ABSTRACT Title of dissertation: QUANTUM SIMULATION OF INTERACTING SPIN MODELS WITH TRAPPED IONS Kazi Rajibul Islam, Doctor of Philosophy, 2012 Dissertation directed by: Professor Christopher Monroe Joint Quantum Institute, University of Maryland Department of Physics and National Institute of Standards and Technology The quantum simulation of complex many body systems holds promise for understanding the origin of emergent properties of strongly correlated systems, such as high-Tc superconductors and spin liquids. Cold atomic systems provide an almost ideal platform for quantum simulation due to their excellent quantum coherence, initialization and readout properties, and their ability to support several forms of interactions. In this thesis, I present experiments on the quantum simulation of long range Ising models in the presence of transverse magnetic fields with a chain of up to sixteen ultracold 171Yb+ ions trapped in a linear radiofrequency Paul trap. Two hyperfine levels in each of the 171Yb+ ions serve as the spin-1=2 systems. We detect the spin states of the individual ions by observing state-dependent fluorescence with single site resolution, and can directly measure any possible spin correlation function. The spin-spin interactions are engineered by applying dipole forces from precisely tuned lasers whose beatnotes induce stimulated Raman transitions that couple virtually to collective phonon modes of the ion motion. The Ising couplings are controlled, both in sign and strength with respect to the effective transverse field, and adiabatically manipulated to study various aspects of this spin model, such as the emergence of a quantum phase transition in the ground state and spin frustration due to competing antiferromagnetic interactions. Spin frustration often gives rise to a massive degeneracy in the ground state, which can lead to entanglement in the spin system. We detect and characterize this frustration induced entanglement in a system of three spins, demonstrating the first direct experimental connection between frustration and entanglement. With larger numbers of spins we also vary the range of the antiferromagnetic couplings through appropriate laser tunings and observe that longer range interactions reduce the excitation energy and thereby frustrate the ground state order. This system can potentially be scaled up to study a wide range of fully connected spin networks with a few dozens of spins, where the underlying theory becomes intractable on a classical computer. Quantum Simulation of Interacting Spin Models with Trapped Ions by Kazi Rajibul Islam Dissertation submitted to the Faculty of the Graduate School of the University of Maryland, College Park in partial fulfillment of the requirements for the degree of Doctor of Philosophy 2012 Advisory Committee: Professor Christopher Monroe, Chair/Advisor Professor Steve Rolston Dr. Ian Spielman Professor Christopher Jarzynski Professor Dianne O'Leary c Copyright by Kazi Rajibul Islam 2012 To my parents ii Acknowledgments First of all, I thank my advisor Prof. Chris Monroe for giving me the op- portunity to work on this project. Instead of writing a long essay on his amazing capabilities as a physicist, and as an advisor, I would just say that he is very close to the ideal advisor I could have hoped for. I appreciate his constant encouragement to engage in fruitful conversations with other people, particularly the theorists, and the independence that he gave me to pursue my experimental ideas. Research in experimental physics is surely a group effort, and this is so true in our group. I have been fortunate to work with a pool of great postdocs and fellow graduate and undergraduate students. Ming-Shien Chang, Kihwan Kim, Emily Edwards and Wes Campbell were all very gifted postdocs, and I learned a lot from them. Thanks to you all. I enjoyed lots of stimulating physics conversations with Wes Campbell over the years. I wish him all the best for his new position as a faculty member at UCLA. It was fun to work with the fellow graduate students, Simcha Korenblit, Jake Smith and Crystal Senko; and the undergrads, Andrew Chew, Aaron Lee, who decided that he loved this experiment too much to leave, and is continuing as a grad student, and most recently with the new undergrad in the team Geoffrey Ji. It was a privilege working with smart theorists like Luming Duan and his students Guin-Dar Lin and Zhexuan Gong, Jim Freericks and his postdoc Joseph Wang, Howard Carmichael and his student Changsuk Noh, and Dvir Kafri. David Huse and Rajdeep Sensarma taught me many aspects of the quantum Ising model. iii Thanks to all of you. I enjoyed interacting with the other members of the group immensely, though I did not directly work with them. Thanks to Taeyoung Choi, Susan Clark, Charles Conover, Shantanu Debnath, Brian Fields, Ilka Geisel, Dave Hayes, David Hucul, Volkan Inlek, Kale Johnson, Kenny Lee, Le Luo, Andrew Manning, Dzmitry Mat- sukevich, Peter Maunz, Jonathan Mizrahi, Steve Olmschenk, Qudsia Quraishi and Jon Sterk. A special thanks to Crystal, Wes and Emily for going over my thesis manuscripts and suggesting important corrections. I thank all my thesis committee members (Steve Rolston, Ian Spielman, Chris Jarzynski, Dianne O'Leary, and of course Chris Monroe) for their support in schedul- ing the defense talk, and accommodating my delays. Thanks to Victor Galitski for serving on my PhD candidacy committee two and a half years back. The Joint Quantum Institute provided an excellent environment for research, and I learned a lot from almost all the members in the basement and on the second floor of the CSS building during our random interactions. Thanks to all of you. Thanks to all the staff members of JQI for helping me with non-academic tasks most efficiently, and always in a timely manner. I thank all the funding agents for making my graduate student life smoother. In particular, the DARPA Optical Lattice Emulator program has been a wonderful experience for the last five years. I thoroughly enjoyed many intellectually invigo- rating discussions in all the OLE meetings. My friends were a constant source of support over all these years. Fortunately, iv they are too numerous to name here. Last but not the least, my family members were always there whenever I needed them. I am eternally grateful to all of you. v Table of Contents List of Tables viii List of Figures ix List of Abbreviations xi 1 Introduction 1 2 Trapped Ions as a Platform for Quantum Simulation 8 2.1 Overview . .8 2.2 Ion Trapping . .8 2.2.1 Trapping 171Yb+ in our Paul trap . 15 2.3 Manipulation of 171Yb+ spin and motional states . 18 2.3.1 Hyperfine states . 18 2.3.2 Doppler cooling . 20 2.3.3 Detection of the spin states . 22 2.3.4 State initialization by optical pumping . 26 2.3.5 Coherent manipulation of the spin states . 27 2.3.6 Raman sideband cooling . 45 2.4 Vibrational normal modes of trapped ions . 46 2.5 Simulating the quantum Ising model . 53 2.5.1 Ising interactions . 54 2.5.2 Adiabatic quantum simulation . 66 2.6 Experimental Apparatus . 68 2.6.1 Ti:Sapphire laser . 68 2.6.2 Generating 369.5 nm light by frequency doubling . 78 2.6.3 369.5 nm optics schematics . 78 2.6.4 Mode-locked 355 nm laser . 81 2.6.5 Optical set up for the Raman transitions . 91 2.7 Quantum simulation recipe for experimentalists . 97 2.8 Troubleshooting with 174Yb+ ....................... 107 3 Simulation of the ferromagnetic quantum Ising model 109 3.1 Overview . 109 3.2 Symmetries of the Hamiltonian . 111 3.3 Low energy eigenstates at T=0 . 112 3.3.1 States near B=J =0 ....................... 112 3.3.2 States near B=J ! 1 ...................... 117 3.3.3 Quantum phase transition at B = J ............... 118 3.4 Experiment: onset of a quantum phase transition . 121 3.4.1 Engineering the ferromagnetic Ising couplings . 121 3.4.2 Experimental protocol and order parameters of the transition 124 3.4.3 Results . 127 vi 3.4.4 Sources of error in the quantum simulation . 130 3.5 Scaling up the simulation to N = 16 with 355 nm mode locked laser . 139 4 Three frustrated Ising spins on a triangle 142 4.1 Overview . 142 4.2 Frustrated quantum Ising model . 145 4.2.1 States near B=J =0 ....................... 145 4.2.2 Preparing the entangled state in adiabatic quantum simulation 147 4.3 Frustration and entanglement . 148 4.4 Experimental methods . 151 4.5 Experimental Results . 154 4.6 Summary and outlook . 160 5 Frustrated magnetic ordering with tunable range antiferromagnetic couplings161 5.1 Overview . 161 5.2 Some features of the long range antiferromagnetic quantum Ising model163 5.2.1 Ground and low energy eigenstates . 163 5.2.2 Frustration and the range of the interactions . 166 5.3 Experimental simulation of the model . 169 5.3.1 Tuning the range of Ising interactions . 171 5.3.2 Experimental protocol and the order parameters . 174 5.4 Results of the quantum simulation . 177 5.4.1 Onset of antiferromagnetic correlations in quantum simulation for N = 10 and N = 16 spins . 177 5.4.2 Frustration of the AFM order with increasing range of inter- actions . 180 5.5 Discussions and conclusion . 184 6 Outlook 185 6.1 Scaling up the system - large numbers of equally spaced ions in a Paul trap . 185 6.2 Creating an arbitrary lattice geometry . 187 6.3 Other interesting spin physics . 188 A Quantum trajectory calculations 190 B Detection of spin states 192 C Relevant Frequencies for 171Yb+ and 174Yb+ 194 Bibliography 195 vii List of Tables 2.1 Phases of various pulses used in quantum simulation. 107 3.1 Symmetries of the eigenstates .