Motivation: Probe topological phases of matter in the presence of large magnetic fields (e.g. quantum Hall effect)

Ultracold atoms in optical lattices as a clean and well controlled model system to study physics in regimes not accessible in typical condensed matter systems

Implement artificial magnetic fields for ultracold neutral atoms

Topological charge pumping as a tool to study higher dimensional quantum Hall physics

Topological Charge Pumping and Artificial Gauge FieldsWith Ultracold Atoms in Optical Superlattices

Michael Lohse, C. Schweizer, M. Aidelsburger, M. Atala, I. Bloch

Ludwig-Maximilians-Universität, München & MPI für Quantenoptik, Garching, Germany

[1] M. Aidelsburger et al., Phys. Rev. Lett. 111, 185301 (2013)[2] M. Atala et al., Nature Physics 10, 588 (2014)[3] M. Aidelsburger et al., Nature Physics 11, 162 (2015)[4] M. Lohse et al., Nature Physics 12, 350 (2016)[5] C. Schweizer et al., PRL (in press, 2016)

Simulate 4D quantum Hall effect by charge pumping in a tilted 2D superlattice

→ non-linear Hall response characterized by 2nd Chern number

Small atom cloud as local probe of charge transport

Measure transverse response when pumping along orthogonal direction

Meissner Effect in Bosonic Ladders

Topological Charge Pumping Transport of charge through adiabatic periodic variation of the underlying Hamiltonian (even in insulating systems)

For filled bands, the transported charge is quantized and related to a topological invariant, the Chern number, of the pumping process

The transported charge is purely determined by the topology of the pump cycle and robust against perturbations

n=1/2 Mott insulator in a 1D superlattice potential: pumping by adiabatic variation of the superlattice phase ϕ

Adiabatic evolution of eigenstate

→ anomalous velocity

Homogeneously populated band

Harper-Hofstadter Model

Artificial Gauge Fields

References

Outlook: Charge Pumping in 2D

Introduction

Charge neutrality prevents direct application of Lorentz force in an external magnetic field

Implementation of artificial gauge potentials by engineering of position-dependent complex tunneling amplitudes (Aharonov-Bohm phase) using laser-assisted tunneling in a tilted optical potential

Effective time-averaged Hamiltonian in high-frequency limit :

Cyclotron orbits of single atoms in isolated 2x2 plaquettes

2_

y

xB’x B’x

dx

dy

∆

k1, ω1

k2, ω2

−∆

|↑> |↓>

π2_π+ -

J

K

On-site modulation with position-dependent phases

ϕmn = q ·R

ω Jx/y

H = −m,n

Keiϕm,n a†m+1,nam,n + J a†m,n+1am,n + h.c.

Uniform effective magnetic flux per 2x2 plaquette

-0.4 -0.2 0 0.1-0.1-0.3

0.1

0

-0.1

-0.2

<X>/dx

<Y >/d

y

0.1

0

-0.1

-0.2-0.1 0.1 0.3 0.40.20

<X>/dx

<Y >/d

y

Time (ms)0 2.1 Time (ms)0 2.1

|↑> |↓>A

D C

B K

iK

J J

yx

2π+_

Quasi-1D ladder systems in the presence of a uniform artificial gauge field

Ground state exhibits chiral current in analogy to the Meissner effect in a type-II superconductor

Phase diagram of the flux ladder

Measurement of currents: Projection onto isolated double wells

Oscillation between even and odd sites with amplitude propotional to current

y

x

J

K

k2, ω2

k1, ω1

∆

i 0

i 1

i 2

i 3

i 0

i 1

i 2

i 3

i 0

i 1

i 2

i 3

i 0

i 1

i 2

i 3

i 0

i 1

-1 -0.5 0 0.5 1

-2

-1

0

1

2

Ener

gy (J

)

q (π/dy)

K/J=0.8

-1 -0.5 0 0.5 1

-3

-2

-1

0

1

2

3

Ener

gy (J

)

q (π/dy)

K/J=2

φ = π/2

( K/J

) C

jC (a.u.)

Chiral current jC (a.u.)

0

0.5

1

1.5

2

2.5

3

3.5

K/J

j C (a

.u.)

0 0.2 0.4 0.6 0.8 1

Flux φ (π)

K/J=√2

φC

K/J

1

2

3

4

Exp

erim

ent

0

1

1

23

4

Meissnerphase

Vortexphase

Vortex phase

Meissner phase

Atom density (a.u.)0

0 1

1

0 1

L

R

J

K

φ

φ

φ

Ground state

Project

Isolateddouble wells

n od

d;L

n o

dd

;R

0.6

0.5

0.4

0 0.5 1 1.5 2Time (ms)

φ = π/2

n od

d;L

n o

dd

;R

0.6

0.5

0.4

0 0.5 1 1.5 2Time (ms)

φ = −π/2

n od

d;L

n o

dd

;R

0.6

0.5

0.4

0 0.5 1 1.5 2

Time (ms)

φ = 0

π/2

π

3π/2

2π

π/2

π

3π/2

2π

π/2

π

3π/2

2π

ΦΦΦ

Φ Φ Φ

ΦΦΦD C

BA

ν1=1

ky (π/a) kx (π/a)-0.5 -0.5

0.5 0.5

ν3=1

ν2=-2

2D square lattice in a uniform artificial magnetic field

Magnetic unit cell contains multiple sites and the lowest Bloch band splits into separate sub-bands which are topologically non-trivial

Measurement of the transverse Hall response by applying a longitudinal gradient

Atoms perform Bloch oscillations and acquire anomalous velocity in the orthogonal direction which is proportional to the Berry curvature

Quantized transport for filled/uniformly populated bands characterized by Chern number

Atomcloud

F

Anomalousveolcity

2

0 50 100 150 200

Diff

eren

tial s

hift

2x(

t)/a

-4

-2

0

2

4

no flux

staggered flux

50

-50

0

50-50 0

Position (a)

Density (a.u.)

50

-50

0

50-50 0

-1 1

50

-50

0

50-50 0

1

3

1

2

3

+F

-F

Bloch oscillation time (ms)

+F

+F

with

νexp = 0.99(5)

∆

J1

J2

ϕ

dsdl

H(ϕ) = −∑m

(J1(ϕ)b

†mam + J2(ϕ)a

†m+1bm + h.c.

)+∆(ϕ)

2

∑m

(a†mam − b†mbm

)

x (dl )-1 10

classical quantum ϕ

π

π/2

0

J1-J2

∆2π

0

ϕ

|un〉 − i∑

n′ =n

|un′〉 〈un′ |∂tun〉εn − εn′

∂εn(kx)

∂kx︸ ︷︷ ︸vgr

− i

[⟨∂un

∂kx

∣∣∣∣∂un

∂φ

⟩−

⟨∂un

∂φ

∣∣∣∣∂un

∂kx

⟩]

︸ ︷︷ ︸Ω(kx,φ)

∂φ

∂t

x = νndl

Quantized displacementνn =

1

2π

∫

FBZ

∫ 2π

0

Ωn(kx, ϕ) dϕdkxwith

ϕ (2π)

x (d

l )

0 1

0

1

1

0.5

0

ϕ (2

π)

x (dl )2-1 10

0.5

1.5

0.5 1.5-0.5

x (d

l )

-1

0

0 1 1.50.5ϕ (2π)

1

-1

n o –

n e

ϕ (2

π)

x (dl )

1

0

-1 1

ν = + 1 ν = −12nd band:1st band:

Charge pumping in 1D superlattice → dynamical version of the 2D integer quantum Hall effect: variation of ϕ equivalent to perpendicular electric field

Spin Pumping Pumping with spin-dependent modulation: transport of spins without charge transport

Hardcore bosons in two hyperfine states: spin chain with dimerized superexchange coupling and spin-dependent tilt

Realization of a similar model with a global magnetic field gradient that is topologically equivalent in the limit of isolated double wells

Direct measurement of spin currents in optical lattices by projection onto static double wells

→ oscillations of the spin imbalance

Integrated spin current inde- pendent of exchange coupling unlike instantaneous current

Separation of the spins‘ center-of-mass position without charge transport

H =− 1

4

∑

m

J + (−1)mδJ) (

S+mS−

m+1 +)+

∆

2

∑

m

(−1)mSzm

I(t′) = A sin

(E2(t )− E1(t )

t′)+ I

0.00

0.06

0.12

0.18

A

II

III

I

-500 -250 0 250 500

∆s/h (Hz)

0.6

0.7

0.8

0 1 2

t´ (ms)

III

-0.1

0.0

0.1

0 2.5 5

II

0 3 6

-0.5

-0.4

-0.3I

Imb

alan

ce

with A

=

= −2 a2(t ) 〈1t | I |2t 〉

j(t )E2(t )− E1(t )

2

0 0.5 1 1.5 2

φ (2π)

-3

-2

-1

0

1

2

3

x (d

s )

pum

ping

α

y

x

y (d

s)

08

4-4

-8

x (ds)-1 1 1-1

left-right fraction

1

-1

0 2000 4000

# pump cycles

n LR

-5 0 5

ϕX (2π)

0.55

0.50

0.45

LR fr

actio

n

PRELIMIN

ARY