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UNIVERSITY OF SOUTHAMPTON DOCTORAL THESIS Propagation in media as a probe for topological properties Author: Supervisor: Samuel MUGEL Dr. Carlos LOBO Prof. Maciej LEWENSTEIN A thesis submitted in fulfillment of the requirements for the degree of Doctor of Philosophy in the Faculty of Social, Human and Mathematical Sciences June 21, 2017 ii “If you haven’t found something strange during the day, it hasn’t been much of a day.” John Archibald Wheeler iii UNIVERSITY OF SOUTHAMPTON Abstract Faculty of Social, Human and Mathematical Sciences Mathematical Sciences Doctor of Philosophy PROPAGATION IN MEDIA AS A PROBE FOR TOPOLOGICAL PROPERTIES by Samuel MUGEL The central goal of this thesis is to develop methods to experimentally study topological phases. We do so by applying the powerful toolbox of quantum simulation techniques with cold atoms in optical lattices. To this day, a complete classification of topological phases remains elusive. In this context, experimental studies are key, both for studying the interplay between topology and complex effects and for identifying new forms of topological order. It is therefore crucial to find complementary means to mea- sure topological properties in order to reach a fundamental understanding of topological phases. In one dimensional chiral systems, we suggest a new way to construct and identify topologically protected bound states, which are the smoking gun of these materials. In two dimensional Hofstadter strips (i.e: systems which are very short along one dimension), we suggest a new way to measure the topological invariant directly from the atomic dynamics. In one dimensional optical lattices, topological bound states are difficult to generate due to the absence of sharp boundaries, and harder still to identify unambiguously. By periodically driving a one dimensional dilute gas of atoms with a pair of Raman lasers, we find that a system analogous to the two-step quantum walk can be realised. This system can host two flavours of topologically protected bound states, meaning that it escapes the standard classification of topological phases. This study details the considerations and many of the relevant experimental tools to design a topologically non-trivial system. In particular, we show that we can build a topological boundary by using the lasers’ finite beam width, and that the topologically protected states which live at this boundary can be identified, and differentiated, by studying their spin distribution. The bulk-boundary correspondence states that a system’s bulk and edge properties are indissociable. It is unclear, however, how this principle ex- tends to systems with vanishingly small bulks, as for instance the Hofstadter strip, which was recently realised using a one dimensional gas of spinful atoms. We define a topological invariant for this system which accurately counts the number of topological bound states. This suggests that, even in such an extreme situation, the bulk-boundary correspondence applies. We suggest a method for experimentally measuring this invariant from the atomic dynamics which relies on three main ingredients: the adiabatic load- ing of a well localised wavepacket in the ground state of the lattice, the application of a weak force along the axis of the strip, and the measurement of the centre of mass position after a Bloch oscillation. v Contents Abstract iii Declaration of Authorship xiii Acknowledgements xv 1 General introduction1 1.1 The integer quantum Hall effect..................1 1.2 Review of systems in which topology appears........2 1.2.1 Table of topological phases...............3 1.2.2 Beyond the topological table..............4 1.3 Experimental study of topology.................4 1.3.1 Review of realisations..................4 1.3.2 Quantum simulators applied to topology.......5 1.3.3 Detection of topological phases.............5 1.4 Topics addressed in this thesis..................6 2 Review of the manifestation of topological properties9 2.1 The Berry phase.......................... 10 2.2 One-dimensional example: the SSH model........... 11 2.2.1 The model.......................... 11 2.2.2 Chiral symmetry..................... 12 2.2.3 Winding number..................... 13 2.2.4 Topological bound states................ 15 2.2.5 Alternative view: SSH model in the continuum... 16 2.3 Two-dimensional example: the Hofstadter model...... 17 2.3.1 The model......................... 18 2.3.2 Hofstadter model on a torus............... 19 The Chern number.................... 20 Particle dynamics in a topological insulator...... 22 The FHS algorithm.................... 24 2.3.3 Hofstadter model on a cylinder............. 24 Dispersion of the Hofstadter Hamiltonian on a cylinder 25 The number of edge branches is a topological invariant 26 2.4 Beyond the topological classification: Floquet systems.... 28 2.4.1 The two-step quantum walk.............. 28 2.4.2 Introduction to Floquet theory............. 29 2.4.3 Chiral symmetry in the two-step quantum walk... 30 2.4.4 Two topological invariants............... 32 2.4.5 Winding numbers of the two chiral time frames... 32 2.4.6 Spin structure of the topological bound states.... 34 vi 3 Review of selected experimental techniques in cold atoms 37 3.1 The atom-light interaction.................... 38 3.1.1 Off-resonant optical trapping.............. 38 3.1.2 Stimulated Raman scattering.............. 39 3.2 Optical lattices........................... 40 3.3 Artificial gauge fields....................... 42 3.3.1 Periodically modulated lattices............. 42 3.3.2 Raman coupling and the Hofstadter strip....... 43 4 Topologically non-trivial quantum walk with cold atoms 45 4.1 The atomic quantum walk.................... 46 4.1.1 Experimental proposal.................. 46 4.1.2 Shift operation on wavepackets............. 48 4.1.3 Rotation operation using the Raman pulses...... 50 Quasimomentum kick.................. 50 Energy difference between spin states......... 50 Effective coin operation.................. 51 4.1.4 Quantum walk with cold atoms............ 52 Complete sequence.................... 52 Simulations........................ 53 A hidden unitary symmetry............... 54 Gauge transformation for a smaller unit cell..... 55 4.1.5 Double atomic quantum walk.............. 56 4.2 Topological properties of the atomic quantum walk..... 58 4.2.1 Symmetries of the atomic quantum walk....... 58 4.2.2 Phase diagram...................... 60 Computation of the winding numbers......... 60 Derivation of the phase boundaries.......... 62 4.2.3 Mapping to the Creutz ladder.............. 63 4.3 Detection of the topological bound state............ 66 4.3.1 Experiment suggested.................. 66 4.3.2 Simulation......................... 68 4.3.3 Identification of the topological protection...... 69 4.4 Pair of momentum separated Jackiw-Rebbi states....... 71 4.4.1 Symmetries........................ 72 4.4.2 Topologically trivial bound states........... 72 4.4.3 Approximate Floquet Hamiltonian........... 72 ˆ 4.4.4 Low energy eigenstates of Happrox0 ........... 74 4.5 Conclusions............................ 75 5 Measure of the Chern number in a Hofstadter strip 77 5.1 The Hofstadter strip....................... 79 5.1.1 The Hofstadter model with a synthetic dimension.. 79 5.1.2 The Hofstadter strip on a cylinder........... 80 5.2 Measure of the Chern number................... 81 5.2.1 A linear potential gradient causes adiabatic transport. 81 5.2.2 Semiclassical equations of motion............ 81 5.3 Charge pumping in the Hofstadter strip............ 83 5.4 Pumping Dynamics........................ 84 5.4.1 Weak spin tunnelling amplitude limit......... 85 5.4.2 Hofstadter strip with isotropic tunnelling amplitudes 88 vii 5.5 Edge effects............................ 90 5.5.1 Bloch wavefunctions to second order in perturbation theory........................... 90 5.5.2 Comparison to the measured Chern number...... 91 5.6 Higher Chern number measurement.............. 92 5.6.1 Hofstadter strip with Φ = 4π=5 ............. 93 5.6.2 Generalisation to q > Ny ................. 94 5.6.3 Comparison to the FHS algorithm results....... 94 5.7 Generalisation to non-interacting atomic gases........ 95 5.7.1 Cloud preparation and Brillouin zone population.. 96 5.7.2 Chern number, measured from the mean displacement 97 5.7.3 Chern number, measured from the mean velocity.. 98 5.8 Perturbations which break translational invariance...... 99 5.8.1 Static disorder....................... 99 5.8.2 Harmonic trap....................... 101 5.9 Conclusions............................ 102 6 Conclusion and outlook 103 Bibliography