Quantum simulation Engineering quantum systems atom-by-atomMonika Aidelsburger
Ludwig-Maximilians Universität München
Munich Center forQuantum Science & Technology
www.sqm.physik.lmu.deYb lab

Cs labMotivation

Quantum simulation with ultracold atoms in optical lattices:•Atoms confined in periodic potentials

UlaserJ• Quantum simulation of Hubbard models• Access to local observables using
quantum gas microscopes

Cs atoms

Ĥ =I. Bloch et al. Rev. Mod. Phys. 80, 885 (2008); C. Gross and I. Bloch, Science 357 (2017)

JX

hi,jiXU†âi âj +n̂i (n̂i2 i1)

Quantum simulation with neutral atoms
Anti-ferromagnetic correlations in theFermi-Hubbard model:

Thermalization of isolated
quantum-many body systems:

Ground states

Dynamics

A. Maruzenko, ... M. Greiner, Nature (2017)

J. Y. Choi, ..., I. Bloch, Science 352, 1547 (2016)

Direct implementation of Hamiltonian

Quantum Gas Microscopy

Quantum Gas Microscopy
Single 2D
layer

Fluorescence imaging in deep lattices• Single-site resolved observables(correlations, full counting statistics, …)• Site-selective addressing
High-NA objective

W. S. Bakr et al., Nature 462, 74 (2009); J. F. Sherson et al. Nature 467, 68 (2010)
L. Cheuk et al. PRL 114, 193001 (2015); E. Haller et al. Nat. Phys. 11, 738 (2015); M. F. Parsons et al., PRL 114, 213002 (2015), ...

Quantum Gas Microscopy
Selected examples:

Rb atoms, Greiner

Yb atoms, Kozuma

Rb atoms, BlochLi atoms, Choi

Review: C. Gross and W. Bakr, Nature Phys. (2021)

K atoms, Zwierlein

Li atoms, Schauss

K atoms, Kuhr

K atoms, Thywissen• Large homogeneous systems• High fidelity preparation &detection• Novel model Hamiltonians• Large energy scales

Realization of large homogeneous systems

Potential shaping
using a digital
micromirror device~ 50 × 50 sites, 2500 atoms~ 0.98 filling

A. Impertro, J. F. Wienand, S. Häfele, H. von Raven, S. Hubele, T. Klostermann, C. R. Cabrera, I. Bloch, MA, arXiv:2212.11974
T. Klostermann et al, Phys. Rev. A 105, 2022; PhD Theses Klostermann & von Raven; Master thesis Hubele

Benchmarking via thermalization dynamics

Hard-core bosons in 1D
Initial state: CDW

4J (Lieb-Robinson bound)

Ballistic spreading of density-density correlations over large distances!

ilerPnim!
yraQuantum Gas Microscopy
Selected examples:

Rb atoms, Greiner

Rb atoms, BlochK atoms, Zwierlein

K atoms, Kuhr

K atoms, Thywissen• Repetition rate
• Local control
Yb atoms, Kozuma

Li atoms, Choi

Review: C. Gross and W. Bakr, Nature Phys. (2021)

Li atoms, SchaussRepetition rate ~20s

Transport

Magneto-optical trap& Raman cooling: ~μKScience cell:~10-11mbar / ~nKdegenerate quantum gas

Atomsource
Bose-Einsteincondensate

Degenerate
Fermi Gas

Tweezer-assisted preparation

Tweezer programmable quantum walks• Fast cycle times by direct laser coolingin deep optical traps• Initial states require rearrangement of atoms

Young,..., Kaufman, Science 377, 885 (2022)

see also work by C. Regal and W. Bakr

Ebadi, ..., Lukin, Nature 595, 227 (2021)

What can we simulate?

Topological phases of matter

Integer & fractional
quantum Hall insulators

Weyl semimetals
Topological insulatorsin 2D & 3D

M. König et al., Science (2007)
A. Roth et al., Science (2009)
K. Klitzing, Rev. Mod. Phys. (1986)
Stormer et al., Rev. Mod. Phys. (1999)

L. Lu et al., Science (2015)
S.-Y. Xu et al., Science (2015)...
Review: X.-L. Qi & S.-C. Zhang, Rev. Mod. Phys. (2011)

Realizing artificial magnetic fields
Non-interactingĤ =lattice Hamiltonian:

X†Jij âi âj + h.c.

hi,jiByCharged particles in magnetic field→ acquire geometric phase
φ3Jeiφφ4

ΦPeierls substitution: Jij ! Jij eZ xiqAdl , B = r ⇥ Aj =~ xj

φ2φ1

Phase aroundclosed loop:=Xj = 2⇡B0xi jΦB: magn. flux
Φ0= h/q: magn. flux quantumRealizing artificial magnetic fields
Non-interactingĤ =lattice Hamiltonian:X†
Jij âi âj + h.c.

hi,ji

B
y

Charged particles in magnetic field→ acquire geometric phasex

φ3

Jeiφφ4ΦLarge magnetic fields on the order of
one flux quantum (>1000 T)

φ2φ1

Phase aroundclosed loop:=X

j = 2⇡

B0

ΦB: magn. flux
Φ0= h/q: magn. flux quantum

Floquet engineering• Time-periodic driven HamiltonianĤ(t) = Ĥ(t + T )• Stroboscopic time evolution governed byF
effective Floquet Hamiltonian Ĥ

Û (T, 0) = exp✓iFT Ĥ~◆Engineer Ĥ F with topological properties!

N. Goldman et al. PRX (2014); M. Bukov et al. Adv. in Phys. (2015); A. Eckardt, Rev. Mod. Phys. (2017)

Lattice 'cyclotron' orbits

Mean atom position during the evolution

e-0.2

Y /dy

0.1

5.5

e0-0.1Time (ms)BB-0.2-0.4-0.2

0X /dxMA, ..., Bloch, Phys. Rev. Lett. 107, 255301 (2011)0.2

0Dy in synthetic
dimensions!

Chalopin, ...,Nascimbene,Nat. Phys. (2020)

Topological lattice models
Hofstadter model

Haldane model

Harper, Proc. Phys. Soc., Sect.A 68, 874 (1955)
Azbel, Zh. Eksp. Teor. Fiz. 46, 929 (1964)
Hofstadter, PRB 14, 2239 (1976)

Haldane, PRL 61, 2015 (1988)

2π2π2πΦΦΦ

3π/2

3π/23π/2ΦΦΦ

πJĤ =JX⇣

Φ

π/2πΦ

π/2πΦJeinĤ =π/2††âm+1,n âm,n + âm,n+1 âm,n + h.c.

m,nReview: N. Cooper et al. Rev. Mod. Phys. 91, 015005 (2019)

Xhiji⌘†tij ĉi ĉj +Xhhijiieiij0 †tij ĉi ĉj +X †ĉi ĉi
ABi2A

MA et al., PRL (2013); H. Miyake et al., PRL (2013)
E. M. Tai et al., Nature (2017)
G. Jotzu et al., Nature (2014) ;Tarnowski et al., Nat. Comm. (2019)

𝜇Topological invariants

Chern number:<latexit sha1_base64="2Dm2l3CDaxpUaB8ShKS1WXDjG7w=">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</latexit>

Berry curvature:

1
C =2⇡
µZ

µ 2

⌦ d q
BZ

⌦µ = i h@qx uµ |@qy uµ i

|uµ (q)i : periodic Bloch function: band index

h@qy uµ |@qx uµ i

M. Atala, et al., Nat. Phys. (2013); L. Duca et al., Science (2015)
G. Jotzu et al., Nature (2014); M. A. et al., Nature Phys. (2015)
N. Fläschner, Science (2016); T. Li, Science (2016)
Tarnowski et al., Nat. Comm. (2019);
L. Asteria et al., Nat. Phys. (2019);
B. Rem et al., Nat. Phys. (2019); .....
Weitenberg/Sengstock(selected) experimental results
Bosonic flux ladders

Hall deflection

2-particle Laughlin state

Externalforce F
AtomcloudJ

Keiφ

Δ

Anomalous
velocity

Observation of chiral
Meissner-like currents

Chern-number measurementwith bosonic atoms

Towards fractional quantum
Hall physics

Atala, MA, ..., Bloch, Nat. Phys. (2014)

MA, ..., Bloch, Nat. Phys. (2015)

Léonard, ..., Greiner, arXiv:2210.10919

Reviews: N. Cooper et al. Rev. Mod. Phys. 91, 015005 (2019); MA et al. C. R. Physique 19, 394-432 (2018),

How to generate topologicaledge modes?

Synthetic dimensions

Mancini, ..., Fallani, Science 349 (2015)
Stuhl, ..., Spielman, Science 349 (2015)

See also: Chalopin, ..., Nascimbène, Nature Physics (2020)

Edge dynamics
Realizing a sharp edge:
NA=0.5DMD
1064nm~1.1μm waist
a=287nmedge+=638nm light (repulsive)

loc. wavepacket

Width of the edge: 2-3 lattice sites!

iler
Pnim!
yraEdge dynamics in anomalous regime

10af=7kHz, m=0.25iler
Pnim!
yra

Edge dynamics in anomalous regime

f=7kHz, m=0.25iler
Pnim!
yr
a

Application:
Quantum Simulation ofgauge theories

Engineering novel Hamiltonians
Topological phases /artificial magnetic fields:
2π2π2π

Φ

Φ

Φ

3π/2

3π/23π/2

Φ

Φ

Φ

Φ

Φ

Φ

π

Jπ/2

ππ/2

Density-dependent
gauge fields:

ππ/2J
Hofstadter/Haldane model

Zurich, Harvard, MIT, Hamburg, NIST, LENS, Chicago, ....

Engineered field dependson site occupation

But: No
Gauss's law!

Gauge theories

Lattice gauge
theories:

Ux-1,x ψxGauge fieldψx+1
Matter field (fermionic)
K. G. Wilson

High Energy Physics

Condensed Matter Physics

Top. Quantum Computation

Gauge theories

Lattice gauge
theories:

Ux-1,x ψxGauge fieldψx+1
Matter field (fermionic)
K. G. Wilson

Challenges for Quantum simulation:• Implement matter and gauge fields• Realize local symmetries (Gauss's law)

U.-J. Wiese et al. Ann. Phys. 525, 777-796 (2013); E. Zohar et al. Rep. Prog. Phys. 79, 014401 (2015);
M. Dalmonte et al. Contemp. Phys. 57, 388-412 (2016); M. Banuls et al. Eur. Phys. J. D 74, 165 (2020)

U(1) lattice gauge theory in 1D

Quantum electrodynamics in 1D
lattice Schwinger model

gauge-invariant matter-gauge couplingnearest-neighbortunnelingmass of "positrons"and "electrons"
electric field operator

Kogut & Susskind, PRD 11, 395 (1975)
Chandrasekharan & Wiese, Nucl. Phys. B 492, 455 (1997)

U(1) lattice gauge theory in 1D

Quantum electrodynamics in 1D
lattice Schwinger model

Local charge:even
occupied
Kogut & Susskind, PRD 11, 395 (1975)
Chandrasekharan & Wiese, Nucl. Phys. B 492, 455 (1997)

oddempty

U(1) lattice gauge theory in 1D

Quantum electrodynamics in 1D
lattice Schwinger model

Local charge:

Gauss's law:

Physical states:
Kogut & Susskind, PRD 11, 395 (1975)
Chandrasekharan & Wiese, Nucl. Phys. B 492, 455 (1997)

U(1) lattice gauge theory in 1D

Quantum electrodynamics in 1D
lattice Schwinger model

Spin-1/2 quantum linkmodel (QLM):

reduced Hilbert-spacefor link operators

Kogut & Susskind, PRD 11, 395 (1975)
Chandrasekharan & Wiese, Nucl. Phys. B 492, 455 (1997)

U(1) lattice gauge theory in 1D

Basic dynamics:

Spin-1/2 quantum linkmodel (QLM):

reduced Hilbert-spacefor link operators

Kogut & Susskind, PRD 11, 395 (1975)
Chandrasekharan & Wiese, Nucl. Phys. B 492, 455 (1997)

State-of-the-art

Few-ion quantum simulation
particle-antiparticle creation processes

E. A. Martinez et al. Nature 534, 516-519 (2016);
N. H. Nguyen et al. PRX Quantum 3, 020324 (2022)

Gauge-fields are eliminated ↔ exotic long-range interactions

State-of-the-art

Rydberg atom arrays

Building blockaf

H. Bernien et al. Nature 551, 579 (2017);
F. M. Surace et al. Phys. Rev. X 10, 021041 (2020)

Matter-fields are eliminatedZ2 LGT:
C. Schweizer,..., MA, Nat. Phys. 15, 1168-1173 (2019)
U(1) LGT:
A. Mil et al. Science 367, 1128-1130 (2020)

Bosonic atoms in tilted optical superlattices

B. Yang et al. Nature 587, 392-396 (2020)
Z.-Y. Zhou Et al., Science 377, 311 (2022)
H.-Y. Wang Et Al., arXiv:2210.17032 (2022)

Ourgoal:• Simulate gauge field & fermionic matter• Simulation of 2D QLMs
• Extension to non-AbelianProposed experimental scheme

The scheme• State-dependent triplewell lattice

t|g〉
Ueg

|e〉

t

Ux-1,x ψxt2/Ueg• Building block: correlated
hopping of fermionsψx+1

gauge fieldδ

S=1/2 quantum
link model

The scheme|g〉• State-dependent triplewell lattice

Ueg

|e〉

t

Ux-1,x ψx• Ab initio calculations:

N. Darwah Oppongt

F. Surace

P. Fromholz

F. M. Surace, P. Promholz, N. Darkwah Oppong, M. Dalmonte, MA, arXiv:2301.03474

ψx+1

M. Dalmonte

S=1/2 quantum
link model

Experimental platform & current status

Novel hybrid tweezer-lattice platform

Optical lattices:large-scale systems,
defect free

Local state-dependent
control of tunneling

Optical tweezers:
local dynamicalcontrol

Ux-1,x ψx

ψx+1

M. Endres, Science (2016)
D. Barredo, Science (2016)

Implementation
ground- and excited clock state of Yb

Why Alkaline-earth(-like) atoms

1388 nm

399 nm29 MHz556 nm
182 kHz578 nm~ 10 mHzUse optical “clock”
qubit states

State-dependent potentials

11S
03P0

Magic
g-Tune-out
e-Tune-out

V. A. Dzuba and A. Derevianko, J. Phys. B: At. Mol. Opt. Phys. 43 074011 (2010)

3S0 ≡ | g⟩ and P0 ≡ | e⟩

State-dependent potentials

11S03P0

Magicg-Tune-out
e-Tune-out

V. A. Dzuba and A. Derevianko, J. Phys. B: At. Mol. Opt. Phys. 43 074011 (2010)

3S0 ≡ | g⟩ and P0 ≡ | e⟩Experimental setup

174Yb MOT

Goal: Direct loading of lattice & rearrangement using optical tweezer
A. Young,..., A. Kaufman, arXiv:2202.01204

Optical clock spectroscopy

opticalclocke-repumper

1D lattice:
759nm, ~600Er,~105 atoms @ 15μK

MOT30 HzOPObeammagiclattice
759nm

Summary & Outlook

New platform for simulating U(1)
LGTs with fermionic Yb

Ux-1,x ψx

Loading atoms into tweezer

ψx+1•Extended lattices in 1&2D
•Possible extension to non-Abelianusing SU(N) sym. interactions
171Yb (I=1/2), 173Yb (I=5/2)

Optical Test Setup, Strehl ratio >0.85

Recent Update174Yb atoms in tweezer array• Short cycle times < 0.5s
• Good imaging fidelityYb hybrid lattice-tweezer lab

U(1) lattice gauge theories with fermionic Yb
in 1D and 2D
Ux-1,x ψx

Etienne Staub,
Tim Höhn,Clara Bachorz
Nelson Darkwah Oppong
Bharath Hebbe
Madhusudhana
Dalila Robledo,
Guillaume BrochierDavid Gröters

F. M. Surace, P. Promholz, N. Darkwah Oppong, M. Dalmonte, MA, arXiv:2301.03474

ψx+1

Clock spectroscopy

30 HzCs quantum gas microscope

Simon
KarchaeW

Alexander Impertro,
Cesar Cabrera
Hendrik von Raven,
Julian Wienand
Till Klostermann,MA, Immanuel Bloch!
grinihre>2000 atoms
98%

Ignacio
Perez

ChristianSchweizer

Scott HubeleSophie Häfele

T. Klostermann, ..., MA, Phys. Rev. A 105, 043319 (2022); Alexander Impertro, ..., MA, arXiv:2212.11974

Cs atoms

Thank you