Exploring topological states with cold atoms and photons

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

Harvard-MIT

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

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

Universality in condensed matter physicsSpontaneous symmetry breaking and order

keV MeV GeV TeVfeV peV µeV meV eV

pK nK µK mK K

neV

roomtemperature

LHCCold atoms experiments10‐11 ‐ 10‐10 K

Higgs mode in ultracold atoms, 2012Higgs mode of the standard model, 2012

Universality ofphysics

Order beyond symmetry breaking

What is the “quantum protectorate” of such precise quantization?

Current along x, measure voltage along y.On a plateau

In 1980 the first ordered phase beyond symmetry breaking was discovered

Integer Quantum Hall Effect: 2D electron gas in strong magnetic field show plateaus in Hall conductance

with an accuracy of 10-9

Topological orderIn a topologically ordered state some physical quantity is given by a discreet “topological invariant”. Some physical response function is determined by this quantized invariant.

Topological invariant: quantity that does not change under continuous deformations

Example of topological invariant in geometry

Gaussian curvature at every point on a surface

Gauss-Bonnet theorem for closed surfaces

g – integer genus of a surface g=0 g=1

How to define topological invariant for electrons in solids?

What kind of curvature can exist for electrons in solids?

Bloch’s theorem and Brillouin zoneOne electron wavefunction in a crystal (periodic) potential can be written as

As k changes, we map an “energy band”. Set of all bands is a “band structure”.

But … lattice momentum is periodic

k is “crystal momentum” restricted to Brillouin zone,a region of k-space with periodic boundaries.Function is periodic (same in every unit cell)

The Brillouin zone can playthe role of the surface. Important property of quantummechanics, the Berry phase, gives us the “curvature”.

Berry phaseConsider a quantum-mechanical system in a nondegenerate ground state, e.g. spin ½ particlein a magnetic field. The adiabatic theoremsays that if the Hamiltonian is changed slowly, thesystem remains in its instantaneous ground state.

Berry phase: when the Hamiltonian goes around a closed loop in parameter space, the system acquires a geometrical phase relative to initial state (in addition to the usual dynamical phase).

“Gauge transformation” of the Berry phase

Gauge invariant quantities are Berry curvature and closed loop integrals

From Berry phase to Chern numberThe change in the electron wavefunction within the Brillouin zone leads to a Berry connection and Berry curvature

Brillouinzone

Kx

Ky

Integral of F is quantized to be integer: first Chern number. It is like Gauss-Bonnet theorem for the Brillouin zone.

TKNN quantization of Hall conductivity for IQHEThouless et al., PRL 1982

Topological order and edge states

TKNN quantization exists only forinsulators with completely filled bands.

Conductance goes through gapless edge states.

Existence of topological invariant requires edge states

Topological invariant cannot change without closing of the insulating gap

Topology in one dimension:Berry phase and electric polarization

Polarization as Berry phase

Vanderbilt, King-SmithPRB 1993

Su-Schrieffer-Heeger Model

B A B BA

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

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

Absorption spectra onneutral and doped trans‐(CH)x

Topological states of matter

Integer and FractionalQuantum Hall effects

Exotic properties:quantized conductance (Quantum Hall systems, Quantum Spin Hall Sysytems)fractional charges (Fractional Quantum Hall systems, Polyethethylene)

This talk:How to explore topology of band structures with synthetic matter: cold atoms and photons

Extend to dynamics. Unique topological properties of dynamics

Quantum Spin Hall effect

3D topological insulators

Magnetization ‐ order parameter in ferromagnets

Nematic order parameter in liquid crystals

Order parameters can be measured

Outline

Exploring edge states in topological phases with photonsT. Kitagawa et al., PRA 82:33429 (2010)

Phys. Rev. B 82, 235114 (2010) Nature Comm. 3:882 (2012)

Theory + Experiments by MPQ groupPhys. Rev. Lett. 110:165304 (2013)

Nature Physics 9, 795 (2013)

Zak/Berry phase measurements as a probe of band topology in OLBloch+Ramsey interferenceexperiments with cold atoms

Probing band topology with Ramsey/Bloch interference

C. Salomon et al., PRL  (1996) 

Tools of atomic physics:Bloch oscillations 

/2 pulse

Evolution

Tools of atomic physics:Ramsey interference

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

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

EvolutionEvolution

Zak phase probe of band topology in 1d

One dimensional superlatticesSu‐Schrieffer‐Heeger model

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

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

Phys. Rev. Lett. 110:165304 (2013) Nature Physics 9, 795 (2013)

SSH model of polyacetylene

Analogous to bichromatic optical lattice potential 

I. Bloch et al.,LMU/MPQ

B A B BA

Su, Schrieffer, Heeger, 1979

Dimerized model

A B A AB

Characterizing SSH model using Zak phase Two hyperfine spin states experience the same optical potential

/2a/2a

a

Zak phase is equal to 0

Problem: experimentally difficult to control Zeeman phase shift

Dynamic phases due todispersion and magnetic field fluctuations cancel.Interference measuresthe difference of Zakphases of the two bands in two dimerizations.Expect phase

Spin echo protocol for measuring Zak phase 

Bloch oscillations measurements in LMU/MPQWith -pulse but no swapping of dimerization

Bloch oscillations measurements in LMU/MPQWith p-pulse and with swapping of dimerization

Zak phase measurements in LMU/MPQ

Zak phase measurements can be used to probetopological propertiesof Bloch bands in 2D and 3D

D. Abanin, T. Kitagawa, I. Bloch, E. DemlerPhys. Rev. Lett. 110:165304 (2013) 

F. Grusdt, D. Abanin, E. DemlerPhys. Rev. A 89, 043621 (2014) 

Integral of the Berry phase around the Dirac point

Measuring Berry curvature in 2d and Chern num

Manifestation of Berry phaseof Dirac points in grapheme:IQHE plateaus are shifted by 1/2

Interferometric probe of Berry curvature and Chern number in 2d systems

Extension to more exotic states: Quantum Spin Hall Effect states and Topological Insulators in 3D. Grusdt et al., Phys. Rev. A 89, 043621 (2014)

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, 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 platesTranslation by bi-refringentcalcite crystals that displace only horizontally polarized light

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

A. White’s group in QueenslandT. Kitagawa et al., Nature Comm. 3:882 (2012)

From discreet timequantum walks to

Topological Hamiltonians

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

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 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 photonsKitagawa et al., Nature Comm. 2012

Rotation is implemented by half-wave platesTranslation by birefringentcalcite crystals that displace only horizontally polarized light

Topological Hamiltonians in 2D with quantum walk

Schnyder et al., PRB (2008)Kitaev (2009)

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

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 2/T

Topological properties of evolution operator

Floquet operator

Time dependent periodic Hamiltonian

Example of topologically non-trivial evolution operator

and relation to Thouless topological pumpingSpin 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 1 describes topological pumping of particles. This is another way to understand Thouless quantized 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

Floquet states beyond in 2dbeyond quantum walk

Topological Floquet states in 2d

Kitagawa et al., PRB (2010)

Topological Floquet states in 2d Oka and Aoki, PRB (2009)Kitagawa  et al., PRB (2010)Lindner, Refael, Galiski, Nat. Phys. (2011)

Observation of Floquet-Bloch states on the surface of a topological insulator

Gedik et al., Science (2013)

Summary

Observation of edge states in topological phases realized with photons

First direct measurementof Zak phase of a 1d band

Prospect of measuringtopological propertiesof 2d bands

How to measure Berry phase of Bloch states Naïve approach:

Move atom on a closed trajectoryaround Dirac pointMeasure accumulated phase

Problems with this approach: Need to move atom on a complicated curved trajectoryNeed to separate dynamical phase

C

Integral of the Berry phase is only well defined on a closed trajectory

Brillouin zone is a torus. There are two types of closed trajectories

is not gauge invariantC

gauge invariantintegral ofBerry curvature

Zak

From Berry phase to Zak phase

Zak phase: integral of Berry phase overreciprocal lattice vector

Zak phase measurements in LMU/MPQ

Universality of collective modes

keV MeV GeV TeVfeV peV µeV meV eV

pK nK µK mK K

neV

roomtemperature

LHC

He N

current experiments10-11 - 10-10 K

first BECof alkali atoms

M. Enders et al.,Observation of Higgs modein 2D superfluid in ultracoldatomsNature 2012