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\DOI{10.5802/crphys.243}
\datereceived{2024-09-17}
\daterevised{2025-02-12}
\dateaccepted{2025-02-24}
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\begin{document}

\begin{noXML}

\CDRsetmeta{articletype}{congress}

\title{Simulating quantum Hall physics in ultracold atomic gases:
prospects and challenges}

\alttitle{Simuler l'effet Hall quantique dans les gaz d'atomes
ultrafroids: perspectives et d\'efis}

\author{\firstname{Sylvain} \lastname{Nascimbene}\CDRorcid{0000-0002-3931-9436}}
\address{Laboratoire Kastler Brossel, Coll\`ege de France, CNRS,
ENS-PSL University, Sorbonne Universit\'e, 11 Place Marcelin Berthelot,
75005 Paris, France}
\email{sylvain.nascimbene@lkb.ens.fr}

\begin{abstract}
This proceedings presents recent advancements in the realization of
topological states of matter using ultracold atomic gases. We focus on
the simulation of the quantum Hall effect, originally discovered in 
two-dimensional electron gases subjected to a magnetic field. Given
that ultracold atoms are neutral, simulating the effect of a magnetic
field requires an ad-hoc technique. We review various methods developed
for this purpose, highlighting their respective advantages and
limitations. Furthermore, we explore the extension of these techniques
to interacting quantum gases within topological bands, aiming to create
analogs of fractional quantum Hall states. These many-body states are
distinguished by their topological order, which is characterized by
long-range entanglement among particles and the emergence of low-lying
excitations exhibiting fractional quantum statistics. The discussion
will encompass the current state of research and potential future
directions in the field.
\end{abstract}

\begin{altabstract}
Cet acte de congr\`es pr\'esente les avanc\'ees r\'ecentes dans la
r\'ealisation d'\'etats topologiques de la mati\`ere \`a l'aide de gaz
atomiques ultrafroids. Nous nous concentrons sur la simulation de
l'effet Hall quantique, d\'ecouvert dans des gaz d'\'electrons
bidimensionnels soumis \`a un champ magn\'etique. \'Etant donn\'e que
les atomes ultrafroids sont neutres, la simulation de l'effet d'un
champ magn\'etique n\'ecessite une technique ad hoc. Nous passons en
revue diverses m\'ethodes d\'evelopp\'ees \`a cette fin, en soulignant
leurs avantages et leurs limites respectifs. En outre, nous explorons
l'extension de ces techniques aux gaz quantiques en interaction dans
des bandes topologiques, dans le but de cr\'eer des analogues d'\'etats
de Hall  fractionnaires. Ces \'etats \`a \tralicstex{\textit{N}}{$N$} corps se distinguent par
leur ordre topologique, qui se caract\'erise par une  intrication \`a
longue port\'ee entre les particules et  l'\'emergence d'excitations de
basse \'energie pr\'esentant des statistiques quantiques
fractionnaires. La discussion portera sur l'\'etat actuel de la
recherche et sur les orientations futures potentielles dans ce domaine.
\end{altabstract}

\keywords{\kwd{Topological phase}\kwd{Quantum simulation}\kwd{Quantum gases}}

\altkeywords{\kwd{Phase topologique}\kwd{Simulation quantique}\kwd{Gaz quantiques}}

\thanks{European Union (grant TOPODY 756722 from the European Research
Council), Institut Universitaire de France}

\maketitle

\end{noXML}

\def\eqref#1{(\ref{#1})}

{\vspace*{1pc}}

\xsection{}

The quantum Hall effect was discovered in 1980 by von
Klitzing~\cite{klitzing_new_1980} by studying the properties of 
condensed-matter systems in which the electron gas is constrained to
evolve in a \mbox{two-dimensional} (2D) plane $xy$. At low temperature and in
the presence of a strong magnetic field $\Bbold=B\ebold_z$, he observed
a quantization of the Hall conductance according to
{\[
 I_y = \sigma _{xy}V_x, \quad \sigma _{xy} = n \sigma _0,
\]}\unskip
where $n$ is an integer, $\sigma_0 = e^2/h$ is the quantum of
conductance, with $\E$ representing the electron charge and $h$ the
Planck constant. This discovery was pivotal in the field of quantum
many-body physics and laid the groundwork for the exploration of
topological states of matter. We will discuss here the current state of
research in ultracold atomic gases towards the simulation of the
quantum Hall effect and the realization of various topological states
of matter associated with it.

\section{Topological bands and the integer quantum Hall effect}
\subsection{Landau levels}\label{section:LL}

The integer quantum Hall effect can be understood in terms of a special
topological structure of single-particle quantum eigenstates of motion.
Before delving into more complex systems, we examine the idealized
model of a charged particle evolving in 2D and subjected solely  to a
perpendicular magnetic field~\cite{landau_vol3}. Its dynamics is
determined by the Hamiltonian
{\[
H = \dfrac{(\pbold-q\Abold)^2}{2M},
\]}\unskip
where $q$ is the particle charge ($-e$ for an electron, but in the
subsequent discussion we will assume $q&gt;0$) and $M$ is its mass. The
selection of the gauge influences the inherent symmetries of the
system. In our discussion, we will primarily utilize the symmetric
gauge, defined as
{\[
\Abold = \frac{1}{2} \Bbold\times\rbold=\frac{B r}{2}\ebold_\phi.
\]}\unskip
While this choice explicitly breaks translational invariance, it
preserves rotational symmetry around the origin.

The Hamiltonian can be expanded as
{\begin{eqnarray}
H &amp;=&amp; H_0 - \frac{\omc}{2}L_z, \label{eq:HLandau} \Seqnsplit
H_0 &amp;=&amp; \frac{p_x^2+p_y^2}{2M} + \frac{1}{8}M\omc^2(x^2+y^2),
\end{eqnarray}}\unskip
where we introduce the cyclotron frequency $\omc=qB/M$ and the angular
momentum projection $L_z=xp_y-yp_x$. The Hamiltonian $H_0$ corresponds
to an isotropic harmonic oscillator in 2D, of frequency $\omc/2$. We
can exhibit an eigenbasis $\mathop{\lvert} n_0,m \mathop{\rangle}$ of
$H_0$ in common with the angular momentum $L_z$, corresponding to the
energy spectrum  $E^{0}_{n_0,m} = (n_0 + 1) ({\hbar\omc}/{2})$ where
$n_0\in\mathbb{N}$ and $m\in\{-n_0,-n_0+2,\ldots,n_0-2,n_0\}$ 
(see Figure~\ref{fig:LLL}(a)).

\begin{figure}
\includegraphics{fig01}
\caption{\label{fig:LLL}(a) Spectrum of the isotropic quantum harmonic
oscillator in two dimensions. Offseting the states by  ${-}
({\omega_{\text{c}}}/{2})L_z$ (red arrows) yields the spectrum of a
charged particle subjected to a magnetic field, corresponding to Landau
levels shown in~(b).}
\end{figure}

Together with the term $({-\omc}/{2})L_z$, we get the energy spectrum
of a charged particle in a magnetic field $ E_{n_0,m}=E^{0}_{n_0,m}-m
({\hbar\omc}/{2})$, which can be written as
{\[
E_{n,m} = (n+\tfrac{1}{2})\hbar\omc,
\]}\unskip
with $n=(n_0-m)/2\in\mathbb{N}$ and $m\geq-n$. The energy spectrum
consists of infinitely degenerate flat bands known as Landau levels, as
illustrated in Figure~\ref{fig:LLL}(b). 

In the following we will focus on properties of the ground band, the
lowest Landau level (LLL). Its eigenstates are described by the
wavefunctions
{\[
\psi_m(x,y)\propto r^m\E^{\I m\phi}\E^{-r^2/4\ell^2},
\]}\unskip
where $m\geq0$. Here, we introduced the polar coordinates $(r,\phi)$
and  the cyclotron length $ \ell=\sqrt{\hbar/qB}. $ The density
probability of $\psi_m$ is concentrated in a ring of radius
$r_m\simeq\sqrt{2m}\ell$, of width $\Delta r\simeq \ell$ independent of
$m$. The area enclosed by this ring is thread by a magnetic flux
{\[
\Phi_m=B\uppi r_m^2\simeq m \Phi_0,
\]}\unskip
where $\Phi_0=h/q$ is the magnetic flux quantum.

In the following, we will consider a finite geometry, assuming that 
the motion is restricted to a disk  $r\leq R$ by a circular hard wall.
This restriction only weakly affects the so-called bulk states $\psi_m$
with $r_m\leq R$ (i.e.\ $ m\leq\mmax=R^2/2\ell^2$) for which the density
probability is almost fully contained within the disk.  For $m\geq
\mmax$, the eigenstates become squeezed by the confinement hard wall
close to $r=R$, leading to an increase of (kinetic) energy (see
Figure~\ref{fig:edge}(a)). As a result, these states acquire a non-zero
azimuthal velocity $\langle v_\phi\rangle$, in contrast with the
dispersionless bulk modes. More precisely, we write the azimuthal
velocity component as
{\begin{eqnarray*}
v_\phi &amp;=&amp; \frac{1}{M}\left(\frac{\hbar}{\I r}\partial_\phi-q A_\phi\right)\\
&amp;=&amp; \frac{1}{M}\left(\frac{\hbar m}{r}-\frac{qB}{2} r\right)\\
&amp;=&amp; \omc\frac{r_m^2-r^2}{r}.
\end{eqnarray*}}\unskip
For bulk states, the azimuthal velocity almost vanishes since  $r\simeq
r_m$. Conversely, for edge modes the density is localized around $r=R$,
hence $v_\phi\propto(r_m^2-R^2)\propto (m-\mmax)$, corresponding to a
chiral dispersive mode~\cite{halperin_quantized_1982} (see
Figure~\ref{fig:edge}(b)).

\begin{figure}
\includegraphics{fig02}
\caption{\label{fig:edge}(a) Spectrum of a quantum Hall system
restricted to a disk for radius $R=10\,\ell$. (b)~Mean azimuthal
velocity for the quantum states of the ground energy band. }
\end{figure}

Introducing the complex coordinate $z=r\E^{\I\phi}=x+\I y$, a generic
bulk state of the Landau level can be expanded over the $\psi_m$ basis
as
{\[
\psi(z) = \sum_{m\geq0}^{\mmax} c_m z^m \E^{-r^2/4\ell^2},
\]}\unskip
which is the product of a gaussian factor in $r$ and a complex
polynomial $P(z)$.  This polynomial is fully characterized by its roots
$z_k$, as
{\[
P(z)\propto\prod_{k=1}^{\mmax}(z-z_k).
\]}\unskip
One checks that the complex phase of $P(z)$ winds once around every
non-degenerate root $z_k$, corresponding to a unit-charge quantum
vortex.

We show in Figure~\ref{fig:states} the density probability of a few
typical states of the LLL, namely  $\psi_{m=0}$,  $\psi_{m=10}$, and a
state defined by a random complex polynomial of degree 25, which could
be realized using a thermal Bose gas restricted to the 
LLL~\cite{castin_quantized_2006}. We also show for each of them the
position of the quantum vortices as red dots.

\begin{figure}
\includegraphics{fig03}
\caption{\label{fig:states}Probability density distribution of a few
states of the LLL: the state $\psi_{m=0}$ (a), the state $\psi_{m=10}$
(b), and a state defined by a random complex polynomial of degree 25
(c). Vortex point defects are shown as red dots.} 
\end{figure}

\subsection{Quantization of the Hall conductance}

The first explanation of the Hall conductance quantization was given by
Laughlin based on a particle pumping
argument~\cite{laughlin_quantized_1981}. For this, we consider a
slightly modified geometry, coined Corbino disk $\rmin\leq r\leq
\rmax$~\cite{corbino_azioni_1911,halperin_quantized_1982} (see
Figure~\ref{fig:pump}(a)). In a fermionic band insulator, the states
with 
{\[
\mmin = \frac{\rmin^2}{2\ell^2}\leq m\leq\mmax = \frac{\rmax^2}{2\ell^2}
\]}\unskip
are occupied with one particle. We consider a solenoid, placed within
the inner radius, which can introduce an additional magnetic flux
$\Phi$. By slowly increasing this  flux by one unit of flux quantum
$\Phi_0$, the adiabatic eigenstates undergo a spectral flow in which
the state $\psi_m$ (which originally encloses a magnetic flux
$m\Phi_0$)  is transferred to the next state $\psi_{m+1}$ (enclosing a
magnetic flux $(m+1)\Phi_0$). Globally, such a pump cycle transfers one
particle from the inner edge (state $\mmin$) to the outer edge (state
$\mmax$), as shown in Figure~\ref{fig:pump}(b). 

\begin{figure}
\includegraphics{fig04}
\caption{\label{fig:pump}(a) Scheme of the Corbino disk used for
Laughlin's pump experiment. (b) Single-particle spectrum of a Corbino
disk defined by the radii $\rmin=2\ell$ and $\rmax=10\ell$. Starting in
a fermionic insulator with a Fermi level
$E_{\text{F}}=\hbar\omega_{\text{c}}$, the adiabatic increase of the
solenoid flux $\Delta\Phi=\Phi_0$ transfers each state $m$ to $m+1$
(red arrows), or equivalently a state from one edge to the other.}
\end{figure}

This pump process can be interpreted in terms of a quantized Hall
conductance. Indeed, the flux variation $\dot\Phi$ gives rise to an
``induced voltage'' $V_\phi=-\dot\Phi$ along any circular path within the
Corbino disk, while the transfer of particles from the inner to outer
edges can be interpreted as a radial current 
{\[
I_r = q\frac{\dot\Phi}{\Phi_0} = -\frac{q^2}{h}V_\phi.
\]}\unskip
This relationship aligns with the quantization of the Hall conductance.

\subsection{Topological bands in lattice
systems}\label{section:lattice}

The quantization of Hall conductance is not limited to ideal Landau
levels; it can also be extended to Bloch bands in lattice systems.

We begin by examining the natural extension of Landau levels to a
discrete square lattice, known as the Hofstadter
model~\cite{harper_single_1955,hofstadter_energy_1976}. This model
features a square lattice with nearest-neighbor hopping, where the
positions of the lattice sites are defined as $
\rbold_{i,j}=i\,a\,\ebold_x+j\,a\,\ebold_y, $ where $a$ is the lattice
spacing and $i$, $j$ are integers. In this framework, the magnetic
field is incorporated through the Peierls substitution, which
introduces complex phases into the hopping amplitudes. These phases
correspond to the Aharonov--Bohm phase acquired along the path
connecting neighboring sites. To simplify the description, we adopt the
Landau gauge, where $A_x=-By$. In this gauge, the hopping terms along
the $y$-direction remain real, while the hopping terms along the
$x$-direction acquire a Peierls phase given by
{\begin{equation}
\phi_{(i,j)\rightarrow(i+1,j)} = 
\frac{q}{\hbar}\int_{\rbold_{i,j}}^{\rbold_{i+1,j}}\,\dd\rbold\, 
A_x(\rbold)=-2\uppi\frac{\Phi}{\Phi_0}j,\label{eq:Peierls}
\end{equation}}\unskip
where $\Phi=B a^2$ is the magnetic flux through a unit cell.

In the general case, the position-dependent Peierls phases disrupt
translational symmetry along the $y$-direction, making it impossible to
define $y$-momentum. However, when the ratio $\Phi/\Phi_0$ takes on
rational values $p/q$, discrete translation invariance by the distance
$qa$ along $y$ is recovered (in addition to the discrete invariance
along $x$ by the distance $a$). As a result, one can introduce a
conserved quasi-momentum $(q_x,q_y)$ defined over a magnetic Brillouin
zone $(0,2\uppi/a)\times(0,2\uppi/qa)$ (see the case $p/q=1/3$ in
Figure~\ref{fig:lattice_models}(a)). 

\begin{figure}
\includegraphics{fig05}
\caption{\label{fig:lattice_models}(a) Scheme of the Hofstadter model,
a square lattice model with complex-valued hopping matrix elements as
defined in Equation~(\ref{eq:Peierls}). For a ratio $p/q=1/3$, the lattice
exhibits three inequivalent sites and a magnetic Brillouin zone defined
by three square unit cells.  (b) Scheme of the Haldane model, a
hexagonal lattice model with real-valued nearest-neighbour hoppings and
complex-valued next-nearest neighbour hoppings.}
\end{figure}

As a result of the magnetic translation symmetries discussed above,
quantum levels are organized into a set of $q$ magnetic Bloch bands. A
given band, provided it is separated in energy from other bands, gives
rise to a quantized Hall effect.  Each band, when sufficiently
separated in energy from others, can give rise to a quantized Hall
effect. Specifically, the transverse linear response of a band
insulator manifests as a quantized Hall conductance expressed by $
\sigma_{xy}=\mathcal{C}\sigma_0, $ where $\mathcal{C}$ is an integer
known as the Chern number. This integer characterizes the topological
properties  of the Bloch band, analogous to the genus of a closed
surface in geometry. For instance, in the case where $p=1$, the lowest
Bloch band consistently exhibits a Chern number of $\mathcal{C}=1$,
resembling the behavior of a Landau level.  We do not discuss here the
concept of Chern number in more detail, since it is very well explained
in several review articles (see for
example~\cite{cooper_topological_2019} for a review in a context of
ultracold quantum gases).

The concept of the Chern number is not confined to lattice models that
can be viewed as discretized versions of continuous systems with
uniform magnetic fields. It can be defined for any isolated Bloch band
within a given lattice model. A Chern insulator emerges when fermionic
insulators fill Bloch bands that possess a non-zero Chern number, which
requires that  the configuration of Peierls phases breaks time-reversal
symmetry. A notable example of a Chern insulator is the Haldane
model~\cite{haldane_model_1988}, illustrated in
Figure~\ref{fig:lattice_models}(b).

\section{Realizing quantum Hall systems with ultracold atoms}

Since magnetic fields do not exert mechanical effects on neutral
particles, simulating quantum Hall physics relies on analogies. In this
section, we will review some of these analogies, discuss their current
limitations, and explore potential strategies to overcome these
challenges in the future.

\subsection{Rotating gases}
\subsubsection{Landau levels in rotating gases}

Historically, the first protocol to create an effective magnetic field
involves rotating gases~\cite{madison_vortex_2000}. Consider an atom
evolving in two dimensions under an isotropic harmonic trap, described
by the Hamiltonian
{\[
H = \frac{p_x^2}{2M} + \frac{p_y^2}{2M} + \frac{1}{2}
M\omega^2(x^2+y^2).
\]}\unskip
In a reference frame that rotates around the center of the harmonic
trap at a frequency $\Omega$, the Hamiltonian transforms to
{\[
H' = \frac{\pbold'^2}{2M} + \frac{1}{2}M\omega^2\rbold'^2 - \Omega L_z',
\]}\unskip
where $L_z'=x'p_y'-y'p_x'$ is the angular momentum and $(x',y')$ are
the rotated coordinates.  When the rotation frequency $\Omega$ matches
the confinement frequency $\omega$, we recover  the Landau Hamiltonian
given in Equation~(\ref{eq:HLandau}), corresponding to a particle of charge $q=1$
and  a cyclotron frequency $\omc=2\omega$ (see
Figure~\ref{fig:rotating_frame}(a)).

\begin{figure}
\includegraphics{fig06}
\caption{\label{fig:rotating_frame}(a) Schematics of the analogy
between an atom held in an isotropic harmonic trap in a rotating frame
and a charged particle subjected to a magnetic field. (b) Image of an
atomic Bose--Einstein condensate set in rotation, which exhibits a
triangular lattice of quantized vortices
(from~\cite{madison_vortices_2000}).}
\end{figure}

\subsubsection{Abrikosov vortex lattices in Bose--Einstein condensates}

Most experiments with rotating gases have focused on Bose--Einstein
condensates containing a large number of atoms $N$. These systems can
be effectively described by a classical field $\psi(x,y)$ representing
the macroscopic wavefunction shared by all atoms. As previously
discussed, a generic wavefunction in the LLL is characterized by its
zeros, which correspond to unit-charged quantum vortices.

In the absence of interactions, any configuration of vortices can be
expected due to the flat-band nature of the LLL. However, interactions
lead to a well-defined many-body ground state. In dilute gases, atoms
primarily interact at short range, resulting in a ground state that
minimizes the interaction energy~\cite{abrikosov_magnetic_1957}
{\[
E_{\text{int}}=g\frac{N(N-1)}{2}\int\,\dd r^2\,|\psi(r)|^4,
\]}\unskip
where $g$ is the interaction coupling constant. Without a magnetic
field, such as in a free Bose gas within a box potential of area
$\mathcal{A}$, the energy is minimized with a uniform density
$n=|\psi|^2=N/\mathcal{A}$ (ignoring boundary effects). When a magnetic
field is applied, distributing the density over the area $\mathcal{A}$
necessitates the insertion of $N_\Phi=B\mathcal{A}/\Phi_0$ quantum
vortices. As shown in~\cite{kleiner_bulk_1964}, the interaction energy
is minimized with a regular arrangement of vortices in a triangular
lattice.

Vortex lattices have been realized and extensively studied in the early
years of atomic BECs, although most cases did not restrict to the LLL
(see Figure~\ref{fig:rotating_frame}(b))
\cite{madison_vortex_2000,abo-shaeer_observation_2001,bretin_fast_2004,schweikhard_rapidly_2004}. 
However, exploring quantum Hall physics through rotating gases presents
significant challenges, as these systems are highly sensitive to any
static defect potential in the lab frame, which induces time-dependent
perturbations in the rotating frame.

\subsubsection{Recent experiments in the LLL}

Recent experiments conducted by the group of Zwierlein have
rekindled interest in rotating gases~\cite{fletcher_geometric_2021}.
Compared to experiments from the 2000s, these new studies benefit from
advanced techniques such as high-resolution in situ imaging and
potential imprinting. This allows fine potential defect diagnosis, as
well as the capability to project a hard-wall potential, enabling the
investigation of uniform gases in a circular box. The rotation was
induced by a rotating quadrupolar potential given by $V(x,y) = -Qxy$
with $Q &gt; 0$, as illustrated in Figure~\ref{fig:elongated_gas}(a) (we
will omit the prime notation in the following discussion).

\begin{figure}
\includegraphics{fig07}
\caption{\label{fig:elongated_gas}(a) In situ image of an atomic gas in
the momentum state $p_x=0$ of the LLL
(from~\cite{fletcher_geometric_2021}). The right panel indicates as
dashed lines the equipotential lines of the rotating saddle potential. 
(b) In situ image of the atomic gas in the presence of a sharp circular
box potential (green circle). The atoms populate a chiral edge mode
propagating along the boundary (from~\cite{yao_observation_2023}).}
\end{figure}

The dynamics induced by the saddle potential can initially be
understood through classical orbits. In the lowest Landau level, the
slow motion of atoms---characterized by the drift of guiding centers of
cyclotron orbits---is governed by the equation of motion
{\[
\vbold= \frac{1}{q}\frac{\boldsymbol\nabla V\times\Bbold}{B^2},
\]}\unskip
which simplifies here to
{\[
\vbold=\frac{Q}{2M\omega}(x\ebold_x-y\ebold_y).
\]}\unskip
Starting from a rotationally symmetric Gaussian distribution, we expect
the gas to contract along the $y$-direction while elongating along
the $x$-direction. Eventually, the extent along $y$ saturates at
the cyclotron length $\ell$, beyond which classical dynamics ceases
to apply.

This dynamics accounts for the formation of a highly elongated gas in
the MIT experiments, as illustrated in
Figure~\ref{fig:elongated_gas}(a). This state can be interpreted as the
occupation of a well-defined wavefunction from the LLL, specifically a
momentum state expressed in the Landau gauge $\mathbf{A} = -By
\mathbf{e}_x$. 

In this gauge, rotational symmetry is broken, and the system exhibits
continuous translation symmetry along the $x$-direction. The
Hamiltonian is given by
{\[
H = \frac{(p_x + qBy)^2}{2M} + \frac{p_y^2}{2M},
\]}\unskip
where the momentum $p_x$ acts as a fixed parameter, resulting in a
one-dimensional harmonic oscillator along $y$. The LLL, parameterized
by $p_x$, corresponds to the ground state:
{\[
\psi_{p_x}(x,y) = \E^{\I p_x x/\hbar} \exp\left[-\frac{(y - p_x
\ell^2/\hbar)^2}{2\ell^2}\right].
\]}\unskip
Returning to the MIT experiments, the elongated gas is very close to
the state $p_x = 0$\footnote{The momentum state $p_x = 0$ is the
only momentum state unaffected by the saddle potential.}.

The control of rotating gases at the level of a single LLL wavefunction
laid the groundwork for subsequent experiments, where the population of
chiral edge modes localized near the boundary of a sharp box potential
was observed (see
Figure~\ref{fig:elongated_gas}(b))~\cite{yao_observation_2023}.
Additionally, the effects of interactions in the flat-band limit were
investigated, revealing a redistribution of momentum within the LLL due
to interactions, which led to vortex crystallization as a precursor to
the formation of an Abrikosov
lattice~\cite{mukherjee_crystallization_2022}.

\subsection{Lattice systems}
\subsubsection{Peierls phase engineering}

Atomic gases in optical lattices provide a versatile platform for
simulating a wide range of discrete lattice
models~\cite{gross_quantum_2017}. Here, we focus on the realization of
lattices with non-trivial topological bands, which require
complex-valued hopping amplitudes. In a standard optical lattice
created by the dipole trapping of an off-resonant standing wave, atoms
hop through spontaneous tunneling events characterized by real-valued
transition amplitudes. The proposal by Jaksch and
Zoller~\cite{jaksch_creation_2003} established the framework for
simulating Peierls phases.

We present a simplified version of this scheme that aligns more closely
with actual experimental implementations. By introducing a linear
potential, we create an energy offset $\hbar \Delta$ between
neighboring wells along the $x$-direction, effectively suppressing
tunneling (see Figure~\ref{fig:shaken_lattice}(a)). The $x$-dynamics
can be restored by employing a pair of laser beams to induce a
two-photon optical transition.

\begin{figure}
\includegraphics{fig08}
\caption{\label{fig:shaken_lattice}(a) Scheme of the lattice
configuration and shaking protocol used to engineer the Hofstadter
model with a magnetic flux $p/q=1/4$
(from~\cite{aidelsburger_realization_2013}). A linear potential induced
by a magnetic field gradient suppresses tunneling along the
$x$-direction. A pair of Bragg laser beams induces resonant tunneling
along $x$, imprinting the complex phase profile of the Bragg process
onto the hopping matrix elements. (b) Scheme of the transverse Hall
response measurements and measured Hall drift as a function of time
(adapted from~\cite{aidelsburger_measuring_2015}). The solid line
represents the quantized drift expected for a Chern number
$\mathcal{C}=1$.}
\end{figure}

The hopping amplitude inherits the electric field amplitudes $E_1
E_2^* \propto \E^{\I \delta \mathbf{k} \cdot \mathbf{r}}$, where
$\delta \mathbf{k}$ represents the relative light momentum. The
complex phases of these amplitudes function as the Peierls phases
associated with a magnetic field, yielding a flux per plaquette given
by
{\[
\Phi = \Phi_0 \frac{\delta k_y a}{2\uppi}.
\]}\unskip
Since the lattice spacing $a$ is comparable to the laser wavelength,
the flux is typically on the order of 1 and can be adjusted by
modifying the lattice geometry or the angle between the two driving
beams. This scheme, along with similar protocols, has been successfully
implemented to simulate both the Hofstadter and Haldane models
discussed in Section~\ref{section:lattice}
\cite{aidelsburger_realization_2013, miyake_realizing_2013,
jotzu_experimental_2014}.

\subsubsection{Probing the band topology}

Various protocols have been developed to probe band topology
(see~\cite{cooper_topological_2019} for a review). Here, we focus on
one such method that examines the transverse linear response of an
atomic gas evolving under the Hofstadter model with a flux  $p/q =
1/4$~\cite{aidelsburger_measuring_2015}. Instead of filling the lowest
energy band with a fermionic band insulator, a thermal Bose gas samples
the ground band uniformly and incoherently. Applying a force $F_x$
along the $x$-direction induces a transverse drift along the
$y$-direction of the center of mass (see
Figure~\ref{fig:shaken_lattice}(b)), as described by
{\[
y(t) = \frac{4a^2 F_x}{\hbar} \mathcal{C} t,
\]}\unskip
where $\mathcal{C}$ is the Chern number of the occupied band. This
technique has provided the first experimental evidence of a quantized
Chern number in ultracold atomic gases.

\subsubsection{The issue of Floquet heating}

Experiments utilizing shaken optical lattices face a common challenge:
the limited duration of coherence due to Floquet heating. This
phenomenon arises because the laser-assisted hopping involves a
time-dependent potential 
$V(x,t)\propto\cos(\delta\kbold\cdot\rbold-\Delta t)$, where the
angular frequency $\Delta$ is determined by the energy difference
$\hbar\Delta$ between neighbouring wells. While this time modulation
prevents the definition of stationary states, the Floquet formalism
allows for the characterization of time-periodic
eigenstates--eigenstates of the unitary evolution over one modulation
period. These states are defined by a quasi-energy that is considered
modulo $\hbar\Delta$ (analogous to the quasi-momentum of Bloch states)
\cite{holthaus_floquet_2015,eckardt_colloquium_2017}. 

In the Floquet picture, the notion of ground band no longer makes
sense. However, an effective ground band can still be identified when
the modulation frequency significantly exceeds the typical tunneling
rates and band gaps to higher bands. Despite this, the effective ground
band remains coupled to higher bands, which can be brought into
resonance by the absorption of quanta of energy $\hbar\Delta$ (see
Figure~\ref{fig:Floquet_heating}(a) and (b)). These detrimental
couplings can be mitigated by appropriately selecting the modulation
frequency~\cite{weinberg_multiphoton_2015}. An additional source of
heating arises when considering interacting atoms, which can absorb
quanta of the modulation energy during two-body collisional
processes
\cite{reitter_interaction_2017,boulier_parametric_2019,wintersperger_parametric_2020,viebahn_suppressing_2021}. 

\begin{figure}
\includegraphics{fig09}
\caption{\label{fig:Floquet_heating}(a) Scheme of a typical
bandstructure, with low-energy bands of interest separated in energy
with respect to higher dispersive bands. (b) When targeting the
bandstructure (a) using time-modulated optical lattices, the modulation
frequency $\Delta$ gives rise, in a Floquet picture, to an infinite
number of replicas separated in energy by integer multiples of the
energy quantum $\hbar\Delta$ (two of them are shown in red and green).
The bands of interest may become resonantly coupled with higher bands,
leading to heating of the system. (c)~Reproduction of the scheme of the
protocol of Jaksch and Zoller~\cite{jaksch_creation_2003}, which
involves spin flips upon hopping between lattice sites. Such a scheme
involves a much higher modulation frequency, which could solve the
issue of Floquet heating.}
\end{figure}

These heating processes have not hindered the success of remarkable
experiments, including the realization of various types of topological
bands and initial studies of strongly interacting systems, as discussed
later. However, Floquet heating remains a significant challenge for the
future realization of complex topological states, which require
extended experimental timescales for their generation.

Future experiments could address the issue of Floquet heating by
employing more complex protocols that utilize an internal degree of
freedom of the atom, as originally proposed by Jaksch and
Zoller~\cite{jaksch_creation_2003} (see
also~\cite{gerbier_gauge_2010}).  By ensuring that the laser-assisted
tunneling is accompanied by a transition between two internal states
(see Figure~\ref{fig:Floquet_heating}(c)), the modulation frequency
$\hbar\Delta$ would become a hyperfine or electronic resonant
frequency, which is several orders of magnitude larger than the
typical energy scale of motion. In this scenario, an effective static
Hamiltonian can be accurately defined, and Floquet heating is not
expected to occur.

We also note that an off-resonant laser field coupled to an internal
degree of freedom can be utilized to simulate a magnetic field in the
continuum. In this scenario, the laser-dressed states become
position-dependent due to the spatial variation of the laser electric
field. An atom that adiabatically follows a given dressed state
experiences a geometrical Berry phase, similar to the Aharonov--Bohm
phase experienced by a charged particle in a magnetic field. The group
led by Spielman demonstrated the nucleation of quantized vortices in
a Bose--Einstein condensate using this
technique~\cite{lin_synthetic_2009}.

\subsection{Synthetic dimensions}

The last technique we discuss involves the concept of synthetic
dimensions~\cite{boada_quantum_2012}.  In this framework, a quantum
Hall system is defined where the $xy$ plane is replaced by a synthetic
plane composed of one spatial dimension $x$ and one synthetic
dimension~\cite{celi_synthetic_2014,mancini_observation_2015,
stuhl_visualizing_2015}. The spatial coordinate $y$ is substituted with
another degree of freedom, which can be a momentum state or an internal
degree of freedom~\cite{ozawa_topological_2019,fabre_atomic_2024}. In a
well-developed scheme, the discrete  projection $m$ of an internal spin
$J$ (with ${-}J\leq m\leq J$) is
utilized~\cite{mancini_observation_2015,stuhl_visualizing_2015,
chalopin_probing_2020}. The dynamics along $m$ is induced by two-photon
Raman transitions involving a pair of lasers counter-propagating along
$x$ (see Figure~\ref{fig:synthetic_dimension}(a)).

\begin{figure}
\includegraphics{fig10}
\caption{\label{fig:synthetic_dimension}(a) Scheme of the laser
configuration used to simulate a magnetic field in a hybrid system
involving a spatial dimension $x$ and a synthetic dimension encoded in
the spin projection $m$. The dynamics along $m$ is induced by resonant
two-photon optical transitions. An effective magnetic field arises from
the velocity kicks that occur along xx during spin transitions.  (b)
Measured Hall response resolved in magnetic projection $m$, which is
consistent with a quantized value in the bulk of the synthetic
dimension.}
\end{figure}

The occurrence of an effective magnetic field arises from the
correlated dynamics in the spin projection $m$ and velocity $v_x$.
Specifically, a spin transition $m\rightarrow m+1$  is accompanied by a
velocity kick $v_x \rightarrow v_x-2\hbar k_{\text{laser}}/M$ due to
the momentum exchange  with light. This relationship leads to the
equation of motion
{\[
M\dot v_x=-2\hbar k_{\text{laser}}\dot m,
\]}\unskip
which can be interpreted as the dynamics of a charged particle under
the influence of a magnetic field
{\[
M\dot v_x=qB\dot y,
\]}\unskip
where $q=1$ and $\Bbold=-2\hbar k_{\text{laser}}\ebold_z$. This
effective magnetic field results in a quantum Hall effect, which is
evidenced by a quantized transverse response in the bulk of the
synthetic dimension~\cite{chalopin_probing_2020} (see
Figure~\ref{fig:synthetic_dimension}(b)). Synthetic dimensions are
particularly advantageous for revealing the occurrence of topological
edge modes due to their finite
nature~\cite{mancini_observation_2015,stuhl_visualizing_2015}. They
also facilitate the engineering of complex geometries, such as
cylinders~\cite{han_band_2019,yan_emergent_2019,luo_tunable_2020,liang_coherence_2021,fabre_laughlins_2022} or systems in dimensions greater than three ~\cite{lohse_exploring_2018,bouhiron_realization_2024}.

Synthetic dimensions also come with some peculiarities and limitations.
First, synthetic dimensions can be limited in size, particularly when
encoding them in the internal spin of an atom. Exploring topological
effects in the thermodynamic limit then requires using large-spin
lanthanide atoms~\cite{chalopin_probing_2020}. Alternatively, synthetic
dimensions can be encoded in momentum states of an
atom~\cite{an_correlated_2018} or rotational states of a
molecule~\cite{sundar_synthetic_2018}, with no fundamental limitation
in size. 

So far, synthetic dimensions have primarily been employed to engineer
various types of topological bands, typically studied in the absence of
significant interaction effects. Notable exceptions include two works
that reported the observation of non-linear effects in a momentum-state
lattice~\cite{an_nonlinear_2021} and a universal Hall response in  a
ladder geometry~\cite{zhou_observation_2023}. More generally,
interactions in systems with synthetic dimensions are expected to
exhibit a peculiar structure. Since atoms on different synthetic sites
can interact, the interactions are anticipated to be of infinite range
along synthetic dimensions. This characteristic may significantly alter
the phenomenology of strongly-interacting topological systems
\cite{barbarino_magnetic_2015,cornfeld_chiral_2015,lacki_quantum_2016,barbarino_synthetic_2016,bilitewski_synthetic_2016,junemann_exploring_2017,ozawa_topological_2019}, 
and the conditions for the occurrence of genuine two-dimensional
fractional Hall states in systems with synthetic dimensions remain to
be elucidated.

\section{Towards fractional quantum Hall states}

In the context of a Bose--Einstein condensate, we have primarily
considered the many-body ground state where each LLL orbital is filled
with a large number of atoms, resulting in a high filling factor 
$\nu=\Nat/\Norb\gg1$. In the regime of lower fillings,  the many-body
system can no longer be described by a macroscopic wavefunction,
leading to the emergence of more complex correlations among the
particles.

The strongly-correlated regime poses significant challenges for
numerical modeling. Even in bosonic systems, the presence of gauge
fields results in complex-valued many-body wavefunctions, which
introduces a sign problem in Monte Carlo simulations. In many cases,
exact diagonalization remains the only viable approach, although it is
constrained by the size of the systems that can be studied. Recently,
advancements in computational techniques have allowed for the
simulation of larger system sizes using methods such as DMRG
~\cite{zaletel_topological_2013,motruk_density_2016,motruk_phase_2017}
or  other tensor network computations~\cite{gerster_fractional_2017}.
Variational projected entangled-pair states have also  been developed
to study the bosonic Harper--Hofstadter model in the thermodynamic
limit~\cite{weerda_fractional_2024}. 

\subsection{The bosonic Laughlin state}

We focus our discussion on the bosonic Laughlin state at filling factor
$\nu=1/2$, a prominent example of a fractional quantum Hall 
state~\cite{laughlin_anomalous_1983}.  For an ensemble of $N$ particles, the
many-body wavefunction is given by
{\[
\psi_L(z_1,z_2,\ldots,z_N) = \prod _{1\leq i&lt; j\leq N}
(z_i-z_j)^2 \prod _{i=1}^N\E^{-|z_i|^2/4\ell^2}.
\]}\unskip
This wavefunction being expressed as a symmetric polynomial of the
complex coordinates $z_i$ (aside from a Gaussian factor that we will
disregard for simplicity), it belongs to the LLL introduced in
Section~\ref{section:LL}. A notable feature of this wavefunction is
that it forbids two particles from occupying the same point, as
evidenced by the second-order correlation function $g_2(r)$ plotted in
Figure~\ref{fig:Laughlin}(b)). This property leads to the remarkable
conclusion that the Laughlin state is the exact many-body ground state
for zero-range repulsive interactions, a scenario particularly relevant
for rotating atomic Bose gases. 

\begin{figure}
\includegraphics{fig11}
\caption{\label{fig:Laughlin}Density profile $\rho(r)$ (a) and
correlation function $g_2(r)$ (b) of a bosonic Laughlin state with 400
particles, computed from a Monte Carlo
simulation~\cite{ciftja_monte_2006}.}
\end{figure}

Any wavefunction derived by multiplying the Laughlin wavefunction by
another symmetric polynomial also exhibits a zero interaction energy.
Among this family of degenerate ground states, the Laughlin state
uniquely minimizes the angular momentum projection $\mtot=N(N-1)$,
effectively leading to a minimal spatial extent centered around
$\rbold=\boldsymbol0$. The maximum (single-particle) angular momentum
$m=2(N-1)$ occupied in the Laughlin state corresponds to a radius
$R\simeq 2\sqrt{N}\ell$, which results in an area of $\mathcal{A}=\uppi
R^2=2\times(2\uppi\ell)^2 N$. The proportionality $\mathcal{A}\propto
N$ reflects the behavior of an incompressible liquid. The bulk density
of the Laughlin state is expressed as
{\begin{equation}\label{eq:rho_Laughlin}
\bar\rho=\rho_0/2,
\end{equation}}\unskip
where $\rho_0=1/(2\uppi\ell^2)$ is the density expected when each
orbital is filled with a single particle, akin to a fermionic Hall
insulator. In practice, the density is nearly uniform throughout most
of the disk $r&lt;R$, with a slight ripple around
$r=R$~\cite{ciftja_monte_2006}, see Figure~\ref{fig:Laughlin}(a). 

Since $\rho_0\propto B$, the uniform density given in
Equation~(\ref{eq:rho_Laughlin}) is proportional to the magnetic field strength.
According to the St\v{r}eda formula
$\sigma_{xy}=\partial\bar\rho/\partial B$, which relates the Hall
conductance to the variation of density with respect to the magnetic
field~\cite{streda_theory_1982}, we have
{\[
\sigma_{xy}=\frac{1}{2}\frac{1}{h},
\]}\unskip
i.e.\ a half-quantized value (we recall our convention of a unit charge
$q=1$). Furthermore, the excitation of density modulations is
suppressed in the bulk by a finite energy gap (roughly given by the
interaction energy experienced by a pair of atoms in the  state
$\psi_{m=0}$~\cite{regnault_quantum_2003}). 

\subsection{State-of-the art: Laughlin state with 2 particles}

A recent experiment conducted by the group of Greiner has
successfully demonstrated the realization of a Laughlin state with just
two atoms~\cite{leonard_realization_2023}. In this experiment, the
researchers utilized an optical lattice to simulate the Hofstadter
model with a flux of $p/q=1/3$ per plaquette. They confined the
dynamics to a $4\times4$ lattice, resulting in a total flux through the
9 plaquettes amounting to $3\,\Phi_0$. This configuration corresponds
to $\Norb=3$ orbitals in the ground band, which is precisely the number
needed to define the Laughlin state.

The interacting two-body ground state was adiabatically prepared by
slowly ramping local potential tilts. The resulting state exhibited the
expected anti-bunching between the atoms (see
Figure~\ref{fig:Laughlin_Greiner}(a)). Furthermore, by measuring the
change in density  in response to variations in the artificial magnetic
field, they were able to determine a Hall response conductance of
$0.6(2)/h$ via the St\v{r}eda formula. This result is consistent with a
half-quantized Hall response  (see
Figure~\ref{fig:Laughlin_Greiner}(b)). 

\begin{figure}
\includegraphics{fig12}
\caption{\label{fig:Laughlin_Greiner}Correlation function
$g_2(\mathrm{d})$ measured for a 2-particle Laughlin state
in~\cite{leonard_realization_2023} (b) Variation of the density with
magnetic flux measured in the same system, consistent with a
half-quantized Hall conductance.}
\end{figure}

A recent preprint from the group of Jochim also reported the
realization of a 2-atom Laughlin state composed of a pair of rapidly
rotating fermionic atoms~\cite{lunt_realization_2024}.  Notably, other
platforms have successfully produced two-photon Laughlin
states~\cite{clark_observation_2020,wang_realization_2024}. 

\subsection{Towards larger Laughlin states}

While robust quantization of the Hall conductance only occurs for large
enough systems, increasing the number of atoms in a Laughlin droplet
remains a significant hurdle. Several approaches can be taken to
address this
\begin{itemize}
\item Starting with independent one-dimensional chains of atoms,
ramping up inter-chain couplings in an adiabatic
manner~\cite{he_realizing_2017}. 

\item A patchwork construction~\cite{palm_growing_2024}, which 
involves preparing multiple $4\times4$ plaquettes, each containing 2
atoms in a Laughlin state, and then adiabatically coupling these
plaquettes together. 

\item By adjusting a parameter that controls a phase transition from a
non-topological to a topological band, similar to methods using local
energy offsets~\cite{aidelsburger_measuring_2015}, one can transition
from a trivial phase (like a Bose superfluid) to a Laughlin
state~\cite{motruk_phase_2017}.
\end{itemize}
In these methods, maintaining an adiabatic process is crucial for
achieving a low-entropy Laughlin state. The importance of the adiabatic
path was highlighted in earlier proposals for creating small Laughlin
states in rotating systems~\cite{popp_adiabatic_2004}. However, as the
number of atoms increases, it becomes increasingly difficult to
maintain adiabatic conditions.

Another potential approach is to create a more complex system with a
central region hosting a Laughlin droplet surrounded by a large
reservoir in a gapless non-topological
phase~\cite{wang_cold-atom_2024}. By tuning the energy of the
reservoir, it may be possible to transfer entropy from the central
region to the reservoir, resulting in a low-entropy Laughlin droplet.
This technique has been successfully used in lattice Fermi gases to
produce antiferromagnets with long-range magnetic
order~\cite{mazurenko_cold-atom_2017}.

Finally, we mention proposals for preparing topological many-body
states using dissipative
dynamics~\cite{diehl_topology_2011,budich_dissipative_2015}, for
instance fractional Hall states by pumping interacting bosons into the
lowest Chern band of a topological lattice~\cite{liu_dissipative_2021}.

\subsection{Probing long-range entanglement}

In contrast to an integer quantum Hall phase, a fractional Hall state
exhibits long-ranged entanglement associated with what is known as
topological order~\cite{wen_topological_1990}. This topological order
is characterized by the existence of degenerate many-body ground states
in a toroidal geometry that cannot be distinguished using local
observables. However, this property poses challenges for experiments,
as implementing periodic boundary conditions can be difficult.

Topological order can be more easily revealed by analyzing long-range
entanglement through the entanglement entropy 
$S_A=-\Tr(\rho_A\log\rho_A)$ upon spatial partition of the system
between two subregions $A$ and $B$ (where $\rho_A$ is the reduced
density matrix obtained by tracing out the $B$ subregion). For a
partition with a perimeter $L$, systems with short-range entanglement
follow the area law, scaling linearly with $L$ according to
$S_A(L)=\alpha L+o(1)$~\cite{eisert_colloquium_2010}, whereas systems
with topological order are expected to modify this law to
$S_A(L)=\alpha L-\gamma +o(1)$. The subleading term $\gamma$, known as 
the topological shift,  quantifies a global entanglement feature that
persists over long distances~\cite{kitaev_topological_2006}.

The group led by Greiner has successfully measured the Renyi
entanglement entropy defined as $S_A^{(2)}=-\log\Tr(\rho_A^2)$, using a
protocol based on many-body interference between two copies of a
many-body system~\cite{daley_measuring_2012,islam_measuring_2015}  (see
Figure~\ref{fig:topo_entropy}(a)). In this approach, two copies $1$ and
$2$ of a bosonic many-body state undergo a local beam-splitter operation 
$\mathop{\lvert} 1 \mathop{\rangle} \rightarrow (
\mathop{\lvert} 1 \mathop{\rangle} + 
\mathop{\lvert} 2 \mathop{\rangle})/\sqrt2$ and 
$\mathop{\lvert} 2 \mathop{\rangle} \rightarrow (
\mathop{\lvert} 1 \mathop{\rangle} -
\mathop{\lvert} 2 \mathop{\rangle})/\sqrt2$.
The purity of the system reduced to subregion $A$ is linked to the mean
parity of the number of particles $N_A^{(1)}$ found in  region $A$ of
system 1 after the splitting as
{\[
\Tr(\rho_A^2) = \langle\text{Parity}(N_A^{(1)})\rangle.
\]}\unskip

If the original many-body state is pure, an even number of atoms is
expected in system 1, similar to the Hanbury Brown and Twiss effect. As
the size of $A$ increases from zero, the mean parity should quiclky
drop to zero on a lengthscale $\ell$. Taking the logarithm gives the
Renyi entropy, whose scaling with $L$ can reveal the topological shift
$\gamma$. Numerical simulations on small Laughlin droplets confirm this
scenario~\cite{zozulya_bipartite_2007,haque_entanglement_2007,morris_topological_2009}
(see Figure~\ref{fig:topo_entropy}(b)). 

\begin{figure}
\includegraphics{fig13}
\caption{\label{fig:topo_entropy}(a) Scheme of the method for measuring
the (Renyi) entanglement entropy of a subregion $A$ within a many-body
system. The process involves interfering two identical copies of the
system using a 50/50 beam splitter. The purity of the reduced density
matrix $\rho_A$ can then be determined from the mean parity of the atom
count in region $A$~\cite{daley_measuring_2012,islam_measuring_2015}.
(b) Expected variation of the (von Neumann) entanglement entropy $S_A$
as a function of the radius $R_A$ of subregion $A$. The $y$-intercept
of the linear fit represents the entanglement entropy $\gamma$
(from~\cite{morris_topological_2009}).}
\end{figure}

\subsection{Anyonic quasi-hole excitations}

A key feature of fractional quantum Hall states is the unique nature of
their elementary excitations. We recall that a wavefunction  $
\psi(z_1,z_2,\ldots,z_N)=P(z_1,z_2,\ldots,z_N)\psi_L(z_1,z_2,\ldots,z_N),
$ where $P$ is a symmetric polynomial, exhibits zero interaction energy
and is therefore degenerate with the Laughlin ground state. A
polynomial $P$ with a degree less than the number of particles
corresponds to edge excitations of the Laughlin droplet. Here, we focus
on a bulk excitation, known as a quasi-hole excitation, represented by
{\[
\psi_\eta(z_1,z_2,\ldots,z_N) = 
\prod_{i=1}^N(z_i-\eta)\psi_L(z_1,z_2,\ldots,z_N).
\]}\unskip
It exhibits a density hole at $z=\eta$, corresponding to a deficit of
1/2 particle (see Figure~\ref{fig:quasihole}(b)). This can be
understood by considering two quasi-holes at the same point,  leading
to the wavefunction
{\[
\psi_{\eta,\eta}(z_1,z_2,\ldots,z_N) = 
\prod_{i=1}^N(z_i-\eta)^2\psi_L(z_1,z_2,\ldots,z_N).
\]}\unskip
It can be identified with the state obtained by removing from a
Laughlin wavefunction of $(N+1)$ particles a single atom at 
$z_{N+1}=\eta$. 

\begin{figure}
\includegraphics{fig14}
\caption{\label{fig:quasihole}(a) Scheme of the protocol used to
nucleate a quasi-hole excitation by focusing a laser beam producing a
repulsive potential. (b) Density profile of a Laughlin state with a
quasi-hole excitation at $r=0$ (from~\cite{comparin_measurable_2022}).
The atom number deficit corresponds to a fractional charge of 1/2 atom,
and the r.m.s.\ size of the hole is linked with the quasi-hole spin
$S=1/4$. (c) Scheme of the braiding operation by moving adiabatically
one quasi-hole around another, which gives rise to a geometrical phase
$\phi_{\text{geom}}=2S(2\uppi)=\uppi$.}
\end{figure}

In practice, a quasi-hole state can be generated by applying a
repulsive laser beam focused at $z=\eta$~\cite{wang_measurable_2022}
(see Figure~\ref{fig:quasihole}(a)). This technique is already
well-established for injecting and controlling quantum vortices in
Bose--Einstein
condensates~\cite{samson_deterministic_2016,kwon_sound_2021}.

Beyond the fractional charge, the quasi-hole density profile provides
further insights~\cite{macaluso_fusion_2019}. The relationship between
the mean square radius and angular momentum~\cite{ho_rotating_2002}
indicates that the size of the quasi-hole is linked to its angular
momentum, according to
{\[
\delta L_z = \int\,\dd^2 z
\left(\frac{|z-\eta|^2}{2\ell^2}-1\right)(\rho_\eta(z)-\bar\rho).
\]}\unskip
This excess angular momentum can be interpreted as the internal spin
$S=\delta L_z$ of the quasi-hole~\cite{comparin_measurable_2022}. For
the bosonic Laughlin state, numerical studies suggest $S=1/4$. This
fractional spin challenges the conventional understanding of
half-integer and integer spins for fermions and bosons, respectively.
According to the spin-statistics theorem, this hints at the existence
of a fractional statistical phase, leading to the term anyons for such
quasi-particles~\cite{wilczek_quantum_1982}. Real-space probes of
quasi-hole excitations can also be generalized to Chern insulators in
lattice systems~\cite{umucalilar_real-space_2018}.

\subsection{Braiding of anyonic excitations}

The fractional statistical phase is defined as the geometrical phase
acquired by the system upon an exchange of two particles.  To probe
this statistical phase, two key requirements must be met (i) the
ability to create and spatially manipulate quasiholes, which can be 
achieved using focused laser beams  (see
Figure~\ref{fig:quasihole}(c)). When a quasihole encircles another, the
many-body system acquires a statistical phase given by
$\phi_{\text{geom}}=2S(2\uppi)=\uppi$. (ii) an interferometric protocol
to reveal the value of this phase. We do not discuss this point here
and refer to the first proposal for a statistical phase interferometer
in the context of cold atomic gases, by Paredes 
et~al.~\cite{paredes_12-anyons_2001}.

Successful measurement of the statistical phase hinges on minimizing
various imperfections.  First, the geometrical phase should not be
significantly affected by a dynamical phase that may arise during the
adiabatic braiding process. This dynamical phase can occur if there is
cross-talk between the quasiholes when they are too close together or
if they are positioned near the confining edge potential. To keep
dynamical phases small, uncontrolled energy offsets must be maintained
well below the bulk excitation gap.  Second, the Laughlin droplet must
be prepared with very low residual entropy to ensure that the average
number of thermal quasiholes remains negligible over the area encircled
by the quasiholes during the braiding operation. These stringent
requirements impose significant constraints on the control over the
system, making the measurement quite challenging.

\subsection{Extension to other fractional quantum Hall states}

So far, our discussion has focused on the bosonic Laughlin state at
filling $\nu=1/2$. This focus is motivated by the fact that the
Laughlin state emerges as the exact many-body ground state for
short-range binary interactions, making it the most experimentally
accessible fractional quantum Hall state in quantum gas systems. Other
fractional quantum Hall states could also be realized using more
sophisticated experimental protocols. While each fractional Hall state
is characterized by a unique topological order, certain key
properties---such as topological entropy or the existence of
fractionalized quasi-particles---can be probed using protocols similar
to those developed for the Laughlin state. 

Among the various fractional Hall states, the bosonic
Moore-Read/Pfaffian state, which occurs at filling $\nu=1$, is
particularly noteworthy due to its exotic properties. This state
involves the pairing of bosonic particles and is characterized by
three-body antibunching. As a result, it could be stabilized in the
presence of three-body repulsive
interactions~\cite{greiter_paired_1991}, although engineering such
interactions remains a significant experimental challenge.
Alternatively, it has been proposed that the Pfaffian state could be
stabilized through strong three-body
dissipation~\cite{roncaglia_pfaffian_2010} or by carefully tuning
two-body interactions~\cite{palm_bosonic_2021}. The Pfaffian state is
especially intriguing because its elementary excitations exhibit
non-Abelian exchange statistics~\cite{moore_nonabelions_1991}.
Demonstrating non-Abelian braiding of these excitations---a critical
step toward realizing topological quantum
computing~\cite{nayak_non-abelian_2008}---remains one of the most
ambitious challenges in experimental physics across platforms.

\section{Conclusion}

In this proceedings, we have discussed the current status of simulating
quantum Hall physics using ultracold atomic gases. Various techniques
have been explored to replicate the effects of a magnetic field on
charged particles, including rotating gases, lattice systems, and
synthetic dimensions. While the realization of topological bands is
well-established at the single-particle level, achieving
strongly-interacting topological systems remains a significant
challenge. Recent advancements, leveraging a high degree of control
over local potentials and individual particles, have shown promising
progress toward realizing fractional quantum Hall systems in ultracold
atomic gases.

While our focus has been primarily on the quantum Hall effect, the
field of ultracold atomic gases has also seen the emergence of diverse
topological systems, such as 1D SSH chains~\cite{atala_direct_2013},
spin-orbit coupled systems~\cite{lin_spinorbit-coupled_2011} and 3D
Weyl semi-metals~\cite{wang_realization_2021}. The exploration of
interacting gases within these topological band structures opens up
exciting possibilities for discovering a wide range of intriguing
strongly-correlated systems.

\section*{Declaration of interests}

The authors do not work for, advise, own shares in, or receive funds
from any organization that  could benefit from this article, and have
declared no affiliations other than their research organizations.

\section*{Acknowledgements}

The author acknowledges fruitful discussions with Jean Dalibard while
preparing the talk at the Institut Henri Poincar\'e. The author also
acknowledges support from European Union (grant TOPODY 756722 from the
European Research Council) and Institut Universitaire de France. 

\CDRGrant[ERC]{TOPODY 756722}

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