Christof Weitenberg

Workshop „Geometry and Quantum Dynamics“

Natal 29.10.2015

arXiv:1509.05763 (2015)

Experimental reconstruction of the Berry curvature

in a topological Bloch band

Topological Insulators

• Topology of the bulk leads to chiral edge states • Experimental access is mostly limited to the edge states

Quantum Hall effect, Von Klitzing, PRL (1980).

Solid State

Silicon photonics Hafezi ,Nat. Photon. 7, 1001 (2013).

Unidirectional backscattering in Polariton system Wang, Nature 461, 772 (2009).

Helical waveguides Rechtsman, Nature 496, 196 (2013).

Topological RF circuit Ningyuan, PRX 5, 021031 (2015).

Mechanical ‘topological insulator’ Süsstrunk, Science 349, 47 (2015).

Model systems

Wavefunction microscope

• To see the bulk topology, we need a wavefunction microscope! • Experiments with cold atoms might provide new insight

See also work with superconducting qubits Roushan, Nature (2014)

Our wavefunction microscope

Topological bands are a hot topic in cold atom research!

• Soltan-Panahi et al., Nat. Phys (2011). • Soltan-Panahi et al., Nat. Phys (2012). • Jo et al., PRL 108, 045305 (2012). • Struck et al. PRL 108, 225304 (2012). • Cheuk et al. PRL 109, 095302 (2012). • Struck et al. Nature Phys. 9, 738 (2013). • Parker et al. Nature Phys. 9, 769 (2013). • Atala et al. Nature Phys. 9, 795 (2013). • Aidelsburger et al. PRL 111, 185301

(2013). • Miyake et al. PRL 111, 185302 (2013). • Jotzu et al. Nature 515, 237 (2014). • Atala et al. Nature Phys. 10, 588 (2014). • Aidelsburger et al. Nature Phys. 11, 162

(2015). • Kennedy et al. Nature Phys. (2015). • Stuhl et al., Science (2015). • Mancini et al., Science (2015). • Jotzu et al. PRL 115, 073002 (2015). • Duca et al. Science 347, 288 (2015). • Li et al. arXiv:1509.02185 (2015). • Taie et al. arXiv:1506.00587 (2015). • Nakajima et al. arXiv:1507.02223

(2015). • Lohse et al. arXiv:1507.02225 (2015). • Lu et al. arXiv:1508.04480 (2015). • …

Experiments/Lattice

Engineering of topological bands: • Jaksch, Zoller, New J. Phys. 5, 56 (2003). • Kitagawa et al. PRB 82, 235114 (2010). • Dalibard et al. Rev. Mod. Phys. 83, 1523

(2011). • Cooper, PRL 106, 175301 (2011). • Rudner et al. PRX 3, 031005 (2013). • Goldman, Dalibard, PRX 4, 031027

(2014). • Baur et al. PRA 89, 051605(R) (2014). • Bukov et al. Adv. Phys. 64, 139 (2015). • …

Detection of topology: • Alba et al. PRL 107, 235301 (2011). • Price, Cooper, PRA 85, 033620 (2012). • Goldman et al. PRL 108, 255303 (2012). • Dauphin, Goldman, PRL 110, 135302

(2013). • Wang et al. PRL 110, 166802 (2013). • Goldman et al. PNAS 110, 6736 (2013). • Price, Cooper, PRL 111, 220407 (2013). • Hauke et al. PRL 113, 045303 (2014). • …

Non-Abelian Gauge fields: • Osterloh et al. PRL 95, 010403 (2005). • Nayak et al. Rev. Mod. Phys. 80, 1083

(2008). • Goldman et al. PRL 103, 035301 (2009). • Hauke et al. PRL 109, 145301 (2012). • …

Topology and Interactions: • Raghu et al. PRL 100, 156401 (2008). • Rachel, Le Hur, PRB 82, 075106 (2010). • Neupert et al. PRL 106, 236804 (2011). • Cooper, Dalibard, PRL 110, 185301

(2013). • Bergholtz et al. Intern. J. Mod. Phys. B

27, 1330017 (2013). • Grushin et al. PRL 112, 156801 (2014). • …

Theory

Experiments with topological bands (I)

Chern number of Hofstadter bands Aidelsburger, Nature Phys. (2015) Bloch group

Cyclotron orbits in Hofstadter Model Aidelsburger, PRL (2013) Bloch group

Meissner effect Atala, Nature Phys. (2014) Bloch group

Condensation in Hofstadter Model Kennedy, Nature Phys. (2015) Ketterle group

Chiral edge states Stuhl, Science (2015) Spielman group

Chiral edge states Mancini, Science (2015) Inguscio group

Hofstadter model

Chiral edge states

Chern number in Haldane Model Jotzu, Nature (2014) Esslinger group

Haldane model

Experiments with topological bands (II)

1D Gauge potential Struck, PRL (2012) Sengstock group

Ferromagnetic domains Parker, Nature Phys. (2013) Chin group

Spin-dependent driving Jotzu, PRL (2015) Esslinger group

Wilson lines Li, arXiv (2015) Bloch group

Aharonov-Bohm interferometer Duca, Science (2015) Bloch group

Zak and Berry phase Spin-orbit coupled

lattice

Spin-orbit coupled lattice Cheuk, PRL (2012) Zwierlein group

Magnetism via lattice driving

Ising XY spin-models Struck, Nature Phys. (2013) Sengstock group

Zak phase Atala, Nature Phys. (2013) Bloch group

1D Gauge potentials

,Berry-ology‘*

Would be nice to see this Berry curvature

• But how are all the different properties related to one-another?

• Berry connection:

• NOT gauge invariant

• Berry curvature:

Berry phase

* Fuchs et al., EPJB 77, 351 (2010)

Chern Number

Calculated Berry Curvature for different systems

Boron nitride (tight-binding model) Fuchs et al., Euro Phys. J B 77, 351 (2010)

Strained graphene Guinea et al., Nature Phys. 6, 30 (2009)

Monolayer MoS2

Feng et al., Phys. Rev. B 86, 165108 (2012)

Ferromagnetic bcc Fe Yao et al., Phys. Rev. Lett. 92, 037204 (2004)

Map of the full Berry curvature

Fläschner et al., arXiv:1509.05763 (2015) related work: Li et al. arXiv:1509.02185 (2015)

How do we do it?

• Tunable hexagonal lattice for fermionic 40K

• Offset between A and B

• Boron-nitride

• Massive Dirac points

• Well separated flat s-bands

• Tomography

• Berry curvature flattens out

B

A See: Soltan-Panahi et al.,Nat.Phys 7, 434 (2011) Baur et al., PRA 89, 051605(R) (2014).

Floquet engineering of dressed bands

• Circular shaking

• Breaks time-reversal symmetry

• k-dependent coupling

• Berry curvature engineering

• Dirac point at K annihilated

• Dirac point at Γ created

• Three-fold symmetry

Qu

asi-

ener

gy

Eigenstates in Bloch sphere representation

• Two-band Hamiltonian allows for a Bloch sphere representation

• States of flat bands lie at the north- and south-pole

• For each k, the ground state is given by:

• is given by:

k

Eigenstate reconstruction

We follow the proposal by P. Hauke, M. Lewenstein, A.Eckardt, PRL 113, 045303 (2014): Related proposal Alba et al., PRL 107, 235301 (2011)

Reconstruction of full Hamiltonian

• The oscillations become visible in k-space after time-of-flight

Momentum space density

• Fitting the oscillations gives and for each momentum (pixel)

Now we can reconstruct the Berry curvature using:

Berry curvature

Amplitude + Phase = Berry curvature

N. Fläschner, B. Rem, M. Tarnowski, D. Vogel, D. Lühmann, K. Sengstock. , C. Weitenberg, arXiv:1509.05763 (2015)

Engineering of Berry Curvature

Increasing amplitude

Ber

ry c

urv

atu

re (

1/|

b|²

) 200 nm

100 nm

Conclusion

• Full state tomography

• Full measurement of Berry curvature

• Allows us to determine

• Chern number

• Engineering of Berry curvature

• Annihilation and creation of Dirac points

• Localization

Outlook

What next?

We want to further explore topological bands…

• Study interacting Fermions, Bosons, or mixtures in these bands

• How can we prepare a Floquet topological Insulator? What can we learn from quenches into the nontrivial regime?

• We still have the spin degree of freedom: study high-spin systems or engineer additional spin-orbit-coupling

• Explore other interesting geometries using the tunable lattice

B

A

The BFM-team

André Eckardt

Maciej Lewenstein

Ham

bu

rg

Ludwig Mathey

In collaboration with

Dre

sde

n

Bar

celo

na

Dirk-Sören Lühmann

Matthias Tarnowski

Christof Weitenberg

Nick Fläschner

Benno Rem

Dominik Vogel

Klaus Sengstock