# The cold-atom elevator:
From edge-state injection to the preparation of fractional Chern insulators

Botao Wang<EMAIL_ADDRESS>CENOLI, Université Libre de Bruxelles, CP 231,
Campus Plaine, B-1050 Brussels, Belgium  Monika Aidelsburger  Faculty of
Physics, Ludwig-Maximilians-Universität München, Schellingstr. 4, D-80799
Munich, Germany  Max-Planck-Institut für Quantenoptik, 85748 Garching, Germany
Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4,
D-80799 Munich, Germany  Jean Dalibard  Laboratoire Kastler Brossel, Collège
de France, CNRS, ENS-PSL University, Sorbonne Université, 11 Place Marcelin
Berthelot, 75005 Paris, France  André Eckardt<EMAIL_ADDRESS>Technische
Universität Berlin, Institut für Theoretische Physik, Hardenbergstr. 36, 10623
Berlin, Germany  Nathan Goldman<EMAIL_ADDRESS>CENOLI, Université
Libre de Bruxelles, CP 231, Campus Plaine, B-1050 Brussels, Belgium

###### Abstract

Optical box traps for cold atoms offer new possibilities for quantum-gas
experiments. Building on their exquisite spatial and temporal control, we
propose to engineer system-reservoir configurations using box traps, in view
of preparing and manipulating topological atomic states in optical lattices.
First, we consider the injection of particles from the reservoir to the
system: this scenario is shown to be particularly well suited to activate
energy-selective chiral edge currents, but also, to prepare fractional Chern
insulating ground states. Then, we devise a practical evaporative-cooling
scheme to effectively cool down atomic gases into topological ground states.
Our open-system approach to optical-lattice settings provides a new path for
the investigation of ultracold quantum matter, including strongly-correlated
and topological phases.

#### Introduction.

Optical box traps have been demonstrated as a powerful tool in cold-atom
experiments [1]. Boxes of different shapes and dimensionalities have been
realized for ultracold atoms or molecules [2, 3, 4, 5, 6], which led to novel
observations including the quantum Joule–Thomson effect [7] and the
recurrences of coherence in a quantum many-body system [8]. The gas
homogeneity also facilitates the probe of density-related quantities,
including the quantum depletion of atomic condensate [9], the low-energy
excitation spectrum of ultracold Fermi gases [10], and sound speed in
superfluids [11, 12, 13, 14, 15, 16, 17, 18]. Besides, box traps have been
used for state preparation, leading to the discovery of a novel breather in a
2D Bose gas [19], the deterministic preparation of a Townes soliton [20], and
the demonstration of the transition between atomic and molecular condensates
[21].

Combined with optical lattices, box potentials allow to study a well-
controlled number of atoms trapped in a few lattice sites. This exquisite
control opens up new possibilities, such as measuring the growth of
entanglement upon a quench [22], or revealing fundamental properties of the
Fermi-Hubbard model [23, 24, 25, 26, 27, 28, 29, 30, 31] and many-body
localization [32, 33, 34]. More recently, programmable box traps enabled the
generation of large homogeneous systems of more than 2000 atoms, leading to
large-scale quantum simulation of out-of-equilibrium dynamics [35, 36, 37]. In
the context of topological matter, a Laughlin-type fractional quantum Hall
(QH) state has been recently realized in a small box filled with strongly
interacting bosons [38]. Isolating 1D lattices also allowed for the
observation of the symmetry-protected Haldane phase [39] and 1D anyons [40].
Furthermore, the ability of creating optical boxes with sharp boundaries
offers an ideal framework to study topological edge modes [41, 42].

Figure 1: Sketch of the cold-atom elevator. (a) Protocol for chiral edge state
injection. Setting the reservoir energy on resonance with the system’s edge
modes, particles are continuously injected into edge states in an energy-
selective manner, and chiral edge currents appear in the system without
populating the bulk. (b) Injection protocol for state preparation. Starting
from a trivial state in the reservoirs, the latter are slowly lifted so as to
adiabatically inject particles into the system until an insulating state, e.g.
a quantum Hall (QH) state, is formed. (c) Cooling protocol for state
preparation. A proper tuning of the reservoir energy can be used to retrieve
excitations (hot atoms) from the system. Removing the particles from the
reservoir, and repeating this lift-removal process over many cycles, leads to
the preparation of the desired insulating (QH) state in the system.

Inspired by the possibility of shaping box potentials of arbitrary geometries,
combined with the ability to control these dynamically, we propose to use box
traps to partition a lattice system into different subregions, separating a
“reservoir” region from a “system” of interest. We explore how dynamically
tuning the relative energy between these two regions allows for the controlled
preparation of interesting states within the system, a scheme coined “cold-
atom elevator”. We investigate two main scenarii: (i) injection of particles
from the reservoir to the system, so as to populate edge states in an energy-
resolved manner [Fig. 1(a)], or to prepare a strongly-interacting topological
ground state in the bulk [Fig. 1(b)]; (ii) controlled removal of particles
from an excited state (e.g. a thermal metal), performed in a repeated “vacuum-
cleaner” manner, in view of cooling the system down to a topological
insulating ground-state [Fig. 1(c)]; see also Refs. [43, 44].

#### Edge-state injection.

Partitioning boxes naturally provides sharp boundaries between a system and
reservoirs, an ideal platform to realize and probe topological edge states.
Hallmark of topologically nontrivial states, chiral edge states have been
observed in photonic systems [45, 46, 47] and in cold atoms using synthetic
dimensions [48, 49, 50, 51, 52, 53, 54]. Despite various proposals [55, 56,
57, 58, 59, 60, 61, 62], the realization of real-space atomic chiral edge
modes has only been reported recently [41, 42]. We now show how our sub-box
geometry can be used to activate topological edge currents within an empty
system, in an energy-selective manner and without populating the bulk.

The general idea consists in coupling an empty lattice system (potentially
hosting QH states) to a reservoir, as sketched in Fig. 1(a). Particles are
initially prepared in the reservoir, in a state that can be chosen trivial
[63]. We then perform a sudden lift of the reservoir sub-box energy
$\epsilon_{R}$ to a proper position, such that the energy of the particles in
the reservoir becomes resonant with that of the edge mode in the system. In
this way, energy-selective edge states will be populated in the initially
empty system, allowing for the observation of chiral transport on a dark
background.

Figure 2: Edge state injection in the HH model. (a) Spectrum as a function of
the eigenstate index $j$. The blue and red dots correspond to the Hamiltonian
describing the reservoir (with $\epsilon_{R}=1$) and the system, respectively.
$E_{F}$ denotes the Fermi energy. (b) Population of HH eigenstates $\rho_{j}$
as a function of their energy $E_{j}$, at time $t=28$. (c) Edge-mode
population as a function of $\epsilon_{R}$ for different initial particle
number $N$ in the reservoir. Snapshots of spatial density distribution for (d)
$\epsilon_{R}\\!=\\!1$, (e) $\epsilon_{R}\\!=\\!2.5$, (f)
$\epsilon_{R}\\!=\\!4$ at times $t\\!=\\!7,21$. Here, a system of size
$13\times 12$ and with flux $\phi\\!=\\!\pi/2$ per plaquette is coupled to a
reservoir of size $7\times 12$. Except for (c), the number of particles is
$N\\!=\\!19$. Energy and time are in units of $J$ and $\hbar/J$, respectively.
The arrows in (d) and (f) are a guide to the eye for the chiral motion.

As a concrete example, we consider the Harper-Hofstadter (HH) model [64], a
square lattice with magnetic flux $\phi\\!=\\!2\pi\alpha$ per plaquette,
coupled to reservoirs:

$\displaystyle\hat{H}=$
$\displaystyle-{\displaystyle\sum_{\left\langle\ell\ell^{\prime}\right\rangle}}\left(J_{\ell\ell^{\prime}}e^{i\phi_{\ell\ell^{\prime}}}\hat{a}_{\ell}^{\dagger}\hat{a}_{\ell^{\prime}}+\text{h.c.}\right)+{\displaystyle\sum_{\ell}}\epsilon_{\ell}\hat{n}_{\ell},$
(1)

where $\hat{a}_{\ell}(\hat{a}^{\dagger}_{\ell})$ are the annihilation
(creation) operators on site $\ell$ and
$\hat{n}_{\ell}\\!=\\!\hat{a}^{\dagger}_{\ell}\hat{a}_{\ell}$. We consider
nearest-neighbor tunneling amplitudes $J_{\ell\ell^{\prime}}$ and Peierls
phases $\phi_{\ell\ell^{\prime}}$, and set
$\epsilon_{\ell}\\!=\\!\epsilon_{R}$ in the reservoir (zero otherwise).
Whether the reservoir is also subjected to the flux or not does not
qualitatively change our findings [63], hence, for simplicity, we suppose that
the entire system-reservoir setting is described by the HH model: we set
$J_{\ell\ell^{\prime}}\\!=\\!1$ and choose Peierls phases
$\phi_{\ell\ell^{\prime}}\\!=\\!\phi n$ (resp. $0$) for hopping along $x$
(resp. $y$), where $n$ is the lattice index along $y$.

The HH Hamiltonian is a paradigmatic model of Chern insulators (CIs): it hosts
topologically nontrivial energy bands, which are characterized by nonzero
Chern numbers [65]. Setting open boundary conditions (OBC), the model hosts
chiral edge modes within the bulk energy gaps [58, 59, 63]. We show the energy
spectrum for a system of size $13\times 12$ and a reservoir of size $7\times
12$ in Fig. 2(a); the regions of low density of states (steeper slopes),
correspond to chiral edge states. When setting the lift energy to the value
$\epsilon_{R}\\!=\\!1$, the states populated in the reservoir become resonant
with the chiral edge states located within the lowest bulk gap of the system.

Based on this observation, we show how to populate edge states in an energy-
selective manner. We start with $N\\!=\\!19$ particles in the reservoir, which
corresponds to a complete filling of its nearly-flat lowest Bloch band; the
flatness allows for energy-selective population of system states. We
investigate the quench dynamics obtained by solving the time-dependent
Schrödinger equation, using different values of $\epsilon_{R}$. For
$\epsilon_{R}\\!=\\!1$, a clockwise chiral edge current is clearly observed in
Fig. 2(d), where we plot the spatial density distribution at different times.
When the box potential is lifted to $\epsilon_{R}\\!=\\!4$, i.e. when the
reservoir is resonant with the edges states located in the upper gap, an
opposite chiral motion occurs [Fig. 2(f)]. Setting $\epsilon_{R}\\!=\\!2.5$,
the populated reservoir states are resonant with the middle Bloch band of the
system, in which case bulk states are populated in the system [Fig. 2(e)].

To quantify our edge-state injection scheme, we define the mean occupation
$\rho_{j}(t)$ of an individual single-particle eigenstate $j$ in the HH
system. As shown in Fig. 2(b), we find dominant populations in the bulk gaps
for $\epsilon_{R}=1$ (lower gap) and $\epsilon_{R}=4$ (upper gap).
Furthermore, we define the total edge-state population
$\mathcal{P}_{\text{edge}}=\sum_{j\in\text{Edge}}\rho_{j}$, where the state
index $j$ runs over all edge modes. Figure 2(c) shows the population
$\mathcal{P}_{\text{edge}}$ as a function of $\epsilon_{R}$ at time
$t\\!=\\!28$ for different particle numbers $N$. By lowering the number of
fermions in the reservoir, one observes a smaller edge-mode population. In any
case, the peak positions clearly indicate the energetic location of the edge
modes (strong signal) and bulk modes (weak signal). As a corollary, our edge-
state injection scheme can be used as a spectroscopic tool for atomic QH
systems.

#### FCI preparation based on particle injection.

A natural question concerns the possibility of using the injection scheme to
form an insulating (QH) ground state within the bulk of the system. We first
explored this scheme for a system of non-interacting fermions in view of
forming a CI, and we present our findings in [63]. Here, we demonstrate the
applicability of this scheme to realize a fractional Chern insulator (FCI): a
lattice analogue of a fractional QH state [66, 67]. Several schemes have been
proposed for realizing FCIs with cold atoms, based on the adiabatic variation
of various system parameters [68, 69, 70, 71, 72, 73, 74, 75, 76, 77]. Such a
scheme was recently implemented to form an FCI state of two strongly-
interacting bosons in a $4\times 4$ lattice [38]. We now show that an open-
system approach, based on dynamically tuning box potentials, offers an
alternative, potentially simpler and better, approach to prepare an FCI ground
state with hard-core bosons.

Figure 3: Preparing a fractional Chern insulator based on injection. (a)
Spatial density distribution of the initial state with $\epsilon_{R}=-3$. (b)
Density distribution of the prepared state using the ramp shown in the inset
of panel (d) and $\tau=160$. (c) Bulk density and the local Středa marker as a
function of $\epsilon_{R}$. The shadow indicates the FCI regime. (d) Many-body
energy gap as a function of $\epsilon_{R}$. Inset: the ramping protocol of
$\epsilon_{R}(t)$ from $-3$ to $-2$ within time $\tau$. (e) Bulk density as a
function of $\epsilon_{R}$ for different $\tau$ and for the instantaneous
ground state. (f) Local marker as a function of $\tau$. Inset: linear fit of
the density versus flux at $\tau=160$, which gives
$C_{\text{str}}\\!=\\!0.51$. Here, we consider $N\\!=\\!12$ hard-core bosons;
the system of size $4\times 4$ is coupled to two reservoirs of size $4\times
4$, with $J_{R}\\!=\\!0.15$. The error bars denote the standard error of the
regression slope used to extract $C_{\text{str}}$ in Eq. (2).

We consider the sub-box configuration depicted in Fig. 1(b): the system is
connected to two reservoirs (without flux). The initial state is an easily-
prepared trivial state with all (interacting) particles in the reservoirs. The
system region, which is initially empty, is described by the Hofstadter-Bose-
Hubbard model with hard-core interactions, which is known to host a $\nu=1/2$
Laughlin-type ground state [78, 79, 80, 81, 82, 83]. We aim at gently
injecting particles from the reservoirs to the system, by slowly lifting the
reservoirs energy, in view of building up an FCI ground state in the system.
Here, we set the hopping $J_{\ell\ell^{\prime}}\\!=\\!J_{R}\\!<\\!1$ within
the reservoirs and the connecting interface, to limit excitations during
preparation.

We first analyze the (static) ground-state properties of our system-reservoir
setup, as a function of the reservoirs’ energy $\epsilon_{R}$. Figure 3(c)
shows the bulk density $n_{\text{B}}$, as evaluated within the central
$2\times 2$ sites. The incompressible nature of the FCI state clearly
manifests as a plateau in the bulk density. In contrast to more conventional
closed-system schemes, the present system automatically chooses the ideal
number of bosons to form the FCI state (for the given flux value and number of
lattice sites). The density reaches $n_{\text{B}}\\!\approx\\!0.18$ on the
plateau, and we verified that it converges towards the thermodynamic
prediction $n_{\text{B}}\\!=\\!1/8$ for increasing system sizes [63]. As
another hallmark signature of the FCI, we evaluate the fractionally-quantized
Hall conductivity $\sigma_{\text{H}}$, which is encoded in the density
distribution via Středa’s formula [84, 85, 86, 87, 83],

$C_{\text{str}}=\frac{\partial
n_{\text{B}}}{\partial\alpha}=\frac{\sigma_{\text{H}}}{\sigma_{0}},$ (2)

where $\sigma_{0}\\!=\\!1/2\pi$ is the conductivity quantum. For a
$\nu\\!=\\!1/2$ Laughlin state, the Středa marker is expected to take the
value $C_{\text{str}}\\!=\\!1/2$, which is the many-body Chern number of the
state. In our case, we find $C_{\text{str}}\\!\simeq\\!0.46$ at
$\epsilon_{R}\\!=\\!-2$, hence indicating the precursor of a fractional Hall
response [Fig. 3(c)]. It is interesting to compare this result to the value
$C^{\prime}_{\text{str}}=0.61$, which is obtained in the experimental closed-
box configuration of Ref. [38], where an exact number of bosons ($N=2$) is
loaded in $4\times 4$ sites; this comparison supports the idea that the system
optimizes the formation of an FCI state when coupled to reservoirs.

The bulk density and the Středa marker both show an interesting behavior
across the transition that occurs within the system, as particles enter the
system and eventually form the FCI state. Indeed, in the vicinity of
$\epsilon_{R}\\!\approx\\!-2.5$, one notices an abrupt increase in
$n_{\text{B}}$ and a breakdown of Středa’s formula [Fig. 3(c)], accompanied
with a sudden drop in the many-body gap [Fig. 3(d)]. The minimal many-body gap
associated with this transition is $\Delta\\!=\\!0.016$, which suggests a
realistic ramping time $\tau\\!\sim\\!100$ for adiabatic state preparation,
compatible with recent experiments [38].

We now analyze how an FCI ground state can be dynamically prepared by slowly
ramping up the reservoir energy to the ideal value
$\epsilon_{R}\\!\approx\\!-2$. To optimize adiabatic preparation, we adjust
the ramp according to the many-body gap; see the inset in Fig. 3(d). By
tracking the bulk density during the ramp [Fig. 3(e)], one recovers the
formation of a plateau for a sufficiently long ramping time $\tau$, in
agreement with the adiabatic-limit prediction of Fig. 3(c). As further
confirmed by the local Středa marker, an FCI ground state with
$C_{\text{str}}\\!\approx\\!0.5$ is prepared for adiabatic times $\tau\gtrsim
140$ [Fig. 3(f)]. The comparison with a less efficient, linear ramp is
presented in Ref. [63].

#### State preparation via repeated cleaning.

So far, we have discussed a protocol by which particles are injected from a
reservoir into an empty system. Motivated by the ability of easily preparing
an empty reservoir (a trivial zero-entropy state), we now explore the
possibility of using the reservoir as a vacuum-cleaning resource in view of
preparing ground states in the system. As sketched in Fig. 1(c), one considers
a ‘dirty’ (excited) initial state within our system. The cleaning cycle is
then as follows: (i) we slowly lower the reservoir energy, such as to retrieve
excitations (hot atoms) from the system in a controlled manner; (ii) after
this cleaning process, one rapidly lifts the reservoir until it becomes
decoupled from the system; and (iii) one completely empties the reservoir.
This cleaning cycle is then repeated $n_{\text{cyc}}$ times, until convergence
is reached towards a target insulating (QH) state within the box. The
advantage of this scheme is two-fold: the empty reservoir state can be viewed
as a perfect and easy-to-prepare zero-temperature state of holes; the
difficulty in removing particles that are located deep in the bulk is
compensated by several repetitions.

Figure 4: Preparing a Chern insulator by using the lift-removal strategy. (a)
Energy spectrum of the HH model coupled to a trivial reservoir. Here, we
partition a lattice of size $20\times 20$ into a target $12\times 12$ system
(central box) and a surrounding reservoir. $E_{F}$ denotes the Fermi energy.
We set a flux $\phi\\!=\\!\pi/2$ in the system only. Inset: Simplified three-
level model; $\left\langle E_{0}\right\rangle$ and $\left\langle
E_{1}\right\rangle$ denote representative energies of the lowest two bands.
(b) Population in HH eigenstates $\rho_{j}$ for different $n_{\text{cyc}}$.
Inset: the HH orbital entropy versus $n_{\text{cyc}}$. We set
$J_{R}\\!=\\!0.15$ and $\tau\\!=\\!90$ per cycle. (c) The local Středa marker
of the evolved state at $n_{\text{cyc}}\\!=\\!16$, as a function of the site
index along the middle row. Inset: the corresponding spatial density
distribution of the evolved state. (d) The local marker as a function of
$\tau$ for different cycles $n_{\text{cyc}}$. The local marker is averaged
over a disk at the center with a radius of $r\\!=\\!2$. The error bars denote
the standard error of the regression slope used to extract $C_{\text{str}}$.

We apply this scheme to a concrete preparation sequence, designed to prepare
Chern insulators in atomic HH systems. Inspired by Ref. [88], we start from a
trivial metal realized by loading non-interacting fermions in a square lattice
at half-filling in the presence of a staggered potential [63]. We then ramp up
the flux in the lattice to the value $\phi\\!=\\!\pi/2$, while reducing the
staggered potential, hence changing the topological nature of the bands: at
the end of this sequence, the target lowest band has a Chern number
$C\\!=\\!1$. Due to the occupation of higher bands in the initial metallic
state, the target (lowest) band remains perfectly filled during the whole
duration of this sequence, despite the gap closing
($C\\!=\\!0\\!\rightarrow\\!1$). The (irregular) band populations, obtained at
the end of this sequence, are shown by blue dots in Fig. 4(b).

Our aim is to remove atoms from higher bands, while leaving the lowest Chern
band ($C\\!=\\!1$) almost perfectly filled, in view of forming a CI in the
system. To achieve this goal, we now apply our vacuum-cleaning protocol by
dynamically tuning the reservoir energy $\epsilon_{R}$. During each cycle of
duration $\tau$, we vary $\epsilon_{R}(t)$ with a saturation function [63],
using a large initial value $\epsilon_{R}^{i}=4$. At the end of this first
cycle, $\epsilon_{R}(t)$ reaches the value $\epsilon_{R}^{f}\\!=\\!-1.14$,
which is located right below the first excited Bloch band of the system. After
lowering the reservoir to the final value $\epsilon_{R}^{f}$, we then quickly
lift it up until it becomes effectively decoupled from the system; we then
empty the reservoir and complete one cycle. We then repeat this cleaning
sequence, but for the sake of efficiency, we progressively increase the final
value $\epsilon_{R}^{f}$ at each cycle to properly address all the higher
bands [63]. We note that this cleaning scheme can be understood through a
simplified 3-level toy model [inset of Fig. 4(a)], which can serve as a guide
in view of optimizing the control parameters [63].

Figure 4(b) demonstrates the efficient depletion of the excited bands (and the
resulting decrease of entropy) as a function of the cycle number
$n_{\text{cyc}}$. In this process, the bulk states of the lowest band remain
almost perfectly filled, and we find that a satisfactory CI ground state is
formed after 16 cycles of duration $\tau\\!=\\!90$. We plot the local (single-
site) Středa marker in Fig. 4(c), which confirms the topological nature of the
bulk, $C_{\text{str}}\\!\approx\\!1$. We plot $C_{\text{str}}$ as a function
of the ramping time per cycle for different $n_{\text{cyc}}$ in Fig. 4(d).
This shows that an efficient cleaning is reached for
$n_{\text{cyc}}\\!\approx\\!12$ cycles of duration $\tau\\!\gtrsim\\!200$, and
for $n_{\text{cyc}}\\!\approx\\!16$ cycles of duration $\tau\\!\gtrsim\\!50$.

#### Concluding remarks.

This work explored different possibilities offered by the design of tunable
boxes in cold-atom experiments, setting the focus on the realization of
topological states. This approach offers substantial advantages: it relies on
the preparation of a simple initial state in the reservoir and the ability to
dynamically tune the latter’s energy relative to the system region. In this
sense, our open-system approach does not require any fine-tuning nor
complicated time-dependence of the system parameters, and it is readily
applicable to create a broad class of (bosonic or fermionic) many-body states
of interest, including exotic Mott insulators and antiferromagnetic states in
Hubbard-type models. While we considered a spatial separation between the
system and reservoir regions on the 2D plane, we note that a double-layer
configuration could also be envisaged to further enhance the transfer of
particles between the two regions; this could be realized using a bilayer
optical lattice or by exploiting two laser-coupled internal states of an atom.
Finally, it would be interesting to combine the injection and cleaning schemes
presented in this work, in view of realizing large FCI states or to explore
quantum thermodynamics.

_Acknowledgments._ The authors thank Julian Leonard, Yanfei Li, Nir Navon,
Cecile Repellin, Raphael Saint-Jalm, Perrin Segura, Boye Sun, Amit Vashisht
and Christof Weitenberg for discussions. J.D. acknowledges the support of the
Solvay Institutes, within the framework of the Jacques Solvay International
Chairs in Physics. Work in Brussels is also supported by the FRS-FNRS
(Belgium), the ERC Starting Grants TopoCold and LATIS, and the EOS project
CHEQS. M.A. and A.E. acknowledge support from the Deutsche
Forschungsgemeinschaft (DFG) via the Research Unit FOR 2414 under Project No.
277974659. M.A. also acknowledges funding from the DFG under Germany’s
Excellence Strategy – EXC-2111 – 390814868.

## References

  * Navon  _et al._ [2021] N. Navon, R. P. Smith, and Z. Hadzibabic, Quantum gases in optical boxes, Nature Physics 17, 1334 (2021).
  * Gaunt  _et al._ [2013] A. L. Gaunt, T. F. Schmidutz, I. Gotlibovych, R. P. Smith, and Z. Hadzibabic, Bose-einstein condensation of atoms in a uniform potential, Phys. Rev. Lett. 110, 200406 (2013).
  * Chomaz  _et al._ [2015] L. Chomaz, L. Corman, T. Bienaimé, R. Desbuquois, C. Weitenberg, S. Nascimbene, J. Beugnon, and J. Dalibard, Emergence of coherence via transverse condensation in a uniform quasi-two-dimensional bose gas, Nature communications 6, 6162 (2015).
  * Mukherjee  _et al._ [2017] B. Mukherjee, Z. Yan, P. B. Patel, Z. Hadzibabic, T. Yefsah, J. Struck, and M. W. Zwierlein, Homogeneous atomic fermi gases, Phys. Rev. Lett. 118, 123401 (2017).
  * Hueck  _et al._ [2018] K. Hueck, N. Luick, L. Sobirey, J. Siegl, T. Lompe, and H. Moritz, Two-dimensional homogeneous fermi gases, Phys. Rev. Lett. 120, 060402 (2018).
  * Bause  _et al._ [2021] R. Bause, A. Schindewolf, R. Tao, M. Duda, X.-Y. Chen, G. Quéméner, T. Karman, A. Christianen, I. Bloch, and X.-Y. Luo, Collisions of ultracold molecules in bright and dark optical dipole traps, Phys. Rev. Res. 3, 033013 (2021).
  * Schmidutz  _et al._ [2014] T. F. Schmidutz, I. Gotlibovych, A. L. Gaunt, R. P. Smith, N. Navon, and Z. Hadzibabic, Quantum joule-thomson effect in a saturated homogeneous bose gas, Phys. Rev. Lett. 112, 040403 (2014).
  * Rauer  _et al._ [2018] B. Rauer, S. Erne, T. Schweigler, F. Cataldini, M. Tajik, and J. Schmiedmayer, Recurrences in an isolated quantum many-body system, Science 360, 307 (2018).
  * Lopes  _et al._ [2017] R. Lopes, C. Eigen, N. Navon, D. Clément, R. P. Smith, and Z. Hadzibabic, Quantum depletion of a homogeneous bose-einstein condensate, Phys. Rev. Lett. 119, 190404 (2017).
  * Biss  _et al._ [2022] H. Biss, L. Sobirey, N. Luick, M. Bohlen, J. J. Kinnunen, G. M. Bruun, T. Lompe, and H. Moritz, Excitation spectrum and superfluid gap of an ultracold fermi gas, Phys. Rev. Lett. 128, 100401 (2022).
  * Navon  _et al._ [2016] N. Navon, A. L. Gaunt, R. P. Smith, and Z. Hadzibabic, Emergence of a turbulent cascade in a quantum gas, Nature 539, 72 (2016).
  * Ville  _et al._ [2018] J. L. Ville, R. Saint-Jalm, E. Le Cerf, M. Aidelsburger, S. Nascimbène, J. Dalibard, and J. Beugnon, Sound propagation in a uniform superfluid two-dimensional bose gas, Phys. Rev. Lett. 121, 145301 (2018).
  * Baird  _et al._ [2019] L. Baird, X. Wang, S. Roof, and J. E. Thomas, Measuring the hydrodynamic linear response of a unitary fermi gas, Phys. Rev. Lett. 123, 160402 (2019).
  * Patel  _et al._ [2020] P. B. Patel, Z. Yan, B. Mukherjee, R. J. Fletcher, J. Struck, and M. W. Zwierlein, Universal sound diffusion in a strongly interacting fermi gas, Science 370, 1222 (2020).
  * Bohlen  _et al._ [2020] M. Bohlen, L. Sobirey, N. Luick, H. Biss, T. Enss, T. Lompe, and H. Moritz, Sound propagation and quantum-limited damping in a two-dimensional fermi gas, Phys. Rev. Lett. 124, 240403 (2020).
  * Garratt  _et al._ [2019] S. J. Garratt, C. Eigen, J. Zhang, P. Turzák, R. Lopes, R. P. Smith, Z. Hadzibabic, and N. Navon, From single-particle excitations to sound waves in a box-trapped atomic bose-einstein condensate, Phys. Rev. A 99, 021601 (2019).
  * Christodoulou  _et al._ [2021] P. Christodoulou, M. Gałka, N. Dogra, R. Lopes, J. Schmitt, and Z. Hadzibabic, Observation of first and second sound in a bkt superfluid, Nature 594, 191 (2021).
  * Zhang  _et al._ [2021a] J. Zhang, C. Eigen, W. Zheng, J. A. P. Glidden, T. A. Hilker, S. J. Garratt, R. Lopes, N. R. Cooper, Z. Hadzibabic, and N. Navon, Many-body decay of the gapped lowest excitation of a bose-einstein condensate, Phys. Rev. Lett. 126, 060402 (2021a).
  * Saint-Jalm  _et al._ [2019] R. Saint-Jalm, P. C. M. Castilho, E. Le Cerf, B. Bakkali-Hassani, J.-L. Ville, S. Nascimbene, J. Beugnon, and J. Dalibard, Dynamical symmetry and breathers in a two-dimensional bose gas, Phys. Rev. X 9, 021035 (2019).
  * Bakkali-Hassani  _et al._ [2021] B. Bakkali-Hassani, C. Maury, Y.-Q. Zou, E. Le Cerf, R. Saint-Jalm, P. C. M. Castilho, S. Nascimbene, J. Dalibard, and J. Beugnon, Realization of a townes soliton in a two-component planar bose gas, Phys. Rev. Lett. 127, 023603 (2021).
  * Zhang  _et al._ [2021b] Z. Zhang, L. Chen, K.-X. Yao, and C. Chin, Transition from an atomic to a molecular bose–einstein condensate, Nature 592, 708 (2021b).
  * Islam  _et al._ [2015] R. Islam, R. Ma, P. M. Preiss, M. E. Tai, A. Lukin, M. Rispoli, and M. Greiner, Measuring entanglement entropy in a quantum many-body system, Nature 528, 77 (2015).
  * Parsons  _et al._ [2016] M. F. Parsons, A. Mazurenko, C. S. Chiu, G. Ji, D. Greif, and M. Greiner, Site-resolved measurement of the spin-correlation function in the fermi-hubbard model, Science 353, 1253 (2016).
  * Mazurenko  _et al._ [2017] A. Mazurenko, C. S. Chiu, G. Ji, M. F. Parsons, M. Kanász-Nagy, R. Schmidt, F. Grusdt, E. Demler, D. Greif, and M. Greiner, A cold-atom fermi–hubbard antiferromagnet, Nature 545, 462 (2017).
  * Chiu  _et al._ [2018] C. S. Chiu, G. Ji, A. Mazurenko, D. Greif, and M. Greiner, Quantum state engineering of a hubbard system with ultracold fermions, Phys. Rev. Lett. 120, 243201 (2018).
  * Chiu  _et al._ [2019] C. S. Chiu, G. Ji, A. Bohrdt, M. Xu, M. Knap, E. Demler, F. Grusdt, M. Greiner, and D. Greif, String patterns in the doped hubbard model, Science 365, 251 (2019).
  * Koepsell  _et al._ [2019] J. Koepsell, J. Vijayan, P. Sompet, F. Grusdt, T. A. Hilker, E. Demler, G. Salomon, I. Bloch, and C. Gross, Imaging magnetic polarons in the doped fermi–hubbard model, Nature 572, 358 (2019).
  * Vijayan  _et al._ [2020] J. Vijayan, P. Sompet, G. Salomon, J. Koepsell, S. Hirthe, A. Bohrdt, F. Grusdt, I. Bloch, and C. Gross, Time-resolved observation of spin-charge deconfinement in fermionic hubbard chains, Science 367, 186 (2020).
  * Gall  _et al._ [2021] M. Gall, N. Wurz, J. Samland, C. F. Chan, and M. Köhl, Competing magnetic orders in a bilayer hubbard model with ultracold atoms, Nature 589, 40 (2021).
  * Hirthe  _et al._ [2023] S. Hirthe, T. Chalopin, D. Bourgund, P. Bojović, A. Bohrdt, E. Demler, F. Grusdt, I. Bloch, and T. A. Hilker, Magnetically mediated hole pairing in fermionic ladders of ultracold atoms, Nature 613, 463 (2023).
  * Xu  _et al._ [2022] M. Xu, L. H. Kendrick, A. Kale, Y. Gang, G. Ji, R. T. Scalettar, M. Lebrat, and M. Greiner, Doping a frustrated fermi-hubbard magnet, arXiv:2212.13983 (2022).
  * Lukin  _et al._ [2019] A. Lukin, M. Rispoli, R. Schittko, M. E. Tai, A. M. Kaufman, S. Choi, V. Khemani, J. Leonard, and M. Greiner, Probing entanglement in a many-body-localized system, Science 364, 256 (2019).
  * Rispoli  _et al._ [2019] M. Rispoli, A. Lukin, R. Schittko, S. Kim, M. E. Tai, J. Léonard, and M. Greiner, Quantum critical behaviour at the many-body localization transition, Nature 573, 385 (2019).
  * Léonard  _et al._ [2023] J. Léonard, S. Kim, M. Rispoli, A. Lukin, R. Schittko, J. Kwan, E. Demler, D. Sels, and M. Greiner, Probing the onset of quantum avalanches in a many-body localized system, Nature Physics 19, 481 (2023).
  * Wei  _et al._ [2022] D. Wei, A. Rubio-Abadal, B. Ye, F. Machado, J. Kemp, K. Srakaew, S. Hollerith, J. Rui, S. Gopalakrishnan, N. Y. Yao, _et al._ , Quantum gas microscopy of kardar-parisi-zhang superdiffusion, Science 376, 716 (2022).
  * Impertro  _et al._ [2022] A. Impertro, J. F. Wienand, S. Häfele, H. von Raven, S. Hubele, T. Klostermann, C. R. Cabrera, I. Bloch, and M. Aidelsburger, An unsupervised deep learning algorithm for single-site reconstruction in quantum gas microscopes, arXiv preprint arXiv:2212.11974 (2022).
  * Wienand  _et al._ [2023] J. F. Wienand, S. Karch, A. Impertro, C. Schweizer, E. McCulloch, R. Vasseur, S. Gopalakrishnan, M. Aidelsburger, and I. Bloch, Emergence of fluctuating hydrodynamics in chaotic quantum systems, arXiv preprint arXiv:2306.11457 (2023).
  * Léonard  _et al._ [2022] J. Léonard, S. Kim, J. Kwan, P. Segura, F. Grusdt, C. Repellin, N. Goldman, and M. Greiner, Realization of a fractional quantum hall state with ultracold atoms, arXiv:2210.10919 (2022).
  * Sompet  _et al._ [2022] P. Sompet, S. Hirthe, D. Bourgund, T. Chalopin, J. Bibo, J. Koepsell, P. Bojović, R. Verresen, F. Pollmann, G. Salomon, _et al._ , Realizing the symmetry-protected haldane phase in fermi–hubbard ladders, Nature 606, 484 (2022).
  * Kwan  _et al._ [2023] J. Kwan, P. Segura, Y. Li, S. Kim, A. V. Gorshkov, A. Eckardt, B. Bakkali-Hassani, and M. Greiner, Realization of 1d anyons with arbitrary statistical phase, arXiv:2306.01737 (2023).
  * Braun  _et al._ [2023] C. Braun, R. Saint-Jalm, A. Hesse, J. Arceri, I. Bloch, and M. Aidelsburger, Real-space detection and manipulation of topological edge modes with ultracold atoms, arXiv:2304.01980 (2023).
  * Yao  _et al._ [2023] R. Yao, S. Chi, B. Mukherjee, A. Shaffer, M. Zwierlein, and R. J. Fletcher, Observation of chiral edge transport in a rapidly-rotating quantum gas, arXiv:2304.10468 (2023).
  * Bernier  _et al._ [2009] J.-S. Bernier, C. Kollath, A. Georges, L. De Leo, F. Gerbier, C. Salomon, and M. Köhl, Cooling fermionic atoms in optical lattices by shaping the confinement, Phys. Rev. A 79, 061601 (2009).
  * Ho and Zhou [2009] T.-L. Ho and Q. Zhou, Squeezing out the entropy of fermions in optical lattices, Proceedings of the National Academy of Sciences 106, 6916 (2009).
  * Rechtsman  _et al._ [2013] M. C. Rechtsman, J. M. Zeuner, Y. Plotnik, Y. Lumer, D. Podolsky, F. Dreisow, S. Nolte, M. Segev, and A. Szameit, Photonic floquet topological insulators, Nature 496, 196 (2013).
  * Hafezi  _et al._ [2013] M. Hafezi, S. Mittal, J. Fan, A. Migdall, and J. Taylor, Imaging topological edge states in silicon photonics, Nature Photonics 7, 1001 (2013).
  * Ozawa  _et al._ [2019] T. Ozawa, H. M. Price, A. Amo, N. Goldman, M. Hafezi, L. Lu, M. C. Rechtsman, D. Schuster, J. Simon, O. Zilberberg, and I. Carusotto, Topological photonics, Rev. Mod. Phys. 91, 015006 (2019).
  * Mancini  _et al._ [2015] M. Mancini, G. Pagano, G. Cappellini, L. Livi, M. Rider, J. Catani, C. Sias, P. Zoller, M. Inguscio, M. Dalmonte, and L. Fallani, Observation of chiral edge states with neutral fermions in synthetic Hall ribbons, Science 349, 1510 (2015).
  * Stuhl  _et al._ [2015] B. K. Stuhl, H.-I. Lu, L. M. Aycock, D. Genkina, and I. B. Spielman, Visualizing edge states with an atomic bose gas in the quantum Hall regime, Science 349, 1514 (2015).
  * Livi  _et al._ [2016] L. F. Livi, G. Cappellini, M. Diem, L. Franchi, C. Clivati, M. Frittelli, F. Levi, D. Calonico, J. Catani, M. Inguscio, and L. Fallani, Synthetic dimensions and spin-orbit coupling with an optical clock transition, Phys. Rev. Lett. 117, 220401 (2016).
  * An  _et al._ [2017] F. A. An, E. J. Meier, and B. Gadway, Direct observation of chiral currents and magnetic reflection in atomic flux lattices, Sci. Adv. 3, 10.1126/sciadv.1602685 (2017).
  * Lustig  _et al._ [2019] E. Lustig, S. Weimann, Y. Plotnik, Y. Lumer, M. A. Bandres, A. Szameit, and M. Segev, Photonic topological insulator in synthetic dimensions, Nature 567, 356 (2019).
  * Chalopin  _et al._ [2020] T. Chalopin, T. Satoor, A. Evrard, V. Makhalov, J. Dalibard, R. Lopes, and S. Nascimbene, Probing chiral edge dynamics and bulk topology of a synthetic Hall system, Nature Physics , 1 (2020).
  * Ozawa and Price [2019] T. Ozawa and H. M. Price, Topological quantum matter in synthetic dimensions, Nature Reviews Physics 1, 349 (2019).
  * Goldman  _et al._ [2010] N. Goldman, I. Satija, P. Nikolic, A. Bermudez, M. A. Martin-Delgado, M. Lewenstein, and I. Spielman, Realistic time-reversal invariant topological insulators with neutral atoms, Physical review letters 105, 255302 (2010).
  * Stanescu  _et al._ [2010] T. D. Stanescu, V. Galitski, and S. D. Sarma, Topological states in two-dimensional optical lattices, Physical Review A 82, 013608 (2010).
  * Liu  _et al._ [2010] X.-J. Liu, X. Liu, C. Wu, and J. Sinova, Quantum anomalous hall effect with cold atoms trapped in a square lattice, Physical Review A 81, 033622 (2010).
  * Goldman  _et al._ [2012] N. Goldman, J. Beugnon, and F. Gerbier, Detecting chiral edge states in the hofstadter optical lattice, Physical review letters 108, 255303 (2012).
  * Goldman  _et al._ [2013] N. Goldman, J. Dalibard, A. Dauphin, F. Gerbier, M. Lewenstein, P. Zoller, and I. B. Spielman, Direct imaging of topological edge states in cold-atom systems, Proc. Natl. Acad. Sci. U.S.A. 110, 6736 (2013).
  * Reichl and Mueller [2014] M. D. Reichl and E. J. Mueller, Floquet edge states with ultracold atoms, Physical Review A 89, 063628 (2014).
  * Goldman  _et al._ [2016] N. Goldman, G. Jotzu, M. Messer, F. Görg, R. Desbuquois, and T. Esslinger, Creating topological interfaces and detecting chiral edge modes in a two-dimensional optical lattice, Phys. Rev. A 94, 043611 (2016).
  * Wang  _et al._ [2018] B. Wang, F. N. Ünal, and A. Eckardt, Floquet engineering of optical solenoids and quantized charge pumping along tailored paths in two-dimensional Chern insulators, Phys. Rev. Lett. 120, 243602 (2018).
  * [63] See Supplementary Materials .
  * Hofstadter [1976] D. R. Hofstadter, Energy levels and wave functions of bloch electrons in rational and irrational magnetic fields, Phys. Rev. B 14, 2239 (1976).
  * Cooper  _et al._ [2019] N. R. Cooper, J. Dalibard, and I. B. Spielman, Topological bands for ultracold atoms, Rev. Mod. Phys. 91, 015005 (2019).
  * Bergholtz and Liu [2013] E. J. Bergholtz and Z. Liu, Topological flat band models and fractional Chern insulators, International Journal of Modern Physics B 27, 1330017 (2013).
  * Parameswaran  _et al._ [2013] S. A. Parameswaran, R. Roy, and S. L. Sondhi, Fractional quantum Hall physics in topological flat bands, Comptes Rendus Physique 14, 816 (2013).
  * Popp  _et al._ [2004] M. Popp, B. Paredes, and J. I. Cirac, Adiabatic path to fractional quantum Hall states of a few bosonic atoms, Phys. Rev. A 70, 053612 (2004).
  * Yao  _et al._ [2013] N. Y. Yao, A. V. Gorshkov, C. R. Laumann, A. M. Läuchli, J. Ye, and M. D. Lukin, Realizing fractional Chern insulators in dipolar spin systems, Phys. Rev. Lett. 110, 185302 (2013).
  * Kapit  _et al._ [2014] E. Kapit, M. Hafezi, and S. H. Simon, Induced self-stabilization in fractional quantum Hall states of light, Phys. Rev. X 4, 031039 (2014).
  * Grusdt  _et al._ [2014] F. Grusdt, F. Letscher, M. Hafezi, and M. Fleischhauer, Topological growing of Laughlin states in synthetic gauge fields, Phys. Rev. Lett. 113, 155301 (2014).
  * Barkeshli  _et al._ [2015] M. Barkeshli, N. Y. Yao, and C. R. Laumann, Continuous preparation of a fractional Chern insulator, Phys. Rev. Lett. 115, 026802 (2015).
  * Repellin  _et al._ [2017] C. Repellin, T. Yefsah, and A. Sterdyniak, Creating a bosonic fractional quantum Hall state by pairing fermions, Phys. Rev. B 96, 161111 (2017).
  * Motruk and Pollmann [2017] J. Motruk and F. Pollmann, Phase transitions and adiabatic preparation of a fractional Chern insulator in a boson cold-atom model, Phys. Rev. B 96, 165107 (2017).
  * He  _et al._ [2017] Y.-C. He, F. Grusdt, A. Kaufman, M. Greiner, and A. Vishwanath, Realizing and adiabatically preparing bosonic integer and fractional quantum Hall states in optical lattices, Phys. Rev. B 96, 201103 (2017).
  * Hudomal  _et al._ [2019] A. Hudomal, N. Regnault, and I. Vasić, Bosonic fractional quantum Hall states in driven optical lattices, Phys. Rev. A 100, 053624 (2019).
  * Andrade  _et al._ [2021] B. Andrade, V. Kasper, M. Lewenstein, C. Weitenberg, and T. Graß, Preparation of the 1/2 Laughlin state with atoms in a rotating trap, Phys. Rev. A 103, 063325 (2021).
  * Sørensen  _et al._ [2005] A. S. Sørensen, E. Demler, and M. D. Lukin, Fractional quantum Hall states of atoms in optical lattices, Phys. Rev. Lett. 94, 086803 (2005).
  * Hafezi  _et al._ [2007] M. Hafezi, A. S. Sørensen, E. Demler, and M. D. Lukin, Fractional quantum Hall effect in optical lattices, Phys. Rev. A 76, 023613 (2007).
  * Gerster  _et al._ [2017] M. Gerster, M. Rizzi, P. Silvi, M. Dalmonte, and S. Montangero, Fractional quantum Hall effect in the interacting Hofstadter model via tensor networks, Phys. Rev. B 96, 195123 (2017).
  * Rosson  _et al._ [2019] P. Rosson, M. Lubasch, M. Kiffner, and D. Jaksch, Bosonic fractional quantum Hall states on a finite cylinder, Phys. Rev. A 99, 033603 (2019).
  * Wang  _et al._ [2022] B. Wang, X.-Y. Dong, and A. Eckardt, Measurable signatures of bosonic fractional Chern insulator states and their fractional excitations in a quantum-gas microscope, SciPost Phys. 12, 095 (2022).
  * Repellin  _et al._ [2020] C. Repellin, J. Léonard, and N. Goldman, Fractional Chern insulators of few bosons in a box: Hall plateaus from center-of-mass drifts and density profiles, Phys. Rev. A 102, 063316 (2020).
  * Widom [1982] A. Widom, Thermodynamic derivation of the hall effect current, Physics Letters A 90, 474 (1982).
  * Streda [1982] P. Streda, Theory of quantised hall conductivity in two dimensions, Journal of Physics C: Solid State Physics 15, L717 (1982).
  * Streda and Smrcka [1983] P. Streda and L. Smrcka, Thermodynamic derivation of the hall current and the thermopower in quantising magnetic field, Journal of Physics C: Solid State Physics 16, L895 (1983).
  * Umucalılar  _et al._ [2008] R. O. Umucalılar, H. Zhai, and M. O. Oktel, Trapped fermi gases in rotating optical lattices: Realization and detection of the topological hofstadter insulator, Phys. Rev. Lett. 100, 070402 (2008).
  * Aidelsburger  _et al._ [2015] M. Aidelsburger, M. Lohse, C. Schweizer, M. Atala, J. T. Barreiro, S. Nascimbène, N. Cooper, I. Bloch, and N. Goldman, Measuring the Chern number of Hofstadter bands with ultracold bosonic atoms, Nature Physics 11, 162 (2015).

Supplementary Material

##  S1 I. Brief reminder of the Harper-Hofstadter band structure:
Edge states and Chern numbers

In this work, we illustrate the cold-atom elevator scheme using the
paradigmatic Harper-Hofstadter (HH) model, which describes hopping of
particles on a two-dimensional square lattice with a homogenous magnetic flux
$\phi$ per plaquette. The HH Hamiltonian can be written in the Landau gauge as

$\hat{H}=-J\sum_{m,n}\left(e^{-i\phi
n}\hat{a}_{m+1,n}^{\dagger}\hat{a}_{m,n}+\hat{a}_{m,n+1}^{\dagger}\hat{a}_{m,n}\text{+h.c.}\right),$
(S1)

where the operators $\hat{a}_{m,n}$ ($\hat{a}^{\dagger}_{m,n}$) annihilate
(create) a particle on the lattice site $\ell=(m,n)$, where $m$ and $n$ denote
the site indices along the $x$ and $y$ directions, respectively. The uniform
flux per plaquette $\phi=2\pi\alpha$ is defined modulo $2\pi$. For a rational
flux $\alpha=p/q$, with $(p,q)$ prime numbers, the single-particle spectrum
splits into $q$ subbands. When represented as a function of the flux $\alpha$,
these energy bands form a fractal structure known as the Hofstadter butterfly.

The finite (uniform) flux per plaquette $\phi$ breaks time-reversal symmetry,
hence leading to non-trivial topological properties: the Hofstadter bands are
associated with a non-zero (first) Chern number. Considering the flux
$\alpha=1/4$ as an example, the HH model exhibits 4 subbands, with the middle
two bands touching at singular (Dirac) points [Fig. S1(a)]. The lowest and the
highest bands share the same Chern number $C=1$; while the middle degenerate
band is associated with the Chern number $C=-2$.

Under open boundary conditions, the model hosts chiral edge modes whose
energies are located within the bulk energy gaps; these edge states can be
identified in the red spectrum displayed in Fig. S1(b), where a steeper slope
reflects the low density of (edge) states within the bulk gaps. Considering
the flux $\alpha=1/4$ (main text), the two bulk gaps each hosts a single edge
mode. The chirality of these two edge modes are opposite: the edge mode
located in the lowest (resp. highest) bulk gap propagates in a clockwise
(resp. anti-clockwise) manner; see also Fig. 2(d) and (f) in the main text,
where these two opposite chiral motions are revealed using the elevator
scheme.

Figure S1:  Edge state injection with a trivial reservoir. (a) Energy spectrum
as a function of quasimomenta $k_{x}$ and $k_{y}$. We set the lattice constant
$a=1$. The Chern numbers of the separated bands are $C=1,-2,1$, respectively.
(b) Energy spectrum as a function of the state indices. The blue and red dots
correspond to the Hamiltonian describing the trivial reservoir (with chemical
potential $\epsilon_{R}=1.5$) and the Hofstadter system, respectively. We
consider a trivial reservoir of size $7\times 16$ which is coupled to a system
of size $16\times 16$ with flux $\phi=\pi/2$ per plaquette. A high wall (one
site wide) is placed between the reservoir and the system [blue lines in (f)],
which leaves a door of width $d$ open in the middle. (c) Population of HH
eigenstates $\rho_{j}$ as a function of their energy $E_{j}$ for different
door width $d$, at time $t=40$. (d) Number of the injected particles in the
edge as a function of the time for different door width $d$. Inset: The number
of particles in the edge as a function of $d$ at $t=100$. Snapshots of spatial
density distribution for (e) $d=16$, (f) $d=4$. The green dashed line
indicates the region used to count the number of particles at the edge, and
the blue bonds represent the high wall. Initially there are $N=10$ particles
in the reservoir, and the on-site potential in the reservoir is suddenly
lifted to the value $\epsilon_{R}\\!=\\!1.5$. Energy and time are in units of
$J$ and $\hbar/J$, respectively. The arrows in (e) and (f) are a guide to the
eye for the chiral motion.

##  S2 II. Edge-state injection using a trivial reservoir

In the main text, we have shown that chiral edge states can be populated in an
energy-selective manner, using a system-reservoir setting immersed in a
uniform magnetic flux. In this Appendix, we show that a trivial reservoir
(i.e. a reservoir without flux) can also be used to inject particles into the
edge states of a quantum Hall system.

Similarly to the sketch in Fig. 1(a) in the main text, we couple a system
described by the HH model to a reservoir without magnetic flux. Initially, the
reservoir is filled with free fermions, and the system is empty. Suddenly
lifting the reservoir on-site potential $\epsilon_{R}$, such that the energy
of the particles in the reservoir becomes resonant with an edge mode of the
system [Fig. S1(b)], is found to generate chiral edge currents in the system
[Fig. S1(e)]. Even though only a small fraction of particles are injected into
the edge states in this case, we find that applying a high wall between the
reservoir and the system, but leaving a small door open in the middle of the
wall, can improve the efficiency. In Fig. S1(f), we plot the snapshots of
density distributions of the time-evolved state using a small door opening of
width $d\\!=\\!4$, from which we observe stronger chiral edge currents
appearing in the system. We attribute the better efficiency of the door-
opening configuration to the fact that a wider range of quasi-momenta is
offered by a smaller door width, thus allowing for the population of more edge
states in the system; see Fig. S1(c) where we plot the population of HH
eigenstates for different $d$.

By monitoring the number of particles $N_{\text{edge}}$ injected in a specific
region of the edge [depicted by the dashed lines in Fig. S1(f)], we find that
tuning the door width $d$ manifests as a way of controlling the number of
particles injected into the edge states, as shown in Fig. S1(d). In the inset
of (d), we plot $N_{\text{edge}}$ as a function of the door width $d$ at time
$t=100$, and the optimal fraction of injected particles is found at a door
width of intermediate size.

##  S3 III. Chern insulator preparation using the injection scheme

Here, we investigate the possibility of preparing a CI based on injecting
particles from a reservoir into an empty system. As an example, we partition a
large box of size $12\times 12$ into a sub-box of size $8\times 8$ at the
center (the “system”) and we define the rest as a trivial reservoir (without
magnetic flux). The middle sub-box is our target system, described by the HH
model with flux $\phi=\pi/2$ per plaquette. We first analyze the ground-state
properties of such a system-reservoir setup. Figure S2(c) shows the bulk
density $n_{\text{B}}$ as a function of the reservoir potential
$\epsilon_{R}$. The incompressible nature of the CI ground state manifests as
a plateau in the bulk density. Its topological nature can be confirmed from
the local Středa’s marker shown by the red dots in Fig. S2(c), which forms a
plateau around 1 in the region with bulk density saturation, which is in
agreement with the Chern number $C=1$ of the occupied lowest band.

Figure S2:  Preparing a Chern insulator based on injection. (a) Energy spectra
of the entire setup as a function of $\epsilon_{R}$. The red line indicates
the Fermi level. Considering a box of size $12\times 12$ with a target system
being a $8\times 8$ sub-box at the center, the Fermi level corresponds to the
80th eigenvalue. (b) Energy gap (difference between the 80th and 81st
eigenvalues) for $J_{R}\\!=\\!0.2$. The minimal energy splitting is found to
be $\delta E_{M}\\!=\\!0.0013$. Inset: the minimal energy splitting $\delta
E_{M}$ as a function of the system length $L_{\text{sys}}$; here, we consider
square boxes of increasing size $L\times L$, and keep the ratio
$L/L_{\text{sys}}\\!\simeq\\!1.6$ constant. (c) The bulk density and the local
Středa marker as a function of $\epsilon_{R}$. The bulk density is evaluated
within the central $2\times 2$ sites. The shadow indicates the regime of Chern
insulator ground states in the system. (d) The bulk density as a function of
$\epsilon_{R}$ for different $\tau$. The black dashed line is the ground state
density shown in (c), which represents the adiabatic limit. Here we linearly
ramp up $\epsilon_{R}$ from $-4$ until $0$ within time $\tau$, with the
initial state being a fully filled trivial reservoir. (e) The population of
the eigenstates of the Hofstadter model. Here we consider a bigger box of size
$20\times 20$ partitioned into the target system of size $12\times 12$ and a
reservoir. We ramp up $\epsilon_{R}$ from $-4$ until $-2.5$ by using a
saturation function with time $\tau$. Inset: zoom-in of heavy depletion region
indicated by the dashed box. (f) The spatial density distribution of a typical
isolated bulk state with index $j=3$. Here, the tunneling strength is
$J_{R}=0.2$.

Based on this ground-state property, one can envisage to adiabatically prepare
a CI by slowly ramping up the reservoir potential $\epsilon_{R}$. In our
setup, such an adiabatic limit will be determined by the energy gap $\delta E$
just above the Fermi level. A typical avoided-crossing is shown in the inset
of Fig. S2(a), and the energy gap $\delta E$ as a function of $\epsilon_{R}$
is plotted in Fig. S2(b). We find that, in order to build up a CI by ramping
up $\epsilon_{R}$, one has to pass through an energy gap as small as $\delta
E=0.0013$, which complicates the preparation in practice. The bulk density is
plotted in Fig. S2(d) for different ramping times $\tau$, upon using a linear
ramping up of $\epsilon_{R}$. It shows that thousands of tunneling times
($\sim 1/\delta E$) would be required to approach the adiabatic limit for this
system size; see also the scaling shown in the inset of Fig. S2(b).

As regards the preparation of a large CI through particle injection, another
difficulty concerns the presence of isolated eigenstates deep in the bulk of
the system. Taking a box of size $20\times 20$ as an example, we partition it
into a target HH system of size $12\times 12$ and define the rest as the
reservoir. Considering an improved ramp (based on a saturation function), we
obtain the eigenstate populations $\rho_{j}$ displayed in Fig. S2(e). From the
zoom-in plot in the inset, we identify several dips in the eigenstate
populations: those dips correspond to eigenstates that are located deep in the
bulk, and which are essentially isolated from the reservoir. The spatial
density distribution of such a typical isolated eigenstate is shown in Fig.
S2(e), with the red box indicating the boundary between the target system and
the reservoir.

The existence of eigenstates located deep in the bulk, which are essentially
decoupled from the reservoir, complicate the perfect filling of the target
topological band. This issue can be solved by improving the system-reservoir
coupling (e.g. by increasing the system-reservoir interface or considering a
double-layer configuration), or by considering systems with (strong)
interactions; see main text and Section IV below.

##  S4 IV. Fractional Chern insulator preparation using the injection scheme

###  S4.1 A. Using a linear ramp

Despite the difficulties mentioned above, injecting particles into an empty
system can still be used to prepare a strongly-correlated FCI state, as
demonstrated in Fig. 3 in the main text. The reasons are twofold. First,
considering a small FCI droplet, the many-body gap across the injection
process can potentially remain (reasonably) large due to finite size effects.
Second, the strong interactions potentially prevent the isolation of bulk
states, as previously illustrated in Figs. S2(e)-(f). As a further
demonstration, here we show that a linear ramp of $\epsilon_{R}$ can also be
used to prepare a small FCI droplet.

Figure S3:  Preparing a fractional Chern insulator by using a linear ramp. The
spatial density distribution of (a) the initial state with $\epsilon_{R}=-3$,
(b) the target state with $\epsilon_{R}=-2$ and (c) the prepared state at time
$t=\tau=200$. (d) The bulk density as a function of $\epsilon_{R}$ for
different $\tau$. The black dashed line corresponds to the adiabatic limit.
(e) The local Středa marker $C_{\text{str}}(t)$ as a function of
$\epsilon_{R}(t)$ for different $\tau$. (f) The final Středa marker as a
function of $\tau$. Inset: linear fit of the density versus flux at
$\tau=180$, which gives $C_{\text{str}}\approx 0.59$. Here, we consider $N=12$
hard-core bosons, and the system of size $4\times 4$ is coupled with two
reservoirs with $J_{R}=0.15$, each of which has size $4\times 4$.The error
bars denote the standard error of a regression slope.

We start from a trivial state with 6 hard-core bosons in each reservoir, as
shown in Fig. S3(a), leaving the system empty. After linearly ramping up the
reservoir potential $\epsilon_{R}(t)$ from $-3$ until $-2$ within a duration
$\tau$, we track the bulk density, as evaluated within the central $2\times 2$
sites, as a function of $\epsilon_{R}(t)$; see Fig. S3(d). One observes that
the bulk density approaches the adiabatic-limit prediction for a sufficiently
long ramping time. The sudden increase of the bulk density indicates a
transition. As further confirmed by the local Středa marker in Fig. S3(e), the
obvious dip (breakdown) of $C_{\text{str}}$ indicates a change in the
topological properties. By using sufficiently long ramping times,
$C_{\text{str}}$ gets close to the expected value $1/2$. The final Středa
marker as function of total ramping time is plotted in Fig. S3(f), with the
inset showing that a ramping time $\tau=180$ leads to the marker’s value
$C_{\text{str}}\approx 0.59$.

###  S4.2 B. Finite size effects

After demonstrating the validity of our injection scheme for the preparation
of a small FCI, we now analyze how the efficiency of this scheme scales with
the size of the reservoir, as well as with the size of the target system.
Focusing on the sub-box configuration introduced in Fig.3(a) of the main text,
we consider a target system of size $L_{S}\times L_{S}$ coupled with two
identical reservoirs of size $L_{R}\times L_{S}$, where $L_{S}$ and $L_{R}$
denote the lengths of the system and reservoir, respectively.

We first investigate the effects of the reservoir length $L_{R}$ by fixing
$L_{S}\\!=\\!4$ and the total particle number $N\\!=\\!12$. We calculate the
many-body gap $\Delta E$ as a function of the reservoir energy $\epsilon_{R}$,
within the range $-3\leq\epsilon_{R}\leq-2$, where a transition is known to
occur (see Fig. 3 in the main text). From this, we extract the minimal many-
body gap $\Delta E_{M}$, which sets the time-scale for adiabatic state
preparation. The resulting quantity $\Delta E_{M}$ is plotted as a function of
$L_{R}$ in Fig. S4(a). We find that for $L_{R}\geq 3$, the minimal gap $\Delta
E_{M}$ linearly decreases with $L_{R}$. Besides, by fixing
$L_{S}\\!=\\!L_{R}\\!=\\!4$, we find a non-linear increase of $\Delta E_{M}$
as a function of the total particle number $N$; see Fig. S4(b). Since the
value of the minimal gap $\Delta E_{M}$ determines the time scale for a
possible adiabatic state preparation, the results presented here indicate that
it is favorable to consider a reservoir of small size with many particles.

We now analyze how the size of the system $L_{S}$ influences the efficiency of
the scheme, by fixing the reservoir length $L_{R}\\!=\\!4$ and the particle
number $N\\!=\\!12$. As shown in Fig. S4(c), the minimal gap decreases with
$L_{S}$, which signals the potential difficulty of preparing an FCI in a
substantially larger lattice system. This limitation could be circumvent by
optimizing the reservoirs (see above), or possibly, by combining the injection
scheme with the cleaning method described in the main text. Another route for
improvement could be offered by a double layered system-reservoir
configuration; in practice, this could also be realized using two different
internal states of an atom for the system and reservoir, respectively.

Last but not least, we analyze the (ideal) particle density expected within
the bulk of the system $n_{B}$ as a function of the system size $L_{S}$, for a
fixed reservoir energy $\epsilon_{R}\\!=\\!-2$; see Fig. S4(d). It is
interesting to note that the particle density $n_{B}$ already approaches its
thermodynamic-limit value $n_{B}=\nu\alpha=1/8$ (expected for a
$\nu\\!=\\!1/2$ Laughlin state) for a system size $L_{S}\gtrsim 8$.
Specifically, we find $n_{B}\simeq 0.127$ and $n_{B}\simeq 0.124$ for
$L_{S}\\!=\\!8$ and $L_{S}\\!=\\!10$, respectively.

Figure S4:  Finite size effects in the FCI-reservoir setup. (a) The minimum
gap $\Delta E_{M}$ as a function of the length of each reservoir $L_{R}$ for a
fixed total number of particles $N=12$. Inset: The many body gap $\Delta E$ as
a function of $\epsilon_{R}$ for $L_{R}$=3 (blue) and 4 (orange). (b) The
minimum gap $\Delta E_{M}$ as a function of particle number $N$ for a fixed
$L_{R}=4$. Inset: $\Delta E$ as a function of $\epsilon_{R}$ for $N$=4 (blue)
and 12 (orange). In (a,b) we fix the system length $L_{S}=4$. (c) The minimum
gap $\Delta E_{M}$ as a function of the system length $L_{S}$ for fixed
$L_{R}=4,N=12$. Inset: $\Delta E$ as a function of $\epsilon_{R}$ for
$L_{S}$=4 (blue) and 5 (orange). (d) The bulk density $n_{B}$ as a function of
$L_{S}$ at $\epsilon_{R}=-2$ for $L_{R}=4,N=12$. Inset: $n_{B}$ as a function
of $\epsilon_{R}$ for $L_{S}=4$ (blue) and $L_{S}=8$ (orange). The dotted line
indicates the expected $n=1/8$ in the thermodynamic limit. The bulk density is
averaged over a disk of radius $r=1$ at the center of the system. In (a-d), we
choose the width of the whole setup to be $W=L_{S}$ and use
$J_{R}=0.15,\alpha=1/4$. The dashed lines are the guide to the eye.

##  S5 V. State preparation via repeated cleaning

###  S5.1 A. Rabi cycles

The aim of this Appendix is to estimate optimal parameters for the vacuum-
cleaning scheme, by analyzing a simplified few-level model.

In Fig. 4(a) of the main text, we presented a sketch of a simplified 3-level
toy model for our system-reservoir setup. Our aim is to gain more intuition on
our repeated cleaning protocol by analyzing the Rabi oscillations that occur
between these levels upon coupling them. To be specific, we introduce the
Level-0, Level-1, and Level-$R$ to denote our (target) lowest Chern band, the
first excited band and the reservoir band, respectively. The corresponding
characteristic energies are represented by $E_{0}$, $E_{1}$ and $\epsilon_{R}$
in Fig. 4(a). Level-0 and Level-1 have the same coupling strength $J_{R}$ with
Level-$R$.

Starting from the situation where both the lowest and the excited bands are
fully occupied, we aim to activate a coupling that keeps the lowest band
occupation essentially unaffected, while cleaning off the population in the
excited bands as much as possible. Specifically, in terms of the 3-level toy
model, we analyze the occupation in Level-0 when the transition from Level-1
to Level-$R$ is maximal. Here, we treat the 3-level toy model as two
independent two-level systems: one describing the coupling of the reservoir to
Level-0 and the other describing the coupling of the reservoir to Level-1.

In a two-level quantum system, it is well-known that the Rabi formula
describes the transition from one level to the other. Starting from a state
initially occupying the first level (denoted by $a$), the transition
probability to the second level (denoted by $b$) reads

$P_{ab}(t)=\frac{\gamma^{2}}{\gamma^{2}+(E_{a}-E_{b})^{2}/4}\sin^{2}\left(\sqrt{\frac{(E_{a}-E_{b})^{2}}{4}+\gamma^{2}}t\right),$
(S2)

where $E_{a}$, $E_{b}$ represent the level energies and $\gamma$ is the
strength of coupling between these two levels.

We first analyze the transition from Level-1 to Level-$R$, and we establish
its optimal regime by finding the maximal value
$P_{1R}^{\text{Max}}=P_{1R}(t^{*})$ using Eq. S2; the time $t^{*}$ will denote
the optimal duration of the coupling. We plot $P_{1R}^{\text{Max}}$ for
different values of $\epsilon_{R},J_{R}$ in Fig S5(a). When $\epsilon_{R}$
becomes resonant with $E_{1}\simeq-0.6$, which is taken as the mean value of
the excited Hofstadter band, the transition probability $P_{1R}$ can take a
value as large as 1. In order to study the remaining population in the Level-0
when $P_{1R}$ is maximal, we plot $1-P_{0R}$ at time $t=t^{*}$ in Fig S5(b).

Now, to identify an optimal regime for our control parameters, we define a
figure of merit $\mathcal{P}$ as

$\mathcal{P}=1-P_{0R}+P_{1R}.$ (S3)

In the ideal case where the population in Level-1 is completely removed
($P_{1R}=1$) while Level-0 remains perfectly unaffected ($P_{0R}=0$), one has
the maximal value $\mathcal{P}=2$; vice versa one has the minimal value
$\mathcal{P}=0$. As shown in Fig. S5(c), it is preferable to set
$\epsilon_{R}$ close to resonance with the excited band $E_{1}$ and to use a
small coupling strength $J_{R}$, in order to empty the Level-1 while
protecting the Level-0.

Figure S5:  Rabi oscillations and quench dynamics. (a) The maximal transition
$P_{1R}^{\text{Max}}$, (b) the transition left in Level-0 $1-P_{0R}$ at
$t=t^{*}$, and the characteristic population $\mathcal{P}$ of the simplified
3-level toy model. (d) The excited band transition $P_{1R}^{HH}$, (e) the
remaining lowest band population $1-P_{0R}^{HH}$, and the characteristic
population $\mathcal{P}_{HH}$ of the quench dynamics of our system-reservoir
configuration at $t=100$.

To make a further step towards our cleaning protocol, we now check the above-
mentioned optimal parameter regime for a quench dynamics of our system-
reservoir setup. Considering an initial state with $1/2$-filling per site
within the HH system (the central sub-box), we investigate the quench dynamics
by suddenly setting $\epsilon_{R}$ to a given value. Similarly, we now define
the figure of merit $\mathcal{P}_{HH}$ for our HH system as,

$\mathcal{P}_{HH}=1-P_{0R}^{HH}+P_{1R}^{HH},$ (S4)

where we can express the possibility of transition from band $i$ to the
reservoir in terms of the eigenstate populations $\rho_{j}$ as

$P_{iR}^{HH}=\sum_{j=1}^{N_{i}}(1-\rho_{j})/N_{i},$ (S5)

with $i=0,1$ and $N_{i}$ being the number of bulk states belonging to band
$i$. By plotting the figure of merit $\mathcal{P}_{HH}$ for different values
of $J_{R},\epsilon_{R}$ in Fig. S5(f), one identifies an optimal parameter
regime, where $\epsilon_{R}$ is close to the mean energy value $E_{1}$ of the
excited band and where the coupling $J_{R}$ is relatively weak. In contrast,
when $\epsilon_{R}$ is set close to the mean energy of the lowest band
$E_{0}\simeq-2.7$, a small $\mathcal{P}_{HH}$ is found due to a heavy
depletion of the lowest band. When $\epsilon_{R}$ is set to a very large value
(off resonance), no transition takes place between the system and the
reservoir, and thus $\mathcal{P}_{HH}$ saturates to 1. This analysis shows a
qualitative agreement with the simplified Rabi model [Fig. S5(c)].

###  S5.2 B. Dynamical cleaning scheme and scaling

Despite the identification of an optimal parameter regime (see above), a
simple quench dynamics does not lead to a satisfactory preparation of the CI,
i.e. a complete depletion of the excited bands while leaving the lowest band
perfectly filled. This is essentially due to the finite dispersion of the HH
bands. To solve this issue, we consider tuning $\epsilon_{R}$ in a time-
dependent fashion, as we now explain.

Figure S6:  Preparing a Chern insulator based on cleaning. (a) The repeated
ramping protocol for $\epsilon_{R}$ according to a saturation function. The
red dashed line is the guide to the eye. (b) Population in HH eigenstates for
different $n_{\text{cyc}}$ with $\tau=70$ per cycle. (c) The local Středa
marker of the evolved state with $n_{\text{cyc}}=10,\tau=70$ as a function of
lattice indices along the middle row. Inset: the corresponding spatial density
distribution of the evolved state. (d) The characteristic population
$\mathcal{P}_{HH}$ as a function of $\tau$ for different cycles
$n_{\text{cyc}}$. (e) The local marker as a function of $\tau$ for different
cycles $n_{\text{cyc}}$ and $L_{S}$. The local marker is averaged over a disk
at the center with a radius of $r=2$. The error bars denote the standard error
of a regression slope. (f) The typical time $\tau^{*}$ as a function of the
system length $L_{S}$. During the scaling, we consider square boxes and keep
the length ratio between the whole set up and the system as $L/L_{S}\simeq
1.6$. For (b-e), we partition a lattice of size $20\times 20$ into a target
$12\times 12$ system (central box) and a surrounding reservoir with particle
number $N=72$. We set a flux $\phi=\pi/2$ in the system and $\phi=0$ in the
reservoir, and use $J_{R}=0.15,\xi=3$.

We choose to dynamically tune the reservoir potential according to a
saturation function,

$\epsilon_{R}(t)=\epsilon_{R}^{i}-(\epsilon_{R}^{i}-\epsilon_{R}^{f})t/(\sqrt{c\tau^{2}+t^{2}}),$
(S6)

with the initial value $\epsilon_{R}^{i}=4$, a constant parameter $c=0.02$ and
the total ramping time $\tau$. As shown in Fig. S6(a), after reaching the
final value $\epsilon_{R}^{f}$, we quickly lift $\epsilon_{R}$ to a high
value. Since the system is now almost completely decoupled from the reservoir,
one can easily remove the particles from the reservoir. The ability of
preparing such an empty reservoir makes it possible to have several cycles. To
increase the efficiency, one can design the final value of $\epsilon_{R}^{f}$
for each cycle $n_{\text{cyc}}$ as

$\epsilon_{R}^{f}(n_{\text{cyc}})=\epsilon_{R}^{f}(1)+J_{R}\xi\frac{n_{\text{cyc}}-1}{N_{\text{cyc}}-1},$
(S7)

with $n_{\text{cyc}}=1,\cdots,N_{\text{cyc}}$ and $N_{\text{cyc}}$ is the
total cycle number. The coefficient $\xi$ controls the final value of
$\epsilon_{R}$. For the very first cycle, we take $\epsilon_{R}^{f}(1)=-1.14$
which is located right below the excited band.

Considering a many-body state at $1/2$-filling as our initial state, we now
apply our repeated cleaning protocol to prepare a CI in the system: we aim at
removing the higher band populations but leaving the lowest Chern band almost
perfectly filled. As shown in Fig. S6(b), we find an efficient depletion of
the excited bands with the increase of the cycle number. During the ramp, the
bulk states of the lowest band remain almost perfectly filled. This already
points towards the realization of a CI state in the system. To further confirm
its topological nature, we plot the local Středa marker (averaged over the
surrounding 4 sites) for $n_{\text{cyc}}=10$ with $\tau=70$ per cycle in Fig.
S6(c), which shows a uniform bulk with $C_{\text{str}}\approx 1$. By plotting
the figure of merit $\mathcal{P}_{HH}$, and the Středa marker in Fig. S6(d)
and (e), respectively, we find an efficient cleaning for $\tau\gtrsim 70$ and
$n_{\text{cyc}}\approx 10$ cycles, or $\tau\gtrsim 100$ and
$n_{\text{cyc}}\approx 8$ cycles. This illustrates how using more cycles would
help reducing the total preparation time. Besides, in order to appreciate the
scaling behavior, we plot a typical time $\tau^{*}$ corresponding to a figure
of merit $\mathcal{P}_{HH}$ above 1.95 in Fig. S6(f). We find that the typical
time per cycle increases more slowly with the system size as one increases the
number of cycles.

###  S5.3 C. Full preparation protocol using the cleaning scheme

In the main text, we apply our repeated cleaning protocol to a realistic
state, which has been prepared starting from a trivial metal (i.e. starting
from a band structure with zero Chern numbers). Here we give more details
about the full ramping protocol.

We start from a trivial metal in the presence of a large staggered potential,
$\delta_{\ell}=[(-1)^{m}+(-1)^{n}]\delta/2$, with $(m,n)$ being the lattice
indices along $x$ and $y$ direction, respectively. We then linearly ramp up
the flux in the lattice from 0 to the value of $\pi/2$. Since the system is
still within a trivial regime (guaranteed by the large staggered potentials
$\delta=20$), one can ramp up the flux rather quickly. After that, we slowly
reduce the staggered potential in order to change the topological nature of
the bands; this transition generates excitations in the higher bands for a
realistic quasi-adiabatic ramp; see the sequence in Fig. S7(a). The density
distribution of the evolved state, across the topological phase transition, is
shown in Fig. S7(b)-(d); this latter situation corresponds to the blue dots in
the band populations shown in Fig. 4(b) in the main text.

With this realistic (but highly excited) state in hand, we now apply our
repeated cleaning protocol. In order to progressively address the higher bands
at each cycle, we set $\epsilon_{R}^{f}(1)=-1.14$ and $\xi=14$ for the final
value of $\epsilon_{R}^{f}$ defined in Eq. (S7); see also Fig. S7(a). In this
case, the final value of $\epsilon_{R}^{f}$ in the last cycle reads 0.94,
which is located near the top of the excited band. Note that even though we
only consider a linear increase of $\epsilon_{R}^{f}$ at each cycle (for
simplicity), a more efficient protocol could be designed by fine tuning
$\epsilon_{R}^{f}$ within each cycle, e.g. by means of optimal control or even
machine learning.

Figure S7:  Full protocol for preparing CI based on cleaning (a) The full
protocol for ramping system parameters. The flux is linearly increased from 0
to $\pi/2$ within $\tau_{1}$, and then we decrease the staggered potentials
$\delta$ from 20 to 0 within $\tau_{2}$. After that, we apply our cleaning
protocol cycle by cycle with $\tau$ being the ramping time per cycle. (b-d)
The spatial density distributions at $t=0,\tau_{1}$ and $\tau_{1}+\tau_{2}$,
respectively. Here, we use time $\tau_{1}=\tau_{2}=100$ for tuning $\phi$ and
$\delta$.