exploring topological phases with quantum walks
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Takuya Kitagawa, Erez Berg, Mark Rudner Eugene Demler Harvard 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. $$ NSF, AFOSR MURI, DARPA, ARO. - PowerPoint PPT PresentationTRANSCRIPT
Exploring Topological Phases With Quantum Walks
$$ 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)