Takuya Kitagawa, Dima Abanin, Immanuel Bloch, Erez Berg, Mark Rudner, Liang Fu, Takashi Oka, Eugene Demler

Transcription

1 Exploring topological states with synthetic matter Takuya Kitagawa, Dima Abanin, Immanuel Bloch, Erez Berg, Mark Rudner, Liang Fu, Takashi Oka, Eugene Demler Harvard-MIT $$ NSF, AFOSR MURI, DARPA OLE, MURI ATOMTRONICS

2 Outline Zak phase as a probe of band topology in 1d Bloch+Ramsey interference experiments with cold atoms D. Abanin, T. Kitagawa, IBl I. Bloch, hed E. Demler Exploring edge states in topological phases with photons T. Kitagawa et al., PRA 82:33429 (2010) Phys. Rev. B 82, (2010) Nature Comm. 3:882 (2012)

3 Measuring topological properties of band structures directly. Berry and Zak phases. Bloch+Ramsey interference experiments with cold atoms D. Abanin, T. Kitagawa, I. Bloch, E. Demler

4 Example of band structure with Berry phase. Hexagonal l( (graphene) )lattice Tight binding model on a hexagonal lattice Dirac fermions in optical lattices, Tarruell et al., Nature 2012

5 Berry phase in hexagonal lattice Eigenvectors lie in the XY plane Around each Dirac point eigenvector ector makes 2p rotation Integral of the Berry phase is p

6 How to measure Berry phase in hexagonal lattice Naïve approach: Move atom on a closed trajectory around Dirac point Measure accumulated phase Problems with this approach: Need to move atom on a complicated curved trajectory Need to separate dynamical phase

7 From Berry phase to Zak phase Integral of the Berry phase is only well defined on a closed trajectory is not gauge invariant C gauge invariant integral of Berry curvature Brillouinzone is a torus. There are two types of closed trajectories

8 How to measure Zak phase using Ramsey interference sequence Two hyperfine spin states experience the same optical potential Advantages Requires only straight trajectory Dynamical phase cancels between two spin states

9 How to measure Zak phase using Ramsey interference sequence It is possible to measure the Chern number of a 2D band using Ramsey interferometry measurement of the Zak phase

10 Zak phase probe of band topology in 1d One dimensional superlattices Su Schrieffer Heeger Shiff model dl

11 Su Schrieffer Heeger model of polyacetylene Analogous to bichromatic optical lattice potential LMU/MPQ

12 A B A B A Dimerized model Topology of the band shows up in the winding of the eigenstate Zak phase is equal to p

13 Characterizing SSH model using Zak phase Two hyperfine spin states experience the same optical potential il a -p/2a 0 p/2a Zak phase is equal to p

14 Spin echo protocol for measuring Zak phase Dynamic phases due to Dynamic phases due to dispersion and magnetic field fluctuations cancel. Interference measures the difference of Zak phases of the two bands in two dimerizations. Expect phase p

15 Discreet time quantum walk with photons Observing edge states on topological domain boundaries Topological properties of dynamics Theory: T. Kitagawa et al., Phys. Rev. A 82:33429 (2010) Phys. Rev. B 82, (2010) Experiments: T. Kitagawa et al., Nature Comm. 3:882 (2012)

16 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

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18 Quantum walk with photons Rotation is implemented by half-wave plates Translation by bi-refringent calcite crystals that displace only horizontally polarized light A. White s group in Queensland T. Kitagawa et al., Nature Comm. 3:882 (2012) Earlier realization of QW with photons: A. Schrieber et al., PRL (2010)

19 From discreet time quantum walks to Topological l Hamiltonians i T. Kitagawa et al., Phys. Rev. A 82, (2010)

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

21 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

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

23 Split-step DTQW

24 Split-step DTQW Phase Diagram

25 Detection of Topological phases: localized states at domain boundaries

26 Phase boundary of distinct topological phases has bound states t Bulks are insulators Topologically distinct, so the gap has to close near the boundary a localized state is expected

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

28 Experimental demonstration of topological quantum walk with photons Kitagawa et al., Nature Comm Rotation is implemented by half-wave plates Translation by birefringent i calcite crystals that displace only horizontally polarized light

29 Topological Hamiltonians in 2D with quantum walk Schnyder et al., PRB (2008) Kitaev (2009)

30 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

31 Topological properties unique to dynamics

32 Topological properties of evolution operator Time dependent d periodic Hamiltonian Floquet operator Floquet operator U k 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

33 Example of topologically non-trivial evolution operator and relation to Thouless topological l pumping Spin ½ particle in 1d lattice. Spin down particles il do not move. Spin up particles move by one lattice site per period group velocity n 1 describes average displacement per period. Quantization of n 1 describes topological lpumping of particles. This is another way to understand Thouless quantized pumping

34 Experimental demonstration of topological quantum walk with photons Kitagawa et al., Nature Comm. 3:882 (2012) Boundary with topologically similar evolution operators Boundary with topologically different evolution operators

35 Summary Zak phase as a probe of band topology in 1d DAb D. Abanin, TKit T. Kitagawa, I. Bloch, E. Demler Exploring edge states in topological phases p g g p g p with photons T. Kitagawa et al., PRA 82:33429 (2010) Phys. Rev. B 82, (2010) Nature Comm. 3:882 (2012)

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