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Exploring Topological Phases With Quantum Walks

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Takuya Kitagawa, Erez Berg, Mark Rudner

Eugene DemlerHarvard University

Exploring Topological Phases With Quantum Walks

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

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

Harvard-MIT

$$ NSF, AFOSR MURI, DARPA, ARO

Topological states of electron systems

Robust against disorder and perturbations

- Geometrical 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

Discreet time quantum walk

From quantum walk to topological Hamiltonians

Edge states as signatures of

topological Hamiltonians.

Experimental demonstration with photons

Topological properties unique to dynamics

Experimental demonstration with photons

Discreet time quantum walk

1D lattice, particle starts at the origin

Spin rotation

Spin-dependent Translation

Analogue of classical random walk.

Introduced in quantum information:

Q Search, Q computations

PRL 104:100503 (2010)

Also Schmitz et al.,

PRL 103:90504 (2009)

PRL 104:50502 (2010)

From discreet time

quantum walks to

Topological Hamiltonians

Discrete quantum walk

Spin rotation around y axis

Translation

One step

Evolution operator

Effective Hamiltonian of Quantum Walk

Interpret 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

k-dependent

“Zeeman” field

Winding Number Z on the plane defines the topology!

Winding number takes integer values.

Can we have topologically distinct quantum walks?

Split-step DTQW

Split-step DTQW

Phase Diagram

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)

Phase boundary of distinct topological phases has bound states

Topologically distinct,

so the “gap” has to close

near the boundary

Bulks are insulators

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

Chern Number

This is the number that characterizes the topology of the Integer Quantum Hall type states

brillouin zone

chern number, for example counts the number of edge modes

Chern number is quantized to integers

2D triangular lattice, spin 1/2

“One step” consists of three unitary and translation operations in three directions

big points

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 implementation

of different types of single particle Hamiltonians

By changing parameters of the quantum walk protocol

we can obtain effective Hamiltonians which correspond

to different topological classes

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

Topological properties of evolution operator

Time dependent

periodic Hamiltonian

Floquet operator

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

Spin ½ particle in 1d lattice.

Spin down particles do not move.

Spin up particles move by one lattice site per period

- group velocity

n1 describes average displacement per period.

Quantization of n1 describes topological pumping of particles.

This is another way to understand Thouless quantized pumping

Experimental demonstration of

topological quantum walk with photons

A. White et al., Univ. Queensland

Topological properties of evolution operator

Dynamics in the space of m-bands

for a d-dimensional system

Floquet operator is a mxm matrix

which depends on d-dimensional k

New topological invariants

Example:

d=3

Harvard-MIT

Summary

Quantum walks allow to explore a wide range

of topological phenomena. From realizing known

topological Hamiltonians to studying topological

properties unique to dynamics.

- First evidence for topological Hamiltonian
- with “artificial matter”

Topological Hamiltonians in 1D

Schnyder et al., PRB (2008)

Kitaev (2009)