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Exploring Topological Phases With
Quantum Walks
Takuya Kitagawa, Erez Berg, Mark Rudner
Eugene Demler
Harvard University
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
Outline
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
Definition of 1D discrete Quantum Walk
1D lattice, particle
starts at the origin
Spin rotation
Spindependent
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)
Detection of Topological phases:
localized states at domain boundaries
Phase boundary of distinct topological
phases has bound states
Topologically distinct,
Bulks are insulators
so the “gap” has to close
near the boundary
a localized state is expected
Split-step DTQW with site dependent rotations
Apply site-dependent spin
rotation for
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 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 unique to
dynamics
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
Example of topologically non-trivial evolution operator
and 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
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
Summary
Harvard-MIT
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)