ABSTRACT
 Title of dissertation: QUANTUM SIMULATIONS OF THE
 ISING MODEL WITH TRAPPED IONS:
 DEVIL?S STAIRCASE AND
 ARBITRARY LATTICE PROPOSAL
 Simcha Korenblit, Doctor of Philosophy
 2013
 Dissertation directed by: Professor Christopher Monroe
 Joint Quantum Institute
 University of Maryland Department
 of Physics and
 National Institute of Standards
 and Technology
 A collection of trapped atomic ions represents one of the most attractive plat-
 forms for the quantum simulation of interacting spin networks and quantum mag-
 netism. Spin-dependent optical dipole forces applied to an ion crystal create long-
 range e ective spin-spin interactions and allow the simulation of spin Hamiltonians
 that possess nontrivial phases and dynamics.
 We trap linear chains of 171Yb+ ions [1] in a Paul trap [2], and constrain the
 occupation of energy levels to the ground hyper ne clock-states, creating a qubit
 or pseudo-spin 1/2 system. We proceed to implement spin-spin couplings between
 two ions using the far detuned M lmer-S renson scheme[3] and perform adiabatic
 quantum simulations of Ising Hamiltonians with long-range couplings. We then
demonstrate our ability to control the sign and relative strength of the interaction
 between three ions. Using this control, we simulate a frustrated triangular lattice,
 and for the  rst time establish an experimental connection between frustration and
 quantum entanglement. We then scale up our simulation to show phase transitions
 from paramagnetism to ferromagnetism for nine ions, and to anti-ferromagnetism
 for sixteen ions.
 The experimental work culminates with our most complicated Hamiltonian - a
 long range anti-ferromagnetic Ising interaction between 10 ions with a biasing axial
  eld, which I have led.
 Theoretical work presented in this thesis shows how the approach to quan-
 tum simulation utilized in this thesis can be further extended and improved. It is
 shown how appropriate design of laser  elds can provide for arbitrary multidimen-
 sional spin-spin interaction graphs even for the case of a linear spatial array of ions.
 This scheme uses currently existing trap technology and is scalable to levels where
 classical methods of simulation are intractable.
QUANTUM SIMULATIONS OF THE ISING MODEL WITH
 TRAPPED IONS:
 DEVIL?S STAIRCASE AND ARBITRARY LATTICE PROPOSAL
 by
 SIMCHA KORENBLIT
 Dissertation submitted to the Faculty of the Graduate School of the
 University of Maryland, College Park in partial ful llment
 of the requirements for the degree of
 Doctor of Philosophy
 2013
 Advisory Committee:
 Professor Christopher Monroe, Chair/Advisor
 Professor Edo Waks
 Professor Ted Jacobson
 Professor Eite Tiesinga
 Professor Ian Spielman
 Professor Gretchen Campbell
c Copyright by
 Simcha Korenblit
 2013
Dedication
 I dedicate this work to my family, who are my rock.
 ? ?????????????????????,???????????????,????????????????????
 . ????????????,??????????????
 How?manifold?are?Thy?works,?O?LORD!?In?wisdom?hast?Thou?made?them
 all?????the?earth?is?full?of?Thy?creatures.
 Psalms?104:24.
 ii
Acknowledgments
 A doctoral program is rarely pursued in a vacuum. There are many people
 and organizations without whom this work would not be possible.
 Firstly, I would like to thank my advisor, Chris Monroe, for giving me the
 opportunity to work at the cutting edge of quantum information. None of this would
 have happened if he did not happen to be in his o ce the day I knocked on his door,
 uninvited, displaying way more con dence than warranted for someone who has not
 yet been accepted to the Michigan Physics Department. Chris is probably one of
 the most hands on advisors out there, the kind of advisor who helps his group put
 up optic table \clouds" and electronics room shelving, while not micro-managing
 group members, and allowing us room to grow as independent scientists.
 Throughout the years, I have bene ted greatly from the advice and discussions
 of scientists outside our group. Here I will only mention a few, and I ask forgiveness
 in advance from whoever I may have forgotten. Luming Duan has collaborated
 with us intensively over the years, and has guided us on many issues and directions
 worth pursuing. His students Guin-Dar Lin, Zeh-Xuan Gong and Chao Shen have
 worked closely with us and supported our work. We have also had many useful
 discussions and collaboration with Jim Freericks and his post-doc Joseph Wang,
 especially regarding the e ect of the transverse  eld on ion chain phonons. We have
 also received numerical support from Howard Carmichael and his student Changsuk
 Noh on the extremely intensive quantum trajectory simulations of our experiment.
 Working in a world class lab means working with world class people. I have
 iii
had the privilege to work with extremely talented and dedicated post doctoral re-
 searchers. When I arrived in Maryland I knew nothing about ion trapping and had
 no knowledge in atomic physics. I received a patient and dedicated introduction
 from Ming-Schien Chang and Kihwan Kim, who were friends and teachers to me.
 Later I had the privilege of working with the post-docs Emily Edwards and Wes
 Campbell. From them I have learned much as well. Soon after arriving I had a an
 amazing graduate student join me - Rajibul Islam, and he shared the trenches with
 me for most of my time at Maryland.
 Later other students have joined our project and contributed as well - Andrew
 Chew, an undergrad set up the iodine lock. Aaron Lee joined us as an undergrad
 and signi cantly improved my messy control program. He enjoyed this work so
 much he later joined us a grad student. Crystal Senko switched over from the fast
 gate project and was instrumental in trouble shooting our messy MBR problem.
 Jake Smith has joined as well and is quickly getting up to speed. Phil Richerme has
 joined the team as a post-doc and quickly began to contribute. I have also bene ted
 from working and playing with many grad students, undergrads and post-docs not
 directly on the project. I list them here in no particular order. Peter Maunz,
 Dzimitry Matsukevich, Susan Clark, Dan Stick, Yisa Ramula, David Moehring,
 Kathy-Ann Brickman, Steve Olmschenk, Qudsia Quraishi, Jon Sterk, Mark Acton,
 Le Luo, Charles Conover, Brian Fields, Kenny Lee, Ilka Geisel, Jonathan Mizrahi,
 Brian Neyenhuis, Andrew Manning, Geo rey Ji, Kale Johnson, Volkan Inlek, David
 Hucul, Dave Hayes, Ken Wright, Shantanu Debnath, Caroline Figgatt, Taeyoung
 Choi, Chenglin Cao, Daniel Brennan, and Volkan Inlek. It has been a pleasure and
 iv
a terri c learning experience working with you.
 I would like to thank Crystal, Phil and Jake for reviewing my thesis.
 I thank my thesis committee for taking upon themselves this service to the
 department and to greater science community.
 I would also like to thank the Physics department sta , particularly Lorraine
 Desalvo and Jane Hessing, who have been very warm and welcoming to me when I
 made the transition from Michigan.
 A man cannot live on professional relationships alone. I would like to thank
 the close friends who were there for me during my time here, but they are too
 numerous to mention. I would like to send a special thanks to Rob Feldmier who
 has been like a brother to me, and Rabbi and Rebbetzin Teitelbaum, who have been
 amazing to me. I would like to thank MesorahDC and the entire Jewish community
 of DC, who have guided the spiritual journey that paralleled my intellectual journey
 in this program.
 This work is supported by the US Army Research O ce (ARO) with funds
 from the DARPA Optical Lattice Emulator (OLE) Program and the IARPA MQCO
 Program, the NSF Physics at the Information Frontier Program, the NSF Physics
 Frontier Center at JQI.
 Thank all of you and please forgive me if I omitted you.
 v
Table of Contents
 List of Figures ix
 List of Abbreviations xi
 1 Introduction 1
 1.0.1 Classical Computation . . . . . . . . . . . . . . . . . . . . . . 2
 1.0.2 Quantum Information Processing . . . . . . . . . . . . . . . . 3
 1.1 The DiVincenzo Criteria . . . . . . . . . . . . . . . . . . . . . . . . . 5
 1.2 Quantum Mechanics . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
 1.3 Adiabatic quantum simulation . . . . . . . . . . . . . . . . . . . . . . 8
 1.3.1 Hydrogen-Like Atom . . . . . . . . . . . . . . . . . . . . . . . 9
 1.3.2 Transition Rules . . . . . . . . . . . . . . . . . . . . . . . . . 14
 1.3.3 Interaction Hamiltonian . . . . . . . . . . . . . . . . . . . . . 14
 1.3.4 E ective Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . 15
 1.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
 2 The 171Yb+ ion 17
 2.1 Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
 2.2 Dopppler Cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
 2.3 Optical Pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
 2.4 Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
 3 Quantum Control of The Ion 28
 3.1 Light-Ion Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
 3.2 Raman Transitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
 3.3 Coupling Spin to Motion . . . . . . . . . . . . . . . . . . . . . . . . . 31
 3.4 Multi-Ion Gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
 4 Experimental Setup 40
 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
 4.2 Ion trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
 4.2.1 Normal Modes . . . . . . . . . . . . . . . . . . . . . . . . . . 43
 4.3 Micromotion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
 4.3.1 The Linear Trap . . . . . . . . . . . . . . . . . . . . . . . . . 50
 4.3.2 Trap Voltages and Control . . . . . . . . . . . . . . . . . . . . 52
 4.3.3 Oscillating Trap Voltage . . . . . . . . . . . . . . . . . . . . . 54
 4.4 Optical Setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
 4.4.1 Ionization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
 4.4.2 935 nm Repump . . . . . . . . . . . . . . . . . . . . . . . . . . 57
 4.4.3 638 nm Repump . . . . . . . . . . . . . . . . . . . . . . . . . . 58
 4.4.4 Cooling, Pumping, and Detecting with 369 nm . . . . . . . . 58
 4.4.5 Iodine Spectroscopy Lock . . . . . . . . . . . . . . . . . . . . 65
 vi
4.4.6 Vanguard 355 nm Laser . . . . . . . . . . . . . . . . . . . . . . 68
 4.4.7 Driving Raman and M lmer-S renson Transitions with the
 Vanguard . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
 4.4.8 Florescence Collection . . . . . . . . . . . . . . . . . . . . . . 78
 4.5 Arbitrary Waveform Generator (AWG) . . . . . . . . . . . . . . . . . 87
 5 Experimental Procedure 90
 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
 5.2 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
 5.3 Frequencies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
 5.4 Loading and Alignment . . . . . . . . . . . . . . . . . . . . . . . . . . 92
 5.5 Raman Beams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
 5.6 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
 5.7 Imaging Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98
 6 Summary of Experiments 99
 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
 6.2 Experimental Demonstration of the Ising interaction . . . . . . . . . 100
 6.2.1 Two spin case . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
 6.2.2 Three spin case . . . . . . . . . . . . . . . . . . . . . . . . . . 103
 6.3 Adiabatic quantum simulation . . . . . . . . . . . . . . . . . . . . . . 105
 6.4 Phase diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
 6.5 Spin frustration and entanglement . . . . . . . . . . . . . . . . . . . 112
 6.6 Scalability of the quantum simulation . . . . . . . . . . . . . . . . . 117
 6.7 Scaling of imperfections . . . . . . . . . . . . . . . . . . . . . . . . . 121
 6.7.1 Spin-motion coupling . . . . . . . . . . . . . . . . . . . . . . . 121
 6.7.2 Diabaticity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
 6.7.3 Spontaneous Emission . . . . . . . . . . . . . . . . . . . . . . 122
 6.7.4 Intensity  uctuations . . . . . . . . . . . . . . . . . . . . . . . 123
 6.7.5 Detection Errors . . . . . . . . . . . . . . . . . . . . . . . . . 123
 6.8 Conclusion and Outlook . . . . . . . . . . . . . . . . . . . . . . . . . 128
 7 Quantum Simulation of the Devil?s Staircase 130
 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
 7.2 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
 7.3 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
 7.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
 8 Simulating the Ising Model with Arbitrary Control of the Couplings 153
 8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
 8.2 Control of an Arbitrary Lattice Hamiltonian . . . . . . . . . . . . . . 154
 8.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
 8.4 Implementation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
 8.5 Scalability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
 8.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
 vii
A Linear Trap External Pi Filter 167
 B Optical Diagram Legend 169
 C AWG Function Library 170
 D Mathematica ROI and Discriminator Selection Script 181
 Bibliography 186
 viii
List of Figures
 2.1 171Yb+ States and transition wavelengths . . . . . . . . . . . . . . . . 18
 2.2 Doppler cooling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
 2.3 Optical pumping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
 2.4 171Yb+ Detection Scheme . . . . . . . . . . . . . . . . . . . . . . . . . 25
 2.5 Single ion histogram . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
 2.6 Idealized histograms of multiple ions . . . . . . . . . . . . . . . . . . 27
 3.1 Raman . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
 3.2 RSB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
 4.1 Paul Trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
 4.2 Transverse Modes For 10 Ions . . . . . . . . . . . . . . . . . . . . . . 47
 4.3 Spin-Spin Couplings for 10 Ions . . . . . . . . . . . . . . . . . . . . . 48
 4.4 Linear trap 3D diagram . . . . . . . . . . . . . . . . . . . . . . . . . 50
 4.5 Trap Control Program . . . . . . . . . . . . . . . . . . . . . . . . . . 53
 4.6 Trap Control Program - Electrodes Tab . . . . . . . . . . . . . . . . . 55
 4.7 MBR-110 Laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
 4.8 369 nm Laser and Optics Setup . . . . . . . . . . . . . . . . . . . . . 63
 4.9 Chamber and Resonant Beams . . . . . . . . . . . . . . . . . . . . . . 64
 4.10 Iodine Spectroscopy Setup . . . . . . . . . . . . . . . . . . . . . . . . 67
 4.11 MLUV Main Window . . . . . . . . . . . . . . . . . . . . . . . . . . . 69
 4.12 MLUV Oscillator Service . . . . . . . . . . . . . . . . . . . . . . . . . 70
 4.13 MLUV Ampli er Settings . . . . . . . . . . . . . . . . . . . . . . . . 71
 4.14 MLUV Oven Settings . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
 4.15 MLUV Position Controls . . . . . . . . . . . . . . . . . . . . . . . . . 72
 4.16 Frequency Combs for Raman . . . . . . . . . . . . . . . . . . . . . . . 74
 4.17 Vanguard Raman Setup . . . . . . . . . . . . . . . . . . . . . . . . . 75
 4.18 Raman Beams Close Up . . . . . . . . . . . . . . . . . . . . . . . . . 79
 4.19 Raman Pro le . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
 4.20 Florescence Collection . . . . . . . . . . . . . . . . . . . . . . . . . . 81
 4.21 10 Bright Ions, Binned . . . . . . . . . . . . . . . . . . . . . . . . . . 82
 4.22 Single Ion on ICCD . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
 4.23 ICCD Histograms of 10 Ions . . . . . . . . . . . . . . . . . . . . . . . 86
 5.1 Trap Corner on ICCD . . . . . . . . . . . . . . . . . . . . . . . . . . 94
 5.2 10 Ion Raman Spectrum . . . . . . . . . . . . . . . . . . . . . . . . . 96
 5.3 Automatic Side-Band Finder . . . . . . . . . . . . . . . . . . . . . . . 97
 6.1 Two Spins Coupled . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102
 6.2 Three Spins Coupled . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
 6.3 Energy Level Diagram for 3 Spins . . . . . . . . . . . . . . . . . . . . 107
 6.4 Measured Ferromagnetic Order . . . . . . . . . . . . . . . . . . . . . 108
 6.5 Phase Diagram for Three Ions . . . . . . . . . . . . . . . . . . . . . . 110
 ix
6.6 Spin Frustration on Triangular Lattice . . . . . . . . . . . . . . . . . 113
 6.7 Entanglement Witness . . . . . . . . . . . . . . . . . . . . . . . . . . 115
 6.8 FM Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
 6.9 FM Data vs Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
 6.10 FM Magnetization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
 7.1 6 Ion Spin-Spin Coupling Decay . . . . . . . . . . . . . . . . . . . . . 134
 7.2 Controlling the Phases of Simulation Pulses . . . . . . . . . . . . . . 137
 7.3 Magnetization Phase Diagram for 6 Ions . . . . . . . . . . . . . . . . 139
 7.4 Axial Simulation for 6 spins - States . . . . . . . . . . . . . . . . . . . 141
 7.5 6 Ion Staircase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
 7.6 6 Ion Complete Energy Spectrum . . . . . . . . . . . . . . . . . . . . 143
 7.7 6 Ion Low Energy Spectrum . . . . . . . . . . . . . . . . . . . . . . . 143
 7.8 6 Ion Exponential Fit . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
 7.9 Filtered 6 Spin Staircase . . . . . . . . . . . . . . . . . . . . . . . . . 146
 7.10 Power Filter VS Adiabaticity . . . . . . . . . . . . . . . . . . . . . . 147
 7.11 10 Ion Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148
 7.12 Axial Simulation for 10 spins - States . . . . . . . . . . . . . . . . . . 149
 7.13 10 Ion Exponential Fit . . . . . . . . . . . . . . . . . . . . . . . . . . 150
 7.14 Power Filtered 10 Ion Data . . . . . . . . . . . . . . . . . . . . . . . 151
 8.1 Spectrum of N Ions with N Detunings . . . . . . . . . . . . . . . . . . 155
 8.2 Example Solutions for Arbitrary Lattice Generation . . . . . . . . . . 159
 8.3 Micromirror Array Simulator . . . . . . . . . . . . . . . . . . . . . . . 160
 8.4 AOM Array Simulator . . . . . . . . . . . . . . . . . . . . . . . . . . 161
 8.5 Phase Array Simulator . . . . . . . . . . . . . . . . . . . . . . . . . . 162
 8.6 1D Ising chain Scaling for Simulator . . . . . . . . . . . . . . . . . . . 164
 A.1 Pi Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
 B.1 Optical Diagram Legend . . . . . . . . . . . . . . . . . . . . . . . . . 169
 x
List of Abbreviations and Symbols
  h Reduced Plank Constant
 KB Boltzman Constant
 AFM Antiferromagnetic
 AOM Acousto Optic Modulator
 BSB Blue Sideband
 COM Center of Mass
 CW Continuous Wave
 EOM Electro Optic Modulator
 FPGA Field Programmable Gate Array
 FM Ferromagnetic
 PCB Printed Circuit Board
 QE Quantum E ciency
 SE Schr odinger Equation
 SHG Second Harmonic Generator
 THG Third Harmonic Generator
 RF Radio Frequency
 ROI Region of Interest
 RSB Red Sideband
 UHV Ultra High Vacuum
 VCA Voltage Controlled Attenuator
 IREAP Institute for Research in Electronics and Applied Physics
 NA Numerical Aperture
 NSA National Security Agency
 xi
Chapter 1
 Introduction
 The work presented in this thesis is complex, and covers many subject areas,
 which cannot be given a full, bottom up presentation in this work. Furthermore, I do
 not posses this knowledge. Six years pursuing an experimental result does not allow
 for such a pursuit. Instead, I will present mostly information needed to understand
 the motivation for this work and the minimal amount of information needed to
 follow it as it is exposed. A deeper understanding can be pursued by the interested
 reader by reading the cited works. Many of the results presented here have already
 been presented in previously published works to which I have contributed in work
 and writing, but all the results presented are the fruits of a team e ort.
 Quantum systems are notoriously di cult to model e ciently using classical
 computers, owing to the exponential complexity in describing a general quantum
 state as the system grows in size. In the 1980s, Richard Feynman proposed to
 circumvent this problem by employing a control quantum system with tailored in-
 teractions and logic gates between quantum bits (qubits) in order to simulate the
 quantum system under investigation [4, 5].
 Quantum spin models such as the Ising model have become a proving ground
 for Feynman?s proposal, with systems of qubits behaving as e ective spins and
 interactions engineered with external electromagnetic  elds. The quantum Ising
 1
model is the simplest spin Hamiltonian that exhibits nontrivial aspects of quantum
 magnetism such as spin frustration, phase transitions [6], and poorly understood spin
 glass and spin liquid phases [7, 8]. Indeed, solving for the ground state con guration
 of spins subject to a general fully-connected Ising interaction is known to be an
 NP-complete problem [9]. At the conclusion of this thesis I will show how a fully-
 connected Ising or more general Heisenberg spin model with arbitrary couplings
 across the spin network can be generated in a scalable system of trapped atomic
 ions, even for a one-dimensional chain in space. This may allow quantum simulations
 with hundreds of spins, where the physics cannot generally be predicted otherwise.
 1.0.1 Classical Computation
 Over the last century computers and electronics have made enormous advances
 and transformed our world our and lives. The theoretical foundations of modern
 computers are rooted in the work of Charles Babbage in the mid 19th century.
 It was in the late 19th century, when the need to handle large amounts of data,
 and the emergence of the vacuum tube, spurred the creation of real world, albeit
 primitive computers.
 The conception of solid state transistors in the 1920?s and their technological
 application in the 1950?s enabled a breakneck acceleration in the development of
 computers. The constant rate of transistor miniaturization led to the observation
 that the number of transistors on a processor doubles every two years (\Moore?s
 Law")and processor performance doubles every 18 months. The average smartphone
 2
that a consumer carries in their back pocket today is more powerful than a 1990?s
 Cray-2 supercomputer!
 Although the available computational power is increasing, this growth cannot
 continue forever, as technological hurdles such as thermal management and impu-
 rities become more severe as the density of transistors and processors continues to
 increase, and physical obstacles such as unwanted quantum e ects due to shrinking
 component size threaten reliability. Currently, the smallest transistors fabricated in
 mass are 22 nm in length [10].
 Furthermore, there are more fundamental limits to computation speed. A
 computational algorithm can be analyzed abstractly, and a relationship between
 the number of steps or other resources needed to solve a problem and the size of the
 problem can be established.
 An important problem in computer science is factoring a large number into
 two prime numbers, as this can be used to decipher encrypted communications
 (using the RSA method). With known methods, the number of steps to  nd the
 prime factors grows exponentially with a linear growth in the number of digits of
 the problem number. This makes this problem practically unsolvable and hence the
 encryption is secure.
 1.0.2 Quantum Information Processing
 However, it is known that a computer with components that remain in a pure
 quantum state (undisturbed by uncontrolled external in uences) only requires a
 3
polynomial increase in steps [11]. This surprising result shows that quantum com-
 puters likely posses computational powers far beyond those of classical computers
 [4] (this is only a hypothesis though as the fundamental theories of computation are
 still in their infancy).
 The other motivation for the development of quantum information processing
 machines is the problem of simulating quantum systems. As Feynman proposed in
 1982 [4, 5], a well-controlled quantum system could e ciently simulate the behavior
 of a complex quantum model that is classically intractable. For example, a quantum
 simulator could be used to determine properties and dynamics derived of poorly un-
 derstood models in condensed matter such as quantum magnetism [12], spin glasses
 [7], spin liquids [13] and high temperature superconductors [14, 15].
 As the resources required to simulate quantum systems on a classical computer
 grow exponentially with linear growth of the quantum system, a machine that is
 itself in a quantum state analogous to a system we wish to investigate will naturally
 bene t from this growth in its computational space as its number of components
 grows linearly. This is likely the more important application of quantum informa-
 tion processing, as the number of digital quantum computing algorithms known is
 limited, but the number of quantum systems whose understanding can bene t are
 unlimited [16].
 Furthermore, it has been shown that a universal quantum simulator is equiv-
 alent in computational power to a universal quantum computer up to a polynomial
 factor in the time needed for computation [17], and as such cannot simply be dis-
 counted as a more limited form of quantum computing.
 4
1.1 The DiVincenzo Criteria
 In order to implement the ideas discussed above, one needs a practical quantum
 system. DiVicenzo presented the grocery list of characteristics of such a system :
  Identi cation of well-de ned qubits
  Reliable state preparation
  Low decoherence
  Accurate quantum gate operations and
  Strong quantum measurements
 Although originally speci ed for a circuit quantum computer, the requirements
 for a quantum simulator are identical. The 171Yb+ system has all of these charac-
 teristics. The qubits are well de ned in the sense that each 171Yb+ ion is exactly the
 same. The only variation between these qubits is due to local gradients in electric
 and magnetic  elds. The technical details of the 171Yb+ qubit will be discussed in
 detail in chapter 2. For now, it will be simply stated that in comparison to compet-
 ing implementations (mainly solid state Josephson junction based systems [18, 19])
 of qubits, the ion systems are superior in all of these criteria except one that is
 not listed - scalability. The storage of ions for long periods of time is di cult as it
 requires strong electric  elds and very low pressures. In this work, we were limited
 to working with 16 ions. Although we faced multiple sources of errors that would
 prevent us from controlling more than 16 ions, the main limiter is our inability to
 5
store in a stable crystal the ions for more than  10 minutes. This is likely due to
 background atoms colliding with the chain, even at our working pressure of 10 12
 Torr.
 1.2 Quantum Mechanics
 The development of quantum mechanics was spurred by false predictions and
 contradictions encountered by physicists in the late 19th and early 20th century.
 These problems spanned several di erent aspects of physics.
 First, there was the famous ultraviolet catastrophe, which predicted the radi-
 ation of in nite energy from a blackbody at the limit of short wavelengths. Second
 there was the photoelectric e ect, where light shining on a metal would cause a
 current that would inexplicably cease when the light frequency was reduced. Third,
 the double-slit experiment demonstrated wave-like behavior for light, except when a
 detector was placed at one of the slits leading to the ba ing appearance of particle-
 like behavior. The fourth and most problematic discovery was that atoms were most
 likely composed of electrons in orbit around a small, positive nucleus. A classical
 description of this would seem to make matter itself impossible due to the electrons
 completely radiating away their energy, unless they were limited to discrete orbitals
 for an (at the time) inexplicable reason. These problems were resolved by the intro-
 duction of the photon, the de Broglie wave equation, and  nally the development of
 quantum mechanics.
 In the quantum mechanical description of physical systems the state is de-
 6
scribed by a vector in a Hilbert space. This vector and its associated conserved
 quantities (\quantum numbers") describe all that can be known about the system.
 The dynamics of such a system is described by the Schr odinger equation
 H j i = i h _j i (1.1)
 Where H is the Hamiltonian, an operator in Hilbert space that is the sum of the
 energy terms in the system, j i is the wavefunction, a vector in Hilbert space,
 and  h is Planck?s constant, de ned as the ratio between photon energy and angular
 frequency.
 The most revolutionary aspect of this new theory is the wave-function descrip-
 tion of nature: although particles are discrete (they are always observed as particles),
 some of their properties can only be described probabilistically. This probability is
 given by the probability amplitude h j i =
 R
    dx integrated over the region of
 interest (this is the inner product of the vector in Hilbert space). The other ob-
 servables of the system are extracted from the wavefunction using an operator, such
 that the expectation value for quantum mechanical observable A is
 hAi = h j A j i (1.2)
 where A is a Hermitian matrix (so as to always have real expectation values).
 7
If we assume the Hamiltonian is time independent, we can rewrite Eq. 1.1
 E j i = H j i (1.3)
 by assuming that j i has an oscillatory form
 j (t)i = e iEt= h j (0)i (1.4)
 and E is the energy of the eigenvector j (t)i .
 In general, we de ne j (t)i = U(t) j (0)i where
 U(t) = Te i
 R
 Hdt= h (1.5)
 is the evolution operator, which simpli es to U(t) = e iHt= h for time independent
 Hamiltonians.1
 1.3 Adiabatic quantum simulation
 One method of quantum information processing is adiabatic quantum simula-
 tion. This approach is motivated by the adiabatic quantum computation algorithm
  rst proposed as a method to solve NP-complete satis ability problems [20]. The
 process of quantum adiabatic computation works as follows: a quantum system is
 initialized to the ground state of a trivial Hamiltonian. Next, the Hamiltonian is
 1T is Dyson?s time ordering operator. This operator handles the non-commutation of H(t) for
 di erent t.
 8
adiabatically deformed into the Hamiltonian of interest, whose ground state encodes
 the solution of a problem that has been mapped to the  nal Hamiltonian. If success-
 ful, the system will remain in the ground state and can be directly probed once the
 system arrives at the desired Hamiltonian. For quantum simulation of magnetically
 interacting spins, this approach allows the determination of ground states where
 the Hamiltonian can easily be written, yet the spin ground state cannot always be
 predicted, even with just a few dozen spins [21].
 A precondition for the use of this state is satisfying the adiabaticity criterion[22]
 2:
 T >>
  
g2min
 (1.6)
 where T is total simulation time, gmin = min(E1(t) E0(t)) is the minimum energy
 gap between the ground state (or any other state we can trivially initialize) and clos-
 est coupled state, where  = max(jh1; tj dHdt j0; ti j) is the change in the Hamiltonian
 at time t for the coupling between those energy levels.
 1.3.1 Hydrogen-Like Atom
 In this work we be using 171Yb+ ions as qubits. To understand their behavior,
 we will model them using a quantum mechanical model,  rst developed to describe
 the simplest atom - Hydrogen. In order to model the Hydrogen atom and other
 atoms that have only one valence electron, we assume a simple model of a negative
 electron orbiting a stationary positive nucleus in a spherically symmetric electric
 2This is the criterion usually encountered in the literature. However, this criterion is not in
 true in general [23]
 9
potential. Then we write our Hamiltonian
 H = V + T =  
1
 4  0
 Ze2
 r
 +
 p2
 2m0
 (1.7)
 where Z is the number of positive nuclear charges, e is the charge of the electron, r is
 the distance of the electron from the nucleus,  0 is the electric permittivity constant,
 and p
 2
 2m0
 is the kinetic energy of the electron (m0 is its mass):  h
 2
 2m0
 r2.
 We  nd the eigenfunctions that satisfy the Schr odinger equation:
  (r;  ;  ) = R(r)Y ( ;  ) (1.8)
 Y ml ( ;  )   
s
 (2l + 1)(l  jmj)!
 4 (l + jmj)!
 eim Pml (cos ) (1.9)
 R(r)nl  
s
  
2Z
 na
  3 (n l  1)!
 2n[(n+ l)!]3
 e rZ=na
  
2Zr
 na
  l
 [2L2l+1n l 1(2Zr=na)] (1.10)
 a  
4  0 h
 2
 mRe2
 (1.11)
 Where Pml are the associated Legendre functions, 2L
 2l+1
 n l 1 are the associated La-
 guerre polynomials, and mR is the reduced mass of the nucleus and electron.
 n; l and m are quantized numbers that label orthogonal eigenfunctions. These
 numbers are described as \good" quantum numbers. Their goodness is a result of
 the simplifying assumptions of the model. In the 171Yb+ ion we will see that these
 assumptions are only roughly true, leading to metastable states de ned by other
 quantum numbers. For Hydrogen-like atoms, n determines the energy of the sta-
 10
tionary state, l (the azimuthal quantum number) orbital angular momentum., and
 m (the magnetic quantum number) the direction of angular momentum vector.
 The de nition of the Legendre polynomials restricts the values of l and m:
 l = 0; 1; 2:::;m =  l; l + 1; :::; 0; 1; ::l (1.12)
 The Laguerre polynomial solution is found when one restricts the radial solution
 R(r) to a non-diverging series. The same restriction truncates l at n 1. Intuitively,
 we can understand that a  nite kinetic energy implies  nite angular momentum.
 Using the standard de nition of angular momentum, L = r  p and our pre-
 vious implicit de nition of the momentum operator, we can show the commutation
 relations for the Cartesian components of L to be
 [Li; Lj] = i h ijkLk (1.13)
 and if we de ne the raising and lowering operators as
 L+ = Lx + iLy L = Lx  iLy (1.14)
 11
then we  nd that these are ladder operators for the atomic eigenstates
 L+ jl;mi =  h
 p
 (l  m)(l +m+ 1) jl;m+ 1i (1.15)
 L jl;mi =  h
 p
 (l +m)(l  m+ 1) jl;m 1i (1.16)
 There are additional complications to this simple model we must address before
 we can discuss the 171Yb+ ion and its energy structure. First, the electron posses an
 innate angular momentum, \Spin", which is described by exactly the same math as
 angular momentum, however l for the electron is  xed at 1=2 and so m only spans
  1=2; 1=2. We label the operators for spin ~S   h2~ where ~ is the Pauli spin vector
 - a vector of operators with components
  x  
0
 B
 B
 @
 0 1
 1 0
 1
 C
 C
 A ;  y  
0
 B
 B
 @
 0  i
 i 0
 1
 C
 C
 A ;  z  
0
 B
 B
 @
 1 0
 0  1
 1
 C
 C
 A (1.17)
 Complicating thins further, the proton(s) in the nucleus of the atom also have
 spin. These spins give the proton and electron a magnetic moment, and these
 experience a magnetic dipole force due to the magnetic  eld created by the orbiting
 electron. These add additional terms to the the total atom Hamiltonian, of the form
 Hd =    B. We de ne two new quantities, the total orbital angular momentum
 J  L + S and F  I + J , where S is the total electronic spin and I is the total
 nuclear spin.
 For most stable atomic states, where we ignore coupling between the outer
 electron and the inner electrons (LS coupling), we will label the atomic states using
 12
the ?Russel-Saunders? convention
 2S+1LJ (1.18)
 The interaction of the electron with the magnetic  eld created by the rotating
 nucleus (in the electron?s rest frame) gives rise to the  ne structure splitting of the
 energy levels, and the interaction of the nucleus? dipole moment with the magnetic
  eld created by the electron leads to a smaller perturbation of the energy levels,
 called the hyper ne splitting.
 In atoms such as 171Yb+ , where the nuclear spin magnetic moment is non-zero
 (due to the odd number of nucleons), the hyper ne coupling will be strong, and so it
 will be necessary to add an energy term to the Hamiltonian for the dipole coupling
 between the nucleus? magnetic moment  and the magnetic  eld from the electron,
 HHF =   I  (Bl +Bs).
 However, the dominant coupling term will be that of the total angular momen-
 tum, so we will write the eigenstates of the atom as superpositions of our previously
 uncoupled basis:
 jJ;Mi =
 X
 ml+ms=mj
 CL;S;L+Sml;ms;mj jl;mli js;msi (1.19)
 where the coe cients CL;S;L+Sml;ms;mj are the Clebsch-Gordan coe cients. These will be
 necessary to calculate transition strengths.
 13
1.3.2 Transition Rules
 What is the signi cance of all this? Well, from here we arrive at the rules
 which tell us what atomic transitions are allowed and which are forbidden. The
 restriction of atomic transitions will allow us to construct a useful qubit.
 When the atom absorbs a photon, angular momentum is conserved. We con-
 sider a perturbation H1 to the atom from an external electromagnetic  eld. The
 probability for a transition from a stationary state  1 to  2 will depend on the
 matrix element h 1jH1 j 2i . Considering only the dipole component of the multi-
 pole expansion of the  eld, the perturbation to the potential of the electron will be
 / e~r  ~E which requires  l =  1 and  m =  1; 0 for a non-zero transition prob-
 ability. We will see how these rules come in to play when we examine the allowed
 transitions of 171Yb+ in chapter 2.
 1.3.3 Interaction Hamiltonian
 For many systems, the Hamiltonian is a sum of a static term HS and an
 interaction term HI . If this is the case we may simplify our analysis by rotating into
 the static frame, so that the time evolution is only a function of HI . So our total
 Hamiltonian in the Schr odinger picture (where all operators are static in time) is
 HT = HS +HI (1.20)
 14
And in the interaction ?picture?
  
  I(t)
  
= eiHSt= h
  
  T (t)
  
(1.21)
 HIS = e
 iHSt= hHSe
  iHSt= h = HS (1.22)
 HII = e
 iHSt= hHIe
  iHSt= h (1.23)
 where the superscript I (T ) indicates interaction (Schr odinger ) picture. We see that
 the static term remains una ected, but now the interaction term in the interaction
 picture is time dependent. Rewriting the Schr odinger equation 1.1 in terms of the
 interaction picture, the static component of the Hamiltonian drops out and we are
 left with
 HII
  
  I(t)
  
= i h _j I(t)i (1.24)
 and now the dynamics only depend on the interacting component of the Hamil-
 tonian.
 1.3.4 E ective Hamiltonian
 For highly complex systems the interaction picture is still too complex to allow
 for analytical analysis. For many of these systems, including the spins in the ion
 trap, simplifying assumptions can be made to rewrite an even simpler Hamiltonian.
 The following treatment, succinctly presenting the formalism developed by D.F.V.
 15
James[24], can be applied to interaction Hamiltonians of the form
 HI(t) =
 NX
 n=1
 Hne
  i!nt +Hyne
 i!nt (1.25)
 where we are summing over N oscillatory terms, and 0 < !1 < !2::!N Then we can
 derive the e ective Hamiltonian given by
 Heff (t) =
 NX
 n;m=1
 1
  h !mn
  
Hm; H
 y
 n
  
ei(!m !n)t (1.26)
 and 1 !mn =
 1
 2(
 1
 !m
 + 1!n ).
 This Hamiltonian is the result of several approximations. First the interaction
 evolution operator 1.5 is time averaged, so that all fast changes are neglected (general
 case of rotating wave approximation). The second approximation is neglecting the
 higher order terms of the averaged evolution operator?s time ordered expansion.
 Applying a second rotating wave approximation leads to 1.26. We will utilize this
 powerful tool when we treat light-ion interactions in 3.
 1.4 Conclusion
 This brief introduction to quantum information processing with ions and quan-
 tum mechanics has laid down the ground work for understanding the quantum as-
 pects of manipulating a 171Yb+ ion, that will be introduced in the following chapter.
 16
Chapter 2
 The 171Yb+ ion
 Now that we have discussed the basic physics necessary to understand atomic
 transitions and energy states, we will introduce our atomic system of choice: the
 171Yb+ ion[1]. We will discuss in 4.2 how we actually trap and create these ions
 from a Yb metal.
 The discussion of how we generate and direct the light on the ion will have to
 wait until chapter 4.
 Many ions are used throughout the world in quantum information experi-
 ments, mostly from the alkaline earth metals. These are chosen so that once singly
 ionized the ion will have a hydrogen-like electronic structure, which is more easily
 understood and controlled.
 The second important feature is that the energy structure is such that the ion
 can be forced through optical pumping and cycling transitions to be constrained to
 two energy levels, so that it behaves as an e ective spin (a qubit). Our ion has a
 nuclear spin of 1/2, giving it the simplest hyper ne structure possible in its ground
 state.
 For our qubit, we will be using the ground state 2S1=2 energy manifold, and
 the clock states jF = 0;mf = 0i as our \down" (j#i ) state and jF = 1;mf = 0i
 as our \up" (j"i ) state. These states are the eigenstates of the  z operator with
 17
369.5262nm (99.5%
 ) 
638.6151n
 m
  
638.6102n
 m
  
F=1 
F=0 
F=0 
F=1 
F=1 
F=2 
F=1 
F=0 
F=3 
F=4 
329n
 m
  
12.642812118466 GHz+310.8B2Hz 
2.105 GHz 
2P1/2 
935.1879n
 m
  
2S1/2 
2P3/2 3D[3/2]1/2 
3D3/2 
2D5/2 
2F7/2 
1D[5/2]5/2 F=3 
F=2 
100 THz 
F=3 
F=2 
Figure 2.1: The 171Yb+ ion states with transition wavelengths.
 eigenvalues 1 and -1 respectively. The clock states are so named as they are used
 in atomic clocks due to only having a second order Zeeman energy shift, making
 them robust to magnetic noise. Correspondingly, this ensures the qubits a long T2
 coherence time - 2.5 s [1] without magnetic shielding, orders of magnitude longer
 than non-clock state qubits with magnetic shielding or magnetic- eld feedback.
 We use the resonant 2S1=2 to 2P1=2 transition for cooling, state initialization
 and detection. As we scatter on this transition, there is a 0:5% chance of decaying
 out of this cycle to the 2D3=2 state. In order to correct for this we irradiate the ion
 with 935:1879 nm light modulated at 3:07 GHz in order to remove the population
 from 2D3=2 jF = 1i and 2D3=2 jF = 2i to the 3D[3=2]1=2 state. This is a metastable
 state (?Bracket State?) where the usual ?good? quantum numebrs are no longer valid
 18
1. The ion has a 98:5% chance of decaying back to 2S1=2 jF = 1i from this state, so
 it is quickly restored to the correct qubit state (this will however create an error in
 the phase of one qubit relative to other qubits in the chain).
 A much less likely event is the ion transitioning to the 2F7=2 state. This
 happens once in about an hour per ion. It is likely caused by a collision with a
 background atom in the Ultra High Vacuum Chamber (UHV), as this is a forbidden
 octopole transition [26]. Previously, we have used 638:61 nm light that is scanned
 between the two resonance frequencies. We have found that our intense 355 nm
 Raman beams are actually better at retrieving the ion from this long lived ( 6
 years) dark state, but it is possible that this is due to YbH+ formation rather than
 transition to the 2F7=2 state [27]. The 355 nm radiation would then disassociate this
 ionic molecule and return the ion to a coolable state.
 2.1 Ionization
 The source of our ions is a nearly pure source of neutral 171Yb+ . This source
 is a solid metal, sliced into small shards that are then forced into a narrow stainless
 steel tube, which is clamped o at one end. A wire is spot welded to the tube, and
 the tube is fed into a titanium holder, designed to have a high electric resistance
 and low thermal conductivity so as to isolate the Ohmic heating to the 171Yb+ oven.
 A current of roughly 2.3 Amps is run through this circuit (the holder is grounded
 1In this state the outer electron couples to the core electrons, and the state is written as
 2s0+1[K]J , Where s0 is the spin of the outer electron, K is the total angular momnetum of the core
 and the angular momentum of the outer electron, and J is K and the spin of the outer electron.
 The letter preceding the bracket is not part of the notation but is L for the core in the usual LS
 notation. [25]
 19
to the chamber). The oven is oriented towards the trapping region, where a 399 nm
 beam will excite the neutral Yb from 1S0 to 1P1. A powerful 355 nm beam or 369 nm
 beam will then excite an electron to the continuum. Some of these 171Yb+ ions will
 be su ciently cooled by Doppler cooling to be trapped.
 2.2 Dopppler Cooling
 Doppler cooling is achieved by scattering red detuned light from the resonant
 transitions [28] for the 2S1=2 to 2P1=2 transitions, as illustrated in Fig. 2.2. The light
 is tuned to a wavelength of 369.521525 nm , and detuned 25 MHz from resonance,
 while the transition has a 20 MHz linewidth. To ensure that the ion is cooled for
 both F=0 and F=1 states of the 2S1=2 manifold, the light is modulated by an Electro
 Magnetic Modulator (EOM) at half the combined hyper ne splitting of the 2S1=2
 and 2P1=2 states. The second order sideband allows the o resonant cooling of
 the j#i state. Doppler cooling utilizes the Doppler shift of the resonant frequency
 in the frame of the moving ion, such that when the ion is moving towards the
 oncoming light it is blue shifted towards resonance. The ion will then spontaneously
 emit a photon into a random direction, so on average losing momentum in the
 direction of the oncoming beam. The temperature limit of this random process
 is the ?Doppler cooling limit?, h =2KB, where  is the linewidth of the transition
 and KB Boltzmann?s constant. For the S-P transition linewidth of 19.7MHz, this
 corresponds to 5  10 3 K. The relation KBT =  n h! [29] implies for our transverse
 direction of motion an average of  12 phonons after Doppler cooling.
 20
F=1 
F=0 
F=1 
2P1/2 
2S1/2 
F=0 
mF=- 1 
mF=1 
mF=0 
Figure 2.2: Doppler cooling the 171Yb+ ion. Note that the 935 nm
 repump transition is not shown here. The wide lines represent the
 lasers driving transitions. The narrow lines represent the allowed
 decays.
 21
2.3 Optical Pumping
 Once the ion chain is su ciently cooled (after Doppler cooling and sideband
 cooling), we must prepare the ions in a pure and known quantum state. This can
 be done with high  delity. We apply light resonant with a transition from the j"i
 state to the 2P1=2 F=1 state (as shown in 2.3), leading to a cycle that with our
 beam power of  10 W pumps the ion to the j#i state within 1.5 s with over 99%
  delity.
 2.4 Detection
 The detection scheme of the 171Yb+ ion relies on the large separation of j"i
 and j#i states versus the linewidth of the detection transition. We use light reso-
 nant with the j"i to the 2P1=2 jF = 0i transition, and polarize the light so as to be
 resonant with all the mf states (i.e. both linear and circularly polarized light). The
 linewidth of this transition is 19:7 MHz , while the laser light linewidth is  100
 KHz wide, and the separation between the j"i and j#i states is 12.6 GHz , so o -
 resonant scattering is small (but not negligible). Also, note that transitions from
 2P1=2 jF = 0i to j#i are forbidden by transition laws.
 This allows us to create a closed cycle of absorption and emission - a cycling tran-
 sition. There is a small complication here - without a magnetic  eld to break the
 degeneracy, a coherent dark state may occur [30]. In this situation, the bright states
 interfere destructively, and detection fails. However, we apply a magnetic  eld via
 an electromagnet under the UHV chamber. This magnet serves multiple purposes -
 22
F=1 
F=0 
F=1 
2P1/2 
2S1/2 
F=0 
mF=- 1 
mF=1 
mF=0 
Figure 2.3: Optical pumping of the 171Yb+ ion.
 23
it de nes the quantization axis (the Z-spin axis corresponding to the  z operator),
 and it breaks the degeneracy of the 2S1=2 manifold, thus preventing this coherent
 dark state. We estimate the  eld to be roughly 5 Gauss, providing a Zeeman shift
 of  7 MHz (310  52 Hz ). This can be measured by taking a Raman spectrum of
 the excitable sidebands.
 As the ion  uoresces, the distribution of detected scattered light should ide-
 ally follow a Poissonian. However, the possibility of o resonant excitation of the
 dark state forces us to convolute this distribution with an exponential function[31].
 A similar analysis is appropriate for the dark state. The atomic physics derived
 distributions are deformed by complicating factors. First, there is unwanted scatter
 o the electrodes of the trap. Second, there is the device physics of the imaging
 device, which will lead to spurious counts from false photon detections and noise
 during electrical readout. I will discuss this more in depth in 4.4.8. When detecting
 the state of a single ion, we can expect a histogram of counts as in  gure 2.5. As
 is seen in the  gure, there are two distributions, one centered around the average
 dark count, and one around the average bright state count of roughly 10 counts.
 In this case we can use a discriminator value that leads to an equal probability of
 mistaking j"i for j#i and vice versa. i.e the overlap of the distributions determines
 our single spin detection error.
 However, when we must detect multiple ions, we encounter two problems when
 simply collecting the light into a Photo Multiplying Tube (PMT). First, we do not
 have the ability to detect the true eigenstate of the chain, only how many ions are
 bright. This is acceptable in some experiments, where total number of bright ions
 24
F=1 
F=0 
F=1 
2P1/2 
2S1/2 
?+ ?? 
? 
F=0 
F=1 
F=2 
F=1 
F=0 
9
 3
 5.1
 8
 7
 9
 n
 m
  
3D[3/2]1/2 
3D3/2 0.86 GHz 
2.2095GHz 
mF=- 1 
mF=0 
mF=1 
mF=0 
Figure 2.4: The 171Yb+ ion detection scheme. The additional 3D F=2
 to F=1 resonant light is needed for the optical pumping, since the
 ion may occupy either 3D3=2 hyper ne states. Only pertinent states
 and transitions are shown.
 is the quantity of interest. Second, as the number of spins increases, the average
 number of bright counts per ions bright should naively increase as Nions x average
 count for one ion, assuming a uniform detection beam intensity across the chain (we
 will see this is not exactly the case). But this leads to increasing overlap between the
 di erent distributions, so even for the task of just detecting the total number of ions
 bright the PMT will under-perform compared to a device with spatial information,
 such as a Charge Coupled Device (CCD) camera.
 25
Figure 2.5: Measured count histogram after Doppler cooling with
 typical experimental conditions. The ion is Doppler cooled for 3ms,
 during which the ion scatters 10 photons on average into the light
 collecting Photo Multiplying Tube (PMT). Here, we are not detect-
 ing the scattered Doppler cooling light. Rather, we are following the
 cooling by a 800 s resonant detection pulse and light collection. This
 is repeated 100 times and then a histogram is constructed from the
 number of counts collected by the PMT via the imaging optics for
 each detection. As can be seen from the graph, the ion is not com-
 pletely bright following cooling. This screen shot from the control
 program also shows the Poissonian  ts for the dark and bright states.
 The mean value of the  ts are entered as parameters. The  tted am-
 plitudes are interpreted as probabilities for the number of ions bright.
 26
0 20 40 60 80
 0
 20
 40
 60
 80
 Photon Counts
  
o
 fEvent
 s
 Figure 2.6: Poissonians for the dark state and up to  ve ions bright.
 These are idealized bright states as there is no leakage to the dark
 state. The leftmost distribution is the dark state, and the distribution
 peaking at 10 counts is a single bright ion. The other distributions
 are centered at multiples of 10 (N  10), corresponding to N ions
 bright. Realistic distributions would have worse overlap due to this
 e ect, including a broadening of the dark histogram as well.
 27
Chapter 3
 Quantum Control of The Ion
 3.1 Light-Ion Interaction
 The last ingredient necessary to make the ion a useful qubit or simulated
 spin is the ability to control its internal state. For the purpose of this exposition
 I will utilize the e ective Hamiltonian theory introduced in 1.3.41. The simplest
 application of this theory is to the analysis of the AC Stark shift [24]. Applying an
 o -resonant  eld, detuned by  from the hyper ne splitting of the qubit states !hf
 with coupling coe cient  , we derive the interaction Hamiltonian
 HI =
  h 
2
 (j"ih#j e i h t + j#ih"j ei h t) (3.1)
 Applying equation 1.26 to the single harmonic term H1 to get
 HStark =
  h2 2
 4
 1
 2 h
 ( j"ih#j j#ih"j
 2
  
 j#ih"j j"ih#j
 2
  
) (3.2)
 resulting in
 HStark =  
 h 2
 4 
(j"ih"j  j#ih#j ) (3.3)
 1I am still following here the compact approach presented by James, which simpli es the treat-
 ment but somewhat obscures the physical justi cations. For a di erent and lengthier presentation,
 see [32].
 28
We see that the laser is dressing the qubit states and shifting their e ective
 energy in opposite directions and by equal amounts. This is equivalent to a phase
 gate on a single spin - as the spin is precessing while shifted by the  eld, it will
 become out of phase with other spins not under the in uence of the  eld. Alternately,
 it may become out of phase with an oscillator that was supposed to be locked to the
 spin?s natural precession frequency - a persistent concern for the experiment. We
 will see in 4 how we can mitigate this. It is important to note that this is only the
 simplest A.C. Stark shift possible, as a Stark shift can be a multi-photon process,
 rather than the two photon process presented here.
 3.2 Raman Transitions
 We now add a third, higher energy level to the system, jei , as depicted in
 Fig 3.1. Applying two light frequencies !1 and !2 detuned by  from jei , the beat
 frequency !hf will drive this transition, while only negligibly populating jei .
 We write the interaction Hamiltonian
 HI =
  h 1
 2 
jeih"j e i t +
  h 2
 2 
jeih#j e i t + h:c: (3.4)
 And apply 1.26:
 Heff =  
 h 21
 4 
(jeihej  j"ih"j ) 
 h 22
 4 
(jeihej  j#ih#j )+
  h 1 2
 4 
(j#ih"j  j"ih#j ) (3.5)
 where  i =  
q
 Ii
 ISat
 is the coupling coe cient for beam i, Ii is the intensity of beam i,
 29
?? 
?? 
? 
?? 
?? 
?? 
??? 
 
Figure 3.1: O resonant coupling of the spins. jei is negligibly pop-
 ulated for  >>   i
 30
ISat is the saturation intensity of the transition for that beam, and  is the radiative
 linewidth for that transition.
 We see the e ect is an AC Stark shift on all levels (a symmetric e ect for
 both qubit levels for equal  elds) and a coupling between the qubit levels. This is
 equivalent to a spin under the in uence of a transverse  eld, as (j#ih"j  j"ih#j ) =
 i y. We will refer to the beatnote that drives this transition from hereon as \carrier".
 3.3 Coupling Spin to Motion
 So far we have neglected the fact that the ion is con ned in a harmonic well.
 This will be treated more in depth in 4.2. Using the standard quantum mechanical
 treatment of the harmonic oscillator, we use the ladder operators to lower and raise
 the number of motional quanta (\phonons") of the ion with mass m and harmonic
 vibration frequency ! with the Hamiltonian H = p^
 2
 2m +
 1
 2m!
 2x^2
 a =
 r
 m!
 2 h
 (x^+
 i
 m!
 p^) (3.6)
 ay =
 r
 m!
 2 h
 (x^ 
i
 m!
 p^) (3.7)
 31
where the state representing the number of phonons is designated jni , and the
 ladder operators give
 a jni =
 p
 n jn 1i (3.8)
 ay jni =
 p
 n+ 1 jn+ 1i (3.9)
 aya jni = n jni (3.10)
 and the commutator [a; ay] = 1.
 Applying two  elds with a beat frequency  detuned to the red (blue) by
 the energy of the phonon, we can couple the spin to the ions motion and remove
 (add) a phonon while  ipping the spin, as in  gure 3.2. This scheme transfers
 momentum from the  elds to the ion. The mode excited is selected spatially by the
 beat frequency wavevector  k, and spectrally by the beat frequency  = !2  !1 =
 !hf + !mode
 This treatment assumes that we are in the resolved sideband limit, i.e. the
 exposure of the ion is long enough and low power enough that the driven transition
 is not broadened to overlap with the carrier or other modes. We can illuminate this
 e ect by writing the interaction Hamiltonian for this situation
 HI =
  h 
2
 (e i( k ~r ( !)t   ) + + e
 +i( k ~r ( !)t   )  ) (3.11)
 where  is the coupling strength of the qubit states,  k is the wavevector
 di erence of the two beams,   is the phase di erence of the beams, ~r is the location
 32
???? 
?? 
? 
?? 
?????? 
?? 
???? ???? ?????? 
?????? 
?????? 
???? 
Figure 3.2: Driving a Red Side Band (RSB). The beat frequency
  ! = !2  !1 = !hf  !COM imparts a spin-dependent momentum
 kick that takes the ion from j"; ni to j#; n 1i by absorbing a photon
 from !1 and emitting a photon into !2.
 33
of the ion, and  ! = !2  !1   , i.e. the beat frequency. Simplifying this general
 description to account for only motion along the transverse-x direction, and writing
 the coordinate as x^ = x0 +
 qp
 2
 (a+ ay), and q =
 q
  h
 M! we rewrite 3.11 as
 HI =
  h 
2
 (e i( k(x0+
 qp
 2
 (a+ay)) ( !)t   ) + + (e
 i( k(x0+
 qp
 2
 (a+ay)) ( !)t   )  ) (3.12)
 where  k now only refers to the wavevector di erence in the x direction. Assuming
 the spatial spread of the ion relative to the beat wavelength is small, we make the
 Lamb-Dicke approximation  =  kq=
 p
 2 << 1, and keep leading order terms in the
 phonon operators
 HI =
  h 
2
 (e i( kx0 ( !)t   )(1 i (ae i!t + ayei!t)) +
 +(ei( kx0 ( !)t   )(1 + i (ae i!t + ayei!t))  )
 (3.13)
 where we have rotated into the frame of the vibrational modes ! [32].
 If we set  ! =  !,
 HBSB =
  h  
2
 e i( kx0   )ay + + h:c: (3.14)
 HRSB =
  h  
2
 e i( kx0   )a + + h:c: (3.15)
 The RSBs will allow us to cool our ion chain beyond the Doppler cooling limit
 by coherently removing phonons.
 34
3.4 Multi-Ion Gates
 Now that we have introduced a way to couple the ion spin to its motion, we can
 show a way to create gates between ions, using their motion as a bus to transport
 information between disparate spins [3, 33, 34, 35]. By applying beatnotes symmet-
 rically detuned from carrier we will virtually excite normal modes, and induce sign
 dependent Stark shifts on ions, where the phase accrued on each spin will depend
 on the mode coupled and the ion location in the chain.
 Generalizing equation 3.13 to multiple ions, while still only coupling to x
 direction phonons, we rewrite equation 3.13
 HI =
 NX
 i
  h i
 2
 [(e i( kx
 i
 0 ( !)t   i)(1 
NX
 m=1
 i i;m(ame
  i!mt + ayme
 i!mt)) i+
 +(ei( kx
 i
 0 ( !)t   i)(1 +
 NX
 m=1
 i i;m(ame
  i!mt + ayme
 i!mt)) i )]
 (3.16)
 where i is an index summed over N ions, and m is an index summed over N modes
 (for a linear chain of ions con ned in an harmonic trap, there are N modes per axis
 [36], as we shall see in 4.2.1. !m is the frequency of the mth x direction mode, and
 now  i;m =  kqi;m=
 p
 2, where qi;m = bi;m
 q
  h
 M!m
 and bi;m is the normalized matrix of
 ion displacements for ion i due to mode m, as described in 4.2.1.
 We will be applying two global, symmetrically detuned from carrier beats, so
 35
we now sum equation 3.16 over two beatnotes   !:
 HI =
  h
 2
 NX
 i
 (1 
NX
 m
 i i;m(ame
  i!mt + ayme
 i!mt)) i
 [e i( k
 Rxi0  !t   
R
 i ) + e i( k
 Bxi0  !t   
B
 i )] i+ + h:c:
 (3.17)
 Where  kR(B);   R(B) refer to the wavevector and phase di erence for the two beams
 tuned to the red (blue) of carrier. Distributing the evolving coordinate and tem-
 porarily dropping the terms that go as ei(!m+ !)t to simplify the equations we are
 left with
 HI =
  h
 2
 NX
 i
 X
 m
  i;m(ame
  i(!m  !)te i( k
 Rxi0   
R
 i )+ayme
  i(!m  !)te i( k
 Bxi0   
B
 i )) i 
i
 ++h:c:
 (3.18)
 which can be more concisely written as 2
 HI =
  h
 2
 NX
 i
 X
 m
  i i;m(ame
 i(!m  !)t i im + ayme
  i(!m  !)t+i im) i  is + h:c: (3.19)
 where  im = ( k
 Rxi0    
R
 i   k
 Bx0;i +   Bi )=2 is the phase of the force on the i
 th
 spin,  is =  ( k
 Rxi0   
R
 i +  k
 Bx0;i   Bi )=2 is the spin phase of the i
 th spin and
   iS =  
i
 +e
  i iS +  i e
  i iS
 In our experimental setup the wavevectors of the BSB and RSB are counter
 oriented, i.e. ~ k
 R
   ~ k
 B
 , causing the spin phase to be insensitive to beam path
 length  uctuations. As both follow the same path, any  uctuation will cause equal
 and opposite phase change in both, i.e.   iS =  ( k
 R +  kB)xi0=2 = 0
 2The interested reader is welcome to compare this to the special case of 2 ions presented in [32]
 36
Although the Hamiltonian above can create displacements in phase space while
 coupling to the ion spins, we are only interested in driving transitions where motion
 is only weakly excited and can be ignored. This happens when the beatnotes are
 su ciently far from the side-bands so to not drive them o -resonantly, i.e.   <<
 !m   !. This is known as the \slow" M lmer-S renson gate [3, 33].
 Applying 1.26 to 3.19, we will get many terms, however, we can observe that
 there are three types of commutators that could possibly contribute to the e ective
 Hamiltonian:
 1. Same mode, same ion
 2. Same mode, di erent ions
 3. Di erent modes, di erent ions
 Assuming we set the spin phase so that we only have spin operators in the x-basis,
 only the second option contributes to the e ective Hamiltonian.
 1. [an i; ayn 
i] = an iayn 
i ayn ian i = ( 
i)2(anayn a
 y
 nan) = I So this term does
 not e ect the e ective Hamiltonian.
 2. [an i; ayn 
j] = an iayn 
j  ayn 
jan i =  i j(anayn  a
 y
 nan) =  
i j
 3. [an i; aym 
j] = an iaym 
j  aym 
jan i = 0
 Thus, the resulting e ective Hamiltonian is (here I am including the terms
 evolving at  ! + !m previously omitted)
 HMS =
  h
 4
 X
 i;j;m
 (
  i j i;m i;m
  ! + !m
 +
  i j i;m i;m
  !  !m
 )e i m;ie i m;j i j = (3.20)
 37
HMS =
  h
 2
 X
 i;j;m
  i j i;m i;m!m
  !2  !2m
 e i m;ie i m;j i j (3.21)
 which we can write succinctly as HMS =
 P
 i;j Ji;j 
i j where
 Ji;j =
  h
 2
 X
 m
  i j i;m i;m!m
  !2  !2m
 e i m;ie i m;j (3.22)
 is the coupling strength between spin i and spin j. For the duration of this work, the
 relative modulation phase will be assumed to be equal for all ions, and will only be
 meaningful when compared to other force generating lasers. This is due to the ions
 being addressed with global beams, and the modulation frequencies in the MHz
 regime have phase fronts that do not signi cantly change over the length of the ion
 chain, that is no longer than  30 m .
 This assumption will change in chapter 8, where individual phase control for
 each ion will allow us to control the relative sign of spin-spin couplings.
 3.5 Conclusion
 In this chapter all the necessary components for creating the simulated or
 e ective components of the Hamiltonians we will investigate numerically and exper-
 imentally in 6 have been introduced and succinctly explained. When we simulate
 these Hamiltonians we will be adding in several frequencies to create both spin-spin
 interactions as in 3.21 and single spin simulated magnetic  elds as in 3.5. The hid-
 den assumption there is that combining the needed modulations of the light will
 38
create an e ective Hamiltonian that is a sum of the desired e ective Hamiltonians.
 This is a valid assumption only if the beatnotes do not create cross terms with a low
 or stationary frequency in the e ective Hamiltonian. For a more in depth treatment
 of the possible unwanted e ects of this approach, see [37].
 39
Chapter 4
 Experimental Setup
 4.1 Introduction
 The experimental setup for the quantum simulation experiments reported in
 this thesis is highly complex. In fact, it is so complex that a single grad student
 starting from an empty lab would likely need more time just to set up part of the
 experiment, let alone perform any experiment, than an entire PhD program should
 reasonably require. The setup described here was luckily not started from scratch.
 The linear ion trap [2] was refurbished from its previous role as a Cadmium ion
 trap. We also also greatly bene ted from previous experience in the group [1, 38, 39]
 regarding how to use 171Yb+ . Control circuits for lasers and optical cavities, and
 the FPGA pulser were designed by authors mentioned in the citations above.
 4.2 Ion trap
 The type of ion trap we use for our experiment is the Paul ion trap [40]. As
 ions are electrically charged particles, we wish to create an electric potential with
 a stable minimum to which the ion will relax. Unfortunately, Earnshaw?s theorem
 [41] informs us that a static electric potential cannot have a stable minimum. Paul?s
 trap circumvents this problem by combining a static electric  eld and an oscillating
 40
???/2  
???/2  
???/2  
???/2  
??? 
x  
y  
z  
Figure 4.1: Paul trap with ions. the distance between hyperbolic
 electrodes is 2r0.
  eld to create a time-averaged pseudo-potential that has a stable minimum. The
 simplest example of such a setup is the four rod Paul trap, where the rods have a
 hyperbolic cross-section, as shown in  gure 4.1.
 In order to create a con ning potential, we apply the voltage to the electrodes
   0 = U + V cos(!t) (4.1)
 where U is a DC voltage and V is the amplitude of an RF voltage oscillating at
 frequency !. r0 is the distance from the electrode to the center of the trap, as shown
 41
in  gure 4.1. The resultant potential is
  =
  0
 2r20
 (x2  y2) (4.2)
 After taking the gradient to  nd the electric  eld, we can write the equations of
 motion (known as the Matheiu equations):
 dx2
 d 2
 + (a+ 2qcos(2 ))x = 0 (4.3)
 dy2
 d 2
  (a+ 2qcos(2 ))x = 0 (4.4)
 where
 a =
 4eU
 Mr20!2
 ; q =
 2eV
 Mr20!2
 ;  =
 !t
 2
 (4.5)
 When the a and q parameters are chosen so that neither coordinate grows exponen-
 tially rather than oscillate, the ions are stably con ned. The missing element of the
 con nement is in the z-direction, the trap axis (I will use this convention from here
 on out). That can be corrected for by simply adding a repulsive DC electrode to
 each end of the trap axis.
 42
4.2.1 Normal Modes
 A linear chain of N ions, trapped in a harmonic trap, experience a pseudo-
 potential due to an AC  eld, a static  eld and their electric charge[36]
 V =
 M
 2
 NX
 n=1
 3X
 i=1
 !2i x
 2
 ni +
 e2
 8  0
 NX
 n;m=1
 m6=n
 "
 3X
 i=1
 (xni  xmi)
 2
 # 1=2
 (4.6)
 from which we can  nd their equilibrium positions in terms of a rescaled co-
 ordinate um = x0m=l, where x
 0
 m is the small displacement from equilibrium de ned
 by xni (t) = xni + x0mi, and
 l =
  
e2
 4  0M!23
  1=3
 (4.7)
 where e is the charge of the ion, M is the mass of a single ion, and !3 is the
 longitudinal collective center of mass frequency - the lowest normal mode frequency.
 We can now consider the Lagrangian of the system,
 L = T  V =
 M
 2
 NX
 n=1
 3X
 i=1
  
_x0ni
  2
  
M
 2
 NX
 n=1
 3X
 i=1
 !2i
  
xni + x
 0
 mi
  2
  
e2
 8  0
 NX
 n;m=1
 m6=n
 (
 3X
 i=1
  
xni + x
 0
 ni  xmi  x
 0
 mi
  2
 ) 1=2
 (4.8)
 And expanding to lowest order, neglecting couplings between di erent spatial
 43
directions1
 L  
M
 2
 (
 NX
 n=1
  
_x0ni
  2
  !23
 NX
 m;n=1
 Amnxm3xn3 +
 2X
 i=1
 "
 NX
 n=1
  
_x0ni
  2
  !23
 NX
 m;n=1
 Bmnxm3xn3
 #)
 (4.9)
 and
 Amn =
 8
 >>>>>>>><
 >>>>>>>>:
 1 + 2
 P
 p=1
 p 6=m
 1
 jum upj
 3 if m = n
  2
 jum unj
 3 if m 6= n
 (4.10)
 Bmn =
  
1
  2
 +
 1
 2
  
 mn  
1
 2
 Amn (4.11)
 where A and B are the matrices for the longitudinal and transverse eigenvalue
 equations. We de ne the normal modes bm that describe the oscillatory displace-
 ment of ion i from its equilibrium with longitudinal (transverse) frequency !z
 p
  m
 (!z
 p
  m)
 NX
 n=1
 Amnbi;n =  mbi;m (i = 1; : : : ; N) (4.12)
 NX
 n=1
 Bmnbi;n =  mbi;m (i = 1; : : : ; N) (4.13)
 and assumed the collective transverse mode frequencies are equal. We de ne  , the
 trap anisotropy
 !x =
 !z
  
(4.14)
 1This is a valid assumption as long as the trap anisotropy parameter is small enough, i.e. the
 chain is not close to buckling from a linear chain into a zig-zag con guration.
 44
where !x is the COM frequency for the x-direction transverse mode and !z is the
 COM frequency along the ion chain axis. The relationship between the axial and
 transverse normal-mode eigenvalues is
  p = 1= 
2 + 1=2  p=2 (4.15)
 Transcribing this analysis into a Wolfram Mathematica script generates the
 ion equilibrium positions:
  k =
 p
 2 2 369:5 10 9 ;
 NumofIons = 10;
 !cm = 2 4:863232  106;
 !tilt = 2 4:81294  106;
  =
 q
 1 !tilt
 2
 !cm2
 ;
 !z =   !cm;
 l =
  
qe2
 4  0MYb171!z2
  
1=3;
 Table[uH[m]; fm; NumofIonsg]; (* create position variables *)
 Table
 h
 fH[m] = uH[m] 
Pm 1
 n=1
 1
 (uH[m] uH[n])^2 +
 PNumofIons
 n=m+1
 1
 (uH[m] uH[n])^2 ;uH
 fm; NumofIonsg];
 EqH = Table[fH[m] == 0; fm; NumofIonsg]; (* generate coupled equations *)
 IniH = Table[fuH[m];m=10g; fm; NumofIonsg]; (* generate initial conditions *)condition
 solH = FindRoot[EqH; IniH];FindRFindR
 uHa = Table[uH[m]/.solH[[m]]; fm; NumofIonsg];
 Print[uHa  l  10^6; \ position in microns of ions"]ositioositio
 And we get the ion distance on the z-axis from the center of the trap in  m
 for the trap parameters used in chapter 7:
 f-10.0021,-7.31662,-5.05321,-2.97461,-0.982864,0.982864,2.97461,5.05321,7.31662,10.0021g
 45
With these distances we can calculate the transverse frequencies for the x-
 direction, which we will be using to create our spin-spin interactions:
 TMatrix =
 Table
 h
 If
 h
 i 6= j;TransMat[i; j] = 1Abs[uHa[[i]] uHa[[j]]]3 ;6 6
 TransMat[i; j] = (1= )2  
Pi 1
 j=1
 1
 Abs[uHa[[i]] uHa[[j]]]3  
PNumofIons
 j=i+1
 1
 Abs[uHa[[i]] uHa[[j]]]3
 i
 ; fi; 1; NumofIonsg; fj; 1; NumofIonsg];
 TransEigVal = Eigenvalues[TMatrix];alues[TMatrix]
 TransNormalFreq = Sqrt[TransEigVal];
 Print [TransNormalFreq  !z/ 2/ =106; \ Trap frequencies in MHz"]
 f4.86323,4.81294,4.74058,4.64815,4.53638,4.40509,4.25343,4.07989,3.88217,3.65694g
 Trap frequencies in MHz
 The transverse frequencies displayed here run from the highest, the COM
 mode, to the lowest - the \zig-zag" mode, as shown in  gure 4.2. The second
 highest mode is the \tilt" mode, and the spacing between COM and tilt, unlike
 any other two modes, is  xed at !2 = !1
 p
 1  2 (from here on the normal mode
 frequencies are referred to as !n, running from high to low). When generating the
 spin-spin coupling matrix J we will have  exibility in control of the form of the J
 by setting our detuning close to the mode that has the form similar to our desired
 couplings, as can be seen in  gure 4.3. Using a single detuning will not give us
 complete control, however the modes form a complete basis, and we will use this in
 8 to gain complete control over J.
 46
Figure 4.2: Transverse mode components for 10 ions. The arrows
 represent the maximum displacement of the ions from equilibrium
 while oscillating at mode frequency. The modes are ordered from
 highest frequency (COM) to lowest (zig-zag).
 47
-1500 -1000 -500 0
 -3
 -2
 -1
 0
 1
 2
 3
 DetuningHKHzL
 J i,
 jHKH
 zL
 J89,10<
 J88,10<
 J88,9<
 J87,10<
 J87,9<
 J87,8<
 J86,10<
 J86,9<
 J86,8<
 J86,7<
 J85,10<
 J85,9<
 J85,8<
 J85,7<
 J85,6<
 J84,10<
 J84,9<
 J84,8<
 J84,7<
 J84,6<
 J84,5<
 J83,10<
 J83,9<
 J83,8<
 J83,7<
 J83,6<
 J83,5<
 J83,4<
 J82,10<
 J82,9<
 J82,8<
 J82,7<
 J82,6<
 J82,5<
 J82,4<
 J82,3<
 J81,10<
 J81,9<
 J81,8<
 J81,7<
 J81,6<
 J81,5<
 J81,4<
 J81,3<
 J81,2<
 Figure 4.3: The mess of couplings for 10 ions. The x-axis is detuning
 in KHz from the COM. As can be seen in the plot, near the COM
 all the couplings are nearly equal, as can be expected from the form
 of b1. Near each mode the couplings diverge to in nity, however this
 behavior is non-physical as we must maintain the weak excitation of
 phonons condition.
 48
4.3 Micromotion
 In an actual trap, the alignment of the DC saddle point and the RF node
 will be imperfect. This will alter the form of the equations of motion adding an
 extra term of excess micromotion - oscillatory motion around the psuedopotential
 minimum, in addition to the inherent micromotion seen in 4.4.
 This unwanted motion creates an oscillating Doppler shift on the ion?s reso-
 nance frequencies, reducing scatter rates for cooling and detection and broadening
 the transitions. We have found that ions are less stable in the trap when this condi-
 tion exists. To correct this condition, the DC voltages must be altered as changing
 the RF node is di cult (or impossible, as in our setup). There are several indica-
 tors useful for reducing the excess micromotion[42]. One is measuring the Doppler
 cooling scattering linewidth. The second is measuring a correlation between the
 trap frequency phase and the scatter rate using a Time to Digital Converter. For
 micromotion along an axis parallel to an imaging device, one can also reduce V to
 see a displacement of the ion from DC saddle-point. In our trap we have found
 that it was very di cult to reduce micromotion, especially in the transverse direc-
 tion not coupled to our gates (y), as reducing micromotion would rotate the trap
 axis, creating unwanted components of the Raman beams along the y axis. This
 would require us to change the incident angle of our Raman beams on the ion chain,
 which is quite di cult. Instead, we would reduce micromotion by maximizing the
 Rabi rate. It is likely that reducing the micromotion along the x-axis leads to more
 e cient Doppler cooling on that axis, leading to a higher Rabi frequency.
 49
4.3.1 The Linear Trap
 The trap used in the work in this thesis is of a di erent design than the one
 discussed above. It is constructed of three layers of thin alumina, with electrodes
 segmented by laser machining of the alumina and formed by gold coating, as shown
 in 4.4.
 (a) (b)
 250 ?m 
dc 
dc 
dc 
dc 
rf rf 
375 ?m 
200 ?m 
Figure 4.4: (a)Linear trap schematic with ions in trapping region.
 The RF electrode is a single electrode with a slit of equal dimensions
 to the DC electrodes central slit. The top and bottom DC layers
 are identical. (b) Linear trap photograph. Here the capacitors and
 resistors in a pi- lter con guration on the DC electrodes can be seen.
 These prevent capacitive RF signal coupling to the DC electrodes.
 Although this con guration is quite di erent from that of the four rod con g-
 uration, the analysis is quite similar. As there is no easily found analytical solution
 for the potential generated by these boundary conditions, the approach we take is
 modeling the potential numerically with a commercial electromagnetic modeler -
 50
Charged Particle Optics. For each electrode, we set all other electrodes to ground,
 and set said electrode to some voltage. Thus the linear combination of scaled po-
 tential maps gives us the total potential for the trap. Using this numerical solution,
 a ponderomotive psuedo-potential can be calculated
  =
 e2
 4m!2
 rV (x; y; z)2 (4.16)
 and treating the linear trap as we would treat the hyperbolic Paul trap - a  eld
 exerting a harmonic restoring force on the ion, we  nd the secular frequency (the
 axial COM frequency) [43, 44, 45]
 !2z =
 e2
 4m2!2
 @2
 @2z
 (jrV (x; y; z)j)2 (4.17)
 Then the secular frequency of the trap is
 !z =
 e2V 2 
4
 p
 2m!2r4
 @2
 @2z
 r(jV (x; y; z)j)2 (4.18)
 which is the identical to the secular frequency of the hyperbolic trap, but with a
 rede ned e ective distance from the electrode to the ion r and a geometric factor
  . This factor is the ratio between the quadrapole portion of the linear trap (or any
 other trap) to the oscillating term of the hyperbolic trap.
 51
4.3.2 Trap Voltages and Control
 As shown in 4.4(a), the trap has six electrodes on both of the DC layers. Wiring
 all electrodes would give more control than necessary to control the trap axis and
 minimize micromotion, and would increase signi cantly the di culty of assembly
 and wiring. Therefore, only 3 DC electrodes in each layer are wired to external
 leads. The remaining three are wired to trap ground. These electrodes are wired to
 spot welded on board pi- lters. The leads running from these  lters then connect
 to an external breakout box, which itself has additional pi- lters for each electrode,
 as discussed in 8.6. The trap voltages are supplied by an Iseg EHS-80-05XK3 high
 voltage module. This module is powered and controlled by an Iseg MPOD mini-
 crate. The crate provides a network based control of the voltages. The module has
 eight channels, all with SHV connectors. These are not to be confused with BNC
 connectors, which are very similar in appearance. We constructed custom cables to
 connect between the SHV connectors on the ISEG box to the BNC connectors on
 the breakout box.
 It is important to note that the channels have a single polarity (dual polarity
 requires a  oating ground that reduces stability)
  Channels 0-5: up to +500V /15mA
  Channels 6-7: down to -500V
 The control program will automatically switch the driven channels from 4 and/or 5
 to 6 and/or 7 if the set voltage on electrodes DC 3 and/or DC 4 is negative.
 52
Figure 4.5: Trap control program. The upper right corner provided
 settings for a timed relaxation and tightening of the trap RF, via
 the LabJack controlled VCA. Upper right side controls the DC trap
 voltages. Lower half indicates sensed DC voltages and currents for
 tab displayed, as well as controls power to individual channels and
 the Iseg chasis (Main Power).
 53
It is assumed the user has switched the cables on the breakout box
 if this is the case.
 4.3.3 Oscillating Trap Voltage
 The RF voltage V is provided via a helical RF resonator [46]. This resonator
 has a quality factor Q of  300 and is resonant at 38.86 MHz . As the power deliv-
 ered through the resonator is changed, the ohmic losses change and the equilibrium
 temperature of the resonator and perhaps the trap itself changes. This leads to
 geometric changes that change the resonant frequency of the resonator and the RF
 voltage of the trap. As this leads to a change in the secular frequency of the ion, we
 have implemented a stabilization scheme. The  rst stage of this stabilization scheme
 is thermal stabilization of the RF resonator. This is accomplished by a Thorlabs
 TC200 Heater controller. This heater senses the temperature of the resonator using
 a thermistor, and stabilizes with a resistive coil. The stabilization is improved by in-
 sulating the resonator and the thermal feedback system with an adhesive foam with
 a re ective aluminum layer. We  nd that this temperature is stable to less than a
 degree when the set point temperature is 250C, slightly above room temperature.
 The second stage of this stabilization scheme is active frequency stabilization
 of the HP-8640B driving the trap RF to resonance. This helps stabilize the delivered
 power. It is helpful, but not su cient, to stabilize the trap transverse modes.
 We pass the driving RF through a Minicircuits BDC ZFBDC-62HPS bidirec-
 tional coupler, and mix the re ected signal with a phase shifted reference signal
 54
Figure 4.6: Trap control program electrodes tab. Here the sensed
 voltages on each electrode are displayed. The electrodes are dis-
 played without the RF layer, facing towards the imaging optics. The
 electrodes labeled ?GND? are grounded.
 from the HP-8640B (this signal is impedance matched by passing through a 50 MHz
 low-pass  lter) frequency generator providing the driving signal (which is ampli ed
 to  26dBm). The phase is adjusted using a potentiometer providing a set-point
 voltage to a Minicircuits JSPHS-51+ narrow band phase shifter (this component
 must be selected carefully for the resonant driving frequency of the trap). This
 mixed signal is low pass  ltered so only the DC component remains, and is fed back
 as a DC frequency shift to the driving 8640B (typical modulation peak of 320KHz).
 55
4.4 Optical Setup
 4.4.1 Ionization
 There are three lasers used for excitation of the neutral Yb, ionization, cooling,
 optical pumping, and detecting. These are all DL-100 Toptica lasers - temperature
 and current controlled laser diodes. Aligning the diode with a re ection from an
 external optical grating allows for mostly single mode selectivity for the modes
 selected by the diode cavity. In general, once the grating is aligned to generate
 a frequency in the neighborhood of the desired frequency (for the minimal lasing
 threshold current) frequency adjustment should be done by adjusting the grating
 angle and diode current. If mode hopping occurs, then one may try a sensitive
 adjustment of the diode temperature.
 As discussed in 2.1, it is necessary to excite the neutral 171Yb before it may
 be ionized by a 369 nm or 355 nm beam. As the beam exiting the laser has an
 elliptical pro le, we  rst circularize the beam with a prism pair. Next we protect
 the laser from back re ections from any of the downstream optical elements, as these
 create unwanted optical cavities with the laser and unnecessary mode hopping (this
 is true for all the lasers in this experiment except the Vanguard). A mechanical
 shutter controlled by the trap control program exposes or conceals the trap to the
 beam, which after passing through the  nal lens is roughly 50 m vertically and
 100 m horizontally, at roughly 2mW . When the 171Yb+ oven is driven at 2.3A
 after warming up for about a minute, the light intensity is powerful enough to load
 an ion roughly every 10s when using the Raman beams for ionization of the excited
 56
Yb. This is achieved with good isotopic selectivity: the 171Yb oven is roughly 90%
 pure, but loading unwanted isotopes occurs less frequently than 10%. As the oven is
 perpendicular to the excitation beam, the atoms do not experience a doppler shift
 with a thermal distribution.
 The desired frequency of the excitation beam is 751.527640 MHz , however we
 do not lock this laser, rather we tune it to the desired frequency and load before it
 drifts too far o resonance. The rate of frequency drift depends on the temperature
 of the room and how long the laser has been operating at given temperature and
 current settings. The laser takes roughly an hour to equilibrate.
 4.4.2 935 nm Repump
 The 935 nm laser providing the repump light discussed in 2 is provided by a
 DL-100 as well. As this light is needed to repump both the F=1 and F=2 manifolds
 of the 3D3=2 state, the 320.56922THz carrier is modulated by an EOSpace  ber
 EOM at 3.07 GHz .
 This laser is only stabilized for long timescales, by a feedback signal to its
 grating angle provided by a software PID. This PID generates a feedback signal
 based on the frequency di erence between the laser wavelength, measured by a
 HighFinesse wavemeter via optical  ber and a software controlled set point. The
 wavemeter accuracy may drift by up to 20 MHz , depending on room temperature,
 and is calibrated to our iodine spectroscopy setup, discussed below. Its precision is
 roughly a MHz .
 57
4.4.3 638 nm Repump
 As discussed in 2, repumping the ion from the long lived but unlikely 2F7=2
 state requires two close frequencies of 638.61 nm light. As this light is not often
 required, and when it is required the ion chain has likely decrystalized (from a
 collision) and data taking cannot immediately continue, the laser is only stabilized
 by the HighFinesse wavemeter ( 5 MHz short time scale stability). To provide both
 wavelengths of 638.6103 nm and 638.6151 nm , the PID feedback to the laser grating
 setpoint is alternated between the two desired wavelengths, once every minute. DL-
 100 laser is  rst passed through a prism pair to circularize the beam pro le, and
 then coupled into a  ber optic. The output of the  ber is directed at the ion chain
 with several mW and a large waist at the ion chain, so careful alignment is not
 required.
 4.4.4 Cooling, Pumping, and Detecting with 369 nm
 To generate the 369 nm light we require, we use a MBR-110 from Coherent.
 This is a tunable Continuous Wave (CW) Ti:Sapphire laser, pumped by a Coher-
 ent 18W 532 nm Verdi-V18. It outputs roughly 1.5W at our desired wavelength of
 739.052526 nm . The MBR-110 is a made of a single block of aluminum, and is thus
 passively stable. The bow tie con guration passes through an optical diode that
 prevents backward re ections. The wavelength of the laser is selected by controlling
 the gain pro le for multiple laser mode selecting elements. These include the laser
 cavity length, an etalon (that oscillates around a rotation angle at 80 KHz to gener-
 58
Ti:Sapphire crystal 
Reference Cavity 
532nm Pump Light 
Etalon 
Verdi 
Figure 4.7: The MBR-110 laser.
 59
ate an error signal used to lock its angle to the laser cavity length), and a birefringent
  lter (with a wavelength dependent response). The Free Spectral Range (FSR) of a
 bow-tie cavity is FSR = CL where L is the length of the cavity. As the largest cavity
 is the laser itself, the mode-hop free frequency range tuning and locking is controlled
 by laser cavity length ( C1M = 300 MHz ). The laser has a linewidth of  75 KHz .
 The 739 nm output power of the MBR-110 is  ber coupled into a high power
  ber, except  100mW that is  ber coupled to the wavemeter, and  40mW to an
 EOSpace  ber EOM, used for the iodine lock, described in section 4.4.5. The high
 power  ber delivers 800mW to a WaveTrain frequency doubler, as shown in  gure
 4.8.
 After exiting the doubler the beam is 430MHz red detuned from resonance.
 It is divided in three, by a piece of glass. The front surface re ection is modulated
 by an New Focus EOM driven at 2.105GHz to provide the optical pumping, and is
 shuttered by an AOM that up shifts the beam by 424 MHz , into resonance with the
 transitions discussed in 2.2. This beam is then combined with the detection beam
 before being  ber coupled. The back surface re ection is modulated by an EOM
 driven at 7.374GHz. It is then passed through a Brimrose AOM. The  rst order
 di racted beam is up shifted by 400MHz and used for cooling (30 MHz detuned
 from resonance). The zeroth order is used as a \protection beam". This powerful
 beam is used to recrystallize the ion chain when a decrystallization is detected by
 the cooling light scatter PMT. As it is modulated for cooling, it is resonant with
 both the 2S1=2 jF = 1i and 2S1=2 jF = 0i to 2P1=2 transition, and we have found this
 to improve the speed and probability of recovering all the ions. The transmission
 60
through the glass is used as the \protection" beam, which is shuttered by an Intra-
 Action AOM and further red detuned by 200 MHz . This beam is more powerful
 than the three other beams, but does not cool both transitions as the previous beam.
 All three beams are coupled into optical  bers to reduce unwanted scatter light in
 the direction of the  rst order beam. This scatter was found to limit coherence time
 in long experiments. The added bene t is clean laser modes and ease of aligning
 and overlapping the beams.
 A small portion of the transmission is siphoned o by a beam splitter, to be
 shuttered and shifted into resonance by a Brimrose AOM. The \protection+" beam
 was combined with the 399 nm beam in order to simplify alignment. Unfortunately,
 we have found that overlap is di cult and the 399 nm beam alignment drifts wildly
 from day to day due the need to change the laser grating angle and the long beam
 path to the chamber. Therefore, alignment with the cooling or protection beam
 at the ion position does not allow perfect overlap along the entire beam path. Be-
 fore entering the chamber window, the detection, cooling and protection beams are
 horizontally stretched by cylindrical lenses, and then focused at the ion chain to a
 horizontal beam waist of  100 m . However, as shown in  gure 4.9, the beams
 enter the chamber at 450 to the ion chain, giving an e ective beam width at the
 chain of  150 m . This gives a roughly equal detection beam pro le across our
 ion chain, that for 16 ions is 26 m for our current operating parameters for the ion
 trap.
 As our beam pro les are roughly Gaussian, the beam dimensions and power are
 limiting factors on the detection homogeneity. As the power in the beam is limited,
 61
handling longer ion chains with optimal exposure and detection beam powers for
 all ions requires a larger horizontal beam pro les. In order to achieve this with a
  xed laser power, we are required to squeeze the vertical beam waist. This leads to
 more stringent requirements on beam alignment, and more noise on the e ective laser
 power at ions, due to beam steering by air turbulence. Currently, with our detection
 beam vertical waist of  10 m this is not a limiting factor on our detection error,
 rather we su er from power  uctuations of  5%. In order to remedy this issue we
 are currently investigating the use of a feedback stabilization system (\noise eater").
 In fact, it is questionable if we can widen the beam much more without increasing
 background scatter from the trap o the imaging window into the imaging optics.
 A possible remedy is to create a beam with a \top hat" pro le using di ractive
 optics, and thus avoid wasted beam power in the wings of the beam and ensure
 equal detection conditions for all the ions.
 Note: for some of the earlier results reported in 6, we used the MBR-110 to
 drive the Raman transition, and a Toptica TA-100 ampli ed diode laser to drive all
 resonant transitions. In that setup, we only generated  200 mW of 739 nm light
 and su ered as a result for larger numbers of ions. The TA-100 was locked using
 the iodine spectroscopy setup described in 4.4.5. The major di erence is the use of
 a home built confocal cavity, that was locked by length to the transition, and the
 use of a Pound-Drever-Hall feedback signal to lock the laser frequency to the cavity
 length. This feedback signal was separated into a low (< 100KHz) component that
 controls the laser grating angle, and a high frequency component (> 10MHz) for
 the laser diode current, controlled via a Bias-T [47].
 62
MBR  
Wave Meter  
Monitor  
cavity  
protection  
7.374GHz Protection+  
Chamber  
Verdi - V18  
532nm Iodine lock  
2.105GHz 
X2  
LBO  
39
 9
 n
 m
  
To Chamber  
Figure 4.8: A schematic diagram of our 369 nm and (399 nm excita-
 tion for neutral Yb) optical setup. The  nal cylindrical lens, mounted
 on a micrometer positioning stage, is a vertical lens with focus 150
 mm. The micrometer is used for aligning the vertical direction of the
 detection beam, which is the most sensitive. Chamber is shown in  g
 4.9
 63
Figure 4.9: A schematic diagram of chamber and  nal stage of res-
 onant beams. The cylindrical lens the beam pass through focuses
 the vertical direction of the beams, with an 80 mm focal length. It
 demagni es the intermediate focus by 5. The dashed line represents
 the rough trajectory of the high purity  90%171Yb oven. As it is
 perpendicular to the excitaiton beam, our ability to selectivly load
 the desired isotope (rather than the impurities) is higher than the
 purity of the oven. Although the beams exit through the front view
 port, from which the ions? state detection  ourescence is collected,
 the 638 nm , 935 nm and roomlight do not create unwanted signals on
 the ICCD or PMT as they are  ltered. As the waists of these beams
 is large at the ions, alignment requirements are lax.
 64
4.4.5 Iodine Spectroscopy Lock
 As the linewidth of the S-P transition is 20 MHz , we must stabilize the laser
 on long time scales to a much narrower center frequency to have consistent detection
 and cooling. This is accomplished using a spectroscopic feature of iodine narrower
 than our linewidth, to which an external cavity (part of the MBR-110 laser) is
 stabilized. An optical  ber carrying 40 mW of the MBR-110 output is coupled into
 a  ber EOM, modulated at 13.315 GHz , so that the  rst order sideband of this
 modulation ( 4mW) excites the iodine transition of interest[47, 1].
 Three beams are derived from the  ber - a reference beam directed at the
 Nirvana Auto-Balanced Photoreceiver, a probe beam which is sent through the
 iodine cell and out into the signal photodiode of the Nirvana, and a pump beam
 transmitted through an 80 MHz Neos AOM into the iodine cell, overlapping with the
 probe. The iodine cell is insulated with  berglass and aluminum foil, and heated
 to 5000c. This excites the higher energy rovibrational states of the iodine molecules
 and strengthen the absorption signal. A cold  nger maintains the pressure in the
 cell and prevents pressure broadening of the transitions [48]. The Nirvana detector
 is designed to subtract the reference from the signal, thus making it a sensitive
 detector able to remove 50dB of noise. We have encountered two major problems
 with this setup:
  Background light from the pump beam can leak onto the signal photodiode.
 As this background light level can drift, the detector fails to subtract the noise.
  The Nirvana detector is designed to balance the attenuation of the photodi-
 65
odes. However, we have witnessed that it is limited in this ability. As the
 reference beam is much more intense than the probe beam, it must be atten-
 uated for the detector to work properly.
 The 80 MHz modulation of the pump beam is itself modulated at 10 KHz , enabling
 the use of a lock-in-ampli er to demodulate and amplify the Nirvana error-signal
 (the derivative of the Doppler free hyper ne transitions in iodine). As there are
 two strong transitions for the above EOM frequency near to the resonant laser
 frequency (for the 739 nm , non-doubled light), there will be three erro-signal peaks.
 We stabilize the MBR-110 to the peak corresponding to the highest frequency of
 the three at 405.644321 THz , which is  20 MHz above the nearest feature. This
 can lead to confusion, as the wavemeter displaying the laser frequency drifts in the
 same range. However, the incorect frequency is close enough to the desired one that
 ions will in fact be visible on the ICCD if this occurs - but they will appear dim.
 We route the error-signal from the lock-in-ampli er to a home built PID. The
 PID output is then routed to the MBR-110 External Lock Input. I have connected an
 attenuator to this input, as it cannot take an input larger than 10V. This signal
 is fed to the piezo controlling the length of the MBR-110 external reference cavity
 (an internal component of the laser, as shown in  gure 4.7). The peizo mounted
 tweeter mirror of the laser locks the length of the laser cavity to the external cavity
 length. This lock can be stable on long time scales to  2 MHz , which is su cient
 for the linewidths at hand.
 66
Iodine Cell  
 
Nirvana  
Pump  
Reference  
P
 robe
  
Monitor  
cavit y  
Figure 4.10: Schematic of iodine setup. The angle between the beams
 in the cell is exaggerated.
 67
4.4.6 Vanguard 355 nm Laser
 The Vanguard Diode Pumped Solid State Laser is intended to be a turn key
 laser. The lasing medium is a Nd:YVO4 crystal (Neodymium Doped Yttrium Or-
 thovanadate) pumped by a solid state diode lasing 12W of 808 nm through a 19 core
  ber bundle. The crystal (\oscillator") should then lase roughly 5W at 1064 nm .
 Unfortunately, we are currently overdriving the oscillator to 6W, for reasons to be
 explained below. The output of the oscillator is then ampli ed by another Nd:YVO4
 crystal, pumped by two 20W 808 nm diodes, to roughly 20W of 1064 nm laser light.
 This light impinges on a Second Harmonic Generator (SHG) that produces an 8W
 beam of 532 nm . A semiconductor saturable absorber re ector mode-locks the laser
 at 80 MHz , where the pulses have a 10 ps duration [49]. This mirror has a re ection
 that decreases with increasing light intensity, thereby allowing the creation of laser
 pulses at high gain.
 The 1064 nm and 532 nm beams are mixed on a Third Harmonic Generator
 (THG) to produce the desired 355 nm light. Although Spectra-Physics is not willing
 to disclose this information, the SHG and THG are likely type 2 Lithium triborate
 crystals [50, 49]. The output of the laser according to its speci cations should
 be 4W, as in  gure 4.11. Unfortunately, the power we observe is lower and it is
 dropping, as the laser is old and bought second-hand. The laser is controlled by
 a desktop computer via serial connection, using the control program MLUV. After
 initial warm-up ( 30 min), the power of the laser and its stability may be optimized.
 As the components of the laser are old, the built in optimization function of MLUV
 68
Figure 4.11: MLUV main control window. The laser has two operat-
 ing modes: one where the laser power is stabilizaed and one where the
 diode currents are stabilized. In order to run in power stabilization
 mode there must be some disposable power. As our power is low,
 we typically run in current stabilization mode. Although we have
 not found the power to be less stable in this mode on the short time
 scale, the overall power is slowly dropping.
 does not work.
 The initial optimization should focus on the diode currents and temperatures.
 Lowering diode currents can surprisingly increase power. Usually the most sensitive
 parameter is the current to \Diode 1", the oscillator pump, as shown in 4.12. These
 settings can only be changed when the laser is operating in Current Mode. If
 this procedure fails to raise the power back to its expected level, the next set of
 parameters to tweak are the ampli er pump diodes currents and temperatures, as
 69
Figure 4.12: MLUV oscillator pump current control.
 shown in 4.13. If these parameters yet again fail to raise the laser power, then it may
 be attempted to change the oven temperatures of the SHG and THG, as shown in
  gure 4.14. This is a slow process, as each change must be given time to equilibrate.
 Each change on the  ne knob (roughly in steps of  ve units) should be given at least
 2 minutes to asses the e ect. If this attempt fails as well, then there is a possibility
 that the THG and/or the SHG have degraded at the current laser path. It may be
 necessary to move the THG or SHG. This may be done via MLUV for the SHG and
 only for the x-direction for the THG, as shown in 4.15. If all the X positions on the
 THG for a Y position have been degraded, then the Y position may be manually
 changed via two screws that lock into 6 di erent orientations, via a side panel on the
 Vanguard. Both screws must be set to the same position, otherwise vertical beam
 70
Figure 4.13: MLUV ampli er pump current control. In this  gure
 the individual current controls are grayed out as the mode has not
 been set to individual diode control.
 Figure 4.14: Control window for THG and SHG oven temperatures.
 The  ne tunning settings should be used  rst.
 71
Figure 4.15: MLUV position controls for the SHG and THG. The
 THG Y-position is only for record keeping - MLUV does not control
 it.
 steering will result.
 Although the methods above have worked in the past to raise the laser power,
 the power has been dropping constantly and can not be recovered (the experiments
 in chapters 6 and 7 have been performed with powers ranging from 3.7W to 3.4W).
 This is likely due to the degrading of the ampli er pump diodes.
 4.4.7 Driving Raman and M lmer-S renson Transitions with the Van-
 guard
 In 3 we discussed how spin-spin couplings can be created using global beams
 modulated to drive virtual transitions in the phonon subspace while driving real spin
 transitions. In the work reported here, this theoretical treatment is realized with the
 Vanguard DPSS. Although this is a pulsed laser, we will be operating in the weak
 72
pulse regime - each pulse pair radiating the ion will contribute a small change to the
 ion state (also referred to as the weak pulse regime). Thus we will be able to treat
 the pulse trains as CW lasers [51]. In order to generate the desired beat frequencies,
 we will interfere the pulses on the ions, absorbing a photon from one frequency
 component of the pulse train and emitting into another. The beam exiting the laser
 will be split into two arms, and both arms will be modulated by AOMS. This will
 give us control of the beat frequency di erence, as schematically represented in  gure
 4.16. For this scheme, the absolute frequency o set and the absolute phase of the
 pulse train of each pulse train will have no e ect[51]. We can choose to subtract or
 add the frequency di erence between AOMs. The beat frequency of interest between
 teeth in the two o set combs is (80:6(n + 157) + AOM2)  (80:6n + AOM1) =
 80:6 157 +  AOM = 12654 MHz  12 = 12642 MHz , which drives the carrier.
 A prerequisite for this scheme is an acceptable and stable laser pulse repetition
 frequency. Our qubit splitting is  xed at 12.6428 GHz . As shown in  gure 4.17, the
 532 nm output of the Vanguard (attenuators not shown) is measured by a Electro-
 Optics ET-4000 fast photodiode. The output of this photodiode is band pass  ltered
 by a tunable mechanical microwave  lter so that only the frequency component
 close to the qubit splitting is transmitted. This signal is ampli ed and mixed with a
 microwave frequency source, allowing us to feed forward a frequency o set to AOM1,
 thus stabilizing the frequency di erence between the Raman arms.
 An acceptable frequency must satisfy two conditions:
 1. It may not drive the carrier transition with just one arm, nor the Zeeman
 73
f 
f 
frep 
n 
n+157 
fc+AOM1+? 
 fc+AOM2 
fc=C/355nm=844.5THz 
fBW 
fBW~1/10ps~100THz 
fw= frep /N 
Figure 4.16: Two pulse trains interfering at the ions represented in
 frequency space. The two AOMs shift the frequency combs, and allow
 us to control the beat frequency. AOM1 modulates at  225 MHz + ,
 where  is a fed forward frequency o set to stabilize the beat fre-
 quency. AOM2 modulates at  213 MHz when we drive the carrier
 transition (and a more complex waveform with multiple frequencies
 for the simulation). The frequency of di erence of  13 MHz is sub-
 tracted from the beat of of the teeth. The pulse width in frequency
 space is 1Nfrep , where N is the number of pulses involved in a interac-
 tion.
 74
PID  
Vanguard  
AWG  
Mach -
 Zender  
?/4 
?/4 
?/2 
?/2 
12.6GHz f rep  
? f AOM1 +? 
f AOM1  
AOM1  
AOM2  
a  
c  
c  
d  
d  
objective  
v  
v  h1  
h2  
b  
X  
X  
Figure 4.17: Vanguard with lock and Raman beam setup. Raman
 arm 1 is modulated by AOM 1, which is driven by an HP8640B with
 a frequency o set controlled by an error signal from the frequency dif-
 ference of the microwave reference and the measured repetition rate.
 Raman 2 is the 0th order beam of AOM 1, and is modulated by AOM
 2, with an RF signal created by the AWG. Lenses c(200mm) and d
 (91mm) image and demagnify the intermediate focus (X) of the verti-
 cal(lens v,50mm) and horizontal lenses h1(500mm) and h2(150mm).
 Lens a(200mm) is intended to slowly focus the beam at AOM1 so
 that is still mostly focused at AOM2. Both Brimrose AOMs func-
 tion best when the beam is focused in the crystal. Lens b(300mm)
 recollimates the beam.
 75
carrier transition (mf = 0 ! mf =  1) at roughly 7 MHz from carrier (de-
 pending on magnetic  eld strength).
 2. It may not drive the M lmer-S renson gate with both Raman beams on (but
 without the requisite modulations) due to a beat note between the arms that
 is equally detuned to the blue and red from the carrier.
 As the pulse width is  10 ps , the bandwidth of the laser is  100 GHz .
 With a repetition rate of 80 MHz , that indicates 1180 comb lines in the frequency
 domain. Enforcing these parameters and above conditions, with the more strin-
 gent requirement that the rate not lie between carrier and Zeeman transition, the
 following Mathematica code reports allowed repetition frequencies :
 ranges = Table [frep; fn; 1; 1180g] ;
 For[n = 1; n  1180; n++;
 ranges[[n]] = Reduce [ffrep  n=2 < (12642:8199 + 7)&&frep  n=2Reduc
 > 12642:8199 7&&frep < 81&&frep > 79g]]
 ranges = DeleteCases[ranges;False]
 And the the frequency ranges of the repetition rate frep that satisfy the con-
 ditions for the possible values between 79 and 81 MHz are reduced to: f80:9988 <
 frep < 81:; 80:7401 < frep < 80:8295; 80:4829 < frep < 80:5721; 80:2274 < frep
 < 80:3163; 79:9735 < frep < 80:0622; 79:7213 < frep < 79:8096; 79:4706 < frep <
 79:5586; 79:2214 < frep < 79:3092; 79: < frep < 79:0614g
 Our Vanguard has a measured rep rate of 80.6 MHz , which seems to be just
 outside one of the permissible ranges (there is no way to control this for this laser,
 as it depends on laser cavity length that cannot be adjusted). The beat frequency
 76
between two comb teeth separated by 157 teeth is closest to the qubit frequency:
 frep  157 = 80:6 157 = 12654:2 MHz . Our fastest Rabi transitions driven by the
 Vanguard have been measured to be  1 MHz , which implies the number of pulses
 needed per Rabi cycle is at least 80. Therefore the width of a comb tooth is no more
 than 1 MHz when we drive Rabi transitions at full power, and  1 KHz for M lmer-
 S renson transitions at our typical transition strength. Another consequence of the
 10 ps pulse width is that pulse is 3mm long in space. As the pulses must overlap at
 the ion at the same time, we adjust the Raman2 arm using a beam cube, named in
 4.17 as Mach-Zender (the setup is similar to a Mach-Zender interferometer as the
 ion measures the phase di erence between the arms).
 As we are utilizing long chains of ions, aligned along the z-axis of the trap (as
 illustrated in  gure 4.18). We are also using global beams to address the entire ion
 chain at once. Therefore, the most power e cient beam shape is a sheet of light. To
 approximate this, we use two sets of cylindrical lenses, one vertical and one horizon-
 tal for each Raman arm. This allows us to compress the vertical beam waist down
 to  7 m and the horizontal beam waist down to 70 m (Raman2) and 200 m
 (Raman1) at the ion. In order to measure these beam pro les, we position a mirror
 in the beam path before the ion, and image the beam at the estimated distance
 to the ion with a Guppy CCD, as in  gure 4.19. Alternatively, one may place the
 Guppy at the intermediate focus (marked with an X in  gure4.17), and estimate
 the beam waists at the ion based on the demagni cation of the telescope (roughly
 1/4). It is important to note that the telescope for Raman2 images AOM2. As
 AOM2 is modulated by multiple frequencies, the de ection point in the AOM must
 77
be imaged so as to refocus all frequencies at the ion chain (the AOM de ects with
 an angle sin 1(m 2 ), where m is the order of the de ected beam,  the wavelength
 of light, and  the sound wave wavelength).
 This optical setup is far from ideal. The 0th order beam of a driven AOM1 has
 a poor beam pro le, and the power in the beam depends on how strongly AOM1
 is driven, thus coupling the beam strengths of both arms. Also, the beam pro le
 seems to be coupled to the imaging plane, and we were unsuccessful at changing
 the beam pro le while maintaining the imaging plane at AOM2. This imaging has
 also been diagnosed by us to be imperfect, possibly leading to a Stark shift across
 the chain, that appears as a weak biasing axial  eld in quantum simulations. An
 additional issue in this setup is that Raman2 just barely grazes the objective, and
 has a re ection o the imaging view port that also passes very close to the objective,
 as seen in 4.17. This limits our possible numerical aperture (NA) to its current value,
 as discussed in 4.4.8.
 4.4.8 Florescence Collection
 The state detection via  orescence, as discussed in 2.4 is achieved by collecting
 the spontaneous emission light at 369 nm from the ions using a microscope objective
 from CVI (UVO-20.0-10.0-355-532). This objective of NA 0.23 is situated roughly
 3mm from the imaging port, and 13mm from the ion chain in e ective optical dis-
 tance, as illustrated in 4.20. This setup is very similar to that described in [52]. With
 an apex angle of 2 , the solid angle  = 2 (1  cos ) = 2 (1  cos(sin 1NA)) =
 78
z
 x
 y
 Raman 1
 ?k
 Raman 2
 Figure 4.18: Schematic of ion trap apparatus and the geometry of
 Raman laser beams for the spin-dependent force. Two Raman beams
 uniformly address the ions, with  k along the transverse x-direction.
 The spin states are de ned with respect to a magnetic  eld of  5 G
 along the y-axis. A photomultiplier tube (PMT) and CCD camera
 are used to measure the spin state of each ion through standard spin-
 dependent  uorescence.
 79
Figure 4.19: Pro le of beam as taken with CCD at estimated ion
 distance. Our home built Labview program  ts for the horizontal
 (wide) and vertical beam waists
 0:168, i.e. only  1:3% of the scattered light. The collected light then passes
 through a home built  400 m pinhole (a washer thinned by a mill and drilled
 through) at the image plane to reduce o axis background scatter. The image is
 further magni ed by a doublet pair to  130 and  ltered by a pair of Semrock  l-
 ters (FF01-370/36-25 and Hg01-365-25) that transmit  95% of the light at 369 nm
 , and block the other lasers and roomlight completly (unfortuantely, they do not
 block 355 nm ). The light is split into two paths - one for imaging, and one for cool-
 ing scatter collection. The latter is part of an automatic decrsytalization detection
 scheme. The imaging path has two possible con gurations - one where a  ipper
 mirror diverts the light to the imaging PMT (Hamamatsu H10682-210), and one
 where the mirror is folded back and the image proceeds directly to the ICCD. The
 PMT beam path has a lens to demagnify the image, as long ion chain images are too
 80
obje ctive  
PMT  
PMT  
Figure 4.20: Objective collects  1:3% of spontaneous emission from
  orescing ions. Then collected light passes through pinhole, dou-
 ble and  lters(not shown) that are all mounted in lenstube atop a
 micrometer positioning stage for alignment. First PMT is for decrys-
 talization detection. PMT or ICCD imaging is selected via remote-
 controled  ipper mirror, in box(dashed lines).
 81
Figure 4.21: The averaged image of 1000 exposures of 10 bright ions.
 The pixels are binned in 4 64 super-pixels (hence the vertical com-
 pression). This image was produced by the script in appendix D for
 the data taken in 7.
 large to  t the PMT aperture otherwise. In retrospect, we have overly magni ed
 the image. The only advantage to magni cation is ease of diagnosing the focus.
 As long as magni cation is enough that adjacent ions do not overlap on the same
 ICCD pixels the magni cation is su cient. Magnifying beyond that point leads to
 using more pixels than necessary, slowing down readout, adding readout noise and
 limiting  eld of view. The binned image of 1000 3 ms exposures (produced by the
 script in appendix D) bright can be seen in  gure 4.21.
 As discussed in the 2.4, the PMT Quantum E ciency (QE) is  30% at
 369 nm , and only 20% for the PiMax3:1024i \Super Blue". When we detect the
 ion state with the PMT for 0.8 s , we collect on average 10 photons for a single
 bright ion, and this is su cient for detection with 97%  delity. Due to the lower
 QE and readout noise of the ICCD, we must use alonger exposure time of 3 ms for
 the same detection light power. In addition, we bin the 1024 1024 pixel CCD into
 4  64 superpixels to reduce readout time and noise[53]. With our parameters the
 detection  delity for a single ion on the ICCD is 95%. For our magni cation and
 di raction limit, the ion occupies a Gaussian circle of 250 m on the ICCD (found
 from an unbinned 12:8 12:8 m pixel size) as seen in  gure 4.22.
 When collecting data via PMT during experiments we  t the histograms to
 simple Poissonians, with an average set by hand. When analyzing such data, we
 82
Figure 4.22: High resolution video still of single ion, taken from our
 control program. Airy ring is somewhat visible. Most of collected
 light is from cooling. The pinhole is barely visible, outlined by light
 scattered from the trap and view port.
 83
re t the histograms with more realistic distributions[31]. When collecting data with
 the camera the treatment is more complex. For real time data taking, we collect
 1000 images of the ions when they are pumped dark, followed by 1000 images of
 the ions rotated to bright by a Raman pulse. The bright images are then averaged,
 and the image discriminated into dark and bright pixels according to a preselected
 brightness cuto . The bright pixels are used to  nd the Regions of Interest (ROI)
 for each ion. The brightness of the entire region of each shot is then analyzed to
 derive histograms for the dark and bright state for each ion. A discriminator that
 gives equal state detection error for both histograms is then determined. The ROIs
 and discriminators are then returned to the Labview control program to be used in
 real time state discrimination. The script is presented in appendix D.
 Due to the spot size of the ions and their typical seperation of 2-5 m , we
 su er from cross talk. This can be seen in the results returned from the ROI script
 in table 4.1. Here we see that the detection error  is lower at the center of the chain.
 This is due to the ions higher proximity at the center the chain. The spillover light
 from neighboring ions causes the bright histogram to be brighter and hence the
 lower overlap with the dark histogram seemingly reduces  . To compensate for this
 cross talk and bias we post process our data. Using the same 1000  2 exposures
 used previously, we derive new histograms, this time by  tting Gaussians centered
 at each ion to vertically summed data, as in  gure 4.23. By counting frequency of
 the Gaussian  tted amplitudes, we construct a histogram that accounts for ion cross
 talk as the  t is found for the entire ICCD width with  xed Gaussian waists and
 centers. Although this process discards one spatial dimension of the images, it does
 84
Y1 Y2 X1 X2  Discriminator
 1 5 45 48 7. 10650
 1 5 56 59 5. 10650
 1 5 65 69 4.9 14450
 1 5 74 78 4.7 14350
 1 5 82 86 4.3 14600
 1 5 90 93 4.9 11100
 1 5 98 102 4.8 14850
 1 5 107 110 4.7 11300
 1 5 116 120 4.6 14800
 1 5 127 131 6. 14600
 Table 4.1: ROIs and detection error %  for 10 ions, as returned by
 the script in appendix D for the 10 ion data in 7.
 .
 not have much of an a ect on the discrimination between signal and background, as
 there is severe vertical image smearing at our ICCD clock speed setting (16 MHz ).
 We further enhance our data by correcting for state bias due to the detection
 error [54]. We construct the matrix M that describes the distortion of the probabil-
 ities for N ions of each real spin state gi into the detected probability for spin state
 fj =
 P2N
 i=1Mjigi
 Mji = (1 p0)
 n0  (1 p0)
 n1  p 0p
  
1 (4.19)
 Where p0 the error probability for one dark ion is equal (or should be) to the error
 probability p1 for one bright ion, n0 (n1 = N n0) is number of down (up) spins,and
  ( ) is number of down (up) spin  ips to up (down) to misidentify state gi as state
 fj. We then simply invert this matrix to  nd state probabilities closer to the real
 ones
 gj =
 2NX
 i=1
 M 1ji fi (4.20)
 85
2000 4000 6000 8000
 200
 400
 600
 800
 Histograms Bright and Dark of Ion  1
 0 1000 2000 3000 4000 5000 6000
 200
 400
 600
 800
 Histograms Bright and Dark of Ion  2
 1000 2000 3000 4000 5000 6000
 200
 400
 600
 800
 Histograms Bright and Dark of Ion  3
 0 1000 2000 3000 4000 5000 6000
 200
 400
 600
 800
 Histograms Bright and Dark of Ion  4
 0 2000 4000 6000
 200
 400
 600
 800
 Histograms Bright and Dark of Ion  5
 1000 2000 3000 4000 5000 6000 7000
 200
 400
 600
 800
 Histograms Bright and Dark of Ion  6
 0 2000 4000 6000
 200
 400
 600
 800
 Histograms Bright and Dark of Ion  7
 0 2000 4000 6000
 200
 400
 600
 800
 Histograms Bright and Dark of Ion  8
 0 1000 2000 3000 4000 5000 6000 7000
 200
 400
 600
 800
 Histograms Bright and Dark of Ion  9
 0 1000 2000 3000 4000 5000 6000 7000
 200
 400
 600
 800
 Histograms Bright and Dark of Ion  10
 Figure 4.23: The background subtracted Gaussian  tted histograms
 for 10 ions, for the data in chapter 7. The units are arbitrary. The
 red line is the found discriminator between down and up states, and
 is somewhat di erent between ions.
 86
The algorithm above assumes that we have correctly diagnosed the state detection
 error, that the error is the same for all ions in the chain, and that the error is
 independent between ions. All of these assumptions are likely only approximately
 true, so we can expect improvement but not complete correction of detected state
 probabilities.
 4.5 Arbitrary Waveform Generator (AWG)
 In order to generate the complex waveforms that drive the simultaneous M lmer-
 S renson and Raman gates, we drive AOM 2 with a a DA12000-16 from Chase Sci-
 enti c. This is a standard PCI expansion card that outputs waveform at up 2 GHz
 and has 16MB of on-board waveform memory. We have setup the AWG to operate
 at 1 GHz , and it is phase locked to a PTS3200 Direct Digital Synthesizer (DDS).
 This oscillator itself is phase locked to the same 10 MHz Rubidium clock as AOM
 1, thus locking the phase between the modulations on both beams. Unlike all other
 oscillators used in the experiment, the AWG does not run continuously - it outputs
 the stored waveform when it receives a TTL trigger, and then stops. This makes
 the phase jitter on this trigger critical, as the jitter is transformed into noise on
 the phase relationship between the Raman AOMs. When a trigger is received, the
 AWG reads each byte in sequence from memory, and converts each byte with a 12
 bit Digital to Analog Converter (DAC), such that 2047 is equivalent to 0V, 4096
 is the max voltage, and 0 is the minimum voltage. This limitation on the vertical
 resolution does not create noticeable e ects in the frequency domain as the AWG is
 87
bandpassed. However it limits the resolution when dealing with waves of di erent
 magnitudes with physical e ects of the same order of magnitude, i.e when we drive
 the M lmer-S renson gate we drive sidebands > 10 times stronger than the wave
 that drives the Raman coupling. Thus, our resolution for the Raman coupling will
 be limited to much less than 212.
 As the resonant frequency of the AOMs is 210MHz, the AWG will typically
 have only 4-5 points per wave cycle. This is su cient to generate a frequency that
 can be bandpass  ltered of its alias at fclock  fsignal  800 MHz . Therefore, we
 chose to run the AWG at half its possible clock frequency and double the duration
 of the waves we can generate. At a clock rate of 1 GHz , a byte is converted to an
 analog voltage every 1ns, allowing us a total waveform time of 16 ms . This is more
 than su cient for our experiments. Unfortunately, it is not su cient for all of our
 diagnostic procedures, such as measuring the spin coherence time.
 Besides the waveform duration limitation, there is also a limitation on wave-
 form upload speed. For longer experiments (3-4 ms ), the upload time becomes
 signi cant. The DA12000 has a standard PCI type connector. Naively, this should
 imply a waveform upload speed of 133MB/s at least, so 120 ms upload time for the
 entire wave. Unfortunately, we have characterized the upload time to be approx-
 imately one hundred times slower. As a result, we are currently in the process of
 replacing this AWG with a more advanced model.
 The AWG is controlled by functions provided in a DLL by Chase. As we in-
 terface the AWG via Labview, it was necessary to construct a DLL to interface the
 function library (da12000 dll.dll) and the card. The chief function of this library
 88
is the translation of data structure from Labview into the data structure expected
 (\SegmentStruct", as de ned in appendix C) by the chapter loading function. Cur-
 rently, the chapters run consecutively, and only the  rst chapter is triggered. A pre-
 vious attempt to have multiple triggered waveforms results in the occasional missing
 of a trigger (likely due to a trigger being sent before a chapter was completely gen-
 erated). This error a ects all following experiments as the AWG chapter execution
 becomes mismatched with the other triggers/oscillators from the  rst AWG chapter
 being generated.
 There is also an option for a chapter to loop for a  xed number of times. This
 may be useful for increasing the AWG e ective time (for instance, for experiments
 where the AWG should generate a short pulse, a long duration of no output, and
 then another short pulse - i.e. Ramsey experiment).
 89
Chapter 5
 Experimental Procedure
 5.1 Introduction
 Before each experiment with our apparatus, we tend to perform the same
 preparatory tests and calibrations. In this chapter I will go through these proce-
 dures and some of the trouble shooting options when the system is more severely
 miscalibrated than usual.
 5.2 Temperature
 Before preparing to trap ions, all lasers must be correctly set and stable.
 Therefore, the  rst action of the day is turning on the lasers and allowing them to
 cool-down (or warm up) and equilibrate. For the DL-100 and Vanguard lasers this
 takes approximately half an hour. For the Verdi and MBR-110 this may take two
 hours, so generally we do not turn these lasers o when taking data. The Verdi and
 MBR require chillers for their operation, so these must be activated before the lasers.
 For the Vanguard, this is done automatically by MLUV. The MLUV program will
 also display the percent warm up of the THG 4.11.
 The Pi:Max ICCD must be cooled down to its operating temperature as well
 ( 250C). The ICCD is cooled by a water circulating CPU cooler, operating at room
 90
temperature. This is due to the crowding on the table preventing proper cooling by
 the ICCD fan. The ICCD is ready for operation in minutes.
 RF power to the trap should be set to the operating power used for the ex-
 periment (typically around 26dBm), as the delivered power changes the resonant
 frequency of the trap. The set point of the TC200 heater for the resonator should
 be checked.
 5.3 Frequencies
 Now that the lasers are hopefully stable, their frequencies should be set. As-
 suming the lasers are well aligned, we set the laser frequencies for 171Yb+ trapping,
 as indicated in table 5.1. These frequencies are measured by a HighFinesse solid
 state wavemeter, via optical  bers, and displayed by a homebuilt web client 1. If
 loading fails, it might be necessary to attempt to load 174Yb+, as it does not have
 hyper ne structure and hence scatters more light, and does not require all the laser
 modulation needed by 171Yb+ nor can it have a coherent dark state [1]. The wave-
 lengths for 174Yb+ are listed in table 5.1 as well. When alternating to 174Yb+, the
 loading rate may be highly increased by switching over to the natural isotope dis-
 tribution Yb oven, although it is possible to load from our usual high purity 171Yb
 oven.
 After setting the laser frequencies, the lasers must be stabilized, as discussed
 in chapter 4, as well as the wavetrain frequency doubler for the MBR. This often
 requires slight adjustments of the input coupling lens.
 1Created by Peter Maunz.
 91
Laser 171Yb+ 174Yb+
 MBR-110 405.644318 THz 405.645530 THz
 935 nm DL-100 320.56922 THz 320.57190 THz
 399 nm DL-100 751.52764 THz 751.52680 THz
 638 nm DL-100 469.445-469.442 THz 469.439 THz
 Table 5.1: Laser frequencies for trapping. The 638 nm can be locked
 anywhere in above range as long as it is mode-hope free for the entire
 range.
 The Vanguard shutter should be opened and emission engaged (via MLUV).
 The Vanguard frequency cannot be set, rather the feed forward lock on AOM2 is
 engaged. For the RF resonator, the reference signal is adjusted to close to 0 by
 slightly adjusting the RF oscillator, and then engaging the RF frequency lock. A
 moderate gain must be set here (by tuning the HP8640B modulation strength knob)
 as recrystallization will severely change the resonance frequency. Also, the Labview
 voltage control program should be checked to see that there is no attenuation of the
 trap RF power. Once all lasers are locked one may turn on the Yb oven, and after
 a minute of oven warm up (at  2:4A) one way drop the trapping voltages and RF
 power using the recrystallization settings, while radiating the trap with the 399 nm
 light and alternating pulses of 355 nm light, while waiting for an ion or ion cloud to
 appear in the Pi:Max3 viewing program ( gure 4.22).
 5.4 Loading and Alignment
 If an ion is successfully loaded, then beam alignments must only be slightly
 adjusted to maximize cooling and detection counts. If loading is slow one may
 attempt to adjust the 399 nm beam, as that is the most likely to be misaligned. If
 92
the detection or cooling seem to be unstable, one may try adjusting the 935 nm
 beam alignment and polarization.
 In general, as all 369 beams share the same  nal path, it is reasonable to
 assume that if they are all overlapping then the beams are close to the correct
 alignment.
 Another sanity check is the scatter patterns of the trap electrodes. By diverting
 the beams slightly up or down from the center of the trap, one may  nd the gaps
 between the trap electrodes by viewing the pro le of the beam on exit through the
 view port. If all beams seem reasonably aligned and loading is not seen on ICCD,
 then the PMT counts should be viewed instead. If this fails as well, then Yb oven
 temp should be increased. If loading still fails, then the trap may have shifted
 relative to the imaging system. In order to realign the imaging system, one must
 detect a trap corner (as shown in  gure 5.1), and thus reorient the imaging system
 micrometer settings relative to the trap. Vertical alignment of the beams based on
 the clipping pro le of the beams is trivial. It is important to remember that the
 beams are entering the trap at a 450 angle, and the horizontal lens alignment of the
 beams is measuring the displacement of the beams at that angle rather than parallel
 to the trap. When viewing the corners of the trap, it is most likely the corners of
 the electrode furthest from the camera that are being viewed. When attempting to
 load over long periods, it is important to occasionally shut o the trap RF so as to
 release possible trapped ion clouds. These do not crystallize nor cool e ciently and
 are thus less visible than single ions.
 Once the ions are loaded it is important to verify that the cooling and detecting
 93
Figure 5.1: (a) Lower left trap corner. (b) Upper right corner. Pin-
 hole is clearly visible due to high scatter o trap. Burs are due to
 imperfect drilling.
 beams are centered on the ion chain. This can be done by moving the ions across the
 camera (using the trap voltages to push them in the z-direction), but PMT detection
 counts are more exact. If one intends to take data using the ICCD, it is important
 to verify that all ions fall on the active PMT surface as the camera and PMT are
 independently aligned. There seem to be certain regions on the PMT surface that
 are not active. If it is suspected that the protection beams require alignment, then
 the iodine lock must be disengaged and the MBR tuned 400 MHz blue so as to bring
 those beam frequencies closer to transition resonance.
 5.5 Raman Beams
 Rough alignment of Raman beams may be done using an independent Ramsey
 experiment for each beam using a qubit frequency microwave pulse, or an AC Stark
 shift measurement. For the Ramsey stark shift measurement, the Vanguard feed
 94
forward is unlocked, and an RF switch diverts the microwave reference to via an
 ampli er to microwave horn (the microwave source frequency is slightly adjusted so
 as to be exactly on carrier). First, the Rabi oscillation  =2 time is measured, i.e. the
 time it takes for the qubit to reach half of maximum brightness (the qubit is then
 in the x-y plane of spin space). Second, using the Experimental control program
 we schedule three experimental pulse sequences (\chapters"): a  =2 pulse, a 355 nm
 pulse, and another  =2 pulse. By scanning the duration of the 355 nm pulse, we can
 see \Ramsey fringes" - oscillations of the qubit due to the Stark shift of the qubit
 frequency. This method is very sensitive as the pulses can be 10?s of milliseconds
 long.
 Once both beams are roughly aligned we can drive Raman transitions with
 both beams. At that stage maximizing the Rabi frequency is the fastest way to
 improve alignment. Finally, we return to Stark shift Ramsey measurements on the
 beams. We can then perform a long Ramsey experiment with no 355 nm light to
 measure accurately the carrier frequency, as any mismatch between the  =2 pulses
 and the qubit frequency will lead to oscillations where there should be none.
 Now we use the Ramsey experiment to validate that the beams are truly cen-
 tered. We shift the ion a distance on the order of the length of long ion chain, and
 validate that the Ramsey frequency is symmetric, adjusting the horizontal microm-
 eter to achieve this. Barring suppression by micromotion, coupling to the y-mode,
 or incorrect polarization our Rabi carrier frequency is now maximized. Both beams
 should be linearly polarized, with electric  elds parallel to table so as to drive the
 Raman transition with  +;  + polarization in the ion?s frame of reference.
 95
0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5
 Raman beatnote detuning from carrier (MHz) 
Ram an spectru m for 10 ions, with 355 nm pulse d lase r  
tilt  
COM  
Carrier  
Figure 5.2: 10 ion Raman spectrum. The x-axis is in MHz detuning
 from carrier, that is centered at 0 MHz . The dominant motional
 modes are the x-transverse modes. As can be seen here, their width
 is only on the order of  100 KHz , vs the carrier which is on the
 order of MHz . The highest frequency mode is the COM, and second
 is tilt. The lowest is the zig-zag mode, only a MHz away from carrier
 for these trap settings. Some of the weaker features are axial and
 y-modes, which have a weak coupling to our beam geometry.
 5.6 Calibration
 As we have identi ed our qubit frequency exactly, we are now interested in
 mapping out the motional modes, as can be seen in  gure 5.2. If we do not know
 the general location of the motional modes (due to changes in trapping voltages,
 for instance), we may load two ions and measure the COM and tilt modes. That is
 su cient to calculate all other modes. This can be done by scanning the modulation
 frequency of Raman beam 2 with a  xed pulse time. If the pulse duration is long
 enough and the intensity high enough to drive several oscillation periods for the
 weakest transition of interest then we can expect to see all resonances.
 96
Figure 5.3: Results for two di erent intensities of the automatic side-
 band  nder. The program  nds increasingly accurate RSB and BSB
 frequencies by  nding upper and lower limits for decreasing inten-
 sity and lengthier pulse times. Before  nding the RSB, a  pulse is
 applied.
 Once we identify the modes, we may preemptively apply two sidebands, roughly
 100 KHz red detuned from the COM, with the intensity we expect to use for our
 M lmer-S renson coupling gate. This will ensure that the following measurements
 will be immune to higher order Stark shifts. However we must shift all frequencies up
 by the measured Raman 2 Stark shift ( 200 KHz , vs  600 KHz for Raman 1), as
 they were performed at the lowest power possible. After applying side-band cooling
 to the COM mode, we may activate the automatic sideband  nding functionality of
 the control program, as shown in  gure 5.3.
 After measuring Rabi oscillations frequencies for the BSB and RSB, we set
 them equal (equal AWG amplitudes for both frequencies do not necessarily drive
 equal strength sidebands), and detune them equally from resonance. We then add
 a carrier frequency, and measure its frequency for three di erent powers, for AWG
 amplitudes  80 times lower than the sideband amplitudes. Thus, we have cali-
 brated the AWG amplitudes to real frequencies. The relationship is linear for these
 low values.
 We now add a second ion, and remove the carrier. We optimize the spin-spin
 97
coupling by alternately scanning the amplitude of one the sidebands or a sideband
 frequency. This should lead to decent (0.1-1.7 ions bright) contrast, and roughly
  ve coherent oscillations at our typical operating parameters. If the coherence is
 bad it is likely due to the vertical alignment of the Raman beams.
 5.7 Imaging Calibration
 We may now load all the ions needed for the experiment. After allowing
 the RF resonator to settle from the power drop needed for loading, we prepare
 an experiment sequence where the ions will cooled, then pumped dark, and then
 detected. The number of experiments per point is set to 1000. Then the camera
 calibration scan is activated, and we select the \Ions Dark?? option. We add a  
pulse to rotate all ions to bright after the optical pumping, and repeat the calibration
 procedure, this time selecting \Ions Bright??. The Mathematica script of appendix D
 will then automatically start. The control program will display the averaged bright
 ion images, with ROI boxes and detection parameters.
 We are now ready to select our experimental chapters and parameters, and to
 begin experimenting.
 98
Chapter 6
 Summary of Experiments
 6.1 Introduction
 In this chapter I will summarize the experimental results we have achieved in
 our group leading up to the work reported in chapter 71. The  rst work I will report
 was our demonstration and control of spin-spin couplings between the pseudo-spins
 for two and three ions using global Raman beams. Once that ability was mastered,
 we were able to proceed with the adiabatic quantum simulation experiments: Mea-
 suring the phase diagram of the ground state Hamiltonian for three spins; Creating
 a frustrated triangle of AFM coupled spins; Creating a phase transition from the
 paramagnetic state to FM and AFM states with 16 spin. For most of this work,
 our Raman beams were driven by an the MBR, detuned only 2.7 THz from the
 2S1=2  2 P1=2 transition, and with only 200 mW of UV light. These gates were en-
 abled by microwave frequency combs of the EOM modulated laser beams, at roughly
 half the qubit frequency. These beams were then replaced by the 33 THz detuned
 355 nm laser beams, with 4W of UV. This scheme o ers the same coupling strengths
 with much lower unwanted scatter rates.
 Using the simulated gates I introduced in chapter 3, we will simulate the
 1This chapter follows closely parts of [55]
 99
Hamiltonian
 H =
 X
 i<j
 Ji;j 
i
 x 
j
 x +By
 X
 i
  iy (6.1)
 The axis of the interactions will be selected by controlling the relative phase of
 beatnotes generating the interactions.
 6.2 Experimental Demonstration of the Ising interaction
 We trap 171Yb+ ions in a linear chain along the trap Z axis, and the axial
 COM trap frequency is controlled in the range  z = 0:6  1:7 MHz. The COM
 mode is set to around  4 MHz. The other transverse Y modes are su ciently far
 away from the X modes and do not overlap. Moreover the coupling to these modes
 is suppressed by a factor of > 10 by rotating the principal Y -axis of motion to be
 nearly perpendicular to the laser beams, as described in chapter 4.4.6.
 We direct two Raman laser beams onto the ions to drive spin-dependent forces,
 with their wavevector di erence aligned along the transverse x-axis of ion motion
 ( k = k
 p
 2, where k = 2 = ). The Raman beams are detuned 0:5  2:7 THz to
 the red of the transition at a wavelength of  = 369:76 nm . The Raman beams are
 phase modulated at a frequency nearly half of the 171Yb+ hyper ne splitting with a
 6:32 GHz resonant EOM, and each of the two Raman beams have independent AOM
 shifters in order to select appropriate optical beatnotes to drive Raman transitions
 [56]. The Raman beams are focused to a waist of approximately 30  100 m with
 a power of up to 10  20 mW in each beam. When their beatnote is adjusted to
 drive the carrier transition at the hyper ne transition  HF , we observe a carrier Rabi
 100
frequency of  i=2  1 MHz. For the spin-dependent force, we set  i=2  0:4
 MHz for each pair of Raman beam and for the transverse  eld,  i=2 is less than 0.1
 MHz. The resonant transition generates the e ective transverse  eld by adjusting
 the phase with respect to the spin-dependent force. For up to nine ions in a chain,
 we observe that the outer two ions experience  2% lower Rabi frequency, and the
 variation in the di erential AC-Stark shift in each qubit of < 1%.
 In the experiment, we  rst Doppler laser cool 171Yb+ ions for 3 ms using a laser
 tuned red of the 2S1=2  2 P1=2 transition at a wavelength of 369:53 nm. We then
 Raman sideband cool all m modes of transverse motion along x to mean vibrational
 indices of  nm < 1 in about 0:5 ms, well within the Lamb-Dicke limit. Next, the ions
 are each initialized to the j#i state through standard optical pumping techniques
 [1]. We then apply the optical spin-dependent force on the ions for a duration  
by impressing the bichromatic beatnotes at  HF   . Afterward, the spin states
 are measured by directing resonant laser radiation having all polarizations on the
 2S1=2(F = 1) 2 P1=2(F 0 = 0) transition following standard state-dependent  uores-
 cence techniques [1]. We use a charge-coupled-device (CCD) imager (the detection
 e ciency is, 95 % per spin). We determine the probability of each spin con guration
 (for example P""# ) by repeating the above procedure more than 1,000 times. We
 also measure the probability Pn of having n spins in state j"i by using the PMT,
 which is useful for higher-e ciency measurements of certain symmetric observables
 such as entanglement  delities and witness operators.
 101
J
  (
 k
 Hz
 )
 (a) (b)
 (d) (e)
 (b)
 (c)
 (d)
 (e)
 ms
 Pavg
 Pavg
 t (ms)
 1
 p
 x1
 p
 x
 1
 p
 x 1
 p
 x
 t (ms)
 (c)
 t (ms)
 t (ms)
 Pavg
 Pavg
 Theory
 Experiment
 Figure 6.1: (a) Measured coupling J for two ions as a function of
 detuning  overlaid with theory (lines) from Eq.(3.22) with no free
  t parameters. The detuning is scaled to the axial ( z) and transverse
 ( 1) COM normal-mode frequencies of motion such that COM, and
 tilt modes of transverse motion occur at  S  ( 2   21)= 
2
 z = 0 and
  1, respectively. (b) Time evolution of the average number of ions
 in the j"i state under the in uence of the bichromatic force in the
 far-detuned limit, showing the secular oscillation of the Ising spin-
 spin coupling, where the detuning  =2 is 80 KHz from the COM
 motional sideband. (c) Measurements with    1 = 2
 p
 3 1 =2  
56 KHz . Here the small oscillations from the motional excitation
 and the coupling to spin states are noticeable on top of the sinusoidal
 oscillations of the Ising interactions. (d) Measurements at    1 =
 2
 p
 2 1 =2  45 KHz . (e) Measurements at    1 = 2 1 =2  
32 KHz . The insets of (b)-(e) show the respective wavepackets in
 phase space and the areas enclosed are shaded.
 6.2.1 Two spin case
 Figure 6.1(a) shows the theoretical values of J = J1;2 from Eq.(3.22) and
 measurements at various detunings  for two spins with J =   1 2(
  21 1
  2  22
   
2
 2 2
  2  22
 ),
 where  1 =  k=
 p
 4m 1 and  2 =  k=
 p
 4m 2 are the Lamb-Dicke parameters for
 COM and tilt modes. The solid theoretical curve is plotted with no adjustable
 parameters, as the motional mode frequencies and the sideband Rabi frequencies
 are independently measured.
 We measure the strength of J by observing the time evolution between j##i z
 and j""i z after applying the spin-dependent forces on j##i z states. The evolution
 102
for the two spins is simply described by U(t) j##i z = exp( iJ 
(1)
 x  
(2)
 x t) j##i z =
 cos(Jt) j##i z+i sin(Jt) j""i z when we neglect the couplings between internal states
 and the motion. As shown in  gure 6.1(b), the oscillations of the populations are
 sinusoidal, with frequency 2J . To detect the sign of the Ising coupling, we applied
 the same force on the initial state j""i and observe the phase of the oscillations.
 When the beatnote detuning  is close to a vibrational mode, or j   1;2j is
 within a few sideband linewidths  1;2 , the coupling between motion and spin states
 modulates the spin state evolution. Figure 6.1(e) shows the spin state evolution at
 a detuning  =    1 = 2 1 [3]. In this case, the motional state is displaced in
 phase space by no more than j j = 1, and at particular times during the evolution
  = n= (n = 1; 2:::) the motional degree of freedom is decoupled from the internal
 state, enabling the generation of pure spin-spin entanglement. As the detuning  
increases as shown in  gure 6.1(b)-(d), the maximum displacement decreases and
 the evolution approaches a pure sinusoid indicative of pure spin-spin interactions.
 Typical experiments are performed with   4  , where the largest displacements
 in a  2% modulation in the spin evolution.
 6.2.2 Three spin case
 For three spins addressed uniformly with the Raman laser beams, we have
 J1 = J1;2 = J2;3 as the nearest-neighbor (NN) interaction and J2 = J1;3 as the next-
 nearest-neighbor (NNN) interaction as shown in  gure 6.2(a). Since the bandwidth
 of the transverse mode spectrum is relatively small, all modes can be addressed from
 103
J
  (
 k
 Hz
 )
 ms
 (a) (b)
 (c)
 P0
 4 J 1 2J 1 - 2J 2
 2J 1 +2J 2
 t (ms)
 Freq uenc y (kHz)
 J1
 J2
 Calculation
 Experiment
 Figure 6.2: (a) Measured couplings J1 = J1;2 = J2;3; J2 = J1;3 for
 three ions as a function of detuning  overlaid with theory (lines)
 from Eq.(3.22) with no adjustable parameters. At the scaled detuning
  S, COM, tilt and zig-zag modes of transverse motion occur at  S  
( 2   21)= 
2
 z = 0; 1 and  2:4, respectively. (b) Time evolution
 of the probability P0 of all spins j#i under the bichromatic force in
 the far-detuned limit. Here, the two couplings J1 and J2 are clearly
 visible. The solid line is a  t to the time evolutions of Eq.(6.1) with
 an empirical exponential decay. The measurements are performed
 at the indicated detuning in (a), or -50 kHz from tilt mode. (c)
 The Fourier transform of the experimental curve shown in (b), where
 three peaks originate from the frequency components of 4J1, 2J1 2J2
 and 2J1 + 2J2. The orange bars represent the calculated values from
 Eq.(3.22). The peak near the origin comes from the overall decay of
 the oscillation due to decoherence with  100 Hz.
 104
a single laser beatnote and the signs and the strengths of J1 and J2 are under great
 control as shown in  gure 6.2(a) and Eq.(3.22) [57]. In the region  >  1, both have
 anti-ferromagnetic (AFM) interactions (J1;2 > 0), and in the region of  2 <  <  1,
 both have ferromagnetic (FM) couplings (J1;2 < 0). In the region  3 <  <  2, the
 NN interaction is FM (J1 < 0) and the NNN is AFM (J2 > 0). When  is near the
 tilt mode, J2 overpowers J1 and when  is closer to the zig-zag mode, J1 is stronger
 than J2. Finally, for  <  3, all interactions are FM again.
 We measure the J1 and J2 couplings by observing the oscillations in the pop-
 ulation of state j###i z after applying spin-dependent force on the three spins. This
 population oscillates as cos2 J1t cos J2t i sin2 J1t sin J2t, as shown in  gure 6.2(b).
 We use Fourier analysis on the oscillations and  nd certain frequency combinations
 of the couplings 4J1, 2(J1  J2), and 2(J1 + J2), as shown in  gure 6.2(c). In this
  gure, we use theoretical values for the signs of J1 and J2.
 6.3 Adiabatic quantum simulation
 In this section, we describe the adiabatic quantum simulation of the transverse
 Ising model with three spins, where the exact solution is known, and discuss the
 criteria for adiabaticity. We then present experimental results for two example Ising
 interactions strengths and signs.
 We experimentally investigate this adiabaticity criterion for the two di erent
 types of NNN coupling that were introduced above. We initialize the spins along
 the By-direction through optical pumping ( 1  s) and a  =2 rotation about the
 105
 x-axis of the Bloch sphere. Figure 1.3 shows the adiabatic simulation protocol.
 The simulation begins with a simultaneous and sudden application of both By and
 J1; J2 where By overpowers J1 (By=jJ1j  10). As the spins are aligned along the
 y-axis, the sudden turn on will not cause diabatic e ects. A typical experimental
 ramp of By decays as By(t) = ae t= + b with a time constant of   30  s, varying
 from a  10 kHz to a  nal o set of b  500 Hz after t = 300  s. By varying the
 power in only one of the Raman beams, this procedure introduces a change in the
 di erential AC Stark shift of less than 2 Hz. We turn o the Ising interactions and
 transverse  eld at di erent By=jJ1j endpoints along the ramp. We then measure the
 magnetic order along the x-axis of the Bloch sphere by  rst rotating the spins by
  =2 about the y-axis, and detecting the z-component of the spins.
 In  gure 6.4(a) all interactions are FM and J2=jJ1j   2 4(as in  gure. 6.3(a)).
 The dashed lines in the top panel are the adiabaticity parameter from Eq.(1.6) cal-
 culated over the trajectory for the two coupled excited states (recall  gure 6.3).
 Due to the 500 Hz  nal o set of By, the simulation stops at By=jJ1j  0:5. To
 examine the behavior extended below this value, we calculate the criteria for an
 exponential ramp with a 100  s time constant. This pro le was chosen to overlap
 with experimental parameters for large By=jJ1j and also reach By = 0 in a typical
 simulation time ( 300 s). The results indicate that Eq.(1.6) is satis ed over the
 trajectory; _By(t) = 2ge remains much less than one even with a maximum occurring
 at By=jJ1j  1. To demonstrate the simulation is indeed adiabatic for these parame-
 ters, we plot the measured probability of occupying a FM state P(FM) = P"""+P###
 (solid dots) in  gure 6.4(a). The black line represents the adiabatic ground state
 106
(a) (b)
 J2 / |J1|~2.1 J2 / |J1|~ - 0.9
 |????y|????, |???? |????y
 |????, |????
 Figure 6.3: Energy level diagrams for Eq.(6.1) with two di erent
 types of spin-spin interactions. For both panels, the NN interactions
 are FM (J1 < 0). (a) The NNN interaction is FM with J2=jJ1j  
-2 and (b) AFM with J2=jJ1j  0.9. The arrow in both diagrams in-
 dicates the trajectory in the simulation, initialized at By=jJ1j  10.
 Under this condition, the initial ground state is an eigenstate of sec-
 ond term in Eq.(6.1), a polarized state along By. In both examples,
 at B  J1 some high energy states cross, but the ground state (black
 solid line) has no level crossings with any excited state. Likewise, the
 highest energy state does not cross any other levels, allowing one to
 also adiabatically follow this state. The dotted lines represent excited
 states which are most strongly coupled to the ground state along the
 path. In the large  eld limit, the energy di erence between ground
 and excited states  ge (here, scaled by
 q
 B2y + J
 2
 1 ) is proportional to
 By, but as By=jJ1j decreases the spin-spin couplings determine the
 energy di erence and the form of the ground state. In both (a) and
 (b), the  nal ground state is FM (de ned along the x-axis of the
 Bloch sphere), however in the case of (a), the minimum  ge is  20
 times larger.
 107
P(F
 M)
 (a) (b)
 J 2 / |J 1 | ~ 2.1 J 2 / |J 1 | ~  - 0.9
 B y / |J 1 | B y / |J 1 |
 Ground State
 Experimental ramping
 t =1 00 ms ramping
 Data
 Ground State
 Experimental ramping
 t =1 00 ms ramping
 Data
 Figure 6.4: (a) The theoretical order from the exact experimental
 ramp with a 35  s time constant and  nal o set value given in the
 text (gray solid line) is in reasonable agreement with the order in the
 true ground state (black solid line) for By=J1 > 0:5. The dotted line is
 the expected state evolution for a pure exponential decay ramp with
 a 100  s time constant, allowing By ! 0. (b) The data also matches
 well to theory, as we avoided the regions where diabatic transitions
 are expected for By=J1  1. According to the calculations, the
 duration of three-spin experiments near the special point should be
 on the order of milliseconds.
 and the grey line is the theoretical expected probability including the experimental
 ramp. The dotted line in this  gure is the theoretical state evolution using a By-
  eld ramp that reaches zero. The predicted evolution does not signi cantly deviate
 from the ideal ground state and the data is in good agreement with all three theory
 curves.
 Figure 6.4(b) presents the case when theNNN interaction is AFM and J2=jJ1j  0.9
 (as in  gure 6.3(b)). When By=jJ1j  1, _By(t) = 2ge reaches a maximum value of
  0:6, indicating that the probability for excitations will likely increase. This di er-
 ence is because in this case the gap  ge at the ?critical? point is  15 times smaller
 than that in  gure 6.3(a). In contrast to the FM J2 case, the theoretical probability
 curves shown in the lower panel of  gure 6.4(b) predict signi cant diabatic e ects
 108
when using this By- eld pro le for simulations near the critical point. In fact, to
 successfully evolve to the true ground state near By = 0, the simulation time (as-
 suming same initial conditions and an exponential ramp of By) should be at least a
 factor of ten longer.
 Because all the data lies outside of the region where the energy gaps are small,
 the diabatic excitations are minimal, but further experimental study is needed to
 precisely quantify this e ect. One method to probe excitations, which may also be
 useful as N  1, is to perform and then reverse the experimental ramp and measure
 the probability of returning to the initial state [58].
 6.4 Phase diagram
 Assuming adiabaticity as described above, we can generate an experimental
 phase diagram for the transverse Ising model. We will  rst describe this for the
 three spin case, and then discuss speci c features and scalability in sections 6.6 and
 6.7. For the three spin Hamiltonian in Eq. (6.1), the competition between the two
 spin-spin couplings and the transverse  eld gives rise to a rich phase diagram. Here
 we label the 23 possible spin con gurations as the two FM states, j"""i and j###i ,
 two symmetric AFM states, j#"#i and j"#"i , and four asymmetric AFM states,
 j""#i , j"##i , j#""i and j##"i , all de ned along the x-axis of the Bloch sphere.
 In  gure 6.5(a), we plot a part of the theoretical phase diagram where the
 nearest-neighbor interactions are always FM (J1 < 0). The order parameter is
 the probability of occupying a FM state, P(FM) = P""" + P###. For regions where
 109
B
 y 
/ |J
 1
 |
 B y / |J 1 |
 B y / |J 1 |
 J 2 / |J 1 |
 P(FM)
 P(F
 M)
 J 2 / |J 1 |
 J2 / |J 1 |
 (a) (b)
 (c) (d)
 P(FM
 )
 (c)
 (d)
 Figure 6.5: (a) Theoretical phase diagram for Eq.(6.1). The color
 scale indicates the amplitude of the FM order parameter, P (FM) =
 Pj"""i +Pj###i . Here, J1 is always negative, yielding FM order in that
 coupling. In the region where J2=jJ1j <0, there is a crossover to FM
 order as By=jJ1j is lowered. When J2=jJ1j >0, the AFM and FM
 interactions compete. When J2=jJ1j =1 and By =0 the ground state
 is comprised of 6 states: four asymmetric AFM and two FM states.
 This creates a special sharpened point where all lines of equiprobable
 FM order converge. (b) Experimental measurements of the phase
 diagram for Eq. 6.1 (solid bars) compared to the theoretical pre-
 diction from Fig. 6.5 (surface) . The vertical amplitude is the FM
 order parameter P(FM)= Pj"""i + Pj###i . The ratio of By=jJ1j was
 varied from  10 to  0.1 for J2=jJ1j values of -1.3,-2.0, -3.6, 4.2, 2.0,
 1.3, 0.92, 0.74, and 0.62. J1 < 0 for all traces. (c) As By=jJ1j ! 0
 in the region where J2=jJ1j <  1 , we observe a smooth crossover to
 FM order. The  lled circles and solid line are the data and theory for
 J2=jJ1j =  1:3, respectively. (d) When changing J2 for a  xed and
 small value of By=jJ1j the system undergoes a sharp transition. The
 data ( lled circles) shown is for a scan of By=Jy = 0:57. The average
 deviation per scan of By=jJ1j from the exact ground state is  0:09.
 110
By=jJ1j  1, the ground state is polarized along By with P(FM)= 2=2N = 1=4, as
 all states in the x-basis are equally populated and there are two FM states. As
 By=jJ1j decreases, di erent magnetic phases arise. When the NNN interaction is
 also FM (J2 < 0), and By=jJ1j  1 the ground states are the two degenerate FM
 states. In the region where the NNN interaction is AFM and J2 overpowers J1
 (J2=jJ1j > 1), the asymmetric AFM states are lowest in energy. A special point
 appears at J2=jJ1j = 1 and By = 0, where all the contours of constant FM order
 meet. Here, the ground state will be a superposition of the FM and asymmetric
 AFM states. This e ect arises because the pairwise interaction energy cannot be
 minimized individually, leading to a highly degenerate, or frustrated, ground state
 [59].
 To be clear, when J1 = J2, both the FM and AFM ground states are degener-
 ate. As can be seen in  gure 6.3(b), as the transverse  eld is reduced the energy gap
 will go to zero, and adiabaticity will be impossible. For the FM case, there is a two
 fold degeneracy. For the AFM case, there is a six fold degeneracy. However, for our
 experiments we always have a slight transverse  eld remaining and the degeneracy
 is lifted, creating the non-degenerate, superposed ground states we discuss in this
 paper.
 This procedure is performed for nine di erent combinations of J1 and J2 de-
 termined by the beatnote detuning  from Eq.(3.22). In  gure 6.5(b) we present the
 results as a 3D plot of the FM order parameter, with the theoretical phase diagram
 (surface) from  gure 6.5(a) superimposed on the data. The data is in good agree-
 ment with the theory (average deviation per trace is  0:09) and shows many of the
 111
essential features of the phase diagram. As By=jJ1j decreases, a smooth crossover
 from a non-ordered state to FM order occurs in the region where J2=jJ1j < 1 (Fig.
 6.5(c)). The data (e.g.  gure 6.5(c) show small amplitude oscillations in the initial
 evolution due to the sudden application of the spin-spin interaction, which is held
 constant during the simulation to minimize variation in the di erential AC stark
 shifts. As the number of spins increases, this is an example of a quantum phase
 transition. A  rst order transition due to an energy level crossing is apparent ( g-
 ure 6.5(d) when changing J2 for a  xed and small value of By=jJ1j = 0:57. This
 transition is sharp, even in the case of three spins [60].
 6.5 Spin frustration and entanglement
 We also study the properties of the ground states in the case of a frustrated
 Ising Hamiltonian. Frustration in spin systems occurs when spins cannot  nd a
 ground state that minimizes the energy of each pairwise interaction [61, 12]. As
 shown in  gure 6.6(a), this can be simply illustrated by three spins with AFM inter-
 actions on a triangular lattice [62]. The situation gives rise to a large ground state
 degeneracy, leading to magnetic analogues of liquids and the crystal arrangement
 of ice [63, 64]. For quantum spins, the frustrated ground states are expected to be
 directly related to entanglement [65, 66].
 We realize the textbook example of spin frustration in a unit triangular cell
 with AFM interactions by setting the Raman beatnotes  to the blue side of COM
 mode. For comparison, we also study the ground state property of all FM interac-
 112
J>0
 1 2
 3
 J>0
 J>0
 1 2 3
 J1 J1
 J2
 (a)
 (b)
 (c) (d)
 t (ms) t (ms)
 B y / J rms B y / J rms
 |???? |???? |???? |????|???? |???? |????|????
 P
 opula
 tions
 P
 opula
 tions
 Ground State
 Experimental ramping
 Ground State
 Experimental ramping
 Figure 6.6: (a) Simplest case of spin frustration in a triangular ge-
 ometry with AFM interactions. (b) Image of three trapped atomic
 171Yb+ ions in the experiment, taken with an intensi ed CCD cam-
 era. The spins in the linear ion string have nearest-neighbor (J1) and
 next-nearest-neighbor (J2) interactions mediated through the collec-
 tive vibrational modes. (c),(d) Evolution of each of the eight spin
 states, measured with a CCD camera, plotted as By=Jrms is ramped
 down in time, with each plot corresponding to a di erent form of the
 Ising couplings. The dotted lines correspond to the populations in
 the exact ground state and the solid lines represent the theoretical
 evolution expected from the actual ramp, including nonadiabaticity
 from the initial sudden switch-on of the Ising Hamiltonian. (c) All in-
 teractions are AFM. The FM-ordered states vanish and the six AFM
 states are all populated as By ! 0. Because J2  0:8J1, a population
 imbalance also develops between symmetric and asymmetric AFM.
 (d) All interactions are FM, with evolution to the two ferromagnetic
 states as By ! 0.
 113
tions by setting  to the red side of COM mode. A linear chain of three ions can
 have NN and NNN interactions through the collective normal modes discussed in
 section 3.4 ( gure 6.6(b)). The experimental procedure to prepare the ground states
 of the Hamiltonian with all AFM interactions and all FM interactions is the same
 as the description in the section 4. Figure 6.6(c) shows the time evolution for the
 Hamiltonian frustrated with nearly uniform AFM couplings and gives almost equal
 probabilities for the six AFM states (three-quarters of all possible spin states) at
 By  0. Because J2 < 0:8J1 for this data, a population imbalance also develops
 between symmetric and asymmetric AFM states. Figure 6.6(d) shows the evolution
 to the two ferromagnetic states as By ! 0, where all interactions are FM.
 The adiabatic evolution of the ground state of Hamiltonian (6.1) from By  
Jrms to By  Jrms should result in an equal superposition of all classical ground
 states and therefore carry entanglement. For instance, for the FM case, we ex-
 pect a GHZ ground state j###i - j"""i . For the isotropic AFM case, we expect
 the ground state to be j##"i + j"##i + j#""i - j""#i - j#"#i - j"#"i . We char-
 acterize the entanglement in the system at each point in the adiabatic evolution
 by measuring particular entanglement witness operators [67]. This is accomplished
 by performing various global rotations to the three spins before measurement, and
 combining the results of many identical experiments. When the expectation value
 of such an operator is negative, this indicates entanglement of a particular type
 de ned by the witness operator. For the FM case, we measure the expectation of
 the symmetric GHZ witness operator WGHZ = 9I^=4  J^ 2x   
(1)
 y  
(2)
 y  
(3)
 y [68, 67],
 where I^ is the identity operator and J^i  12( 
(1)
 l +  
(2)
 l +  
(3)
 l ) is proportional to
 114
(a) (b)
 B y /J rms B y /J rms
 ?W
 GHZ
 ?
 ?W
 W
 ?
 t (ms ) t (ms )
 Gro u n d  Sta te
 Exp e ri m e n ta l  ra m p i n g
 In c l u d i n g  s p i n - m o ti o n  c o u p l i n g
 Gro u n d  Sta te
 Exp e ri m e n ta l  ra m p i n g
 In c l u d i n g  s p i n - m o ti o n  c o u p l i n g
 Figure 6.7: (a) Entanglement generation through the quantum simu-
 lation for the all FM interactions. We measure the entanglement by
 observing the expectation of a particular operator that indicates the
 presence of entanglement, called the entanglement witness [67]. For
 the case of all FM interactions we use the GHZ witness [67], sensitive
 to the state (j###i  j"""i )=
 p
 2. We  nd that entanglement occurs
 when jByj=Jrms < 1. (b) Entanglement generation for the case of all
 AFM interactions. Here, we use the symmetric W-state witness, sen-
 sitive to the state (j##"i +j#"#i +j"##i  j#""i  j"#"i  j""#i )=
 p
 6
 and we  nd that entanglement emerges for By=Jrms <1.1. In both (a)
 and (b) the error bars represent the spread of the measured expec-
 tation values for the witness, likely originating from the  uctuations
 of experimental conditions. The black solid lines are theoretical wit-
 ness values for the exact expected ground states, and the black dashed
 lines describe theoretically expected values at the actual ramps of the
 transverse  eld By. The blue lines reveal the oscillation and suppres-
 sion of the entanglement due to the remaining spin-motion couplings,
 showing better agreement to the experimental results. Note that the
 residual spin-motion couplings do not appear to impact on the FM
 order of each state, as shown in Fig. 6.6. In the theoretical curves
 we do not include other possible errors such as state detection in-
 e ciency or errors due to spontaneous scattering or  uctuations in
 control parameters.
 115
the lth projection of the total e ective angular momentum of the three spins. For
 the AFM (frustrated) case, we measure the expectation of the symmetric W state
 witness, WW = (4 +
 p
 5)I^  2(J^ 2x + J^
 2
 y ) [67]. In both cases, as shown in  gure 6.7,
 we  nd that entanglement of the corresponding form develops during the adiabatic
 evolution.
 In macroscopic systems, the global symmetry in the Ising Hamiltonian (6.1) is
 spontaneously broken, and ground-state entanglement originating from this symme-
 try is expected to vanish for the non-frustrated FM case [6]. However, for the
 frustrated AFM case, the resultant ground state after symmetry-breaking (e.g.,
 j""#i + j"#"i + j#""i ) is still entangled. While spontaneous symmetry-breaking
 does not occur in a small system of three spins, we can mimic its e ect by adding a
 weak e ective magnetic  eld  Bx
 P
 i  
(i)
 x to the Hamiltonian during the adiabatic
 evolution[69, 70]. We experimentally observed that the frustrated ground state car-
 ries entanglement even after global symmetry is broken by using appropriate witness
 in the Ising model, and thereby establishes a link between frustration and an extra
 degree of entanglement [59].
 In the presence of transverse  eld, however, the disentanglement between mo-
 tional states and spin states becomes imperfect and is accumulated during the adi-
 abatic evolution. Fortunately, the residual entanglement does not have an in uence
 on the probabilities of spin product states measured in the direction of the Ising
 model axis [37]. Therefore we do not see the e ects on the experiments generat-
 ing the phase diagram shown in section 6.4. The in uence of spin-motion coupling
 becomes noticeable in the witness measurements, since the motional degrees of free-
 116
dom are traced out during the spin state detections. As shown in the blue curves of
  gure 6.7, the entanglement of the spin states is suppressed because of the remaining
 spin coupling to motions.
 6.6 Scalability of the quantum simulation
 As the number of spins N grows, the technical demands on the apparatus are
 not forbidding [59, 57]. In particular, the expected adiabatic simulation time for the
 spin models is inversely proportional to the ?critical? gap in the energy spectrum;
 for instance, in a fully-connected uniform ferromagnetic transverse Ising model in a
  nite-size system, this gap decreases as N 1=3 [71]. Scaling this system to accom-
 modate long ion chains will allow the investigation of critical behavior depending
 on the the system size, which is intractable in classical numerical simulation.
 We perform a benchmarking experiment where all interactions Ji;j are ferro-
 magnetic regardless of number of spins in the system by tuning the Raman beatnote
 detuning close to the COM mode. We carefully investigate deviations of experimen-
 tal simulations from theoretical predictions as the system size increases and discuss
 possible solutions overcoming the limitations. We observe a crossover from para-
 magnetic to ferromagnetic spin order, and the crossover sharpens as the number
 of spins is increased, prefacing the expected quantum phase transition [6] in the
 thermodynamic limit. We  nd that particular order parameters of the system can
 be quite insensitive to the imperfections of the quantum simulations, and the ex-
 traction of intensive variables such as the magnetization are much less susceptible
 117
to decoherence compared with full tomographic characterizations of the resulting
 quantum state.
 In the experiment, we produce the strength of Ji;j close to 1 kHz by setting
 the Raman beatnote detuning    1 + 4 1 , where  1 are the Lamb-Dicke param-
 eter and the frequency of COM motional mode, respectively. The strength of the
 couplings are pretty uniform among the pairs, since the COM mode dominates the
 interactions. The non-uniformity in the Ising couplings arises from other vibrational
 modes, which produce around 30% di erences in the strength at the most. We note
 that for larger detunings, the range of the interaction falls o even further with
 distance, approaching the limit Ji;j  1=ji jj3 for    1 [81, 72].
 The experiment is performed according to the adiabatic quantum simulation
 protocol, as described in section 1.3. We initially start with strong e ective trans-
 verse  eld By  5NJrms (N = 2, 3, ..., 9) after preparing the ground state of the
 By Hamiltonian. We transfer it to the Ising Hamiltonian with weak transverse  eld
 by exponentially ramping down By with time constant  = 80  s. We observe the
 evolution of state step by step as we proceed the experiment. For the measurements,
 we use the PMT and obtain the probability Ps of having s spins in state j"i from a
 histogram of  uorescence counts, constructed by the more than  1000  N times
 repetition of the experiments [73]. The  nal states of the adiabatic evolution are
 the superposition of two perfect FM states j""    "i and j##    #i , called GHZ
 state, since we implement FM couplings for all the pairs of spins and begin with
 the ground state of the transverse  eld. Therefore we measure the density matrix
 of the GHZ state as the simulation evolves. We also use other observables such as
 118
B y / NJ rms
 P(F
 M)
 (a) (b)
 P
 a
 rity
 ? (rad)
 N=2
 N=3
 N=4
 N=5
 N=2
 N=3
 N=4
 N=5
 N=6
 Figure 6.8: Experimental results of adiabatic quantum simulation
 depending on the system size. Here all pairs of spins have FM in-
 teractions. (a) The FM order P (FM) evolutions as the system size
 increases. Initially P (FM) starts 2=2N , since P (FM) is the prob-
 abilities of two states Pj""   "i and Pj##   #i over equally distributed
 2N states. As the spin-spin interactions J overpowers By (By ! 0),
 P (FM) are developed. Here J is the average strength of all inter-
 actions. The red, orange, yellow, green and blue dots represent the
 experiments for the total number of spins N=2, 3, 4, 5, and 6, re-
 spectively. Ideally P (FM) should be close to 1 at the end. However,
 P (FM) clearly reduces as the number of spins increases in the sys-
 tem from 2 to 6. The sources and amounts of errors are discussed in
 the text. (b) The parity oscillations for the  nal states of the simula-
 tion depending on the number of spins, obtained from the population
 di erence between the even number of j"i state and the odd num-
 ber of j"i state after applying analysis  =2 pulse and swipping its
 phase  . The contrast of the oscillations provide the lower bound of
 the coherence, the o -diagonal element of the density matrix for the
 GHZ state (j""    "i  j##    #i )=
 p
 2. The coherences decrease to
 0.8, 0.47, 0.35, 0.27 much faster than P (FM) as the number of spins
 increase from 2 to 5, because of spin-motional couplings during the
 simulation as discussed in the text.
 119
magnetization to characterize the time evolutions and the phase transitions.
 We analyze the reliability of the simulation depending on the system size by
 using the  delity of GHZ state, jh SIM jGHZi j2 = 12(P##   # + P""   ") + jC##   #;""   "j.
 Here P (FM) = P##   # + P""   " and the GHZ coherence jC##   #;""   "j is the coe cient
 of the j##    #i h""    "j in the density matrix [68]. We measure coherence of
 GHZ state by observing the contrast of the oscilating parity signals [Figure 6.8(b)],
 obtained by applying analysis  =2 pulse with di erent phase  and taking the dif-
 ferences in populations of the even number bright states and odd number bright
 states (P (0) + P (2) + ::: P (1) P (3) :::).
 Figure 6.8 shows the experimental results of the quantum simulation as the
 number of spins increases in the system. Initially P (FM) starts 2=2N , since P (FM)
 is the total probabilities of two states (j""    "i , j##    #i ) and the ground state
 of By
 P
  (j)y , j##    #i y are equally distributed in a total 2N states in the x-basis.
 Ideally P (FM) should be close to 1 at the end of the simulation. However, we
 observe that the  nal states are increasingly deviated from the ideal situation as
 the number of spins grows as shown in  gure 6.8(a). We also observe that the GHZ
 coherence decreases much more rapidly than P (FM) shown in  gure 6.8(b). After
 6 spins, we did not measure any signi cant GHZ entanglement for the  nal state
 due to the large suppression of the coherence compared to the populations. In the
 following subsections, we discuss the reason of these deviations and we summarize
 the expected experimental imperfections in quantum simulations as the system size
 grows.
 120
6.7 Scaling of imperfections
 6.7.1 Spin-motion coupling
 As discussed in section 3.4, the e ective spin-spin coupling Hamiltonian that
 we assume to describe the system is only valid when the detuning of the beatnote
 is much larger than the sideband strength. As our detuning is limited by the coher-
 ence time of the system, i.e. the need for strong coupling strengths, the spin-motion
 entanglement does not completely vanish during the simulation. In fact, the spin-
 motion coupling increases in the presence of the transverse  eld. The transverse
  eld mixes the spin states along the axis where the spin-dependent force is applied,
 therefore this coupling induces phonon excitations, modifying the  nal state from
 the ideal GHZ state to (j""    "i j i  j##    #i j  i )=
 p
 2. Here j i is a mo-
 tional coherent state and hni = j j2 increases as the amount of state mixing grows.
 According to numerical calculations corresponding to the experiment, hni increases
 to  0:5 for the  ve spin experiment. The e ective phonon excitation occurs pri-
 marily in the early stages of the simulation evolution, where the strength of the
 transverse  eld is much larger than the frequency of spin rotations. In the exper-
 imental conditions, the Raman beatnote is detuned  4 1 from the COM mode,
 which results in an increment of j j, because of the required large initial transverse
  eld. However, the population P (FM) and the evolution is not sensitive to these
 spin-motion couplings.
 121
6.7.2 Diabaticity
 The  nite ramping speed of parameters in the Hamiltonian leads to excitations
 out of the ground state and can lead to oscillations in the observed order parameter.
 This diabaticity in the evolution, along with errors in the initialization to the original
 ground state is estimated to suppress the  nal value of P (FM) by  4% for N = 5
 shown in the orange line of  gure 6.9(a). As discussed in section 6.7.2, the diabaticity
 is related to the minimum gap over the trajectory of the Hamiltonian as well as the
 ramping time. We note that the gap between the ground and  rst-excited state of
 the fully-connected uniform FM model scales as N 1=3, implying that the simulation
 time by a factor of ten when the number of spins grows by a factor of one thousand.
 6.7.3 Spontaneous Emission
 One of major error sources is the spontaneous emission from Raman beams
 which amounts to a  10% spontaneous emission probability per spin in 1 ms for a
 detuning of   2.7 THz [74]. Spontaneous emission dephases and randomizes the
 spin state, and thus introduces entropy into the system. In addition, each sponta-
 neous emission event populates other states outside of the Hilbert space of each spin
 with a probability of 1=3. Spontaneous emission errors grow with increasing system
 size, which also suppresses P (FM) order with increasing N , as seen in  gure 6.9.
 We theoretically estimate the suppression of P (FM) due to spontaneous emission
 by averaging over quantum trajectories and solving density matrix equations to be
  5% for N = 2 spins and  13% for N = 5 spins. The error contributions are the
 122
same for both populations and coherences.
 6.7.4 Intensity  uctuations
 Intensity  uctuations on the Raman beams during the simulation induce an
  uctuating AC Stark shift on the spins. The AC stark shifts produce an imbal-
 ance on the blue and red sideband detuning, which give rise to imperfections in the
 spin-spin interactions. According to our numerical calculations, the imbalance  uc-
 tuations of  150 Hz ( 1:5% intensity  uctuations in our experimental conditions)
 can explain the suppressions of P (FM) at the end of the experimental simulations
 shown in  gure 6.9. We investigate the non-uniformity of the laser beams that in-
 duces di erent AC shifts on the spins by measuring the AC shift on each location,
 which is introducing position dependent Bz- eld with at most  200 Hz di erence.
 According to the numerical calculation, the additional small Bz- eld does not no-
 ticeably suppress P (FM) .
 6.7.5 Detection Errors
 Imperfect spin detection e ciency contributes 5 10 % uncertainties in P (FM).
 Fluorescence histograms for P (0) and P (1) have a  1% overlap (in detection time
 of 0.8 ms) due to o -resonant coupling of the spin states to the 2P1=2 level. This
 prevents us from increasing detection beam power or photon collection time to sep-
 arate the histograms. In the experiment, the average photon number from a single
 bright spin is 12. The uncertainty in  tting the observed  uorescence histograms to
 123
B y / NJ rms
 P(F
 M)
 (a) (b) (c)
 Ground State
 Experimental ramping
 + Spontaneous emissions
 + Intensity fluctuations
 Data
 N=5
 P(FM) Coherence
 N=2 N=3 N=4 N=5
 Errors
 Data
 Diabaticity
 Spontaneous Emission
 Intensity Fluctuations
 Spin - Motion coupling
 Errors
 Data
 Diabaticity
 Spontaneous Emission
 Intensity Fluctuations
 N=2 N=3 N=4 N=5
 Figure 6.9: (a) The comparison of experimental results of P (FM)
 to the theoretical expectations including various imperfections of ex-
 periments for the 5 spins. The black line represents the evolution of
 the perfect adiabatic evolution, and the dashed line shows the actual
 time evolution including the actual ramping of the transverse  eld
 with the imperfections of initial state preparations. The red line is
 obtained from the theory with spontaneous emissions and the green
 line is calculated by adding intensity  uctuations of 1.5 %. The over-
 all populations of P (FM) are in agreement with the green theoretical
 curve including all the above mentioned imperfections. The horizon-
 tal shift By=J comes from the inaccuracy of the calibration that is
 not fully understood yet. (b) The amount of errors in P (FM) (the
 diagonal parts of the density matrix) from the ideal ground states for
 the case of N=2,3,4 and 5. (c) The amount of errors in coherence (the
 o -diagonal parts). In both (b) and (c), the green squares represent
 the total errors measured in the experiment and the bars illustrate
 the numerically estimated errors from the various sources of experi-
 mental imperfections. The black, red and green bars show the error
 amounts from non-perfect adiabatic evolutions, spontaneous emis-
 sions and intensity  uctuations. The blue bars stand for deviations
 from the spin-motion couplings. The spin-motion coupling reduces
 the coherence signi cantly as the system size grows, while it dose not
 have any in uence on the P (FM).
 determine P (s) increases. The histograms are also a ected by the intensity  uctua-
 tions of detection laser beams and the  nite widths. This problem can be eliminated
 by detecting each spin individually with an imaging detector.
 Figure 6.9 (a) shows the experimental results and the theoretical calculations
 including a few steps of imperfections discussed above for N=5 spin case as an exam-
 ple. In the population P (FM) also shown in  gure 6.9 (b), the main deviations come
 124
from the spontaneous emissions and the intensity  uctuations. As discussed, small
 amounts of intensity  uctuations degrade the performance of the experimental sim-
 ulations. One of the solutions for those imperfections is to implement a high power
 laser with a detuning far from the 2P energy levels, which would minimize sponta-
 neous emission while maintaining the same level of Ising couplings. This would also
 allow versatility in varying the Ising interaction (together with the e ective external
  eld) during the simulation, as the di erential AC stark shift between spin states
 is negligible for a su ciently large detuning. The coherence time increases in the
 absence of spontaneous emission, allowing for a longer simulation time necessary to
 preserve adiabaticity as the system grows in size. Recently Raman transitions have
 been driven using a mode-locked high power pulsed laser at a wavelength of 355
 nm, which is optimum for 171Yb+ wherein the ratio of di erential AC Stark shift to
 Rabi frequency is minimized and spontaneous emission probabilities per Rabi cycle
 are < 10 5 per spin [51, 75].
 Figure 6.9(c) shows the measured and numerically estimated errors in the mag-
 nitude of coherence, jC##   #;""   "j, at the  nal state of the quantum simulation. We
 can clearly see that the errors of the coherence much more rapidly increase than
 those of P (FM). According to the numerical study, the dominant source for the
 errors of jC##   #;""   "j is the non-vanishing spin-motion couplings in the experimental
 simulation, shown as the blue area of the  gure 6.9(c). In principle, we can elim-
 inate the e ect of the spin-motion coupling by alternating the transverse  eld and
 Ising interactions [76]. The adiabatic evolution can be discretized by the Trotter
 125
expansion written by
 UTIM( ) = T exp
  
 i
 Z  
0
 dt0 [HI(t
 0) +HB(t
 0)]
  
 [exp ( iHI( =N)) exp ( iHB( =N))]
 N ; (6.2)
 where HI and HB are the Ising Hamiltonian and the transverse  eld Hamiltonian,
 respectively. In the experiment, we can choose  =N as the special duration (1/ ),
 where the spin-motion couplings vanish. We can also reduce the errors of the Trotter
 expansion by increasing the Raman beatnote detuning  from the motional mode.
 In condensed matter, phase transitions are typically described in terms of
 order parameters or correlations instead of the density matrices of particular states.
 We use a absolute magnetization hjmji =
 P
 m jm=N jP (m) per site along the Ising
 direction. Actually we rescale the magnetization hjmji from 0 to 1 regardless of the
 number of spins to make a fair comparison even for small size systems. We  nd
 that the deviation between experiment and theory for this order parameter does
 not grow substantially as the system is scaled up in size. Figure 6.10 shows the
 scaled magnetization, hjmSji for N = 2 to N = 9 spins, showing a  nal value of
  80% ( gure 6.10) regardless of number of spins. Moreover as shown in  gure
 6.10(a) we observe the sharpening of the crossover curves from paramagnetic to
 ferromagnetic spin order with increasing system size. The continued sharpening of
 this transition is of great interest to the understanding of  nite size e ects in phase
 transitions and can be used to compare various numerical techniques in studying
 critical phenomena.
 126
B y / NJ rms
 B y / NJ rms
 ?|
 m
 S
 |?
 ?|m
 S |?
 (a) (b)
 N=2
 N=5
 N=9
 N = 2 Data
 N = 9 Data
 N = 2 Theory
 N = 9 Theory
 Figure 6.10: (a) Scaled average absolute magnetization per site,
 < jmSj > vs B=NJrms is plotted for N = 2 to N = 9 spins. As
 B=NJrms is lowered, the spin ordering undergoes a crossover from a
 paramagnetic to ferromagnetic phase. The crossover curves sharpen
 as the system size is increased from N = 2 to N = 9, anticipating a
 QPT in the limit of in nite system size. The oscillations in the data
 arise due to imperfect initial state preparation and non-adiabaticity
 due to  nite ramping time. (b) Magnetization data for N = 2 spins
 (circles) is contrasted with N = 9 spins (squares). The data deviate
 from unity at B=NJrms by  20%, predominantly due to spontaneous
 emissions in Raman transitions and intensity  uctuations of Raman
 laser beams, as discussed in the text. Here, the theoretical time evo-
 lution curves (red line for N = 2 and black line for N = 9 spins) are
 calculated by averaging over 10,000 quantum trajectories.
 127
6.8 Conclusion and Outlook
 Trapped atomic ions represent a promising platform for the quantum sim-
 ulation of intractable Hamiltonian systems. There have been several theoretical
 proposals in this direction, largely following schemes in the realm of quantum com-
 puting, and recent experiments have shown that the system can be scaled to a
 degree where all classical simulations become impossible [59, 70, 77]. This paper
 has shown both theoretically and experimentally that the quantum simulation of
 quantum magnetism and the emergence of spin order can be controlled through
 external laser beams up to 9 spins and can further be scaled to much larger num-
 bers of spins. The stable con nement of larger numbers of ions may require novel
 ion trap architectures such as anharmonic axial potentials [78] for a linear chain
 or two-dimensional trap geometries [79, 80], but there are no known fundamental
 limitations in this scaling. As discussed here, for a  xed level of total laser power,
 the errors associated with decoherence from spontaneous Raman scattering from
 the lasers is expected to grow only as N1=3 for the linear chain, holding errors from
 phonon creation and diabatic transitions to excited states at  xed levels. Alterna-
 tively, all of these errors can be held at a  xed value independent of N so long as the
 laser power increases by N1=3. In either case the required time for adiabatic ramping
 grows as N1=3, so slowly drifting errors such as (real) magnetic  elds and motional
 heating of the ions must be kept under control for very large N as discussed in the
 supplementary information of Ref. [59]
 This system can also be extended to Heisenberg or XYZ spin models [81] or
 128
spin-1 systems by adding a few more laser beams. As the system grows, the trans-
 verse motional modes that mediate the Ising couplings can give rise to higher levels
 of frustration and complex phases of magnetic ordering. For instance, by preparing
 a ground state of a highly frustrated collection of trapped ion spins, it should be
 possible to create localized topological excitations and guide their transport through
 the system [82]. This example of topological matter is of great interest for the ro-
 bust representation and manipulation of quantum information [83, 84]. But more
 generally, the trapped ion system is poised to be the  rst to determine ground state
 features of Hamiltonians where no solution can be obtained otherwise.
 129
Chapter 7
 Quantum Simulation of the Devil?s Staircase
 7.1 Introduction
 Recent e orts in quantum simulation, motivated by its promise as a tool for
 advancing understanding of condensed matter systems, solving di cult optimization
 problems, and simulating other physical systems while providing an in-situ insight
 previously unattainable have seen signi cant advances. These include simulation
 of the nearest -neighbor anti-ferromagnetic Ising model [85], and the long-range
 anti-ferromagnetic Ising model. Recent theoretical interest in the simulation of the
 complete Devil?s Staircase [86] using cold, trapped ion quantum simulators [87] have
 motived us to simulate this system for six and ten spins.
 In this chapter I report a quantum simulation experiment1, similar to those in
 chapter 6. In this experiment a linear chain of trapped, cold ions are irradiated by
 lasers far detuned from resonant transitions to simulate and control an Hamiltonian
 of AFM coupled Ising spins with long range, frustrated interactions. We add to
 this e ective interaction a combined axial and transverse simulated  elds, and by
 scanning the strength of the axial  eld we explore N/2 phase transitions in the
 ground state of the zero transverse  eld spin chain of N spins. Site resolved imaging,
 as described in section 4.4.8 allows us to extract ground state order parameters and
 1Manuscript in preparation.
 130
state probabilities, ampli ed over the background by the post-processing techniques
 of section 4.4.8. We compare the experimental results to theoretical predictions.
 7.2 Overview
 We perform a quantum simulation of the AFM Ising model with a tunable
 axial biasing  eld Ba:
 H =
 NX
 i;j
 Jij 
i
 x 
j
 x +Bx
 NX
 i
  ix +By
 NX
 i
  iy (7.1)
 Where H is the simulated Hamiltonian of the system,  ix the Pauli spin operator
 on spin i, and Ji;j is the spin-spin coupling between spin i and spin j as de ned in
 3.22. The j"i and j#i states are de ned as presented in 2.4, and the ions are radially
 con ned with a transverse collective COM mode frequency !1 = 2 4:863 MHz ,
 and a tilt frequency !2 = 2 4:813 MHz . This Hamiltonian has been Identi ed
 as a physical model for several real world systems, such as graphite intercalation
 compounds, and exhibits the much investigated complete Devil?s staircase behavior
 for a macroscopic number of spins [88].
 Just as in the experiments described in chapter 6, we will initialize the spin
 system in the ground state of a trivial Hamiltonian, where the magnetic  elds over-
 whelm the J component of the Hamiltonian. Unlike the simulations of chapter 6, we
 cannot simply rotate the spins to align with transverse  eld By, rather we will have
 to apply a Raman pulse that rotates the spins into the superposition of Bx + By.
 Thus, for each value of Bx the phase of the initialization pulse relative to the simu-
 131
lation frequency phases will be di erent. For our selected simulation pulse phases,
 the phase of the initialization pulse will be
  = 1800  tan 1(Bx=By) (7.2)
 As all components of J are positive for our red beatnote detuning from the transverse
 COM, initializing in the ground state (along the total B  eld) will simulate AFM
 coupling.
 After initializing, we will apply all components of the Hamiltonian simultane-
 ously, and proceed to ramp down By with an exponential envelope (with no o set).
 The duration of the time constant of the envelope is a compromise - the ideal time
 constant would satisfy the adiabatic condition [89] with a large safety margin. How-
 ever, we are constrained by the coherence time of the ions, which for us was found
 to be roughly 3 ms . Therefore, we chose our time constant so as to achieve as low
 a B- eled as possible in the available time without causing severe oscillations or
 excitations towards the end of the simulation.
 Finally, we apply a Raman rotation to measure the spin state in the mea-
 surement basis (the z-axis of the Bloch sphere). Repeating this experiment with
 di erent Bx values allows us to map the phase diagram of this Hamiltonian.
 7.3 Experiment
 As described in section 3.4, we control the dynamics [57] by applying far
 detuned global Raman beams with a wavenumber di erence  k along the transverse
 132
Coupling Strength
 J1;2 660 Hz
 J1;3 368 Hz
 J1;4 235 Hz
 J1;5 158 Hz
 J1;6 106 Hz
 J2;3 621 Hz
 J2;4 373 Hz
 J2;5 242 Hz
 J2;6 158 Hz
 J3;4 609 Hz
 J3;5 373 Hz
 J3;6 235 Hz
 J4;5 621 Hz
 J4;6 368 Hz
 J5;6 660 Hz
 Table 7.1: Calculated spin-spin couplings for 6 ions with our trap,
 detuning, and laser intensity parameters
 (perpendicular to chain alignment - \x" direction) direction of the ion chain.
 To create the the spin-spin coupling matrix J, we detune two Raman beams
 80 KHz to the blue and red of the COM mode, so that the beatnote  is red of
 the COM mode. This is done while verifying that the strengths of the beams drive
 sideband transitions on a single ion with equal strength, are symmetrically detuned
 from the carrier transition, and the beams are centered on the center of the spin
 chain (the position of a single ion in the trap).
 To diagnose the intensity of our 355 nm center wavelength 80 MHz pulsed laser
 Raman beams, we measure a two ion spin-spin coupling strength of 66.5 KHz . As-
 suming equal intensity across our ion chain, the spin-spin coupling matrix J is
 calculated and presented in table 7.1. The single ion RSB or BSB strength for this
 intensity on resonance is calculated to be 23 KHz , so our detuning   3  .
 133
2 3 4 5 6
 0.0
 0.1
 0.2
 0.3
 0.4
 0.5
 0.6
 Ion Index i
 J 1
 ,
 iHKH
 zL
 Spin-Spin Coupling Decay
 Figure 7.1: Power law  t for the for the  rst ion. The curve was
  tted with only the exponent as a parameter.
 As in our other experiments where we tuned to the red of the COM, the
 couplings are decaying according to a power law with exponent  
Ji;j  
Ji;i+1
 ji jj 
(7.3)
 This exponent  depends only on trap conditions - it is a feature of the motional
 mode bandwidth. For our current trap conditions and six ions,  = 0:976, as
 shown in  gure 7.1. In previous work, we have explored the e ect of changing this
 parameter on the correlations of the  nal AFM state for Bx = 0.2  characterizes
 the range of the spin-spin coupling, and for longer range AFM interactions we can
 expect to see more frustration in the ground state with smaller energy gaps.
 2Accepted for publication in \Science", R. Islam et al Emergence and Frustration of Magnetism
 with Variable-Range Interactions in a Quantum Simulator
 134
The characteristic power law decay arises due to the non-negligible contribu-
 tion of the tilt mode as well as smaller contributions from the remaining trans-
 verse x-modes. y-modes are suppressed by beam alignment and control of the trap
 principal-axes, so that the y-mode sidebands are at least 10 times suppressed com-
 pared to the x-modes. We further suppress the y COM mode by lowering its fre-
 quency relative to the x COM mode, by adjusting trap anisotropy.
 As described in 4.4.6, to generate the Raman beams we use two lasers, mod-
 ulated by two AOMs. The  rst AOM is driven by an oscillator stabilized by an
 error signal from the beating of a Microwave source and the frequency component
 of the pulsed laser near the qubit frequency. The second AOM is driven by an AWG.
 During the simulation, the AWG drives the AOM with three di erent frequencies:
 one to drive the qubit Rabi oscillations, and two symmetrically detuned from this
 central frequency to drive the M lmer-S renson spin-spin coupling gate. It is only
 when these frequencies interfere on the ion with the light modulated by the  rst
 AOM that transitions are driven.
 The AWG and the driver of the stabilized AOM are phase locked to a Rubidium
 clock, outputting a reference signal at 10 MHz . We select and de ne the Pauli vector
 component via the relative phase of the di erent beatnotes. We set the phases as
 elucidated in table 7.2
 As demonstrated in section 3.4 it is only the phase di erence of the BSB
 and RSB  im that a ects the M lmer-S renson spin-spin coupling. Therefore, we
 arbitrarily set the phase of the RSB to 0, and set only the phase of the BSB to
 control the axis of the interaction. As  im is equal to half the di erence of BSB and
 135
Pulse Phase Result
 Initialization  Bx xi +By 
y
 i
 Bx 900  xi
 By 00  
y
 i
 RSB 00 Combined with BSB we get
 BSB 1800 Jij xi  
x
 j
 Final 900  yi
 Table 7.2: Pulses for axial simulation experiment. Initial pulse  is
 de ned in 7.2. For example, with no axial  eld  = 1800. The  nal
 rotation of the spins into the measurement basis can alternatively be
 set to 900, as this will only result in  ipping the spins in the  nal
 state and does not change the physics.
 RSB phases, setting  to 1800 sets the interaction axis to be the x-axis. The control
 panel for the AWG chapter phases, frequencies and amplitudses is shown in  gure
 7.2.
 All the lasers used in the experiment are global - we do not use individual
 addressing. This simpli es optical setup and reduces sensitivity to beam steering,
 however this prevents us from measuring the elements of the density matrix that
 require operations that are not a global rotation.
 The experiment proceeds as follows. The ions are Doppler cooled for  
3 ms , and optically pumped to the j#i state by a 8 s laser pulse resonant with
 the 2S1=2 jF = 1i ! 2P1=2 jF = 1i transition, as shown in  gure 2.3.
 30 cycles of alternating red-sidebands resonant with the x COM mode and 3 s
 long optical pumping pulses reduce the occupation number of this phonon mode to
  0:1 phonons, so that state is initialized in the spin as well as motion sub-spaces.
 We then rotate our spins to the trivial ground state of a Hamiltonian domi-
 nated by a transverse magnetic  eld, where our initial transverse  eld By is 5 times
 136
Figure 7.2: Control panel for quantum simulation AWG waveform
 in the Labview control program. The phase of J is the phase of the
 BSB.
 137
larger than the largest spin-spin coupling JMAX .
 We ramp down the transverse  eld By with an exponential time constant of
 600 s . This value was found by numerically evolving the Schr odinger equation
 (SE) for our experimental parameters and maximizing the resemblance of the mag-
 netization order parameter of the evolving state to that of the adiabatic perfect
 ground state of Hamiltonian 7.1 at time t of the simulation. The magnetization
 order parameter is de ned here as M = 1=2 +
 nj"i  nj#i
 2N .
 When the ramping is complete, we rotate our spins back to the measurement
 basis, and measure each spin using a resonant beam with a Princeton PI-MAX 3
 ICCD with a 3ms exposure time, as described in 4.4.8. Thus, we are able to measure
 the distribution of spin eigenstates for this process, with a  93% readout  delity
 per ion. This  delity is reduced from the theoretically attainable  delity of  99:5%
 by electronic readout noise and the 0.23 numerical aperture of the imaging system.
 In order to correct the biasing of the state probabilities, we redistribute them as
 described in 4.4.8.
 For 6 ions, we map out a phase diagram by stopping the simulation at three
 di erent  nal times of 306 s , 960 s and 3 ms corresponding to By=JMAX values of
  3:00; 1:01 and 0.034. We measure the  nal state 4000 times for 61 equally spaced
 values of Bx=JMAX between 0 and 4. Figure 7.3 displays the phase diagram of
 the magnetization order parameter. The surface plot is the perfect adiabatic order
 parameter for the ground state, whereas the points are experimental results. The
 theoretical plot shows three  rst-order [88] phase transitions at By=JMAX = 0. We
 have also attempted to repeat the experiment for points close to the phase transition
 138
0 0.5 1
 Magnetization
 Figure 7.3: Magnetization Phase Diagram for 6 Ions.
 139
once our measurements indicated their location. However, our control of the system
 parameters did not seem to be stable enough over the duration of the data taking.
 Our dominant measured states at By=JMAX  0 are displayed in  gure 7.4 and
 further investigated. Comparing only the dominant states to theoretical evolution
 of Schr odinger equation (solid lines) we see the data for the dominant states closely
 follows. However, the simulation is not su ciently adiabatic and the data is too
 noisy to clearly show the sharpness of the phase transitions.
 This is accentuated when we compare the magnetization parameter for the
 adiabatic solution, the evolved solution and the data, as in  gure 7.5.
 The lack of adiabaticity at the phase transition can be investigated theoreti-
 cally from multiple aspects. One aspect is that of the adaibaticity parameter, which
 for our system roughly corresponds to the inverse of the minimum energy gap be-
 tween the ground state and the  rst coupled excited state for the trajectory of the
 simulation [22]. For the Hamiltonian 7.1, there are eigen-energy crossings zero at
 the phase transitions, as can be seen in  gure 7.6 and 7.7.
 Another interesting feature of the phase transitions is its relation to an in-
 verse pseudo-temperature. Currently, there is ongoing theoretical and experimental
 investigation of thermalization in closed, coherent quantum systems [90]. Accord-
 ing to the currently developing Eigenstate Thermalization Hypothesis (ETH)3 [91],
 some classes of closed quantum systems display steady-state observables that corre-
 spond to the micro-canonical expected observables for analogous classical systems.
 For the Hamiltonian at hand, we will see that in the low axial  eld limit the state
 3I would like to thank Chao Shen for introducing me to this work.
 140
0 1 2 3 4
 0.0
 0.2
 0.4
 0.6
 0.8
 1.0
 Bx  JMAX
 Probabilit
 y
 States vs Bx
  
Probabilit
 y
 fl ? fl fl ? fl
 fl fl fl fl fl fl
 fl ? fl ? fl ? + ? fl ? fl ? fl
 fl fl ? fl ? fl + fl ? fl ? fl fl
 fl fl fl ? fl fl + fl fl ? fl fl fl
 fl fl fl fl fl ? + ? fl fl fl fl fl
 Figure 7.4: Results and simulation of dominant states. The phase
 transitions in the adiabatic limit are marked with dashed lines. The
 non-adiabaticity causes the states to bleed out of their phase in the
 adiabatic case. Some excited states are populated signi cantly. The
 red and green curves and data are noticebaly excited states for their
 respective regimes. The  rst excited state for each stair is a kink in
 the ground state - the spins are all  ipped relative to the ground state
 at a point in the spin chain that requires the least energy. Typically
 this is close to the edge chain, as the couplings there are weakest.
 141
0 1 2 3 4
 0.0
 0.2
 0.4
 0.6
 0.8
 1.0
 Bx  JMAX
 Magnetizatio
 n
 Figure 7.5: 6 ion staircase. Black solid line is perfect adiabatic magne-
 tization, red solid line is numerically integrated Schr odinger equation,
 and dots are calculated from data.
 142
0 1 2 3 4
 - 10
 - 5
 0
 5
 10
 15
 20
 Bx JMAX
 EHKH
 zL
 Figure 7.6: Complete eigen-energy spectrum for our Hamiltonian. On
 the left hand side, the pure AFM case, the lowest energy states are
 the AFM states, which are degenerate, and the highest states are the
 FM states. As the axial bias is increased, the FM states degeneracy
 is lifted. The FM state counter aligned to the  eld increases in energy
 and remains the highest, while the FM state aligned with the  eld
 becomes the lowest energy state at high axial  eld limit. At the high
 axial  eld limit, the energy states converge into 6 degenerate levels,
 corresponding to number of spins aligned with the  eld.
 (a) (b)
 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5- 8
 - 7
 - 6
 - 5
 - 4
 - 3
 - 2
 Bx JMAX
 EHKH
 zL
 0 1 2 3 4
 0.0
 0.1
 0.2
 0.3
 0.4
 0.5
 Bx  JMAX
 DE
 HKH
 zL
 Figure 7.7: (a) Low lying energy levels and and crossing points. The
 dashed lines mark the energy crossings. (b) Energy di erence between
 the two lowest lying states. These lead to the  rst-order transitions
 seen in 7.5
 143
probabilities will be occupied with a probability that is roughly proportional to an
 exponential function of the energy of the eigenstate:
 Pi = e
  (Ei E0)T (7.4)
 Where E is the energy of the state, E0 is a normalizing factor, and T is the inverse
 pseudo-temperature. By  tting to the simulated histograms, we can extract T for
 the  nal states at all Bx values and By = 0, as shown in  gure 7.8. For six ions,
 this treatment is not very convincing, as only the  rst peak aligns itself with the
  rst phase transition. Nevertheless, I will use this treatment to de ne a pseudo-
 temperature lowering  lter. The  lter will succeed to amplify the dominant ground
 state in the regime where the pseudo-temperature picture is not valid as well, as
 long as the most dominant state is the true ground state - a reasonable assumption
 when By = 0.
 We de ne an amplifying parameter A, which for A=1 leaves the probabilities
 unchanged, and for increasing A will suppress the weaker states compared to the
 most dominant states. For each state probability for a given Bx and By, we de ne
 a new probability P 0i = P
 A
 i . Then
 PN
 i=1 P
 0
 i is renormalized to 1. The e ect can be
 compared to removing excess energy from the system, as can be seen in  gure 7.10.
 In this  gure, we compare the e ect of increasing A for a simulated  nal evolved
 state to increasing the total simulation time while keeping the ratio of simulation
 time to exponential decay time constant  xed. Starting with a simulated state from
 the evolved Schr odinger equation with 0.8 s time constant and our experimental
 144
(a) (b)
 -2 0 2 4
 10-7
 10-6
 10-5
 10-4
 0.001
 0.01
 0.1
 EHKHzL
 Probabilit
 y
 -10 -5 0 5 10 15 20
 10-8
 10-6
 10-4
 0.01
 EHKHzL
 Probabilit
 y
 (c)
 0 1 2 3 4
 0
 2
 4
 6
 8
 Bx  JMAX
 Invers
 e
 Temperatur
 eH10
 -3
 K
 B-
 1 L
 Figure 7.8: (a) Exponential  t of probability of state as a function
 of state energy, for Bx = 0. The  t slope is the inverse pseudo-
 temperature. (b) Exponential  t for Bx=JMAX = 4. (c) The temper-
 ature coe cient for all  ts. The  t fails rather quickly after showing
 a correspondence to the  rst phase transition. Phase transitions are
 marked with dashed lines.
 J, we gradually increase A, and then compare the individual states for increasing
 Alpha and decay time. Scaling the A axis for all states and values of Bx equally, we
 see that for most values of Bx the ampli ed and the slowed evolution states match
 well.
 Applying the  lter to our data with increasing ampli cation (or \cooling"),
 we can retroactively slow down the simulation. Figure 7.9 demonstrates this e ect.
 We repeat the same experiment, this time for 10 ions, and only for a  nal
 simulation time of 3 ms . Our initial value of By is set to 3Jmax, and we perform the
 145
0 1 2 3 4
 0.0
 0.2
 0.4
 0.6
 0.8
 1.0
 Bx  JMAX
 Magnetizatio
 n
 A=1
 0 1 2 3 4
 0.0
 0.2
 0.4
 0.6
 0.8
 1.0
 Bx  JMAX
 Magnetizatio
 n
 A=2
 0 1 2 3 4
 0.0
 0.2
 0.4
 0.6
 0.8
 1.0
 Bx  JMAX
 Magnetizatio
 n
 A=3
 0 1 2 3 4
 0.0
 0.2
 0.4
 0.6
 0.8
 1.0
 Bx  JMAX
 Magnetizatio
 n
 A=4
 Figure 7.9: Power  lter with increasing strength on 6 ion magnetiza-
 tion for Bx = 0. When  lter strength is A=1, the state probabilities
 are unchanged. When A=4, the adiabatic magnetization staircase
 emerges from the data.
 adiabatic simulation procedure for Bx=JMAX = 0! 5:5, in step of 0.05, and repeat
 each simulation 4000 times. As J is of the same order of magnitude and range as
 before, as shown in  gure 7.11, we can expect to see smaller energy gaps. As we
 cannot extend simulation time, the expected e ect will be much worse diabaticity,
 i.e. stronger excitation of low lying energy states. We can also expect a lower ratio
 of signal to noise, as decoherence and detection error for overall state will increase
 for a larger number of spins. This can be seen in a comparison of our detected states
 to simulated states in  gure 7.12. Here we must rescale the y-axis for the data to
 compensate for di erent scales for data and theory.
 Interestingly, for 10 ions the pseudo-temperature treatment works far better.
 As seen in  gure 7.13, the exponential  t lies next to many more points, improving
 146
1 2 3
 0.30
 0.35
 0.40
 0.45
 0.50
 1 2 3
 t HmsL
 Probabilit
 y
 o
 fStat
 e
 A
 fl ? fl ? fl ? , Bx  JMAX = 0
 1 2 3
 0.00
 0.02
 0.04
 0.06
 0.08
 1 2 3
 t HmsL
 Probabilit
 y
 o
 fStat
 e
 A
 fl ? fl ? ? fl , Bx  JMAX = 0
 1 2 3
 0.20
 0.22
 0.24
 0.26
 1 2 3
 t HmsL
 Probabilit
 y
 o
 fStat
 e
 A
 fl fl fl ? fl fl , Bx  JMAX = 2.1
 1 2 3
 0.00
 0.01
 0.02
 0.03
 0.04
 0.05
 0.06
 0.07
 1 2 3
 t HmsL
 Probabilit
 y
 o
 fStat
 e
 A
 fl fl ? fl ? fl , Bx  JMAX = 2.1
 Figure 7.10: Figure above demonstrates e ect of power  lter versus
 improved adiabaticity, for two states for two values of Bx. Red line
 is power  lter on 0.8 s evolved state. Blue line is evolved state with
 increasing exponential decay constant  , with  xed ratio of  Tfinal .
 For some values of Bx this procedure will fail, as close to a phase
 transition there may be population oscillations.
 147
(a) (b)
 2 4 6 8 10
 0.0
 0.1
 0.2
 0.3
 0.4
 0.5
 0.6
 Ion Index i
 J 1
 ,
 iHKH
 zL
 Spin-Spin Coupling Decay
 0
 5
 10
 i
 j
 0.0
 0.2
 0.4
 0.6
 Ji, j KHz
 Figure 7.11: (a) Power law decay for 10 ions, shown for spin 1. (b)
 The J coupling matrix. Note that it is symmetric across the chain
 and across the diagonal. The diagonal elements are meaningless.
 the validity of the  t. When we extract all the pseudo-temperature coe cients for
 the Bx domain of the simulation, we see more temperature cusps corresponding to
 the adiabatic regime phase transitions.
 Applying the same  ltering method to our 10 ion magnetization data, we see
 how the staircase emerges again. However, for this data the  ltering is much more
 aggressive, as could be expected for a more diabatic simulation. It is less successful
 as well, which is likely due to increased overall noise and decoherence [92].
 7.4 Conclusion
 Despite the technical challenges faced by this experiment, the sharp phase
 transitions at near zero transverse  eld can be observed. Here we rely on the fact
 that when our transverse  eld is extinguished, the dominant state will be the ground
 state. As can be seen in  gure 7.14, the staircase signal can be extracted from the
 148
0 1 2 3 4 50.0
 0.1
 0.2
 0.3
 0.4
 0.5
 0.6
 0.7
 0
 0.1
 0.2
 0.3
 Bx  JMAX
 Pro
 bHSimulatio
 nL
 States vs Bx
 Pro
 bHDat
 aL
  
Pro
 bHSimulatio
 nL
 fl ? fl ? fl fl ? fl ? fl
 fl fl ? fl fl fl fl ? fl fl
 fl fl fl fl fl fl fl fl fl fl
 ? fl ? fl ? fl ? fl ? fl + fl ? fl ? fl ? fl ? fl ?
 fl ? fl fl ? fl ? fl ? fl + fl ? fl ? fl ? fl fl ? fl
 fl ? fl fl fl ? fl fl ? fl + fl ? fl fl ? fl fl fl ? fl
 fl fl ? fl fl ? fl fl ? fl + fl ? fl fl ? fl fl ? fl fl
 fl fl fl ? fl fl fl ? fl fl + fl fl ? fl fl fl ? fl fl fl
 fl fl fl fl fl ? fl fl fl fl + fl fl fl fl ? fl fl fl fl fl
 fl fl fl fl fl fl fl fl fl ? + ? fl fl fl fl fl fl fl fl fl
 Figure 7.12: Results and simulation of dominant states. The phase
 transitions in the adiabatic limit are marked with dashed lines. The
 non-adiabaticity causes the states to bleed out of their phase in the
 adiabatic case. Dashed lines mark phase-transitions, found from en-
 ergy crossings. Here the y-axis of the data and theory are to di enret
 scales, as the data is suppressed by noise and increased decoherence
 [92]
 149
(a) (b)
 -2 0 2 4 6
 10-6
 10-4
 0.01
 EHKHzL
 Pro
 b
 Prob VS EHKHzL, Bx = 0.
 -5 0 5 10
 10-10
 10-8
 10-6
 10-4
 0.01
 EHKHzL
 Pro
 b
 Prob VS EHKHzL, Bx = 2.2
 (c) (d)
 -15 -10 -5 0 5
 10-10
 10-8
 10-6
 10-4
 0.01
 1
 EHKHzL
 Pro
 b
 Prob VS EHKHzL, Bx = 4.95
 0 1 2 3 4 5
 0
 2
 4
 6
 8
 Bx  JMAX
 Invers
 e
 Temperatur
 eH10
 -3
 K
 B-
 1 L
 Figure 7.13: (a) Exponential  t of probability of state as a function
 of state energy, for Bx = 0. The  t slope is the inverse pseudo-
 temperature. (b) Exponential  t for Bx=JMAX = 2:2. The higher
 energy states are beginning to separate into energy bands. (c) Ex-
 ponential  t for Bx=JMAX = 4:95. The higher energy states are now
 strongly separated into gaps, corresponding to the number of spins
 aligned with the axial  eld. This causes the psuedo-temperature pic-
 ture to fail.(d) The temperature coe cient for all  ts. This time the
  t shows reliably the phase transition points, and only followng the
 second to last transition. Phase transitions are marked with dashed
 lines.
 150
0 1 2 3 4 5
 0.0
 0.2
 0.4
 0.6
 0.8
 1.0
 Bx  JMAX
 Magnetizatio
 n
 A=1
 0 1 2 3 4 5
 0.0
 0.2
 0.4
 0.6
 0.8
 1.0
 Bx  JMAX
 Magnetizatio
 n
 A=6
 0 1 2 3 4 5
 0.0
 0.2
 0.4
 0.6
 0.8
 1.0
 Bx  JMAX
 Magnetizatio
 n
 A=11
 0 1 2 3 4 5
 0.0
 0.2
 0.4
 0.6
 0.8
 1.0
 Bx  JMAX
 Magnetizatio
 n
 A=16
 Figure 7.14: 10 ion data power  ltered with power A, increasing from
 no  ltering to A=16. A=16 plot has a solid black curve that is the
 adiabatic ground state solution. Dashed lines are phase transition
 location for adiabatic ground state.
 151
data. However, this  lter will fail to amplify the correct states close to a phase
 transition, as the exact ground state will not dominate for all simulation times and
 diabatic parameters.
 As quantum simulators are scaled to higher numbers of spins and more complex
 Hamiltonians, extraction of the desired signal[93] will be challenged by increased de-
 coherence [92], readout error for the total state, and diabatic e ects. However, these
 issues may be overcome by data processing methods catered to the Hamiltonian sim-
 ulated and guided by theoretical understanding of certain classes of Hamiltonians.
 This work was supported by the US Army Research O ce through funds from
 the DARPA Optical Lattice Emulator Program and the IARPA MQCO Program;
 the NSF Physics at the Information Frontier Program, and the NSF Physics Frontier
 Center at JQI.
 152
Chapter 8
 Simulating the Ising Model with Arbitrary Control of the Couplings
 8.1 Introduction
 In previous chapters I have discussed work were we applied global spin-
 dependent optical dipole forces to generate trivial forms of the spin couplings, such
 as a uniformly decaying ferromagnet or anti-ferromagnet. Now I will present my pro-
 posal1 for how to tailor optical forces to generate arbitrary fully-connected networks
 of N spins that uniquely specify each of the N(N  1)=2 pairwise interactions.
 The scheme is independent of the spatial geometry of the ion crystal and is
 compatible with one-dimensional arrays of trapped ions used in current experiments.
 We start with the arbitrary fully-connected Ising Hamiltonian on N spins,
 H =
 X
 i<j
 Ji;j 
(i)
 x  
(j)
 x ; (8.1)
 with the same conventions we have used in previous chapters.
 As in the previous work in this thesis, the spins are coherently manipulated
 through a pair of counter-propagating laser beams that drive stimulated Raman
 transitions in the far detuned limit between the spin states while also coupling
 o -resonantly to the collective motion of the atomic chain [95, 96]. The atoms
 1The discussion in this chapter closely follows [94]
 153
are arranged in a linear array, as is typical in a linear radiofrequency ion trap,
 although this scheme can also apply to other spatial geometries [97, 98, 99]. When
 the di erence frequency between the Raman  elds is bichromatic, with two spectral
 components tuned symmetrically at !s   with   !s, the e ective spin-spin
 interaction of Eq. 8.1 emerges, mediated by the Coulomb-coupled motion of the
 atomic ions crystal [3, 33, 100, 32]. Just as we drove transverse modes previously,
 we will assume the Raman lasers have wave vector di erence  k along the principal
 X axis of transverse motion of the ion crystal [101], and that we are cool the ions
 su ciently for us to operate in the Lamb-Dicke limit [95, 102], and the symmetric
 detuning  is set su ciently far from all motional sidebands (j!m   j >>  i;m i),
 so that the phonon states can be adiabatically eliminated, leaving the pure spin-spin
 coupling above [3, 33, 57].
 8.2 Control of an Arbitrary Lattice Hamiltonian
 The above expression has N +1 control parameters in the set of Rabi frequen-
 cies f ig and the global beatnote detuning  . In order to generate an arbitrary Ising
 coupling matrix Ji;j however, it is necessary to have at least N(N 1)=2 independent
 controls [103]. Additional control parameters can be introduced by adding multiple
 spectral beatnote detunings to the Raman beams, one near each motional mode
 (see Fig. 8.1), with a unique pattern of spectral components on each ion. There are
 several ways to achieve this, all involving some form of individual ion addressing.
 For simplicity, we retain the same set of N Raman beatnote detunings  m on each
 154
?
 2  ? N  ? 1  
frequency 
?
 N  
?
 1  
?i,N  ?i,1  
Figure 8.1: Spectrum of transverse mode frequencies !m for N = 10
 ions in an anisotropic harmonic linear trap (solid lines), with the
 highest mode frequency !1 corresponding to center-of-mass motion.
 Raman beatnote detunings  m from the qubit frequency !s are de-
 noted by the N = 10 dashed lines, with each spectral feature near a
 given motional sideband. The height of the dashed lines represents
 the intensity of each beatnote for ion i. In general each ion will be
 illuminated with a di erent set of intensities.
 ion and allow the spectral amplitude pattern to vary between ions, all characterized
 by the N  N Rabi frequency matrix  i;n of spectral component n at ion i. Note
 that the relative signs of the Rabi frequency matrix elements can be controlled by
 adjusting the phase of each spectral component. This individual spectral amplitude
 addressing provides N2 control parameters, and the general Ising coupling matrix
 becomes
 Ji;j =
 NX
 n=1
  i;n j;n
 NX
 m=1
  i;m j;m!m
  2n  !2m
 (8.2)
  
NX
 n=1
  i;n j;nFi;j;n; (8.3)
 where Fi;j;n characterizes the response of Ising coupling Ji;j to spectral component n.
 An exact derivation of the e ective Hamiltonian given a spectrum of spin-dependent
 forces gives rise to new o resonant cross terms, which can be shown to be negligible
 155
in the rotating wave approximation, as long as the bandwidth is  xed and the
 transverse center of mass mode is set to a frequency smaller than twice the transverse
 zig-zag mode frequency. Thus sums and di erences of beatnotes do not directly
 encroach any sideband features in the motional spectrum of the crystal [101].
 We tune each beat note frequency near a unique normal mode so that Fi;j;n
 has independent contributions for each n. Given a desired Ising coupling matrix Ji;j,
 we use standard constrained nonlinear optimization to  nd the corresponding Rabi
 frequency matrix  i;n, while minimizing the total beam intensity. The deviation
 between the desired and the attained coupling was less than typical round-o errors.
 However, Eq. 8.3 depends nonlinearly on the Rabi frequencies, and it is not
 clear that a solution exists or how the resulting total optical power scales with N .
 We  rst describe a formal method to invert Eq. 8.3, showing the existence
 of a solution. For simplicity we force the control matrix to be lower triangular, or
 set  i;n = 0 for i < n, with  i;i  !s (this still leaves N(N  1)=2 independent
 parameters). Isolating the n = i term in the sum of Eq. 8.3, we  nd
  j;i =
 Ji;j  
P
 n<i  i;n j;nFi;j;n
 !sFi;j;i
 (i < j): (8.4)
 The left hand side de nes column i of the Rabi frequency matrix, while the right
 hand side depends only on matrix elements in columns to the left of column i
 (n < i), resulting in a recursive de nition for successive columns of  i;n. If the re-
 sponse functions Fi;j;n are small however, the resulting Rabi frequencies will become
 unreasonably large and we  nd that the required total optical power in this simple
 156
procedure generally grows exponentially with N .
 We now show how to obtain a solution to inverting Eq. 8.3 that scales e -
 ciently with N . If we neglect the e ect of each beatnode  n on modes with n 6= m,
 Fi;j;n is separable in i and j and we can write Ji;j =
 P
 nRi;nRj;n, or in matrix form,
 J = RRT where the matrix Ri;n =  i;n i;n
 q
 !n
  2n !2n
 . This quadratic equation can
 be inverted by diagonalizing the symmetric matrix J with some orthogonal matrix
 U so that Jdiag = UTJU, then we simply write R = UT
 p
 Jdiag. As long as the
 eigenvalues of J are not too large, the matrix elements Ri;n will be bounded2. In
 practice we can impose an upper bound on the total optical power (proportional to
 P
 i;n j i;nj) and implement numerical optimization techniques. These produce very
 accurately the desired coupling matrix, much more than the approximation above,
 as the the non-close modes for each detuning have a non-negligilble e ect on the
 solution. However, the above approximation can be used to quickly calculate an
 initial guess for R.
 8.3 Examples
 We now present two example solutions for  i;n that produce interesting inter-
 action graph topologies. First we calculate a Rabi frequency matrix that results in
 a 2D square lattice of nearest-neighbor antiferromagnetic couplings with N = 25
 ions (5  5 grid with periodic boundary conditions), shown in Figs. 8.2a-b. Next
 we produce a 2D Kagome lattice of antiferromagnetic interactions, a geometry that
 can support high levels of geometrical frustration [104], shown in Figs. 8.2d-e. In
 2To avoid negative values in Jdiag we may rede ne J0 = J + E1 with no e ect on the physics
 157
both cases we assume the center-of-mass (COM) mode to be !1=2 = 5 MHz , and
 a  xed total optical intensity corresponding to
 P
 i;n j i;nj = 1 MHz . The beatnote
 frequencies  m are each tuned blue of the mode m sideband by a fraction fs of the
 spacing !1 !2 between the most closely-spaced modes (the COM and \tilt" modes,
 see Fig. 8.1), which itself scales as logN=N2. In these examples, the sparse nearest-
 neighbor nature of the interaction graphs require that most of the Ising interactions
 vanish, indicating a high level of coherent control over all of the Ising couplings.
 8.4 Implementation
 In order to generate a unique spectrum of Raman beams for each of N ions,
 some type of individual addressing is necessary. For simplicity, we assume one
 of the two Raman beams is uniform and monochromatic, and the high frequency
 beatnote near the qubit frequency !s or other global o set frequencies can be set
 by tuning this monochromatic beam. We focus attention on providing the requisite
 frequencies of the second beam, spread over a range given by the bandwidth of the
 transverse motional mode frequencies of the ion chain, typically in the range 1 5
 MHz . We suggest three possible methods for providing spatial dependent frequency
 modulation to one of the Raman beams. The  rst method (Fig. 8.3 and Fig.8.4)
 splits a single beam with a linear chain of N individual optical modulators (e.g.,
 acoustooptic or electrooptic devices), driven by N independent arbitrary waveform
 generators.
 The second method splits a single monochromatic beam into a N  N square
 158
(a) (b)
 5
 10
 15
 20 25
 0 5
 10
 15 20 25
 (40
 (30
 (20
 (10
 0
 10
 20
 30
 40
 Beatnote Index
 Spectral components for 5x5 square lattice
  1 MHz total power, J = 27.5905 Hz
 Ion Index
 Required Rabi powers (kHz
 ) 1 2 3 4 5 
6 7 8 9 10 
11 12 13 14 15 
16 17 18 19 20 
21 22 23 24 25 
(c)
 1 2 3 4 5 6 7 8 10  12 14 16 18 20 22 24 9 11 13 15 17 19 21 23 25 
(d) (e)
 10
 20
 300 6 12 18 24 30 36
 (30
 (20
 (10
 0
 10
 20
 30
 Beatnote Index
 Spectral components for 36 element kagome lattice
  1 MHz total power, J = 93.4345 Hz
 Ion Index
 Required Rabi powers (kHz
 )
 1 2 7 8 13 14 
4 
3 
5 10 
9 
11 16 
15 
17 
20 25 
6 
26 31 
12 
32 19 
18 
22 
21 
23 28 
27 
29 34 
33 
35 
24 30 36 
Figure 8.2: (a) Calculated Rabi frequency matrix  i;n to generate
 2D square lattice shown in (b), using the linear chain of N = 25
 ions shown in (c). The ion index refers to the order in the linear
 chain. The attained Ji;j nearest-neighbor is 27:6 Hz for fs = 0:1.
 (d) Calculated Rabi frequency matrix  i;n to generate 2D Kagome
 lattice shown in (e) using a linear chain of N = 36 ions. The attained
 Ji;j nearest-neighbor is 93:4 Hz for fs = 0:03. In both cases the total
 optical intensity corresponds to a Rabi frequency of 1 MHz if focused
 on a single ion, the nearest-neighbor couplings are antiferromagnetic
 and we impose periodic boundary conditions.
 159
f
 1
 f
 2
 f
 3 f
 4
 f
 5
 f
 6
 f
 7
 f
 8
 Figure 8.3: Schematic for individual spectral addressing a linear chain
 of N ions. A laser beam is split into a linear array of spots that each
 traverse N independent AOMs or EOMs, driven by N independent
 AWGs. Alternatively, as discussed in the text, the beam can be
 broken into an array of N2 beams that strike a N  N array of
 micromirrors[105] each independently modulated, or a spatial light
 modulator.
 grid and directs them onto a 2D array of N2 micromirrors [105] that are each indi-
 vidually phase modulated at a single frequency (and phase) [106] and  nally focused
 on the ion chain. The third method again splits the beam into an N  N grid of
 beams, this time with the vertical direction split by a single AOM, correlating beam
 position to frequency. This beam is then directed into a spatial light modulator that
 acts to mask (or phase shift) each of the N  N beams independently, and again
 focused onto the ion chain, as shown in  gure 8.5. In these implementations, it may
 be desirable to work with a uniformly spaced array of ions in the linear trap, so that
 the modulating elements are also uniformly spaced. This can be accomplished by
 using a quartic or higher order linear trap [78, 107].
 160
Figure 8.4: A phase mask splits a single beam into N beams, which
 are then modulated by N idependent AOMs, each one driven with
 N independent amplitudes and phases (but the same frequencies,
 to imprint the same detunings for each mode for each ion). The
 modulated and de ected beams are then imaged onto the ions.
 161
Figure 8.5: Schematic for individual spectral addressing a linear chain
 of N ions. The beam is again split into a N N array, this time with
 a single acoustooptic modulator providing the vertical fanout, cor-
 relating spatial row with optical frequency. The grid of beams then
 enters a spatial light modulator, which can be a static or recon g-
 urable liquid crystal mask, that attenuate and/or phase shift each
 pixel appropriately before the columns are imaged onto the ions.
 162
8.5 Scalability
 As the number of spins N grows, the optical modulation scheme becomes more
 complex, with either N or N2 elements required. However, our schemes restrict most
 of the resources? overhead to the design and fabrication of the micro-mirrors or phase
 modulator. As the ions are equally spaced, the modulator can simply be imaged
 onto the ion chain without changing imaging optics.
 We now estimate how the Ising couplings are expected to scale with the num-
 ber of spins along with errors due to experimental  uctuations, phonon creation and
 spontaneous emission scattering, assuming a  xed transverse mode bandwidth. The
 probability of phonon creation scales as pph =
 P
 i;m
  
 i;m i;m
 !m  m
  2
 . The o -resonant
 optical dipole forces are accompanied by a  nite rate of spontaneous emission scat-
 tering, given by  =  
P
 i;m j i;mj, where   1 is the ratio of excited state linewidth
 to Raman detuning. The scaling of these potential errors depends upon the par-
 ticular graph, so we consider two extremes. A uniform fully-connected interaction
 graph can be trivially generated with a single spectral component tuned close to the
 COM mode with a detuning j!1   j=!1  logN=N2. For a  xed level of phonon
 error, the total optical intensity should be reduced as logN=N , taking into account
 the intensity reduction per ion as the beam is expanded to accommodate the lin-
 early expanding chain in space. In this case the uniform Ising coupling is expected
 to scale as N jJi;jj / logN=N2, and the spontaneous emission rate per spin actually
 decreases with N . For a sparse interaction graph, such as a 1D (nearest-neighbor)
 Ising model, all modes are involved, and this time for a  xed phonon error the total
 163
? ? ?? ?? ? ? ?? ??
 ?
 ?
 ?
 ?
 ?
 ?
 ?
 ??????????????
 ????
 ??????
 ?
 Figure 8.6: Scaling of the nearest-neighbor coupling for the case of the
 production of a 1D Ising chain for N = 3 to N = 33 ions. The total
 optical intensity is  xed with
 P
 i;n j i;nj = 1 MHz , and each spec-
 tral component is detuned from its motional sideband by a fraction
 fs = 0:03 of smallest mode splitting, with a center-of-mass (COM)
 frequency !1=2 = 5 MHz . We  nd that the resulting Ising coupling
 scales roughly as 1=N .
 optical intensity can remain  xed, since the typical mode splitting falls only as 1=N ,
 while spontaneous emission per ion is  xed. The calculation in Fig. 8.6 shows that
 the resulting nearest-neighbor interaction scales as Ji;i+1 / 1=N . In either case of
 fully-connected or local Ising model, we thus expect to be able to support signi cant
 Ising interaction strengths with up to a few hundred spins.
 For a general Ising graph, from Eq. 8.3 we  nd that each pairwise interaction
 Ji;j depends upon a balance of N terms, and errors will accumulate with N from
 164
 uctuations of relative optical intensities of the various spectral components of the
 beam (which should be stable if the spectral components are generated with high
 quality RF sources and modulators as shown in Fig. 8.3) or their detunings from
 the motional sidebands. The most important source will likely be  uctuations in the
 motional trap frequencies, where we expect the fractional error in the Ising coupling
 to grow as
 p
 N( !m=!m), so that a typical fractional  uctuation in the motional
 trapping frequencies of  10 3 might be expected to cause Ising coupling errors at
 a level of about 1% for N  100 ions.
 The  nal limitation on simulation time is e ective spin coherence. Assuming
 the qubits experience Markovian noise, the coherence time of the ensemble will go
 as 1/N, so that a qubit coherence time of  10 s (as is typical for hyper ne qubits)
 will allow for a simulation time on order  100 ms for  100 ions.
 8.6 Conclusion
 The scheme presented here can also be applied to more general Heisenberg
 spin models involving other noncommuting spin-spin interactions, such as the XY
 model or the 2D hexagonal Kitaev model relevant to topological quantum degrees
 of freedom [108]. Here, additional Raman beams that couple to the other axes of
 motion can be exploited This scheme may also be used to study phase transitions
 in a quantum transverse magnetic  eld [109]. Alternatively, a single direction of
 motion can be used as discussed above, with a stroboscopic alternation between
 Raman laser beams with di erent beat note phases as the various Ising interactions
 165
in the Hamiltonian are applied sequentially. Here, we employ the Trotter expansion
 of the evolution operator [5, 110] and switch the various Ising terms rapidly enough
 so that higher order terms in the expansion can safely be neglected. Although this
 discussion concentrated on a linear array of ions in space, these ideas apply in general
 to any stable ion crystal where the motional sidebands are resolved and prepared in
 the Lamb-Dicke limit, and should be useful for higher-dimensional trap geometries
 such as trap arrays [98, 99] or Penning traps [97, 111].
 166
Appendix A
 Linear Trap External Pi Filter
 DC output pads  
DC input pads  RF coupling DIP  
Switch  
C1  Inductor  
C2  
Figure A.1: PCB mask, pads and traces for linear trap pi- lter. C1
 and C2 are high voltage ceramic capacitors of 0.1 F each; the induc-
 tors are 100 H each.
 In A.1 the Printed Circuit Board (PCB) mask is presented. There are several
 things to take note of in this design:
 1. This  lter is based upon a Pi  lter design, where the RF rejecting component
 is an inductor (an alternate design uses a resistor instead). However numerical
 simulations indicated that the addition of a resistor in series with the inductor
 167
gave a sharper frequency response. Therefore a resistor was added to the
 actual circuit, however the mask in A.1 was the one ordered and therefore the
 resistor is not indicated in the silkscreeen.
 2. The circuit includes a DIP switch near the output end. This switch selects
 between coupling the RF pickup on a DC channel to an output, through a
 capacitor to ground, or to an output. This is useful for measuring the RF
 pickup on the DC electrodes through capacitive coupling to the RF electrode.
 168
Appendix B
 Optical Diagram Legend
 Optical fiber  
Cylindrical lens  lens  
mirror  
M icrometer  
O scillator  Beam splitter  
Prism Pair  Diode  
Mixer  Pinhole  
Diagram Symbol Legend  
Las er  
Wavep late  
Cylindrical  
 mirror  
AOM  
E OM  
Figure B.1: This is the legend of all symbols used in optical diagrams
 in this work.
 169
Appendix C
 AWG Function Library
 1 // AWG dll . cpp : Def ines the expor ted f unc t i on s f o r the DLL app l i c a t i o n
 .
 2 #include " s tda fx . h"
 3 #include <windows . h>
 4 #include <math . h>
 5 #include <s t d l i b . h>
 6 #include "AWG dll . h"
 7 #include " da11000 d l l impor t . h"
 8
 9 const double num=2 3.14159265358979323846/1000; / f o r speed , f requnecy
 in MHz  /
 10
 11 struct SegmentStruct //need to b u i l d an array o f t h e s e
 12 f
 13 DWORD SegmentNum ; // Current Segment Number
 14 unsigned short  SegmentPtr ; // Pointer to curren t user segment
 15 DWORD NumPoints ; // Number o f po in t s in segment
 16 DWORD NumLoops ; // Number o f t imes to repea t segment ( a p p l i e s to next
 segment )
 17 DWORD BeginPadVal ; // Pad va lue f o r beg inn ing o f t r i g g e r e d segment
 18 DWORD EndingPadVal ; // Pad va lue f o r ending o f t r i g g e r e d segment
 19 DWORD TrigEn ; // I f > 0 then wai t f o r t r i g g e r b e f o r e going to next
 170
segment .
 20 DWORD NextSegNum ; // Next segment to jump to a f t e r complet ion
 21 // o f curren t segment a c t i v i t i e s
 22 g ;
 23
 24 struct wave
 25 f
 26 double f ;
 27 double phase ;
 28 double amp ;
 29 g ;
 30
 31 struct l inear param
 32 f
 33
 34 double f1 , p1 , f2 , p2 , f3 , p3 ;
 35 f loat a1 , a2 , a3 ;
 36 double s l ope ;
 37 g ;
 38
 39 struct sb param
 40 f
 41 unsigned long modes ;
 42 double  f ;
 43 double  durat ion ;
 44 unsigned long OPtime ;
 45 unsigned short p u l s e s ;
 171
46 g ;
 47
 48 int make64x (unsigned long s i z e )
 49 f
 50 d i v t r e s u l t=div ( s i z e , 6 4 ) ;
 51 i f ( r e s u l t . rem>0)
 52 return(64 r e s u l t . rem) ;
 53 else
 54 return (0 ) ;
 55 g
 56
 57 d e c l s p e c ( d l l e x p o r t ) long s i d e b a n d c o o l i n g s e q u e n t i a l (unsigned
 short  a , unsigned long modes , double  f ,
 58 unsigned long  duration , unsigned long OPtime ,
 unsigned short p u l s e s )
 59 f
 60 unsigned long i , j ,m, t =0;
 61
 62 for ( i =0; i<p u l s e s ; i++)
 63 for (m=0;m<modes ; m++)
 64 f
 65 for ( j =0; j<round ( s q r t ( (double ) ( (double ) p u l s e s /(double ) ( pu l ses i )
 ) )  (double ) durat ion [m] ) ; j++) / t ime i s in NANOSECONDS  /
 66 f
 67 a [ t ] = (unsigned short ) round (2048 2048 s i n (num f [m] t ) ) ;
 68 t++;
 69 g
 172
70 for ( j =0; j<OPtime ; j++)
 71 f
 72 a [ t ]=2048;
 73 t++;
 74 g
 75 g
 76 //Dirty hack to de lay wave to match Raman 1
 77
 78 for ( i =0; i <150; i++)
 79 f
 80 a [ t ]=(unsigned short ) (2048) ;
 81 t++;
 82 g
 83
 84 unsigned long pad=make64x ( t ) ;
 85 for ( j =0; j<pad ; j++)
 86 f
 87 a [ t ]=2048;
 88 t++;
 89 g
 90 return ( t ) ;
 91 g
 92
 93 d e c l s p e c ( d l l e x p o r t ) unsigned long c a l c u l a t e s b c a r r a y s i z e (unsigned
 long modes , double  f , unsigned long  duration , unsigned long
 OPtime , unsigned short p u l s e s )
 94 f
 173
95 unsigned long i , j ,m, t =0;
 96
 97 for ( i =0; i<p u l s e s ; i++)
 98 for (m=0;m<modes ; m++)
 99 f
 100 for ( j =0; j<round ( s q r t ( (double ) ( (double ) p u l s e s /(double ) ( pu l ses i )
 ) )  (double ) durat ion [m] ) ; j++) / t ime i s in NANOSECONDS  /
 101 t++;
 102 t+=OPtime ;
 103 g
 104 t +=150;
 105 t+=make64x ( t ) ;
 106 return ( t ) ;
 107 g
 108
 109 d e c l s p e c ( d l l e x p o r t ) long f l a t (unsigned short  a , unsigned long &t ,
 unsigned long durat ion )
 110 f
 111 for (unsigned long i =0; i<durat ion ; i++)
 112 f
 113 a [ t ]=2048;
 114 t++;
 115 g
 116 return ( t ) ;
 117 g
 118
 119 d e c l s p e c ( d l l e x p o r t ) unsigned long exp s imu la t i on (unsigned short  a ,
 174
double  f , double  p , double  amp,
 120 unsigned long  durat ion )
 121 //amp has 6 va l u e s
 122 // j  j j      n j  j
 123 // 0 Ci : 1 , Cf :2 ,3 ,4 5
 124 // durat ion has 5 : j  j j      n j  j
 125 // 0 1 2 3 4
 126 //CAREFUL! Labview c a l l s a func t i on to f i nd the l en g t h o f t h i s waveform
 . Update t h i s f unc t i on in the con t r o l program i f
 127 //you change t h i s f unc t i on
 128 f
 129 unsigned long i ;
 130 int pad ;
 131 double s l ope= abs (amp [ 2 ] ) ;
 132
 133 unsigned long t =0;
 134
 135 for ( i =0; i<durat ion [ 0 ] ; i++)
 136 f
 137 a [ t ]=(unsigned short ) (2048.5+amp [ 0 ]  s i n (num f [ 0 ]  t+p [ 0 ] ) ) ;
 138 t++;
 139 g
 140 for ( i =0; i<durat ion [ 1 ] ; i++)
 141 f
 142 a [ t ]=2048;
 143 t++;
 144 g
 175
145 for ( i =0; i<durat ion [ 2 ] ; i++)
 146 f
 147 a [ t ]=(unsigned short ) (2048.5+amp [ 7 ]  s i n (num f [ 4 ]  t )+//comp2
 148 amp[6] (1 exp ( i / s l ope ) )  s i n (num f [ 3 ]  t )+//comp1 fo r c a r r i e r
 149 exp ( i / s l ope )  amp [ 1 ]  s i n (num f [ 0 ]  t+p [ 1 ] )+// c a r r i e r
 150 amp [ 3 ]  s i n (num f [ 1 ]  t )+// rsb
 151 amp [ 4 ]  s i n (num f [ 2 ]  t+p [ 2 ] ) ) ; // bsb
 152 t++;
 153 g
 154 for ( i =0; i<durat ion [ 3 ] ; i++)
 155 f
 156 a [ t ]=2048;
 157 t++;
 158 g
 159 for ( i =0; i<durat ion [ 4 ] ; i++)
 160 f
 161 a [ t ]=(unsigned short ) (2048.5+amp [ 5 ]  s i n (num f [ 0 ]  t+p [ 3 ] ) ) ;
 162 t++;
 163 g
 164
 165 for ( i =0; i <150; i++)
 166 f
 167 a [ t ]=(unsigned short ) (2048) ;
 168 t++;
 169 g
 170
 171 pad=make64x ( t ) ;
 176
172 for ( i =0; i <(unsigned long ) pad ; i++)
 173 f
 174 a [ t ]=2048;
 175 t++;
 176 g
 177
 178 return ( t ) ;
 179 g
 180
 181 d e c l s p e c ( d l l e x p o r t ) unsigned short  f i n d a d d r e s s (unsigned short  a
 )
 182 f
 183 return ( a ) ;
 184 g
 185
 186 d e c l s p e c ( d l l e x p o r t ) unsigned long BuildArray ( SegmentHdl  S , unsigned
 short  a )
 187 f
 188 unsigned long i , j , t =0;
 189 / 
190 s t r u c t SegmentStruct //need to b u i l d an array o f t h e s e
 191 f
 192 DWORD SegmentNum ; // Current Segment Number
 193 unsigned shor t  SegmentPtr ; // Pointer to curren t user segment
 194 DWORD NumPoints ; // Number o f po in t s in segment
 195 DWORD NumLoops ; // Number o f t imes to repea t segment ( a p p l i e s to next
 segment )
 177
196 DWORD BeginPadVal ; // Pad va lue f o r beg inn ing o f t r i g g e r e d segment
 197 DWORD EndingPadVal ; // Pad va lue f o r ending o f t r i g g e r e d segment
 198 DWORD TrigEn ; // I f > 0 then wai t f o r t r i g g e r b e f o r e going to next
 segment .
 199 DWORD NextSegNum ; // Next segment to jump to a f t e r comple t ion
 200 // o f curren t segment a c t i v i t i e s
 201 g ;  /
 202
 203
 204 for ( i =0; i <(unsigned long ) (  S) >NumOfSegments ; i++)
 205 f
 206 for ( j =0; j <(unsigned long ) ( (  S) >segment [ i ] . SegmentPtr ) >
 WaveSize ; j++)
 207 f
 208 a [ t ]=( (  S) >segment [ i ] . SegmentPtr ) >wave [ j ] ;
 209 t++;
 210 g
 211 g
 212 return ( (unsigned long ) (  S) >NumOfSegments ) ;
 213 g
 214
 215 d e c l s p e c ( d l l e x p o r t ) unsigned long BuildDA11000Array ( SegmentHdl  S ,
 SegmentStruct  a )
 216 f
 217 unsigned long i ;
 218 / 
219 s t r u c t SegmentStruct //need to b u i l d an array o f t h e s e
 178
220 f
 221 DWORD SegmentNum ; // Current Segment Number
 222 unsigned shor t  SegmentPtr ; // Pointer to curren t user segment
 223 DWORD NumPoints ; // Number o f po in t s in segment
 224 DWORD NumLoops ; // Number o f t imes to repea t segment ( a p p l i e s to next
 segment )
 225 DWORD BeginPadVal ; // Pad va lue f o r beg inn ing o f t r i g g e r e d segment
 226 DWORD EndingPadVal ; // Pad va lue f o r ending o f t r i g g e r e d segment
 227 DWORD TrigEn ; // I f > 0 then wai t f o r t r i g g e r b e f o r e going to next
 segment .
 228 DWORD NextSegNum ; // Next segment to jump to a f t e r comple t ion
 229 // o f curren t segment a c t i v i t i e s
 230 g ;  /
 231 for ( i =0; i <(unsigned long ) (  S) >NumOfSegments ; i++)
 232 f
 233 a [ i ] . BeginPadVal=(  S) >segment [ i ] . BeginPadVal ;
 234 a [ i ] . EndingPadVal=(  S) >segment [ i ] . EndingPadVal ;
 235 a [ i ] . NextSegNum=(  S) >segment [ i ] . NextSegNum ;
 236 a [ i ] . NumLoops=(  S) >segment [ i ] . NumLoops ;
 237 a [ i ] . SegmentNum=(  S) >segment [ i ] . SegmentNum ;
 238 a [ i ] . TrigEn=(  S) >segment [ i ] . TrigEn ;
 239 a [ i ] . NumPoints=( (  S) >segment [ i ] . SegmentPtr ) >WaveSize ;
 240 a [ i ] . SegmentPtr =( (  S) >segment [ i ] . SegmentPtr ) >wave ;
 241 g
 242 return ( (unsigned long ) (  S) >NumOfSegments ) ;
 243 g
 244
 179
245 d e c l s p e c ( d l l e x p o r t ) bool LoadSegmentHdl ( SegmentHdl  S)
 246 f
 247 bool e r r o r=fa l se ;
 248 SegmentStruct  a= new SegmentStruct [ (  S) >NumOfSegments ] ; // don ? t
 f o r g e t d e l e t e [ (  S) >NumOfSegments ]
 249 BuildDA11000Array (S , a ) ;
 250 // INITIALIZE BOARD
 251 da11000 SetTriggerMode (1 , 0 , 0 ) ;
 252 da11000 CreateSegments (1 , 1 , (  S) >NumOfSegments , a ) ;
 253
 254 delete [ (  S) >NumOfSegments ] a ;
 255 return ( e r r o r ) ;
 256 g
 180
Appendix D
 Mathematica ROI and Discriminator Selection Script
 The following script is called after taking 1000 shots of the ions pumped dark
 and then 1000 shots of the ions rotated to bright.
 It is called from the Labview control program automatically after the shots
 have been taken, run as a command line script with the following parameters -
 \dark.csv" - \bright.csv" - Nions - Nimages - cuto (0 - 1)
 The script (\ROI auto selection.m")is created and run from the directory of the
 data  les. example :
 math - script \ROI auto selection.m" \dark Shots.csv" \bright Shots.csv" 4 1000
 0.5
 DarkDataT = Transpose[Import[$CommandLine[[4]]]];ort[$CommandLine[[4]]]]
 dataT = Transpose[Import[$CommandLine[[5]]]];
 Nions = ToExpression[$CommandLine[[6]]];oExpression[$CommandLine[[6]]]oExpression[$CommandLine[[6]]]
 MaxDataT = Max[dataT];
 Nimages = ToExpression[$CommandLine[[7]]];oExpression[$CommandLine[[7]]]
 width = Dimensions[dataT][[2]]=Nimages;
 height = Dimensions[dataT][[1]];
 This section sums all the frames and then normalizes the image. It also cal-
 culates the average value of both the dark and bright pixels.
 NormImageDark = Sum[DarkDataT[[All; widthx+ 1;;(x+ 1)width]]; fx; 0; Nimages 1g];
 NormImage = Sum[dataT[[All; widthx+ 1;;(x+ 1)width]]; fx; 0; Nimages 1g];
 181
NormImageSubtracted = NormImage NormImageDark;
 NormImageSubtracted/=Max[NormImageSubtracted];NormImageSubtracted/=Max[NormImageSubtracted]NormImageSubtracted/=Max[NormI
 IonImage = NormImageSubtracted/.x /;x < ToExpression[$CommandLine[[8]]]->0;
 Here we  nd the vertical Region for ion1, VertStart and VertEnd that de ne
 the vertical edges. We  nd VertStart by starting at 0, and then scan the horizontal
 pixels, looking for brightness > threshold. If no non-zero pixels are found, VertStart
 is shifted by 1 and we scan again ....VertEnd starts from the height, and we scan
 along the horizontals, if no bright pixel is found we shift to height - 1 and scan again
 and so on ...
 VertStart = 0;
 VertEnd = height;
 For[n = 1; n < height + 1; n++;
 For[i = 1; i < width + 1; i++;
 If[IonImage[[n; i]] > 0;VertStart = n; i = width + 1;n = height + 1]]]
 For[n = height; n > VertStart; n{;
 For[i = 1; i < width + 1; i++;
 If[IonImage[[n; i]] > 0;VertEnd = n; i = width + 1;n = 1]]]
 If[VertEnd > height;VertEnd = height];t]
 VertStart = If[VertStart > 0;VertStart; 1];
 Export[\uncuto image " <> DateString[f\Year";\Month";\Day"g] <> \.jpg";
 Image[IonImage[[All; All]]]]
 HEdge = Join[ff1; 0gg;Table[f1; 1g; fi; 1; Nionsg]]
 center = VertStart + Floor[(VertEnd VertStart)=2];ertStarertStar
 182
For[i = 1; i<=Nions; i++;
 For[j = HEdge[[i; 2]] + 1; j < width; j++;
 If[Count[Table[N[IonImage[[v; j]]]; fv;VertStart;VertEnd; 1g]; 0:]
 < (3(VertEnd VertStart))=4; HEdge[[i+ 1; 1]] = j;
 For[k = j; k < width; k++;
 If[Count[Table[N[IonImage[[v; k]]]; fv;VertStart;VertEnd; 1g]; 0:] >
 (3(VertEnd VertStart))=4; HEdge[[i+ 1; 2]] = k  1; k = width; j = width]
 ]]; ]]
 HEdge = Delete[HEdge; 1]
 dataFrames = Table[dataT[[VertStart;;VertEnd; widthx+ 1;;(x+ 1)width]];ertEnrtEn
 fx; 0; Nimages 1; 1g];
 darkFrames = Table[DarkDataT[[VertStart;;VertEnd; widthx+ 1;;(x+ 1)width]];ertEn
 fx; 0; Nimages 1; 1g];
 HistogramDataBright = Table[Total[Flatten[dataFrames[[j; All; HEdge[[i; 1]];;otal[Flatten[dataotal[Flatten[data
 HEdge[[i; 2]]]]]]; fi; Nionsg; fj; 1; Nimagesg];
 HistogramDataDark = Table[Total[Flatten[darkFrames[[j; All; HEdge[[i; 1]];;able[
 HEdge[[i; 2]]]]]]; fi; Nionsg; fj; 1; Nimagesg];
 Export[\Histograms " <> DateString[f\Year";\Month";\Day"g] <> \.pdf";
 Table[Histogram[fHistogramDataDark[[i]]; HistogramDataBright[[i]]g;t[
 f500g; ImageSize->300]; fi; Nionsg]//TableForm]
 Export[\Histogram tables dark" <> DateString[f\Year";\Month";\Day"g] <> \.xls";
 Table[Table[f#[[1; i]]; #[[2; i]]g; fi; Length[#[[2]]]g]&
 [HistogramList[HistogramDataDark[[j]]]]; fj; Nionsg]]
 Export[\Histogram tables bright" <> DateString[f\Year";\Month";\Day"g]
 <> \.xls";Table[Table[f#[[1; i]];
 183
#[[2; i]]g; fi; Length[#[[2]]]g]&[HistogramList[HistogramDataBright[[j]]]]; fj; Nionsg]]
 ErrorThreshold = Table[f0; 0g; fi; Nionsg];
 Steps = 50;
 For[i = 1; i < Nions + 1; i++;
 DL = HistogramList[HistogramDataDark[[i]]; fStepsg];HistogramList[HistogramDataDark[HistogramList[HistogramDataDark[
 BL = HistogramList[HistogramDataBright[[i]]; fStepsg];HistogramList[HistogramDataBrigHistogramList[HistogramDataBrig
 We match the histogram dimensions
 Di L = Min[BL[[1]]] Min[DL[[1]]];
 Di U = Max[BL[[1]]] Max[DL[[1]]];Max[BL[[1]]
 If[Di L>=0;
 fBLNew1 = Join[Table[Min[BL[[1]]] Abs[Di L] + (n 1)  Steps;able[Min[BL[[1]]
 fn; 1; Floor[Abs[Di L]=Steps]g]; BL[[1]]];
 BLNew2 = Join[Table[0; fn; 1; Floor[Di L=Steps]g]; BL[[2]]];
 BLNewInt = Join[fBLNew1g; fBLNew2g];
 DLNewInt = DL; g;
 fDLNew1 = Join[Table[Min[DL[[1]]] Abs[Di L] + (n 1)  Steps;able[Min[DL[[1]]
 fn; 1; Floor[Abs[Di L]=Steps]g]; DL[[1]]];
 DLNew2 = Join[Table[0; fn; 1; Floor[Abs[Di L]=Steps]g]; DL[[2]]];
 DLNewInt = Join[fDLNew1g; fDLNew2g];
 BLNewInt = BL; g];
 If[Di U>=0;
 fDLNew3 = Join[DLNewInt[[1]];Table[Max[DLNewInt[[1]]] + n  Steps;
 fn; 1; Floor[Di U=Steps]g]];
 DLNew4 = Join[DLNewInt[[2]];Table[0; fn; 1; Floor[Di U=Steps]g]];
 DLNew = Join[fDLNew3g; fDLNew4g];
 BLNew = BLNewInt; g;BLNewI
 184
fBLNew3 = Join[BLNewInt[[1]];Table[Max[BLNewInt[[1]]] + n  Steps;
 fn; 1; Floor[Abs[Di U]=Steps]g]];
 BLNew4 = Join[BLNewInt[[2]];Table[0; fn; 1; Floor[Abs[Di U]=Steps]g]];
 BLNew = Join[fBLNew3g; fBLNew4g];
 DLNew = DLNewIntg; ];
 TDL = Table[Total[DLNew[[2; 1;;j]]]; fj; 1; Dimensions[DLNew[[2]]][[1]]g];
 TBL = Table[Nimages Total[BLNew[[2; 1;;j]]];
 fj; 1; Dimensions[BLNew[[2]]][[1]]g];
 Di = TDL TBL;
 Now we  nd the unbiased thresholds and export our  ndings for the control
 program to use
 x = Position[Abs[Di ]; Min[Abs[Di ]]][[1; 1]];osition[Abs[Di osition[Abs[Di 
Error = N[(Sum[BLNew[[2; n]]; fn; 1; xg]=Nimages)  100];
 ErrorThreshold[[i]] = fVertStart 1;VertEnd; HEdge[[i; 1]] 1;
 HEdge[[i; 2]]; Error; BLNew[[1; x]]g; ]
 Export[\ROI Error and Thresholds.csv"; ErrorThreshold]an
 ErrorThreshold = Prepend[ErrorThreshold; f\Lower Vertical Edge";\Upper Vertical Edge";EdgeEdg
 \Left Edge"; \Right Edge"; \State detection error"; \Threshold"g]
 Export[\ROI Error and Thresholds " <> DateString[f\Year";\Month";\Day"g]an
 <> \.xls"; ErrorThreshold]
 185
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