Topological matter

with arrays of Rydberg atoms

Laboratoire Charles Fabry,Institut d’Optique, CNRS, Palaiseau, France

GdR COMPLEXEDecember 1st 2020

V. Lienhard, S. de Léséleuc, P. Scholl, D. Barredo,

K.N. Schymik, H. Williams, T. Lahaye and A. Browaeys

Open problems in condensed-matter physics 1/21

Quantum many-body systems, prominent effect of interactions

Quantum magnetism

Herbertsmithite

Nature 492, 406 (2012)

Superconductivity

Charles O'Rear Getty Images

Many-body localization

Science 352, 1547 (2016)

Open questions : effect of spin-statistics, emergence of exotic phases, out-of-

equilibrium dynamics…

Synthetic quantum matter

Ion trapsSuperconducting

qubits

Rydberg atoms

Optical tweezers

Quantum gas

microscopes

Well-controlled assemblies of interacting particles

Implement a many-body Hamiltonian to mimic condensed-matter problems

Cre

dit

: Hél

oïs

e C

ho

cho

is Georgescu et al., RMP 86, 153 (2014)

Quantum simulation

o Study of spin systems

o Study of transport phenomena

VL et al., PRX 8, 021070 (2018)

H. Labuhn et al., Nature 534, 667 (2016)

S. de Léséleuc et al., Science 365, 7757 (2019)

VL et al., PRX 10, 021031 (2020)

2/21

Our platform

Arrays of single atoms

10 µm

Interaction between Rydberg atoms

+

e-

+

e-

D. Barredo et al., Science 354, 1021 (2016)

A. Browaeys and T. Lahaye, Nat. Phys. 16, 132 (2020)

3/21

Topological phases

Non-interacting

fermions

With interactions?Fractional Chern insulators

Exotic statistics

No demonstration

on artificial systemsBosons or fermions

Effect of spin-statistics

M. König et al., Science 318, 766 (2007)

Topological insulatorsBand structure

- Excitation gap

- Edge states

Conduction

band

Valence

band

Fermi

level

Edge

states

4/21

Outline

Implementation

of the SSH model

Observation of

edge states

Towards 2D

edge states

Outline

Implementation

of the SSH model

Observation of

edge states

Towards 2D

edge states

The SSH model

Normal

Chain of dimers with or without weakly coupled edges: two configurations

No coupling between subchains: sub-lattice symmetry

Infinite 1D-chain

of dimers

Strong link Weak link

x

Finite-size chain cases:

Topological

Electronic transport

in polyacetylene

Reviews: Asboth, arXiv:1509.02295; Cooper, arXiv:1803.00249

Su, Schrieffer and HeegerPRL 42, 1698 (1979)

5/21

The SSH model, single-particle spectrum

Normal Topological

Edge

states

6/21

Exchange of a P excitation Hopping of a particle

Resonant dipole-dipole interaction

0 particle

1 particle

Mapping to a bosonic problem

Microwave field:o Adds/removes particles coherently in the chain

o Probe energy spectrum

Atom 1 Atom 2

7/21

Spatial dependence 8/21

The SSH model, experimental realization

TopologicalNormal

θMθM

x x

14-atom chain 14-atom chain

9/21

Outline

Implementation

of the SSH model

Observation of

edge states

Towards 2D

edge states

Mic

row

ave d

etu

nin

g (

MH

z)

Site

Lower band

Site

Edge

states

Band gap

Single-particle regime

Vacuum

Single-particle

spectrum

Normal Topological

Observation of edge states,

signature of topology

Microwave probe

10/21

Hybridization of edge states

Prepare one excitation on left edge site

and

Coherent exchange between left and right

Chain length

Nearest-neighbour

Long-range

Time (µs)

Site

Site

Site

11/21

Going to the many-body regime

S. de Léséleuc et al., Science 365, 775 (2019)

0 particle

1 particle

Mapping to a bosonic problem

2 particles

System of hard-core bosons

Adiabatic passage

Spectroscopy from

a half-filled bulk Observation of a

symmetry protected

topological (SPT) phase

12/21

Outline

Implementation

of the SSH model

Observation of

edge states

Towards 2D

edge states

Peierls phases

Complex-valued hopping

(Peierls phase)

Topological band structure

(Haldane model)

Effective magnetic field for neutral particles

Artificial gauge fieldsN. R. Cooper et al., RMP 91, 015005 (2019)

J. Dalibard et al., RMP 83, 1523 (2011)

D. Jaksch et al., NJP 5, 56 (2003)

Laser-assisted

tunneling

M. Aidelsburger et al.,

PRL 107, 255301 (2011)

Driven lattices

J. Struck et al.,

PRL 108, 255304 (2012)

Superconducting

qubits

P. Roushan et al.,

Nat. Phys. 13, 146 (2017)

On our platform : engineer a Peierls phase using the intrinsic

spin-orbit coupling of the dipolar interaction

VL et al., PRX 10, 021031 (2020)

13/21

Spin-orbit coupling and dipole-dipole interaction

Atom 1 Atom 2

Atomic plane

B

Intrinsic spin-orbit coupling

Perturbative regime Virtual spin-flip hopping to pick up a phase

14/21

Perturbative approach

Atomic plane

B

1

2

3

Energy levels

Two ways to go from site 1 to site 3:

- Direct hopping

- Off-resonant hopping

Combination of the two processes

Implementation of a

complex hopping amplitudeArtificial gauge field

(Peierls phase) Cold atom platforms: Spielman, Bloch, Sengstock, Esslinger, Beugnon, Fallani,…

15/21

Required flux for a chiral motion

Coupling P. Roushan et al., Nature Physics 13, 146 (2017)

Total flux1

2

3

16/21

Chiral motion on a triangle

B

1

2

3 B

1

2

3

VL et al., PRX 10, 021031 (2020)

17/21

Many-body regime?

1

2

3B

1

2

3B

Interacting hopping particles

Fourth-order process

18/21

A density-dependent Peierls phase

1

2

3

Hopping amplitude

between 1 and 3

o If complex-valued

o If real

Mapping to an anyonic statistics (with an additional particle-hole transformation)

1 anyon = 1 hole = 2 particles

Single-anyon Hamiltonian has real hopping amplitudes

No chirality for the propagation of the hole

The Hamiltonian for two anyons has complex hopping amplitudes

due to their exchange statistics

Chirality for the propagation of one particle

19/21

Summary and outlookSummary

A platform to build synthetic matter : arbitrary geometries and tunable interactions

Implementation of complex hopping amplitude using spin-orbit coupling

Observation of a 1D topological phase

20/21

Summary and outlook

Chiral edge states on

honeycomb lattices

Many-body regime?- Density-dependent Peierls phase

- Fractional bosonic Chern insulator?

Outlook

Summary

A platform to build synthetic matter : arbitrary geometries and tunable interactions

Observation of spin-ordered phases

Implementation of complex hopping amplitude using spin-orbit coupling

21/21

The Rydberg team in Palaiseau

Antoine

Browaeys

Sylvain de

Léséleuc

Thierry

Lahaye

Daniel

Barredo

Vincent

Lienhard

Florence

Nogrette

Pascal

Scholl

H.-P. Büchler N. Lang S. WeberTheory:

https://atom-tweezers-io.org/

Kai-Niklas

Schymik

Hannah

Williams

M. Fleischhauer