Exploring topological states with cold atoms and photons 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) Harvard-MIT $$ NSF, AFOSR MURI, DARPA OLE, MURI ATOMTRONICS
Universality in condensed matter physics Spontaneous symmetry breaking and order
Universality of physics fev pev nev µev mev ev kev MeV GeV TeV pk nk µk mk K Cold atoms experiments 10 11 10 10 K room temperature LHC Higgs mode in ultracold atoms, 2012 Higgs mode of the standard model, 2012
Order beyond symmetry breaking 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 Current along x, measure voltage along y. On a plateau with an accuracy of 10-9 What is the quantum protectorate of such precise quantization?
Topological order In 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 zone One electron wavefunction in a crystal (periodic) potential can be written as k is crystal momentum restricted to Brillouin zone, a region of k-space with periodic boundaries. Function is periodic (same in every unit cell) As k changes, we map an energy band. Set of all bands is a band structure. But lattice momentum is periodic The Brillouin zone can play the role of the surface. Important property of quantum mechanics, the Berry phase, gives us the curvature.
Berry phase Consider a quantum-mechanical system in a nondegenerate ground state, e.g. spin ½ particle in a magnetic field. The adiabatic theorem says that if the Hamiltonian is changed slowly, the system 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 number The change in the electron wavefunction within the Brillouin zone leads to a Berry connection and Berry curvature Ky Brillouin zone Kx 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 IQHE Thouless et al., PRL 1982
Topological order and edge states TKNN quantization exists only for insulators 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-Smith PRB 1993
Su-Schrieffer-Heeger Model B A B A B When d z (k)=0, states with t>0 and t<0 are topologically distinct.
Domain wall states in SSH Model An interface between topologically different states has protected midgap states Absorption spectra on neutral and doped trans (CH) x
Topological states of matter Integer and Fractional 3D topological insulators Quantum Hall effects Quantum Spin Hall effect 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
Order parameters can be measured Magnetization order parameter in ferromagnets Nematic order parameter in liquid crystals
Outline Zak/Berry phase measurements as a probe of band topology in OL Bloch+Ramsey interference experiments with cold atoms Theory + Experiments by MPQ group Phys. Rev. Lett. 110:165304 (2013) Nature Physics 9, 795 (2013) Exploring edge states in topological phases with photons T. Kitagawa et al., PRA 82:33429 (2010) Phys. Rev. B 82, 235114 (2010) Nature Comm. 3:882 (2012)
Probing band topology with Ramsey/Bloch interference
Tools of atomic physics: Bloch oscillations C. Salomon et al., PRL (1996)
Tools of atomic physics: Ramsey interference /2 pulse Evolution /2 pulse + measurement ot S z gives relative phase accumulated by the two spin components Evolution Used for atomic clocks, gravitometers, accelerometers, magnetic field measurements
Zak phase probe of band topology in 1d One dimensional superlattices Su Schrieffer Heeger model Theory: Takuya Kitagawa (Harvard), Dima Abanin (Harvard/Perimeter), Eugene Demler (Harvard) Experiments Marcos Atala, Monika Aidelsburger, Julio Barreiro, Immanuel Bloch (LMU/MPQ) Phys. Rev. Lett. 110:165304 (2013) Nature Physics 9, 795 (2013)
SSH model of polyacetylene Su, Schrieffer, Heeger, 1979 B A B A B Analogous to bichromatic optical lattice potential I. Bloch et al., LMU/MPQ
A B A B A Dimerized model
Characterizing SSH model using Zak phase Two hyperfine spin states experience the same optical potential a /2a 0 /2a Zak phase is equal to Problem: experimentally difficult to control Zeeman phase shift
Spin echo protocol for measuring Zak phase 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
Bloch oscillations measurements in LMU/MPQ With -pulse but no swapping of dimerization
Bloch oscillations 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 and 3D D. Abanin, T. Kitagawa, I. Bloch, E. Demler Phys. Rev. Lett. 110:165304 (2013) F. Grusdt, D. Abanin, E. Demler Phys. Rev. A 89, 043621 (2014)
Measuring Berry curvature in 2d and Chern num Integral of the Berry phase around the Dirac point Manifestation of Berry phase of 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 Spin rotation Spindependent Translation Analogue of classical random walk. Introduced in quantum information: Q Search, Q computations
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)
From discreet time quantum walks to Topological Hamiltonians T. Kitagawa et al., Phys. Rev. A 82, 033429 (2010)
Discrete quantum walk Spin rotation around y axis Translation One step Evolution operator
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
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. Can we have topologically distinct quantum walks?
Split-step DTQW
Split-step DTQW Phase Diagram
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
Split-step DTQW with site dependent rotations Apply site-dependent spin rotation for
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
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 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
Topological properties of evolution operator Time dependent periodic Hamiltonian Floquet operator 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 2 /T
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 group velocity n 1 describes average displacement per period. Quantization of 1 describes topological pumping of particles. This is another way to understand Thouless quantized pumping
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
Floquet states beyond in 2d beyond 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 First direct measurement of Zak phase of a 1d band Prospect of measuring topological properties of 2d bands Observation of edge states in topological phases realized with photons
How to measure Berry phase of Bloch states C 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
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 Brillouin zone is a torus. There are two types of closed trajectories Zak Zak phase: integral of Berry phase over reciprocal lattice vector
Zak phase measurements in LMU/MPQ
Universality of collective modes M. Enders et al., Observation of Higgs mode in 2D superfluid in ultracold atoms Nature 2012 fev pev nev µev mev ev kev MeV GeV TeV pk nk µk mk K current experiments 10-11 -10-10 K first BEC of alkali atoms He N room temperature LHC