Topological quantum dynamics of artificial...

Topological quantum dynamicsof artificial matter

Eugene Demler Harvard University

Collaborators:

Takuya Kitagawa, Erez Berg, Mark Rudner, Liang Jiang, Jason Alicea, Anton R. Akhmerov, David Pekker, Gil Refael, J Ignacio Cirac Mikhail D Lukin Peter Zoller Takashi OkaJ. Ignacio Cirac, Mikhail D. Lukin, Peter Zoller, Takashi Oka, Arne Brataas, Liang Fu

C ll b ti ith A Whit ’ U i f Q l dCollaboration with A. White’s group, Univ. of QueenslandObservation of topological states with “artificial matter”Topological models realized with photons

$$ NSF, AFOSR MURI, DARPAHarvard-MIT

Topological models realized with photons

Paradigms in equilibrium quantum many-body physics

Is there universality in quantum many-body d i ?dynamics?

What are universality classes and paradigms of nonequilibrium many body dynamics?nonequilibrium many-body dynamics?This talk: topological properties of periodically driven systems(universal, beyond those known for static systems)

Can we use nonequilibrium quantum dynamics for technological applications?

Control material properties through dynamics

T. Kitagawa et al., arXiv:1104.4636

Dynamically create robust topological qubits

T. Kitagawa et al., arXiv:1104.4636

L. Jiang, T. Kitagawa, D. Pekker, et al., PRL (2011)

Topological states of electron systems

R b t i t di d d t b tiRobust against disorder and perturbationsGeometrical character of ground states

Realizations with cold atoms: Jaksch, Zoller, Sorensen, Lewenstein, Gurarie,Das Sarma , Spielman, Hemmerich, Mueller , Duan, Gerbier,Dalibard , Cooper, Morais Smith, and many others

Can dynamics possess topological properties ?Ca dy a cs possess opo og ca p ope es

One can use dynamics to make stroboscopic implementations of static topological Hamiltonians

D i it i t l i lDynamics can possess its own unique topological characterization

Both can be realized experimentally and studied with “artificial matter”: ultracold atoms and photonsp

“Lessons” for traditional condensed matter systems.Example: applying circularly polarized light to graphene

OutlineFrom quantum walk to topological Hamiltonians

Edge states as signatures of topological Hamiltonians.Experimental demonstration with photonsExperimental demonstration with photons

Topological properties unique to dynamicsExperimental demonstration with photons

Topological phases of periodically driven hexagonalp g p p y glattice

Photo induced quantum Hall effect in graphenePhoto-induced quantum Hall effect in graphene

Floquet Majorana fermions with ultracold atoms

Discreet time quantum walk

Definition of 1D discrete Quantum Walk

1D lattice particle1D lattice, particle starts at the origin

Spin rotation

Spin-dependent pTranslation

Analogue of classical random walk.

Introduced in quantumIntroduced in quantum information:

Q Search, Q computations

PRL 104:50502 (2010)

PRL 104 100503 (2010)PRL 104:100503 (2010)

Also Schmitz et alAlso Schmitz et al.,PRL 103:90504 (2009)

From discreet timeFrom discreet timequantum walks to

T l i l H il iTopological Hamiltonians

T. Kitagawa et al., Phys. Rev. A 82, 033429 (2010)

Discrete quantum walkSpin rotation around y axisSpin rotation around y axis

Translation

One stepOne stepEvolution operator

Effective Hamiltonian of Quantum WalkInterpret evolution operator of one step

as resulting from Hamiltonian.

Stroboscopic implementation of p pHeff

Spin-orbit coupling in effective Hamiltonianp p g

From Quantum Walk to Spin-orbit Hamiltonian in 1d

k-dependent“Zeeman” field

Winding Number Z on the plane defines the topology!

Winding number takes integer valuesWinding number takes integer values.Can we have topologically distinct quantum walks?

Split-step DTQW

Split-step DTQWPhase Diagram

Symmetries of the effective HamiltonianChiral symmetry

Particle-Hole symmetry

For this DTQW, Time-reversal symmetry

For this DTQW,

Topological Hamiltonians in 1DTopological Hamiltonians in 1D

Schnyder et al PRB (2008)Schnyder et al., PRB (2008)Kitaev (2009)

Detection of Topological phases:localized states at domain boundaries

Phase boundary of distinct topological h h b d t tphases has bound states

Bulks are insulators Topologically distinct, so the “gap” has to close

near the boundarynear the boundary

a localized state is expected

Split-step DTQW with site dependent rotationsApply site-dependent spin

rotation for

Split-step DTQW with site dependent rotations: Boundary Staterotations: Boundary State

Experimental demonstration of topological quantum walk with photonstopological quantum walk with photonsT. Kitagawa et al., arXiv:1105.5334

Quantum Hall like states:Quantum Hall like states:2D topological phase

with non-zero Chern number

Chern NumberThis is the number that characterizes the topologyThis is the number that characterizes the topology

of the Integer Quantum Hall type states

Chern number is quantized to integers

2D triangular lattice, spin 1/2“One step” consists of three unitary and translation operations in three directions

Phase Diagram

Topological Hamiltonians in 2DTopological Hamiltonians in 2D

Schnyder et al PRB (2008)Schnyder et al., PRB (2008)Kitaev (2009)

C bi i diff t d f f d lCombining different degrees of freedom one can also perform quantum walk in d=4,5,…

What we discussed so farWhat we discussed so far

Split time quantum walks provide stroboscopic implementationSplit time quantum walks provide stroboscopic implementationof different types of single particle Hamiltonians

By changing parameters of the quantum walk protocolwe can obtain effective Hamiltonians which correspond to different topological classesto different topological classes

T. Kitagawa et.al, PRB(2010) Related theoretical work N Lindner et al Nature Physics (2011)Related theoretical work N. Lindner et al., Nature Physics (2011)

Topological properties unique toTopological properties unique to dynamics

T. Kitagawa et.al, PRB(2010)

Topological properties of evolution operatorTi d dTime dependent periodic Hamiltonian

Floquet operator Uk(T) gives a map from a circle to the space of

Floquet operator

q p k( ) g p punitary matrices. It is characterized by the topological invariant

This can be understood as energy winding.This is unique to periodic dynamicsThis is unique to periodic dynamics. Energy defined up to 2p/T

Topological properties of evolution operatorSi l lSimple example

Quantum walk: during one cycle spin up atoms do not move spin done atomsatoms do not move, spin done atoms move right by one lattice constant

After introducing coupling between spins

Over one period the average position Q anti ation of impliesOver one period the average positionof a particle in the spin-up band is shiftedby one unit cell, spin down does not shift

Quantization of n1 impliesquantization of pumped chargeThouless, PRB (1983)

Experimental demonstration of topological quantum walk with photonstopological quantum walk with photons

T. Kitagawa et al., arXiv:1105.5334

Topological properties of evolution operatorD i i th f b dDynamics in the space of m-bandsfor a d-dimensional system

Floquet operator is a mxm matrixwhich depends on d-dimensional k

New topological invariants

Example:d 3d=3

Topological dynamics beyond quantum walk

Dynamically induced topological phases in a hexagonal lattice T Kitagawa et alin a hexagonal lattice T. Kitagawa et al.,

Phys. Rev. B 82, 235114 (2010)

Dynamically induced topological phases in a hexagonal latticein a hexagonal latticeCalculate Floquet spectrum on a strip

Edge states indicate theEdge states indicate theappearance of topologicallynon-trivial phases

Photo-induced quantum Hall insulator in grapheneHall insulator in graphene

Control material properties through dynamicsControl material properties through dynamics

Photo-induced quantum Hall insulator in graphene

T Kit t l Xi 1104 4636T. Kitagawa et al., arXiv:1104.4636

Consider circularly polarized off resonant light

Photo-induced quantum Hall insulator in graphene

s and t correspond to sublatticed ll d f f dand valley degrees of freedom

Each band is characterized by aEach band is characterized by a non-zero Chern number

We find quantum Hall insulatorof the type discussed by Haldane PRL (1988)

Photo-induced quantum Hall insulator in graphene

Spectrum on a strip

Photo-induced quantum Hall insulator in graphene

Consider right circularly polarized lightOff-resonant light with sufficiently strong intensityturns graphene into a quantum Hall insulatorturns graphene into a quantum Hall insulator.

Sign of Hall conductance can be reversed by changing light polarization

Realizing Majorana fermions with ultracold atoms.D i l l i l biDynamical topological qubits

Realizing Majorana fermions with ultracold atomsL. Jiang, T. Kitagawa, D. Pekker, et al., Phys. Rev. Lett. (2011) J a g, taga a, e e , et a , ys e ett ( 0 )

• Optically trapped fermionic atoms form a 1D quantum wire. • Two Raman beams create coupling between two fermionic• Two Raman beams create coupling between two fermionic

states and create spin-orbit like term• RF induced conversion between molecular BEC andfermionic atoms

Realizing Majorana fermions with ultracold atoms

After spin-dependent Galilean transformation

Topological and trivial phases of effective Hamiltonian

Majorana fermions can be created atboundaries between topological and trivial phases

Floquet Majorana fermionsModulate RF frequency detuning for molecule to atoms conversion

States at E=0 and E=p/T are Majorana states (particle hole conjugates of themselves)(particle -hole conjugates of themselves)

SummarySummary

“Artificial matter” allo s to e plore a ide range“Artificial matter” allows to explore a wide range of topological phenomena. From realizing known topological Hamiltonians to studying topologicaltopological Hamiltonians to studying topologicalproperties unique to dynamics.

Experimental demonstration of topological dynamicswith quantum walk protocols for photonswith quantum walk protocols for photons

$$ NSF, AFOSR MURI, DARPA Harvard-MIT

Example of topologically non-trivial evolution operator

d l i Th l l i l iand relation to Thouless topological pumpingSpin ½ particle in 1d lattice. S i d ti l d tSpin down particles do not move. Spin up particles move by one lattice site per period

group velocity

n1 describes average displacement per period.Q ti ti f d ib t l i l i f ti lQuantization of n1 describes topological pumping of particles. This is another way to understand Thouless quantized pumping

Topological Hamiltonians in 1DTopological Hamiltonians in 1D

Schnyder et al PRB (2008)Schnyder et al., PRB (2008)Kitaev (2009)

Topological aspects of dynamicsTopological aspects of dynamics

T k Kit E B M k R dTakuya Kitagawa, Erez Berg, Mark RudnerEugene Demler Harvard University

References:PRA 82:33429 and PRB 82:235114 (2010)

Collaboration with A. White’s group, Univ. of Queensland

First observation of topological states with “artificial matter”Topological models realized with photons

$$ NSF, AFOSR MURI, DARPA, AROHarvard-MIT

Symmetries of the effective HamiltonianChiral symmetry

Particle-Hole symmetry

For this DTQW, Time-reversal symmetry

For this DTQW,

Equilibrium quantum many-body systems:paradigms and universalityparadigms and universality

Fermi liquids of interacting electrons

Well defined quasiaprticlesBand strucures

Luttinger liquids in one dimension

Po er la correlationsPower law correlationsSpin-charge separation

Equilibrium quantum many-body systems:paradigms and universalityparadigms and universality

Spontaneous symmetry breaking Topological phases

Classification of Topological insulators in 1DClassification of Topological insulators in 1D