1. Exploring Topological Phases With Quantum Walks $$ NSF, AFOSR MURI, DARPA, ARO Harvard-MIT 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)

2. 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

3. 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

4. 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

5. Discreet time quantum walk

6. 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

8. PRL 104:100503 (2010) Also Schmitz et al., PRL 103:90504 (2009)

9. PRL 104:50502 (2010)

10. From discreet time quantum walks to Topological Hamiltonians

11. Discrete quantum walk One step Evolution operator Spin rotation around y axis Translation

12. Effective Hamiltonian of Quantum Walk Interpret evolution operator of one step as resulting from Hamiltonian. Stroboscopic implementation of H eff Spin-orbit coupling in effective Hamiltonian

13. 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

14. Split-step DTQW

15. Phase Diagram Split-step DTQW

16. Symmetries of the effective Hamiltonian Chiral symmetry Particle-Hole symmetry For this DTQW, Time-reversal symmetry For this DTQW,

17. Topological Hamiltonians in 1D Schnyder et al., PRB (2008) Kitaev (2009)

18. Detection of Topological phases: localized states at domain boundaries

19. 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

20. Apply site-dependent spin rotation for Split-step DTQW with site dependent rotations

21. Split-step DTQW with site dependent rotations: Boundary State

22. Experimental demonstration of topological quantum walk with photons A. White et al., Univ. Queensland

23. Quantum Hall like states: 2D topological phase with non-zero Chern number

24. Chern Number This is the number that characterizes the topology of the Integer Quantum Hall type states Chern number is quantized to integers brillouin zone chern number, for example counts the number of edge modes brillouin zone chern number, for example counts the number of edge modes

25. 2D triangular lattice, spin 1/2 “One step” consists of three unitary and translation operations in three directions big points

26. Phase Diagram

27. 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,…

28. 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

29. Topological properties unique to dynamics

30. Floquet operator U k (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

31. 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 n 1 describes average displacement per period. Quantization of n 1 describes topological pumping of particles. This is another way to understand Thouless quantized pumping group velocity

32. Experimental demonstration of topological quantum walk with photons A. White et al., Univ. Queensland

33. Topological properties of evolution operator Dynamics in the space of m-bands for a d-dimensional system Floquet operator is a m x m matrix which depends on d-dimensional k Example: d=3 New topological invariants

34. 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”

36. Topological Hamiltonians in 1D Schnyder et al., PRB (2008) Kitaev (2009)