Exploring Topological Phases With Quantum Walks

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

Takuya Kitagawa, Erez Berg, Mark RudnerEugene Demler Harvard University

Also collaboration with A. White’s group, Univ. of Queensland

PRA 82:33429 and arXiv:1010.6126 (PRA in press)

Topological states of electron systems

Robust against disorder and perturbationsGeometrical character of ground states

Realizations with cold atoms: Jaksch et al., Sorensen et al., Lewenstein et al.,Das Sarma et al., Spielman et al., Mueller et al., Dalibard et al., Duan et al., and many others

Can dynamics possess topological properties ?

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

Dynamics can possess its own unique topological characterization

Focus of this talk on Quantum Walk

OutlineDiscreet time quantum walk

From quantum walk to topological Hamiltonians

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

Topological properties unique to dynamicsExperimental demonstration with photons

Discreet time quantum walk

Definition of 1D discrete Quantum Walk

1D lattice, particle starts at the origin

Analogue of classical random walk.Introduced in

quantum information:

Q Search, Q computations

Spin rotation

Spin-dependent Translation

PRL 104:100503 (2010)

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

PRL 104:50502 (2010)

From discreet timequantum walks to

Topological Hamiltonians

Discrete quantum walk

One stepEvolution operator

Spin rotation around y axis

Translation

Effective Hamiltonian of Quantum WalkInterpret evolution operator of one step as resulting from Hamiltonian.

Stroboscopic implementation of Heff

Spin-orbit coupling in effective Hamiltonian

From Quantum Walk to Spin-orbit Hamiltonian in 1d

Winding Number Z on the plane defines the topology!

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

k-dependent“Zeeman” field

Split-step DTQW

Phase DiagramSplit-step DTQW

Symmetries of the effective Hamiltonian

Chiral symmetry

Particle-Hole symmetry

For this DTQW, Time-reversal symmetry

For this DTQW,

Topological Hamiltonians in 1D

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

Detection of Topological phases:localized states at domain boundaries

Phase boundary of distinct topological phases has bound states

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

near the boundary

a localized state is expected

Apply site-dependent spin rotation for

Split-step DTQW with site dependent rotations

Split-step DTQW with site dependent rotations: Boundary State

Experimental demonstration of topological quantum walk with photons

A. White et al., Univ. Queensland

Quantum Hall like states:2D topological phase

with non-zero Chern number

Chern Number This 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 2D

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

Combining different degrees of freedom one can also perform quantum walk in d=4,5,…

What we discussed so far

Split 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 classes

Related theoretical work N. Lindner et al., arXiv:1008.1792

Topological properties unique to dynamics

Floquet operator Uk(T) gives a map from a circle to the space of unitary matrices. It is characterized by the topological invariant

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

Topological properties of evolution operator

Floquet operator

Time dependent periodic Hamiltonian

Example of topologically non-trivial evolution operatorand relation to Thouless topological pumping

Spin ½ particle in 1d lattice. Spin down particles do not move. Spin up particles move by one lattice site per period

n1 describes average displacement per period.

Quantization of n1 describes topological pumping of particles. This is another way to understand Thouless quantized pumping

group velocity

Experimental demonstration of topological quantum walk with photons

A. White et al., Univ. Queensland

Topological properties of evolution operatorDynamics in the space of m-bandsfor a d-dimensional system

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

Example:d=3

New topological invariants

Summary Harvard-MIT

Quantum walks allow to explore a wide range of topological phenomena. From realizing known topological Hamiltonians to studying topologicalproperties unique to dynamics.

First evidence for topological Hamiltonianwith “artificial matter”

Topological Hamiltonians in 1D

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