Simulating Topological and Disordered Systems …Random dimer models, n-mer models • quasidisorder and quasicrystal lattices • Off-diagonal disorder and/or diagonal disorder •

Simulating Topological and Disordered Systems with Resonantly Driven Atomic Matter Waves

Fangzhao Alex AnEric J. Meier

Bryce GadwayUniversity of Illinois at Urbana-Champaign

publish.illinois.edu/gadwaylab/

Interacting Quantum Systems Driven Out of EquilibriumMay 06, 2016

Rice Center for Quantum Materials, Houston, USA

The Gadway Group

Eric J. MeierFangzhao Alex An Jackson Ang’ong’a

Local theory friends: Smitha Vishveshwara, Taylor Hughes

87Rb BECsAug 2015

Er/Rb quantumgases project (1/2016-)

Rb/K quantumgases project (8/2014-)

• resonant Hamiltonian engineering

• topological matter

• Emergent behavior in quantum spin systems

• Nonequilibrium spin transport

A short and biased history of cold atom simulation

Let the atoms (or molecules) act naturally- superfluid behavior of atomic Bose-Einstein condensates- BEC/BCS crossover physics of atomic Fermi gases- atoms in standard optical lattices – Hubbard models w/ on-site

interactions & emergent magnetism (� t2/U)- long-range interactions with dipolar atoms/molecules, Rydberg

atoms, ions, and cavity-mediated interactions of atoms

Still natural behavior, but added complexity- additional potentials for new physics

�disordered Hubbard models�optical superlattices, etc.

- spin-dependent potentialsMostly non-native behavior- Lorentz forces (classical gauge fields), spin-orbit coupling- “Gauss’ Law” in dynamical gauge field simulations

Ketterle group, 2001

Greiner, Mandel, Esslinger,Hansch & Bloch, 2002 Hulet group, 2015

Jin & Ye groups, 2013

Esslinger group, 2015







- spin-dependent potentials

Mostly non-native behavior- Lorentz forces (classical gauge fields), spin-orbit coupling- “Gauss’ Law” in dynamical gauge field simulations

Esslinger group, 2012

DeMarco group, 2011

Sengstock group, 2015







- spin-dependent potentials

Mostly non-native behavior- Lorentz forces (classical gauge fields), spin-orbit coupling- “Gauss’ Law” in dynamical gauge field simulations

Top-down Hamiltonian engineering

Esslinger group, 2014Sengstock group, 2012

Bloch group(also Ketterle)

cf. also,Chu group (2005)

Arimondo group (2007)etc.

Bottom-up approaches to Hamiltonian design

Create a large set of available modes& connect them in a controlled fashion

• arbitrary spatial variations• well-defined system boundaries

Szameit group

Segev group / Rechtsman Silberhorn group

Hafezi group

cf. also Zilberberg, Silberberg, Christodoulides, Simon, White, etc.




Regal group (cf. also Jochim, Greiner, etc.)

Browaeys group(cf. also Saffman, Bloch, etc.)




Field-driven transitions between the modes – “driven tunneling”• spectroscopic control over system parameters• tunneling phase control• arbitrary time-dependence

Celi, et al. PRL 112 043001 (2014)Mancini, et al. Science 349 1510 (2015);Stuhl, et al. ibid. 1514 (2015)

In the spirit of employing“synthetic dimensions”

Outline

Bottom-up Hamiltonian design with resonant control- realization with lattice-driven matter waves- long-range interactions, higher dimensions, non-Abelian gauge fields

First studies: dynamics, disorder, and topological defect states

Near-term prospects: higher dimensions, coupled topological wires, and quantum chaotic dynamics

Atom-optics simulator of tight-binding lattice models

atomic diffraction, Chin group, Chicago laser diffraction, dragonlasers.com

p2/2ME

p

modes = plane-wave momentum states of a BEC

Ln knp 2=

Rn EnEnE 21

2 4==

Rn EnE 4)12( +=∆


BG. PRA 92 043606 (2015)

momentum states are coupled through global Bragg laser fields,and this coupling is energy-selective, and thus state-selective

BECLaserField 1

LaserField 1

( )..1†† chccetccH nn

i

nnnn

nn

n ++= +∑∑ ϕεpotential landscape tunneling amplitudes and phases

• Detunings determine potential landscape• Fields strengths control tunneling amplitude• Laser phase directly controls tunneling phase• (Rabi freq. << 8ER/ħ to address individual links)


BG. PRA 92 043606 (2015)


i

nnnn

nn


δω

BECLaserField 1

LaserField 1


BG. PRA 92 043606 (2015)


i

nnnn

nn


δω

BECLaserField 1

LaserField 1


BG. PRA 92 043606 (2015)


i

nnnn

nn


δω

BECLaserField 1

LaserField 1


BG. PRA 92 043606 (2015)


i

nnnn

nn


δω

BECLaserField 1

LaserField 1

ϕieϕie


Completely out-of-equilibrium

Initial temperature scales are irrelevant

p-modes


Continuous-time random walk

Image mode populations in time-of-flight

Natural decoherence mechanism- atoms fly off from “near field” region

K 42~ µ

Why pursue this matter-wave approach?

Easily extendable to higher dimensions

time

nx nx nx nx

ny

Local, arbitrary, and time-dependent control of parameters (also, multi-range hopping through higher-order Bragg)


Easily extendable to higher dimensions

Non-Abelian gauge fields

Local, arbitrary, and time-dependent control of parameters (also, multi-range hopping through higher-order Bragg)

( )h.c.ˆˆˆ†1eff +≈ +∑ nnn

nn

I cUctH


We’re using matter waves – interactions!

short-ranged (almost zero-ranged) in real space& long-ranged (almost infinite-ranged) in momentum-space

( )( )2

2

18

akakrel

rel +=

πσ

Future studies of interacting topological fluids & influence of interactionson localization physics

krel a ≈ 0.063Δn

For our latticeλ = 1064 nmkrel = 4πΔn/λ

For our 87Rb atomsa ≈ 5.3 nm

Nearly “all-to-all” for current parameters – no clear influence yet

0 10 20 30 40

1.0

0.0

0.5

Δn

σ/σ0

current expt.2 � krela5 � krela

First demonstrations of new control technique


i

nnnn

nn

n ++= +∑∑ ϕε

Continuous-time quantum walks on finite-sized lattices- ballistic spreading and reflection from boundaries

Eric J. Meier, F. Alex An, and BG. PRA(R) forthcoming


Continuous-time quantum walks on tilted lattices- Bloch oscillations


i

nnnn

nn

n ++= +∑∑ ϕε


1/ =∆ tε



i

nnnn

nn

n ++= +∑∑ ϕε

Global tunneling phase inversion (band inversion)- rotary echo


Prospects for simulating disordered systems

• Arbitrary potential landscapes• Varying disorder correlation lengths• Comparing different disorder distributions• Random dimer models, n-mer models• quasidisorder and quasicrystal lattices

• Off-diagonal disorder and/or diagonal disorder

• Disordered topological systems

• Time-fluctuating disorder – annealed vs. quenched disorder

• Disorder in higher dimensions• Random flux lattices• Bond percolation networks

Nonequilibrium disordered quantum walks

Quantum walks in a pseudo-disordered potential

( ) ( )∑∑ +++∆= +n

nnnnn

chcctccnbH ..cos 1††φπ

Aubry-André model

2/)51( +=bgolden ratio

InguscioRolstonSchnebleBlochetc.



( ) ( )∑∑ +++∆= +n

nnnnn


Aubry-André model





( ) ( )∑∑ +++∆= +n

nnnnn


Aubry-André model





( ) ( )∑∑ +++∆= +n

nnnnn


Aubry-André model



Quantum annealing into disorder

Slowly modify Hamiltonian to increase t / Δ

( ) ( )∑∑ +++∆= +n

nnnnn


time

Δ

t

1.5 ms



( ) ( )∑∑ +++∆= +n

nnnnn


time

Δ

t

1.5 ms



( ) ( )∑∑ +++∆= +n

nnnnn


time

Δ

t

1.5 ms

theory predictionfor transition inInfinite system

AndersonLocalized

Delocalized

Dynamically fluctuating (annealed) disorder

Mimic coupling to thermal reservoir through randomly varying system parameters

( )∑ += +n

nni chccetH n ..1

†ϕ

( ) ( ) ( )nii

in TJT φωωϕ +=∑ cos ( ) TkBeJ /ωωω −∝random tunneling phase jumps [with an Ohmic-like spectrum]

for some effective temperature T

Observe a changefrom ballistic spreadingto diffusive-like ( )transport

time

10

8

6

4

2

0 8006004002000Time [μs]

Mom

entu

m-w

idth

[ħk] tTk B 64.0=

Prospects for simulating topological systems

• In 2D - can directly implement arbitrary “gauge fields”• Uniform, spatially varying, or random fluxes

• In 1D – can control the symmetry of the Hamiltonian to access 1D topological insulator states of the chiral symmetric class (AIII and BDI)

−+∆+=

2)1(1 j

j tt

BDI tunnelings

1D topological wires – Su-Schrieffer-Heeger model

( )( ) ( )( ).... 1†

1† chcctchcctH nn

oddnnn

evenn+∆−++∆+= +

∈+

∈∑∑

1D chiral symmetric class BDI topological insulator

Electronic properties of polyacetylene(Su, Schrieffer, Heeger 1979)

Recent Thouless pumping expts.Bloch / Takahashi groups

1D topological wires – Su-Schrieffer-Heeger model

( )( ) ( )( ).... 1†

1† chcctchcctH nn

oddnnn

evenn+∆−++∆+= +

∈+

∈∑∑

1D chiral symmetric class BDI topological insulator

Electronic properties of polyacetylene

Adiabatic preparation of edge & defect states

Start with zero “weak” coupling – slowly turn on weak coupling

……

…

t4.0=∆

Nonadiabatic projection onto a midgap edge state

“Inject” all atomic population directly at the system boundary

0 1.2 ms0.6 ms

02468

0

12

Population localized to the edgek

p2

t2.0=∆

Nonadiabatic projection onto a midgap edge state

“Inject” all atomic population directly at the system boundary

0 1.2 ms0.6 ms

kp2

02468

“Inject” population in the bulk of the system

Population localized to the edge

02468

1012

kp2

Phase-sensitive projection onto mid-gap states

In addition to their decaying amplitudes from the edge into the bulk, the BDI midgap states are characterized by a πphase inversion every two sites.

a2π

Phase-sensitive projection onto mid-gap states

Try to match boundary state wavefunction through nonadiabatic transfer of population (partial transfer / π-pulse + phase shift)

In addition to their decaying amplitudes from the edge into the bulk, the BDI midgap states are characterized by a πphase inversion every two sites.

total relative phase [deg]0 180-180

average “distance” from edge [ħk]

after 760 μsevolution

t2.0=∆4

6

8

10

Some near-term experimental prospects

Disordered topological wires

“anneal” into the edge statefor no disorder

What we see when a random (box)disorder strength just exceeds 2Δ

Odd-even parity of edge states is destroyed

Next probe: study robustness of Thouless pumping against added diagonal disorder



Higher dimensions (like 2)• IQHE, Floquet Topological Insulators, coupled 1D wires, etc.







Dynamics of Quantum Chaotic Systems Making kicked atoms look like “kicked tops”

Summary

Developed a new “bottom-up” approach to atomic quantum simulations• well-suited to studying topological systems, disorder, and dynamics

Next major developments:• extend to higher dimensions• harness the long-ranged interaction in k-space

Thanks!

Generating multiple frequency sidebands


Calibration of tunneling strengths


0ħk

2ħk

Documents

Simulating Topological and Disordered Systems …Random dimer models, n-mer models • quasidisorder and quasicrystal lattices • Off-diagonal disorder and/or diagonal disorder •