24) $\displaystyle=\frac{1}{2}(\mathbb{I}+\bm{Q}_{\nu}(t)\cdot\bm{\sigma}),$
(25)

where the polarization of the qubit $\bm{Q}_{\nu}(t)$ reads

$\bm{Q}_{\nu}(t)=\int\differential^{2}\bm{\Phi}~{}\frac{|\chi_{\nu}(\bm{\Phi}+\bm{\omega}t)|^{2}}{W_{\nu}}\bm{b}_{\nu}(\bm{\Phi}).$
(26)

The qubit is in the statistical mixture of the adiabatic states
$\ket{\psi_{\nu}(\bm{\Phi})}$ weighted by the translated normalized phase
density $|\chi_{\nu}(\bm{\Phi}+\bm{\omega}t)|^{2}/W_{\nu}$.

Figure 4:  Purity of the qubit $\gamma_{+}(t)$ and $\gamma_{-}(t)$ in each cat
components $\ket{\Psi_{+}(t)}$ and $\ket{\Psi_{-}(t)}$. The qubit is prepared
on the equator $\theta_{q}=\pi/2$ to create a cat of equal weight
$W_{\pm}=1/2$ for all values of $\Delta\phi$ and $\varphi_{q}$. (a1), (b1) The
phase densities $|\chi_{-}(\bm{\Phi}+\bm{\omega}t)|^{2}$ and
$|\chi_{+}(\bm{\Phi}+\bm{\omega}t)|^{2}$ of each cat components translates to
densities of adiabatic states on the Bloch sphere (respectively in blue and
orange). The qubit state is the statistical mixture weighted according to
these densities with resulting polarizations $\bm{Q}_{\pm}$. (a2) For small
$\Delta\phi$, the phase densities are localized around
$\bm{\Phi}^{0}-\bm{\omega}t$. The adiabatic states have small variations
around $\bm{\Phi}^{0}-\bm{\omega}t_{1}$ and larger variation around
$\bm{\Phi}^{0}-\bm{\omega}t_{2}$, inducing
$\gamma_{-}(t_{1})>\gamma_{-}(t_{2})$. These local variations are quantified
by the quantum metric of the adiabatic states. (b2) (d) For large
$\Delta\phi$, the adiabatic states cover a large part of the Bloch sphere,
corresponding to a high entanglement. The phase densities of the two
components have complementary support on the torus, leading to different
densities on the Bloch sphere. The purity of each component oscillates in
opposite phase around their temporal average (black dotted line). The details
of the oscillations depend on the shape of the phase densities which changes
with the qubit initial state $\varphi_{q}$. (c) Time average of the purity and
temporal fluctuations (errorbars) depending on $\Delta\phi$. A large support
$\Delta\phi$ on the torus translates to a large support on the Bloch sphere
due to the topological nature of the coupling, and thus a large entanglement.

####  III.1.1 Entanglement of quasi-phase states.

Let us first focus on the adiabatic component of a cat obtained from a quasi-
phase state of small $\Delta\phi$. We show below that entanglement between the
qubit and the quantum modes of such a state is set by the quantum metric of
the adiabatic states.

The translated phase density $|\chi(\bm{\Phi}+\bm{\omega}t)|^{2}$ of the modes
is a normalized $2\pi$-periodic Gaussian centered on
$\bm{\Phi}^{0}-\bm{\omega}t$ and of width $(\Delta\phi_{1},\Delta\phi_{2})$.
Plugging expansions of Eqs. (13) and (19), and $\bm{b}_{\nu}(\bm{\Phi})$
around $\bm{\Phi}^{0}-\bm{\omega}t$ in the limit of small
$\Delta\phi_{1},\Delta\phi_{2}$ into Eq. (26) we get

$\bm{Q}_{\nu}(t)=\bm{b}_{\nu}(\bm{\Phi}^{0}-\bm{\omega}t)\\\
+\frac{1}{2}\sum_{i}(\Delta\phi_{i})^{2}\frac{\partial^{2}\bm{b}_{\nu}}{\partial\phi_{i}^{2}}(\bm{\Phi}^{0}-\bm{\omega}t)+\mathcal{O}(\Delta\phi^{4}).$
(27)

Note that in the limit $\Delta\phi_{1}=\Delta\phi_{2}=0$ of the classical
description of the phase, we recover that the qubit follows the instantaneous
adiabatic state $\bm{b}_{\nu}(\bm{\Phi}^{0}-\bm{\omega}t)$. From the
normalization of the adiabatic states $|\bm{b}_{\nu}|^{2}=1$, we deduce the
relation
$\bm{b}_{\nu}\cdot\partial^{2}_{\phi_{i}}\bm{b}_{\nu}=-\partial_{\phi_{i}}\bm{b}_{\nu}\cdot\partial_{\phi_{i}}\bm{b}_{\nu}=-4g_{\nu,ii}$
where the last equation is an expression of the quantum metric of a two-level
system in term of the Bloch vectors [51, 52]. From this we obtain the
development at order $\Delta\phi^{2}$ of the purity

$\displaystyle\gamma_{\nu}(t)=1$
$\displaystyle-2(\Delta\phi_{1})^{2}g_{\nu,11}(\bm{\Phi}^{0}-\bm{\omega}t)$
$\displaystyle-2(\Delta\phi_{2})^{2}g_{\nu,22}(\bm{\Phi}^{0}-\bm{\omega}t)+\mathcal{O}(\Delta\phi^{4}).$
(28)

For a two level system, the quantum metric of the two levels identifies
$g_{+,ij}=g_{-,ij}$ such that the qubit is equally entangled with the modes in
each adiabatic component $\ket{\Psi_{\pm}}$. The topological coupling
corresponds to a non-vanishing average Berry curvature $F_{\nu}$. From the
inequality $g_{\nu,11}+g_{\nu,22}\geq|F_{\nu}|$ originating from the positive
semidefiniteness of the quantum geometric tensor [53] we obtain a lower bound
on the entanglement between the qubit and the modes:

$\left<\gamma_{\nu}\right>_{t}\leq
1-\frac{|{\mathcal{C}}_{-}|}{\pi}(\Delta\phi)^{2}+\mathcal{O}(\Delta\phi^{4}).$
(29)

This demonstrates that a topological pump necessarily entangles the qubit with
the modes, a property only captured by the quantum description provided in
this paper.

The statistical average of the adiabatic states is represented on Fig. 4(a1)
for an initial width of the Gaussian state
$\Delta\phi=\Delta\phi_{1}=\Delta\phi_{2}\simeq 0.09\pi$, corresponding to
$\Delta n_{1}=\Delta n_{2}\simeq 2.6$, with the qubit initialized in
$\varphi_{q}=0$. At a given time $t$, the densities on the torus of each
component $|\chi_{\pm}(\bm{\Phi}+\bm{\omega}t)|^{2}/W_{\nu}$ is centered on
$\bm{\Phi}^{0}-\bm{\omega}t$ with the initial width $\Delta\phi$. This
translates into complementary densities of adiabatic states on the Bloch
sphere (in blue and orange) encoding the statistical mixture of the qubit in
$\ket{\Psi_{\pm}}$. In Fig. 4(a2), the purity of the qubit after the time of
separation $t_{\rm sep}$ is represented respectively in blue and orange for
each component $\ket{\Psi_{\pm}(t)}$. The temporal fluctuations of this purity
follow those of the quantum metric at $\bm{\Phi}^{0}-\bm{\omega}t$ represented
by a black dashed line, as predicted by Eq. (28). As an illustration, we
notice that the quantum metric is smaller at time $t_{1}$ than at $t_{2}$,
manifesting that the evolution of adiabatic states with the phase $\bm{\Phi}$
is weaker at $\bm{\Phi}=\bm{\Phi}^{0}-\bm{\omega}t_{1}$ than at
$\bm{\Phi}=\bm{\Phi}^{0}-\bm{\omega}t_{2}$. This translates into a larger
purity of the qubit at $t=t_{1}$ than at $t=t_{2}$.

####  III.1.2 Entanglement of quasi-Fock state

We now consider states with an increasing initial width in phase $\Delta\phi$.
The dependance on $\Delta\phi$ of the average in time of the of the qubit’s
purity $\langle\gamma_{\pm}\rangle_{t}$ is represented on Fig. 4(c) for each
adiabatic component. The amplitude of its temporal fluctuations are
represented as errorbars. This average purity decreases with $\Delta\phi$,
corresponding to an increase of entanglement: the larger the phase support on
the torus, the larger the support on the Bloch sphere, and thus the smaller
the polarization (26). The large support on the Bloch sphere originates from
the topological nature of the coupling, which imposes that the adiabatic
states $\bm{b}_{\pm}(\bm{\Phi})$ reaches all points of the Bloch sphere as
$\bm{\Phi}$ varies. A topologically trivial coupling would lead to a localized
distribution of adiabatic states on the Bloch sphere corresponding to an
almost pure state of the qubit. This is another manifestation that topological
pumping and entanglement between the qubit and the modes are strongly
intertwined.

The time average of the purity is the same for the two adiabatic components
$\langle\gamma_{+}\rangle_{t}=\langle\gamma_{-}\rangle_{t}$ for every
$\Delta\phi$. A qualitative explanation is the following. Given (26) the
purity $\langle\gamma_{\pm}\rangle_{t}$ corresponds to an average of adiabatic
states with respect to translated phase distributions. The average over time
depends only on the extension of the phase density
$|\chi_{\pm}(\bm{\Phi})|^{2}$, which, from (13), satisfies

$\displaystyle|\chi_{+}(\bm{\Phi}+\bm{\omega}t)|^{2}+|\chi_{-}(\bm{\Phi}+\bm{\omega}t)|^{2}=|\chi(\bm{\Phi}+\bm{\omega}t)|^{2}.$
(30)

Hence they split the phase density of the total system
$|\chi(\bm{\Phi}+\bm{\omega}t)|^{2}$ in two complementary supports, as
illustrated in appendix H. For cat states with equal weight $W_{+}=W_{-}$,
according to (19) these two supports have equal weight, leading to the same
average purity over time.

As discussed above, for quasi phase states the purity of both components
fluctuate temporally in phase Fig. 4(a2). In the opposite limit of an initial
Fock state fully delocalized in phase with $\Delta\phi=\pi$, these two
purities are in opposite phase. This is represented Fig. 4(b2) and (d), where
the qubit is initialized respectively on $\varphi_{q}=0$ and
$\varphi_{q}=\pi/4$. The phase density of the modes is uniform
$|\chi(\bm{\Phi})|^{2}=1/(2\pi)^{2}$, such that according to (30) the phase
density of the two cat components have complementary supports on all the torus
$[0,2\pi]^{2}$. We represent on Fig 4(b1) in blue the density of adiabatic
states $\bm{b}_{-}(\bm{\Phi})$ associated to the support
$|\chi_{-}(\bm{\Phi}-\bm{\omega}t)|^{2}$ and in orange the density of
adiabatic states $\bm{b}_{+}(\bm{\Phi})$ associated to the support
$|\chi_{+}(\bm{\Phi}-\bm{\omega}t)|^{2}$. At $t=t_{3}$, the density
$|\chi_{+}(\bm{\Phi}-\bm{\omega}t_{3})|^{2}$ covers a larger portion of the
sphere than the density $|\chi_{-}(\bm{\Phi}-\bm{\omega}t_{3})|^{2}$,
corresponding to $|\bm{Q}_{+}(t_{3})|^{2}<|\bm{Q}_{-}(t_{3})|^{2}$ and
$\gamma_{+}(t_{3})<\gamma_{-}(t_{3})$ on Fig. 4(b2), while the situation is
opposite at $t=t_{4}$. The details of the temporal variations of the purity
depends on the details of the shape of the densities
$|\chi_{\pm}(\bm{\Phi})|^{2}$, which depends on the initial state
$\varphi_{q}$ of the qubit. Temporal oscillations for $\varphi_{q}=\pi/4$ are
represented on Fig 4(d): the temporal average remains of the same order of
magnitude and the temporal fluctuations of the purity of each components
remains in opposite phase.

###  III.2 Breathing dynamics and Bloch oscillations

Figure 5:  From Bloch oscillations to Bloch breathing. (a) Trajectories of the
average values of number of quanta of each cat component $\ket{\Psi_{\pm}(t)}$
for the example of Fig. 2(a) localized in phase. The two components splits
along $n_{\perp}$ at an average velocity
$\pm\bm{v}=\mp(-\omega_{2},\omega_{1}){\mathcal{C}}/(2\pi)$. The Bloch
oscillations are the temporal fluctuations of the average number of quanta
around this drift. (b) Trajectories for the initial Fock state of Fig. 2(c).
The temporal fluctuations are reduced. (c) Photon number distribution of Fig.
2(c4) zoomed on the component $\ket{\Psi_{-}(t)}$ for $t=t_{a}$ a quasi-
period. (d) Photon number distribution of Fig. 2(c5) zoomed on the component
$\ket{\Psi_{-}(t)}$ for $t=t_{b}$ a time of expansion. (e) Circles: amplitude
$\Delta_{t}\langle\hat{n}_{\scriptscriptstyle E}\rangle$ and
$\Delta_{t}\langle\hat{n}_{\perp}\rangle$ of the Bloch oscillations depending
on the width in phase $\Delta\phi$. Triangle: temporal average of the
spreading $\Delta_{Q}\hat{n}_{\scriptscriptstyle E}$ and
$\Delta_{Q}\hat{n}_{\perp}$, with the amplitude of their temporal fluctuations
in errorbar. In small $\Delta\phi$ limit, the spreading is constant, the state
remains gaussian with $\Delta_{Q}\hat{n}_{i}=1/(2\Delta\phi)$. When
$\Delta\phi$ increases, the amplitude of the Bloch oscillations
$\Delta_{t}\langle\hat{n}_{i}\rangle$ decreases and the temporal fluctuations
of the spreading increase, corresponding to a breathing. (f,g,h) Bloch
oscillations. Temporal fluctuations of the average value of number of quanta
$\langle\bm{\hat{N}}\rangle$ in the component $\ket{\Psi_{-}(t)}$ around the
quantized drift $\bm{v}t$. At quasi-periods $t=t_{a}$ and $t=t_{b}$, the
evolution is almost given by the quantized drift
$\langle\bm{\hat{N}}\rangle=vt$. The increase of $\Delta\phi$ corresponds to a
decrease of the temporal fluctuations. (i) Time evolution of the spreading
$\Delta_{Q}\hat{n}_{i}$ in the component $\ket{\Psi_{-}(t)}$ for an initial
state localized in phase. The state remains Gaussian with constant spreading
$\Delta_{Q}\hat{n}_{i}=1/(2\Delta\phi)$. (j,k) Bloch breathing of the
spreading for small $\Delta\phi$. The wavepacket localizes at quasi-periods
$t=t_{a}$ and $t=t_{c}$, and expand in between ($t=t_{b}$).

We now discuss in more details the oscillations of both the center of mass and
of the width in number of each adiabatic component of cat states that manifest
themselves on the examples of Fig. 2. This dynamics is reminiscent of Bloch
oscillations and Bloch breathing [54, 55, 56, 24]. Bloch oscillations
correspond to temporal oscillations of the center of a wave-packet on a
lattice when submitted to an electric field. Bloch breathing corresponds to
temporal oscillations of the width of this wave-packet. The nature of these
oscillations and breathing depends on the width of the wavepacket’s momentum
distribution. In our context, the lattice correspond to the numbers of quanta
$\bm{N}=(n_{1},n_{2})$ of mode. The first term of the Hamiltonian (1) is
linear in $\bm{N}$ and plays the role of the coupling to an electric field
$\bm{\omega}$, while the second term correspond to a spin-orbit coupling as
discussed in section I.1. Hence the dynamics in $\bm{N}$ representation of the
adiabatic components identifies with the Bloch oscillations and breathing in
the presence of both a longitudinal electric field and an anomalous transverse
topological velocity.

####  III.2.1 Qualitative evolution of an adiabatic component

Two trajectories of the average value of number of quanta
$\langle\hat{\bm{N}}\rangle_{\Psi_{\pm}(t)}$ are represented on Fig. 5(a) and
(b) for two widths $\Delta\phi$. The two adiabatic subspaces $\nu=\pm$ are
associated to opposite anomalous velocities $\mp\bm{v}$ with
$\bm{v}=(\omega_{2},-\omega_{1}){\mathcal{C}}_{-}/(2\pi)$. This induces a
drift of the two wavepackets in opposite directions along $n_{\perp}$, shown
on the figure. We highlight the dynamics of the component $\ket{\Psi_{-}(t)}$
around this drift on Fig. 5(c) and (d), the results for $\ket{\Psi_{+}(t)}$
being similar. In this figure, we note $\langle\hat{n}_{i}\rangle(t)$ the
average values of $\hat{n}_{i}$ in $\ket{\Psi_{-}(t)}$ in the rotated
coordinates (5), $\hat{n}_{i}$ referring to $\hat{n}_{\scriptscriptstyle E}$
or $\hat{n}_{\perp}$. We denote as
$\Delta_{Q}\hat{n}_{i}(t)=[\langle\hat{n}_{i}^{2}\rangle(t)-\langle\hat{n}_{i}\rangle(t)^{2}]^{\frac{1}{2}}$
the quantum fluctuations, or spreading of $\hat{n}_{i}$ in $\ket{\Psi_{-}(t)}$
Fig. 5(c-d).

The numbers of quanta $\langle\hat{\bm{N}}\rangle(t)$ have temporal
fluctuations around the quantized drift $\bm{v}t$ represented on Fig. 5(f-h)
for the three adiabatic cats of Fig. 2. When we increase the width
$\Delta\phi$ of the initial state, the amplitude
$\Delta_{t}\langle\hat{n}_{i}\rangle$ of these temporal fluctuations is
reduced, while the spreading of the wavepacket $\Delta_{Q}\hat{n}_{i}$
increases. For small $\Delta\phi$, $\Delta_{Q}\hat{n}_{i}$ is almost constant
(Fig. 5(i)), the state remains Gaussian with its initial width
$\Delta_{Q}\hat{n}_{i}=1/(2\Delta\phi)$ as illustrated on Fig. 2(a1-a4). When
$\Delta\phi$ increases, $\Delta_{Q}\hat{n}_{i}$ oscillates in time 5(j-k).
This corresponds to breathing: oscillations, such as at $t=t_{b}$ on Fig.
5(d), between relocalization occurring e.g. at $t=t_{a}$ and $t_{c}$ on Fig.
5(c). The corresponding time evolution of the width $\Delta_{Q}\hat{n}_{i}$ is
represented on Fig. 5(i-k).

####  III.2.2 Quasi-periods

In Fig. 5(j-k), we observe, for a large initial $\Delta\phi$, a relocalization
of the wavepacket at specific times such as $t_{a}$ and $t_{c}$. These
fluctuations in time of the adiabatic wave-packet correspond to two
dimensional Bloch oscillations [54, 55, 56]. Historically, Bloch oscillations
were first considered in one dimension [57]. The electric field induces a
constant increase of the Bloch momenta of a semiclassical wavepacket, which
crosses periodically the one dimensional Brillouin zone. As a consequence the
average position of the wavepacket oscillates periodically. In two dimensions,
Bloch oscillations are richer. The Bloch momenta evolves on the two
dimensional Brillouin zone along the direction of the electric field:
$\bm{\Phi}(t)=\bm{\Phi}^{0}-\bm{\omega}t$. Such an evolution is periodic for a
commensurate ratio between $\omega_{1}$ and $\omega_{2}$, corresponding to an
electric field in a crystalline direction. Noting
$\omega_{1}/\omega_{2}=p_{1}/p_{2}$ with $p_{1}$ and $p_{2}$ coprime integers,
the trajectory of the Bloch momenta on the two dimensional Brillouin zone is
periodic with period $T=p_{1}2\pi/\omega_{1}=p_{2}2\pi/\omega_{2}$. In
practice any real number $\omega_{1}/\omega_{2}$ can be approximated by a set
of rational numbers [55, 24]. Each rational approximation leads to a quasi
period $T$ for which $\bm{\Phi}^{0}-\bm{\omega}T\simeq\bm{\Phi}^{0}$. The
times $t_{a}$ and $t_{c}$ on Fig. 5 are two examples of these quasi-periods
for our choice of $\omega_{1},\omega_{2}$.

The periodicity of a trajectory on the Brillouin zone translates into a
periodic motion in the direction of the electric field $n_{\scriptscriptstyle
E}$ but not in the transverse direction $n_{\perp}$, even in the absence of an
anomalous velocity [56, 58]. For an initial phase $\bm{\Phi}^{0}$, the
classical equations of motion in adiabatic space $\nu$ of the center of the
wavepacket can be written as [14, 23]:

$\displaystyle n_{\scriptscriptstyle E}(t)$
$\displaystyle=\int_{0}^{t}\frac{1}{\hbar}\frac{\partial
E_{\nu}}{\partial\phi_{\scriptscriptstyle
E}}(\bm{\Phi}(t^{\prime}))\differential t^{\prime}\ ,$ (31) $\displaystyle
n_{\perp}(t)$ $\displaystyle=\int_{0}^{t}\left(\frac{1}{\hbar}\frac{\partial
E_{\nu}}{\partial\phi_{\perp}}(\bm{\Phi}(t^{\prime}))-|\bm{\omega}|F_{\nu}(\bm{\Phi}(t^{\prime}))\right)\differential
t^{\prime}.$ (32)

where the evolution of the phase reads in the rotated phase coordinates:
$\phi_{\scriptscriptstyle E}(t)=\phi_{\scriptscriptstyle
E}^{0}-|\bm{\omega}|t$ and $\phi_{\perp}(t)=\phi_{\perp}^{0}$. We choose the
origin of the number of quanta set by their initial values, such that
$n_{\scriptscriptstyle E}(0,\bm{\Phi}^{0})=n_{\perp}(0,\bm{\Phi}^{0})=0$. The
time integral can be rewritten as a line integral over
$\phi_{\scriptscriptstyle E}$, leading to the conservation equation
$\hbar|\bm{\omega}|n_{\scriptscriptstyle
E}(t,\bm{\Phi}^{0})=E_{\nu}(\bm{\Phi}^{0})-E_{\nu}(\bm{\Phi}(t))$, which is
vanishingly small at a quasi-period $T$ such that
$\bm{\Phi}(T)\simeq\bm{\Phi}^{0}$. A quasi-period defines an almost closed
trajectory on the torus. The line integral of (32) does not vanish on this
closed trajectory, and reads $n_{\perp}(T,\bm{\Phi}^{0})\simeq vT$. The
approximation gets better for longer quasi-periods $T$, i.e. large $p_{1}$ and
$p_{2}$.

####  III.2.3 Bloch oscillations of the average number of quanta

In the hybrid classical-quantum description of a topological pump [14, 23],
the topological quantization of the pumping rate is recovered under a time-
average of the instantaneous flux of quanta $\dot{n}_{i}(t)$. This time
average of the Berry curvature entering the pumping rate is set by the Chern
number of the adiabatic states over the torus $[0,2\pi]^{2}$. The quantum
nature of the mode induces another source of averaging. The time evolution of
the average number of quanta (16) corresponds to an average of the classical
evolution (32) with respect to the phase density $|\chi_{\nu}(\bm{\Phi})|^{2}$
of the adiabatic component. Thus an increase of the support
$|\chi_{\nu}(\bm{\Phi})|^{2}$ reduces the temporal fluctuations of the average
number of quanta around its average drift. The quantization of topological
pumping is indeed better between quantum than classical modes. This is
represented on Fig. 5(e). We note $\Delta_{t}\langle\hat{n}_{i}\rangle$ the
amplitude of the fluctuations with time of the average value of
$\langle\hat{n}_{i}\rangle$. These temporal fluctuations are reduced when
$\Delta\phi$ increases (circles on Fig. 5(e)).

The reduction of the temporal fluctuations of pumping is set by the width of
the phase density $|\chi_{\nu}(\bm{\Phi})|^{2}$. One would expect that a
complete delocalization in phase, $|\chi_{\nu}(\bm{\Phi})|^{2}=1/(2\pi)^{2}$,
averages instantaneously the classical pumping rate over the whole phase space
such that $\langle\hat{\bm{N}}\rangle(t)\simeq\bm{v}t$ without any temporal
fluctuations. Such a projected state with uniform delocalization in
$\bm{\Phi}$ corresponds to a Wannier state for a particle on a lattice which
is topologically obstructed [59, 60, 61]. First, let us note that the
spreading $\Delta_{Q}\hat{n}_{i}$ is infinite in such obstructed Wannier
state, making them hard to realize experimentally. Moreover, due to the
adiabatic decomposition (11) of the initial (separable) state, such adiabatic
states are never realized: the phase density $|\chi_{\nu}(\bm{\Phi})|^{2}$,
defined in (13) contains the density of projection of the qubit initial state
$|\innerproduct{\psi_{\nu}(\bm{\Phi})}{\psi_{q}}|^{2}$ which necessarily
vanishes on the configuration space for a topological pump, irrespective of
$\ket{\psi_{q}}$. A Wannier state cannot be obtained by the adiabatic
decomposition of a separable state, and the temporal fluctuations of the
pumping remain finite.

####  III.2.4 Bloch breathing of the spreading

When the phase density $|\chi_{\nu}(\bm{\Phi})|^{2}$ is localized around
$\bm{\Phi}^{0}$, the center of mass performs Bloch oscillations following the
classical trajectory $\bm{N}(t,\bm{\Phi}^{0})$ (31), (32). Intuitively, when
the adiabatic state has a large support in phase
$|\chi_{\nu}(\bm{\Phi})|^{2}$, the different trajectories
$\bm{N}(t,\bm{\Phi})$ on this support superpose, inducing a spreading of the
wavepacket. The spreading of the wavepacket is then captured by the variance
$\text{Var}_{|\chi_{\nu}|^{2}}[n_{i}(t,\bm{\Phi})]$ of the classical
trajectories with respect to the (normalized) initial phase distribution
$|\chi_{\nu}(\bm{\Phi})|^{2}/W_{\nu}$:

$\text{Var}_{|\chi_{\nu}|^{2}}[n_{i}(t,\bm{\Phi})]=\int\differential^{2}\bm{\Phi}~{}\frac{|\chi_{\nu}(\bm{\Phi})|^{2}}{W_{\nu}}n_{i}(t,\bm{\Phi})^{2}\
\\\
-\left(\int\differential^{2}\bm{\Phi}~{}\frac{|\chi_{\nu}(\bm{\Phi})|^{2}}{W_{\nu}}n_{i}(t,\bm{\Phi})\right)^{2}.$
(33)

We show in appendix I that the time evolution of the spreading
$\Delta_{Q}\hat{n}_{i}$ in the adiabatic state $\ket{\Psi_{\nu}(t)}$ is indeed
related to this variance, and is the sum of three different terms.

For a quasi-phase state with a narrow distribution
$|\chi_{\nu}(\bm{\Phi})|^{2}$, the classical trajectories for an initial phase
on this support are all similar, such that the variance term (33) vanishes.
The quantum fluctuations weakly evolve in time
$\Delta_{Q}\hat{n}_{i}(t)\simeq\Delta_{Q}\hat{n}_{i}(0)\simeq
1/(2\Delta\phi)$, as shown in Fig. 5(i). In the case of large $\Delta\phi$,
the classical evolution $n_{i}(t,\bm{\Phi})$ follows different trajectories
for the different values of $\bm{\Phi}$ on $|\chi_{\nu}(\bm{\Phi})|^{2}$,
leading to a large $\text{Var}_{|\chi_{\nu}|^{2}}[n_{i}(t,\bm{\Phi})]$ and an
expansion of the wavepacket, seen for example at $t=t_{b}$ on Fig. 5(d)(k). At
a quasi-period $T$ discussed in Sec. III.2.2, the classical trajectories leads
almost to the same quantized drift $\bm{N}(T,\bm{\Phi})\simeq\bm{v}T$ for all
initial phases $\bm{\Phi}$, such that
$\text{Var}_{|\chi_{\nu}|^{2}}[n_{i}(T,\bm{\Phi})]\simeq 0$. The wave-packet
refocuses at these quasi-periods, seen at $t=t_{a}$ and $t=t_{c}$ on Fig.
5(k).

The default of refocusing is usually discussed in the literature in relation
with the default of rephasing
$\text{Var}_{|\chi_{\nu}|^{2}}[n_{i}(T,\bm{\Phi})]\neq 0$ [24, 54], such that
$(\Delta_{Q}\hat{n}_{i})(T)\gtrsim(\Delta_{Q}\hat{n}_{i})(t=0)$. We note an
important point about these rephasing: even if the refocusing is perfect
$(\Delta_{Q}\hat{n}_{i})(T)=(\Delta_{Q}\hat{n}_{i})(t=0)$ the spread at these
refocusing times corresponds to the initial spread of the adiabatic component
$\ket{\Psi_{-}(t=0)}$ and not to the initial state $\ket{\Psi(t=0)}$. This is
the reason why on Fig. 5(c) the state does not refocus into a Fock state.
Indeed, as discussed above the phase distribution $|\chi_{-}(\bm{\Phi})|^{2}$
is not fully delocalized on the torus, such that by Heisenberg inequality the
distribution of number of quanta in $\ket{\Psi_{-}(t=0)}$ is not fully
localized. For the cat state with equal weight, we discussed in section III.1
that $|\chi_{-}(\bm{\Phi})|^{2}$ cover approximately half the torus,
corresponding to a spread of order $\pi/4$, such that
$\Delta_{Q}\hat{n}_{i}(t=0)\geq 2/\pi\simeq 0.63$. This is approximately the
values of the spreading at the refocusing times on Fig. 5(k).

##  IV Discussion

In this work, we have shown that the dynamics of a qubit coupled topologically
to two slow quantum modes generically creates a cat state, a superposition of
two adiabatic states with mesoscopically distinct energy content. For each
adiabatic component of the cat, the topological nature of the coupling induces
an intrinsic entanglement between the qubit and the modes.

The realization of such topological adiabatic cat states opens interesting
perspectives, in particular to elaborate protocols to disentangle the qubit
from the quantum modes, creating an entangled cat state between the modes. One
can build on existing protocols for a superconducting qubit dispersively
coupled to quantum cavities, with cat composed of coherent states non-
entangled with the qubit, of typical form
$(\ket{\alpha_{1},\alpha_{2},\uparrow}+\ket{\beta_{1},\beta_{2},\downarrow})/\sqrt{2}$
[62]. From our analysis of Sec. III.1, similar states are obtained from an
adiabatic cat state in the quasi-phase limit at a time $t$ where a small value
of the quantum metric $g_{ij}(\bm{\Phi}^{0}-\bm{\omega}t)$ is reached.
Besides, it is worth point out that the topological splitting of the two
adiabatic components allows for the experimental preparation of adiabatic
states, by using a projection on the number of quanta $(n_{1},n_{2})$ such
that $n_{1}-n_{2}>n_{1}^{0}-n_{2}^{0}$ after the time of separation. In the
perspective of a superconducting qubit coupled to quantum cavities, such a
measurement protocol can be adapted from the methods of photon number
resolution [63] using an additional qubit dispersively coupled to the two
cavities.

Finally, let us stress that we have focused on the adiabatic limit of a
quantum description of a Floquet system. We characterized the entanglement
between the drives and the driven quantum system in term of the quantum
geometry of adiabatic states. Extending this relation between entanglement and
geometry beyond the adiabatic limit [64, 65] is a natural and stimulating
perspective.

###### Acknowledgements.

We are grateful to Audrey Bienfait, Emmanuel Flurin, Benjamin Huard and Zaki
Leghtas for insightful discussions.

## References

  * Thouless [1983] D. Thouless, Quantization of particle transport, Physical Review B 27, 6083 (1983).
  * Kraus  _et al._ [2012] Y. E. Kraus, Y. Lahini, Z. Ringel, M. Verbin, and O. Zilberberg, Topological states and adiabatic pumping in quasicrystals, Physical review letters 109, 106402 (2012).
  * Verbin  _et al._ [2015] M. Verbin, O. Zilberberg, Y. Lahini, Y. E. Kraus, and Y. Silberberg, Topological pumping over a photonic fibonacci quasicrystal, Physical Review B 91, 064201 (2015).
  * Ke  _et al._ [2016] Y. Ke, X. Qin, F. Mei, H. Zhong, Y. S. Kivshar, and C. Lee, Topological phase transitions and thouless pumping of light in photonic waveguide arrays, Laser & Photonics Reviews 10, 995 (2016).
  * Lohse  _et al._ [2016] M. Lohse, C. Schweizer, O. Zilberberg, M. Aidelsburger, and I. Bloch, A thouless quantum pump with ultracold bosonic atoms in an optical superlattice, Nature Physics 12, 350 (2016).
  * Nakajima  _et al._ [2016] S. Nakajima, T. Tomita, S. Taie, T. Ichinose, H. Ozawa, L. Wang, M. Troyer, and Y. Takahashi, Topological thouless pumping of ultracold fermions, Nature Physics 12, 296 (2016).
  * Lu  _et al._ [2016] H.-I. Lu, M. Schemmer, L. M. Aycock, D. Genkina, S. Sugawa, and I. B. Spielman, Geometrical pumping with a bose-einstein condensate, Physical review letters 116, 200402 (2016).
  * Cerjan  _et al._ [2020] A. Cerjan, M. Wang, S. Huang, K. P. Chen, and M. C. Rechtsman, Thouless pumping in disordered photonic systems, Light: Science & Applications 9, 1 (2020).
  * Grinberg  _et al._ [2020] I. H. Grinberg, M. Lin, C. Harris, W. A. Benalcazar, C. W. Peterson, T. L. Hughes, and G. Bahl, Robust temporal pumping in a magneto-mechanical topological insulator, Nature communications 11, 1 (2020).
  * Xia  _et al._ [2021] Y. Xia, E. Riva, M. I. Rosa, G. Cazzulani, A. Erturk, F. Braghin, and M. Ruzzene, Experimental observation of temporal pumping in electromechanical waveguides, Physical Review Letters 126, 095501 (2021).
  * Dreon  _et al._ [2022] D. Dreon, A. Baumgärtner, X. Li, S. Hertlein, T. Esslinger, and T. Donner, Self-oscillating pump in a topological dissipative atom–cavity system, Nature 608, 494 (2022).
  * Minguzzi  _et al._ [2022] J. Minguzzi, Z. Zhu, K. Sandholzer, A.-S. Walter, K. Viebahn, and T. Esslinger, Topological Pumping in a Floquet-Bloch Band, Physical Review Letters 129, 053201 (2022).
  * Fabre  _et al._ [2022] A. Fabre, J.-B. Bouhiron, T. Satoor, R. Lopes, and S. Nascimbene, Laughlin’s topological charge pump in an atomic hall cylinder, Physical Review Letters 128, 173202 (2022).
  * Martin  _et al._ [2017] I. Martin, G. Refael, and B. Halperin, Topological frequency conversion in strongly driven quantum systems, Physical Review X 7, 041008 (2017).
  * Peng and Refael [2018] Y. Peng and G. Refael, Topological energy conversion through the bulk or the boundary of driven systems, Physical Review B 97, 134303 (2018).
  * Ma  _et al._ [2018] W. Ma, L. Zhou, Q. Zhang, M. Li, C. Cheng, J. Geng, X. Rong, F. Shi, J. Gong, and J. Du, Experimental Observation of a Generalized Thouless Pump with a Single Spin, Physical Review Letters 120, 120501 (2018).
  * Körber  _et al._ [2020] S. Körber, L. Privitera, J. C. Budich, and B. Trauzettel, Interacting topological frequency converter, Physical Review Research 2, 022023 (2020).
  * Chen  _et al._ [2020] Q. Chen, H. Liu, M. Yu, S. Zhang, and J. Cai, Dynamical decoupling for realization of topological frequency conversion, Physical Review A 102, 052606 (2020).
  * Malz and Smith [2021] D. Malz and A. Smith, Topological Two-Dimensional Floquet Lattice on a Single Superconducting Qubit, Physical Review Letters 126, 163602 (2021).
  * Körber  _et al._ [2022] S. Körber, L. Privitera, J. C. Budich, and B. Trauzettel, Topological burning glass effect, Physical Review B 106, L140304 (2022).
  * Schwennicke and Yuen-Zhou [2022] K. Schwennicke and J. Yuen-Zhou, Enantioselective topological frequency conversion, The Journal of Physical Chemistry Letters 13, 2434 (2022).
  * Nathan  _et al._ [2019] F. Nathan, I. Martin, and G. Refael, Topological frequency conversion in a driven dissipative quantum cavity, Phys. Rev. B 99, 094311 (2019).
  * Luneau  _et al._ [2022] J. Luneau, C. Dutreix, Q. Ficheux, P. Delplace, B. Douçot, B. Huard, and D. Carpentier, Topological power pumping in quantum circuits, Physical Review Research 4, 013169 (2022).
  * Long  _et al._ [2022] D. M. Long, P. J. Crowley, A. J. Kollár, and A. Chandran, Boosting the quantum state of a cavity with floquet driving, Physical Review Letters 128, 183602 (2022).
  * Long  _et al._ [2021] D. M. Long, P. J. Crowley, and A. Chandran, Nonadiabatic topological energy pumps with quasiperiodic driving, Physical Review Letters 126, 106805 (2021).
  * Nathan  _et al._ [2021] F. Nathan, R. Ge, S. Gazit, M. Rudner, and M. Kolodrubetz, Quasiperiodic floquet-thouless energy pump, Physical Review Letters 127, 166804 (2021).
  * Kolodrubetz  _et al._ [2018] M. H. Kolodrubetz, F. Nathan, S. Gazit, T. Morimoto, and J. E. Moore, Topological floquet-thouless energy pump, Physical review letters 120, 150601 (2018).
  * Crowley  _et al._ [2019] P. J. Crowley, I. Martin, and A. Chandran, Topological classification of quasiperiodically driven quantum systems, Physical Review B 99, 064306 (2019).
  * Crowley  _et al._ [2020] P. J. Crowley, I. Martin, and A. Chandran, Half-integer quantized topological response in quasiperiodically driven quantum systems, Physical Review Letters 125, 100601 (2020).
  * Psaroudaki and Refael [2021] C. Psaroudaki and G. Refael, Photon pumping in a weakly-driven quantum cavity–spin system, Annals of Physics 435, 168553 (2021).
  * Qi  _et al._ [2021] Z. Qi, G. Refael, and Y. Peng, Universal nonadiabatic energy pumping in a quasiperiodically driven extended system, Physical Review B 104, 224301 (2021).
  * Mead and Truhlar [1979] C. A. Mead and D. G. Truhlar, On the determination of born–oppenheimer nuclear motion wave functions including complications due to conical intersections and identical nuclei, The Journal of Chemical Physics 70, 2284 (1979).
  * Berry [1989] M. V. Berry, The quantum phase, five years after, Geometric phases in physics , 7 (1989).
  * Xiao  _et al._ [2010] D. Xiao, M. C. Chang, and Q. Niu, Berry phase effects on electronic properties, Reviews of Modern Physics 82, 1959 (2010).
  * Wang  _et al._ [2016] D.-W. Wang, H. Cai, R.-B. Liu, and M. O. Scully, Mesoscopic Superposition States Generated by Synthetic Spin-Orbit Interaction in Fock-State Lattices, Physical Review Letters 116, 220502 (2016).
  * Deng  _et al._ [2022] J. Deng, H. Dong, C. Zhang, Y. Wu, J. Yuan, X. Zhu, F. Jin, H. Li, Z. Wang, H. Cai, C. Song, H. Wang, J. Q. You, and D.-W. Wang, Coherent control of quantum topological states of light in Fock-state lattices (2022), arXiv:2208.03452 .
  * Cai and Wang [2021] H. Cai and D.-W. Wang, Topological phases of quantized light, National Science Review 8, nwaa196 (2021).
  * Yuan  _et al._ [2021] J. Yuan, C. Xu, H. Cai, and D.-W. Wang, Gap-protected transfer of topological defect states in photonic lattices, APL Photonics 6, 030803 (2021).
  * Saugmann and Larson [2022] P. Saugmann and J. Larson, A Fock state lattice approach to quantum optics (2022), arXiv:2203.13813 .
  * Carruthers and Nieto [1968] P. Carruthers and M. M. Nieto, Phase and angle variables in quantum mechanics, Reviews of Modern Physics 40, 411 (1968).
  * Fruchart  _et al._ [2014] M. Fruchart, D. Carpentier, and K. Gawedzki, Parallel transport and band theory in crystals, EPL (Europhysics Letters) 106, 60002 (2014).
  * Simon and Rudner [2020] S. H. Simon and M. S. Rudner, Contrasting lattice geometry dependent versus independent quantities: Ramifications for berry curvature, energy gaps, and dynamics, Physical Review B 102, 165148 (2020).
  * Bernevig  _et al._ [2006] B. A. Bernevig, T. L. Hughes, and S.-C. Zhang, Quantum spin hall effect and topological phase transition in hgte quantum wells, Science 314, 1757 (2006).
  * Born and Huang [1954] M. Born and K. Huang,  _Dynamical theory of crystal lattices_ (Clarendon press, 1954) appendix VIII.
  * Messiah [1962] A. Messiah,  _Quantum mechanics: volume II_ (North-Holland Publishing Company Amsterdam, 1962).
  * Emmrich and Weinstein [1996] C. Emmrich and A. Weinstein, Geometry of the transport equation in multicomponent WKB approximations, Communications in mathematical physics 176, 701 (1996).
  * Stiepan and Teufel [2013] H.-M. Stiepan and S. Teufel, Semiclassical approximations for hamiltonians with operator-valued symbols, Communications in Mathematical Physics 320, 821 (2013).
  * Berry [1984] M. V. Berry, Quantal phase factors accompanying adiabatic changes, Proceedings of the Royal Society of London. A. Mathematical and Physical Sciences 392, 45 (1984).
  * Aharonov and Anandan [1987] Y. Aharonov and J. Anandan, Phase change during a cyclic quantum evolution, Physical Review Letters 58, 1593 (1987).
  * Provost and Vallee [1980] J. Provost and G. Vallee, Riemannian structure on manifolds of quantum states, Communications in Mathematical Physics 76, 289 (1980).
  * Piéchon  _et al._ [2016] F. Piéchon, A. Raoux, J.-N. Fuchs, and G. Montambaux, Geometric orbital susceptibility: Quantum metric without berry curvature, Physical Review B 94, 134423 (2016).
  * Graf and Piéchon [2021] A. Graf and F. Piéchon, Berry curvature and quantum metric in n-band systems: An eigenprojector approach, Physical Review B 104, 085114 (2021).
  * Peotta and Törmä [2015] S. Peotta and P. Törmä, Superfluidity in topologically nontrivial flat bands, Nature Communications 6, 8944 (2015).
  * Witthaut  _et al._ [2004] D. Witthaut, F. Keck, H. J. Korsch, and S. Mossmann, Bloch oscillations in two-dimensional lattices, New Journal of Physics 6, 41 (2004).
  * Kolovsky and Korsch [2003] A. R. Kolovsky and H. J. Korsch, Bloch oscillations of cold atoms in two-dimensional optical lattices, Physical Review A 67, 063601 (2003).
  * Mossmann  _et al._ [2005] S. Mossmann, A. Schulze, D. Witthaut, and H. J. Korsch, Two-dimensional Bloch oscillations: A Lie-algebraic approach, Journal of Physics A: Mathematical and General 38, 3381 (2005).
  * Mendez and Bastard [1993] E. Mendez and G. Bastard, Wannier-Stark Ladders and Bloch Oscillations in Superlattices, Physics Today 46, 34 (1993).
  * Zhang and Liu [2010] J. M. Zhang and W. M. Liu, Directed coherent transport due to the Bloch oscillation in two dimensions, Physical Review A 82, 025602 (2010).
  * Marzari  _et al._ [2012] N. Marzari, A. A. Mostofi, J. R. Yates, I. Souza, and D. Vanderbilt, Maximally localized wannier functions: Theory and applications, Reviews of Modern Physics 84, 1419 (2012).
  * Thouless [1984] D. Thouless, Wannier functions for magnetic sub-bands, Journal of Physics C: Solid State Physics 17, L325 (1984).
  * Monaco  _et al._ [2018] D. Monaco, G. Panati, A. Pisante, and S. Teufel, Optimal decay of wannier functions in chern and quantum hall insulators, Communications in Mathematical Physics 359, 61 (2018).
  * Leghtas  _et al._ [2013] Z. Leghtas, G. Kirchmair, B. Vlastakis, M. H. Devoret, R. J. Schoelkopf, and M. Mirrahimi, Deterministic protocol for mapping a qubit to coherent state superpositions in a cavity, Physical Review A 87, 042315 (2013).
  * Schuster  _et al._ [2007] D. I. Schuster, A. A. Houck, J. A. Schreier, A. Wallraff, J. M. Gambetta, A. Blais, L. Frunzio, J. Majer, B. Johnson, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Resolving photon number states in a superconducting circuit, Nature 445, 515 (2007).
  * Ng  _et al._ [2021] N. Ng, S. Wenderoth, R. R. Seelam, E. Rabani, H.-D. Meyer, M. Thoss, and M. Kolodrubetz, Localization dynamics in a centrally coupled system, Physical Review B 103, 134201 (2021).
  * Engelhardt  _et al._ [2022] G. Engelhardt, S. Choudhury, and W. V. Liu, Unified Light-Matter Floquet Theory and its Application to Quantum Communication (2022), arXiv:2207.08558 .
  * Berry and Robbins [1993] M. V. Berry and J. Robbins, Chaotic classical and half-classical adiabatic reactions: geometric magnetism and deterministic friction, Proceedings of the Royal Society of London. Series A: Mathematical and Physical Sciences 442, 659 (1993).
  * Teufel [2003] S. Teufel,  _Adiabatic perturbation theory in quantum dynamics_ (Springer Science & Business Media, 2003).

##  Appendix A Time evolution of phase states

We consider the Hamiltonian $\hat{H}_{\rm
tot}=\hbar\bm{\omega}\cdot\hat{\bm{N}}+H(\hat{\bm{\Phi}})$ where
$\bm{\omega}=(\omega_{1},\cdots,\omega_{N})$,
$\hat{\bm{N}}=(\hat{n}_{1},\cdots,\hat{n}_{N})$,
$\hat{\bm{\Phi}}=(\hat{\phi}_{1},\cdots,\hat{\phi}_{N})$, where the operators
$\hat{n}_{i}$ and $\hat{\phi}_{i}$ are conjugated
$[\hat{n}_{i},\hat{\phi}_{j}]={\rm i}\delta_{i,j}\mathbf{1}$, and
$\bm{\omega}\cdot\hat{\bm{N}}=\sum_{i}\omega_{i}\hat{n}_{i}$. We determine the
time evolution of a phase eigenstate
$\ket{\Psi(t=0)}=\ket{\bm{\Phi}}\otimes\ket{\psi}$ with $\ket{\psi}$ is an
arbitrary state of the two-level system. In the interaction representation
with respect to the Hamiltonian of the modes, the time evolved state is:

$\ket{\Psi_{I}(t)}=\exp\left[{\rm
i}t\bm{\omega}\cdot\hat{\bm{N}}\right]\ket{\Psi(t)}.$ (34)

The dynamics of $\ket{\Psi_{I}(t)}$ is governed by the Hamiltonian in the
interaction representation:

$\displaystyle\hat{H}_{I}(t)$ $\displaystyle=\exp\left[{\rm
i}t\bm{\omega}\cdot\hat{\bm{N}}\right]H(\hat{\bm{\Phi}})\exp\left[-{\rm
i}t\bm{\omega}\cdot\hat{\bm{N}}\right]$ (35)
$\displaystyle=H(\hat{\bm{\Phi}}-\bm{\omega}t)$ (36)

since the operators $\hat{n}_{i}$ are generators of the phase translation. As
a consequence,

$\hat{H}_{I}(t)\left(\ket{\bm{\Phi}}\otimes\ket{\psi}\right)=\ket{\bm{\Phi}}\otimes
H(\bm{\Phi}-\bm{\omega}t)\ket{\psi}$ (37)

such that the time-evolution of the initial state
$\ket{\bm{\Phi}}\otimes\ket{\psi}$ in the interaction representation is

$\displaystyle\ket{\Psi_{I}(t)}$
$\displaystyle=\mathcal{T}\exp\left[-\frac{{\rm
i}}{\hbar}\int_{0}^{t}\differential\tau~{}\hat{H}_{I}(\tau)\right]\ket{\bm{\Phi}}\otimes\ket{\psi}$
$\displaystyle=\ket{\bm{\Phi}}\otimes U(t;\bm{\Phi})\ket{\psi}$ (38)

where $\mathcal{T}$ denotes time-ordering and
$U(t;\bm{\Phi})=\mathcal{T}\exp\left[-\frac{{\rm
i}}{\hbar}\int_{0}^{t}\differential\tau~{}H(\bm{\Phi}-\bm{\omega}\tau)\right]$
is the time evolution operator associated to the time-dependant Hamiltonian
$H(\bm{\Phi}-\bm{\omega}t)$ for classical modes. We then obtain the time-
evolved state in the Schrödinger representation

$\displaystyle\ket{\Psi(t)}$ $\displaystyle=\exp\left[-{\rm
i}t\bm{\omega}\cdot\hat{\bm{N}}\right]\ket{\Psi_{I}(t)}$ (39)
$\displaystyle=\ket{\bm{\Phi}-\bm{\omega}t}\otimes U(t;\bm{\Phi})\ket{\psi}.$
(40)

##  Appendix B Construction of the adiabatic subspaces

We construct the states of the two-level system $\ket{\psi_{\nu}(\bm{\Phi})}$
such that the family of states
$\ket{\bm{\Phi}}\otimes\ket{\psi_{\nu}(\bm{\Phi})}$ with
$\bm{\Phi}\in[0,2\pi]^{2}$ is stable under the dynamics governed by the total
Hamiltonian $H_{\rm
tot}=\sum_{i}\hbar\omega_{i}\hat{n}_{i}+H(\hat{\bm{\Phi}})$. This family of
states corresponds to the image of the associated adiabatic projector

$\displaystyle\hat{P}_{\nu}$
$\displaystyle=\int\differential\bm{\Phi}~{}\ket{\bm{\Phi}}\bra{\bm{\Phi}}\otimes\ket{\psi_{\nu}(\bm{\Phi})}\bra{\psi_{\nu}(\bm{\Phi})}$
(41)
$\displaystyle=\int\differential\bm{\Phi}~{}\ket{\bm{\Phi}}\bra{\bm{\Phi}}\otimes\pi_{\nu}(\bm{\Phi}).$
(42)

We first construct the family of projectors of the two-level system
$\pi_{\nu}(\bm{\Phi})=\ket{\psi_{\nu}(\bm{\Phi})}\bra{\psi_{\nu}(\bm{\Phi})}$.
As detailed in appendix A, for an arbitrary state $\ket{\psi}$ of the fast
quantum degree of freedom, the phase eigenstates evolve according to

$\exp(-i\hat{H}_{\rm
tot}t/\hbar)\left(\ket{\bm{\Phi}}\otimes\ket{\psi}\right)=\ket{\bm{\Phi}-\bm{\omega}t}\otimes
U(t;\bm{\Phi})\ket{\psi}$ (43)

with $U(t;\bm{\Phi})$ the time evolution operator associated to the Floquet
Hamiltonian $H(\bm{\Phi}-\bm{\omega}t)$. Thus the previous family of states is
stable if the projectors $\pi_{\nu}(\bm{\Phi})$ satisfy

$U(t;\bm{\Phi})\pi_{\nu}(\bm{\Phi})U(t;\bm{\Phi})^{\dagger}=\pi_{\nu}(\bm{\Phi}-\bm{\omega}t)\,,$
(44)

or equivalently
$-i\hbar\bm{\omega}\cdot\nabla_{\bm{\Phi}}\pi_{\nu}(\bm{\Phi})=[H(\bm{\Phi}),\pi_{\nu}(\bm{\Phi})].$
This equation can be solved perturbatively, assuming that the eigenstates
evolve slowly. Similarly to [66], we rescale all of the frequencies by a
dimensionless parameter $\epsilon$: $\bm{\omega}\to\epsilon\bm{\omega}$. We
search a projector $\pi_{\nu}(\bm{\Phi})$ expressed a formal series of
$\epsilon$

$\pi_{\nu}(\bm{\Phi})=\sum_{k}\epsilon^{k}\pi_{\nu,k}(\bm{\Phi})=\pi_{\nu,0}(\bm{\Phi})+\epsilon\pi_{\nu,1}(\bm{\Phi})+\dots$
(45)

solution of the equations

$\displaystyle-i\epsilon\hbar\bm{\omega}\cdot\nabla_{\bm{\Phi}}\pi_{\nu}(\bm{\Phi})$
$\displaystyle=[H(\bm{\Phi}),\pi_{\nu}(\bm{\Phi})]$ (46)
$\displaystyle\pi_{\nu}(\bm{\Phi})^{2}$
$\displaystyle=\pi_{\nu}(\bm{\Phi})\,.$ (47)

Such a solution exists as an asymptotic series in $\epsilon$ [46, 47, 67].

We detail the determination of the two first terms. The conditions (46) and
(47) gives for the order 0

$\displaystyle[H(\bm{\Phi}),\pi_{\nu,0}(\bm{\Phi})]$ $\displaystyle=0$ (48)
$\displaystyle\pi_{\nu,0}(\bm{\Phi})^{2}$
$\displaystyle=\pi_{\nu,0}(\bm{\Phi})$ (49)

which is satisfied for a family of projectors
$\pi_{\nu,0}(\bm{\Phi})=\ket{\psi_{\nu}^{0}(\bm{\Phi})}\bra{\psi_{\nu}^{0}(\bm{\Phi})}$
on eigenstates $\ket{\psi_{\nu}^{0}(\bm{\Phi})}$ of $H(\bm{\Phi})$ associated
to the eigen-energy $E_{\nu}^{0}(\bm{\Phi})$,
$H(\bm{\Phi})\ket{\psi_{\nu}^{0}(\bm{\Phi})}=E_{\nu}^{0}(\bm{\Phi})\ket{\psi_{\nu}^{0}(\bm{\Phi})}$.
At order 1, the conditions (46) and (47) read

$\displaystyle[H(\bm{\Phi}),\pi_{\nu,1}(\bm{\Phi})]$
$\displaystyle=i\hbar\bm{\omega}\cdot\nabla_{\bm{\Phi}}\pi_{\nu,0}(\bm{\Phi})$
(50) $\displaystyle\pi_{\nu,1}(\bm{\Phi})$
$\displaystyle=\pi_{\nu,1}(\bm{\Phi})\pi_{\nu,0}(\bm{\Phi})+\pi_{\nu,0}(\bm{\Phi})\pi_{\nu,1}(\bm{\Phi}).$
(51)

When the eigenstate $\ket{\psi_{\nu}^{0}(\bm{\Phi})}$ is non-degenerate for
all $\bm{\Phi}\in[0,2\pi]^{2}$, these equations are satisfied for

$\pi_{\nu,1}(\bm{\Phi})=\sum_{\mu\neq\nu}\ket{\psi_{\mu}^{0}(\bm{\Phi})}\frac{\sum_{i}\hbar\omega_{i}A_{\mu\nu,i}^{0}(\bm{\Phi})}{E_{\mu}^{0}(\bm{\Phi})-E_{\nu}^{0}(\bm{\Phi})}\bra{\psi_{\nu}^{0}(\bm{\Phi})}+\mathrm{h.c.}$
(52)

with
$A_{\mu\nu,i}^{0}(\bm{\Phi})=i\innerproduct{\psi_{\mu}^{0}(\bm{\Phi})}{\partial_{\phi_{i}}\psi_{\nu}^{0}(\bm{\Phi})}$
the components of the non-abelian Berry connection of the eigenstates. Thus
the adiabatic state at order 1 decomposes on the eigenstates similarly to the
usual time dependant perturbation theory

$\ket{\psi_{\nu}(\bm{\Phi})}=\ket{\psi_{\nu}^{0}(\bm{\Phi})}+\\\
\epsilon\sum_{\mu\neq\nu}\left(\frac{\sum_{i}\hbar\omega_{i}A_{\mu\nu,i}^{0}(\bm{\Phi})}{E_{\mu}^{0}(\bm{\Phi})-E_{\nu}^{0}(\bm{\Phi})}\right)\ket{\psi_{\mu}^{0}(\bm{\Phi})}+\mathcal{O}(\epsilon^{2}).$
(53)

##  Appendix C Time evolution of the adiabatic states

In appendix B we constructed the projectors
$\pi_{\nu}(\bm{\Phi})=\ket{\psi_{\nu}(\bm{\Phi})}\bra{\psi_{\nu}(\bm{\Phi})}$
on the states of the two-level system such that the image of the adiabatic
projector $\hat{P}_{\nu}$ is stable under the dynamics. As discussed in the
main text, such states $\ket{\psi_{\nu}(\bm{\Phi})}$ have to satisfy

$U(t;\bm{\Phi})\ket{\psi_{\nu}(\bm{\Phi})}=e^{i\theta_{\nu}(t;\bm{\Phi})}\ket{\psi_{\nu}(\bm{\Phi}-\bm{\omega}t)}$
(54)

with $U(t;\bm{\Phi})$ the time evolution operator associated to the time-
dependant Hamiltonian $H(\bm{\Phi}-\bm{\omega}t)$ and with
$\theta_{\nu}(t;\bm{\Phi})$ a phase factor. We show that this phase factor is
related to the Aharonov-Anandan phase. By definition, the time evolution
operator $U(t;\bm{\Phi})$ satisfies

$\displaystyle i\hbar\frac{\differential}{\differential t}U(t;\bm{\Phi})$
$\displaystyle=H(\bm{\Phi}-\bm{\omega}t)U(t;\bm{\Phi})$ (55)

such that (54) gives

$\frac{1}{i\hbar}H(\bm{\Phi}-\bm{\omega}t)\ket{\psi_{\nu}(\bm{\Phi}-\bm{\omega}t)}=i\frac{\partial\theta_{\nu}}{\partial
t}(t;\bm{\Phi})\ket{\psi_{\nu}(\bm{\Phi}-\bm{\omega})}\\\
-\sum_{i}\omega_{i}\ket{\partial_{\phi_{i}}\psi_{\nu}(\bm{\Phi}-\bm{\omega}t)}.$
(56)

The phase factor $\theta_{\nu}(t;\bm{\Phi})$ is then given by

$\theta_{\nu}(t;\bm{\Phi})=\int_{0}^{t}\differential
t^{\prime}\left(-\frac{1}{\hbar}E_{\nu}(\bm{\Phi}-\bm{\omega}t^{\prime})-\sum_{i}\omega_{i}A_{\nu,i}(\bm{\Phi}-\bm{\omega}t^{\prime})\right)$
(57)

with the energy function

$E_{\nu}(\bm{\Phi})=\bra{\psi_{\nu}(\bm{\Phi})}H(\bm{\Phi})\ket{\psi_{\nu}(\bm{\Phi})}$
(58)

and the (generalized) Berry connection

$A_{\nu,i}(\bm{\Phi})=i\innerproduct{\psi_{\nu}(\bm{\Phi})}{\partial_{\phi_{i}}\psi_{\nu}(\bm{\Phi})}$
(59)

of the adiabatic states $\ket{\psi_{\nu}(\bm{\Phi})}$.

##  Appendix D Pumping rate in an adiabatic state

We derive the time-evolution (16) of the pumping rate
$\langle\hat{n}_{1}\rangle$ for an initial state (11) projected in the
adiabatic subspace $\nu$

$\ket{\Psi_{\nu}(t=0)}=\int\differential^{2}\bm{\Phi}~{}\chi_{\nu}(\bm{\Phi})\ket{\bm{\Phi}}\otimes\ket{\psi_{\nu}(\bm{\Phi})}.$
(60)

From appendix C, the time evolution of this projected state is

$\ket{\Psi_{\nu}(t)}=\int\differential^{2}\bm{\Phi}~{}\chi_{\nu}(\bm{\Phi})e^{i\theta_{\nu}(t;\bm{\Phi})}\ket{\bm{\Phi}-\bm{\omega}t}\otimes\ket{\psi_{\nu}(\bm{\Phi}-\bm{\omega}t)}.$
(61)

with the phase factor $\theta_{\nu}(t;\bm{\Phi})$ given by (57). The Ehrenfest
theorem reads

$\frac{\differential}{\differential
t}\langle\hat{n}_{1}\rangle_{\Psi_{\nu}(t)}=\frac{1}{i\hbar}\langle[\hat{n}_{1},\hat{H}_{\rm
tot}]\rangle_{\Psi_{\nu}(t)}$ (62)

where from the Hamiltonian (1) we have $[\hat{n}_{1},\hat{H}_{\rm
tot}]=i\frac{\partial H}{\partial\phi_{1}}(\hat{\bm{\Phi}})$ such that

$\frac{\differential}{\differential
t}\langle\hat{n}_{1}\rangle_{\Psi_{\nu}(t)}=\int\differential^{2}\bm{\Phi}~{}\frac{|\chi_{\nu}(\bm{\Phi})|^{2}}{W_{\nu}}\\\
\frac{1}{\hbar}\bra{\psi_{\nu}(\bm{\Phi}-\bm{\omega}t)}\frac{\partial
H}{\partial\phi_{1}}(\bm{\Phi}-\bm{\omega}t)\ket{\psi_{\nu}(\bm{\Phi}-\bm{\omega}t)}$
(63)

with $W_{\nu}=\innerproduct{\Psi_{\nu}(t)}{\Psi_{\nu}(t)}$ given by (21). The
average value of the derivative of the Hamiltonian can be written

$\bra{\psi_{\nu}}\frac{\partial
H}{\partial\phi_{1}}\ket{\psi_{\nu}}=\frac{\partial}{\partial\phi_{1}}\left(\bra{\psi_{\nu}}H\ket{\psi_{\nu}}\right)\\\
-\bra{\partial_{\phi_{1}}\psi_{\nu}}H\ket{\psi_{\nu}}-\bra{\psi_{\nu}}H\ket{\partial_{\phi_{1}}\psi_{\nu}}$
(64)

where the dependance on $\bm{\Phi}-\bm{\omega}t$ is implicit. The first term
of this equation gives the term of variation of energy in (16). Using the
expression (56) for $H\ket{\psi_{\nu}}$ and using the normalization condition
$\innerproduct{\partial_{\phi_{1}}\psi_{\nu}}{\psi_{\nu}}=-\innerproduct{\psi_{\nu}}{\partial_{\phi_{1}}\psi_{\nu}}$,
we write the last two terms of (64) in term of the curvature $F_{\nu}$ (14)
such that we obtain the expression (16) of the pumping rate

$\frac{\differential}{\differential
t}\langle\hat{n}_{1}\rangle_{\Psi_{\nu}(t)}=\int\differential^{2}\bm{\Phi}~{}\frac{|\chi_{\nu}(\bm{\Phi})|^{2}}{W_{\nu}}\\\
\left(\frac{1}{\hbar}\frac{\partial
E_{\nu}}{\partial\phi_{1}}(\bm{\Phi}-\bm{\omega}t)+\omega_{2}F_{\nu}(\bm{\Phi}-\bm{\omega}t)\right)$
(65)

with the energy function
$E_{\nu}(\bm{\Phi})=\bra{\psi_{\nu}(\bm{\Phi})}H(\bm{\Phi})\ket{\psi_{\nu}(\bm{\Phi})}$
and the curvature $F_{\nu}$ (14). This expression identifies with the time
evolution obtained within a classical-quantum description [14, 23] weighted by
the normalized phase wavepacket density
$|\chi_{\nu}(\bm{\Phi})|^{2}/W_{\nu}(\Psi)$ of the adiabatic component. The
quantum evolution is recovered from an average of classical trajectories, this
is due to the linearity in $\hat{\bm{N}}$ of the Hamiltonian such that in
phase representation the evolution of the wavepacket (61) is non-dispersive.

##  Appendix E Gaussian phase states

The modes are prepared in a Gaussian state
$\innerproduct{n_{i}}{\chi_{i}}\propto\exp(-(n_{i}-n_{i}^{0})^{2}/(2\Delta
n_{i})^{2}+i\phi_{i}^{0}n_{i})$ centered on $(n_{i}^{0},\phi_{i}^{0})$, and of
width $\Delta n_{i}$. In the case of a quantum harmonic oscillator for which
$\hat{n}_{i}=\hat{a}^{\dagger}\hat{a}$, a coherent state $\ket{\alpha}$ with
$\alpha=\sqrt{n_{i}^{0}}e^{i\phi_{i}^{0}}$ with an average number of quanta
$n_{i}^{0}\gg 1$ reduces to a Gaussian state with $\Delta
n_{i}=\sqrt{n_{i}^{0}}$. By Fourier transform, the phase distribution
$|\innerproduct{\phi_{i}}{\chi_{i}}|^{2}$ of the modes is a periodic
normalized Gaussian centered on $\phi_{i}^{0}$ and of width
$\Delta\phi_{i}=1/(2\Delta n_{i})$ in each direction

$|\innerproduct{\phi_{i}}{\chi_{i}}|^{2}=f(\phi_{i}-\phi_{i}^{0},\Delta\phi)\exp\left(-\frac{(\phi_{i}-\phi_{i}^{0})^{2}}{2(\Delta\phi_{i})^{2}}\right)$
(66)

where $f(\phi_{i}-\phi_{i}^{0},\Delta\phi)$ is a function ensuring
normalization and periodicity. Explicitly
$f(\phi_{i}-\phi_{i}^{0},\Delta\phi)=2(\Delta
n_{i})^{2}\frac{\left|\vartheta_{3}\left(-2{\rm i}\pi(\Delta
n_{i})^{2}(\phi_{i}-\phi_{i}^{0}),~{}e^{-4\pi^{2}(\Delta
n_{i})^{2}}\right)\right|^{2}}{\vartheta_{3}\left(0,~{}e^{-\frac{1}{2(\Delta
n_{i})^{2}}}\right)}$, with $\vartheta_{3}$ the third Jacobi theta function.
Such phase distribution is almost uniform for $\Delta\phi\gtrsim\pi$.

##  Appendix F Numerical method

For the numerical simulation, we diagonalize the Hamiltonian in
$(n_{1},n_{2})$ representation, where
$e^{i\hat{\phi}_{i}}\ket{n_{i}}=\ket{n_{i}-1}$, with the truncation $-59\leq
n_{1}\leq 59$ and $-52\leq n_{1}\leq 52$. We keep only the positions
$(n_{1},n_{2})$ in an rectangle oriented along the directions $n_{\perp}$ and
$n_{\scriptscriptstyle E}$ (5), corresponding to

$\displaystyle|n_{\scriptscriptstyle E}|$
$\displaystyle=\frac{1}{\sqrt{\omega_{1}^{2}+\omega_{2}^{2}}}|\omega_{1}n_{1}+\omega_{2}n_{2}|\leq
30$ (67) $\displaystyle|n_{\perp}|$
$\displaystyle=\frac{1}{\sqrt{\omega_{1}^{2}+\omega_{2}^{2}}}|-\omega_{2}n_{1}+\omega_{1}n_{2}|\leq
50$ (68)

with $\omega_{2}/\omega_{1}=(1+\sqrt{5})/2$. We use open boundary conditions.

We construct numerically the adiabatic projector up to order 1 in the
adiabatic parameter $\epsilon$. The adiabatic projector $\hat{P}_{\nu}$ is
defined by an asymptotic series in the formal dimensionless parameter
$\epsilon$

$\hat{P}_{\nu}=\sum_{r=0}^{\infty}\epsilon^{r}\hat{P}_{\nu,r}=\hat{P}_{\nu,0}+\epsilon\hat{P}_{\nu,1}+\dots$
(69)

such that

$\displaystyle[\hat{H}_{\rm tot},\hat{P}_{\nu}]$ $\displaystyle=0$ (70)
$\displaystyle\hat{P}_{\nu}\hat{P}_{\nu}$ $\displaystyle=\hat{P}_{\nu}$ (71)

with

$\hat{H}_{\rm
tot}=H(\hat{\phi}_{1},\hat{\phi}_{2})+\epsilon(\omega_{1}\hat{n}_{1}+\omega_{2}\hat{n}_{2}).$
(72)

We use the half-BHZ model for the qubit (2) with the gap parameter $\Delta=2$.
The maximum on $(\phi_{1},\phi_{2})$ of the ground state energy of
$H(\phi_{1},\phi_{2})$ for these values of parameters is $E_{-,\max}^{0}=-1$,
and the minimum of excited state energy is $E_{+,\min}^{0}=1$.

At order 0, $\hat{P}_{\nu}$ is a spectral projector of the Hamiltonian
$H(\hat{\phi}_{1},\hat{\phi}_{2})$. We diagonalize numerically the Hamiltonian
at zero frequency $H(\hat{\phi}_{1},\hat{\phi}_{2})$ in the truncated Hilbert
space

$H(\hat{\phi}_{1},\hat{\phi}_{2})\ket{\Psi_{k}^{0}}=E_{k}^{0}\ket{\Psi_{k}^{0}}.$
(73)

The projector at order 0 is the projector on the states of the ground band,
i.e. on the states such that $E_{k}<E_{-,\max}^{0}$

$\displaystyle\hat{P}_{-,0}$ $\displaystyle=\sum_{\begin{subarray}{c}k\\\
E_{k}<E_{-,\max}^{0}\end{subarray}}\ket{\Psi_{k}^{0}}\bra{\Psi_{k}^{0}}$ (74)

Note that we do not take into account the edge states whose energy lie in the
gap for the construction of the projectors.

The conditions (70) and (71) translates in recursive conditions for the
different orders $\hat{P}_{\nu,r}$ of the projector

$\displaystyle\hat{P}_{\nu,r}$
$\displaystyle=\sum_{s=0}^{r}\hat{P}_{\nu,s}\hat{P}_{\nu,r-s}$ (75)
$\displaystyle[H(\hat{\bm{\Phi}}),\hat{P}_{\nu,r}]$
$\displaystyle=[\hat{P}_{\nu,r-1},\hbar\bm{\omega}\cdot\hat{\bm{N}}]$ (76)

from which we can deduce order by order the expression of $\hat{P}_{\nu,r}$ in
the basis of $\ket{\Psi_{k}^{0}}$. At order 1 we obtain

$\hat{P}_{-,1}=\sum_{\begin{subarray}{c}k,l\\\ E_{k}<E_{-,\max}^{0}\\\
E_{l}>E_{+,\min}^{0}\end{subarray}}\ket{\Psi_{k}^{0}}\frac{\bra{\Psi_{k}^{0}}\hbar\bm{\omega}\cdot\hat{\bm{N}}\ket{\Psi_{l}^{0}}}{E_{k}^{0}-E_{l}^{0}}\bra{\Psi_{l}^{0}}+\mathrm{h.c.}$
(77)

which can be constructed numerically.

##  Appendix G Difference between eigenstates and adiabatic states.

Figure 6:  Fidelity $F(\Psi_{<},\Psi_{-}^{0})$ and $F(\Psi_{<},\Psi_{-}^{1})$
between the cat component $\ket{\Psi_{<}(t)}$ and the adiabatic projections at
order 0 and order 1 $|\Psi_{-}^{0/1}(t)\rangle$. After the time of separation
$t_{\rm sep}\simeq 8$, the cat component $\ket{\Psi_{<}(t)}$ is very close to
the lowest order adiabatic approximation $\ket{\Psi_{-}^{0}(t)}$ with a
fidelity of 99.5%. The fidelity further increase with the adiabatic projection
of order 1, such that $\ket{\Psi_{<}(t)}$ identifies with $\ket{\Psi_{-}(t)}$.
Figure 7:  Effect of the difference between eigenstates and adiabatic states
on the weight of the cat. Initial state of the Fig. 3(c) with
$\theta_{q}=\pi/2$. In blue, weight $W_{-}$ of the cat computed dynamically
from the splitting (same as Fig. 3(c)). In orange, weight
$W_{-}^{0}=\innerproduct{\Psi_{-}^{0}}{\Psi_{-}^{0}}$ computed from the
spectral projector $\hat{P}_{-}^{(0)}$. In green, weight
$W_{-}^{1}=\innerproduct{\Psi_{-}^{1}}{\Psi_{-}^{1}}$ from the projector at
order 1, which almost identifies with the weight obtained dynamically.

We show that after the time of separation $t_{\rm sep}$, the cat component
which splits in the direction $n_{\perp}<n_{\perp}^{0}$ identifies with the
adiabatic component $\hat{P}_{-}\ket{\Psi(t)}$ of the total state. The
adiabatic projector $\hat{P}_{-}$ is constructed from the qubit’s adiabatic
states $\ket{\psi_{-}(\bm{\Phi})}$ (10). It is a perturbative correction of
the spectral projector $\hat{P}_{-}^{(0)}$ of the qubit Hamiltonian
$H(\hat{\bm{\Phi}})$ constructed form the eigenstates
$\ket{\psi_{-}^{0}(\bm{\Phi})}$.

We note $\hat{P}_{<}$ the projector on the states
$\ket{n_{1}}\otimes\ket{n_{2}}\otimes\ket{s}$ with
$n_{\perp}=(-\omega_{2}n_{1}+\omega_{1}n_{2})/|\bm{\omega}|<n_{\perp}^{0}$,
$s=\uparrow_{z},\downarrow_{z}$, and
$\ket{\Psi_{<}(t)}=\hat{P}_{<}\ket{\Psi(t)}$ the component of the system on
this region $n_{\perp}<n_{\perp}^{0}$. The adiabatic projector $\hat{P}_{-}$
is constructed numerically at order 0 $\hat{P}_{-}^{(0)}=\hat{P}_{-,0}$ and at
order 1 $\hat{P}_{-}^{(1)}=\hat{P}_{-,0}+\hat{P}_{-,1}$ from the expressions
(74) and (77). We note respectively
$\ket{\Psi_{-}^{0}(t)}=\hat{P}_{-}^{(0)}\ket{\Psi(t)}$ and
$\ket{\Psi_{-}^{1}(t)}=\hat{P}_{-}^{(1)}\ket{\Psi(t)}$ the adiabatic
projections respectively at order 0 and order 1 of the total state
$\ket{\Psi(t)}$.

We note
$F(\Psi_{1},\Psi_{2})=\innerproduct{\Psi_{1}}{\Psi_{2}}/(\innerproduct{\Psi_{1}}{\Psi_{1}}\innerproduct{\Psi_{2}}{\Psi_{2}})$
the fidelity between two states $\ket{\Psi_{1}}$ and $\ket{\Psi_{2}}$. The
figure 6 represents the fidelity $F(\Psi_{<},\Psi_{-}^{0})$ and
$F(\Psi_{<},\Psi_{-}^{1})$ between the cat component $\ket{\Psi_{<}(t)}$ and
the adiabatic projections at order 0 and order 1. After the time of separation
$t_{\rm sep}\simeq 8$, the cat component has a fidelity of approximately
$99.5\%$ with $\ket{\Psi_{-}^{0}(t)}$ and $99.9\%$ with
$\ket{\Psi_{-}^{1}(t)}$. The adiabatic projection at order 0 gives a very good
approximation of the cat component, corrected at higher orders to gives the
full adiabatic component $\ket{\Psi_{-}(t)}$. The slight decrease with time of
the fidelity after the time of separation is due to the successive Landau-
Zener transitions.

The difference between the adiabatic projector $\hat{P}_{-}$ and the spectral
projector $\hat{P}_{-}^{(0)}$ is visible in the weight of the cat state. We
represent on Fig. 7 the weight of the adiabatic projection computed
dynamically from the splitting $W_{-}=\innerproduct{\Psi_{<}(t_{\rm
sep})}{\Psi_{<}(t_{\rm sep})}$, and the weight
$W_{-}^{0}=\innerproduct{\Psi_{-}^{0}}{\Psi_{-}^{0}}$ and
$W_{-}^{1}=\innerproduct{\Psi_{-}^{1}}{\Psi_{-}^{1}}$ computed numerically
from the projection of the initial state respectively at order 0 and order 1.
The initial state is a Gaussian state centered on $\bm{\Phi}^{0}=(0,0)$ and
the qubit in $\varphi_{q}=0$, $\theta_{q}=\pi/2$. As discussed in Sec. II.3.2,
the corresponding average ground state $\bar{\bm{b}}_{-}^{0}$ lies on the
$z$-axis for all $\Delta\phi$ such that $W_{-}^{0}=\frac{1}{2}$. The first
order correction $W_{-}^{1}$ almost identifies with the weight obtained
dynamically $W_{-}$.

##  Appendix H Purity from initial Fock state

Figure 8:  Phase densities $|\chi_{\pm}(\bm{\Phi}-\bm{\omega}t)|^{2}$ of the
cat components $\ket{\Psi_{\pm}(t)}$ translated to densities of adiabatic
states on the Bloch sphere via the map
$\bm{\Phi}\mapsto\bm{b}_{\pm}(\bm{\Phi})$. The polarization of the qubit
$\bm{Q}_{\pm}(t)$ is the average of adiabatic states with respect to these
densities.

We illustrate the role of the phase densities in the entanglement between the
qubit and the modes in the cat components $\ket{\Psi_{\pm}(t)}$ for a quasi-
Fock initial state $\Delta\phi=\pi$, and the qubit initialized in
$\theta_{q}=\pi/2$, $\varphi_{q}=0$. As discussed in Sec. III.1.2, for an
initial quasi-Fock state the phase densities
$|\chi_{\pm}(\bm{\Phi}+\bm{\omega}t)|^{2}$ of the two components of the cat
split the torus in two complementary supports of equal weight. We represent
these densities on Fig. 8 at the time $t=t_{3}$ and $t=t_{4}$ indicated on
Fig. 4(b2). The densities on the torus translate into densities of adiabatic
states on the Bloch sphere via the map
$\bm{\Phi}\mapsto\bm{b}_{\pm}(\bm{\Phi})$.

At $t=t_{3}$, $|\chi_{-}(\bm{\Phi}+\bm{\omega}t_{3})|^{2}$ covers a smaller
part of the Bloch sphere than $|\chi_{+}(\bm{\Phi}+\bm{\omega}t_{3})|^{2}$,
such that the qubit is more entangled with the modes in
$\ket{\Psi_{+}(t_{3})}$ than in $\ket{\Psi_{-}(t_{3})}$:
$|\bm{Q}_{+}(t_{3})|<|\bm{Q}_{-}(t_{3})|$. During the dynamics, the phase
densities are translated on the torus with no dispersion, changing the
densities on the Bloch sphere and the purity of the qubit. At $t=t_{4}$, the
domains have been almost exchanged
$|\chi_{-}(\bm{\Phi}-\bm{\omega}t_{4})|^{2}\simeq|\chi_{+}(\bm{\Phi}-\bm{\omega}t_{3})|^{2}$,
such that $\bm{Q}_{-}(t_{4})\simeq-\bm{Q}_{+}(t_{3})$ and
$\bm{Q}_{+}(t_{4})\simeq-\bm{Q}_{-}(t_{3})$.

For the figures, the densities are computed from the eigenstates
$\ket{\psi_{-}^{0}(\bm{\Phi})}$.

##  Appendix I Time evolution of the quantum fluctuations

We derive the time evolution of the quantum fluctuation, or spreading, of the
modes’ number of quanta in an adiabatic component $\ket{\Psi_{\nu}(t)}$ (12)

$[\Delta_{Q}\hat{n}_{i}(t)]^{2}=\langle\hat{n}_{i}^{2}\rangle_{\Psi_{\nu}(t)}-\langle\hat{n}_{i}\rangle_{\Psi_{\nu}(t)}^{2}$
(78)

with
$\langle\hat{\mathcal{O}}\rangle_{\Psi_{\nu}(t)}=\bra{\Psi_{\nu}(t)}\hat{\mathcal{O}}\ket{\Psi_{\nu}(t)}/\innerproduct{\Psi_{\nu}(t)}{\Psi_{\nu}(t)}$,
and $i=1,2$. We show that due the linearity in $\hat{\bm{N}}$ of the
Hamiltonian we can relate these quantum fluctuations to the variance of the
classical trajectories $n_{i}(t;\bm{\Phi})$ [14, 23] which are obtained in an
hybrid classical-quantum description of the qubit-mode coupling

$\displaystyle n_{i}(t,\bm{\Phi})$ $\displaystyle=\int_{0}^{t}\differential
t^{\prime}\left(\frac{1}{\hbar}\frac{\partial
E_{\nu}}{\partial\phi_{i}}(\bm{\Phi}-\bm{\omega}t^{\prime})+\sum_{j}\omega_{j}F_{\nu,ij}(\bm{\Phi}-\bm{\omega}t^{\prime})\right).$
(79)

According to (61) the time evolution of the adiabatic component is given by

$\ket{\Psi_{\nu}(t)}=\int\differential^{2}\bm{\Phi}~{}\xi_{\nu}(t,\bm{\Phi})\ket{\bm{\Phi}-\bm{\omega}t}\otimes\ket{\psi_{\nu}(\bm{\Phi}-\bm{\omega}t)}$
(80)

with

$\xi_{\nu}(t,\bm{\Phi})=\tilde{\chi}_{\nu}(\bm{\Phi})e^{i\theta_{\nu}(t;\bm{\Phi})},$
(81)

$\tilde{\chi}_{\nu}(\bm{\Phi})=\chi_{\nu}(\bm{\Phi})/W_{\nu}$ the normalized
wave-function of the decomposition (11) of the initial state and
$\theta_{\nu}(t;\bm{\Phi})$ the phase factor (57). This form of the time
evolution is due to the linearity in $\bm{N}$ of the Hamiltonian: the phase
density is translated without dispersion and the acquired phase
$\theta_{\nu}(t;\bm{\Phi})$ is given by the energy function
$E_{\nu}(\bm{\Phi})$ (58) and the connection $A_{\nu,i}(\bm{\Phi})$ (59).

As derived in appendix D, the time evolution of the average number of quanta
reduces to a statistical average of the classical trajectories with respect to
the initial phase density $|\tilde{\chi}_{\nu}(\bm{\Phi})|^{2}$

$\displaystyle\langle\hat{n}_{i}\rangle_{\Psi_{\nu}(t)}=\langle\hat{n}_{i}\rangle_{\Psi_{\nu}(t=0)}+\int\differential^{2}\bm{\Phi}~{}|\tilde{\chi}_{\nu}(\bm{\Phi})|^{2}n_{i}(t,\bm{\Phi}).$
(82)

Using
$\bra{\phi_{i}^{\prime}}\hat{n}_{i}\ket{\phi_{i}}=-i\partial_{\phi_{i}}\delta(\phi_{i}^{\prime}-\phi_{i})$,
we get after a few lines

$\displaystyle\langle\hat{n}_{i}^{2}\rangle_{\Psi_{\nu}(t)}$
$\displaystyle=\int\differential\bm{\Phi}~{}\xi_{\nu}(t,\bm{\Phi})^{*}\left(i\frac{\partial}{\partial\phi_{i}}+A_{\nu,i}(\bm{\Phi}-\bm{\omega}t)\right)^{2}\xi_{\nu}(t,\bm{\Phi})$
$\displaystyle+\int\differential^{2}\bm{\Phi}~{}|\xi_{\nu}(t,\bm{\Phi})|^{2}g_{\nu,ii}(\bm{\Phi}-\bm{\omega}t)$
(83)

with $g_{\nu,ii}$ the quantum metric of the adiabatic states (22). Let us
comment this equation. In a single band approximation of Bloch oscillations,
we ignore the rotation of the states $\ket{\psi_{\nu}(\bm{\Phi})}$ such that
we assume that $\xi_{\nu}(t,\bm{\Phi})$ is the wavefunction of the modes
ignoring the role of the projection $\hat{P}_{\nu}$. The average value of
$\hat{n}_{i}^{2}$ is then given by (I) without the connection $A_{\nu,i}$ and
the metric $g_{\nu,ii}$. Here the first line corresponds to the average value
of the projected observable $(\hat{P}_{\nu}\hat{n}_{i}\hat{P}_{\nu})^{2}$,
where $\hat{P}_{\nu}\hat{n}_{i}\hat{P}_{\nu}$ reduces to a covariant
derivative with the connection $A_{\nu,i}$ in the representation
$\ket{\bm{\Phi}}\otimes\ket{\psi_{\nu}(\bm{\Phi})}$. The second term involving
the quantum metric originates from the difference between the observables and
the projected observables

$\hat{P}_{\nu}\hat{n}_{i}^{2}\hat{P}_{\nu}=(\hat{P}_{\nu}\hat{n}_{i}\hat{P}_{\nu})^{2}+\hat{P}_{\nu}\hat{n}_{i}(1-\hat{P}_{\nu})\hat{n}_{i}\hat{P}_{\nu},$
(84)

such that we show
$\langle\hat{P}_{\nu}\hat{n}_{i}(1-\hat{P}_{\nu})\hat{n}_{i}\hat{P}_{\nu}\rangle=\int\differential^{2}\bm{\Phi}|\xi_{\nu}(t,\bm{\Phi})|^{2}g_{\nu,ii}(\bm{\Phi}-\bm{\omega}t)$.

We express the time evolution (I) it terms of the classical trajectories
$n_{i}(t,\bm{\Phi})$ (I). Using
$A_{\nu,i}(\bm{\Phi}-\bm{\omega}t)-A_{\nu,i}(\bm{\Phi})=\int_{0}^{t}\differential
t^{\prime}\sum_{j}\omega_{j}\partial_{j}A_{\nu,j}(\bm{\Phi}-\bm{\omega}t^{\prime})$
and $F_{\nu,ij}=\partial_{i}A_{\nu,j}-\partial_{j}A_{\nu,i}$, these
trajectories can be expressed in term of the phase factor
$\theta_{\nu}(t,\bm{\Phi})$ (57)

$n_{i}(t,\bm{\Phi})=-\frac{\partial\theta_{\nu}}{\partial\phi_{i}}(\bm{\Phi};t)+A_{\nu,i}(\bm{\Phi}-\bm{\omega}t)-A_{\nu,i}(\bm{\Phi}),$
(85)

such that after developing the time evolution (81) of the wavefunction
$\xi_{\nu}(t,\bm{\Phi})$ we obtain

$\displaystyle\langle\hat{n}_{i}^{2}\rangle_{\Psi_{\nu}(t)}$
$\displaystyle=\int\differential^{2}\bm{\Phi}~{}\tilde{\chi}_{\nu}(\bm{\Phi})^{*}\left(i\frac{\partial}{\partial\phi_{i}}+A_{\nu,i}(\bm{\Phi})\right)^{2}\tilde{\chi}_{\nu}(\bm{\Phi})$
$\displaystyle~{}~{}~{}+\int\differential^{2}\bm{\Phi}~{}|\tilde{\chi}_{\nu}(\bm{\Phi})|^{2}n_{i}(t,\bm{\Phi})$
$\displaystyle~{}~{}~{}+2\int\differential^{2}\bm{\Phi}~{}J_{\nu,i}(\bm{\Phi})n_{i}(t,\bm{\Phi})$
$\displaystyle~{}~{}~{}+\int\differential^{2}\bm{\Phi}~{}|\tilde{\chi}_{\nu}(\bm{\Phi})|^{2}g_{\nu,ii}(\bm{\Phi}-\bm{\omega}t)$
(86)
$\displaystyle=\langle\hat{n}_{i}^{2}\rangle_{\Psi_{\nu}(t=0)}+\int\differential^{2}\bm{\Phi}~{}|\tilde{\chi}_{\nu}(\bm{\Phi})|^{2}n_{i}(t,\bm{\Phi})$
$\displaystyle~{}~{}~{}+2\int\differential^{2}\bm{\Phi}~{}J_{\nu,i}(\bm{\Phi})n_{i}(t,\bm{\Phi})$
$\displaystyle~{}~{}~{}+\int\differential^{2}\bm{\Phi}~{}|\tilde{\chi}_{\nu}(\bm{\Phi})|^{2}(g_{\nu,ii}(\bm{\Phi}-\bm{\omega}t)-g_{\nu,ii}(\bm{\Phi}))$
(87)

with the current density of the initial state

$\displaystyle J_{\nu,i}$
$\displaystyle=\frac{i}{2}\left(\tilde{\chi}_{\nu}^{*}\frac{\partial\tilde{\chi}_{\nu}}{\partial\phi_{i}}-\tilde{\chi}_{\nu}\frac{\partial\tilde{\chi}_{\nu}^{*}}{\partial\phi_{i}}\right)+|\tilde{\chi}_{\nu}|^{2}A_{\nu,i}$
(88)

satisfying $\langle\hat{n}_{i}\rangle_{\Psi_{\nu}(t=0)}=\int
J_{\nu,i}(\bm{\Phi})\differential\bm{\Phi}$.

As a result, the time evolution of the spreading is given by the variance (33)
of the classical trajectories and by two other terms

$\displaystyle[(\Delta_{Q}$
$\displaystyle\hat{n}_{i})(t)]^{2}=[(\Delta_{Q}\hat{n}_{i})(t=0)]^{2}+\text{Var}_{|\chi_{\nu}|^{2}}[n_{i}(t,\bm{\Phi})]$
$\displaystyle+\int\differential\bm{\Phi}~{}|\tilde{\chi}_{\nu}(\bm{\Phi})|^{2}\left(g_{\nu,ii}(\bm{\Phi}-\bm{\omega}t)-g_{\nu,ii}(\bm{\Phi})\right)$
$\displaystyle+\delta C_{i}(t).$ (89)

where $\delta C_{i}(t)$ is a bounded term taking the form of correlations
between the classical trajectories $n_{i}(t,\bm{\Phi})$ and a current density
of the initial state $J_{\nu,i}(\bm{\Phi})$

$\displaystyle\delta C_{i}(t)=$ $\displaystyle
2\int\differential^{2}\bm{\Phi}~{}J_{\nu,i}(\bm{\Phi})n_{i}(t,\bm{\Phi})$ (90)
$\displaystyle-2\langle\hat{n}_{i}\rangle_{\Psi_{\nu}(t=0)}\int\differential^{2}\bm{\Phi}~{}|\tilde{\chi}_{\nu}(\bm{\Phi})|^{2}n_{i}(t,\bm{\Phi})$
(91)

As discussed above, the classical trajectories $n_{i}(t,\bm{\Phi})$
characterize the spreading of the projected observables
$\hat{P}_{\nu}\hat{n}_{i}\hat{P}_{\nu}$, and the quantum metric relates the
spreading of the projected and non-projected observables. Concerning Bloch
oscillations and Bloch breathing, the important feature of this quantum metric
contribution is that it is small compared to the initial value
$[(\Delta_{Q}\hat{n}_{i})(t=0)]^{2}$ in the case of small $\Delta\phi$, and it
is vanishingly small at quasi-periods $T$ such that
$\bm{\Phi}-\bm{\omega}T\simeq\bm{\Phi}$. The last term $\delta C_{i}(t)$ has
the same features. It is vanishingly small at quasi-periods $T$ since
$n_{i}(T,\bm{\Phi})\simeq 0$. It is also small in the small $\Delta\phi$ limit
since it can be written as classical correlations with respect to the density
$|\tilde{\chi}_{\nu}(\bm{\Phi})|^{2}$ between the function
$\partial_{i}\alpha(\bm{\Phi})+A_{\nu,i}(\bm{\Phi})$ and $n_{i}(t,\bm{\Phi})$,
with $\alpha(\bm{\Phi})$ the complex argument of
$\tilde{\chi}_{\nu}(\bm{\Phi})$.

We thus recover the behaviors of Bloch oscillations and Bloch breathing
discuss in Sec. III.2: the spreading of the cat component remains almost
constant $(\Delta_{Q}\hat{n}_{i})(t)\simeq(\Delta_{Q}\hat{n}_{i})(t=0)$ in the
case of localization in phase $\Delta\phi\ll 1$, and it refocuses at quasi-
period $T$, $(\Delta_{Q}\hat{n}_{i})(T)\simeq(\Delta_{Q}\hat{n}_{i})(t=0)$
irrespective to the value of $\Delta\phi$.