Realization of a fractional quantum Hall state with ultracold atoms Julian L eonard1Sooshin Kim1Joyce Kwan1Perrin Segura1 Fabian Grusdt2 3C ecile Repellin4Nathan Goldman5and Markus Greiner1

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Realization of a fractional quantum Hall state with ultracold atoms

Julian L´eonard,1, ∗Sooshin Kim,1Joyce Kwan,1Perrin Segura,1

Fabian Grusdt,2, 3 C´ecile Repellin,4Nathan Goldman,5and Markus Greiner1

1Department of Physics, Harvard University, Cambridge, Massachusetts 02138, USA

2Department of Physics and ASC, LMU M¨unchen, Theresienstr. 37, M¨unchen D-80333, Germany

3Munich Center for Quantum Science and Technology (MCQST), Schellingstr. 4, D-80799 M¨unchen, Germany

4Univ. Grenoble Alpes, CNRS, LPMMC, 38000 Grenoble, France

5CENOLI, Universit´e Libre de Bruxelles, CP 231, Campus Plaine, B-1050 Brussels, Belgium

Strongly interacting topological matter [1] ex-

hibits fundamentally new phenomena with po-

tential applications in quantum information tech-

nology [2, 3]. Emblematic instances are frac-

tional quantum Hall states [4], where the inter-

play of magnetic ﬁelds and strong interactions

gives rise to fractionally charged quasi-particles,

long-ranged entanglement, and anyonic exchange

statistics. Progress in engineering synthetic mag-

netic ﬁelds [5–21] has raised the hope to cre-

ate these exotic states in controlled quantum

systems. However, except for a recent Laugh-

lin state of light [22], preparing fractional quan-

tum Hall states in engineered systems remains

elusive. Here, we realize a fractional quantum

Hall (FQH) state with ultracold atoms in an op-

tical lattice. The state is a lattice version of

a bosonic ν= 1/2Laughlin state [4, 23] with

two particles on sixteen sites. This minimal sys-

tem already captures many hallmark features of

Laughlin-type FQH states [24–28]: we observe a

suppression of two-body interactions, we ﬁnd a

distinctive vortex structure in the density corre-

lations, and we measure a fractional Hall conduc-

tivity of σH/σ0= 0.6(2) via the bulk response to

a magnetic perturbation. Furthermore, by tun-

ing the magnetic ﬁeld we map out the transition

point between the normal and the FQH regime

through a spectroscopic probe of the many-body

gap. Our work provides a starting point for ex-

ploring highly entangled topological matter with

ultracold atoms [29–33].

The FQH eﬀect emerges in two-dimensional electron

gases from the combination of a magnetic ﬁeld and repul-

sive interactions [4]. The magnetic ﬁeld quenches the ki-

netic energy into highly degenerate Landau levels, among

which the particles arrange themselves to minimize their

interaction energy. In many cases FQH states are de-

scribed by Laughlin’s wave function [23], whose char-

acteristic pairwise correlated vortex motion results in a

screening of interactions and strong anti-correlations at

distances below the vortex size (Fig. 1a). FQH states

show a topological robustness with exotic properties that

are unseen in non-interacting systems, including quasi-

particles with fractional charge, non-local topological en-

tanglement, and anyonic exchange statistics [4].

The desire to study these phenomena in a controlled

environment has triggered eﬀort to realize FQH states

in quantum-engineered systems. Since the constituents

of those platforms are typically charge neutral, synthetic

magnetic ﬁelds are introduced through the Coriolis force

in rotating systems [5–8, 35], or by engineering geo-

B = B ez

Normal Fractional quantum Hall (FQH)

Atom

B = 0

Vortex

motion

yMott

insulator

Initial

state

Inverted

preparation

Ground state

overlap

Snapshots

Adiabatic

preparation

Jφ

FIG. 1. Realizing a fractional quantum Hall state

in an optical lattice. a, Without magnetic ﬁeld, a two-

dimensional gas remains in a superﬂuid (normal) state with

weak correlations. In the presence of a strong magnetic ﬁeld,

the system may enter a FQH state with strong correlations,

which (for Laughlin states) manifest through a simultaneous

vortex motion between all pairs of atoms.The system thereby

minimizes interactions while incorporating the angular mo-

mentum induced by the magnetic ﬁeld. b, We realize such

a system with two bosonic 87Rb atoms in an optical lattice

potential with 4 ×4 sites. The system is placed in the focus

of a high-resolution imaging system, which allows us to take

projective measurements of the quantum state with single lat-

tice site resolution. The system is described by the Harper-

Hofstadter model with tunneling rates Kand Jalong xand

y, respectively, magnetic ﬂux φper plaquette and on-site in-

teraction U.c, The experimental protocol begins with a Mott

insulator, from which we prepare the initial state with both

atoms on neighbouring edge sites. We adiabatically change

the Hamiltonian parameters until reaching the FQH state.

We either take snapshots of the ﬁnal state, or we invert the

preparation protocol and map the ﬁnal state back to the ini-

tial state in order to characterize the adiabaticity of the pro-

tocol.

arXiv:2210.10919v2 [cond-mat.quant-gas] 29 Oct 2022

Initial state Tube Coupled wires

2D tilted Delocalize Final state

1 2 3 4 5 6

Theory Experiment

1D system 2D system FQH state

Density

Tilt ∆y (ћ/τ)

0.5

Tunneling J (ћ/τ)

00.51

0.8

1.2

Energy gap (ћ/τ)

2 3 4 5e

Ramp time (τ)

Fidelity Spectrum (ћ/τ)

1 2 3 4 5 6

0 0.1 0.2 0.3 0.4

Flux /2

1.1

1.2

1.3

1.4

Tunneling K (ћ/τ)

Energy gap (ћ/τ)

0.2 0.22 0.24 0.26 0.28 0.30

0.2

0.4

0.6

0.8

Fidelity

K/J = 1.19(3)

K/J = 1.00(3)

0.1

0.2

0.3

Energy gap (ћ/τ)

Normal

K = 1.2J

K = J

Coupled wires (K > J) FQH (K = J)

dequalize

tunneling

φc

Tunneling K (ћ/τ)

Tilt ∆x (ћ/τ)

∆x � 0

Momentum

φ/a Momentum

5 6

0.5

0 20 40 60 80 100

Normal

FQH

Flux φ/2π

FIG. 2. FQH state preparation and gap diagram. a, Adiabatic preparation: (1) The ground state of the initial

Hamiltonian corresponds to two repulsively interacting bosons on neighbouring sites. (2) Enabling tunneling Jalong yand

reducing the gradient ∆yhomogenously delocalizes the particles into one column. (3) Switch on tunneling Kalong xin the

presence of a strong tilt and at ﬂux φ/2π= 0.27. (4) Particles spread over the entire 2D system as the tilt is reduced. (5)

Up to this point K > J, such that the system resembles coupled horizontal wires. (6) Ramp to K=Jto reach the ﬁnal

state. b, Measurements of the site-resolved density conﬁrm the delocalization into the 2D box potential, in agreement with

exact numerical calculations [34]. c, The preparation scheme ensures optimal adiabaticity by avoiding closing of the energy

gap between the ground state and the excited states (shown in units of the inverse tunneling time τ= 4.3(1) ms), as conﬁrmed

by numerical calculations of the many-body spectrum. d, The robustness of the scheme can be understood in a coupled-wire

picture, where the quadratic dispersions of weakly coupled rows are oﬀset by multiples of the momentum φ/a (with athe

lattice constant). While excited single-particle states get shifted for ∆x>0 (blue circles), the ground state in each wire

remains robust. The two-body ground state is reminiscent of a charge density wave and adiabatically connects to a FQH

state as the tunneling ratio approaches K=J.e, We quantify the preparation ﬁdelity through the quantum state overlap

F=hψ0|ˆρFinal|ψ0i, inferred from measurements after inversion of the protocol. Despite a decreasing energy gap to the ﬁrst

(solid line) and higher excited states (shaded lines) in the energy spectrum during the preparation, a signiﬁcant population

remains in the ground state throughout the evolution. f, For K=Jthe numerically calculated energy gap shows a closing,

whereas it remains open for K > J.g, We spectroscopically reveal the gap closing through a loss of adiabaticity, signaled by

a reduction of the ground-state overlap when preparing the system at the ﬂux φc/2π≈0.25 (dark). The reduction is absent

when ending the preparation at step (5) with K > J (light blue). Errorbars denote the s.e.m. and are smaller than the marker

size if not visible.

metric phases [9–13, 15, 18, 19]. Recently, interaction-

induced behaviour has been observed in several systems

[14, 16, 20, 21], including a Laughlin state made of light

[22]. Quantum gases in driven optical lattices [36] con-

stitute a particularly promising platform to study FQH

physics due to their exquisite control and large attain-

able system sizes, yet, reaching the strongly interacting

regime remains a challenge.

Here, we realize a bosonic FQH state in a bottom-up

approach using two particles in a driven optical lattice.

The presence of few-particle FQH states in lattice mod-

els, also called fractional Chern insulators, is numerically

well-established [24–28, 30, 37]. Conceptually they orig-

inate from ﬂat Chern bands that take the role of the

Landau levels. The proposed preparation schemes for

those states, however, have exceeded experimental capa-

bilities so far [38–40]. In our work, we devise and apply

a novel adiabatic state preparation scheme, enabled by

site-resolved control in a quantum gas microscope with

bosonic 87Rb (Fig. 1b) [14, 41]. We verify that the pre-

pared state corresponds to the target FQH state by in-

verting the preparation scheme (Fig. 1c), and we sam-

ple density snapshots from the prepared state to conﬁrm

that it exhibits key properties of a FQH state, including

a screening of two-body interactions, a vortex structure

in the density correlations, and a fractional Hall conduc-

Photoassociation

0.2 0.22 0.24 0.26 0.28 0.3

0.01

0.02

0.03

0.04

pDoublon

0.2 0.21

0.01

0.02

0.03

0.04

pDoublon

0.27 0.28

Flux φ/2πFlux φ/2π

φcφc

Normal FQH Normal FQH

Separation

FIG. 3. Suppression of two-body interactions. We ﬁnd a

reduction of the doublon fraction in the FQH state, revealing

the screening of interactions in the many-body wave function.

a, Doublon fraction measurement by photoassociation, con-

verting the doubly occupied lattice site into an empty one. b,

Doublon fraction measurement by separating the particles on

two diﬀerent lattice sites prior to ﬂuorescence imaging. Solid

lines show exact calculations for the ground state, dashed

lines take into account the ﬁnite ground-state overlap [34].

All errorbars denote the s.e.m.

tivity.

The system is governed by the interacting Harper-

Hofstadter model (Fig. 1c), which describes the motion

of particles on a square lattice in the presence of a mag-

netic ﬁeld. In our setup the eﬀective magnetic ﬁeld is

realized by Floquet engineering complex tunneling ampli-

tudes with Raman-assisted tunneling processes [34, 36].

Within the Floquet-engineered Hamiltonian, we achieve

independent control of the ﬂux φ/2πper unit cell, the

tunneling rates Kand Jalong xand y, respectively, as

well as the gradients ∆xand ∆y. The on-site interac-

tion Uremains constant and large compared to all other

energy scales.

The state preparation begins from an initial state of

two localized atoms in the absence of any tunneling.

First, we increase the tunneling Jalong yin the pres-

ence of a gradient ∆y, while tunneling along xremains

inhibited (Fig. 2a). Since Jremains approximately con-

stant for the remainder of the protocol [34], it sets the

tunneling time τ=~/J = 4.3(1) ms and the interac-

tion strength U= 6.7(1) J. Subsequently ∆yis adiabat-

ically removed and we obtain a one-dimensional system

in its ground state. A similar procedure is then per-

formed along x: tunneling Kis increased at constant

ﬂux φ/2π= 0.27 and gradient ∆x, then the gradient is

adiabatically removed. Up to this point we keep the tun-

neling ratio at K/J = 1.19(3). In a ﬁnal step we bring

the tunneling amplitudes to K=Jto reach the tar-

get state. At each step of the evolution we measure the

system’s density distribution and ﬁnd agreement with an

exact numerical prediction at the respective Hamiltonian

parameters (Fig. 2b).

Our preparation scheme at asymmetric tunneling K >

J(Fig. 2c,d) can be understood in a coupled-wire picture

[34, 42], where the parabolic dispersions of weakly cou-

pled rows are oﬀset in momentum by multiples of φ/a.

While the excited state energies in each parabola are af-

fected by the tilt ∆xdue to their directional momentum,

the ground state energies remain independent. As the

tunneling ratio approaches K=J, anti-correlated densi-

ties on neighbouring sites, reminiscent of a charge density

wave state, are converted into Laughlin orbitals.

The success probability of the state preparation is

given by the ﬁdelity F=hψ0|ˆρFinal|ψ0i, which measures

the overlap between the density operator ˆρFinal describing

the state after the preparation protocol, and the ground

state |ψ0iat the ﬁnal Hamiltonian parameters. Since |ψ0i

is a delocalized and entangled state, measuring Fdirectly

with local observables is not possible. Instead, we map

|ψ0iback to the initial state by following the prepara-

tion with an identical, but reversed protocol. Assuming

that the evolution during both ways is independent, the

ﬁnal ground state population is given by F2, which can

be directly measured because it equals the probability to

measure the initial density distribution. We ﬁnd a dom-

inant ground state population throughout the evolution,

and a ﬁdelity of F= 43(6)% to prepare the ﬁnal state

(Fig. 2e).

The ﬂux φ/2πat which the lowest bosonic Laughlin

state is expected to stabilize corresponds to a ﬁlling fac-

tor of ν=N/Nφ= 1/2, such that the system contains

twice the number of magnetic ﬂux quanta Nφthan the

number of charge carriers N. In systems with sharp edges

this condition is only approximate and a systematic un-

derstanding of the ﬁnite size eﬀects on the ﬁlling factor

is still lacking [28]. In order to map out the transition

between the normal state and the FQH state we use the

adiabaticity of the preparation scheme as a spectroscopic

signature for the energy gap (Fig. 2g). The ﬁdelity Fis

limited by the smallest energy gap during the preparation

protocol, and therefore decreases when the energy of the

ground and excited states approach each other. Repeat-

ing the protocol for diﬀerent ﬂux values shows a break-

down of the adiabaticity at φ/2π≈0.25, indicating the

location of the transition point. The observed transition

point is in agreement with exact numerical calculations of

the gap diagram (Fig. 2f,g). In contrast, when repeating

the measurement at K > J no breakdown of adiabatic-

ity is visible, indicating that the many-body gap remains

open until we reach homogeneous tunneling K=J.

A hallmark of Laughlin-type FQH states is the screen-

ing of on-site interactions due to the pairwise vortex

motion. In our two-particle system, the interaction en-

ergy simpliﬁes to Eint =hψ0|PiUˆni(ˆni−1)/2|ψ0i=

U×pDoublon, where ˆniis the number operator on site i

and pDoublon is the global probability to observe the two

particles on the same lattice site. We measure the dou-

blon probability in two diﬀerent ways. In a ﬁrst set of

measurements we perform photo-association of the dou-

blons into molecular states, whose excess energy ejects

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RealizationofafractionalquantumHallstatewithultracoldatomsJulianLeonard,1,SooshinKim,1JoyceKwan,1PerrinSegura,1FabianGrusdt,2,3CecileRepellin,4NathanGoldman,5andMarkusGreiner11DepartmentofPhysics,HarvardUniversity,Cambridge,Massachusetts02138,USA2DepartmentofPhysicsandASC,LMUMunchen,Theresienstr...

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Realization of a fractional quantum Hall state with ultracold atoms Julian L eonard1Sooshin Kim1Joyce Kwan1Perrin Segura1 Fabian Grusdt2 3C ecile Repellin4Nathan Goldman5and Markus Greiner1

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