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