lecture ead part2 - harvard universitycmt.harvard.edu/demler/2013_talks/lecture_ead_part2.pdf ·...

Exploring topological states with cold atoms and photons

$$ NSF, AFOSR MURI, DARPA OLE, MURI ATOMTRONICS

Harvard-MIT

Theory: Takuya Kitagawa, Fabian Grusdt, Dima Abanin, Erez Berg, Mark Rudner, Liang Fu, Immanuel Bloch, Eugene Demler

Experiments: I. Bloch’s group (MPQ/LMU) A. White’s group (Queensland)

Topological states of maUer

Integer and FracVonal Quantum Hall effects

ExoVc properVes: quanVzed conductance (Quantum Hall systems, Quantum Spin Hall Sysytems) fracVonal charges (FracVonal Quantum Hall systems, Polyethethylene)

This lecture: How to explore topology of band structures with syntheVc maUer: cold atoms and photons

Extend to dynamics. Unique topological properVes of dynamics

Quantum Spin Hall effect

3D topological insulators

MagneVzaVon -‐ order parameter in ferromagnets

NemaVc order parameter in liquid crystals

Order parameters can be measured

Topology of Bloch bands Berry/Zak phase in 1d

Zak, PRL 1989 Vanderbilt, King-Smith, PRB 1993

Su-Schrieffer-Heeger Model

B A B B A

When dz(k)=0, states with dt>0 and dt<0 are topologically distinct.

Domain wall states in SSH Model An interface between topologically different states has protected midgap states

Field theory argument (Jackiw and Rebbi 76)

For small dt focus on low energy states with

Massive Dirac fermions

Zero mode: topologically protected eigenstate a E=0

Take continuum limit

Probing band topology with Ramsey/Bloch interference

g

Tools of atomic physics: Bloch oscillaVons

More than 20,000 Bloch oscillations: Innsbruck, Florence

p/2 pulse

Evolution

Tools of atomic physics: Ramsey interference

Used for atomic clocks, gravitometers, accelerometers, magnetic field measurements

p/2 pulse + measurement ot Szgives relative phase accumulated by the two spin components

Evolution Evolution

Zak phase probe of band topology in 1d

One dimensional superla^ces Su-‐Schrieffer-‐Heeger model

Experiments Marcos Atala, Monika Aidelsburger, Julio Barreiro, I. Bloch (LMU/MPQ)

Theory: Takuya Kitagawa (Harvard), Dima Abanin (Harvard/Perimeter), Immanuel Bloch (MPQ), Eugene Demler (Harvard)

arXiv:1212.0572

SSH model of polyacetylene

Analogous to bichromaVc opVcal la^ce potenVal

I. Bloch et al., LMU/MPQ

B A B B A

Su, Schrieffer, Heeger, 1979

Characterizing SSH model using Zak phase Two hyperfine spin states experience the same opVcal potenVal

p/2a -p/2a

a

Zak phase is equal to p 0

Problem: experimentally difficult to control Zeeman phase shift

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

Spin echo protocol for measuring Zak phase

Bloch oscillaVons measurements in LMU/MPQ With p-pulse but no swapping of dimerization

Bloch oscillaVons measurements in LMU/MPQ With p-pulse and with swapping of dimerization

Zak phase measurements in LMU/MPQ

Zak phase measurements can be used to probe topological properties of Bloch bands in 2D

D. Abanin, T. Kitagawa, I. Bloch, E. Demler Phys. Rev. LeU. 110:165304 (2013)

PolarizaVon and Zak phase in 1D

Bloch states and Wannier functions

Kingsmith and Vanderbilt PRB(1993)

Zak phase as polarization/center of Wannier functions +Q -Q

From Zak phase to Chern number

2D: evolution of Wannier centers as a function of ky

Consider 2d system as a collection of 1d systems at different ky

Chern number and QHE

Hall current

Hall conductance

How to measure Berry/Zak phases in 2D Relation to Chern number

Measure Zak phase for different initial points in the primitive cell

Full winding of z gives Chern number of BZ

This gives topological flux density in momentum space

Probe of the Berry phase of Dirac points

D. Abanin, T. Kitagawa, I. Bloch, E. Demler Phys. Rev. LeU. 110:165304 (2013)

Berry phase in hexagonal la^ce

•  Eigenvectors lie in the XY plane •  Around each Dirac point eigenvector makes 2p rotaVon •  Integral of the Berry phase is p

Berry’s phase of Dirac fermions Shifted positions of Integer quantum Hall plateaus

Measurement of the Berry and Zak phases in 2D with Ramsey/Bloch method

Experimental realizaVon Tarruell et al., Nature (2012)

When f jumps by p Sz changes sign

In Ramsey interference

How to measure the p-Berry phase of Dirac fermions “Parallel” measurements with a cloud of fermions

Measurement of the Berry and Zak phases in 2D Spin echo protocol

Measurement of the Berry and Zak phases in 2D Modified spin echo protocol

Ramsey/Zak extended to probe Z2 order parameter

Z2 invariant and Quantum Spin Hall effect Kane and Mele, PRL (1995)

They can have Spin Hall effect. Simple cartoon: spin and see opposite magnetic field and have opposite Chern numbers.

Time reversal invariant systems can not have Chern number. Under time reversal

Z2 invariant and Quantum Spin Hall effect

Z2 invariant and Quantum Spin Hall effect Degeneracies: Chern numbers are not well defined

How to measure Z2 invariant

Internal state 1 Internal state 2

Start with a p/2 pulse that prepares two hyperfine states in a superposition of two bands and two internal states at point (-p,ky)

For each ky measure extension of Zak phase difference



Perform Bloch oscillations until point (0,ky)




Perform Rabi rotation at point (0,ky) that switches bands and spins




Perform Rabi rotation at point (0,ky) that switches bands and spins.


Rabi rotation is performed by modulating the force at the frequency equal to the band splitting. This adds Frank-Condon phase factor.




Perform Bloch oscillations until point (p,ky)




Close the Ramsey sequence and measure the accumulated phase difference F(ky)

How to measure Z2 invariant For each ky measure extension of Zak phase difference

Measure Ramsey/Zak/Rabi phase difference for many F(ky).

Exploring topological states with photons

Theory: Takuya Kitagawa, Erez Berg, Mark Rudner, Liang Fu, Takashi Oka, Eugene Demler

Experiments: A. White’s group (Queensland)

Theory: T. Kitagawa et al., Phys. Rev. A 82:33429 (2010) Phys. Rev. B 82, 235114 (2010)

Experiments: T. Kitagawa et al., Nature Comm. 3:882 (2012)

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

Quantum walk with photons

Rotation is implemented by half-wave plates Translation by bi-refringent calcite crystals that displace only horizontally polarized light

Earlier realization of QW with photons: A. Schrieber et al., PRL (2010)

A. White’s group in Queensland T. Kitagawa et al., Nature Comm. 3:882 (2012)

From discreet time quantum walks to

Topological Hamiltonians

T. Kitagawa et al., Phys. Rev. A 82, 033429 (2010)

Discrete quantum walk

One step Evolution operator

Spin rotation around y axis

Translation

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

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 Diagram

Split-step DTQW

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

Experimental demonstration of topological quantum walk with photons

Kitagawa et al., Nature Comm. 2012

Rotation is implemented by half-wave plates Translation by birefringent calcite crystals that displace only horizontally polarized light

Quantum Hall like states: 2D topological phase

with non-zero Chern number

2D triangular lattice, spin 1/2

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

big points

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

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

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 operator

and relation to Thouless topological pumping Spin ½ parVcle in 1d la^ce. Spin down parVcles do not move. Spin up parVcles move by one la^ce site per period

n1 describes average displacement per period. QuanVzaVon of n1 describes topological pumping of parVcles. This is another way to understand Thouless quanVzed pumping

group velocity

Experimental demonstration of topological quantum walk with photons

Kitagawa et al., Nature Comm. 3:882 (2012)

Boundary with topologically different evolution operators

Boundary with topologically similar evolution operators

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

Example: d=3

New topological invariants

Dynamically induced topological phases in a hexagonal lattice T. Kitagawa et al.,

Phys. Rev. B 82, 235114 (2010)

Dynamically induced topological phases in a hexagonal lattice

Calculate Floquet spectrum on a strip

Edge states indicate the appearance of topologically non-trivial phases

Photo-induced quantum Hall insulator in graphene

Consider circularly polarized off resonant (around 1000 THz) light

T. Kitagawa et al., arXiv:1104.4636


Consider right circularly polarized light Off-resonant light (around 1000 THz) with sufficiently strong intensity turns graphene into a quantum Hall insulator.

Sign of Hall conductance can be reversed by changing light polarization

Summary

Observation of edge states in topological phases realized with photons

First direct measurement of Zak phase of a 1d band

Prospect of measuring topological properties of 2d bands

lecture ead part2 - harvard universitycmt.harvard.edu/demler/2013_talks/lecture_ead_part2.pdf ·...

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