supplementalMaterials

#  Quantum nonlinear optics on the edge of small lattice fractional quantum
Hall fluids

Alberto Nardin  Université Paris-Saclay, CNRS, LPTMS, 91405 Orsay, France
<EMAIL_ADDRESS>Daniele De Bernardis  INO-CNR
Pitaevskii BEC Center and Dipartimento di Fisica, Università di Trento, 38123
Povo, Italy<EMAIL_ADDRESS>Rifat Onur Umucalılar  Department of
Physics, Mimar Sinan Fine Arts University, 34380 Sisli, Istanbul, Turkey
Leonardo Mazza  Université Paris-Saclay, CNRS, LPTMS, 91405 Orsay, France
Matteo Rizzi  Institute for Theoretical Physics, University of Cologne,
D-50937 Köln, Germany  Forschungszentrum Jülich GmbH, Institute of Quantum
Control,
Peter Grünberg Institut (PGI-8), 52425 Jülich, Germany  Iacopo Carusotto  INO-
CNR Pitaevskii BEC Center and Dipartimento di Fisica, Università di Trento,
38123 Povo, Italy

###### Abstract

We study the quantum dynamics of the edge modes of lattice fractional quantum
Hall liquids in response to time-dependent external potentials. We show that
the nonlinear chiral Luttinger liquid theory provides a quantitatively
accurate description even for the small lattice geometries away from the
continuum limit that are available in state-of-the-art experiments.
Experimentally accessible signatures of the quantized value of the bulk
transverse Hall conductivity are identified both in the linear and the non-
linear response to an external excitation. The strong nonlinearity induced by
the open boundaries is responsible for sizable quantum blockade effects,
leading to the generation of nonclassical states of the edge modes.

####  Introduction —

Since their first observation more than $40$ years ago in solid-state systems
[1], fractional quantum Hall (FQH) liquids have been attracting continuous
experimental and theoretical attention due to the remarkable interplay between
topology and strong correlations, featuring a robust quantization of the
transverse Hall conductivity [2], collective excitations exhibiting fractional
charge and exchange statistics in the bulk of the fluid [3, 4, 5] and gapless
chiral modes on its edge [6, 7, 8, 9, 10, 11, 12]. At low energies, these
latter are well described in terms of an effective one-dimensional chiral
Luttinger liquid ($\chi$LL ) theory [10] and provide a powerful probe of the
topological properties of the fluid [13, 14, 15, 12, 16, 17, 18, 19, 20].

Beyond electronic systems, a strong experimental attention is presently being
devoted to the realization of FQH liquids in synthetic matter systems such as
ultra-cold atoms under synthetic magnetic fields [21, 22, 23, 24] or strongly
interacting photons in nonlinear topological photonic devices [25, 26, 27].
First experimental observations of small FQH clouds have been recently
reported in both the atomic [28, 29, 30] and photonic [31, 32] contexts.
Concurrently, a number of theoretical studies have proposed protocols that
exploit the new manipulation and diagnostic tools that are available for these
systems to obtain insight on both the bulk [33, 34, 35, 36, 37] and the edge
[38, 39, 40] physics of FQH fluids from new points of view.

In state-of-the-art lattice setups [29, 32] however, the lattice size and the
particle number are typically small, the spatial confinement provided by open
boundaries is far from being smooth, and the magnetic field too large for a
continuum description to be accurate. One may therefore be concerned that the
basic FQH physics is distorted and the topological features washed out.

In this Letter we show that the $\chi$LL nature of the FQH edge excitations is
robust and unambiguously visible even in small lattice systems containing a
few particles. In particular, we show that the edge excitations generated in
response to a time-dependent external potential give access to the quantized
Hall conductivity of the FQH bulk. Turning the small spatial size and the open
boundaries of state-of-the-art systems into an advantage, we anticipate that
the non-linearity of the $\chi$LL induced by the spatial confinement [41, 42,
43, 44, 45] mediates efficient three-wave mixing process between Luttinger
quanta in the discrete chiral edge modes around the fluid. This extends to the
FQH edge quantum blockade effects which are well-known in electronic [46],
photonic [47, 48, 49, 50], and opto-mechanical [51, 52] systems and opens a
path towards the generation and control of nonclassical states of the edge
modes.

####  Model —

Figure 1: (a) Scheme of the model under consideration. (b-c) Numerically
calculated spectrum of the many-body eigenstates for $N=4$ atoms with
$\alpha=0.15$ in a (b) $12\times 12$ and (c) $10\times 10$ square lattice
(black horizontal lines). In panel (b) the confinement is dominated by a
harmonic potential
$V_{h}(\mathbf{r})=\omega_{h}^{2}\,(\mathbf{r}-\mathbf{r}_{0})^{2}/2$ of
strength $\omega_{h}=0.14J$ around the lattice center $\mathbf{r}_{0}$. In
panel (c), the only trapping comes from the open boundaries. Comparison with
the linear dispersion of a $\chi$LL theory (blue dashed line) highlights the
stronger anharmonicity of the open boundary case of (c): here, the
nonlinearity-induced splitting of the $l=2$ manifold is indicated by the green
arrows, while the blockade effect is illustrated by the red full/dashed arrows
indicating the resonant/detuned transitions. Orange points show the
eigenenergies obtained from the diagonalization of the nonlinear $\chi$LL
Hamiltonian Eq. (2). The effective parameters $\Omega,\Gamma$ are estimated
from the energy of the first excited state and the splitting between the two
second excited states, respectively.

We consider the set-up sketched in Fig. 1(a), namely a system of $N$ bosonic
particles moving in a $M\times M$ two-dimensional square lattice with open
boundaries and lattice spacing $a$, pierced by a uniform and orthogonal
(synthetic) magnetic field $B$. We describe the system with the Hofstadter-
Bose-Hubbard Hamiltonian [53, 54, 55, 56]

$\hat{H}_{\text{HBH}}=-J\sum_{\braket{ij}}\left(e^{i\phi_{ij}}\hat{a}_{j}^{{\dagger}}\hat{a}^{{\vphantom{\dagger}}}_{i}+{\rm
h.c.}\right)+\frac{U}{2}\sum_{i}\hat{a}_{i}^{{\dagger}}\hat{a}_{i}^{{\dagger}}\hat{a}^{{\vphantom{\dagger}}}_{i}\hat{a}^{{\vphantom{\dagger}}}_{i}$
(1)

where $\hat{a}^{\vphantom{\dagger}}_{i}$ ($\hat{a}^{\dagger}_{i})$ are the
destruction (creation) operators for a particle at site $i$. Both the on-site
interaction energy $U$ and the hopping strength $J>0$ are taken uniform in the
lattice, and we restrict to the $U\gg J$ hard-core limit. The hopping phase
$\phi_{ij}$ from site $i$ to $j$ is related to the magnetic vector potential
through
$\phi_{ij}=\frac{q}{\hbar}\int_{\mathbf{r}_{i}}^{\mathbf{r}_{j}}\mathbf{A}\cdot
d\mathbf{r}$ [57] and gives a gauge-invariant phase $\alpha=2\pi\phi/\phi_{0}$
for hopping around a plaquette. Here, $\phi=Ba^{2}$ is the magnetic flux
piercing a plaquette and $\phi_{0}=h/q$ is the flux quantum. Calculations for
the eigenstates, the matrix elements, and the dynamics in response to external
perturbations are numerically performed by exact diagonalization methods.

This model is known to host a variety of FQH states [55, 58, 59, 60, 61]. In
this Letter, we will focus on the simplest Laughlin $\nu=1/2$-like [3] state
that correspond to the ground state in the region $0.1\lesssim\alpha\lesssim
0.2$ (see the transparency window in Fig. 2(a-d) [62]). The finite extension
of the FQH region is allowed by the flexibility in the value of the density at
the center offered by the density-depleted region at the boundary [56, 36, 63,
62].

Figure 2: Magnetic field dependence of the excited state eigenenergies in the
different $l=1,\ldots,4$ (a-d) sectors. Points are coloured according to the
normalized matrix element $l\mathcal{M}_{n,0}^{(l)}/S_{l}$. The purple
background transparency is proportional to the overlap of the ground state
with a discretized Laughlin state [62]: minimal purple opacity represents
maximal overlap.(e) Normalized static structure factor (full lines) compared
to the $\chi$LL prediction in the thermodynamic limit (lateral arrows) and a
finite-size calculation for a $N=4$ system in the continuum (horizontal dashed
lines). Same parameters as in Fig. 1(c).

####  Nonlinear chiral Luttinger liquid —

In the angular momentum basis labeled by the angular momentum $l>0$, the
effective one-dimensional nonlinear $\chi$LL  theory [64, 65, 44] of the edge
dynamics of a anharmonically trapped FQH fluid has the form

$\hat{H}=\\!\sum_{l>0}\Omega\,l\,\hat{b}_{l}^{\dagger}\hat{b}_{l}^{\vphantom{\dagger}}\,+\\!\sum_{l,l^{\prime}>0}\Gamma\sqrt{ll^{\prime}(l+l^{\prime})}\left[\hat{b}_{l+l^{\prime}}^{\dagger}\hat{b}_{l}^{\vphantom{\dagger}}\hat{b}_{l^{\prime}}^{\vphantom{\dagger}}\\!+\\!\textrm{h.c.}\right]$
(2)

where $\hat{b}_{l}$ ($\hat{b}_{l}^{\dagger}$) are the bosonic destruction
(creation) operators of Luttinger quanta in the $l$ mode. The first term
describes the chiral angular velocity $\Omega$ of edge excitations, as
determined by the spatial confinement. On top of this, the anharmonicity of
the confinement gives the second term describing scattering processes between
Luttinger quanta mediated by a nonlinear term of strength $\Gamma\geq 0$ [44].
Higher-order contributions to the phonon dispersion can be safely neglected as
they are irrelevant for the low-energy processes considered in this work [44,
45].

If the trapping is purely harmonic ($\Gamma=0$), the theory reduces to the
standard chiral Luttinger liquid [66], where the excitations are non-
interacting and have a perfectly linear dispersion. As such, the eigenstates
collapse on the $\omega_{l}=\Omega l$ line [blue line in Fig.1(b,c)] and, at
each $l$, have a high degeneracy given by the number $P_{l}$ of inequivalent
integer partitions of $l$ [10], that is
$P_{1}=1,\,P_{2}=2,\,P_{3}=3,\,P_{4}=5,\,\ldots$. Under a generic confinement
potential, the massive degeneracy of the eigenstates at a given $l$ is lifted
by the nonlinearity in (2).

####  Spectrum of edge excitations —

In a square lattice the full rotational symmetry is not present; a proper
choice of the gauge $\phi_{ij}$ [38] ensures that the discrete rotation
operator $e^{i\hat{L}_{z}\frac{\pi}{4}}$ commutes with the system’s
Hamiltonian Eq. (1), so that the angular momentum is conserved modulo $4$.
Examples of numerically computed excited state spectra are plotted in Fig.
1(b,c) as a function of the reduced angular momentum $C_{4}$. The reduced
rotation symmetry makes the $C_{4}$-dependent spectrum to get folded with a
period $4$. In Fig. 1(b) the trapping is dominated by an additional harmonic
confinement, so that the spectrum is almost linear and resembles the standard
$\chi$LL prediction (blue dashed line) with minor splittings and corrections
due to the residual anharmonicity. In Fig. 1(c), the only confinement is the
one due to the open boundaries: its intrinsic anharmonicity is responsible for
marked nonlinear effects and the wide splitting of the multiplets at given
$l$. The numerical eigenenergies are in remarkable quantitative agreement with
those of the nonlinear $\chi$LL theory Eq. (2), which is is equivalent to a
free fermionic theory [45].

This physics is specially visible in the simplest $l=2$ manifold. From the
form of (2), it is immediate to recognize that the effect of the nonlinearity
is to coherently convert a pair of $l=1$ Luttinger quanta into a single $l=2$
Luttinger quantum and viceversa, so that the eigenstates are non-classical
superpositions of the $\hat{b}_{2}^{\dagger}|0\rangle$ and
$(\hat{b}_{1}^{\dagger})^{2}|0\rangle$ Fock states above the ground state
$|0\rangle$. In analogy with the biexciton Feshbach blockade effect proposed
in the optical context in [67], the anharmonicity of the resulting spectrum
leads to a marked blockade effect indicated by the red arrows in the level
scheme of Fig. 1(c): an external excitation that is resonant with the $l=1$
mode at linear regime will not be able to inject a second quantum, as the
corresponding eigenstates are shifted away in energy. A similar analysis holds
for the higher $l\geq 3$ multiplets, yet is made less straightforward by the
intertwining of the edge-modes splittings with bulk modes, and is further
complicated by the folding effect.

Figure 3:  (a) Time evolution of the dipole moment $d_{x}(t)$ (blue solid
line) and $d_{y}(t)$ (orange solid line). The pulsed excitation temporal
profile $\lambda(t)$ is shown as a red dotted line. (b) Snapshots of the
normalized density variation $\delta\rho/\rho_{\rm n}$ at the different time
instants indicated in (a). We used $N=3$, $M=9$, $\alpha=0.15$,
$\lambda_{d}=10^{-4}J/a$, $\tau J/\hbar=50$ and $\rho_{\rm
n}=4\lambda_{d}/(aJ)$. The dipole frequency is $\hbar\omega_{d}/J\approx
0.027$. (c) Time evolution of the real ${\rm Re}[Q_{z}(t)]$ (yellow solid
line) and imaginary ${\rm Im}[Q_{z}(t)]$ (blue solid line) parts of the
quadrupole moment. The pulsed excitation temporal profile $\lambda(t)$ is
shown as a red dotted line. (d) Snapshots of the normalized density variation
$\delta\rho/\rho_{\rm n}$ at the different time instants indicated in (c). We
used $N=3$, $M=9$, $\alpha=0.15$, $\lambda_{Q}=10^{-5}J/a^{2}$, $\tau
J/\hbar=160$ and $\rho_{\rm n}=50\lambda_{Q}/(a^{2}J)$. The quadrupole
frequency is $\hbar\omega_{Q}/J\approx 0.035$.

####  Linear response —

We now proceed with a study of the response of the fluid to external
perturbations, starting from the linear regime of a weak excitation. We
consider an external potential of angular momentum $l$ and amplitude
$\lambda(t)$ of the form
$\hat{U}(t)=\sum_{i}U(\mathbf{r}_{i},t)\hat{a}_{i}^{\dagger}\hat{a}_{i}$ with
$U(\mathbf{r},t)=\text{Re}\left[z^{l}\lambda(t)\right]$ in terms of the
shorthand $z=x+iy$. Beyond their straightforward realization in synthetic
matter systems [68, 44, 40], such potentials can nowadays be implemented also
for neutral edge excitations in electronic FQH systems [69]. As a consequence
of the transverse Hall response, the component of the force oriented along the
azimuthal direction will induce a flow in the radial direction and, thus, a
spatially periodic modulation of the density along the edge. Within the
$\chi$LL theory [10, 44], the strength of this response is expected to be
proportional to the quantized transverse conductivity and thus to the filling
factor $\nu$ [62].

Each excited state $\ket{C_{4},n}$ (i.e. the $n$-th excited state at angular
momentum $C_{4}$) will give a peak in the response to the external potential
located at frequency $\hbar\omega_{C_{4},n}=E_{C_{4},n}-E_{0,0}$ ($E_{0,0}$
being the ground-state energy) with a strength determined by the matrix
element $\mathcal{M}_{n,0}^{(l)}=\left|\Braket{C_{4},n\left
missing}{\sum\nolimits_{i}z_{i}^{l}\hat{n}_{i}}{0,0}\right|^{2}$ with
$C_{4}=l\pmod{4}$. The frequencies and the strengths of the peaks for the
lowest $l=1\ldots 4$ excitation channels are summarized in Fig. 2(a-d). In
spite of the reduced rotational symmetry, the matrix element weight is
concentrated in those eigenstates that result from the splitting of the
corresponding multiplet at $l$ in the nonlinear $\chi$LL theory and, as first
remarked in [40], excitations to eigenstates folded by Bragg scattering
processes have a negligible matrix element. Moreover, the fact that the number
of visible $l=4$ states does not match the number of partitions $P_{l=4}=5$
confirms the presence of uncoupled dark edge eigenstates [62] first predicted
by the fermionized theory of [45] and here numerically highlighted also in the
lattice geometry.

Summing over all the low-energy eigenstates at a given $C_{4}=l\pmod{4}$ that
are illustrated in Fig. 2(a-d) provides the numerical value for the structure
factor of the edge dynamics $S_{l}=\sum_{n}\mathcal{M}_{n,0}^{(l)}$ that we
display in Fig. 2(e). Here, the exact diagonalization results for the lattice
geometry (full circles) are compared to the numerical ones for a $N=4$
Laughlin state in a continuum geometry [44] (horizontal dashed lines) [62] and
to the $\chi$LL prediction in the thermodynamic limit $S^{\chi LL}_{l}=\nu
lR_{\text{cl}}^{2l}$ (lateral arrows). In this last expression, the classical
radius $R_{\text{cl}}=\sqrt{2N/\nu}\,l_{B}$ of the FQH cloud accounts for the
chosen $r^{l}$ radial dependence of the applied potential and
$l_{B}=a/\sqrt{2\pi\alpha}$ is the usual magnetic length.

For sufficiently small $l$, a clear plateau is visible in the region around
$\alpha\sim 0.15$ of good overlap with the discrete Laughlin state highlighted
in Fig.2(a-d). The height of the plateaus relative to the $l=1$ and $l=2$
modes are in perfect quantitative agreement with the $\chi$LL prediction. For
$l>2$ the corrections due to the small number of particles induce sizeable
deviations between the $N=4$ lattice numerics and $\chi$LL predictions, while
still remaining in quantitative agreement with numerical results for a $N=4$
Laughlin state [62]. This suggests that the observed deviations for the higher
multipole excitations are not a consequence of the lattice discretization but
of the small particle number.

####  Measuring the transverse conductivity —

These numerical results provide evidence of the validity of the $\chi$LL
theory also in small lattice FQH fluids and suggest that the quantization of
the transverse conductivity can be experimentally assessed from a measurement
of the response of the fluid to external potentials. Fig. 3(a) illustrate this
idea for the $l=1$ dipole case. The single dipole mode can be efficiently
excited by means of a spatially uniform and temporally pulsed force,
$U(\mathbf{r},t)=\lambda(t)x$ with
$\lambda(t)=\lambda_{d}\exp\left[-(t/\tau)^{2}\right]$. A few snapshots of the
spatial profile of the density perturbation generated in the fluid at
different times are shown in Fig. 3(b). After the end of the perturbation, the
$x,y$ components of the dipole moment [blue and yellow lines in Fig. 3(a)]
keep oscillating in quadrature at the natural frequency
$\hbar\omega_{d}=E_{1,0}-E_{0,0}$ according to
$d_{x}(t)=\Braket{\sum\nolimits_{i}x_{i}\hat{n}_{i}}(t)\simeq-
A_{d}\sin\left(\omega_{d}t\right)$ and a corresponding formula for $y$. The
amplitude $A_{d}=\frac{S_{1}}{2\hbar}\,\widetilde{\lambda}(\omega_{d})$ is the
same for the $x,y$ components, proportional to $S_{1}$ and the Fourier
component at $\omega_{d}$ of the excitation pulse. From a measurement of
$d_{x}(t)$, it is then possible to estimate the structure factor $S_{1}$ and
thus the transverse conductivity.

Application of a similar protocol to the higher $l>1$ modes requires isolating
the several eigenstates forming the higher multiplets. The quadrupole $l=2$
case is illustrated in Fig.3(c). The two excited states can be separately
addressed using a rotating saddle-shaped pulse of the form
$U(\mathbf{r},t)=\lambda_{Q}\,\text{Re}\left[z^{2}\lambda(t)\right]$ with a
long oscillating pulse $\lambda(t)=e^{-(t/\tau)^{2}}\,e^{-i\omega t}$ [red
dotted line in Fig. 3(c)] on resonance with each natural oscillation frequency
$\hbar\omega_{Q}=E_{2,0(/1)}-E_{0,0}$. At the end of the excitation pulse, the
complex-valued quadrupole moment
$Q_{z}(t)=\braket{\sum_{i}(x_{i}^{2}-y_{i}^{2})\hat{n}_{i}}+i\braket{\sum_{i}(2x_{i}y_{i})\hat{n}_{i}}$
keeps oscillating at the selected bare frequency $\omega_{Q}$ as
$Q_{z}(t)\simeq A_{Q}^{(+)}e^{i\omega_{Q}t}+A_{Q}^{(-)}e^{-i\omega_{Q}t}$.
Here, the counter-rotating terms proportional to $A_{Q}^{(-)}$ stem from the
reduced four-fold rotational symmetry of the lattice which enhances (reduces)
the quadrupole moment when the density perturbation is along the cartesian
(diagonal) directions [Fig. 3(d)]. $A_{Q}^{(+)}$ can be extracted from the
amplitude of the real and imaginary parts of $Q_{z}(t)$ and gives direct
information on the matrix element
$\mathcal{M}_{n,0}^{(2)}=(A_{Q}^{(+)}/\lambda_{Q})(2\hbar/(\sqrt{\pi}\tau))$.
Upon summing over both quadrupole modes, the static factor $S_{2}$ provides an
estimate of the transverse conductivity, to be compared to the quantized value
expected from $\chi$LL theory, Fig. 2(e).

Figure 4: Amplitude of the dipole oscillations under the effect of a
monochromatic excitation of different durations $\tau$ as indicated in the
legend (solid red, brown, yellow lines) as a function of the (normalized)
excitation strength $\lambda_{d}$. Dashed black line shows the theoretical
form (3) of the Rabi oscillations. Same parameters as in Fig. 3.

The brown dotted line is the linear regime prediction.

####  Quantum nonlinear dynamics —

The $\chi$LL theory becomes even richer when one goes beyond the linear
response regime and considers excitations that are strong enough to induce
nonlinear effects. These are mediated by the new interaction terms that appear
in the Hamiltonian Eq. (2) as a consequence of the anharmonic trapping
potential. As a simple yet most illustrative example, we consider a quantum
blockade effect for edge modes. Let us consider a periodically oscillating
potential $U(\mathbf{r},t)=\lambda(t)\cos(\omega_{d}t)x$, resonant with the
dipole transition at frequency $\omega_{d}$, with a Gaussian envelope of
duration $\tau$ and sizable peak amplitude $\lambda_{d}$. Given the strong
anharmonicity of the excitation spectrum highlighted in Fig.1(c), this
perturbation is able to efficiently excite the $l=1$ excited state but remains
well detuned from all two-excitation states in the higher $l=2$ manifold which
therefore remain empty. As a consequence, the dynamics recovers the one of a
resonantly-driven two-level system as first noticed in [40], yet with the key
advantage of the much stronger nonlinearity due to open boundaries. At the end
of the excitation sequence, each dipole component $d_{x,y}$ oscillates at the
fast frequency $\omega_{d}$, with an amplitude $A_{d}$. This is determined by
the coherent transfer between the ground- and excited- state during the
excitation sequence [70], at a Rabi frequency roughly proportional to the
perturbation strength $\lambda_{d}$. As one can see in Fig. 4, the Rabi
oscillations are cleanest and closest to the theoretical prediction

$A_{d}=\frac{\sqrt{S_{1}}}{2}\sin\left(\sqrt{\pi}\tau\,\frac{\lambda_{d}\sqrt{S_{1}}}{2\hbar}\right)$
(3)

for longer-excitation times $\tau$, when the narrower spectrum of the
excitation pulse makes the blockade effect on the $l=2$ manifold more
effective. As typical in cavity quantum optics, a small system size is
beneficial to the blockade effect, as it makes the dynamics faster by
reinforcing the nonlinear splitting, and thus helps taming decoherence
effects. Interestingly, the dependence of the Rabi oscillation frequency in
Eq. (3) on the structure factor offers an alternative way to assess the
quantized transverse conductivity of the FQH fluid.

Looking closer at the quantum state generated by the excitation sequence, we
note that after a $2\pi$ (or multiple) pulse, the system is brought back to
the ground state by the Rabi oscillation, while after a $\pi$ pulse the system
is instead fully transferred in its $l=1$ excited state. In a quantum optical
language [71] this corresponds to the generation of a very non-classical Fock
state with a single Luttinger quantum of excitation in the FQH edge, analogous
to a single photon wavepacket in quantum optics. As another remarkable quantum
effect, the spectral splitting of the $l=2$ response due to the nonlinear
interaction terms of the $\chi$LL theory can be used to convert an $l=2$
Luttinger quantum into a pair of $l=1$ quanta so to obtain a source of
correlated Luttinger pairs for quantum optical experiments on FQH edge modes.

####  Conclusions —

In this work we have shown that the edge dynamics of small lattice fractional
quantum Hall fluids [29, 32] is well captured by a nonlinear generalization of
chiral Luttinger liquid theory [65, 44]. The response to time-dependent
external potentials carries evidence of a quantized value of the transverse
conductivity in the bulk. Turning the small spatial size and the open
boundaries of state-of-the-art systems [31, 29, 32, 30] into an advantage, we
have moved the first steps into quantum nonlinear optics of FQH edge modes:
thanks to the intrinsic nonlinearity stemming from spatial confinement,
quantum blockade effects set in and lead to the generation of nonclassical
states of the edge modes. Future theoretical work will deal with more complex
quantum optical effects, such as the generation of entangled $\chi$LL pairs
[72], solitonic and bunched excitations of the fractional quantum Hall edge
[73, 74, 75] and exploitation of the edge as a information channel between
localized impurities [76, 77].

####  Acknowledgements —

We are thankful to E. Macaluso for continuous collaboration in the first
ground-breaking steps of this work, N. Goldman, J. Kwan and the M. Greiner
group for insightful discussions. A.N. acknowledges financial support from
Fondazione Angelo dalla Riccia, LoCoMacro 805252 from the European Research
Council and LPTMS for warm hospitality. D.D.B. acknowledges funding from the
European Union - NextGeneration EU, ”Integrated infrastructure initiative in
Photonic and Quantum Sciences” - I-PHOQS [IR0000016, ID D2B8D520, CUP
B53C22001750006]. M.R. acknowledges support from the Deutsche
Forschungsgemeinschaft (DFG) via project Grant No. 277101999 within the CRC
network TR 183 (B01) and under Germany’s Excellence Strategy – Cluster of
Excellence Matter and Light for Quantum Computing (ML4Q) EXC 2004/1 –
390534769. I.C. acknowledges financial support from the European Union
H2020-FETFLAG-2018-2020 project “PhoQuS” (n.820392), from the Provincia
Autonoma di Trento, from the Q@TN initiative, and from the PNRR MUR project
PE0000023-NQSTI. This work is part of HQI initiative (www.hqi.fr) and is
supported by France 2030 under the French National Research Agency award
number “ANR-22-PNCQ-0002”.

11footnotetext: See Supplemental Material for a derivation of the structure
factor value in the thermodynamic limit, a brief derivation of Kohn’s theorem,
the link between the structure factor and the transverse conductivity in the
bulk, additional numerical results (different lattice sizes and particle
numbers, as well as the overlaps with discretized model wavefunctions and an
analysis of the energy levels and relevant matrix elements as a function of an
additional harmonic confinement) and for the explicit calculations for the
relevant observables time-evolution, both for the dipolar and quadrupolar
excitations and in the perturbative/non-linear regimes.

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Supplemental Material for
Quantum nonlinear optics on the edge of small lattice fractional quantum Hall
fluids

###### Contents

  1. I Wen’s matrix elements
  2. II Kohn’s theorem
  3. III $\chi$LL commutation relations
  4. IV Lattice-size effects
  5. V Scaling with the number of particles
  6. VI Effect of additional harmonic confinement
  7. VII Dipolar excitation
    1. VII.1 Perturbation theory
    2. VII.2 Rabi oscillations
  8. VIII Quadrupolar excitation
    1. VIII.1 Perturbation theory
    2. VIII.2 Rabi oscillations

##  I Wen’s matrix elements

Figure 5: Low-energy static structure factor $S_{l}$ (see Eq. (5)) for a
continuum Laughlin state as a function of the number of particles $N$. Our
results are normalized to the expected result $S_{l}^{\text{$\chi$LL~{}}}$
(see Eq. (9) per contributing state $l$, $S_{l}^{\text{$\chi$LL~{}}}/l=\nu
R_{\text{cl}}^{2l}$. The points are the results of Monte-Carlo simulations,
while the dashed lines highlight the expected limit $l$.

In this section we review the theory for the static structure factor of the
relevant observable $\sum_{i=1}^{N}z_{i}^{l}$ in the case of a $N$-particle
Laughlin state in the continuum, in the thermodynamic limit $(N\gg 1)$. In
this case the system is fully rotationally symmetric; the angular momentum is
a good quantum number, and we can label the system’s eigenstates as
$\ket{l,n}$, with $l$ being the angular momentum with respect to the one of
the Laughlin ground state. Here $n$ is the additional quantum number which
enumerates the system’s eigenstates at fixed $l$. We are interested in the
matrix elements

$S_{l}=\sum_{n}\left|\Braket{l,n\left
missing}{\sum_{i=1}^{N}z_{i}^{l}\,}{0}\right|^{2}.$ (4)

These matrix elements can be conveniently rewritten as

$S_{l}=\sum_{n}\left|\int r^{l}e^{il\theta}\Braket{l,n\left
missing}{\hat{\rho}(\mathbf{r})}{0}d^{2}\mathbf{r}\right|^{2}.$ (5)

Since the bulk is incompressible, the only contributions can come from a thin
layer at the system’s edge, close to system’s classical radius
$R_{\text{cl}}=\sqrt{2N/\nu}l_{B}$, $l_{B}$ being the magnetic length; in the
large particle number ($N\gg 1$) limit the previous expression can be
approximated by

$S_{l}\approx R_{\text{cl}}^{2l}\sum_{n}\left|\Braket{l,n\left
missing}{\widetilde{\rho}_{l}}{0}\right|^{2}.$ (6)

where $\widetilde{\rho}_{l}=\int e^{il\theta}\widetilde{\rho}(\theta)d\theta$
is the angular Fourier transform of a one-dimensional effective density
$\widetilde{\rho}(\theta)=\int\hat{\rho}(\mathbf{r})rdr$. Following Wen’s
bosonization we can replace the full $\widetilde{\rho}_{l}$ operators with
bosonic fields

$\widetilde{\rho}_{l}=\sqrt{\nu l}\,\hat{b}^{\dagger}_{l},$ (7)

where
$\left[\hat{b}_{l}^{\vphantom{\dagger}},\hat{b}_{l^{\prime}}^{\dagger}\right]=\delta_{l,l^{\prime}}$,
and see the excited states $\ket{l,n}$ as being generated by the current
algebra through

$\ket{l,n}=\prod_{l^{\prime}\in\boldsymbol{\lambda_{l}}}\frac{b^{\dagger}_{l^{\prime}}}{\sqrt{n_{l^{\prime}}!}}\ket{0}$
(8)

where $\boldsymbol{\lambda}_{l}$ is an integer partition with angular momentum
$l$ and $n_{l^{\prime}}$ the multiplicity with which $l^{\prime}$ appears in
$\boldsymbol{\lambda}_{l}$. The only state contributing to the summation in
Eq. (6) is therefore $\ket{l,n}=\hat{b}^{\dagger}_{l}\ket{0}$, and we get

$S_{l}\approx\nu l\,R_{\text{cl}}^{2l}.$ (9)

Some numerical Monte-Carlo results obtained as detailed in [44, 45] for the
edge-excitations on top of a bosonic $\nu=1/2$ state are shown as a function
of the number of bosons $N$ for $l=1,\ldots,4$ in Fig. 5.

Notice that at $l=1$ the result is filling-fraction-independent and compatible
with Kohn’s theorem, as we will detail in the next section, and the numerical
result for a translationally invariant system in Fig. 5 indeed exactly matches
the result of Eq. (9); however it should also be noticed that there is no
a-priori reason for our lattice system to obey Kohn’s theorem and the result
we show for $l=1$ has to be understood as a signature of fractional quantum
Hall physics.

##  II Kohn’s theorem

In this section we briefly review the content of Kohn’s theorem [78]. The
system one considers is a collection of $N$ quantum particles moving in two-
dimensions under the influence of a perpendicular magnetic field, $B$, under
the influence of internal forces alone and at most an external harmonic
potential

$H=\sum_{i=1}^{N}\frac{(\mathbf{p}_{i}-q\mathbf{A}(\mathbf{r}_{i}))^{2}}{2M}+\sum_{i=1}^{N}\frac{1}{2}M\omega^{2}\mathbf{r}_{i}^{2}+\sum_{1\leq
i<j\leq N}V(\mathbf{r}_{i}-\mathbf{r}_{j}).$ (10)

We here use the symmetric gauge, $\mathbf{A}(\mathbf{r})=\frac{B}{2}(-y,x,0)$.
The motion of the center of mass can be decoupled from the one of the $N-1$
remaining relative coordinates; the former is ruled by a simple Hamiltonian:
introducing the center of mass variable
$\mathbf{R}=\frac{1}{N}\sum_{i}\mathbf{r}_{i}$ and its canonically conjugate
momentum $\mathbf{P}=\sum_{i}\mathbf{p}_{i}$ one can separate the Hamiltonian
$H=H_{\text{rel}}+H_{\text{CM}}$ into two parts, a part $H_{\text{rel}}$ which
only depends on the $N-1$ relative coordinates and a second part
$H_{\text{CM}}$ which instead depends only on the center of mass coordinates

$H_{\text{CM}}=\frac{\mathbf{P}^{2}}{2M^{\prime}}-\frac{\omega_{c}}{2}L_{z}+\frac{1}{2}M^{\prime}\Omega^{2}\mathbf{R}^{2},$
(11)

where $L_{z}=(\mathbf{R}\times\mathbf{P})\cdot\hat{z}$ is the center of mass
angular momentum. Here we introduced the cyclotron frequency
$\omega_{c}=\frac{qB}{M}$, the total mass $M^{\prime}=NM$ and an effective
oscillator frequency $\Omega=\sqrt{\omega^{2}+\omega_{c}^{2}/4}$. The
eigenfunctions are those of a harmonic oscillator; introducing radial $R$ and
azimuthal $\theta$ coordinates for the center of mass position, these
eigenfunctions can be written as

$\Phi_{n,m}^{(\text{CM})}(R,\theta)=\frac{1}{\sqrt{\pi\frac{\Gamma\left(n+\frac{|m|+m}{2}+1\right)}{\Gamma\left(n-\frac{|m|-m}{2}+1\right)}\,l_{\text{CM}}^{2}}}\,\left(\frac{R}{l_{\text{CM}}}\right)^{|m|}e^{im\theta}\,L_{n-\frac{|m|-m}{2}}^{(|m|)}\left(\frac{R^{2}}{l_{\text{CM}}^{2}}\right)\,e^{-\frac{R^{2}}{2l_{\text{CM}}^{2}}},$
(12)

where $\Gamma(n)$ is Euler’s Gamma function, $L_{n}^{(\alpha)}(x)$ are
associated Laguerre polynomials and we introduced a center of mass unit of
length $l_{CM}=\sqrt{\frac{\hbar}{M^{\prime}\Omega}}$. These wavefunctions
diagonalize the center of mass angular momentum,
$L_{z}\Phi_{n,m}^{(\text{CM})}=m\,\Phi_{n,m}^{(\text{CM})}$, as well as the
center of mass Hamiltonian Eq. (11), with eigenvalue

$E_{n,m}^{(\text{CM})}=\frac{\hbar\Omega}{2}\,\left[4\left(n+\frac{m+1}{2}\right)-\frac{\omega_{c}}{\Omega}\,m\right].$
(13)

In terms of the principal quantum number $n\in\mathbb{N}$, the angular
momentum quantum number $m$ must satisfy $m\geq-n$. For any value of $\omega$,
the ground state can be seen to correspond to $n=0$, $m=0$.

When the cyclotron frequency is much larger than the harmonic confinement one,
$\omega_{c}\gg\omega$, the energies reduce to Landau level ones
$E_{n,m}^{(\text{CM})}\simeq\hbar\omega_{c}\,\left(n+\frac{1}{2}\right)$,
independent of the angular momentum quantum number $m$. This center of mass
separation is relevant for the evaluation of the matrix elements of certain
observables, such a

$\displaystyle U=$ $\displaystyle\sum_{i}z_{i}=NR\,e^{+i\theta}$ (14)
$\displaystyle U^{*}=$ $\displaystyle\sum_{i}z_{i}^{*}=NR\,e^{-i\theta}$

since they only depend on the center of mass variables and not on the $N-1$
relative coordinates. In particular, $U$ couples the ground state only to
$n=0$, $m=1$. This can be seen to be a low-energy excitation, since it does
not change the Landau level quantum number $n$. $U^{*}$ on the other hand
couples the ground state only to $n=1$, $m=-1$; it therefore represents an
high-energy excitation (whose cost is set by the cyclotron energy
$\hbar\omega_{c}$). This result is known as the Kohn’s cyclotron’s resonance
[78]. It is not difficult to show that

$\bra{\Phi_{0,1}^{(\text{CM})}}U\ket{\Phi_{0,0}^{(\text{CM})}}=N\,l_{\text{CM}}\xrightarrow[\omega_{c}\gg\omega]{}\sqrt{2N}\,l_{B}$
(15)

where $l_{B}=\sqrt{\frac{\hbar}{M\omega_{c}}}$. Since this single state is
contributing at low-energies (in the limit $\omega_{c}\gg\omega$), we get

$S_{1}=|\bra{\Phi_{0,1}^{(\text{CM})}}U\ket{\Phi_{0,0}^{(\text{CM})}}|^{2}=2N\,l_{B}^{2},$
(16)

which is the same result Eq. (9) obtained in the previous section in the case
in which the angular momentum carried by $\sum_{i}z_{i}^{l}$ is $l=1$. Notice
also that information on the relative motion will start to matter for any
$l>1$.

Figure 6: Schematic depiction of a quantum Hall system with an edge. The
yellow region represents the incompressible quantum Hall fluid, whose edge
(the black curly line) is deformed with respect to its ground-state position
(the dashed black line). A force is applied along the $\hat{x}$ direction and
a current flows proportionally to the applied force in the $\hat{y}$ direction
due to the presence of a strong perpendicular magnetic field.

##  III $\chi$LL commutation relations

In this section we want to give an alternative heuristic derivation of the
commutation relations of the density operators of a $\chi$LL , which
highlights the connection with the transverse conductivity of the bulk and
does not rely on a specific form for the edge Hamiltonian.

We consider a quantum Hall region infinite in the $\hat{x}$ direction and
semi-infinite in the $\hat{y}$ direction, as we schematically depict in Fig.
6. The magnetic field is along the $\hat{z}$ direction. If we apply a constant
force in the $\hat{x}$ direction, by imposing an external potential
$U=qE_{x}x$, a quantized current flows in the bulk $J_{y}=\sigma_{xy}E_{x}$
perpendicularly to the edge.

Since charge is conserved, a continuity equation
$\partial_{t}\rho=-\boldsymbol{\nabla}\cdot\boldsymbol{J}/q$ must hold. For
the present discussion, we only focus on the effect of this transverse current
and not on the charge dynamics once the edge is reached (i.e. we neglect the
chiral motion along the boundary caused by the sample confinement) by dropping
the $\hat{x}$ contribution to the current; more explicitly,
$\partial_{t}\rho\simeq-\partial_{y}J_{y}/q$. We now integrate this equation
along the $\hat{y}$ direction, from the bulk region (where the density has not
changed because of the bulk’s incompressibility and
$J_{y}^{\text{bulk}}=\sigma_{xy}E_{x}$) to the outside of the sample (where
both the density and the current vanish)

$\frac{\partial}{\partial t}\int dy\rho(x,y)=-\frac{1}{q}\int dy\frac{\partial
J_{y}}{\partial
y}=\frac{1}{q}J_{y}^{\text{bulk}}=\frac{\sigma_{xy}}{q^{2}}\frac{\partial
U}{\partial x}.$ (17)

On the other hand, the left hand side of the expression can be rewritten by
subtracting off the equilibrium density and $\int
dy\left[\rho(x,y)-\rho_{0}(y)\right]=\delta\rho(x)$ can be seen as an
effective edge density.

From a more microscopic perspective, $U$ couples to the system’s density
$\hat{\rho}=\hat{\Psi}^{\dagger}\hat{\Psi}^{\vphantom{\dagger}}$ through an
interaction Hamiltonian

$\hat{H}=\int dx\,dy\,U(x)(\hat{\rho}-\rho_{0})=\int dx\,U(x)\int
dy(\hat{\rho}-\rho_{0})=\int dx\,U(x)\delta\hat{\rho},$ (18)

and Eq. (17) can be seen as the expectation value of a Heisenberg equation of
motion $\frac{\partial}{\partial
t}\delta\hat{\rho}(x)=\frac{\sigma_{xy}}{q^{2}}\frac{\partial U}{\partial x}$.
In order for this to be compatible with the Hamiltonian Eq. (18), we need

$\frac{i}{\hbar}\left[\hat{H},\delta\hat{\rho}(x)\right]=\frac{\partial}{\partial
t}\delta\hat{\rho}(x)=\frac{\sigma_{xy}}{q^{2}}\frac{\partial U}{\partial x},$
(19)

or

$\frac{i}{\hbar}\int
dy\,U(y)\left[\delta\hat{\rho}(y),\delta\hat{\rho}(x)\right]=\frac{\sigma_{xy}}{q^{2}}\frac{\partial
U}{\partial x}.$ (20)

If we assume
$\left[\delta\hat{\rho}(y),\delta\hat{\rho}(x)\right]=\alpha\delta^{\prime}(y-x)$,
we get $\alpha=i\frac{\hbar\sigma_{xy}}{q^{2}}$ or

$\begin{split}\left[\delta\hat{\rho}(y),\delta\hat{\rho}(x)\right]=&i\frac{\sigma_{xy}}{\hbar
q^{2}}\delta^{\prime}(y-x)\\\
=&i\frac{\nu}{2\pi}\delta^{\prime}(y-x)\end{split}$ (21)

which indeed does coincide with the famous $\chi$LL liquid commutation
relations derived by Wen [6, 7, 8, 11, 10]. This highlights the connection
between the non-Fermi-liquid commutator Eq. (21) (which in turn is responsible
for the static structure factor value in Eq. (9)) and the bulk’s transverse
conductivity.

Figure 7: Energy levels $E_{n}$ (with respect to the system’s ground state
energy $E_{0}$) as a function of the magnetic flux per plaquette $\alpha$, for
$N=4$ hard-core bosons moving in a (a) $7\times 7$, (b) $8\times 8$, (c)
$9\times 9$, (d) $10\times 10$ square lattice. The purple background
transparency is regulated according to the (squared) overlap of the ground
state $\ket{0}$ with the discretized Laughlin state of Eq. (22),
$\ket{\Psi_{L}}$. In order for the data point not to disappear when the
overlap drops, when the squared overlap drops below $0.7$ the transparency is
kept constant.  Figure 8: Ground state densities of the $N=4$ hard-core boson
system in lattices with different sizes, for some selected values of the
magnetic flux per plaquette $\alpha$ which correspond to a ground state in the
Laughlin phase. In particular, the four panels correspond to (a) $7\times 7$
lattice with $\alpha=0.22$, (b) $8\times 8$ lattice with $\alpha=0.21$, (c)
$9\times 9$ lattice with $\alpha=0.17$ and (d) $10\times 10$ lattice with
$\alpha=0.15$. The particular values of $\alpha$ have been chosen so as to
(approximately) maximize the overlap (see Fig. 10) of the given state with the
discretized Laughlin wavefunction in Eq. (22).

##  IV Lattice-size effects

In this section we analyze how the number of lattice sites influences the
energy spectra and the structure factors.

In Fig. 7 we show the excitation spectra for $N=4$ particles on a $M\times M$
lattice, for $M=7,\ldots,10$. The background transparency is regulated
according to the overlap of the ground state with a discretized bosonic
Laughlin wavefunction

$\Psi_{L}=\prod_{1\leq i<j\leq
N}(z_{i}-z_{j})^{1/\nu}\,e^{-\frac{1}{4l_{B}^{2}}\sum_{i=1}^{N}|z_{i}|^{2}}.$
(22)

Here $\nu=1/2$ is the (robust) filling fraction associated to the particular
incompressible phase, while $z_{i}=x_{i}+iy_{i}$ take values on the square
lattice. A large region in which the ground state has a large overlap with
this model-wavefunction can be seen (see also Fig. 10). It can also be seen
from Fig. 7 that as the size of the lattice is increased (from the leftmost
panel to the rightmost), the edge-excitation velocity decreases, as can be
expected from the atoms avoiding the occupation of the boundary lattice sites
(see Fig 8).

Figure 9:  Matrix elements
$\mathcal{M}_{n,0}^{(l)}=\left|\Braket{n}{\sum_{i}z_{i}^{l}\hat{n}_{i}}{0}\right|^{2}$
as a function of the magnetic flux per plaquette $\alpha$, for $N=4$ hard-core
bosons, moving in a (a,e) $7\times 7$, (b,f) $8\times 8$, (c,g) $9\times 9$,
(d,h) $10\times 10$ square lattice. The top (a-d) (/bottom (e-h)) panels are
for $l=1$ (/$l=2$). The purple background transparency is regulated according
to the (squared) overlap of the ground state $\ket{0}$ with the discretized
Laughlin state of Eq. (22), $\ket{\Psi_{L}}$. In order for the data point not
to disappear when the overlap drops, when the squared overlap drops below
$0.7$ the transparency is kept constant. The static structure factor per state
$S_{l}^{\text{$\chi$LL~{}}}/l$ given by Eq. (4) is shown as black dashed
lines.

In Fig. 9 we then plot the matrix elements $\mathcal{M}_{n,0}^{(l)}$ for
$l=1,2$; since in the Laughlin region only $l$ modes have large matrix-
elements, we normalize the results to the integrated structure factor per
state in the continuum, $S^{\text{$\chi$LL~{}}}_{l}/l=\nu R_{\text{cl}}^{2l}$,
$R_{\text{cl}}=\sqrt{2N/\nu}l_{B}$ being the classical radius and $l_{B}$ the
magnetic length. This result can be obtained from a $\chi$LL description of
the edge excitations of a Laughlin ground state in the continuum, as we
explicitly showed in the first section. With such a normalization, summing
over the $l$ non-vanishing matrix elements should therefore give $l$, provided
the geometric factor $R_{\text{cl}}^{2l}$ is not strongly affected by the
shape of the square boundary. As the system size is increased, it can be seen
in Fig. 9(a-d) that the dipole matrix element ($l=1$) does indeed approach the
expected result $S^{\text{$\chi$LL~{}}}_{1}=\nu R_{\text{cl}}^{2}$;
panels(e-h) show instead what happens for the quadrupole matrix elements: in
the Laughlin region (highlighted by the purple transparency) as the number of
lattice sites is increased two states only (full circles) contribute to the
structure factor $S^{\text{$\chi$LL~{}}}_{2}$; it can also be seen how their
magnitude approaches $S^{\text{$\chi$LL~{}}}_{2}/2=\nu R_{\text{cl}}^{4}$,
hinting to the correctness of the sum rule Eq. (6). We will show in greater
detail in the next section (in particular in Fig. 12) how
$S^{\text{$\chi$LL~{}}}_{2}$ changes as a function of the magnetic flux per
plaquette $\alpha$, for various lattice sizes and number of particles.

Finally, in Fig. 10 we plot the overlaps of a few selected eigenstates with
discretized model fractional quantum Hall wavefunctions for the gapless edge
excitations of a Laughlin liquid, for different system sizes. In particular,
the ground state, with symmetry eigenvalue $C_{4}$, which we label
$\ket{C_{4},0}$, is compared to the discretized Laughlin wavefunction of Eq.
(22); a good agreement (black curves) for the squared overlaps
($|\Braket{\Psi_{L}}{C_{4},0}|^{2}$ up to $\sim 0.9$) can be seen over a wide
range of synthetic magnetic flux per plaquette $\alpha$, for various lattice
sizes. The lowest lying excitation $\ket{C_{4}^{\prime},0}$, where
$C_{4}^{\prime}=C_{4}+1\pmod{4}$, is compared instead to the edge excitation

$\Psi_{d}=\sum_{i}z_{i}\prod_{1\leq i<j\leq
N}(z_{i}-z_{j})^{1/\nu}\,e^{-\frac{1}{4l_{B}^{2}}\sum_{i=1}^{N}|z_{i}|^{2}}.$
(23)

Again, a good agreement (red curves) can be found in the same region of
$\alpha$, although this window is slightly narrower.

Figure 10: Overlaps between selected numerical eigenstates and model
fractional quantum Hall wavefunctions for the Laughlin state Eq. 22 and its
edge excitations Eq. (23), Eq. (24), for a $N=4$ system of hard-core bosons
moving on (a) $7\times 7$, (b) $8\times 8$, (c) $9\times 9$ and (d) $10\times
10$ lattices, as a function of the synthetic magnetic flux per plaquette.

Finally, the two quadrupolar modes $\ket{C_{4}^{\prime\prime},0}$ and
$\ket{C_{4}^{\prime\prime},1}$ can be compared to discretized quadrupolar
excitations of a Laughlin state

$\begin{split}\Psi_{q}&=\left(\sum_{i}z_{i}\right)^{2}\prod_{1\leq i<j\leq
N}(z_{i}-z_{j})^{1/\nu}\,e^{-\frac{1}{4l_{B}^{2}}\sum_{i=1}^{N}|z_{i}|^{2}}\\\
\Psi_{q}^{\prime}&=\sum_{i}z_{i}^{2}\prod_{1\leq i<j\leq
N}(z_{i}-z_{j})^{1/\nu}\,e^{-\frac{1}{4l_{B}^{2}}\sum_{i=1}^{N}|z_{i}|^{2}}.\end{split}$
(24)

These two states $\ket{\Psi_{q}}$ and $\ket{\Psi_{q}^{\prime}}$ are linearly
independent but not orthonormal: we therefore orthonormalize them with a
standard Gram-Schmidt procedure; moreover, it does not make sense to compare
the two separately to the two quadrupolar states
$\ket{C_{4}^{\prime\prime},0}$ and $\ket{C_{4}^{\prime\prime},1}$: in
principle we can however expect these to be a mixture of $\ket{\Psi_{q}}$ and
$\ket{\Psi_{q}^{\prime}}$ which minimizes the particular Hamiltonian, provided
mixing with states outside of this manifold is small. We test this idea by
defining a matrix of overlaps

$U=\begin{pmatrix}\braket{\Psi_{q}}{C_{4}^{\prime\prime},0}&\braket{\Psi_{q}^{\prime}}{C_{4}^{\prime\prime},0}\\\
\braket{\Psi_{q}}{C_{4}^{\prime\prime},1}&\braket{\Psi_{q}^{\prime}}{C_{4}^{\prime\prime},1}\end{pmatrix}$
(25)

and testing whether it defines an unitary matrix (i.e. a “rotation” between
two otherwise equivalent bases) by computing an Hilbert-Schmidt norm of
$U^{\dagger}U^{\vphantom{\dagger}}$

$||U^{\dagger}U^{\vphantom{\dagger}}||=\sqrt{\frac{1}{d}\sum_{i,j=1}^{d}\left|(U^{\dagger}U^{\vphantom{\dagger}})_{i,j}\right|^{2}},$
(26)

where $d$ is the dimension of the subspace under consideration; in this case,
$d=2$. Ideally, $U^{\dagger}U^{\vphantom{\dagger}}=\mathbb{1}$ and the
normalization in Eq. (26) has been chosen so that $||\mathbb{1}||=1$. Notice
finally that when $d=1$, Eq. (26) reduces to the squared overlap used in the
two previous cases with the ground- and dipole- states. Again, a good
agreement (yellow curve) can be seen in the relevant region of $\alpha$ in
Fig. 10, with $||U^{\dagger}U^{\vphantom{\dagger}}||\approx 0.9$.

Although these overlaps can be expected to drop to zero in the thermodynamic
limit 111As it is usually the case for the overlaps between model
wavefunctions and “true” ground-state wavefunctions, the overlap can be
expected to go to zero as the number of particles is increased.
($N,L\rightarrow\infty$), the fact that they are significantly close to $1$ in
a limited window of fluxes per plaquette for the lattice sizes we analysed
supports our interpretation of them being the edge modes of a Laughlin state.
Let us finally stress that the fact that we expect these overlaps to drop does
not mean that our conclusions will be altered: on the contrary, Eq. (6) is a
thermodynamic limit statement and a consequence of dealing with a Laughlin
topological order 222Namely, it does not rely on the assumption of dealing
with a Laughlin wavefunction, but rather on having a single chiral edge-mode
satisfying Eq. (21)., and not of exactly having model Laughlin-like
wavefunctions for the ground state and its excitation.

Figure 11:  Matrix elements
$\mathcal{M}_{n,0}^{(l)}=\left|\Braket{n}{\sum_{i}z_{i}^{l}\hat{n}_{i}}{0}\right|^{2}$
as a function of the magnetic flux per plaquette $\alpha$, for (a,e) $N=2$
(b,f) $N=3$, (c,g) $N=4$ and (d,h) $N=5$ hard-core bosons, for different
lattice sizes. The top (a-d) (/bottom (e-h)) panels are for $l=1$ (/$l=2$).
The static structure factor per state $S_{l}^{\text{$\chi$LL~{}}}/l$ given by
Eq. (4) is shown as black dashed lines.

##  V Scaling with the number of particles

In this section we briefly show and discuss the results for the matrix
elements $\mathcal{M}_{n,0}^{(l)}$ for different number of particles.

In particular, in Fig. 11 the results in the $l=1,2$ cases are shown as a
function of the synthetic magnetic flux per plaquette $\alpha$, for different
lattice sizes. In the top panels (a-d) we show the results for
$\mathcal{M}_{n,0}^{(1)}$. It can be seen that a single state has a
significantly non-zero matrix element in the region of $\alpha$ in which the
ground state is a Laughlin state. Only this state, which we identify with the
dipole excitation of Eq. (23), contributes to the integrated structure factor
of Eq. (4); indeed it can be seen that in the Laughlin region the value of
this matrix element collapses to the static structure factor curve Eq. (9)
(black dashed line), for every $N$ and $L$ displayed here.

In the bottom panels (e-h) we instead show the results for
$\mathcal{M}_{n,0}^{(2)}$. It can be seen that only two states contribute
significantly to the static structure factor Eq. (4); we interpret these two
states as the quadrupole modes of a Laughlin state, Eq. (24). The matrix
elements have been compared to the expected static structure factor per state,
$S_{l}^{\text{$\chi$LL~{}}}/l$ (black dashed line); as we already discussed in
the previous section, it is important to notice that there is no a priori
reason why the two matrix elements should be equally contributing to
$S_{l}^{\text{$\chi$LL~{}}}$: in principle only one of them could be carrying
all the spectral weight (see for example Eq. (6), Eq. (7) and Eq. (8) above).
It can however be seen, at least qualitatively, that as the number of
particles is increased, the expected value of $S_{l}^{\text{$\chi$LL~{}}}$ is
reached.

We also performed a more quantitative test. In particular, in Fig. 12 we show
the integrated dynamic structure factor, namely
$S_{l}=\sum_{n}\mathcal{M}_{n,0}^{(l)}$. In the $l=1$ case (panels (a-d)) a
good agreement with the prediction for $S_{l}^{\text{$\chi$LL~{}}}$ given by
Wen’s theory (black dashed line), Eq. (9), can be seen. In the $l=2$ case
(panels e-h)), even though the integrated structure factor $S_{l}$ deviates
from this prediction, as the number of particles is increased the plateau that
$S_{l}$ exhibits gets closer to $S_{l}^{\text{$\chi$LL~{}}}$. Furthermore,
this finite size correction is compatible with the results (red dotted dashed
line) that we obtained for a Laughlin state in the continuum (see Fig. 5).

Figure 12:  Static structure factors $S_{l}=\sum_{n}\mathcal{M}_{n,0}^{(l)}$
as a function of the magnetic flux per plaquette $\alpha$, normalized to the
static structure factor per contributing state $S_{l}^{\text{$\chi$LL~{}}}/l$
given by Eq. (9); in the various panels, different system sizes are shown:
(a,e) $N=2$ (b,f) $N=3$, (c,g) $N=4$ and (d,h) $N=5$. The expected static
structure factor given by Eq. (9) is shown as black dashed lines. The top
(a-d) (/bottom (e-h)) panels are for $l=1$ (/$l=2$). In the $l=2$ cases
(panels (e-h)), the red dashed-dotted lines are the results for the static
structure factor Eq. (5) for a continuum state (see also Fig. 5).

##  VI Effect of additional harmonic confinement

In this section we analyse the effect of an additional harmonic confinement
which does not spoil the fourfold rotational symmetry of the lattice; namely,
we add to the Harper-Hofstadter Hamiltonian an additional potential term

$\hat{U}_{\text{conf}}=\sum_{i,j}U_{\text{h}}a^{2}\left(\left(i-\frac{M-1}{2}\right)^{2}+\left(j-\frac{M-1}{2}\right)^{2}\right)\hat{n}_{ij}.$
(27)

We study the energies and matrix elements $\mathcal{M}_{n,0}^{(l)}$ as a
function of $U_{\text{h}}$ in Fig. 13.

In the absence of the harmonic confinement, when plotted against the fourfold
discrete rotation symmetry eigenvalue $C_{4}$, the level structure cannot be
easily interpreted visually (see Fig. 13(a)) due to the competing energy-
scales of bulk and edge excitations. However, a weak harmonic confinement
isolates in each symmetry sector a low-lying group of states with the counting
predicted by the edge-mode theory for a Laughlin state (see Fig. 13(f)). The
degeneracy is slightly shifted by perturbations of Wen’s Hamiltonian, leading
to a weakly non-linear $\chi$LL  [65, 44, 45].

Figure 13: (a) Energy levels (with respect to the ground state one) of a
system of $N=4$ hard-core bosons moving on a $10\times 10$ lattice, at
$\alpha=0.18$. The energies are plotted against the fourfold discrete rotation
symmetry eigenvalue $C_{4}$.
(b-e) Energy levels (with respect to the ground state one) for the same
system, as a function of the harmonic confinement strength $U_{\text{h}}$ (see
Eq. (27); the various panels refer to the different $C_{4}$ values: (b)
$C_{4}=0$ (c) $C_{4}=1$ (d) $C_{4}=2$ and (e) $C_{4}=3$. The points are
coloured according to the value of the matrix elements
$\mathcal{M}_{n,0}^{(l)}=\left|\braket{n\left
missing}{\sum_{i}z_{i}^{l}\hat{n}_{i}}{0}\right|^{2}$, for $l=C_{4}\pmod{4}$,
i.e. (b) $l=4$, (c) $l=1$, (d) $l=2$ and (e) $l=3$.
(f) Energy levels (with respect to the ground state one) plotted against the
symmetry eigenvalue $C_{4}$ for the same system, in the presence of an
additional harmonic confinement with strength $U_{h}/J=0.02$. Dashed-blue
ellipses highlight the now well isolated group of edge modes with the counting
predicted for a Laughlin ground state [38, 40].

In the mid panels, Fig. 13(b-e), we show that the ground state in the absence
of the harmonic confinement Eq. (27) is adiabatically connected to the ground
state in its presence, which is unambigously identified to be a Laughlin
topological order by the edge-mode counting seen in Fig. 13(f). Furthermore,
the lowest lying states at $l=1$ and $l=2$ can be seen to be connected with
the low-lying edge excitations highlighted in Fig. 13(f). This supports the
results we previously exhibited in Fig. 10 for the overlaps with the
discretized model wavefunctions for the Laughlin state of Eq. (22) and its
edge excitations Eq. (23) and Eq. (24).

The points in Fig. 13(b-e) are also coloured according to the matrix elements
$\mathcal{M}_{n,0}^{(l)}=\left|\braket{n\left
missing}{\sum_{i}z_{i}^{l}\hat{n}_{i}}{0}\right|^{2}$; it is worth
highlighting some features. First of all in the $C_{4}=3$ plot in Fig. 13
(panel (e)) it can be seen that the $3$ states that carry non-zero weight for
the $l=3$ excitation are those that at large $U_{\text{h}}$ can be identified
with the $3$ edge excitations of a continuum Laughlin liquid at the same
angular momentum variation with respect to the ground state; at small
$U_{\text{h}}$ these states cross with other levels, but they do not mix with
them. Secondly, for $N=4$ particles, also the counting of edge modes in the
$C_{4}=4\equiv 0\pmod{4}$ symmetry sector shown Fig. 13(b) is expected to show
the thermodynamic-limit counting [10, 80] (i.e. a group of $5$ states). For
the parameters we used however this is not so clear. Indeed in Fig. 13(b) it
can be seen that around $U_{\text{h}}/J=0.01$ there is a low-lying group of
$5$ states which couple to the density operator in the same way a non-
harmonically confined Laughlin state does [44, 45] \- namely only $4$ states
couple to it while the fifth one is dark; however, as $U_{\text{h}}$ is
further increased there is a level crossing (for which we lack an intuitive
understanding) at around $U_{\text{h}}/J\approx 0.12$, mixing in this
subspace. This makes the interpretation of the low-lying group of $5$ states
at $U_{\text{h}}/J=0.02$ (seen both in Fig. 13(b) and (f)) a bit ambiguous. A
well isolated group of $5$ states can indeed be obtained (in the region
$0.005\lesssim U_{\text{h}}/J\lesssim 0.015$) by slightly increasing the size
of the lattice, as we show in Fig. 14. Also in this case it can be seen that
one of the five edge states is dark (which may be difficult to spot at a fast
glance because “hidden” below the green coloured points), analogously to the
weakly non-linear Luttinger liquid [44, 45].

It is crucial here to notice that, even though removing the harmonic
confinement (as $U_{\text{h}}$ is decreased) introduces many level crossings
of the edge-modes, the eigenvectors do not mix; even though the energetic
structure of the $\chi$LL is lost, the eigenvectors survive these strong
perturbations which do not close the bulk many-body gap. Roughly speaking,
since the topological order does not change, the structure of the edge theory
must still be the same - and carry the information of the quantized transverse
Hall current.

As a final minor comment, notice in Fig. 13(b) that $\mathcal{M}_{0,0}^{(4)}$
is non-zero when $U_{\text{h}}\simeq 0$, as can also be seen in Fig. 2 of the
main text. For the parameters of Fig. 13(b), at $U_{h}=0$,
$4\mathcal{M}_{0,0}^{(4)}/S_{4}^{\text{$\chi$LL~{}}}\simeq 0.38$. This non-
zero transition matrix element is allowed by the reduced four-fold symmetry of
the lattice. We notice here however that $\mathcal{M}_{0,0}^{(4)}$ goes to
zero as the strength of the harmonic confinement is increased, approaching
thus the value expected in the continuum from angular momentum conservation.

Figure 14: Same as Fig. 13(b), but for an $11\times 11$ lattice.

##  VII Dipolar excitation

###  VII.1 Perturbation theory

In this section we explicitly derive the system’s dipole response to a dipolar
pulsed excitation

$\hat{V}(t)=\lambda(t)\sum_{i}x_{i}\hat{n}_{i};$ (28)

we consider the pulse to have a Gaussian temporal profile
$\lambda(t)=\lambda_{d}\exp(-t^{2}/\tau^{2})$. We monitor the time evolution
of the system’s dipole moment

$\hat{O}=\sum_{i}x_{i}\hat{n}_{i}.$ (29)

We consider our system to start in the ground state $\ket{0}$.

To first order in perturbation theory we get

$\braket{\hat{O}}(t)\simeq-\frac{2}{\hbar}\,\sum_{n}\text{Im}\left[\int_{-\infty}^{t}dt^{\prime}\,\lambda(t^{\prime})e^{-i\omega_{n,0}(t^{\prime}-t)}|\braket{n}{\hat{O}}{0}|^{2}\right]$
(30)

where $\omega_{n,0}=(E_{n}-E_{0})/\hbar$ is the energy of the excited state
$\ket{n}$. Here we used the fact that $\bra{0}\hat{O}\ket{0}=0$.

In the Laughlin phase, at low energies a single mode alone contributes to the
summation; namely the low-energy “dipole” edge mode (see Fig. 1(a) in the main
text). We define $\omega_{d}$ to be the frequency of such a transition.
Moreover,
$|\braket{n}{\hat{O}}{0}|^{2}=\left|\left<n\left|\sum_{i}\frac{x_{i}+iy_{i}}{2}\,\hat{n}_{i}\right|0\right>\right|^{2}=S_{1}/4$.
The first equality is a consequence of the lattice fourfold rotational
symmetry; the second one comes from the fact that the only contribution to the
dipolar static structure factor is exhausted by this single matrix element
(see Fig. 1(b,c) in the main text).

If $t\gg\tau$ we can replace the upper integration limit in Eq. (30) with
$t\rightarrow\infty$, obtaining

$\braket{\hat{O}}(t)\simeq-\frac{S_{1}}{2\hbar}\,\text{Im}\left[\widetilde{\lambda}(\omega_{d})e^{i\omega_{d}t}\right]$
(31)

where we defined the Fourier transform
$\widetilde{\lambda}(\omega_{d})=\int_{-\infty}^{\infty}dt^{\prime}\,\lambda(t^{\prime})e^{-i\omega_{d}t^{\prime}}=\lambda_{d}\tau\sqrt{\pi}\,\exp\left[-\left(\frac{\omega_{d}\tau}{2}\right)^{2}\right]$
of the pulsed excitation $\lambda(t)$. This is the equation we quoted in the
main text.

To conclude this section, let us notice that in order to control the strength
of the perturbation $\tau$ cannot be chosen to be too large, otherwise the
pulse becomes adiabatic with respect to the edge modes, as testified by the
Fourier factor
$\propto\tau\exp\left[-\left(\frac{\omega_{d}\tau}{2}\right)^{2}\right]$. This
problem can be easily circumvented in principle by resonantly shaking the
lattice with $\lambda(t)=\lambda_{d}\exp(-t^{2}/\tau^{2})\cos(\Omega t)$,
$\Omega\approx\omega_{d}$. Then, by neglecting the off-resonant term
$\widetilde{\lambda}(\omega_{d}+\Omega)$, one obtains

$\braket{\hat{O}}(t)\simeq-\frac{S_{1}}{4\hbar}\,\text{Im}\left[\widetilde{\lambda}(\omega_{d}-\Omega)e^{i\omega_{d}t}\right].$
(32)

Notice that all these results are valid only within the scope of linear
perturbation theory. Non-linear effects are discussed in the next section in a
two-level approximation.

###  VII.2 Rabi oscillations

Thanks to the strong non-linearity imposed by the hard-wall condition,
contrary to standard chiral Luttinger liquid (or in the presence of weak non-
linearities) - the spectrum of the edge modes is not linearly dispersing with
the excitation momentum. Highly non-linear behaviour can be seen in Fig. 1 in
the main text. The idea is to exploit this non-linearity in order for the
system to perform Rabi oscillations between the ground state and the dipole
mode as if these two levels constituted an isolated two-level system. In this
section we give some mathematical details on the performed calculations.

In Heisenberg’s representation
$\ket{\widetilde{\psi}}=e^{i\hat{H}t/\hbar}\ket{\psi}$, Schrödinger equation
reads

$i\hbar\frac{\partial\ket{\widetilde{\psi}}}{\partial
t}=\widetilde{V}\ket{\widetilde{\psi}},$ (33)

where $\widetilde{V}=e^{i\hat{H}t/\hbar}\hat{V}e^{-i\hat{H}t/\hbar}$ is the
excitation operator, $\hat{V}=\lambda(t)\hat{O}$. The observable we excite is
again dipolar, $\hat{O}=\sum_{i}x_{i}\hat{n}_{i}$. As we briefly commented at
the end of the previous section, we resonantly shake the lattice

$\lambda(t)=\lambda_{d}e^{-t^{2}/\tau^{2}}\,\cos(\Omega t).$ (34)

If the spectrum of edge modes is linearly dispersing with the momentum, as one
would expect for smooth edges in the thermodynamic limit where the chiral
Luttinger liquid description is expected to be valid, long excitations of this
kind will not only drive a transition from the Laughlin ground state to its
dipole moment, but also higher order excitations which on the long run drive
the system out of these two levels. However, as we already anticipated, strong
non-linearities imposed by the hard-wall confinement together with the small
lattices analysed here make these higher-order transitions off-resonant. We
can therefore effectively describe the dynamics as being restricted to two
levels, and we can expand
$\ket{\widetilde{\psi}}=\widetilde{c}_{0}\ket{0}+\widetilde{c}_{1}\ket{1}$:
the two states represent the ground state and the lowest-lying dipole state
respectively, see Fig. 1 in the main text. Taking the scalar products of Eq.
(33) with $\ket{0}$ and $\ket{1}$ we get

$\begin{cases}i\hbar\frac{\partial\widetilde{c}_{0}}{\partial
t}=\widetilde{c}_{1}\bra{0}\widetilde{V}\ket{1}\\\
i\hbar\frac{\partial\widetilde{c}_{1}}{\partial
t}=\widetilde{c}_{0}\bra{1}\widetilde{V}\ket{0}.\end{cases}$ (35)

Here, we used the fact that $\bra{0}\hat{O}\ket{0}=\bra{1}\hat{O}\ket{1}=0$ by
symmetry. On the other hand, $\bra{1}\hat{O}\ket{0}\equiv
d_{x}=|d_{x}|e^{i\alpha}$ is non-zero. The phase can be absorbed into the
wavefunction amplitudes by defining $b_{0}=\widetilde{c}_{0}e^{i\alpha/2}$,
$b_{1}=\widetilde{c}_{1}e^{-i\alpha/2}$.

Since $\Omega\approx\omega_{d}$ we can neglect fast terms, obtaining

$\begin{cases}i\frac{\partial{b}_{0}}{\partial
t}=\Omega_{R}\,{b}_{1}(t)\,e^{i(\Omega-\omega_{d})t}e^{-t^{2}/\tau^{2}}\\\
i\frac{\partial{b}_{1}}{\partial
t}=\Omega_{R}\,{b}_{0}(t)\,e^{i(\omega_{d}-\Omega)t}e^{-t^{2}/\tau^{2}}.\end{cases}$
(36)

Here we defined an effective Rabi frequency

$\Omega_{R}=\frac{\lambda_{d}|d_{x}|}{2\hbar}.$ (37)

For simplicity we consider the driving to be resonant, namely
$\Omega=\omega_{d}$. The solutions with initial conditions $b_{0}(t\ll\tau)=1$
and $b_{1}(t\ll\tau)=0$ read

$\displaystyle\begin{cases}{b}_{0}(t)=\cos\left(\Lambda(t)\right)\\\
{b}_{1}(t)=-i\sin\left(\Lambda(t)\right)\end{cases}$
$\displaystyle\qquad\Lambda(t)=\frac{1}{2}\sqrt{\pi}\tau\,\Omega_{R}\left(1+\text{erf}\left(\frac{t}{\tau}\right)\right).$
(38)

We can finally write down the time evolution of the dipole moment

$\displaystyle\braket{\hat{O}}(t)$
$\displaystyle=2\text{Re}\bigl{(}b_{0}(t)b_{1}^{*}(t)e^{i\omega_{d}t}|d_{x}|\bigr{)}=-|d_{x}|\sin(2\Lambda(t))\,\sin(\omega_{d}t).$
(39)

As we explained in the previous section, $|d_{x}|=\sqrt{S_{1}}/2=|d|/2$.
Therefore, at large times $t\gg\tau$ (since $\text{erf}(t/\tau)\rightarrow 1$)
we can write

$\displaystyle\braket{\hat{O}}(t\gg\tau)=-\frac{|d|}{2}\sin\left(\sqrt{\pi}\tau\,\frac{\lambda_{d}|d|}{2\hbar}\right)\,\sin(\omega_{d}t)$
(40)

which is the result we quoted in the main text. Notice that the dipole
oscillates as a pure sine-wave with an amplitude
$\frac{|d|}{2}\sin\left(\sqrt{\pi}\tau\,\frac{\lambda_{d}|d|}{2\hbar}\right)$:
this can be interpreted as the system performing Rabi oscillations with
frequency $\Omega_{R}=\frac{\lambda_{d}|d|}{\hbar}$, driven over an effective
time-window $T=\sqrt{\pi}\tau$.

Notice finally that when $\lambda_{d}|d|\tau/\hbar\ll 1$ it correctly recovers
the perturbative result obtained in the previous section, Eq. (32). On the
other hand, the strong anharmonicity due to the confinement allows one to
dynamically get larger dipole response

$|\braket{\hat{O}}|\approx\frac{|d|}{2}$ (41)

provided one can perform a $\pi/2$ pulse which drives the system from its
many-body ground state $\ket{0}$ to the coherent and equal superposition
$(\ket{0}+\ket{1})/\sqrt{2}$; this in practice is limited by the pulse
duration $\sqrt{\pi}\tau$, which is limited by the coherence time of the
experiment.

Figure 15: Energies (with respect to the ground state one) for a system of
$N=3$ bosons on a $9\times 9$ lattice, as a function of the magnetic field per
plaquette $\alpha$. The points have been coloured according to the value of
(a) $\mathcal{M}_{n,0}^{(2)}=\left|\braket{n\left
missing}{\sum_{i}z_{i}^{2}\hat{n}_{i}}{0}\right|^{2}$ and (b)
$\left|\braket{n\left
missing}{\sum_{i}(z_{i}^{*})^{2}\hat{n}_{i}}{0}\right|^{2}$.  Figure 16:
Square modulus of the Fourier transform $\tilde{Q}_{z}(\omega)$ of the
quadrupole response $Q_{z}(t)$ at long times (after the pulsed excitation has
been turned off), as a function of the frequency $\omega$ (normalized to the
relevant quadrupolar frequencies (a) $\omega_{q_{1}}$ and (b)
$\omega_{q_{2}}$. In both cases, two narrow peaks at $\pm 1$ can be seen:
Wen’s sum rule Eq. (9) can be studied by analysing the positive frequency
peaks in the two cases. The yellow horizontal line is the linear perturbation
theory prediction (see Eq. (48)); the red one instead is the non-linear two-
level system prediction (see Eq. (49)).

##  VIII Quadrupolar excitation

###  VIII.1 Perturbation theory

Since the non-linearities introduced by the hard-wall confinement strongly
split the two quadrupolar edge-modes (see Fig. 1(c) in the main text), the
quadrupolar static structure factor is most easily measured by separately
addressing the two transitions so as to measure the two transition matrix
elements. In this section, we label these two states as $\ket{q_{1}}$ and
$\ket{q_{2}}$.

We consider the ground state to be excited by a rotating saddle potential

$\hat{V}=\lambda_{Q}\left(\hat{O}_{x^{2}-y^{2}}\cos(\Omega
t)+\hat{O}_{xy}\sin(\Omega t)\right)e^{-t^{2}/\tau^{2}},$ (42)

where

$\displaystyle\hat{O}_{x^{2}-y^{2}}$
$\displaystyle=\sum_{i}(x_{i}^{2}-y_{i}^{2})\,\hat{n}_{i}$ (43)
$\displaystyle\hat{O}_{xy}$ $\displaystyle=\sum_{i}2x_{i}y_{i}\,\hat{n}_{i}.$
(44)

The perturbation can be rewritten in terms of the complex combination
$z_{i}=x_{i}+iy_{i}$ as

$\hat{V}=\frac{1}{2}\lambda_{Q}\left(\hat{O}_{z^{2}}^{\vphantom{\dagger}}e^{-i\Omega
t}+\hat{O}_{z^{2}}^{\dagger}e^{+i\Omega t}\right)e^{-t^{2}/\tau^{2}}$ (45)

with
$\hat{O}_{z^{2}}=\sum_{i}z_{i}^{2}\,\hat{n}_{i}=\hat{O}_{x^{2}-y^{2}}+i\hat{O}_{xy}$.

If the system was fully rotationally symmetric, due to angular momentum
conservation we would have
$\braket{q_{1}}{\hat{O}_{z^{2}}^{\dagger}}{0}=\braket{q_{2}}{\hat{O}_{z^{2}}^{\dagger}}{0}=0$.
However, our system has only fourfold rotational symmetry. As a consequence,
these matrix elements do not vanish, even though we find them to be
significantly smaller (by a factor $\sim 10$). We show a comparison in Fig.
15. The purpose of the modulation at frequency $\Omega$ is to make transitions
driven by $\hat{O}_{z^{2}}^{\dagger}$ from $\ket{0}$ to the quadrupole modes
$\\{\ket{q_{1}},\ket{q_{2}}\\}$ small; this allows one to focus instead on the
transitions driven by $\hat{O}_{z^{2}}^{\vphantom{\dagger}}$, of which we want
to reconstruct the static structure factor.

Using linear-order perturbation theory we write

$Q_{z}(t)\simeq\frac{i}{\hbar}\int_{-\infty}^{t}\braket{\widetilde{V}(t^{\prime})\widetilde{O}_{z^{2}}(t)-\widetilde{O}_{z^{2}}(t)\widetilde{V}(t^{\prime})}\,dt^{\prime}$
(46)

where the tilde denotes Heisenberg representation with respect to $\hat{H}$
and we defined $Q_{z}(t)=\braket{\hat{O}_{z^{2}}}(t)$. Notice that
$\hat{O}_{z^{2}}(t)=\hat{O}_{x^{2}-y^{2}}+i\hat{O}_{xy}$ is a complex linear
combination of two physical observables.

Neglecting off-resonant terms in Eq. (46) we obtain

$\displaystyle
Q_{z}(t)=\frac{i}{2\hbar}\int_{-\infty}^{t}dt^{\prime}\lambda(t^{\prime})\sum_{n}\left(e^{i\omega_{n,0}t}e^{i(\Omega-\omega_{n,0})t^{\prime}}|\braket{n}{\hat{O}_{z^{2}}^{\vphantom{\dagger}}}{0}|^{2}-e^{-i\omega_{n,0}t}e^{-i(\Omega-\omega_{n,0})t^{\prime}}\braket{0}{\hat{O}_{z^{2}}^{\vphantom{\dagger}}}{n}\braket{n}{\hat{O}_{z^{2}}^{\vphantom{\dagger}}}{0}\right).$
(47)

For large enough $\tau$ and $\Omega\simeq\omega_{q}$ only one of the
quadrupole modes $\ket{q}$ will be excited. For $t\gg\tau$ we write

$\displaystyle
Q_{z}(t\gg\tau)=\frac{i}{2\hbar}\left(e^{i\omega_{q}t}\widetilde{\lambda}(\omega_{q}-\Omega)|\braket{q}{\hat{O}_{z^{2}}^{\vphantom{\dagger}}}{0}|^{2}-e^{-i\omega_{q}t}\widetilde{\lambda}^{*}(\omega_{q}-\Omega)\braket{0}{\hat{O}_{z^{2}}^{\vphantom{\dagger}}}{q}\braket{q}{\hat{O}_{z^{2}}^{\vphantom{\dagger}}}{0}\right).$
(48)

By resolving the two frequencies, for example through Fourier analysis, the
relevant amplitude $|\braket{q}{\hat{O}_{z^{2}}^{\vphantom{\dagger}}}{0}|^{2}$
can be extracted; once this is done for the first quadrupole mode
$\ket{q_{1}}$, the same can be done for the second one, $\ket{q_{2}}$ and one
can reconstruct the static structure factor and match it with Wen’s
prediction.

We show the discrete Fourier transforms (taken only for $t\gtrsim\tau$) of
$Q_{z}(t)$ for the two quadrupolar transitions in Fig. 16. Frequencies are
measured in units of the relevant transition frequency $\omega_{q}$. Peaks at
$\omega/\omega_{q}=\pm 1$ can be seen; as expected, the one at negative
frequency, which emerges because of the reduced rotational symmetry of the
lattice, can be seen to be smaller than the relevant one at positive
frequency. The peak-amplitude of the latter is compared with the amplitude
that we obtain from the perturbation theory result (yellow-dashed line) and
with the non-linear two-level system result Eq. (49). It can be seen that the
latter compares successfully with the numerical results and can be used to
estimate the relevant matrix element $|\braket{q}{\hat{O}_{z^{2}}}{0}|^{2}$.

Figure 17: Positive frequency amplitude of $Q_{z}(t\gg\tau)$ as a function of
the pulse angle $\Omega_{R}T$ ($T=\sqrt{\pi}\tau$), for different values of
the pulse duration $\tau$. The resonant two-level Rabi oscillation prediction
(see Eq. (49)) is shown as a black-dashed line, and the linear approximation
(see Eq. (48)) is shown as a dash-dotted brown line.

###  VIII.2 Rabi oscillations

Analogously to the dipolar-excitation case, due to the strong non-linearities
imposed by the lattice the ground-state to quadrupole transition can form an
isolated two-level manifold within the exponentially large set of many-body
states. The quantitative analysis is the same as in the case of the dipolar
excitation, with the additional complication of the reduced rotational
symmetry of the lattice. We find, for a resonant excitation
($\Omega=\omega_{q}$)

$Q_{z}(t\gg\tau)=i\,\underbrace{\frac{\left|Q\right|}{2}\sin\left(\sqrt{\pi}\tau\,\frac{\lambda_{Q}\left|Q\right|}{\hbar}\right)}_{A_{Q}^{(+)}}\,e^{i\omega_{q}t}-i\underbrace{\frac{Q^{\prime}e^{i\alpha}}{2}\sin\left(\sqrt{\pi}\tau\,\frac{\lambda_{Q}\left|Q\right|}{\hbar}\right)}_{A_{Q}^{(-)}}\,e^{-i\omega_{q}t}$
(49)

where $Q=\braket{q}{\hat{O}_{z^{2}}^{\vphantom{\dagger}}}{0}$,
$Q^{\prime}=\braket{0}{\hat{O}_{z^{2}}^{\vphantom{\dagger}}}{q}$ and
$\alpha=\arg\left(Q\right)$. To linear order in perturbation theory, this
result reduces to the perturbative one given in Eq. (48).

We compare the numerical amplitude of the peak at positive frequency,
$A_{Q}^{(+)}$, with the above result Eq. (49) in Fig. 17, for the case of the
lower quadrupolar transition. It can be seen that for long-enough pulses the
predictions nicely match the analytical result.