Exploring topological states with ultracold atoms

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

Harvard-MIT

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

Experiments:  Marcos  Atala,  Monika  Aidelsburger,  Julio  Barreiro,  Immanuel  Bloch  (MPQ/LMU)    

Outline Introduction

Tools of atomic physics for measuring topological properties of Bloch bands.

Theory: how to measure the Berry/Zak phase with Bloch/Ramsey interference

Experimental measurement of the Berry/Zak phase in dimerized lattice

Theory: how to measure topological properties of 2D systems: Dirac points, Chern number.

Magnetization - order parameter in ferromagnets

Order parameters

Berry/Zak phase in 1d

Vanderbilt, King-Smith PRB 1993  

How to measure topological order parameter?

Measure  the  Berry/Zak  phase  itself,  not  its  consequence  

Su-Schrieffer-Heeger Model

B   A   B   B  A  

When dz(k)=0, states with dt>0 and dt<0 are topologically distinct. We can not deform two paths into each other without closing the gap.

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

AbsorpUon  spectra  on  neutral  and  doped  trans-­‐(CH)x  

Probing band topology with Ramsey/Bloch interference

SSH model in bichromatic lattices

Analogous  to  bichromaUc  opUcal  laWce  potenUal    

I.  Bloch  et  al.,    LMU/MPQ  

B   A   B   B  A  

Su, Schrieffer, Heeger, 1979

C.  Salomon  et  al.,  PRL    (1996)    

Tools  of  atomic  physics:  Bloch  oscillaUons    

p/2  pulse  

 EvoluUon  

Tools  of  atomic  physics:  Ramsey  interference    

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

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

EvoluUon    EvoluUon  

Measurements  of  Zak/Berry  phase    in  one  dimensional  Bloch  band  

One  dimensional  superlaWces    Su-­‐Schrieffer-­‐Heeger  model  

M.  Atala  et  al.,  arXIv:1212.0572  

Characterizing  SSH  model  using  Zak  phase    Two  hyperfine  spin  states  experience  the  same  opUcal  potenUal  

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  oscillaUons  measurements  With p-pulse but no swapping of dimerization

             Bloch  oscillaUons  measurements  With p-pulse and with swapping of dimerization

Zak/Berry  phase  measurements  

Zak/Berry  phase  measurements  extended  

Zak/Berry phase measurements of Bloch bands topology in 2D

D.  Abanin,  T.  Kitagawa,  I.  Bloch,  E.  Demler  arXIv:1212.0572  

Geometrical character of ground states: Example: TKKN quantization of Hall conductivity for IQHE Thouless et al., PRL (1982)

Chern Number This is the number that characterizes the topology

of the Integer Quantum Hall type states

Brillouin zone

Kx

Ky

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

Berry  phase  in  hexagonal  laWce  

•   Eigenvectors  lie  in  the  XY  plane  •   Around  each  Dirac  point    eigenvector        makes  2p  rotaUon  •   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  realizaUon  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

Summary

Ramsey/Bloch interference as a probe of band topology in 1d. Theory +Experiment

Ramsey/Bloch interference as a probe of band topology in 2d models. Theory