ABSTRACT Title of dissertation: QUANTUM SIMULATION OF INTERACTING SPIN MODELS WITH TRAPPED IONS Kazi Rajibul Islam, Doctor of Philosophy, 2012 Dissertation directed by: Professor Christopher Monroe Joint Quantum Institute, University of Maryland Department of Physics and National Institute of Standards and Technology The quantum simulation of complex many body systems holds promise for understanding the origin of emergent properties of strongly correlated systems, such as high-Tc superconductors and spin liquids. Cold atomic systems provide an almost ideal platform for quantum simulation due to their excellent quantum coherence, initialization and readout properties, and their ability to support several forms of interactions. In this thesis, I present experiments on the quantum simulation of long range Ising models in the presence of transverse magnetic elds with a chain of up to sixteen ultracold 171Yb+ ions trapped in a linear radiofrequency Paul trap. Two hyper ne levels in each of the 171Yb+ ions serve as the spin-1=2 systems. We detect the spin states of the individual ions by observing state-dependent uorescence with single site resolution, and can directly measure any possible spin correlation function. The spin-spin interactions are engineered by applying dipole forces from precisely tuned lasers whose beatnotes induce stimulated Raman transitions that couple virtually to collective phonon modes of the ion motion. The Ising couplings are controlled, both in sign and strength with respect to the e ective transverse eld, and adiabatically manipulated to study various aspects of this spin model, such as the emergence of a quantum phase transition in the ground state and spin frustration due to competing antiferromagnetic interactions. Spin frustration often gives rise to a massive degeneracy in the ground state, which can lead to entanglement in the spin system. We detect and characterize this frustration induced entanglement in a system of three spins, demonstrating the rst direct experimental connection between frustration and entanglement. With larger numbers of spins we also vary the range of the antiferromagnetic couplings through appropriate laser tunings and observe that longer range interactions reduce the excitation energy and thereby frustrate the ground state order. This system can potentially be scaled up to study a wide range of fully connected spin networks with a few dozens of spins, where the underlying theory becomes intractable on a classical computer. Quantum Simulation of Interacting Spin Models with Trapped Ions by Kazi Rajibul Islam Dissertation submitted to the Faculty of the Graduate School of the University of Maryland, College Park in partial ful llment of the requirements for the degree of Doctor of Philosophy 2012 Advisory Committee: Professor Christopher Monroe, Chair/Advisor Professor Steve Rolston Dr. Ian Spielman Professor Christopher Jarzynski Professor Dianne O?Leary c Copyright by Kazi Rajibul Islam 2012 To my parents ii Acknowledgments First of all, I thank my advisor Prof. Chris Monroe for giving me the op- portunity to work on this project. Instead of writing a long essay on his amazing capabilities as a physicist, and as an advisor, I would just say that he is very close to the ideal advisor I could have hoped for. I appreciate his constant encouragement to engage in fruitful conversations with other people, particularly the theorists, and the independence that he gave me to pursue my experimental ideas. Research in experimental physics is surely a group e ort, and this is so true in our group. I have been fortunate to work with a pool of great postdocs and fellow graduate and undergraduate students. Ming-Shien Chang, Kihwan Kim, Emily Edwards and Wes Campbell were all very gifted postdocs, and I learned a lot from them. Thanks to you all. I enjoyed lots of stimulating physics conversations with Wes Campbell over the years. I wish him all the best for his new position as a faculty member at UCLA. It was fun to work with the fellow graduate students, Simcha Korenblit, Jake Smith and Crystal Senko; and the undergrads, Andrew Chew, Aaron Lee, who decided that he loved this experiment too much to leave, and is continuing as a grad student, and most recently with the new undergrad in the team Geo rey Ji. It was a privilege working with smart theorists like Luming Duan and his students Guin-Dar Lin and Zhexuan Gong, Jim Freericks and his postdoc Joseph Wang, Howard Carmichael and his student Changsuk Noh, and Dvir Kafri. David Huse and Rajdeep Sensarma taught me many aspects of the quantum Ising model. iii Thanks to all of you. I enjoyed interacting with the other members of the group immensely, though I did not directly work with them. Thanks to Taeyoung Choi, Susan Clark, Charles Conover, Shantanu Debnath, Brian Fields, Ilka Geisel, Dave Hayes, David Hucul, Volkan Inlek, Kale Johnson, Kenny Lee, Le Luo, Andrew Manning, Dzmitry Mat- sukevich, Peter Maunz, Jonathan Mizrahi, Steve Olmschenk, Qudsia Quraishi and Jon Sterk. A special thanks to Crystal, Wes and Emily for going over my thesis manuscripts and suggesting important corrections. I thank all my thesis committee members (Steve Rolston, Ian Spielman, Chris Jarzynski, Dianne O?Leary, and of course Chris Monroe) for their support in schedul- ing the defense talk, and accommodating my delays. Thanks to Victor Galitski for serving on my PhD candidacy committee two and a half years back. The Joint Quantum Institute provided an excellent environment for research, and I learned a lot from almost all the members in the basement and on the second oor of the CSS building during our random interactions. Thanks to all of you. Thanks to all the sta members of JQI for helping me with non-academic tasks most e ciently, and always in a timely manner. I thank all the funding agents for making my graduate student life smoother. In particular, the DARPA Optical Lattice Emulator program has been a wonderful experience for the last ve years. I thoroughly enjoyed many intellectually invigo- rating discussions in all the OLE meetings. My friends were a constant source of support over all these years. Fortunately, iv they are too numerous to name here. Last but not the least, my family members were always there whenever I needed them. I am eternally grateful to all of you. v Table of Contents List of Tables viii List of Figures ix List of Abbreviations xi 1 Introduction 1 2 Trapped Ions as a Platform for Quantum Simulation 8 2.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2 Ion Trapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.2.1 Trapping 171Yb+ in our Paul trap . . . . . . . . . . . . . . . . 15 2.3 Manipulation of 171Yb+ spin and motional states . . . . . . . . . . . 18 2.3.1 Hyper ne states . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3.2 Doppler cooling . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.3.3 Detection of the spin states . . . . . . . . . . . . . . . . . . . 22 2.3.4 State initialization by optical pumping . . . . . . . . . . . . . 26 2.3.5 Coherent manipulation of the spin states . . . . . . . . . . . . 27 2.3.6 Raman sideband cooling . . . . . . . . . . . . . . . . . . . . . 45 2.4 Vibrational normal modes of trapped ions . . . . . . . . . . . . . . . 46 2.5 Simulating the quantum Ising model . . . . . . . . . . . . . . . . . . 53 2.5.1 Ising interactions . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.5.2 Adiabatic quantum simulation . . . . . . . . . . . . . . . . . . 66 2.6 Experimental Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . 68 2.6.1 Ti:Sapphire laser . . . . . . . . . . . . . . . . . . . . . . . . . 68 2.6.2 Generating 369.5 nm light by frequency doubling . . . . . . . 78 2.6.3 369.5 nm optics schematics . . . . . . . . . . . . . . . . . . . . 78 2.6.4 Mode-locked 355 nm laser . . . . . . . . . . . . . . . . . . . . 81 2.6.5 Optical set up for the Raman transitions . . . . . . . . . . . . 91 2.7 Quantum simulation recipe for experimentalists . . . . . . . . . . . . 97 2.8 Troubleshooting with 174Yb+ . . . . . . . . . . . . . . . . . . . . . . . 107 3 Simulation of the ferromagnetic quantum Ising model 109 3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 3.2 Symmetries of the Hamiltonian . . . . . . . . . . . . . . . . . . . . . 111 3.3 Low energy eigenstates at T=0 . . . . . . . . . . . . . . . . . . . . . 112 3.3.1 States near B=J = 0 . . . . . . . . . . . . . . . . . . . . . . . 112 3.3.2 States near B=J !1 . . . . . . . . . . . . . . . . . . . . . . 117 3.3.3 Quantum phase transition at B = J . . . . . . . . . . . . . . . 118 3.4 Experiment: onset of a quantum phase transition . . . . . . . . . . . 121 3.4.1 Engineering the ferromagnetic Ising couplings . . . . . . . . . 121 3.4.2 Experimental protocol and order parameters of the transition 124 3.4.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 vi 3.4.4 Sources of error in the quantum simulation . . . . . . . . . . . 130 3.5 Scaling up the simulation to N = 16 with 355 nm mode locked laser . 139 4 Three frustrated Ising spins on a triangle 142 4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142 4.2 Frustrated quantum Ising model . . . . . . . . . . . . . . . . . . . . . 145 4.2.1 States near B=J = 0 . . . . . . . . . . . . . . . . . . . . . . . 145 4.2.2 Preparing the entangled state in adiabatic quantum simulation 147 4.3 Frustration and entanglement . . . . . . . . . . . . . . . . . . . . . . 148 4.4 Experimental methods . . . . . . . . . . . . . . . . . . . . . . . . . . 151 4.5 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 154 4.6 Summary and outlook . . . . . . . . . . . . . . . . . . . . . . . . . . 160 5 Frustrated magnetic ordering with tunable range antiferromagnetic couplings161 5.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 5.2 Some features of the long range antiferromagnetic quantum Ising model163 5.2.1 Ground and low energy eigenstates . . . . . . . . . . . . . . . 163 5.2.2 Frustration and the range of the interactions . . . . . . . . . . 166 5.3 Experimental simulation of the model . . . . . . . . . . . . . . . . . . 169 5.3.1 Tuning the range of Ising interactions . . . . . . . . . . . . . . 171 5.3.2 Experimental protocol and the order parameters . . . . . . . . 174 5.4 Results of the quantum simulation . . . . . . . . . . . . . . . . . . . 177 5.4.1 Onset of antiferromagnetic correlations in quantum simulation for N = 10 and N = 16 spins . . . . . . . . . . . . . . . . . . 177 5.4.2 Frustration of the AFM order with increasing range of inter- actions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 5.5 Discussions and conclusion . . . . . . . . . . . . . . . . . . . . . . . . 184 6 Outlook 185 6.1 Scaling up the system - large numbers of equally spaced ions in a Paul trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 6.2 Creating an arbitrary lattice geometry . . . . . . . . . . . . . . . . . 187 6.3 Other interesting spin physics . . . . . . . . . . . . . . . . . . . . . . 188 A Quantum trajectory calculations 190 B Detection of spin states 192 C Relevant Frequencies for 171Yb+ and 174Yb+ 194 Bibliography 195 vii List of Tables 2.1 Phases of various pulses used in quantum simulation. . . . . . . . . . 107 3.1 Symmetries of the eigenstates . . . . . . . . . . . . . . . . . . . . . . 119 5.1 Experimental parameters used to generate long range Ising model with variable range . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174 C.1 Frequency lock points for various lasers . . . . . . . . . . . . . . . . . 194 viii List of Figures 2.1 Schematics of the three layer linear Paul trap . . . . . . . . . . . . . 13 2.2 Ionization beam and oven geometry . . . . . . . . . . . . . . . . . . . 17 2.3 171Yb+ level diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 2.4 Detection of the spin states . . . . . . . . . . . . . . . . . . . . . . . 23 2.5 Fluorescence histograms of the spin states . . . . . . . . . . . . . . . 25 2.6 Optical pumping to the j#zi state . . . . . . . . . . . . . . . . . . . . 26 2.7 Two photon stimulated Raman transition in a system . . . . . . . 28 2.8 Adiabatic elimination of the excited state . . . . . . . . . . . . . . . . 32 2.9 Resonant hyper ne (Carrier) Rabi oscillations in 171Yb+ spin states . 41 2.10 Ramsey interferometry in a 171Yb+ ion . . . . . . . . . . . . . . . . . 42 2.11 Carrier and sideband transitions . . . . . . . . . . . . . . . . . . . . . 44 2.12 Raman sideband cooling . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.13 Image of ten bright 171Yb+ ions in a linear con guration . . . . . . . 47 2.14 Axial and transverse vibrational modes of trapped ions . . . . . . . . 49 2.15 Transverse mode eigenvectors for N = 10 ions . . . . . . . . . . . . . 51 2.16 Nineteen ions in a zig-zag con guration . . . . . . . . . . . . . . . . . 52 2.17 M lmer-S rensen transition in a system of two spins . . . . . . . . . . 55 2.18 Ising couplings for various M lmer-S rensen detuning . . . . . . . . . 63 2.19 Ising oscillations between spin states . . . . . . . . . . . . . . . . . . 64 2.20 Experimental sequence in a quantum simulation . . . . . . . . . . . . 66 2.21 Schematics of the MBR-110 Ti:Sapphire laser . . . . . . . . . . . . . 70 2.22 MBR modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 2.23 MBR-110 Ti:Sapphire laser . . . . . . . . . . . . . . . . . . . . . . . 77 2.24 Schematics of the 369.5 nm beams . . . . . . . . . . . . . . . . . . . . 79 2.25 Schematics of a two photon Raman transition using a mode-locked laser . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 2.26 Radio frequency comb-teeth used in the two photon Raman transitions 84 2.27 Drift in the repetition rate of the 355 nm mode-locked laser . . . . . . 86 2.28 Schematics of the repetition rate lock . . . . . . . . . . . . . . . . . . 87 2.29 The role of the Phase locked loop in the repetition rate stabilization scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 2.30 Raman transition set up . . . . . . . . . . . . . . . . . . . . . . . . . 93 2.31 The Clebsh-Gordan coe cients relevant for the two photon hyper ne Raman transition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 3.1 Energy spectrum of the ferromagnetic transverse Ising model for N = 2 spins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 3.2 Ground state degeneracy splitting by the transverse eld . . . . . . . 116 3.3 A few low energy eigenstates of Eq. (3.1) for N = 5 spins . . . . . . . 120 3.4 Raman spectrum of vibrational modes and Ising coupling pro le for N = 9 spins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 3.5 Binder cumulant and magnetization for the adiabatic theory . . . . . 126 ix 3.6 Emergence of the ferromagnetic spin order . . . . . . . . . . . . . . . 128 3.7 Onset of a quantum phase transition- sharpening of the crossover curves with increasing system size. . . . . . . . . . . . . . . . . . . . 129 3.8 Suppression in the ferromagnetic delity with increasing system size . 131 3.9 Spontaneous emission from the Raman beams . . . . . . . . . . . . . 133 3.10 Detection with a PMT { overlapping photon count histograms . . . . 136 3.11 Suppression in the GHZ coherence with increasing system size . . . . 137 3.12 Quantum simulation of the ferromagnetic Ising model with N = 16 spins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 4.1 Three frustrated Ising spins on a triangle . . . . . . . . . . . . . . . . 143 4.2 Increased sensitivity to quantum uctuations in presence of frustration146 4.3 Energy diagram for J1 6= J2 . . . . . . . . . . . . . . . . . . . . . . . 153 4.4 Population of spin states with ferromagnetic and antiferromagnetic Ising couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 4.5 Entanglement generation through the quantum simulation . . . . . . 157 4.6 Entanglement from the frustration . . . . . . . . . . . . . . . . . . . 158 5.1 Energy of creating spin excitations in the long range antiferromag- netic Ising model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165 5.2 Dependence of the spectrum of Hamiltonian (5.1) on frustration . . . 167 5.3 Ising coupling pro le and t to a power law . . . . . . . . . . . . . . 170 5.4 Dependence of the range of Ising interactions on the M lmer-S rensen detuning and the bandwidth of vibrational modes . . . . . . . . . . . 172 5.5 Onset of the antiferromagnetic ordering with 10 spins . . . . . . . . . 178 5.6 Antiferromagnetic spin ordering with 16 spins . . . . . . . . . . . . . 179 5.7 CCD image of N = 10 antiferromagnetically ordered spins . . . . . . 181 5.8 Frustration of antiferromagnetic spin ordering with increasing range of interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 5.9 Plausible decoherence in our quantum simulation . . . . . . . . . . . 183 x List of Abbreviations AFM Antiferromagnetic AOM Acousto Optic Modulator bsb Blue sideband CCD Charged Coupled Device COM Center of Mass CW Continuous Wave (lasers) DC Direct Current (here used to refer to static voltages) DMRG Density Matrix Renormalization Group EOM Electro Optic Modulator FM Ferromagnetic GHZ Greenberg-Horne-Zeilinger (entanglement) ICCD Intensi ed Charged Coupled Device PMT Photo Multiplier Tube QPT Quantum Phase Transition RF Radio frequency rsb Red sideband RWA Rotating Wave Approximation Yb Ytterbium xi Chapter 1 Introduction A quantum system can be in a superposition of several possible states at the same time, a feature distinctly di erent from the classical superposition [1]. This bizarre property results in many counter-intuitive phenomena, such as non-local correlations or entanglement between di erent parts of a quantum system, leading to the famous Einstein-Podolsky-Rosen paradox [2]. Quantum superposition presents an outstanding challenge in computing, as we have to keep track of an exponentially large number of coe cients to describe the time evolution of a quantum system. As an example, a system consisting of N two level objects (referred to as the qubits, or spins) is represented by a wavefunction j i, which can be written as, j i = X s1=f";#g ::: X sN=f";#g as1;s2;:::;sN js1; s2; :::sNi: (1.1) Here js1; s2; :::; sNi is the state where the rst spin is in state s1 (either ", or #) and so on. A full speci cation of the wavefunction requires 2N of the amplitudes as1;s2;:::;sN . A classical system, on the other hand, requires only N coe cients to describe the probability of an outcome. Conventional computers used for computing today are composed of classical bits, and hence require resources exponentially large in N to faithfully simulate the properties of an arbitrary N qubit quantum system. In his 1981 seminal lecture on ?Simulating Physics with Computers? [3], Feyn- 1 man suggested using quantum logic gates to simulate the behavior of a quantum system. He envisioned the idea of a quantum computer that will \do exactly the same as nature." In 1996, Lloyd showed that quantum computers can be programmed to simulate any local quantum system [4]. Although building a full scale universal quantum simulator or a quantum computer [5] with error correcting codes [6] is a distant dream as of now, experimental e orts to use a controlled quantum system for manipulating speci c quantum information are underway. These systems include trapped ions [7, 8, 9, 10], neutral atoms in optical lattices [11, 12], resonator coupled superconducting qubit arrays [13, 14], electron spins in quantum dots and nitrogen vacancy centers in ultra-pure diamond crystals [15, 14], and photons [16, 17, 18, 19]. A quantum simulator based on Feynman?s original idea may be called a digi- tal quantum simulator [20], where the desired Hamiltonian, H is constructed from piecewise application of local Hamiltonians, H = Pl i=1Hl, following the Trotter expansion, e iHt e iH1t=ne iH2t=n::: e iHlt=n n : (1.2) The error in simulating the Hamiltonian can be kept under a given value by properly choosing the number of steps, n. Another class of quantum simulators, known as analog simulators, continuously follow the Hamiltonian evolution of a physically di erent but mathematically equivalent system [21]. Analog quantum simulators are restricted to simulating a few classes of Hamiltonians, but they prove to be very useful to study some non-trivial many body physics so far. Understanding the physics of many body emergent phenomena is one of the 2 main areas of research in modern physics. Despite many minimalistic models [22, 23, 24] to understand the behavior of the strongly correlated systems [25, 26], such as high temperature superconductors [27], heavy fermion materials [28], colossal magneto-resistance materials [29], frustrated spin liquids [30], and quasi-low dimen- sional materials [31], the mechanism behind some of the exotic properties remains mostly unknown. Numerical techniques such as quantum Monte Carlo [32] and den- sity matrix renormalization group (DMRG) [33] provide valuable insights into the many body physics of a quantum system, but they do not work very well when the underlying model involves frustration [34] and long range interactions [35]. A quan- tum simulator takes a bottom-up approach, where the behavior of a quantum system under a well understood Hamiltonian is experimentally studied, and complexities are added piece by piece. The search is for the minimal interactions between the fundamental building blocks required to explain the many body emergent properties of the macroscopic system. Cold atomic systems provide a nearly ideal platform for quantum simulation, due to their long coherence time, near perfect initialization and detection delities and ability to support many classes of interactions. In recent years, a number of cold atom experiments, with both neutral atoms and ions, have simulated and studied interesting many-body physics, such as transition from super uid to Mott insulator in Bosonic systems [36, 37, 38], quantum phase transitions between spin phases or quantum magnetism [39, 40, 41, 42, 43, 44], Bardeen-Cooper-Schrie er (BCS) pairing [45] and BEC-BCS crossover physics [46], investigation of quantum criticality [47, 48, 49], synthetic gauge elds to simulate quantum Hall physics and 3 topological insulators [50], quantum simulation of relativistic dynamics [51, 52], and long range spin models involving spin frustration [43, 53]. Cold atom systems o er some distinct advantages over the condensed matter systems, some of which are, Cold atom systems show remarkable tunability, and often can access a much larger parameter range than the condensed matter systems. As we shall discuss in this thesis, the e ective spin interactions in a simulated spin model can be changed in sign, magnitude, and range by changing the frequency of a laser beam [54]. In solid state systems, the sign of the magnetic interactions cannot be changed, and altering the strength of the couplings by a modest amount may require application of large hydrostatic pressures [55]. Another example is the control of interactions in a cold atomic system provided by Feshbach resonances [56, 57]. Cold atomic systems have a very low level of defects compared to the solid state systems. For example, the extreme low entropy in a spin-polarized Mott insulator state allows the study of quantum magnetism [40]. Defects can be controllably added to study the physics of disordered systems such as Anderson localization [58, 59, 60]. They have ultra-low densities compared to the solid state systems. Typical spacing between the neighboring trapped ions or optical lattice sites is of the order of a micron, much larger than the typical electronic spacing of a few Angstroms in solid state materials. This allows optical imaging of individual components, and direct observations of spin ordering in quantum magnetism [43, 61]. The time scale of the dynamics in a cold atom system is longer and more experi- 4 mentally accessible than their solid state counterpart. Trapped ions have been in the forefront of quantum information processing since experimental investigation began [62], demonstrating universal quantum gates [63, 64, 65, 66], and quantum teleportation [67, 68, 69, 70]. The long coherence time of trapped ion systems, and the easy access to long range tunable interactions [71, 54] make them an outstanding choice to simulate long range spin models that demonstrate quantum phase transitions [47] and spin frustrations [72]. In this work, we simulate a long range quantum Ising model with a chain of up to sixteen 171Yb+ ions in a radio-frequency trap. Two hyper ne states of each 171Yb+ ion constitute an e ective spin-12 system. The collective vibrational modes of the trapped ions, excited o -resonantly with stimulated Raman transitions by precisely tuned laser elds, act as an ?information bus?, and mediate the two body spin interactions [71]. An e ective transverse external eld, simulated by stimulated Raman transitions between the spin levels, introduces quantum uctuations in the system. We study the ground state properties and excitations in the system by preparing the spins in an eigenstate of a trivial Hamiltonian and tuning it to the more complicated one to be simulated, following the adiabatic quantum simulation protocol [73]. With our quantum simulator we investigate various non-trivial many body physics, such as a quantum phase transition between magnetic phases [41], spin frustration leading to entanglement [43], and the observation of spin ordering in a system of sixteen spins (two qubytes), the largest number of spins used to process quantum information in a linear trap to this date. The present work serves as a 5 benchmark for quantum simulation with larger systems, where the underlying theory becomes intractable with conventional computers. This thesis is divided into the following chapters. The second chapter gives the necessary theoretical and experimental background on quantum simulation with trapped ions. This includes a brief discussion on the collective vibrational motion of a chain of 171Yb+ ions in a radio-frequency Paul trap, that are used to generate the spin interactions. We derive the e ective Ising Hamiltonian starting from the basic interactions of the ions with the Raman laser eld. Control over the Ising couplings, and the quantum simulation protocol are discussed. In the experimental section, we discuss the hardware used in the set up, in particular the lasers that drive the stimulated Raman transitions. Section 2.7 is devoted to discussing the experimental procedure step by step. The third chapter discusses and presents some results on the quantum simulation of the ferromagnetic Ising model. We observe the onset of ferromagnetic spin ordering in a system of up to sixteen spins. As the e ective external eld is tuned with respect to the Ising interactions, the system undergoes a crossover from the paramagnetic to the ferromagnetic phases. The sharpness of this crossover increases with the system size, consistent with the expected quantum phase transition in the thermodynamic limit. We simulate the long range antiferromagnetic quantum Ising model in the fourth and the fth chapters with up to sixteen spins. The long range interactions in the trapped ion system gives rise to a fully connected spin network. The interactions 6 can be suitably tailored to simulate a higher dimensional lattice geometry, in this one dimensional chain of ions. In chapter 4, we simulate a frustrated spin network with three spins interacting antiferromagnetically on the corners of a triangle, and study the many body ground state. Frustration leads to a large degeneracy, with six of the eight (= 23) basis spin states belonging to the ground state manifold. We detect and characterize the entanglement coming out of this frustration induced degeneracy, and contrast it to the entanglement coming out of the symmetries in the Hamiltonian (section 3.2). The amount of frustration in the long range antifer- romagnetic Ising model is controlled by varying the range of the antiferromagnetic couplings. We compare the spin order for various degrees of frustration in chapter 5. The sixth chapter gives an outlook for future directions. 7 Chapter 2 Trapped Ions as a Platform for Quantum Simulation 2.1 Overview This chapter gives a brief overview of the theoretical and experimental tools needed in our quantum simulation experiment with trapped 171Yb+ ions. I brie y describe the working principle of an ion trap, without going into too much detail. Two hyper ne states of the 171Yb+ ions are used as the pseudo spin-12 states, which are manipulated by various standard atomic physics techniques, such as Doppler cooling, optical pumping, stimulated Raman transitions, and Raman sideband cool- ing. We give a short introduction to each of these topics. Finally we describe some of our experimental apparatus in detail, particularly focusing on the lasers used to drive the stimulated Raman transitions. 2.2 Ion Trapping A charged particle cannot be trapped in space by electrostatic forces alone, as pointed out by Samuel Earnshaw [74] in 1842. This is due to the fact that, an electrostatic potential V (X; Y; Z) has to satisfy the constraint r2V (X; Y; Z) = 0 (The Laplace equation, r2 @ 2 @X2 + @2 @Y 2 + @2 @Z2 ) everywhere in the free space, and the properties of the Laplace equation prevents its solution to admit any local extremum. 8 Ion traps work by applying either an oscillating electric eld (Paul traps [75]), or a static magnetic eld (Penning traps [76]) in conjunction with static electric elds to create an extremum in the e ective time averaged potential. We use a radio frequency Paul trap for our quantum simulation experiments. An electrostatic potential created by applying static voltages on metallic elec- trodes cannot create a potential extremum in space, but by properly choosing the boundary conditions, it is possible to create a potential saddle. As an example, the general form of a quadrupole potential is, V (X; Y; Z) = X2 + Y 2 + Z2: (2.1) The Laplace equation puts the constraint + + = 0, which admits the solution = 2; = 1; = 1 for some speci c boundary conditions. The potential has a saddle point at (X = 0; Y = 0; Z = 0), i.e., V (X; Y; Z) is con ning along the Y and the Z directions, but anti-con ning along the X direction for a positive charge. On top of the electrostatic eld, we superpose a spatially inhomogeneous os- cillating electric eld, the time average of which will generate an e ective con ning potential. Let?s consider the motion of a charged particle in an oscillating eld [77] alone. The force equation is, m X = FX(t) = eE0(X) cos rf t: (2.2) Here we assume that the oscillating electric eld with frequency rf and amplitude 9 E0(X) couples to the X component only, e and m are the charge and the mass of the particle respectively. If E0(X) = E0 is a homogeneous eld, the average force hFX(t)it = 0, since hcos rf tit = 0, and the solution to Eq. (2.2) is, X(t) = eE0 m 2rf cos rf t+X0: (2.3) Here X0 = hX(t)it is an integration constant and represents the time averaged position of the charged particle, and we have assumed that at t = 0 the particle was at rest, i.e., _X(t = 0) = 0. Here h:::it denotes averaging over a time long compared to the time period of the oscillating eld. Hence, under a homogeneous oscillating electric eld, a charge particle does not experience any con ning potential, and oscillates with the driving eld. The oscillation of the charged particle at the frequency of the driving eld is known as the micromotion. Now, we add a small inhomogeneity in the driving eld amplitude E0(X), maintaining its value at X = X0, and expand it about X = X0, E0(X) = E0(X0) + @E0(X) @X X=X0 (X X0): (2.4) Let (t) X(t) X0 be the displacement of the charged particle about the mean position (we assume that the mean position of the charged particle is unchanged with this added inhomogeneity. For a detailed discussion see ref [77]). 10 From Eq. (2.2), the time averaged value of the force is, FX(t) = eE0(X) cos rf t = e E0(X0) + @E0(X) @X X=X0 (t) ! cos rf t = eE0(X0) cos rf t e2 m 2rf E0(X0) @E0(X) @X X=X0 (cos rf t) 2: (2.5) Here we have approximated (t) by the displacement of the charge particle in a homogeneous eld (Eq. (2.3)) in the last line. The time average of this force is, F (X0) hFX(t)it = e2 2m 2rf E0(X0) @E0(X) @X X=X0 = e2 4m 2rf @ @X0 E20(X0) = e @ @X0 " e 4m 2rf E20(X0) # : (2.6) We identify the quantity inside the square bracket in the last line of Eq. (2.6) as a pseudo-potential, known as the ponderomotive potential, pond(X0) e 4m 2rf E20(X0); (2.7) and the time averaged force in Eq. (2.6) is known as the ponderomotive force. Note that the force is independent of the sign of the charge, and hence the same ponderomotive potential can trap both a positive and a negative charge. It can be shown that the kinetic energy in the micromotion of the charged particle is Ekinetic = e pond(X0). This kinetic energy vanishes for E0(X0) = 0, i.e., where the 11 amplitude of the oscillating eld vanishes. The region of E0(X0) = 0 is referred to as a ?radio-frequency null?. Depending on the geometry of the trap, the radio- frequency null may be a point, a collection of discrete points, or a line. In a quantum information experiment, the static potentials are adjusted so that the equilibrium positions of the ions lie on a radio-frequency null to minimize the micromotion [78], which may couple to the vibrational modes of the ion chain, and result in quantum decoherence by heating up the modes. A more rigorous analysis of the motion of a charged particle in an ion trap involves solving the Mathieu equation, discussed in Ref. [75]. The ponderomotive potential overcomes the decon ning potential due to static electric elds along the X direction in our example, and the superposition of the static and the pondero- motive potentials result in a three dimensional con nement of the charged particle. The e ective potential takes the form, V (X; Y; Z) = 1 2 m !2XX 2 + !2Y Y 2 + !2ZZ 2 ; (2.8) where (!X ; !Y ; !Z) are called the secular frequencies along the three trap axes. While constructing an ion trap for an experiment, the desired trapping po- tential is simulated with commercially available softwares, such as the 3-D Charged Particle Optics Program (CPO-3D) produced by CPO Ltd. The software solves the Laplace equation for a set of boundary conditions, given in terms of voltages on the electrodes. Ref. [70] gives a detailed instruction on how to use CPO-3D for electrostatics simulations. 12 Figure 2.1: Schematics of the three layer linear Paul trap: A linear chain of 171Yb+ ions are con ned in a three layer radio-frequency (Paul) trap. The electrodes are gold-coated on alumina substrates. The top and the bottom sets of electrodes are approximately 250 m thick, and carry static voltages (DC), and the middle ones are approximately 125 m, and carry a radio frequency (rf) voltage at a frequency of rf=2 = 38:6 MHz. The ions (shown in dots) form a linear crystal along the Z axis of the trap. The electrodes labeled GND are grounded. 13 Fig 2.1 shows a schematic of the radio-frequency trap used to trap Ytterbium ions (Yb+) in our experiment. The top and the bottom electrodes carrying static voltages of up to a hundred volts are segmented into three zones each. A radio- frequency signal at about 38.6 MHz is coupled to the middle electrodes through a helical resonator (transformer) [79], with a resonance around 38.6 MHz and a Q- factor of about 200. The input power to the helical resonator is approximately 27 dBm, or 500 mW. This generates a radio-frequency voltage of about 200-300 Volts, leading to secular frequencies of !X !Y 2 5 MHz. The three layer trap is housed in a vacuum chamber with a pressure of < 10 11 Torr (reading EO3 on a SenTorr vacuum gauge). The static voltages DC1 through DC6 (along with the radio frequency) in g 2.1 control the trapping potential. The static voltages are provided by an 8 channel high precision HV module from the Iseg Spezialelektronik GmbH company. Linear combinations of the static voltages are used to manipulate the position of the ions and properties of the trapping potential, such as The end average voltage Vend = (V1 + V2 + V5 + V6)=4 and the central average Vcentral = (V3 + V4)=2 control the overall strength of the trapping potential, where Vi is the static voltage applied on the electrodes labeled DCi (i = 1; 2; :::; 6). The ratio of these two voltages also controls the principal axes of the trap along the transverse directions. Principle axes are the Cartesian coordinate axes X and Y such as the trapping potential does not involve any term coupling both the coordinates, i.e., the coe cient XY = 0 in the 14 generalized form of a quadratic potential, V (X; Y ) = XXX2 + XYXY + Y Y Y 2. The Z-push voltage VZ = [(V1 + V5) (V2 + V6)]=2 controls the ions position along the Z axis. The displacement of the ion for a given change in the Z push voltage depends on Vend as well. For example, a change of 3 Volts in VZ results in a single ion displacement of about 20 m when the end average voltage Vend = 10 Volts. The end vertical di erence V end vert = [(V1 + V2) (V5 + V6)]=2 and the central di erence V central = V3 V4 are used to minimize the radio frequency micromotion of the ions [78]. 2.2.1 Trapping 171Yb+ in our Paul trap The vacuum chamber housing the trap contains two ovens made of stainless steel hypodermic needles (in a Titanium holder) packed with neutral Ytterbium metal, one with naturally abundant Yb, dominated by 174Yb (30%), and the other with isotopically enriched 171Yb ( 90% isotope purity). Ytterbium ions (171Yb+ ) are loaded into our linear Paul trap from the isotopically enriched oven by photoion- ization. A current of 2.4-2.8 Ampere is sent through the oven, and the Joule heating produced by the current sublimates Yb atoms into a directional spray. Neutral Yb is ionized into 171Yb+ in two steps. Exciting an electron from the 1S0 level to the 1P1 level of the neutral Yb. 15 Ionizing that electron by supplying 1P1-energy continuum or more. Another radiation at 369.5 nm or 355 nm ionizes the atom into a 171Yb+ ion. The rst step is done by a 1 mW laser beam of 398.911 nm radiation from a semicon- ductor diode laser (Toptica DL-100). This beam is focused cylindrically with beam waists (1=e2 radius in intensity) of 100 m in the horizontal direction and 50 m in the vertical direction. The second step can be performed by any light with a wavelength under 394.1 nm. We use about 1 mW of 369.5 nm light beam focused into a cylindrical spot with waists 100 m 50 m to ionize. It takes about half a minute to trap a single ion after the oven warms up (a couple of minutes after the current source is turned on). The 369.5 nm light is generated by frequency doubling 739 nm light (generated by a Ti:Sapphire laser or a semiconductor diode laser with a tapered ampli er system) by an LBO crystal in a cavity (WaveTrain, made by Spectra-Physics). The loading process can be expedited greatly with a brief pulse (under a second) of 355 nm light (about 1 W light focused into 100 m 7 m, this beam is also used for stimulated Raman transitions) providing the ionizing energy. Isotope selectivity - The oven used for loading 171Yb+ contains 90% pure 171-isotope. The remaining is mostly 174Yb+ isotope, 1% other isotopes of Yb and other impurities. The isotope shift between the 171Yb+ and 174Yb+ in the 1S0 1P1 transition frequency is about 800 MHz [80], more than the power broadened transition width (about 200 MHz, found empirically). We send the ionization beams approximately perpendicular to the direction of the atomic spray from the oven (Fig. 2.2), thus eliminating rst order Doppler broadening. Empirically, we load 171Yb+ 16 Figure 2.2: Ionization beam and oven geometry: The ionization beam is sent nearly perpendicular to the atomic beam jet sprayed from the 171Yb+ enriched (90% isotopically pure) oven. This nearly eliminates the Doppler broadening of the 1S0 1P1 transition line. The gray rectangles denote the vacuum chamber windows. This gure is not to scale. with more than 98% success rate. While loading 171Yb+ ions we shine the loading region with the Doppler cooling beam and the 935 nm repump (along with the two loading beams), and look for a uorescence signal on a charged coupled device (CCD) imager. The integration time of the CCD is set to about 200 ms. A wrong isotope of Ytterbium or another atomic species does not uoresce from the Doppler cooling beam, and appears as a dark spot in the ion chain. We experimentally nd that a lower radio frequency and static voltage on the trap electrodes make the trapping easier. An additional cooling beam, called the protection beam, which is detuned from the 2S1=2 !2 P1=2 resonance by 600 MHz is also kept on during the loading process. When we use 369.5 nm beam for the second step in the ionization, the ions load one by one, but with the 355 nm beam, multiple ions load at the same time. 17 Melting of the ion crystal and re-capturing - One of the most important problems of dealing with a long chain of ions is collisions with background gases present in the vacuum chamber, which result in melting of the crystal. The rate of the background collisions increase with increasing system size, and is typically one melting event per ve minutes on an average for a chain of 10 ions (this is approximate, we did not investigate the statistics of these collision events). Once the ion chain melts, we try to re-capture them by turning on the Doppler cooling and the protection beam, and lowering the trap depth. The radio frequency power going into the trap is lowered by about 11 dB (from 26 dBm to 15 dBm), and the average static voltage (End average DC) is reduced to about 4 Volts. These settings are empirically found. In our quantum simulation experiments, we monitor the Doppler cooling uorescence from the ions on a PMT to check for melting. If the re-crystallization process is initiated soon enough after the melting occurs, all or most of the ions come back into the crystalline structure with a decent probability (works > 50% of the time). 2.3 Manipulation of 171Yb+ spin and motional states 2.3.1 Hyper ne states 171Yb+ has a spin-12 nucleus, resulting in hyper ne structure in the electronic ground state. Figure 2.3 shows the ne structure levels with their hyper ne sublevels in 171Yb+ . The two hyper ne states of 2S1=2jF = 1;mF = 0i and 2S1=2jF = 0;mF = 0i form an e ective spin-12 system, identi ed as j "zi and j #zi respectively. Here 18 Figure 2.3: 171Yb+ level diagram 19 ~ p F (F + 1) is the total angular momentum of the atom, and ~mF is its projection along the quantization axis, in our experiments de ned by the externally applied magnetic eld of BY 5 G, where ~ = h=2 , h being the Planck?s constant. This magnetic eld is not to be confused with any e ective magnetic eld in the spin models that we want to simulate with this system. As we shall discuss later, we simulate the e ective transverse eld (B) in the quantum Ising model by laser induced stimulated Raman transitions. The spin states are not sensitive to the Zeeman shift in the leading order of the applied magnetic eld BY , making them useful in precision atomic clocks [81]. The hyper ne frequency splitting between them is !hf=2 = 12 642 812 118:5 Hz + B2Y 310:8 (Hz=G 2) [82]. 2.3.2 Doppler cooling The 171Yb+ ions are Doppler cooled on the 2S1=2 2 P1=2 line, with a Gaussian beam at a wavelength = 369:521525 nm, red detuned from the resonance by about 25 MHz. Each time an atom absorbs a photon, it acquires an ~k recoil momentum from the radiation eld, where ~k is the momentum vector of the cooling light. When an atom moves opposite to the direction of beam propagation, i.e., towards the light source, the frequency of the light as observed from the atom?s rest frame is up-shifted from its laboratory frame frequency, due to the Doppler e ect. Since the beam is red detuned from the 2S1=2 2P1=2 resonance, the up-shifted frequency is closer to atomic resonance, and the atom absorbs more photons. On the other hand, when the atom moves away from the source, it sees a down-shifted frequency of the cooling beam, 20 and absorbs less. Thus on an average, the atom experiences more momentum kick (and more radiation pressure) while moving opposite to the beam propagation than while moving along the beam propagation direction. Thus the atom slows down, on average, by absorbing photons from the radiation eld. Once in the excited state, the atom emits the photon back to the eld via spontaneous emission, but this photon is emitted in a random direction with zero average momentum, and the average momentum transfered to the atom from the eld vanishes. Thus the beam slows down the atom. For trapped ions, the con nement is achieved by electrical voltages, and thus Doppler cooling works provided the cooling beam couples to motion along all the principle axes. In order to cool both the spin states (j"zi and j#zi), we frequency modulate the Doppler cooling beam at 7.37 GHz by using an Electro Optic Modulator (EOM). The second sideband ( 1% of the carrier strength) generated by this EOM at 14.74 GHz couples the 2S1=2jF = 0;mF = 0i state (or the spin j #zi state) to 2P1=2jF = 1;mF = 0i state. Thus both the spin states scatter from the Doppler cooling beam. The optical power in the cooling beam used is approximately 25 W focused to a spot size of approximately 100 m 10 m at the ion , and we cool the ions for about 3 ms. To e ciently cool the ion, we use an additional laser at 935.2 nm (Toptica DL-100) sent through an EOM driven at 3.07 GHz to re-pump the 2D3=2 levels that 2P1=2 states leak into with a probability of about 0.005 [70]. This laser pumps the atom to the 3D[3=2]1=2 state, from which it returns promptly to the 2S1=2 states [83], without mixing between the j"zi and the j#zi states, as the transition 3D[3=2]1=2jF = 21 0;mF = 0i ! j#zi is forbidden. The 935 nm laser beam has about 20 mW power, and is not focused tightly (with about a few hundred microns Gaussian beam waist at the ion location) to make the alignment process easier. We frequency stabilize this laser by feeding back to the grating in the laser cavity and the diode current, using a slow software lock that compares the frequency of this laser measured by a wavemeter (WS Ultimate Precision by High Finesse GmbH) and a set frequency (320.56922 THz). This software lock is technically easier than implementing other cavity based locks, such as a Pound-Drever-Hall lock [84], and empirically found to be su cient for the current application. The atom may also leak into the long lived low lying 2F7=2 states with a life- time of about 10 yrs, presumably in a non-radiative process involving collisions with the background atoms in the vacuum chamber [85, 86], at a rate of approximately one every couple of hours for a single atom. A laser at 638.6 nm is scanned be- tween the two transitions near the wavelengths of 638.6151 nm (2F7=2jF = 4i ! 1D[5=2]5=2jF = 3i) and 638.6102 nm (2F7=2jF = 3i ! 1D[5=2]5=2jF = 2i) (Fig. 2.3), again by using the wavemeter frequency lock. 2.3.3 Detection of the spin states The spin states are detected by a spin dependent uorescent technique. The spin j "zi state is excited with a 369.53 nm beam, on resonance with the 2S1=2jF = 1;mF = 0i $ 2P1=2jF = 0;mF = 0i transition. Once excited to the 2P1=2jF = 0;mF = 0i state, the ion spontaneously decays to one of the three 2S1=2jF = 1;mF = 22 Figure 2.4: Detection of the spin states: A near resonant laser beam at 369.5 nm couples the j "zi state to the 2P1=2jF = 0;mF = 0i state, which can uoresce back into the j "zi state, or to the Zeeman states 2S1=2jF = 1;mF = 1i, but not to the spin j#zi state. The detection beam has all the polarization components ( ; +; ) and hence the Zeeman states are excited back to the 2P1=2jF = 0;mF = 0i state, and the atom can uoresce again. This light appears o -resonant to the j#zi state by 12.6 GHz, and hence hardly scatters from the j#zi state. We collect the uorescence by a di raction limited optics on a PMT (Fig. 2.5) or a CCD imager. Here the solid black arrows represent stimulated absorption of the detection laser beam by the ion, and the red fuzzy lines show the spontaneous emission channels. The gray arrow shows the same detection light as appears to the j#zi state (the dark state). 23 0; 1i states after about 10 ns [87]. Since 2P1=2jF = 0;mF = 0i $ 2S1=2jF = 0;mF = 0i transition is forbidden by having a zero matrix element of the dipole moment operator between these two states, this forms a cycling transition, and spin state j "zi (and the Zeeman states) appears as a ?bright? state. Figure 2.4 shows the spontaneous channels used in the detection. We collect the uorescence with di raction limited optics (NA=0.25) on a photomultiplier tube (PMT) or a CCD imager. The overall imaging magni cation is about 130, and the set up is similar to that explained in Ref. [88]. The number of emitted photons from a bright ion in a given time interval is distributed according to a Poisson distribution. We collect about 10 photons on average with the PMT, when the detection beam is on for 800 s. The same beam appears o -resonant to the spin state j #zi, by about 12.643 GHz, and hence does not excite this state. Spin j #zi appears as a ?dark? state on the PMT or the CCD imager. To repump from all the Zeeman states of 2S1=2 manifold, the detection beam contains all the three polarizations ( ; +; ) w.r.t. the external magnetic eld of about 5 G, required to de ne the quantization axis and to avoid coherent population trapping [89]. The two spin states may get mixed up if the 2P1=2jF = 1;mF = 0; 1i states are populated, as they couple to both the spin states. The detection beam is 2.105 GHz detuned from the 2S1=2jF = 1;mF = 0i $ 2P1=2jF = 1;mF = 0; 1i states, and the probability of o -resonantly populating the states is 10 5. This o - resonant excitation alters the emitted photon histogram only slightly [90, 91]. We 24 Figure 2.5: Fluorescence histograms of the spin states: A single 171Yb+ ion is excited by a laser beam which is nearly on resonance with the 2S1=2jF = 1;mF = 0i $ 2P1=2jF = 0;mF = 0i transition. we collect the uorescence of the ion on a photomultiplier tube (PMT) for 800 s. A histogram of the photon counts is shown for the bright state (j "zi) in red. The spin state j #zi appears dark (blue histogram), as the detection laser beam is o -resonant to the 2S1=2jF = 0;mF = 0i $ 2P1=2jF = 0;mF = 0i transition. Here we prepared the dark state by optical pumping, and the bright state by applying a carrier Raman pulse. 25 Figure 2.6: Optical pumping to the j #zi state: A laser beam resonant with the 2S1=2jF = 1;mF = 0i ! 2P1=2jF = 0;mF = 0i (the line used for detecting the spin states, Fig. 2.4) is frequency modulated by a 2.105 GHz EOM. The rst order sideband couples the j "zi state to the 2P1=2jF = 1;mF = 0; 1i states, that can spontaneously decay to the j#zi state. The ion gets trapped in this state. Here the solid black lines represent stimulated absorption of the detection laser beam by the ion, and the red fuzzy lines show the spontaneous emission channels. We have only shown the stimulated absorption and spontaneous emission channels responsible for trapping the system in the dark state, and not shown any decay into the bright states. get a spin detection delity of 98:5% by a PMT, and about 93% by a CCD imager. The reduced delity with the CCD imager is primarily due to the presence of electronic noise on the CCD [91]. 2.3.4 State initialization by optical pumping The spin state of an ion is initialized in the j#zi state by an incoherent optical pumping technique. For this, we frequency modulate the beam on resonance with 26 the 2S1=2jF = 1;mF = 0i $ 2P1=2jF = 0;mF = 0i transition by 2.105 GHz using an EOM. The rst order sideband generated by the EOM couples the 2S1=2jF = 1;mF = 0i state to the 2P1=2jF = 1;mF = 0; 1i states, which can decay into the spin j#zi state (Fig. 2.6). Once the atom is in the j#zi state, the optical beam is o - resonant from the 2P1=2 states by 12.643 GHz, and hence hardly scatters, and gets trapped in that state. The probability of trapping into this dark state increases with number of scattering events, and eventually almost all the population is transfered to the spin j #zi state. In our set up, the optical pumping e ciency is more than 99% for a 3 s optical pumping time. 2.3.5 Coherent manipulation of the spin states Once the pure state j #zi is prepared using spontaneous emission induced optical pumping, it can be coherently manipulated either with microwave magnetic elds or with two photon laser induced stimulated Raman coupling. Microwave radiation couples to the magnetic dipole moment matrix element between two states, and induces coherent Rabi oscillations between them. While microwave radiation has been used to perform quantum information experiments in 171Yb+ , it is not ideal for quantum simulation of spin models, as the spin interactions are mediated by the phonon modes, and the microwave eld does not have su cient momentum to excite the vibrational modes of an ion chain. We use stimulated Raman transition induced by optical elds for coherent manipulation of spins. The laser eld couples the spin states to an excited state 27 Figure 2.7: Two photon stimulated Raman transition in a system: Two laser beams with frequencies !La and ! L b o -resonantly couple the low lying energy states jai and jbi to the excited state jei, with single photon Rabi frequencies g (de ned in the text). If the system initially is in one of the states jai or jbi, and the detuning from the excited state is much larger than the Rabi frequency g, this system can be approximated as a two level system, with coherent Rabi opping between the states jai and jbi at a rate = g g=2 . (2P1=2 states). In order to understand the physics of two photon stimulated Raman transition, let?s consider a system, shown in Fig. 2.7. The low lying energy states jai and jbi with energies ~!a and ~!b respectively are coupled to the excited state jei via the two continuous wave (CW) laser elds at frequencies !La and ! L b , that are detuned from the excited states by a frequency . 28 We de ne, !ea !e !a (2.9a) !eb !e !b (2.9b) !ba !b !a (2.9c) From Fig. 2.7, we get the relation between the frequencies, !ea ! L a = (2.10a) !ea ! L b = + ! (2.10b) !eb ! L a = !ba (2.10c) !eb ! L b = + ! !ba: (2.10d) In the absence of the coupling elds, the atomic Hamiltonian (~ = 1) is, H0 = !ajaihaj+ !bjbihbj+ !ejeihej: (2.11) Let us assume that the wavefunction at time t, (t) = Ca(t)jai+ Cb(t)jbi+ Ce(t)jei: (2.12) Schr odinger?s equation i@ @t = H0 gives us, C (t) = C (0)e i! t ( = a; b; e) (2.13) 29 where the coe cients C (0) are integration constants obtained from the initial con- ditions. The interaction Hamiltonian between the laser eld and the atom is [92], HI = E(t) = h aejaihej+ eajeihaj+ bejbihej + ebjeihbj i E0 h cos(!La t) + cos(! L b t) i : (2.14) Here E(t) is the total electric eld. Both the laser elds are assumed to have the same electric eld amplitude, E0. We ignore the spatially varying term (kX) and the o -set in the phases of the electric elds for now, and treat the problem in one dimension. Here is the dipole moment of the atom, with the matrix elements, ae = haj jei (2.15a) be = hbj jei (2.15b) ea = ae = hej jai (2.15c) eb = be = hej jbi: (2.15d) We get the relations connecting Ca(t); Cb(t) and Ce(t) in Eq. (2.12) by equating the coe cients of j i ( = a; b; e) on both sides of the Schr odinger?s equation, 30 i@ (t)@t = (H0 +HI) (t) to each other, i _Ca(t) = !aCa(t) gCe(t) h cos(!La t) + cos(! L b t+ ) i (2.16a) i _Cb(t) = !bCb(t) gCe(t) h cos(!La t) + cos(! L b t+ ) i (2.16b) i _Ce(t) = !eCe(t) g h Ca(t) + Cb(t) ih cos(!La t) + cos(! L b t+ ) i ; (2.16c) where the single photon Rabi frequency, g = aeE0 = beE0, assuming ae = be. We want to see how the interaction Hamiltonian modi es Eqs. (2.13). Thus, we de ne the slowly varying amplitudes ~Ca(t); ~Cb(t) and ~Ce(t) through the equations C (t) ~C (t)e i! t ( = a; b; e): (2.17) Note that for HI = 0, ~C (t) = C (0) ( = a; b; e). We use Eqs. (2.17) in Eqs. (2.16), expand the cosine terms to get the relations between ~Ca(t); ~Cb(t) and ~Ce(t), i _~Ca(t) = g 2 ~Ce(t)e i!eat h ei! L a t + e i! L a t + ei! L b t + e i! L b t i (2.18a) i _~Cb(t) = g 2 ~Ce(t)e i!ebt h ei! L a t + e i! L a t + ei! L b t + e i! L b t i (2.18b) i _~Ce(t) = g 2 h ~Ca(t)e i!eat + ~Cb(t)e i!ebt i h ei! L a t + e i! L a t + ei! L b t + e i! L b t i : (2.18c) Next, we make a rotating wave approximation (RWA), where we ignore fast oscillating terms in the exponentials, by neglecting exponents that involve a sum of two large (optical) frequencies. Using Eqs. (2.10) in Eqs. (2.18), we get, after 31 Figure 2.8: Adiabatic elimination of the excited state: Solution of Eqs. (2.19) for a. = 5jgj, b. = 10jgj, c. = 20jgj d. = 100jgj. The red curve shows the probability that the system will be in state jai at time t, Pa(t) = j ~Ca(t)j2 = jCa(t)j2, similarly the blue and the black curves show Pb(t) and Pe(t). The curves are obtained by numerically solving the di erential equations (Eqs. (2.19)) by Wolfram Mathematica, with the initial conditions ~Ca(0) = 1; ~Cb(0) = 0; ~Ce(0) = 0. The frequency splitting !ba between the states jai and jbi is assumed to be very small compared to g and . As the detuning is increased compared to the single photon Rabi frequency g, the three level system behaves more like a two level system, composed of the low lying states jai and jbi. 32 RWA, i _~Ca(t) = g 2 ~Ce(t) e i t + e i( + !)t (2.19a) i _~Cb(t) = g 2 ~Ce(t) e i( !ba)t + e i( !ba+ !)t (2.19b) i _~Ce(t) = g 2 h ~Ca(t) ei t + ei( + !)t + ~Cb(t) ei( !ba)t + ei( !ba+ !)t i (2.19c) These coupled di erential equations can be numerically solved on a computer. Fig 2.8 shows the time evolution of the co-e cients j ~C (t)j2 ( = a; b; e) for four di erent ratios of =jgj (we set !ba = ! = 0 in this calculations), with the initial conditions, ~Ca(0) = 1; ~Cb(0) = 0; ~Ce(0) = 0. Adiabatic elimination of the excited state - Since the sates jai and jbi are not connected directly by a eld, we expect the coe cients ~Ca(t) and ~Cb(t) to vary much more slowly compared to the exponentials in Eq. (2.19c) ( jgj). 33 Thus we integrate Eq. (2.19c), keeping ~Ca(t) and ~Cb(t) constant, ~Ce(t) i 2 g " ~Ca(t) h Z t 0 ei tdt+ Z t 0 ei( + !)tdt i + ~Cb(t) h Z t 0 ei( !ba)tdt+ Z t 0 ei( !ba+ !)tdt i # = i 2 g " ~Ca(t) hei t 1 + ei( + !)t 1 + ! i + ~Cb(t) hei( !ba)t 1 !ba + ei( !ba+ !)t 1 !ba + ! i # g 2 " ~Ca(t) h ei t + ei( + !)t 2 i + ~Cb(t) h ei( !ba)t + ei( !ba+ !)t 2 i # : (2.20) In the last line, we have approximated all the denominators by , as !ba; !. Using Eq. (2.20) in Eq. (2.19a), we get, _~Ca(t) = i jgj2 4 e i t + e i( + !)t " ~Ca(t) h ei t + ei( + !)t 2 i + ~Cb(t) h ei( !ba)t + ei( !ba+ !)t 2 i # = i jgj2 4 " ~Ca(t) h 2 + ei !t + e i !t 2e i t 2e i( + !)t i + ~Cb(t) h 2e i!bat + e i(!ba !)t + e i(!ba+ !)t 2e i t 2e i( + !)t i # i jgj2 4 " ~Ca(t) h 2 + ei !t + e i !t i + ~Cb(t) h 2e i!bat + e i(!ba !)t + e i(!ba+ !)t i # ; (2.21) 34 where we have thrown out the fast oscillating terms e i t and e i( + !)t, as they average out to zero over a time period of oscillations of the other terms. Eq. (2.21) involves only the two states jai and jbi and looks similar to Eq. (2.16a). We can further apply RWA to Eq. (2.21), and ignore terms that are oscillating at !ba and !. This is a good approximation for the hyper ne transition between the 171Yb+ j "zi and j #zi states, as !ba = !hf ! = 2 12.6 GHz, and jgj2=2 2 1 MHz. Thus Eq. (2.21) becomes, _~Ca(t) i jgj2 2 ~Ca(t) + i 1 2 jgj2 2 ~Cb(t)e i(!ba !)t: (2.22) Similarly the coe cient for jbi obeys the equation, _~Cb(t) i jgj2 2 ~Cb(t) + i 1 2 jgj2 2 ~Ca(t)e i(!ba !)t: (2.23) Eqs. (2.22) and (2.23) describe the dynamics of the two level system composed of the states jai and jbi under the e ective Hamiltonian, Heff = jgj2 2 jaihaj jgj2 2 jbihbj jgj2 4 e i( k r+[!ba !]t )jaihbj jgj2 4 ei( k r+[!ba !]t )jbihaj = jgj2 2 I 2 e i( k r+[!ba !]t ) 2 ei( k r+[!ba !]t ) +: (2.24) Now, we have inserted the spatial dependence and an o set phase in the phases. Here, k is the di erence in momenta of the laser beams, and r is the atom?s 35 position vector. + = jbihaj and = jaihbj are the atomic raising and lowering operators. In the last line of Eq. (2.24), we identify the two photon A.C. Stark shift of each of the states jai and jbi as jgj2=2 ( jgj2=4 from each of the beams), and the two photon Rabi oscillation strength as = jgj2=2 . For lasers with unequal intensities, and hence unequal single photon Rabi frequencies g1 and g2, the two photon Rabi frequency = g 1g2 4 + g1g 2 4 . The di erential A.C. Stark shift - The A.C. Stark shift Eq. (2.24) is negative as the detuning is negative in Fig. 2.7 (we use to denote the absolute value here.) Note that the di erential A.C. Stark shift between the states jai and jbi cancels to the leading order in 1= . This is due to approximating all the denom- inators in the second sub-equation of Eq. (2.20) as . The leading non-zero order in the di erential A.C. Stark shift between the two states can be found by treating the beams separately. The two photon A.C. Stark shift on state j i ( = a; b) measures the strength of j i ! jei ! j i transition, where the intermediate state jei is only virtually excited, since jgj. The A.C. Stark shift experienced by the state jai from the beam with frequency !La is a = jgj 2=4 , and the A.C. Stark shift experienced by the state jbi from the same beam is b = jgj2=4( !ba). 36 The di erential A.C. Stark shift AC = b a = jgj2 4( !ba) + jgj2 4 = jgj2 4 1 1 !ba= 1 = jgj2 2 h 1 + !ba 1 i = !ba jgj2 4 1 2 : (2.25) Note that AC < 0, i.e., the states jai and jbi get closer together while interacting with the beams. AC / jgj2 / I, where I is the intensity of the beam. AC / 1 2 , while the two photon Rabi frequency / 1 . Thus the ratio of the di erential A.C. Stark shift to the Rabi frequency can be reduced by increasing the detuning . The two photon di erential A.C. Stark shift, AC is independent of the sign of the detuning , as AC / 1 2 . As an example, we can show that for ! L a > !ea (Fig. 2.7), the di erential A.C. Stark shift AC = jgj2 +!ba jgj 2 < 0, same as when !La < !ea. Thus, the di erential A.C. Stark shift from di erent levels do not cancel each other. The di erential A.C. Stark shift can be nulled by going to a rotating frame 37 at frequency AC , which means that we adjust the beat-note frequency of the two lasers to account for the shift in the jai $ jbi resonance due to the Stark shift. Hyper ne carrier transition - In our experiment, the states jai and jbi refer to the spin states j #zi and j "zi respectively, and hence !ba = !hf is the hyper ne splitting. (we assume that we have already accounted for the di erential A.C. Stark shift in de ning !hf ). For the resonant (or near resonant) hyper ne transition, the motional state of the ion is unchanged. Thus k r in the phase in Eq. (2.24) is a constant c-number, and can be absorbed in . Setting dw = !hf + carr, we get the e ective Hamiltonian for the spin ip, or the carrier transition, Hcarr = 2 +e i( carrt+ ) + ei( carrt+ ) ; [Carrier] (2.26) For carr = 0, Eq. (2.26) generates the resonant carrier transition between the spin states, at a Rabi rate of . The phase sets the ?axis of rotation? of the spin vector. Hcarr = 2 +e i + ei = 2 1 2 [ x + i y]e i + 1 2 [ x i y]e i = 2 x ei + e i 2 i y ei e i 2 = 2 ( x cos + y sin ) = 2 ; (2.27) where cos x + sin y. Thus the e ective Hamiltonian is 2 x for = 0 38 and 2 y for = =2. The unitary evolution operator of this Hamiltonian acts on the spin state j#zi according to U(t)j#zi = e iHcarrtj#zi = ei t 2 j#zi = cos t 2 I+ i sin t 2 j#zi = cos t 2 j#zi+ ie i sin t 2 j"zi: (2.28) The spin state precesses between j#zi and j"zi at a rate =2, with a phase . Note that the population precesses at a rate of , as the probability of detecting j "zi at time t is P"z(t) = sin 2 t=2 = (1 sin t)=2. The Bloch Sphere - The state of a two level system can be represented as a vector, called the Bloch vector moving on the surface of a sphere, known as the Bloch sphere. Two angles ( ; ) completely specify a general state j ( ; )i = cos 2 j#zi+ ie i sin 2 j"zi of the two level system. The action of the unitary operator U(t) on the Bloch vector is speci ed in terms of the rotation operator R( ; ) = e i 2 = cos 2 I i sin 2 . For a resonant carrier transition (t) = t. =2 pulse - A resonant Raman carrier transition of duration =2 is known as a =2 pulse if ( =2) = =2 ) =2 = 2 : (2.29) 39 Under a =2 pulse, the state j #zitransforms to j #zi iei j "zi, and the state j "zi transforms to j"zi iei j"zi. pulse - A Raman carrier pulse of duration = = is known as a pulse. It ips the spin states, j#zi ! iei j"zi; j"zi ! iei j#zi. E ective magnetic eld - From Eq. (2.28), we see that a Bloch vector precesses about the axis set by the Bloch vector angle under a resonant carrier transition. Thus the resonant transition simulates an e ective magnetic eld, and the phase sets the direction of it. In our experiment, we control the carrier Rabi frequency (by varying the intensity or jgj2) to control the magnitude of this e ective eld. In our trapped 171Yb+ system, the spin states j "zi and j #zi are coupled through the excited 2P1=2 and 2P3=2 states. We create the beat-note at ! = !hf by sending a CW laser beam through an electro optic modulator (EOM) that generates frequency modulated sidebands near !hf . Another way is to use the frequency comb generated by a mode-locked laser. We describe the operation of the mode-locked laser used in our experiment in section 2.6.4. Typical order of magnitude for the single photon Rabi frequency, g=2 in our experiment is 1 GHz, and the detuning =2 from the excited 2P1=2 states range from 2.7 THz to 33 THz for two di erent lasers used. Fig 2.9 shows Rabi oscillation between the two spin states. Ramsey interferometry - The coherence time of the spin states is es- timated by a standard Ramsey interferometric technique. A single spin is rst prepared in the optically pumped state j #zi. Then a =2 pulse is applied with a phase , either by a microwave or by the stimulated carrier Raman transition 40 Figure 2.9: Resonant hyper ne (Carrier) Rabi oscillations in 171Yb+ spin states: Two level Rabi oscillation between the spin j"zi and j#zi states of a single spin induced by two photon stimulated Raman transition from 355 nm laser light. The blue curve is a sinusoidal t to the data (blue points), and its frequency is about 2 1 MHz. The spin is rst prepared in the optically pumped j #zi state. Each point is an average over about 200 experimental points. 41 Figure 2.10: Ramsey interferometry in a 171Yb+ ion: The oscillation in the probability of the spin states at the frequency di erence between the oscillator driv- ing the =2 pulses and the atomic transition frequency denotes the coherence present in the spin states or the qubits. The =2 pulses are provided by microwave radiation from a microwave horn antenna at 12.642819 GHz. The dots are data, and the solid line is a sinusoid t with an exponential decay in amplitude. The decoherence time is estimated to be more than 800 ms from this data. 42 at frequency !osc, followed by a delay. Finally another =2 pulse is applied from the same oscillator, and at the same phase, and the spin state is detected. This method compares the two clocks, namely the hyper ne splitting of the ion, and the oscillator used to drive the =2 pulses. As the duration of the delay is scanned, a fringe is obtained in the observed probability of the bright state (j"zi) at frequency j!osc !hf j, where !hf is the hyper ne frequency. Fig 2.10 shows the experimental sequence and a typical Ramsey fringe data in the experiment. Coupling to the motional states - The carrier transition ips the spin from j#z; ni to the j"z; ni state (n represents the number of vibrational quanta in a given mode), and does not change the phonon excitations in the system. To excite the vibrational mode at frequency !X along with the spin ip (we assume that the Raman beat-note momentum k is along the X direction), we set the detuning ! in Eq. (2.24) to ! = !hf + !X + . We assume that the temperature of the ion is cold enough so that the motion can be expressed in terms of the vibrational mode phonon annihilation (a^) and the creation (a^y) operators, and the position coordinate X(t) can be written as, X^(t) = X0 a^e i!X t + a^yei!X t : (2.30) X0 = q ~ 2m!X is the characteristic length scale of the motional mode. Thus the e ective Hamiltonian of Eq. (2.24), ignoring the constant A.C. Stark 43 Figure 2.11: Carrier and sideband transitions: The harmonic oscillator energy ladder with the spin states (not to scale). A carrier transition induces coherent oscillation between the spin states without any change in the motional state, a blue sideband transition is higher in energy than a carrier transition, and involves the j #z; ni $ j "z; n + 1i transition. The red sideband transition transition is lower in energy than the carrier, and induces coherent oscillation between j #z; ni $ j "z ; n 1i. shift becomes, Hbsb = 2 e i( k r+[!ba !]t ) 2 ei( k:r+[!ba !]t ) + = 2 e i( kX0[ae i!Xt+ayei!Xt] [!X+ ]t ) 2 ei( kX0[ae i!Xt+ayei!Xt] [!X+ ]t ) + 2 1 i [ae i!X t + ayei!X t] ei[(!X+ )t+ ] 2 1 + i [ae i!X t + ayei!X t] e i[(!X+ )t+ ] + i 2 a ei( t+ ) ay + e i( t+ ) ; [blue side band] (2.31) where we have expanded the exponential up to rst order in the dimensionless 44 Lamb-Dicke parameter kX0 1 in the third line , and used the RWA to throw away terms rotating at or near !X in the last line. Hbsb makes the transition j #z; ni $ j "z; n + 1i, which is higher in energy than the carrier transition. For = 0, this results in a resonant stimulated Raman blue sideband transition. Similarly, if we set the beat-note between the lasers ! = !hf !X , we get a stimulated Raman red sideband transition, with the e ective Hamiltonian, Hrsb = i 2 ay e i( t ) a + ei( t ) ; [red side band] (2.32) We see that the transition strength between the states j #z; n 1i and j "z; ni is p n, where the factor of p n is contributed by the creation operator (a^yjn 1i = p njni). Fig 2.11 illustrates the carrier, red sideband and the blue sideband processes. 2.3.6 Raman sideband cooling Doppler cooling brings the average phonon occupation to n = 2!m , where is the natural linewidth of the cooling transition. The ions can be further cooled by mapping the motional degree of freedom to the spins, and then removing the spin entropy from the system by the optical pumping technique, described previously. Fig 2.12 describes the Raman cooling scheme. Let us assume that a single ion is in the Fock state j #z; ni. We apply a red sideband pulse, which annihilates a More rigorously, the exponential can be expanded if h kXi is small. This is equivalent to the Lamb-Dicke approximation p n+ 1 1. In our system 0:06 and average phonon occupation is < 3 after the Doppler cooling, and < 0:1 after side band cooling (the X vibrational mode). Thus the Lamb-Dicke approximation holds good. 45 Figure 2.12: Raman sideband cooling: The spin system is initialized in the optical pumped state, i.e., in the spin state j #z; ni, where n denotes the motional state of a vibrational mode with frequency !m. A red sideband pulse (solid red arrow) transfers the spin to the j "z; n 1i state, and annihilates a phonon. This is followed by an optical pumping pulse (gray fuzzy line), which ips the spin back to the j#zi state. This process is repeated, and the system rolls down the harmonic oscillator ladder, until it reaches n = 0. phonon, while ipping the spin, i.e., it takes the system to the state j "z; n 1i. We then optically pump the spin to j#z; n 1i state without changing the motional state, and thus extract a quantum of vibration from the system. This process can be repeated to reach the motional ground state. Since the strength of the sideband Rabi frequency depends on the motional state, the duration of the pulse has to be adjusted accordingly. In our set up, we apply about 30 pulses to cool the COM mode to near zero point motion. 2.4 Vibrational normal modes of trapped ions The con ning potential created by the static and ponderomotive electric forces try to bring the ions closer together at the trap center. The Coulomb repulsion 46 Figure 2.13: Image of ten bright 171Yb+ ions in a linear con guration: Fluo- rescence light of ten 171Yb+ ions, induced by the Doppler cooling beam, is captured by the imaging optics, and the signal is integrated for about 200 ms on a CCD cam- era. The ions form a linear con guration due to the high anisotropy in the trapping potential. The plus sign in the middle of the chain is a cursor used on the camera interface. between the ions tend to push them away from each other. A compromise between the attractive and the repulsive components is reached when the ions are at a certain distance apart from each other, and they form a Coulomb crystal. In order to avoid micromotion, we make the trap anisotropic with !X !Y !Z , so that the ions form a linear crystal along the Z axis, and lie on the radio-frequency null in the trap. Figure 2.13 shows a crystal of ten 171Yb+ ions held in a linear con guration. The spacing between the ions in a crystal depend on the axial con nement strength (characterized by the secular frequency, !Z) and the number of ions, and varies between 2 and 5 microns for the range of trap settings used in our experiments. A small perturbation from the equilibrium positions of the ions makes them oscillate about the equilibrium. In general the ions oscillate in complicated patterns, but any oscillation of the system (with small amplitudes) can be Fourier decomposed into collective vibrational modes, known as the normal modes of vibration [93]. For 47 a system of N ions, there are N normal modes along each of the three dimensions. The nature of the axial and the transverse normal modes are very di erent in a Coulomb crystal. Axial Normal modes - The axial normal modes are lower in frequency compared to the transverse modes, due to the anisotropy in the trapping potential required to keep the ions in a linear con guration. The lowest mode is the center of mass (COM) mode, at a frequency of !Z independent of the number of ions N . All the ions move back and forth uniformly in this mode, with an eigenvector component of 1= p N for all the ions. The COM mode has the longest spatial wavelength. The next mode is the ?breathing? mode, with a frequency of p 3!Z independent of the number of ions, N . The frequencies of the axial normal modes increase monotonically with decreasing wavelength. Transverse Normal modes - The transverse normal modes are higher in frequency due to stronger con nement along the transverse direction, and bunched closer together compared to the axial modes. Contrary to the axial vibrational motion, the transverse COM modes (at frequencies !X and !Y for all N) have the highest frequencies among all the modes along that speci c transverse direction. In our trap, the X modes are slightly higher in frequency than the Y modes, and the splitting between the X and Y COM modes depends on the axial con nement. For a stronger axial con nement, the X and Y COM modes are separated more. The next lower frequency mode is the ?tilt? mode, where the two halves of the chain oscillate with opposite phases, and the amplitude of vibration increases away from 48 Figure 2.14: Axial and transverse vibrational modes of trapped ions: Fre- quencies of the Axial and the transverse modes (along the X direction) of N = 10 trapped ions, for a trap anisotropy of X = 10. Each solid line represents a normal mode at frequency m. The horizontal axis shows the ratio of the mode frequencies to the axial COM frequency. The Axial modes (black) are almost equispaced, but the transverse modes (red) are not. They are closely bunched together at the high frequencies. The axial COM mode is the lowest in frequency of all the axial modes, while the transverse COM mode is the highest in frequency of all the transverse modes. 49 the center of the chain. This mode occurs at a frequency of !tilt = q !2X ! 2 Z (2.33) for all N . For an odd system size, the center ion does not take part in this mode. As we shall discuss later, the spin interactions in our simulated Ising Hamiltonian are mediated by the phonons [71], and the center spin does not interact with any other spin if only the tilt mode is used to generate the interactions in a system with odd number of ions. The frequencies of the transverse normal modes decrease monotonically with decreasing wavelength, and hence the transverse modes show anomalous dispersion. As seen from Eq. (2.33), the splitting between the COM and the tilt modes is dependent on the axial con nement. In general the bandwidth of the transverse modes increases with increasing axial con nement. Fig 2.14 compares the axial and the X transverse modes for an anisotropy X = 10. While the neighboring axial modes maintain an almost xed separation in frequency, the transverse modes are bunched together, especially at the higher frequencies. Figure 2.15 shows the eigenvectors of the transverse normal modes for a system of 10 ions. Transition to a zig-zag con guration - As the anisotropy in the trap- ping potential, de ned by the dimensionless parameters, X = !X !Z and Y = !Y !Z is decreased, by increasing the axial con nement strength keeping the transverse con nement xed, the ions come closer to each other, and the bandwidth of the transverse vibrational modes increases. 50 Figure 2.15: Transverse mode eigenvectors for N = 10 ions: The dots repre- sent the equilibrium position of the ions, and the arrows represent the eigenvector components for the mode (enumerated on the left). The simulated Ising coupling between spins i and j depend on the product of the eigenvector components of the normal mode(s) excited to generate the spin interactions, according to Eq. (2.53). 51 Figure 2.16: Nineteen ions in a zig-zag con guration: The ion chain undergoes a structural phase transition from a linear con guration to a zig-zag con guration at a certain trap anisotropy. The central region of the chain undergoes this transition before the outer region. The 8th ion from the left is either another isotope of Yb+ or another species, and hence it does not uoresce from the cooling and detection beams. The low uoresce count from the rightmost ion is due to the nite aperture size of a pinhole used in the imaging system. This con guration of the ions is equal in energy to its spatially re ected (about the Z axis of the chain) con guration. These two con gurations are referred to as the ?zig? and the ?zag? con gurations. Eventually, the frequency of the smallest wavelength mode, the zig-zag vi- brational mode reaches zero and the ion chain undergoes a structural phase tran- sition to a zig-zag con guration [94, 95]. This structural phase transition oc- curs at approximately X;Y = !X;Y !Z = 0:73N0:86, as found numerically [96]. For a large system (N 1), the transition point can be found analytically [95] as X;Y = !X;Y !Z = 0:77 Np logN . Ions near the center of the chain are closely packed together compared to the ions near the edge, and these central ions break into the zig-zag con guration before the others. In our trap the Y vibrational modes are slightly lower in frequencies than the respective X vibrational modes, and hence the zig-zag transition occurs in the Y direction rst, as the axial con nement strength is tuned keeping the transverse frequencies the same. For slightly tighter con ne- ment along the axial direction, the X zig-zag mode reaches zero frequency as well, as the ions form a helix in the X Y plain. The zig-zag vibrational mode must not be 52 confused with the zig-zag con guration, the former is a pattern of vibration where the equilibrium positions of the ions are still on a one dimensional chain, while in the latter the equilibrium position of the ions form a zig-zag con guration resulting in an entirely new set of vibrational modes. In the zig-zag phase, there are two degenerate con gurations for the ions, the ?zig? and the ?zag? con gurations forming a double well potential. This opens up the possibility of observing many interesting physical phenomena, such as simulating non-linear lattice models, the defect formation while traversing a phase transition (Kibble-Zurek mechanism [97, 98]), and coherent tunneling between the two wells in the double well potential [99]. 2.5 Simulating the quantum Ising model In this work, we simulate the quantum Ising model, with the Hamiltonian, H = NX i=1 i 1X j=1 Ji;j i x j x +B NX i=1 iy; (2.34) where Ji;j is the Ising coupling between the spins i and j (i; j = 1; 2; :::; N), B is an e ective transverse magnetic eld, and ?s are the spin-12 Pauli matrices ( = x; y; z). As described previously, a resonant carrier transition between the spin states j "zi and j #zi with a phase = =2 in Eq. (2.26) acts as an e ective transverse magnetic eld. In this section we describe the simulation of the Ising interactions. 53 2.5.1 Ising interactions The ising interactions are simulated by following M lmer-S rensen scheme [71], where we apply Raman laser beams to excite the vibrational modes o -resonantly. Figure 2.17 shows the schematics of a M lmer-S rensen transition, in the case of two spins interacting through a single vibrational mode at frequency !m = 2 m. Two laser beat-notes are applied, with their frequencies symmetrically detuned from the carrier. We shall call these beat-notes bsb and rsb , tuned near a blue sideband transition and a red sideband transition with frequencies hf respectively. We also de ne the M lmer-S rensen detuning m. Let?s focus on the spin state j #z#z; ni. If it absorbs a rsb photon, it may get o -resonantly excited to the state j "z#z; n 1i or j #z"z; n 1i with equal probabilities. Since the frequencies of the rsb and the bsb add up to 2 hf , it may make a transition to the j"z"z; ni by absorbing a bsb photon. If the M lmer-S rensen detuning is kept larger than the sideband Rabi frequency p n, the intermediate single spin excited states can be adiabatically eliminated, and the amplitude of the j#z#z; ni ! j"z"z; ni transition is j#z#z ;ni!j"z"z ;ni; path 1 = j#z#z ;ni!j"z#z ;n 1i j"z#z ;n 1i!j"z"z ;ni 4 = ( p n)( p n) 4 = n ( )2 4 : (2.35) Here the minus sign denotes that the Raman beat-note is tuned red to the red 54 Figure 2.17: M lmer-S rensen transition in a system of two spins: O - resonant red and blue sidebands, symmetrically detuned about the carrier transition simulate the two body spin interactions in our experiment [71]. Here we show the spin and the relevant motional states (labeled by the number of phonon n). The red (blue) solid lines show an o -resonant red (blue) sideband transition. The four di erent pathways connecting j #z#z; ni ! j"z"z; ni partially interfere destructively to cancel the dependence on the phonon quantum number n, resulting in a pure spin model in the Lamb-Dicke limit. The Raman beat-note frequency controls the nature of the Ising couplings, as discussed in the text. 55 sideband transition. This amplitude interferes with another path where the spins rst absorb a photon from the bsb beam, and then from the rsb beam, and makes a transition to the j"z"z; ni state with an amplitude j#z#z ;ni!j"z"z ;ni; path 2 = j#z#z ;ni!j"z#z ;n+1i j"z#z ;n+1i!j"z"z ;ni 4 = ( p n+ 1)( p n+ 1) 4 = (n+ 1) ( )2 4 : (2.36) Thus the total amplitude from these two paths becomes, path 1;2 = ( )2(n+ 1) 4 ( )2n 4 = ( )2 4 : (2.37) Similarly there are two more paths where the second spin absorbs the rsb and the bsb photons rst, and they add an equal contribution as path 1;2 to the overall Ising amplitude. From Eq. (2.37) we see that the amplitude of the j #z#z; ni ! j"z"z; ni is independent of the motional state n, as long as the intermediate states with n 1 and n+ 1 photons are not populated, resulting in a pure spin Hamiltonian, HIsing;eff = J1;2 1 2 ; (2.38) where J1;2 = ( )2=2 is the total amplitude from the interference of all the paths, and is the Ising coupling between the two spins. Here = cos x + sin y, and hence by properly choosing by adjusting the beat-note phase of the Raman beams, 56 we generate a x x interaction between the spins. If we include the o -resonant excitation of the blue sideband from the rsb beam and the o -resonant excitation of the red sideband from the bsb beams, the e ective Ising coupling Ji;j becomes, Ji;j = ( )2 2( !m) ( )2 2( + !m) = ( )2!m 2 !2m : (2.39) Contribution from o -resonant carrier transitions from the bsb and the rsb beams cancel between path 1 and path 2. Alternative derivation - We may get the e ective Ising Hamiltonian start- ing from the e ective Hamiltonians for the red sideband and the blue sideband, Eqs. (2.32) and (2.31). We shall get an expression for the e ective Hamiltonian for N spins, subject to a red sideband beat-note at a frequency !hf with a phase = r and a blue sideband beat-note at !hf + with a phase = b. Let the normal mode frequencies for the N ions along the transverse direction that couples to the Raman beat-note wave vector k be !m (m = 1; 2; :::; N). Analogous to the single ion case, we expand the position coordinate of each ion into normal mode coordinate, a^m and a^ym. Thus k:x^i = X m i;m a^me i!mt + a^yme i!mt ; (2.40) 57 where the Lamb-Dicke parameters are now de ned as, i;m = bi;m k r ~ 2m!m : (2.41) Here bi;m are the normal mode eigenvector components between ion i and mode m. We de ne M lmer-S rensen detunings, m !m: (2.42) Thus the e ective Hamiltonian, using Eqs. (2.32) and (2.31), is Heff = Hbsb +Hrsb = i NX i=1 NX m=1 i;m i 2 h am i e i( mt+ b) aym + i e i( mt+ b) +aym i e i( mt r) am + i e i( mt r) i = NX i=1 NX m=1 i;m i 2 ame i mtei M + ayme i mte i M i s ; (2.43) where the motional phase M b r 2 and the spin phase s 2 + b+ r 2 . The spin operator i s = cos s i x + sin s i y: We set r = 0, b = , then M = =2 and s = , and the e ective Hamiltonian involves i s = i x. Heff = i NX i=1 NX m=1 i;m i 2 ame i mt ayme i mt ix: (2.44) The time evolution operator for this Hamiltonian, using the second-order Magnus 58 formula, is U(t; 0) = T^ e i R t 0 dt1Heff (t1) = e i R t 0 dt1Heff 1 2 R t 0 dt2 R t2 0 dt1[Heff (t2);Heff (t1)] + ::: (2.45) Z t 0 dt1Heff = i Z t 0 dt1 NX i=1 NX m=1 i;m i 2 ame i mt1 ayme i mt1 ix = i NX i=1 NX m=1 i;m i 2 am(ei mt 1) i + aym(e i mt 1) i ix = i NX i=1 NX m=1 i;m i m sin mt 2 ame i mt=2 ayme i mt=2 ix:(2.46) And, Z t 0 dt2 Z t2 0 dt1[Heff (t2); Heff (t1)] = NX i;j j !X . Long range antiferro- magnetism introduces frustration in the system, which we study in chapters 4 and 5. On the other hand if we tune the Raman beat-note close to the COM mode with frequency !X , but keeping < !X , all the Ising couplings are negative or ferromag- netic. We shall study the long range ferromagnetic Ising model in the presence of a transverse eld in chapter 3. The range of the interactions can be tuned by varying the M lmer-S rensen detuning !m, and also by varying the bandwidth of the vibrational modes. We shall discuss this in chapter 5. As pointed out earlier, the center ion in a chain with odd number of spins do not couple to the tilt mode, hence bmiddle ion;2 = 0, and thus Jmiddle ion;j = 0 for 62 Figure 2.18: Ising couplings for various M lmer-S rensen detuning: Top - Transverse normal mode spectrum (black solid lines) for N = 10 ions, with axial frequency Z = !Z=2 = 1 MHz, and transverse COM frequency X = !X=2 = 4:8 MHz. a-f. Bar chart of the Ising couplings Ji;j (divided by 2 ) vs i and j for various detunings (mentioned on top of each gure). The sideband Rabi frequencies are taken to be =2 = 35 KHz. 63 Figure 2.19: Ising oscillations between spin states: Two spins are optically pumped to the j #z#zi state, and made to interact with the Raman beat-notes gen- erating the Ising couplings, according to Eq. (2.53). The sideband Rabi frequency is i;1 i=2 35 KHz for the ion i (i = 1; 2) and the COM mode (m = 1), and the beat-note =2 1 + 105 KHz. The red points are the data and the blue solid line is a t with an Gaussian decay in the amplitude of oscillations. The contrast of the oscillations decays to 1=e in about 3.3 ms, presumably due to decoherence induced by intensity uctuations in the Raman beams. The Raman beams are generated from the mode-locked tripled Vanadate laser at 355 nm. all other ion j, from Eq. (2.53). Near the tilt mode long range antiferromagnetic couplings compete with the short range ferromagnetic couplings, and lead to a rst order phase transition [100]. Measurement of the Ising couplings - A system with N = 2 spins, ini- tialized in the state j#z#zi (by optical pumping) will undergo Rabi opping between the states j #z#zi and j "z"zi, under the unitary evolution operator of the e ective 64 Ising Hamiltonian, U(t)j#z#zi = e iHIsingtj#z#zi = e iJtj#z#zi = cos Jtj#z#zi i sin Jtj"z"zi; (2.54) where J J1;2 is the Ising coupling between the spins. Since the spin state j "zi appears as the bright state in our detection scheme (section 2.3.3), average number of bright ions nbright(t) = 2 cos2 Jt would oscillate with a frequency 2J under the Ising interactions. Thus the strength of the Ising interaction is found from the observed oscillation in the average number of bright ions interacting with the Raman beat- notes. Figure 2.19 shows the oscillations of the system between the spin states under the Ising evolution operator. The measured Rabi frequency is 2 1.9 KHz, and this corresponds to an Ising coupling of J = 2 0:95 KHz. We cannot determine the sign of the coupling from this oscillations. If we prepare the spin states along the y axis of the Bloch sphere by a =2 rotation about the x axis (i.e., by applying R( =2; 0) on the optically pumped state j #z#zi), and then turn on the Ising couplings, the spins will oscillate with opposite phases for the positive and negative Ising couplings. The absolute sign of the Ising couplings still remain undetermined. For N > 2 the oscillations in the observed average number of bright ions contain all the Ising couplings. We can Fourier transform such a signal to extract the Ising couplings. However, this method is not very e cient beyond a few spins, 65 Figure 2.20: Experimental sequence in a quantum simulation: Outline of quantum simulation protocol. The spins are initially prepared in the ground state (or the highest excited state) of B P i i y, then the Hamiltonian (Eq. (2.34)) is turned on with starting eld B0 J followed by an adiabatic exponential ramping to the nal value B, keeping the Ising couplings xed. Finally the x component of the spins are detected. Details of the initialization are shown in the dotted box below. Trapped ions are Doppler cooled to an average COM (along theX direction) phonon occupation number of n 2, then optically pumped to the spin state j #z#z :::i. The ions are prepared in their zero-point vibrational energy state by Raman sideband cooling. Finally a coherent =2 pulse around the x axis of the Bloch sphere orients the spins along the magnetic eld. as the Fourier transform contains the sums and di erences of all the frequencies and it becomes hard to resolve and identify the Ising couplings. 2.5.2 Adiabatic quantum simulation In an adiabatic quantum simulation, the simulator is initialized in the ground state of a trivial Hamiltonian. Then, the Hamiltonian is gradually tuned to the more complicated and interesting one. If the ramping is done at a rate slow compared 66 to the excitation energy scale, the population remains in the ground state. Our experiments are performed in the following steps ( g 2.20): The spins are optically pumped to the j#z#z :::i state. They are also sideband cooled to get close to the phonon ground states. The spins are polarized along the y direction of the Bloch sphere, by a co- herent =2 stimulated Raman pulse, about the x axis of the Bloch sphere. The Hamiltonian is turned on with the e ective transverse eld much larger than the maximum Ising coupling. Thus the spins are in the ground state of the total Hamiltonian, to a very good approximation. The e ective eld B is ramped down exponentially with a time constant , keeping the Ising couplings constant, according to, B(t) = B0 exp( t= ): (2.55) Here B0 is the initial e ective eld. Finally, the spins are globally rotated by a =2 pulse about the y axis of the Bloch sphere. This maps the x components of the spins to the z components (our measurement basis). Thus, by performing the nal spin rotation, we are e ectively measuring the spin order along the Ising direction, i.e., the x axis of the Bloch sphere. 67 The transverse eld couples the instantaneous eigenstates of the Hamiltonian. The minimum energy gap, c between the ground and the rst excited state that couples to the ground state sets the adiabaticity criteria [101, 42], _B(t) 2c 1; (2.56) where jhg(t)jdH(t)dB je(t)ij characterizes coupling between the instantaneous ground state jg(t)i and the relevant excited state je(t)i. Exponential ramping is experimentally more convenient than a linear ramping, as this keeps the total duration of the experiment under the time scale set by deco- herence processes. Instead of following the ground state, the highest excited state of the Hamiltonian may as well be followed. Following the highest excited state of H is equivalent to following the ground state of the sign inverted Hamiltonian, H. This is an experimental way to ip the e ective signs of all the Ising couplings from antiferromagnetic (Ji;j > 0) to ferromagnetic (Ji;j < 0), a trick that we follow to simulate the ferromagnetic quantum Ising model, discussed in chapter 3. 2.6 Experimental Apparatus 2.6.1 Ti:Sapphire laser The Titanium:sapphire laser (MBR-110, developed by Coherent Inc) is pumped by an 18 Watt green laser (Verdi V-18, Coherent Inc) at 532 nm. Figure 2.21 shows a schematic of the MBR-110, and Fig. 2.23 shows a photograph of MBR-110 with 68 important parts labeled. The rod shaped Titanium:Sapphire crystal is placed in- side a bow-tie ring cavity made from a single Aluminum block (The acronym MBR stands for a Monolithic Block Resonator), that stabilizes the cavity against the vi- brations of its constituents. The at faces of the crystal are Brewster cut to minimize re ection. The brass unit holding the crystal is water cooled. Mirrors M1 and M2 have a radius of curvature of 10 cm each, and coated to be highly re ecting across a large wavelength range. M1 also focuses the 532 nm pump beam onto the Ti:Sapphire crystal. Mirror M3 is a piezo-mounted highly re ecting square-shaped small planar mirror (called a tweeter mirror). Mirror M4 is a planar mirror, and also the output coupler. In between the mirrors M2 and M3, there is an optical diode, consisting of a Faraday rotator crystal placed in a strong permanent magnetic eld, and a less than 0.5 mm thick Brewster angled retardation plate. The optical diodes ensure that the light is circulating in one direction only, namely M1!M2!M3!M4! M1. The Faraday rotator rotates the polarization by a few degrees in a direction independent of the travel direction of the beam. The retardation plate rotates the polarization in the opposite direction by almost the same amount (undoing the polarization change by the Faraday rotator) for light traveling through it in the ?right? direction, and in the same direction for any light traveling in the ?wrong? direction. Thus after a few passes through the ring cavity, lights circulating in the wrong direction is blocked by the Brewster angled retardation plate, and the cavity supports p -polarization light circulating in one direction. The desired single-frequency operation (Fig. 2.22) of MBR-110 is chosen in 69 Figure 2.21: Schematics of the MBR-110 Ti:Sapphire laser: The MBR-110 (Coherent Inc) is pumped by an 18 Watt green continuous-wave laser at 532 nm (Verdi-18, Coherent Inc.). MBR-110 houses a bow-tie ring cavity made of the four mirrors M1-M4, and a cylindrical Titanium:Sapphire crystal rod with Brewster cut at faces is placed in between the curved mirrors M1 and M2 (both with 10 cm radius of curvature). Mirror M1 also serves as the input coupler, and mirror M2 re ects the infrared radiation only, blocking the majority of the pump beam, which is then blocked behind M2. Mirror M3 is a piezo-mounted highly re ecting square-shaped small planar mirror, and the planar mirror M4 also acts as the output coupler. The optical diode, consisting of a Faraday rotator and a retardation plate (a wave plate) positioned at the Brewster angle selects a particular polarization to circulate in the cavity. The birefringent lter, the etalon, the two galvanometer mounter fused silica Brewster plates and the piezo-mounted mirror M3 select the mode of radiation. The cavity mode can also be locked to the piezo controlled external reference cavity, using the error signal generated by the photodiodes shown here, as explained in the text. The electronics is operated from a separate control box. (Image Credit: Coherent Inc.). 70 Figure 2.22: MBR modes: The radiation mode that builds up in the laser cavity is determined by the gain pro le of the Ti:Sapphire crystal (not shown here), the frequency response of the birefringent lter (blue dashed-dot curve), the etalon modes (black dashed curve) and the cavity modes (red solid curve). The black solid curve shows the lasing threshold where the gain from the Ti:Sapphire crystal equals the loss in the cavity. The birefringent lter has a very broad frequency response (30-40 nm), and the laser output frequency hops between the etalon modes as the lter micrometer is rotated. The etalon has a free spectral range (FSRetalon)of about 225 GHz. The cavity free spectral range (FSRcavity) is about 300 MHz. As the etalon is rotated the laser output hops between the cavity modes. This is a schematic, and not to scale. 71 the following steps: The Ti:Sapphire crystal has a broad gain pro le (few hundred nanometers), that o ers tunability over a large wavelength range [102]. The optics used in the laser limits the width of the gain pro le to under 100 nm. The polarization change through a birefringent material is frequency dependent and thus the birefringent lter further narrows down the bandwidth of the radiation circulating in the cavity to about 30-40 nm. The lter rotates the polarization of light with frequencies outside this band appreciably every time the light passes through it, and in a few passes the polarization is blocked by the Brewster angled optics in the cavity. Thus the birefringent lter acts as a coarse wavelength knob. The intracavity etalon - MBR-110 has a less than 0.5 mm thick etalon placed in between the mirrors M3 and M4, with a nesse of about 25, and a free spectral range of about 225 GHz. As we turn the birefringent lter, it hops between the etalon modes. Rotating the knob clockwise selects a higher frequency etalon mode, and rotating it counter-clockwise selects a lower frequency mode. The etalon is mounted onto a piezo which is mounted onto a galvanometer using an aluminum mount (Fig. 2.23). The galvanometer is mounted onto a rotation stage, with all three translational degrees of freedom. The galvanometer (and hence the etalon) can be rotated from the MBR control box. The etalon should not be oriented such that the normal to its surface is parallel to the incident light. This orientation is known as the ? ash? position, and the power output from the laser will fall to half its normal value (when the etalon is not in the ? ash? position), as the cavity lases in both direction. 72 In the ash position of the etalon, a large fraction of the incident power is re ected back to the exact same direction, and the isolation provided by the optical diode is not su cient to prevent lasing in the wrong direction. The laser cavity modes- the ring cavity is about L = 1 meter long, and the corresponding free spectral range is c=L = 300 MHz. We select a particular cavity mode by turning the angle of the etalon. As we rotate the etalon by turning the ?etalon tune? knob from the control box, the laser output hops between the cavity modes. The etalon lock - The re ected light from the etalon is directed and focused onto a photodiode by a 45 prism mirror and a fast biconvex lens. The intensity of this re ected light can be monitored from a test point in the laser box (?etalon photodiode signal? in g 2.23) using a mini-BNC cable. As the etalon is rotated by turning the etalon knob from the electronic control box this signal increases, from zero, when the knob is fully counter-clockwise, to about 6-8 volts when it is fully clockwise. This signal may also be monitored at the test point TP6 on the analog board in the electronics control box. If the maximum signal is at the fully clockwise position of the etalon knob is low, make sure that the re ected beam is properly hitting the photodiode by aligning the lens. The trim-pot next to the monitor point in the laser box can also be adjusted to increase the voltage. The etalon is modulated at a piezo driving frequency of 82.3 KHz, generated from the electronics control box. This driving signal may be monitored at the test point TP5, and a 90 phase shifted signal may be monitored at test point TP7. The cylindrical piezo has two halves, which are driven with opposite phases. This 73 sets up a standing wave on the etalon, and the light re ected from the etalon is intensity modulated at this frequency as the re ectivity depends on the angle of the incidence. The amplitude of the intensity modulation is the largest when the beam hits a node of the displacement standing wave, which has the largest change in the angle w.r.t. the incident direction of the beam. This modulated signal may be monitored at the test point TP6 on an AC coupled oscilloscope, triggered by the signal from test point TP5, or TP7. As the etalon knob is turned, the signal on the oscilloscope should show a ?breathing mode? with an amplitude of about 100 mV. An error signal is generated in the electronic control box by demodulating this modulated etalon signal at the driving frequency. The error signal can be monitored from the back panel of the control box. When the etalon knob on the control box is turned clockwise, the error signal hops between the cavity modes, and generates a sawtooth signal on an oscilloscope, with a steeper rise and a slower fall. To lock the etalon to a cavity mode, the error signal should be centered around zero volts, by adjusting the etalon o set from the control box. The etalon error signal is approximatey 300 mV in our system. We center the error signal at a desired cavity mode, by turning the etalon knob on the control box, and press the etalon lock button. The etalon lock will not be robust, or it may not even lock if the error signal amplitude is below 100 mV. The amplitude may be reduced due to a drift in the driving frequency, or a drift in the mechanical resonance frequency of the piezo. Adjust the trim-pots PR14 ( ne) and PR15 (coarse, usually not required) to bring the driving frequency back on the piezo resonance. This is achieved by observing the sawtooth etalon error signal on an oscilloscope while turning the etalon knob, and maximizing the amplitude of the signal. 74 In some unfortunate cases, the best error signal may be obtained when the incident light hits a spot on the etalon that does not optically transmit well without a signi cant loss in intensity. Then we may have to compromise between a good error signal and output power. We tested a couple of etalons, and picked the best. The external reference cavity - MBR-110 has an external reference Fabry- Perot cavity with nesse in the range of 25-50. When the etalon is locked to the laser cavity, the laser frequency does not hop between di erent etalon modes, but slow thermal drifts in the laser cavity length changes the frequency. The laser cavity can be locked to the reference cavity by feeding back to the tweeter mirror. This is done by superposing an o set signal to the reference cavity signal, such that the fringes are centered about zero volts, and locking the tweeter mirror (M3) to a side of a reference cavity fringe. The reference cavity fringes, as measured by the ?photodiode A? in Fig. 2.23 may be monitored at the test point in the laser box (?Reference cavity signal?), after setting the reference cavity toggle switch on the back of the electronics control box to the ?dither? position. The o set static voltage, as measured by the ?photodiode B? may be observed at the ?normalization signal? port in the laser box, and adjusted by turning the trim-pot next to it. When the o set is properly adjusted, the reference cavity error signal peaks, observed at the back of the electronics control box, should be about 6-8 V high, and centered around zero volt. For Raman transitions with a two photon detuning of a few THz used in the experiments, the slow drifts in the frequency due to thermal drifts in the reference cavity length is not crucial. However, in order to use the Ti:Sapphire laser for the 75 near resonant operations (Doppler cooling, detection etc), the reference cavity is locked to an absolute frequency reference, provided by a Doppler free saturation absorption line (at about 405.644321 THz) of Iodine molecules, see Ref. [103] for details. The saturation absorption error signal from the Iodine set up is sent to a PID controller, the output of which is fed into the Ext Lock port at the back of the MBR control box. Our MBR-110 generates about 2 Watts of optical power around 740 nm. 76 Figur e 2.23 : MBR-11 0 Ti:Sapphir e laser . 77 for a 2.6.2 Generating 369.5 nm light by frequency doubling We frequency double the infrared Ti:Sapphire output to the ultraviolet using a Lithium Triborate (LBO) non-linear crystal in a frequency doubling cavity (Wave- Train, Spectra-Physics). The frequency doubling e ciency is about 10% per Watt, and we get approximately 490 mW of 370 nm light to be used for Raman transitions from an input light of 2.2 W. We use another WaveTrain frequency doubler to generate 369.5 nm light for Doppler cooling, detection, protection, ionization and optical pumping. The dou- bling e ciency of this doubler is about 6% per Watt, and we get approximately 20 mW of ultraviolet light, with an input of about 600 mW. 2.6.3 369.5 nm optics schematics We use the frequency doubled light at 369.5 nm to generate the Doppler cool- ing, detection, optical pumping and protection beams. Fig 2.24 shows a schematic of the optics set up. The output of the frequency doubler is about 430 MHz red detuned from the 2S1=2 2 P1=2 resonance. A portion of this light is sent to a 7.37 GHz EOM (Model 4851 from New Focus, driven by a Lab Brick Signal generator from Vaunix Corporation, 34 dBm rf power) that generates sidebands required for Doppler cooling the j #zi state. An AOM (made by Brimrose Corp.) driven at 400 MHz up-shifts the frequency to about 30 MHz (red detuned) from the resonance. 78 Figure 2.24: Schematics of the 369.5 nm beams: A Ti:Sapphire output at 740 nm is frequency doubled by an LBO crystal inside a triangular cavity is split into the Doppler cooling, detection, optical pumping and the protection beams. 50/50 beam splitters are used to combine multiple beams. Vertical cylindrical lens V2 images the intermediate focus IF at the ion position, with a magni cation of 1=5. Spherical lenses are shown in blue, horizontal cylindrical lenses are shown in white and the vertical cylindrical lenses are shown in gray. 399 nm beam is combined with the protection beam on a PBS, 935 nm and 638 nm beams are combined with the 369.5 nm and 399 nm beams on mirrors with appropriate coatings. 79 Another part of the beam is sent through an AOM driven at about 424 MHz, and this is used as the detection beam. The optical pumping beam is also derived from the same parent beam, and sent through a 2.105 GHz EOM (Model 4431 Visible Phase Modulator from New Focus Inc.) followed by a 424 MHz AOM. The detec- tion and the optical pumping beams are combined on a 50-50 non-polarizing beam splitter (Model BSW20 from Thorlabs), and then coupled to an optical ber (single mode at 320 nm from Coastal Connections), the output of which is then combined with the Doppler cooling beam on a 50-50 non-polarizing beam splitter (Edmund Optics NT 48-213). The protection (the additional cooling) beam is 200 MHz fur- ther red detuned by an AOM (from Intra-Action Corp.), and is mixed with 399 nm light on a polarization beam splitter. The protection and the 399 nm beams are then combined with the Doppler cooling, optical pumping and the detection beams on a 50-50 beam splitter (Edmund Optics NT 48-213). All the beams are focused in the vertical direction at an intermediate focus (labeled IF), which is then imaged at the ion position by an imaging lens (V2) of focal length 80 mm. The magni cation in this imaging is 1=5. The nal lens V2 is mounted tightly on a xed cylindri- cal lens mount which is attached to a pedestal. The beam is vertically shifted by moving the vertical lens V1. The beam at the ion position moves by a factor of 5 less than the vertical translation in the lens V1, making the beam stable against vibrations in the mount holding the lens V1. The cooling, detection and optical pumping beams are approximately 8 m wide in the vertical direction and about 100 m along the horizontal direction (1=e2 radius in intensity) transverse to the beam propagation. Since this beam enters the vacuum chamber at an angle of 45 80 degrees w.r.t. the axis of the ion chain, the e ective beam waist in the horizontal direction is 100 m= cos 45 140 m. A portion of the Ti:Sapphire (MBR-110) output (about 40 mW) is used for the Doppler free saturation absorption spectroscopy. This light is coupled to a ber EOM (from EOSpace Inc., driven at 13.315 GHz), and the output (about 9 mW) is sent to the saturation spectroscopy set up [103]. We also monitor the MBR modes on a home-made 20 cm confocal cavity. Details of the confocal cavity may be found in Appendix C of Ref. [70]. On a historical note, we used a semiconductor diode laser with a tapered am- pli er system (Toptica TA 100) to generate the 739.5 nm light (and the Ti:Sapphire laser to generate the Raman beams) for the experiments in chapter 3 (with N = 2 to N = 9 spins) and chapter 4. 2.6.4 Mode-locked 355 nm laser We use a mode-locked laser with center wavelength at 355 nm (Vanguard, Spectra-Physics) to drive two photon Raman transitions [104]. The lasing medium is a Neodymium doped Vanadate (Nd:YVO4). This is an industrial laser used pri- marily for semiconductor fabrication, and comes in a closed box with no direct access to the laser cavity. Some parameters of interest are: Average optical power 4 Watt. Repetition rate, rep 80.6 MHz, delay between the pulses 12 ns. 81 Optical bandwidth 100 GHz (estimated). This laser has su cient optical bandwidth to drive the two photon Raman transi- tions. We drive the stimulated Raman transitions by shining two non-co-propagating beams derived from this laser on the ions, as shown in g 2.25a. Each beam generates an optical frequency comb, with comb-teeth spaced regularly by the repetition rate, rep, as shown in Fig. 2.25c. We shift the optical frequency of the rst comb (shown in red) relative to the second (blue) by AOM?s used in the beam path. The AOM frequency di erence j AOM j is tuned such that the beatnote between mth comb-tooth of the red comb and (m + n)th comb-tooth of the blue comb (m and n are positive integers) equals the atomic transition frequency, ab between states jai and jbi (Fig. 2.25b), for all m and a particular n. The atom then absorbs a photon from the (m + n)th tooth of the blue comb and emits to the mth tooth of the red comb to undergo a stimulated Raman transition between the states jai and jbi, via the excited state jei. For example, to drive the hyper ne transition (at the frequency hf ) between the clock states of 171Yb + 2S1=2 ground state hyper ne manifold, ab = hf , and since the repetition rate of our 355 nm mode-locked laser is approximately 80.6 MHz, we use the n = 157th comb-tooth. The beatnote is generated by all pairs of comb-teeth separated by 157 rep in frequency. The mechanism behind exciting an atomic transition with a frequency comb may also be understood by looking at the frequency spectrum of the light using a radio-frequency photodiode that has a response time fast enough to resolve between 82 Figure 2.25: Schematics of a two photon Raman transition using a mode- locked laser: a. Two non-co-propagating beams generated from the same mode- locked laser drive stimulated Raman transition in an atom. b. Level diagram of a three level system. Individual laser beams o -resonantly couple the states jai and jbi to the excited state jei. The beams are detuned from the excited state jei by . In our experiments with 171Yb+ states jai and jbi are the hyper ne (and motional) ?clock? states in the electronic ground state of the 2S1=2 manifold, and the excited states used are the 2P1=2 and the 2P3=2 ne structure states. The beams are detuned by 33 THz blue of the 2P1=2 states for our 355 nm mode-locked laser. c. Each beam generates an optical frequency comb, shown in red and blue. The frequencies of the beams are shifted by AOMs, and at some di erence frequency j AOM j between the shifts, the beatnote between mth comb-tooth of red rst comb and (m + n)th comb-tooth of the blue comb (m and n are positive integers) equals the atomic transition frequency, ab, for all m and a particular n. The atom absorbs a photon from the (m+ n)th tooth of the blue comb and emits a photon to the mth tooth of the red comb to make a transition from state jai to jbi. 83 Figure 2.26: Radio frequency comb-teeth used in the two photon Raman transitions: We use a radio-frequency photodiode to look at the radio-frequency spectrum of the mode-locked laser ( rep 80 MHz) and the beatnote. The light is split into two arms using a beam-splitter, with individual frequency control by the AOMs, and recombined at the second beam-splitter. The radio-frequency photodi- ode averages over the optical cycles. (a) measured spectrum with the beam from AOM2 blocked. (b) measured spectrum with the beam from AOM1 blocked, and (c) measured spectrum with both the beams. Sidebands at frequencies j AOM j appear due to interference of the beams. In the two photon STR, atoms take the place of the nal beam-splitter. the pulses, but slow compared to an optical cycle. We illustrate this in Fig. 2.26 . Here the beam from the mode-locked laser is split into two paths, with the AOMs shifting the frequencies of the individual paths. The two arms constitute a Mach- Zehnder interferometer, with the ion replacing the nal beam splitter. The radio- frequency photodiode averages out the intensity over the optical cycles but shows the radio frequency comb-teeth at frequencies m rep (m = 0; 1; 2; :::) in the measured frequency spectrum of a single beam (Fig 2.26a-b). The e ective time constant of the photodiode circuit (and the speed of the ampli er used) limits the overall bandwidth of the radio-frequency beatnote comb detected electronically. Since the two arms of the Mach-Zehnder interferometer are shifted in frequencies by AOM1 and AOM2 The data were taken with another mode-locked laser with a repetition rate of 80 MHz. 84 respectively by the acousto optic modulators (AOMs), the beatnote generated by the nal beam-splitter is amplitude modulated at j AOM j = j AOM1 AOM2j . This puts sidebands y on the radio frequency comb-teeth at frequenciesm rep j AOM j. We can control the position of the sidebands by controlling the frequency j AOM j, and bring one of the sidebands on resonance with an atomic transition. An atomic transition, at frequency ab is excited when the AOM frequencies are tuned such that ab equals a particular sideband frequency sb = n rep j AOM j, where n refers to some comb-tooth in the radio-frequency comb. Hence, we want to stabilize this particular amplitude modulation sideband at the transition frequency. Repetition rate stabilization The repetition rate drifts and uctuates (Fig. 2.27) due to the thermal drift and uctuations. Since we do not have an easy access to the laser cavity of this laser, we cannot stabilize the repetition rate directly. However, as described in the previous section, our two photon Raman transition is dependent on the beat-note between two beams generated from this laser, and hence we need to stabilize the beat-note only. One intuitive way to stabilize the beat-note is to measure the repetition rate at time t, rep(t) by a frequency counter, and directly change the driving frequency of one (or both) of the AOMs accordingly using a software locking scheme, to keep the sideband frequency sb = n rep(t) j AOM j at a constant value. This requires without loss of generality, we have chosen the AOM shifts to both be positive. yThe carrier is not depleted completely in Fig. 2.26, since the interfering beams are not perfectly mode-matched, and have unequal optical power. 85 Figure 2.27: Drift in the repetition rate of the 355 nm mode-locked laser: The repetition rate of the mode-locked Vanguard laser at 355 nm is measured with a digital frequency counter in regular two minutes intervals. The repetition rate shows a long term drift of about 1 Hz/minute in this case. The uctuations are correlated with the ambient temperature uctuations. Typically the repetition rate approaches a steady state value after about a couple of hours of turning on the laser, provided that the ambient temperature is stable. The two photon Raman transition in 171Yb+ hyper ne states uses the 157th comb-tooth, which drifts by about 157 Hz/minute. We stabilize the uctuations and the long term drift in the repetition rate by feeding forward to an external acousto optic modulator, as described in the text. stabilizing rep to =n, in order to stabilize the sideband frequency to . The software lock would be slow for large n (n = 157 in our experiment) as the integration time of the frequency counter could be comparable or larger than the time scale at which the repetition rate uctuates. In our experiments, to stabilize the sideband within a fraction of 1 KHz, we need to measure rep(t) within a few Hz, which corresponds to an integration time of about a second. We have experimentally found that a software-based lock of this type is insu cient for achieving high- delity transitions. 86 Figure 2.28: Schematics of the repetition rate lock: a. The output of a fast photodiode is mixed with a local oscillator (LO) signal, which is sent to a Phase Locked Loop (PLL) after rejecting the high frequency beatnote by using a low pass lter (LPF). The output of the PLL drives the AOM. b. Details of the repetition rate lock circuit used in our experimental set up. The second harmonic light (at 532 nm) from a mode-locked tripled Vanadate laser (Vanguard, Spectra Physics, repetition rate rep 80:6 MHz) is incident on a radio-frequency photodiode, which generates a radio-frequency comb with comb-teeth at frequencies m rep (m is a positive integer). This signal is ampli ed and passed though a bandpass lter (BPF), which transmits the n = 157th comb-tooth at n rep 12:655 GHz. This is then mixed with a radio-frequency signal at LO = 12:438 MHz generated by an HP8672A synthesizer, and the lower frequency beatnote (at 217 MHz) is sent to the PLL, where an HP8640B is frequency modulated to output a signal that is phase locked with the beatnote. The bandwidth of the output signal depends on the bandwidth BW of the low pass lter LPF2 used in the PLL. Frequency spectrum of the signals at monitoring points MP1 and MP2 are shown in Fig. 2.29. 87 To stabilize the beatnote, we monitor the repetition rate continuously by mea- suring the intensity of the 532 nm light generated in the laser cavity (which has the same repetition rate, and the same uctuations in the repetition rate) by a fast photodiode (a GaAs PIN Detector, ET-4000, made by Electro-Optics Technology, Inc.), and correct for the uctuations by feeding forward to the AOMs. We lter the n(= 157)th comb-tooth by a microwaves mechanical lter, and beat the signal at n rep(t) with a frequency stabilized local oscillator at frequency LO, as shown in g. 2.28(a). Here rep(t) is the repetition rate at time t (we care about the uc- tuation time scale, which is slow compared to the delay between the laser pulses, and hence a repetition rate can be de ned at time t). The beatnote is sent through a low pass lter, which allows the lower frequency component of the beat signal at n rep LO to pass through (we assume that n rep > LO). In principle, we may use this signal directly to drive one of the AOMs, AOM1 for example. Thus at time t, n rep(t) LO = AOM1(t) ) n rep(t) AOM1(t) = LO (2.57) The right hand side of Eq. (2.57) is independent of time, which shows that the time dependence, or the uctuations in the repetition rate is canceled by a time dependent AOM1 driving frequency. Hence the sideband frequency is at sb = n rep(t) j AOM j = n rep(t) AOM1(t)+ AOM2 = LO+ AOM2, independent of time. Note that the uctuations do not cancel for the other sideband (at n rep+ 88 j AOM j) of the nth comb-tooth, or for any other comb-tooth and their sidebands in general. From a practical point of view, this beat-note may not be used to drive the AOM1 for the following reasons. First, the amplitude of this radio frequency signal is dependent on the photodiode output, and hence on the laser power. Second, the thermal and white noise from the ampli ers used makes the signal very noisy, and the noise may drive unwanted transition. This problem is overcome by feeding the beatnote signal into a phase locked loop (PLL), and using its output to drive the AOM1. Beyond the bandwidth of the low-pass lter used in the PLL, the noise pro le is characteristic of the oscillator used, and the amplitude of the beatnote is independent of the laser power. Fig. 2.29a shows the spectrum of the signals that may be used to drive AOM1, with and without the PLL. The noise oor is 30 dB lower when a PLL is used. In g. 2.29b we show the observed Raman spectrum of a single trapped 171Yb+ ion as a function of the AOM2 frequency, for a pulse duration of 40 s. The white noise present in the AOM1 signal may drive unwanted Raman transitions for a range of frequencies of AOM2, thus providing a non-zero background in the observed spectrum, as shown in the red trace. When we drive the AOM1 with the output of the PLL, this background goes away, as seen in the black trace. If the signal to white noise ratio in the AOM driving signal generated from the repetition lock circuit is , the strength of the stimulated transition by the noise eld, noise = = p , as the two photon Rabi frequency is proportional to the electric eld of each beam. Here is the Rabi frequency of the stimulated transition 89 Figure 2.29: The role of the Phase locked loop in the repetition rate stabi- lization scheme: a. Frequency spectra of the beatnote (that may be used to drive the AOM1) at points MP1 and MP2 in Fig. 2.28 respectively. The PLL gets rid of the (white) noise outside the bandwidth of the low pass lter used. b. Probability of two photon Raman excitation of a single trapped 171Yb+ ion vs AOM2 frequency with and without the PLL. Here the system is initialized in the state j #i. If the signal at point MP1 is used to drive the AOM1, the noise excites unwanted tran- sitions at all AOM2 frequencies, as seen in the constant background in the Raman spectrum. The output of the PLL does not have this noise beyond the bandwidth of the low pass lter used, and hence the Raman frequency spectrum is cleaner. Here we show the ?carrier? transition between the hyper ne 171Yb+ ?clock? states at 205 MHz and the vibrational sidebands around 200 MHz and 210 MHz. 90 by the signal at the transition frequency. In order to keep this unwanted transition probability to under in a total experimental time of T , noiseT , which implies that minimum signal to noise ratio ( T= )2. Thus for a transition frequency of =1 MHz, and an experimental duration of 1 ms, we need a signal of noise ratio > 1010, or 100 dB to keep 1%. To characterize the stability of the beatnote frequency, we compared it to the 171Yb+ ?clock? hyper ne qubit by Ramsey interferometric measurements. With the lock engaged we measured a coherence time of 800 ms. This coherence time is limited by the presence of noise a ecting the qubit frequency, such as magnetic eld noise, and not by the noise in the repetition rate. With the repetition rate lock disengaged, the coherence drops to 3ms, showing the usefulness of this lock to achieve high delity quantum operations. 2.6.5 Optical set up for the Raman transitions Stimulated two photon Raman transitions generate the quantum Ising model in our experiments, as discussed previously. We also use stimulated Raman tran- sitions for single qubit manipulation through the Bloch sphere. The two photon Raman transitions require an optical beatnote at the hyper ne transition (12.64 GHz) for simulating the e ective magnetic eld. We use two di erent ways to gen- erate this optical beatnote, Frequency modulate the output of a CW laser beam (a Ti:Sapphire laser, frequency doubled in a second harmonic generation process) using an electro 91 optic modulator (custom made by New Focus Inc.) [105] at !EOM=2 = EOM hf=2 6:32 GHz. Use the suitable comb-teeth pair from an optical frequency comb generated by a mode-locked tripled Vanadate laser with center wavelength at 355 nm. Our mode-locked laser has a repetition rate of !rep=2 = rep 80:6 MHz, and hence we use the comb-teeth pair separated by a frequency of 157 rep 12:655 GHz, which is the closest beatnote to the hyper ne splitting. A ner control of the frequency of the Raman beatnote is achieved by using AOMs. The electric eld of a CW laser beam at a frequency of !L is frequency mod- ulated by the EOM at !EOM as E1 = E0 2 exp [i(kx !Lt)] n=1X n= 1 Jn( ) exp [in [( k)x !EOM t]] + c:c: (2.58) where E0 and nu0 are the incoming (unmodulated) electric eld amplitude and frequency respectively. Jn( ) is the nth order Bessel function with modulation index , and k = !EOM=c, c is the speed of light. The modulation index depends on the radio-frequency power used to drive the EOM, and increases with higher driving power. Thus the EOM puts on equispaced sidebands around the carrier at !L, separated by the modulation frequency !EOM . 92 Figure 2.30: Raman transition set up: The schematics of the Raman set up optics with the mode-locked tripled Vanadate laser. AOM1 and AOM2 generate the frequency shifts necessary to address various stimulated Raman transitions. The 532 nm light (shown in green) is used to monitor the repetition rate, and correct for the uctuations in the repetition rate, as described in the text. The horizontal cylindrical lenses are shown in white, vertical cylindrical lenses are shown in gray and the spherical lenses are shown in blue. 93 The two photon hyper ne transition Rabi frequency, depends on the product of the electric eld amplitudes of the two beams. Thus / 1X n= 1 Jn( )Jn+2( )e i2( k)x; (2.59) as the pair of sidebands separated by 2!EOM !hf contribute to the hyper ne transition. Unfortunately, 1X n= 1 Jn( )Jn+2( ) = 0; (2.60) as J n( ) = ( 1)nJn( ), and hence the Rabi frequency vanishes from destructive interference of sidebands with opposite phases. This problem can be overcome by making the phase in the right hand side of Eq. (2.59) dependent on n, as discussed in Ref. [106]. This is achieved by splitting the frequency modulated beam into two arms of a Mach-Zehnder interferometer and recombining them at the ion position. The delay in the Mach-Zehnder interferometer is adjusted to maximize the Rabi frequency. Even with the properly adjusted interferometer, more than half of the optical power is wasted due to the destructive interference. We simulated the ferromagnetic quantum Ising model for N = 2 to N = 9 spins (Chapter 3), and the frustrated spin network with N = 3 spins (Chapter 4) with this set up. A mode-locked laser provides the desired frequency comb without any phase problem as in the EOM. Figure 2.30 shows the optics schematics of our Raman 94 set up using the mode locked laser. (The set up with the Ti:Sapphire CW laser was almost identical to this one, with the vanguard laser replaced by the frequency doubled Ti:sapphire light sent through the EOM). The two Raman beams, referred to as Raman 1 and Raman 2 in the diagram, intersect perpendicular to each other at the ion position. Thus the wave-vector di erence k = p 2k, where k is the magnitude of the wave-vector of each beam. Raman 1 is generated by the negative rst order de ection of AOM1 (from Brimrose Corp), which is driven by the output MP2 in Fig. 2.28b at approximately 217.7 MHz to to stabilize the repetition rate, as described in section 2.6.4. We image an intermediate focus IF1 at the ion position (relay imaging), by rst collimating the beam in both the horizontal and the vertical directions using a 200 mm spherical lens, followed by focusing down using a 91 mm spherical lens. AOM2 is driven by radio frequencies generated by an arbitrary waveform gen- erator (AWG, Model no. DA-12000 made by Chase Scienti c Company, 12 bit, 1 Gs/sec, 4 MB memory). The carrier transition is excited at an AOM2 driving frequency AOM2 204:819 MHz, while the red and blue sidebands are excited at 199:97 and 209:66 MHz respectively. AOM2 is imaged at the ion position by relay imaging method as follows. A horizontal cylindrical lens H2 (f=150 mm) im- ages the AOM2 aperture at the intermediate focus IF2. This is also approximately the vertical focus of V2. A 200 mm focal length spherical lens is about 200 mm away from the intermediate focus, and hence it collimates the beam in both the horizontal and the vertical directions. The beam is approximately 220 m (1=e2 radius in intensity) in the horizontal direction and 20 m in the vertical direction 95 Figure 2.31: The Clebsh-Gordan coe cients relevant for the two photon hyper ne Raman transition: We show the Clebsh-Gordan (CG) coe cients for relevant states coupled by the + and the polarizations. A linearly polarized light cannot drive the hyper ne transition, as various paths interfere destructively. The transitions j"zi $ 2P3=2jF = 2;mF = 1i contributes to the Stark shift only. at the intermediate focus. This collimated beam is then focused by a spherical 91 mm focal length lens at the ions. The Mach-Zehnder delay stage (MZ) is used to equalize the length of the Raman 1 and Raman 2 arms, to overlap the arrival times of the laser pulses at the ion position through these two paths. The =4 and the =2 wave-plates are used to gain full control of the po- larization of the Raman beams. The polarizations are shown in double headed arrows. The magnetic eld points upwards in this gure. The choice of our beam polarizations are dictated by the Clebsh-Gordan coe cients of the relevant transitions (Fig. 2.31). We avoid driving the Zeeman transitions between the 96 j #zi ! jF = 1;mF = 1i states by minimizing the polarization component in our Raman beams. The Raman beams are linearly polarized perpendicular to each other and perpendicular to the magnetic eld BY . A linear polarization is an equal superposition of + and light (in the atomic basis, w.r.t. to the externally applied magnetic eld BY ), and cannot drive the hyper ne transition alone, since the transition paths interfere destructively due to a relative negative sign of the Clebsh-Gordan coe cients between the paths, as seen from Fig. 2.31. In our set up each beam is linearly polarized and hence the beams do not interfere at the ion position, and the total intensity is constant at all times. However the two photon Rabi frequency is proportional to the excess electric eld in the + polarization over the polarization, which creates a beatnote at the frequency di erence of the two beams. The two photon Rabi frequency is proportional to the product of the electric elds of the two Raman beams. Thus, / p 1 2(1 1) in our set up, where 1 and 2 are the de ection e ciencies of AOM1 and AOM2 respectively. The two photon Rabi frequency is maximized for 1 = 0:5. 2.7 Quantum simulation recipe for experimentalists This section describes the detailed steps of the quantum simulation experi- ments done in our laboratory. Some of the most important instruments required are: Lasers - Ti:Sapphire laser (with 532 nm Verdi-18 pump) and frequency dou- 97 bler for generating the Doppler cooling, detection, optical pumping, and pro- tection/additional cooling light at 369.5 nm, 355 nm mode-locked laser for stimulated Raman transitions, 935 nm and 638 nm (semiconductor diode) repump lasers, 399 nm (semiconductor diode) ionization laser. Imaging - Photomultiplier tube, and ICCD camera. Frequency modulators - 7.37 GHz (Doppler cooling), 2.105 GHz (optical pumping) and 3.1 GHz (935 nm repump) EOM; AOMs for Doppler cooling, de- tection, protection, Raman beams, Iodine saturation absorption spectroscopy. Their supply radio-frequency oscillators, and ampli ers. Trap electrode voltage supplies (DC and radio frequency), and The FPGA (and the control program) running the experimental sequence, and the data acquisition. Here are the steps to be followed. 1. Lock all the laser frequencies. This involves locking the Ti:Sapphire laser (etalon lock, reference cavity lock, iodine saturation spectroscopy lock), locks for the diode lasers (935 nm, 399 nm, any 739 nm laser), repetition rate lock of the 355 nm mode-locked laser. 2. Load a single 171Yb+ ion in the trap (section 2.2.1). 3. Check the alignment and power of the near resonant beams (detection, Doppler cooling, optical pumping, protection/additional cooling), by maximizing the scatter 98 from the detection and the cooling beams, and minimizing the scatter from the optical pumping beam by the ion. Block the 935 nm repump to make sure that the scattering is caused due to the ion, and not by scattering o the electrodes, trap parts or the imaging system. Work on the alignment, and the polarizations of all the beams, if necessary. In our set up, the FPGA detection counters are tied to the Detect window in the experimental sequence. Hence we use the following experimental sequences to check the optimal settings for the near resonant beams. The experimental chapters are shown in di erent colors. We use a PMT to monitor the uorescence from the ion. To check the detection - Doppler Cool ! Detect. We get approximately ten photon counts on average from the detection beam in 800 s. To check the Doppler cooling - same as above, except now block the detection beam with a beam-block, or turn the detection AOM OFF from the radio- frequency switch box, and turn the Doppler cooling AOM ON all the time. The Doppler cooling time in our experiment is usually 3 ms, but we shall detect the scatter from the ions only during the Detect chapter (800 s) in this diagnostic test. The PMT detects about ten photons on an average from the cooling beam. Don?t forget to put the Doppler AOM back to the computer control, and unblock or turn the detection beam back ON. To check the optical pumping - Doppler Cool ! Optical pump ! Detect. The PMT counts should be close to zero (less than one count on an average in our experiment). 99 4. Drive carrier Rabi oscillations with the Raman beams, maximize the Rabi frequency by working on the Raman beam alignments. If the frequency of the carrier transition is unknown, a frequency scan of one of the Raman beam AOMs may be necessary. The experimental sequence is Doppler Cool ! Optical pump ! Raman ! De- tect. Scan the Raman pulse duration to nd the =2 time, and then maximize the scattering by the ion, keeping the Raman pulse duration xed at the =2 time. To get a ner response in the alignment, a longer pulse duration (5 =2, or 9 =2) may be used. 5. Check the spin/qubit coherence. A Ramsey experiment would give us an idea of any decoherence due to magnetic eld and other noises, that a ect the frequency splitting of the qubit state. If we use the 355 nm mode locked laser for the =2 pulses in the Ramsey interferometry, the results would also indicate the e cacy of the beat-note lock. The Ramsey experimental sequence is Doppler Cool ! Optical pump ! Raman ( =2) ! Delay ! Raman ( =2) ! Detect. The intensity noise on the Raman beams due to beam pointing instability or power uctuations can be estimated from the decay in the amplitude of the carrier Rabi oscillation signal. 6. Take a frequency scan to observe all the important features of the system, such as the Zeeman states, and the motional sidebands (the Raman pulse duration should be long enough and adjusted to see the features). Make sure that the Zeeman and the motional transition are at the expected frequencies. Any peak due to the micromotion must be removed by tweaking the static voltages on the trap electrodes. 100 The Zeeman transitions may be suppressed by tweaking the polarizations of the Raman beams, and the direction of the magnetic eld by varying the currents owing though the magnetic coils. 7. Di erential A.C. Stark shift from each of the Raman beams may be measured by turning on that beam in the Delay chapter of the Ramsey experimental sequence. The di erence in the frequencies of the Ramsey fringes with and without the Raman beam present in the Delay chapter is the di erential A.C. Stark shift from the Raman beam. A typical di erential A.C. Stark shift in the qubit states of 171Yb+ with about 1 Watt 355 nm beam focused to a 150 m 7 m waists (cylindrical beam) is about 400 Hz. This measurement can be used to center the Raman beams horizontally on the ion. Move the ion horizontally by changing the static voltages on the trap electrodes, and make the A.C. Stark shift from each beam symmetric about the ion position with normal static voltage settings. 8. Raman sideband cooling - Turn on the sideband cooling sequence (SBCool ), at the sideband frequencies (estimated from the frequency scan). The red sideband peak in a Raman frequency scan should be depleted in the presence of the sideband cooling. In our experimental geometry, the Raman beams predominantly couple to the X transverse modes, and hence it is hard to cool the Y modes appreciably. 9. Measure the sideband frequencies precisely. The experimental sequence is: Doppler Cool ! Optical pump ! SBCool ! Raman ! Detect. Turn the power in one of the Raman beams (we call it Raman 2) to a very low value, and nd the position of the sidebands carefully. The sideband frequencies in the limit of zero optical power 101 do not contain the di erential A.C. Stark shifts, thus measure the ?true? sideband frequencies at this trap con guration. Note that the di erential Stark shift from the other beam (Raman 1) is still present. However, the contribution from Raman 1 in this experiment is primarily to add the two photon di erential A.C. Stark shift, which is typically negligible compared to the four photon di erential A.C. Stark shift, arising due to o -resonant coupling of the Raman beatnote (at the sideband frequency) to the carrier transition. Typical numbers in our set up are 600 Hz for the two photon Stark shift (from the two beams), versus about 30 KHz for the four photon Stark shift (when driving a sideband at about 5 MHz). Once we nd the sideband frequencies, we set the frequency of the carrier tran- sition (that will drive the e ective external magnetic eld in our quantum Ising Hamiltonian), carr = rsb + bsb 2 : (2.61) Our next task is to balance the power in the two Raman beat-notes (referred to as the rsb and the bsb beat-notes), so that the four photon di erential A.C. Stark shift from the o -resonant carrier is canceled, when symmetrically detuned about the carrier (to drive the M lmer-S rensen transition in multiple spins later). We increase the optical power in the beat-notes by increasing the power in the radio- frequency signal driving the AOM 2. As the di erential A.C. Stark shift from the AOM 2 will shift all the three frequencies fcarrier, rsb, bsbg, rst we add/subtract the pre-calibrated di erential A.C. Stark shift from all the frequencies. Next, we detune both the rsb and the bsb beat-notes symmetrically from the carrier by a few 102 sideband line widths (which can be estimated from the width of the sideband peaks in the frequency scan, or by driving a sideband on resonance). If the power in the rsb and the bsb beat-notes are balanced, the four photon di er- ential A.C. Stark shift should cancel, as the beat-notes are detuned symmetrically about the carrier. We check this by a Ramsey experiment, with the following se- quence, Doppler Cool ! Optical pump ! SBCool ! Raman ( =2, 0) ! O - resonant Raman rsb with rf power P1+O -resonant Raman bsb with rf power P2 ! Raman ( =2, =2) ! Detect. The duration of the chapter in italics is scanned. Here Raman ( =2, ) is a =2 pulse with a phase . Thus we initialize the spin along the y axis (by rotating about the x axis of the Bloch sphere, = 0 refers to the x axis.) Then we turn on the detuned rsb and bsb beat-notes. If the A.C. Stark shift is nulled, these beat-notes do not rotate the spin state in the Bloch sphere, and hence the nal rotation about the y axis ( = =2) from the Raman transition leave the spin in the same state. So, we?ll detect a brightness of 0.5, as the detection basis is along the z axis of the Bloch sphere. If, however, there is an uncompensated Stark shift due to unbalanced power in the two beams, we shall observe a Ramsey fringe as the duration of the ?O -resonant Raman rsb with rf power P1+O -resonant Raman bsb with rf power P2? chapter is scanned. Tweak the power P1, and P2 to null this fringe over the maximum duration of the desired quantum simulation experiment. 10. Drive the red sideband on resonance to measure the sideband Rabi frequency more precisely. For this, set the Raman rsb beat-note on resonance with the motional 103 red sideband transition, and detune the bsb beat-note from the blue sideband reso- nance by 4-5 times the estimated sideband Rabi frequency. Since the red sideband transition for the optically pumped spin state j"zi is suppressed due to the Raman sideband cooling, we apply a pulse to ip the spin state to j "zi. We observe the j "z; ni $ j #z; n + 1i Rabi oscillation, with n 0, and measure the frequency of the oscillation. This should be equal to , where is the Lamb-Dicke parameter, and is the carrier Rabi frequency consistent with the optical intensity applied to drive the sideband. The experimental sequence is: Doppler Cool ! Optical pump ! SBCool ! Raman ( ) ! Raman rsb on resonance+o -resonant Raman bsb ! Detect. Similarly, measure the blue sideband Rabi frequency precisely by bringing the Ra- man bsb beat-note on resonance with the motional blue sideband transition, and detuning the rsb beat-note away from the transition. The experimental sequence is, Doppler Cool ! Optical pump ! SBCool ! O -resonant Raman rsb+Raman bsb on resonance ! Detect. Now that we have found the appropriate frequency and power settings for the M lmer-S rensen transitions, we shall calibrate the e ective transverse magnetic eld, B before moving on to multiple ions. 11. The strength of the B eld is given by, B = B 2 ; (2.62) 104 where B is the carrier Rabi frequency. We change the power of the radio frequency signal driving the Raman AOM P to vary B, and hence B. We calibrate the B eld for a given radio-frequency power P , by measuring the B eld strength for a few P values, and following the trend. 12. Load another 171Yb+ ion in the trap. Now we have two ions, and we are ready to observe Ising coupling induced oscillations between spin states. 13. Detune the Raman rsb and bsb beat-notes symmetrically about the carrier, at fre- quencies HF , where is the beat-note frequency, = x + . Choose to be larger than a few times the sideband Rabi frequencies ( ) in order to avoid populating the phonon modes. Now prepare the spins in the j #z#zi states, and apply the Raman beat-notes. The Ising interaction, J 1x 2 x will make the spins os- cillate between the j#z#zi and j"z"zi states. The frequency of the oscillations is the Ising coupling (multiplied by two) between the spins. The experimental sequence is: Doppler Cool ! Optical pump ! SBCool ! Raman rsb and bsb symmetrically detuned from the carrier ! Detect. The probability of j "z"zi state, P (j "z"zi) should reach unity (the average number of spins in the j "zi state should reach 2) in the course of time evolution. At this point, we may also maximize the contrast of this oscillation by ne adjustments of the rsb and bsb beat-note frequencies. For this, set the pulse duration to the time when P (j "z"zi) is at its maximum, and frequency scan the rsb beat-note around = !x + , keeping the bsb beat-note xed in the previous experimental sequence. Find the frequency of the rsb beat-note that maximizes P (j "z"zi). This 105 should almost be identical to the frequency that we set earlier to observe the Ising coupling induced oscillations. If the new frequency is slightly shifted, reset the carrier according to eq 2.61. 14. Load the desired number of ions in the trap. 15. If we are going to use a CCD imager to detect the spatial resolved spin states, at this point we should nd the Region of Interests (ROIs), and discriminator counts of the ions [91] on the CCD. 16. [Optional] Take a Raman frequency scan, and identify the motional transitions. Include other modes in the sideband cooling sequence. The transverse modes used for generating the Ising couplings are tightly packed together in frequency, and hence the modes need not be addressed separately for Raman cooling. Empirically, three Raman beat-notes are su cient to cool down almost all the X modes for a system of 10 spins in our experiment. 17. Now we are ready to run a quantum simulation experiment. Have the parameters ready for the quantum simulation experiment, such as the M lmer-S rensen detun- ing ( ), phases of the carrier =2 pulses for single qubit rotation, the initial B- eld, time constant of ramping the B eld down. Table 2.1 lists the phases of various pulses used in our quantum simulation experiment. 106 Field/pulse phase comments First =2 pulse 180 for polarizing the spins along the y direction B (carrier) 270 for simulating FM quantum Ising model B (carrier) 90 for simulating AFM quantum Ising model rsb 0 bsb 180 Final =2 pulse 270 for rotating the measurement basis Table 2.1: Phases of various pulses used in quantum simulation. 2.8 Troubleshooting with 174Yb+ 174Yb+ has a zero nuclear spin and hence the hyper ne structure is absent in the electronic ground state. Thus the Doppler cooling beam does not need any sideband to cool the atom, unlike 171Yb+ which requires a 14.74 GHz sideband. The lack of hyper ne Zeeman sub-levels also eliminates the coherent population trapping in the dark states [89], and results in an increased uorescence. 174Yb+ can be used to troubleshoot optics alignment problems. To load 174Yb+ we need only one 369.5 nm beam, the 935 nm beam and of course the loading beams. We switch all the lasers to 174Yb+ frequencies (Table C.1), and try to capture 174Yb+ ions by making sure that all the beams are approximately passing through the trapping region. Once we load 174Yb+ ions (either in a crystal or in a cloud), we verify of the alignment of individual 369.5 nm beams (detection, Doppler cooling, optical pumping, and the protection beams), and the 935 nm beam. The EOMs should be turned o to observe maximum uorescence from 107 the 174Yb+ ions. If the alignment was the only problem, we should now be all set to load 171Yb+ in our trap, after switching the current supply to the isotopically enriched oven. If we still cannot load 171Yb+ in the trap, we should verify that the Doppler cooling and 935 nm EOMs are generating appropriate sidebands, and check that the magnetic eld coils are not shorted. 108 Chapter 3 Simulation of the ferromagnetic quantum Ising model 3.1 Overview The ferromagnetic quantum Ising model is one of the simplest models that admit a quantum phase transition (QPT). For N interacting spin 1=2 objects, the Hamiltonian is given by H = X i;j j 0) is the Ising coupling between spins i and j, and B is an e ective transverse magnetic eld. Here the Ising couplings are not limited to the nearest neighbors only, and the range of the interactions can be tuned, as we have discussed in the previous chapter. This Hamiltonian (Eq. (3.1)) shows quantum properties as the Ising interactions and the e ective transverse eld couple to non-commuting spin components. In this chapter , we discuss the simulation of the ferromagnetic quantum Ising Hamiltonian (Eq. (3.1)) in a chain of up to 16 trapped 171Yb+ ions. We tune the e ective transverse magnetic eld while keeping the Ising couplings xed, following Most of the results presented in this chapter are published in Ref. [41]. 109 the adiabatic quantum simulation protocol [73], and directly measure the spin order along the Ising (x) direction. For an e ective transverse magnetic eld much stronger than the Ising interactions, the spins are polarized along the y direction, resulting in a paramagnetic state along the x direction. As the magnitude of the transverse eld is lowered, the spins start to align themselves according to the ferromagnetic Ising couplings, and we observe a crossover between paramagnetic and ferromagnetic spin order. Since a QPT is expected to emerge for a su ciently large system with a system Hamiltonian given by Eq. (3.1), the observed nite system crossover curves should sharpen as we scale the system up. In this chapter we compare the results of quantum simulation with N = 2 to N = 9 spins, and observe that the crossover curves between the phases indeed get sharper with increasing system size, prefacing the expected quantum phase transition in the thermodynamic limit. This experiment serves as a benchmark for simulation of arbitrary fully connected quantum spin models, for which the theory becomes intractable for more than a few dozen spins. As described previously, we use stimulated Raman transitions to engineer Hamiltonian (Eq. (3.1)). The experiments with N = 2 to N = 9 spins are per- formed using Raman beams from a Ti:Sapphire laser, which is 2:7 THz detuned from the 2S1=2 to 2P1=2 resonance in 171Yb + at 369:5 nm, resulting in a small but non-negligible probability of spontaneous emission, which introduces decoherence in the coherent time evolution. We compare the experimental results with numerical calculations using a quantum trajectory method, which becomes ine cient beyond a few dozen spins. We use a far detuned pulsed laser at 355 nm to reduce the 110 decoherence error due to spontaneous emission, and N = 16 ion experiments are performed with this laser. 3.2 Symmetries of the Hamiltonian The quantum Ising Hamiltonian (Eq. (3.1)) has a global spin rotation sym- metry by a Bloch vector angle of about the B eld axis, or the y axis. The single spin unitary rotation operator Ur = exp ( i 2 y) = i y is a symmetry operation, under which the x spin states transform in the following way, Urj"i = j#i (3.2a) Urj#i = j"i: (3.2b) In our experiments we simulate the quantum Ising model by shining laser beams with nearly uniform intensities on all the ions. Thus the e ective transverse magnetic eld is near uniform across the spin chain, and the Ising couplings have a spatial re ection symmetry about the center of the spin chain. The Hamiltonian inherits this re ection symmetry. In the absence of any external biasing eld, a nite system does not sponta- neously break the symmetry of the Hamiltonian. Thus as we tune the dimensionless parameter of the e ective transverse eld to the Ising couplings in the Hamiltonian, the symmetries are preserved. We shall elucidate this for a system of N = 2 spins in the next section. We omit the subscript x from the eigenstates of x throughout this thesis. 111 3.3 Low energy eigenstates at T=0 We de ne the e ective average Ising coupling as J = h P j 6=i Ji;jii, where h:::ii indicates averaging over the index i. With this de nition, J 2J0 (N > 2) when the Ising couplings are limited to the nearest neighbors only (Ji;j = J0 for j = i 1, and 0 otherwise), and J = (N 1)J0 NJ0 for ?in nite range? interactions (Ji;j = J0 for any i and j). For B=J ! 0 the Hamiltonian is dominated by the Ising interactions, and the spins show ferromagnetic order along the x direction. In the other limit of B=J !1 the e ective transverse eld dominates over the Ising interactions, and the spins are polarized along the y directions, or paramagnetic along the x direction (we assume that B > 0). In between these two extremes, the system undergoes a crossover between these two phases, which approaches a quantum phase transition in the large system limit of N !1. 3.3.1 States near B=J = 0 At B=J = 0, the ferromagnetic (FM) states j "" ::: "i and j ## ::: #i form a doubly degenerate ground state manifold, where j "i and j #i are the eigenstates of x. Any quantum superposition of these two states is also a ground state. To nd the ground states for small B=J , we treat the transverse eld part of Hamiltonian (Eq. (3.1)), HB = B P i i y as a perturbation over the Ising Hamiltonian, HI = P i 0, the antisymmetric superposition is the new ground state, separated from the symmetric state ( rst excited state) by an energy of 2B2=J = 2J(B=J)2. The ground state is antisym- metric w.r.t. the spin ip symmetry (Eqs. (3.2)), and symmetric w.r.t. the spatial re ection symmetry. For a system size of up to a dozen spins, the ground state can be exactly calcu- lated by diagonalizing the Hamiltonian (Eq. (3.1)), but the perturbation treatment presented here gives us some insight on how the degeneracy in the ground state splits in the presence of a small transverse eld. We may generalize our discussion to larger spin chains, and note that for a system of N spins, the degeneracy in the ground state is generically broken at the N th order in perturbation, and hence the energy splittings between the perturbed ground states is O((B=J)N). We illustrate this for N = 2 and N = 3 spins in gure 3.2. The ground state is always entangled for small but nite B=J . For very large N , the ground state and the rst excited states are nearly degenerate for B=J < 1 (B > 0). The perturbation theory breaks down when B=J approaches unity. 115 Figure 3.2: Ground state degeneracy splitting by the transverse eld: A non-zero transverse eld B breaks the degeneracy in the ground state manifold of Eq. (3.1) at B = 0. Since the FM ground states in the absence of a transverse eld are connected by a global spin ip only, the degeneracy is broken in the N th order in perturbation for a system of N spins, as explained in the text. Here we plot the energy of the rst excited state (measured from the ground state) as a function of B=J for N = 2 and N = 3 spins. The disks and the triangles are the results from an exact diagonalization of Eq. (3.1), and the solid lines are quadratic (N = 2) and cubic (N = 3) polynomials in B=J . For a system of N spins, the energy splitting is O(B=J)N . When the Ising interactions are antiferromagnetic, the competition between the interactions may lead to increased degeneracy in the ground state manifold, and the transverse eld may break the degeneracy more easily, as discussed in chapter 4. 116 Note that in the presence of long range ferromagnetic interactions the thermo- dynamic limit of HI may not exist, as the energy density may become unbounded from below, unless the Ising couplings themselves implicitly depend on the system size N appropriately. 3.3.2 States near B=J !1 At B=J ! 1 (J = 0), the ground state of Hamiltonian (Eq. (3.1)) is non- degenerate, with all spins pointing along the y direction, namely j "y"y :::i . The ground state transforms to ( i y)( i y)j"y"yi = j"y"yi under the spin ip oper- ation i.e., it is antisymmetric w.r.t. the spin ip symmetry. This state is symmetric under the re ection symmetry of the Hamiltonian. The rst excited states are the singe spin ipped states, j#y"y"y :::i, j"y#y"y :::i, ... . The second excited states are the double spin ipped states and so on. For B=J !1, we may treat HI as a perturbation over HB in the Hamiltonian. HI lifts the degeneracy between the states in the rst excited state manifold, as Jij ix j x directly couples ( rst order in perturbation) the excited state with the i-th spin ipped and the excited state with the j-th spin ipped. So the degeneracy in the rst excited state manifold is generically broken with a nite Ising interaction between the spins. As an example, the rst excited states with N = 2 spins are j"y#yi and j#y"yi 117 at B=J !1. As xj"yi = ij#yi and xj#yi = ij"yi, HI j"y#yi = J j#y"yi (3.6a) HI j#y"yi = J j"y#yi (3.6b) Thus the e ective Hamiltonian up to the rst order in perturbation in the basis formed by the unperturbed states in rst excited state manifold, i.e., j "y#yi and j#y"yi is, 0 B B @ 0 J J 0 1 C C A (3.7) Hence the new eigenstates are (j #y"yi + j "y#yi)= p 2 with energy J and (j "y#y i j #y"yi)= p 2 with energy J . The energy splitting between these states is 2J , independent of B to the rst order in perturbation. Note that the ground state energy is 2B, and hence the energy splittings of these two perturbed states from the ground state j #y#yi are 2B J and 2B + J respectively. We tabulate the symmetries of these and the other eigenstates of Eq. (3.1) for N = 2 spins in table 3.1. Similarly, for a system of N spins, the unperturbed rst excited state manifold contains N single spin ipped states (along y) at B=J ! 1, that are mixed and entangled by the Ising interactions HI . 3.3.3 Quantum phase transition at B = J In between the two extreme limits of B=J ! 0 and B=J ! 1, the Ising interactions and the transverse eld compete to align the spins accordingly. As B=J 118 Parameters Eigenstates Spin ip Spatial Re ection B=J ! 0 1p 2 (j""i j##i) antisymmetric symmetric 1p 2 (j""i+ j##i) symmetric symmetric 1p 2 (j"#i j#"i) symmetric antisymmetric 1p 2 (j"#i+ j#"i) antisymmetric symmetric B=J !1 j"y"yi antisymmetric symmetric 1p 2 (j"y#yi+ j#y"yi) symmetric symmetric 1p 2 (j"y#yi j#y"yi) symmetric antisymmetric j#y#yi antisymmetric symmetric Table 3.1: Symmetries of the eigenstates: The eigenstates of Eq. (3.1) are either symmetric or antisymmetric w.r.t. the two symmetries of the Hamiltonian- the spin ip symmetry and the re ection symmetry, as described in section 3.2. Here we tabulate the symmetries of the eigenstates of the Hamiltonian (Eq. (3.1)) for N = 2 spins in the two extreme limits of B=J ! 0 and B=J !1. is reduced from its in nite value, the system crosses over from the polarized or the paramagnetic state to the ferromagnetic state in a nite system. As the system size increases, a second order phase transition point emerges. The ground state of Eq. (3.1) smoothly connects the ground states in these two limits of B=J ! 0 and B=J !1 for all intermediate values of B=J . The rst excited state in the limit of B=J !1 remains the rst excited state for all B=J 6= 0, and becomes a ground state at B=J = 0. The second excited state undergoes an avoided level crossing with the ground state. As we vary the dimensionless parameter B=J , the instantaneous eigenstates of the Hamiltonian change preserving the symmetries ( though in the in nite system size limit, the system may break the symmetries spontaneously). If the Hamiltonian is changed faster than the adiabatic limit, the system may be excited from one eigenstate to another, but states with opposite symmetries do not mix. As an 119 Figure 3.3: A few low energy eigenstates of Eq. (3.1) for N = 5 spins: Here we plot the energy spectra of six low energy eigenstates of Eq. (3.1) for N = 5. The Ising couplings fall-o with distance as Jij = J=ji jj in this example. E refers to the energy of the eigenstates relative to the ground state. The minimum energy between the ground and the rst excited state having the same symmetry as the ground state is the critical gap, c that sets the relevant energy scale for an adiabatic quantum simulation, as discussed in the text. The cusps on the highest energy states shown here due to level crossings from other states not included in this diagram. 120 example, the ground state for N = 2 couples to the third (the highest) excited state as they are both antisymmetric w.r.t. spin ip and symmetric w.r.t. re ection. This result holds for N > 2 spins, and the minimum gap between the ground and the third excited state is the critical gap, c, which occurs at B J , that sets the time scale of ramping the Hamiltonian in order to be adiabatic. We show a few low energy states of the Hamiltonian (Eq. (3.1)) and the critical gap for a system of N = 5 spins, with ferromagnetic coupling in Fig. 3.3. The critical gap depends on the system size, and vanishes in the limit of an in nite system. This also depends on the range of the interactions. For nearest neighbor interactions, this scales as 1=N with system size, while in the other extreme of perfectly uniform interactions between all the spins it scales more favorably, c N 1=3 [107]. 3.4 Experiment: onset of a quantum phase transition 3.4.1 Engineering the ferromagnetic Ising couplings We simulate the Ising interactions by o -resonant stimulated Raman transi- tions on the transverse vibrational modes of the ion chain, following the M lmer- S rensen scheme [71], as described previously. The Ising coupling Jij between spins i and j is given by Ji;j = i jR NX m=1 bi;mbj;m 2 !2m : (3.8) where bi;m is the component of the normal mode eigenvector between ion i and mode m at frequency !m = 2 m (with 1 = X denoting the COM mode), and 121 i = g2i = is the carrier Rabi frequency on the ith ion. Here gi is the single photon Rabi frequency of the ith ion and is the detuning of the Raman beams from the 2S1=2 $ 2P1=2 transition. R = ~ k2=(2M) is the recoil frequency of a single ion (of mass M) in the stimulated Raman transition process. The M lmer-S rensen beatnote controls the sign and range of the interactions, as di erent normal modes are virtually excited. We tune close to the COM mode, to make all the couplings of the same sign. The COM mode (m = 1) makes all the couplings uniform across the chain, as the eigenvectors bi;1 = 1= p N for all i, and hence Jij is the same for any pair (i; j). The system exhibits in nite range interactions, and the geometry of the spin chain (and the dimensionality) becomes irrelevant. For a su ciently large system, each spin feels a ?mean eld? created by all the other spins. We note that the Ising coupling Jij / 1=N , thus the thermodynamic limit of Eq. (3.1) exists, and the phase transition point B = J = h P j 6=i Ji;jii becomes independent of the system size. Other modes o -resonantly excited by the Raman beatnote suppress the range of the interactions. We use up to N = 9 spins to observe the sharpening of the crossover from the paramagnetic to the ferromagnetic phase. In the experiment 2 2:7 THz, i 2 370 KHz and we expect J=N 2 1 KHz for the beatnote detuning such that !1 4 i;1 i, , as shown in Fig. 3.4a . This beatnote corresponds to The Lamb-Dicke parameter for COM scales with the system size as 1= p N . However, as pointed out in Ref. [53], a more stringent condition to avoid phonon excitations would be to keep the detuning constant for all system size. This is due to the fact that, even if the Lamb-Dicke parameter goes down with increasing system size, there are more spin con gurations that lead to phonon excitations. 122 Figure 3.4: Raman spectrum of vibrational modes and Ising coupling pro- le for N = 9 spins: Transverse [108] vibrational modes are used in the exper- iment to generate Ising couplings according to Eq. (3.8). a. Raman sideband spectrum of vibrational normal modes along transverse X direction for nine ions, labeled by their index m. The two highest frequency modes at 1 (CM mode) and 2 = p 21 2z (\tilt" mode) occur at the same position independent of the number of ions. The dotted and the dashed lines show beatnote detunings of 1+30 KHz and 1 +63 KHz used in the experiment for N = 9 and N = 2 ions respectively. Carrier transition, weak excitation of transverse Y and axial Z normal modes and higher order modes are faded (light grey) for clarity. b. Theoretical Ising coupling pattern (Eq. (3.8)) for N = 9 ions and uniform Raman beams. The main contribu- tion follows from the uniform COM mode, with inhomogeneities given by excitation through the other nearby modes (particularly the tilt mode). Here, J1;1+r / 1=r0:35 (r 1), as found out empirically. For larger detunings, the range of the interaction falls o even faster with distance, approaching the limit Ji;j / 1=ji jj3 for !1 [109]. 123 2 63 KHz blue of the COM mode frequency for 2 ions and 2 30 KHz for 9 ions. The expected Ising coupling pattern for a uniformly illuminated ion chain is shown in Fig. 3.4b for N = 9 ions and the couplings are dominated by uniform contribution of the COM mode. The non-uniformity in the Ising couplings arises from other vibrational modes and variation in i across the ion chain (gaussian Raman beams with a waist of 70 m along the ion chain and 7 m perpendicular to the ion chain were used in the experiment). For N = 9 ions the chain is 14 m long, and the variation in i is 2%. 3.4.2 Experimental protocol and order parameters of the transition In the experiment, we follow the highest excited state of the Hamiltonian H [39, 54], which is formally equivalent to the ground state of Hamiltonian H (Eq. (3.1)). This reduces direct excitation of the tilt mode phonons. The quantum simulation experiment proceeds as follows (Fig. 2.20). We cool all the X transverse modes of vibration to near their ground states, and deep within the Lamb-Dicke regime by standard Doppler and Raman sideband cooling procedures. We initialize the spins to be aligned to the y direction of the Bloch sphere by optically pumping to j #z#z ::: #zi and then coherently rotating the spins through =2 about the Bloch x axis with a carrier Raman transition. Next we switch on the Hamiltonian H with an e ective magnetic eld B0 5J so that the spins are prepared predominantly in the ground state. Then we exponentially ramp down the e ective magnetic eld with a time constant of = 80 s to a nal value B, keeping the Ising couplings xed. 124 We nally measure the spins along the Ising (x) direction by coherently rotating the spins through =2 about the Bloch y axis before uorescence detection on a photomultiplier tube (PMT). We repeat the experiment 1000 N times for a system of N spins and generate a histogram of uorescence counts and t to a weighted sum of basis functions to obtain the probability distribution P (s) of the number of spins in state (j "i), where s = 0; 1; :::N . We can generate several magnetic order parameters of interest from the distribution P (s), showing transitions between di erent spin orders. One order parameter is the average absolute magnetization (per site) along the Ising direction, mx = 1N NP s=0 jN 2sjP (s). Eq. (3.1) has a global time rever- sal symmetry of f ix ! i x, i z ! i z, i y ! i yg as discussed previously and this does not spontaneously break for a nite system, necessitating the use of av- erage absolute value of the magnetization per site along the Ising direction as the relevant order parameter. For a large system, this parameter shows a second or- der phase transition, or a discontinuity in its derivative with respect to B=J . On the other hand, the fourth-order moment of the magnetization or Binder cumulant g = NP s=0 (N 2s)4P (s)= NP s=0 (N 2s)2P (s) 2 [110, 111] becomes a step function at the QPT and should therefore be more sensitive to the phase transition. We illustrate this point by plotting the exact ground state order in the simple case of uniform Ising couplings for a moderately large system (N = 100) in Fig. 3.5. When the spins are polarized along the y direction of the Bloch sphere, the distribution of total spin along the x direction is binomial and approaches a Gaus- 125 Ord er P ara meter s (pe rfe ct ad iaba tic the ory) 0.0 0.5 1.0 N=2 Binder cumulant, N=100 (uniform Ising) N=9 Magnetization, N=100 (uniform Ising) N=9 N=2 0.01 0.1 1 6 g Figure 3.5: Binder cumulant and magnetization for the adiabatic theory: Theoretical values of order parameters mx and g are plotted vs B=J for N = 2 and N = 9 spins with non-uniform Ising couplings as used in the experiment in the case of a perfectly adiabatic time evolution. The order parameters are calculated by directly diagonalizing Hamiltonian (3.1). Order parameters are also calculated for a moderately large system (N = 100) with uniform Ising couplings, to show the di erence between the behaviors of mx and g. In case of uniform Ising couplings the e ective ground state manifold reduces to N + 1 dimensions in the total spin basis. The scaled Binder cumulant g, unlike the scaled magnetization mx, approaches a step function near the transition point B=J 1, making it experimentally suitable to probe the transition point for relatively small systems. 126 sian (with zero mean) in the limit of N ! 1. For system size of N , mx takes on theoretical value of m0x;N = 1 N2N NP n=0 NCnjN 2nj in the perfect paramagnetic phase (B=jJ j ! 1) and unity in the other limit of B=J = 0. We rescale mx to mx = m0x;N mx = m0x;N 1 which should ideally be zero in perfect paramag- netic phase and unity in perfect ferromagnetic phase for any N . This accounts for the ?trivial? nite size e ect due to the di erence between Binomial and Gaussian distribution. Similarly the Binder Cumulant g is scaled to g = (g0N g) = (g 0 N 1), where g0N = 3 2=N is the theoretical value of g for B=J !1. 3.4.3 Results In Fig 3.6a-b we present data for the scaled magnetization mx and the scaled Binder cumulant g as B=J is varied in the adiabatic quantum simulation for N = 2 and N = 9 spins. The data shows the increased steepness for the larger system size. The scaled magnetization mx is suppressed by 25% (Fig. 3.6a) and the scaled Binder cumulant g is suppressed by 10% (Fig. 3.6b) from unity at B=J = 0, predominantly due to decoherence from o -resonant spontaneous emission and ad- ditional dephasing due to intensity uctuations in Raman beams during the simu- lation. Figure 3.7 shows the scaled magnetization, mx for N = 2 to N = 9 spins depicting the sharpening of the crossover curves from paramagnetic to ferromag- netic spin order with increasing system size. The linear time scale indicates the exponential ramping pro le of the (logarithmic) B=J scale. The three dimensional background is unphysical, and is used just to help us visualize this sharpening. 127 Figure 3.6: Emergence of the ferromagnetic spin order: a. Magnetization data for N = 2 spins (circles) is contrasted with N = 9 spins (diamonds) with representative detection error bars. The data deviate from unity at B=J = 0 by 20%, predominantly due to decoherence from spontaneous emission in Raman transitions and additional dephasing from Raman beam intensity uctuation, as discussed in the text. The theoretical time evolution curves (solid line for N = 2 and dashed line for N = 9 spins) are calculated by averaging over 10,000 quantum trajectories. b. Scaled Binder cumulant ( g) data and time evolution theory curves are plotted for N = 2 and N = 9 spins. At B=jJ j = 0 the data deviate by 10% from unity, due to decoherence as mentioned before. 128 Figure 3.7: Onset of a quantum phase transition- sharpening of the crossover curves with increasing system size.: Scaled magnetization, mx vs B=J (and simulation time) is plotted for N = 2 to N = 9 spins. As B=J is lowered, the spins undergo a crossover from a paramagnetic to ferromagnetic phase. The crossover curves sharpen as the system size is increased from N = 2 to N = 9, prefacing 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. The (unphysical) 3D background is shown to guide eyes. 129 We compare the data shown in Fig. 3.6 to the theoretical evolution taking into account experimental imperfections and errors discussed below, including sponta- neous emission to the spin states and states outside the Hilbert space, and additional decoherence. The evolution is calculated by averaging 10; 000 quantum trajectories. This takes only one minute on a single computing node for N = 2 spins and ap- proximately 7 hours, on a single node, for N = 9 spins. Extrapolating from this calculation suggests that averaging 10; 000 trajectories for N = 15 spins would re- quire 24 hours on a 40 node cluster, indicating the ine ciency of classical computers to simulate even a small quantum system. 3.4.4 Sources of error in the quantum simulation In this section we discuss some of the primary sources of errors in our quantum simulation. We characterize errors in the current simulation by plotting the observed parameter P(FM)=P(0)+P(N) for N = 2 to N = 9 spins in Fig. 3.8. Theoretically P(FM) = 2=2N when the spins are transversely polarized i.e.,P(FM) = 0:5 for N = 2 spins and exponentially goes down to 0:004 for N = 9 spins, and unity when there is perfect ferromagnetic order. Since P(FM) involves only two of the 2N basis states, it is more sensitive to errors compared with the order parameters mx and g. For instance, at B=J = 0 in Fig. 3.6a-b and Fig. 3.8a-d we nd that mx and g do not change appreciably with the system size, but P(FM) degrades to 0:55 for N = 9 spins from 0:9 for N = 2. In Fig. 3.8 we compare the data with theory results that include experimental sources of diabatic errors. 130 Figure 3.8: Suppression in the ferromagnetic delity with increasing sys- tem size: a-d. Ferromagnetic order P(FM)=P(0)+P(N) is plotted vs B=J for N = 2 to N = 9 spins. The circles are experimental data and the lines are theo- retical results including decoherence and imperfect initialization. As this quantity includes only two of 2N basis states random spin- ips and other errors degrade it much faster than the magnetization and Binder cumulant. The representative de- tection error bars are shown on a few points for each N . The P(FM) reduces from 0:9 to 0:55 as the system size is increased from two to nine. The principle contribution to this degradation is decoherence, predominantly due to spontaneous emission from intermediate 2P1=2 states in the Raman transition and additional de- phasing primarily due to intensity uctuations in Raman beams. Shown in d is an estimated breakdown of the suppression of P(FM) from various e ects for N = 9 spins. Non-adiabaticity due to nite ramping speed, spontaneous emission and addi- tional dephasing due to uctuating Raman beams suppress P(FM) by 8%, 18% and 24% respectively from unity (B=J ! 0). 131 Some of the primary sources of experimental error are discussed below: Diabaticity Since we are ramping the Hamiltonian at a nite speed, there is a non-zero probability of population transfer to the excited states. As discussed in the previ- ous chapter, the diabaticity is related to the critical gap ( c) between the ground and the relevant excited state. Diabaticity due to nite ramping speed and error in initialization is estimated (by direct numerical integration of the Schr odinger?s equation) to suppress P(FM) by 3% for N = 2 to 8% for N = 9. This also gives rise to the oscillations seen in the data (Fig. 3.6a-b and Fig. 3.8a-d). Spontaneous Emission The experiments with N = 2 to N = 9 are done with Raman beams de- tuned by 2.7 THz from the 2S1=2 2 P1=2 transitions, which corresponds to a 10% spontaneous emission probability per spin in 1 ms. Spontaneous emission during the otherwise coherent time evolutions leads to the following errors ( gure 3.9): Spin ip { When one of the 2P1=2jF = 1;mF = 1i states are populated, spontaneous emission from them may lead to spin ip with probability 1/3. Throwing the system out of the Hilbert space { Spontaneous emission may populate the Zeeman states with a probability of 1=3, which are not included in the Hilbert space of the e ective spin-1/2 system. The Zeeman states, 132 Figure 3.9: Spontaneous emission from the Raman beams: O -resonant excitations from the Raman beams populate the 2P1=2 states, which then sponta- neously decay to the 2S1=2 states. Spontaneous emission introduces entropy in the coherent quantum evolution, by ipping the spin states randomly and resetting the phase of the evolution. Spontaneous emission may also populate the Zeeman states jF = 1;mF = 1i and jF = 1;mF = 1i (shown in red), and thus throws the system out of the Hilbert space. Here the violet arrows denote the + and polarization components of a Raman beam, and the red fuzzy arrows are the spontaneous emis- sion channels. For clarity, we show spontaneous emission channels from one of the hyper ne states in the 2P1=2 manifold only. Similarly 2P1=2jF = 1;mF = 1i can decay to the states jF = 1;mF = 1; 0i and jF = 0;mF = 0i of the 2S1=2 manifold. All the Clebsch-Gordan coe cients have equal magnitudes of 1= p 3 for these transi- tions. The Raman beams are detuned by from the 2P1=2 states. For the frequency doubled CW Ti:sapphire laser used in some of the experiments, = 2:7 THz (red detuned from the 2P1=2 states); and for the mode-locked 355 nm laser, = 33 THz (blue detuned from the 2P1=2 states). 133 however, are detected as bright states i.e., as j"i states. Dephasing { The j "i (j #i) state may return to itself, with a probability of 1/3, after a spontaneous emission event. However, the entanglement between spins may be lost, as the phase of the coherent time evolution is reset. Spontaneous emission loosely leads to a nite ?spin temperature? in this system, though the spins do not fully equilibrate with the ?bath? and the total probability of spontaneous emission increases linearly during the quantum simulation. Another physical way to think about the e ects of spontaneous emission is that it introduces entropy into the quantum evolution. Spontaneous emission errors grow with increas- ing system size, which also suppresses P(FM) order with increasing N , as seen in Fig. 3.8a-d. We theoretically estimate the suppression of P(FM) due to diabaticity and spontaneous emission together by averaging over quantum trajectories to be 7% for N = 2 spins and 26% for N = 9 spins. Intensity uctuations on the Raman beams Intensity uctuations on the Raman beams during the simulation modulate the AC Stark shift on the spins and dephase the spin states, which causes addi- tional diabaticity and degrades the nal ferromagnetic order. When we introduce a theoretical dephasing rate of 0:3 per ms per ion (see Appendix A) in the quantum trajectory computation the predicted suppression of P(FM) increases to 9% for N = 2 and 50% for N = 9. 134 Finite detection e ciency Imperfect spin detection e ciency contributes 5 10% error in P(FM). Fluorescence histograms for P(0) and P(1) have a 1% overlap (in detection time of 800 s) due to o -resonant coupling of the spin states to the 2P1=2 level [90]. This o -resonant coupling prevents us from increasing detection beam power or photon collection time to separate the histograms. Detection error in the data include uncertainty in tting the observed uorescence histograms to determine P (s), intensity uctuations and nite width of the detection beam. A problem with detecting the spin states with a PMT is that the bright state histograms for s bright ions overlap strongly with s+ 1 bright ions for large s. This is illustrated in Fig. 3.10, where we show the photon count histograms for s = 1 to s = 9 bright ions. The increased overlap between the histograms with increasing system size results in detection error in the measured magnetization. We de ne the overlap between the histograms as, Pm;n = P1 k=0Pm(k)Pn(k) [ P1 k=0Pm(k) 2 P1 k=0Pn(k) 2]1=2 ; (3.9) where Ps(k) is the probability of observing n photons in the detection time window from s bright ions. With this de nition, the overlap increases from P1;2 = 2% between s = 1 and s = 2 bright ions, to P8;9 = 80% between s = 8 and s = 9 bright ions. 135 Figure 3.10: Detection with a PMT { overlapping photon count histograms: We use state dependent uorescence to detect the spin state of the ions. If an ion is in the j "zi state, it uoresces when hit by a laser beam resonant with the 2S1=2jF = 1;mF = 0i 2P1=2jF = 0;mF = 0i transition, which we collect using f=2:1 optics (numerical aperture, NA=0.24) on a PMT for about 800 s. Over repeated measurements, the emitted photons from a j "zi spin form a modi ed Poisson distribution, as discussed in the text. A j #zi spin, on the other hand, does not uoresce as the exciting laser is o -resonant (by 14:7 GHz compared to the natural linewidth of 20 MHz) to the 2S1=2jF = 0;mF = 0i 2P1=2jF = 1;mF = 1; 0; 1i transitions. Thus we identify j "zi as the observed ?bright? state, and j "zi as the ?dark? state. Here we show typical uorescence histograms for s = 1 to s = 9 bright spins. n denotes the number of photons collected by the PMT in the detection time, and Ps(n) denotes the probability of observing n photons for s bright ions. The overlap between the photon distributions for s and s 1 bright ions increases with increasing s. Thus the detection error increases as the system size grows. As an example, the overlap between the histograms for s = 8 and s = 9 bright spins is 80%. 136 Figure 3.11: Suppression in the GHZ coherence with increasing system size: GHZ coherence or parity of the observed spin state is shown for N = 2 to N = 5 spins. Here we apply an analysis =2 pulse after ramping the transverse eld B to near zero, and scan the phase of the pulse. The decreasing contrast of the parity oscillation curves denote the decaying GHZ entanglement in the simulation as the system size grows. Phonons play an important role in reducing the GHZ entanglement, as phonon states mix symmetric and antisymmetric FM spin com- binations, and the GHZ entanglement is partly destroyed when we trace over the phonon states during measurement. The amplitude of the parity oscillation is 0.8, 0.47, 0.35 and 0.27 for N = 2; 3; 4 and 5 spins respectively. Other primary souces of decoherence include spontaneous emission from the Raman beams, and dephasing from intensity uctuations on the Raman beams, as discussed in the text. Here the blue circles are the experimental data, and the solid curves are sinusoidal t to the data. 137 Phonons The role of phonons in the results of the quantum simulation is investigated both experimentally and numerically. Phonons destroy the spin ip symmetry of the pure spin Hamiltonian (Eq. (3.1)), as the term in the unitary evolution operator that contains the phonon operators (a and ay), also includes the spin operator x. Thus states with symmetries di erent from the ground state are excited in the adiabatic quantum simulation, and the pure spin entanglement is partly destroyed when we trace over the phonon states during the measurement. The nal FM state achieved in the quantum simulation with the phonon e ects included is of the form (j"" :::ij i j## :::ij i) [112], where j i is a coherent state, with average phonon occupation hni = j j2. The phonon occupation for N = 9 spins was numerically found to be under hni = 1:5, and lower for N < 9 spins. However, these phonons do not alter the ground state spin ordering and hence preserve spin-spin correlation. In Fig. 3.11 we show the loss of entanglement in the simulation with increasing system size for N = 2 to N = 5 spins. We apply an analysis =2 pulse with phase after the quantum simulation, and observe the parity as the phase is scanned from 0 to 2 . The delity of the GHZ state, de ned to be the overlap of the ideal GHZ state with the actual experimentally observed FM state, j FMi is given by, FGHZ = jh FM jGHZij2 = 12(P##:::#+P"":::")+jC##:::#;"":::"j. Here, P (FM) = P##:::#+P"":::" and jC##:::#;"":::"j denotes the GHZ coherence, i.e., the coe cient of j ## ::: #ih"" ::: " j in the density matrix [113]. We measure the GHZ coherence by observing the contrast of the oscillating parity signal, (P (0) + P (2) + :::) (P (1) + P (3) + :::). 138 As shown in Fig. 3.11, the observed amplitude of the parity oscillations is 0:8 for N = 2 spins, and decreases to 0:27 for N = 5 spins. This corresponds to a GHZ delity, FGHZ of 0:95 for N = 2 spins to 0:52 for N = 5 spins. For N > 5 spins, the delity drops below 0.5, indicating the loss of GHZ entanglement, possibly from phonon occupation, along with other sources of decoherence like spontaneous emission discussed in this section. 3.5 Scaling up the simulation to N = 16 with 355 nm mode locked laser Two of the primary sources of decoherence in our experiments with N = 2 to N = 9 spins, spontaneous emission and uctuating di erential AC Stark shift, can be reduced by using a laser detuned farther from the 2P1=2 energy level for the stimulated Raman transitions. The Rabi frequency of the transition and the decoherence e ects (spontaneous emission and the di erential Stark shift) depend on the detuning, as, / 1 (3.10a) spont / 1 2 (3.10b) AC / 1 2 ; (3.10c) 139 Figure 3.12: Quantum simulation of the ferromagnetic Ising model with N = 16 spins: We simulate Eq. (3.1) in a system of N = 16 spins. The Hamiltonian is simulated using a tripled Vanadate mode-locked laser at 355 nm, which virtually eliminates spontaneous emission error in the experiment. We use a PMT to detect the uorescence from the ions, and repeat the experiments about 2000 times to make a histogram of the collected photons. Each spin in the j"i state emits 10 photons on average in the detection time of 800 s. a. At B J , the spins are polarized along the e ective eld B (paramagnetic phase) and hence half of the total number of spins are in state j "i. We observe a histogram centered at 80 photons, as expected. b. As the transverse eld is reduce, the FM correlations start to build up, thus the uorescence distribution broadens. c. At a very low eld (B J), the spins order ferromagnetically, as seen in the bifurcation of the histogram. 140 where spont and AC are the rate of spontaneous emission and the AC Stark shift respectively. Thus increasing the detuning helps reducing the decoherence e ects compared to the Rabi strength in the Raman transitions. We use a mode locked tripled Vanadate laser with the center wavelength at 355 nm and an average optical power of 4 W to alleviate laser-induced decoherence. With this laser, spont= < 10 5 and AC= < 10 5 [114], thus practically eliminating these issues. We ran a quantum simulation experiment with N = 16 spins with this laser. As we tune the e ective external eld B compared to the ferromagnetic Ising cou- plings, we observe the emergence of FM order. In Fig. 3.12 we show the total uorescence counts for this experiment, at three di erent values of the e ective ex- ternal eld. When the spins are polarized along the y axis of the Bloch sphere, we observe a mean uorescence count of about 80. This denotes that roughly half of the spins are in the state j "i (a spin j "i would uorescence with about 10 mean photons in our detection time of 800 s). As B is reduced keeping the Ising couplings constant, the uorescence histogram broadens, and nally bifurcates at B=J ! 0, characteristic of the FM phase in absence of any biasing eld, and any spontaneous symmetry breaking. 141 Chapter 4 Three frustrated Ising spins on a triangle 4.1 Overview A network is said to be frustrated when it is impossible to satisfy all the interactions (?bonds?) individually. Frustrated magnetic systems [72, 115] may lead to non-trivial many body properties, such as massive degeneracy in the ground state, and entanglement [43] in a quantum system. Frustration is believed to be a key ingredient to understand properties of exotic spin systems like quantum spin liquids and spin glasses [116, 47, 117, 118]. Other complex systems and phenomena such as social [119] and neural [120] networks, and protein folding [121] owe their complexity to frustration. Three Ising spins interacting antiferromagnetically with each other on the corners of an equilateral triangle constitute one of the simplest examples of spin frustration. As shown in Fig. 4.1a, for antiferromagnetically oriented spins 1 and 2, spin 3 cannot satisfy both the (antiferromagnetic) interactions with the other spins simultaneously, and hence the ground state is frustrated. Out of 23 = 8 possible basis spin states, six belong to the ground state manifold. Since any quantum superposition of the basis ground states is also a ground state, it is possible to prepare a massive entangled ground state while experimentally simulating this spin model. 142 Figure 4.1: Three frustrated Ising spins on a triangle: a. Three spins inter- acting antiferromagnetically on the corners of a triangular lattice cannot satisfy all the interactions simultaneously, a phenomenon known as the geometric frustration. Here spin 3 cannot simultaneously anti-align to both the spins 1 and 2. For a classi- cal spin system, spin 3 chooses one of the spin con gurations (" or #) at random. A quantum system may be in both the con gurations at the same time, owing to the quantum superposition principle. Frustration in a magnetic system usually leads to an exponentially large number (in the system size) of degenerate ground states, leading to a non-zero entropy at the lowest temperature. b. For a fully connected spin network, the geometry of the lattice is irrelevant. Here we simulate the frus- trated spin network with three spins given by the hyper ne states of trapped 171Yb+ ions in a linear con guration. Spins 1 and 3 interact with Ising coupling J2, which is almost equal to the nearest neighbor interactions J1, e ectively generating the triangular network geometry. Here the bright spots (enumerated as 1, 2 and 3) are the CCD images of three trapped and laser-cooled 171Yb+ ions. 143 In this chapter we describe the simulation of frustrated Ising spins on a trian- gle. We probe the ground state of the transverse eld Ising model by adiabatically ramping the transverse magnetic eld. The ground state reached using this method is an entangled state. Note that the Hamiltonian has a global spin rotation sym- metry by a Bloch vector angle of about the y axis (section 3.2), which may lead to entanglement even when the system is not frustrated. This symmetry is, how- ever, broken easily by applying an axial magnetic eld. Since the degeneracy in the ground state in the case of antiferromagnetic interactions arises from the interac- tions, rather than the symmetry in the Hamiltonian, the entanglement does not go away completely when the axial eld is turned on. This chapter is organized in the following sections: Section 4.2 - we describe the theoretical ground state of the transverse eld antiferromagnetic Ising model with N = 3 spins. Section 4.3 - a brief introduction to the entanglement and entanglement wit- ness that is used to characterize the ground state entanglement. Section 4.4 - we describe the choice of the experimental parameters, and de- scribe the experimental sequence to simulate the frustrated spin model. Section 4.5 - we present data showing the frustrated ground state, and entan- glement. We compare this case with results from the simulation of a ferro- magnetic model that does not show frustration. The results presented in this chapter are published in Ref. [43]. 144 4.2 Frustrated quantum Ising model We simulate [43] the frustrated Ising model in presence of an e ective trans- verse magnetic eld, in a system of N = 3 Ising spins. The Hamiltonian is given by, H = J1( 1 x 2 x + 2 x 3 x) + J2 1 x 3 x B( 1 y + 2 y + 3 y): (4.1) Here the Ising couplings J1 and J2 are antiferromagnetic (> 0), ix and i y are the x and y spin-1/2 Pauli matrices for the ith spin (i = 1; 2; 3), and B is the e ective transverse magnetic eld. Note that the exact geometry of the spin chain is irrelevant in this example, as the network is fully-connected. In fact, we simulate Eq. (4.1) with a linear chain of three spins. For J1 = J2 = J this system is equivalent to the system shown in Fig. 4.1b, and the ground state is highly frustrated. 4.2.1 States near B=J = 0 At B=J = 0, the ground state of Eq. (4.1) lies in the six dimensional Hilbert space formed by the basis states j ""#i; j "#"i; j "##i; j ##"i; j #"#i; j #""i. The two ferromagnetic states j """i and j ###i and all their quantum superpositions belong to the excited state manifold. A nite transverse eld induces couplings between the members of the ground state manifold and hence splits the degeneracy. For B=J ! 0 we treat the coupling to the transverse eld as a perturbation over the Ising spin couplings. Using per- turbation theory, analogous to our treatment in section 3.3.1, we nd that a unique 145 Figure 4.2: Increased sensitivity to quantum uctuations in presence of frustration: In the absence of the e ective transverse magnetic eld B in Eq. (4.1), the ground state manifold contains two fold degenerate FM states (j """ i and j ###i) when the Ising couplings are negative (J1 < 0; J2 < 0), and six fold degenerate AFM states (j ""#i, j "#"i, j #""i, j ##"i, j #"#i, j "##i) when the couplings are equal and positive (J1 = J2 = J > 0). A small transverse eld induces quantum uctuations between the spin states and lifts the degeneracy. a. In case of ferromagnetic couplings, the energy splitting between the ground and the rst excited state is O(B=jJ j)3, as explained in the text. Here the blue circles are exact energy calculated from the direct diagonalization of Eq. (4.1). The blue solid line is a cubic polynomial of the form E=J = 0:75(B=J)3. (Here E is the energy measured from the ground state). b. In case of antiferromagnetic couplings (frustration), the degeneracy is split in the rst order in B=J . The blue circles are the exact results from the diagonalization, and the blue solid line is a polynomial of the form E=J = B + 0:75B2 0:35B3. 146 ground state emerges for non-zero eld, namely j""#i+ j"#"i+ j#""i j"##i j##" i j#"#i. This is a superposition of two W states [122]. This can be contrasted with the case where all the interactions are ferromag- netic (i.e., for J1 < 0; J2 < 0 in Eq. (4.1)). At B=J = 0 (J1 = J2 = J) the ground state manifold is two dimensional, and spanned by the ferromagnetic (FM) states j"""i and j###i. For a nite B=jJ j, the ground state is (j"""i+ j###i)= p 2, and the energy splitting between the ground state and the rst excited state is O((B=J)3). This is due to the fact that the FM states are only related by the global spin ip symmetry (along x) for the ferromagnetic Ising model, and to break the degeneracy between the states, we need to go to N = 3rd order in perturbation, as illustrated previously in section 3.3.1. In the case of antiferromagnetic Ising model, the frustra- tion present in the system leads to extra degeneracy in the ground state manifold, and a small quantum uctuation may break the symmetry of the ground state, and lead to non-trivial phases. 4.2.2 Preparing the entangled state in adiabatic quantum simulation As in the case of the ferromagnetic quantum Ising model, we follow the ground state of Eq. (4.1) (nearly) adiabatically in the experiment. For large B=J , the ground state is the spin polarized state j "y"y"yi. As we change the dimensionless ratio B=J , the ground state evolves, maintaining the symmetry of the Hamiltonian, as discussed in section 3.3.3. Thus we reach the ground state j ""#i+ j "#"i+ j #"" i j "##i j ##"i j #"#i when the e ective transverse eld is ramped down to 147 zero. Clearly this state is entangled, as it cannot be written as a product state of the three spins. The entanglement present in the system can be characterized by measuring the ?entanglement witness operators?. In the next section, we describe the witness operators in brief. 4.3 Frustration and entanglement A genuine tripartite entanglement in a system of three qubits belongs to one of the following two classes: Greenberger-Horne-Zeilinger (GHZ) entangled state, jGHZi = j"""i+ j###i, also referred to as a ?Schr odinger?s cat state?, and W-states- Superposition of states with k number of spins ipped from the state j###i. For N = 3 spin system, there are two W-states, corresponding to k = 1; 2; namely, jW1i = j"##i+j#"#i+j##"i and jW2i = j#""i+j"#"i+j""#i . The constituent basis spin states may have arbitrary phases in both the GHZ and W-states. Note that W1 and W2 are connected by a global spin ip symmetry in the case of a system with three spins. Any tripartite entangled state of the three spins can be transformed to either a GHZ or a W-state by local (single qubit) unitary operations. A general way to characterize entanglement is to experimentally measure the density matrix of the system, also known as quantum tomography [66]. The 148 o -diagonal terms in the density matrix quantify the quantum cohererence in the system. However, measuring the density matrix involves exponentially many mea- surements (in system size) and hence is not always practical or e cient. Measuring the expectation value of an entanglement witness operator provide a practical alter- native to detect and characterize entanglement. The witness operators are de ned to detect entanglement of a particular kind, and it may be non-trivial to construct a witness operator. In general a witness operator W with respect to some entangled state is constructed as, W = I j ih j; (4.2) where the constant is chosen such that Trace W 0 (4.3) if the system shows this particular ( type) entanglement. If Eq. (4.3) is not satis ed, we cannot rule out entanglement in the system, but we may conclude that the system does not exhibit entanglement of the type . In case of three spins, = GHZ or W. In this chapter we prepare the frustrated ground state of the three spin Ising network. The ground state is a superposition of the two W-states. We compare it to the case of ferromagnetic couplings, where there is no frustration present in the system, and the expected ground state is a GHZ state. We use the following witness 149 operators to detect the tripartite entanglement (GHZ or W-state) present in our system [122, 43, 113], WGHZ = 9=4 J^ 2 x (1) (2) (3) (4.4a) WW = 4 + p 5 2(J^ 2y + J^ 2 z ); (4.4b) where J^ 12 P i (i) is proportional to the projection of the total e ective angular momentum of the three spins along the -direction, and is a direction in the yz-plane of the Bloch sphere. The degeneracy in the ground state due to competing antiferromagnetic in- teractions leads to entanglement, which we detect and quantify using the witness WW . By switching all the interactions to ferromagnetic, we observe a GHZ entan- glement, characterized by WGHZ . In presence of an axial magnetic eld, the GHZ entanglement is destroyed, as the magnetic eld destroys the spin ip symmetry in Eq. (4.1) and a non-degenerate ground state emerges (j """i or j ###i, depending on the sign of the axial eld). This is in sharp contrast with the frustrated ground state, which retains some entanglement even after the symmetry is broken. This establishes a connection between frustration and an extra degree of entanglement. We use the bipartite spin-squeezing witness operator, WSS [43, 122] to characterize entanglement of the symmetry broken frustrated antiferromagnetic ground state. The witness is given by, WSS = (J^ 2 x + 3 4 )2 4hJ^xi 2 (J^ 2y + J^ 2 z 3 2 )2; (4.5) 150 which is less susceptible to experimental errors than the W-state witness operator [43]. 4.4 Experimental methods We work with three 171Yb+ ions forming a linear chain along the Z direction of our Paul trap, with a center of mass (COM) trap frequency of Z =1.49 MHz. The three normal modes of transverse vibrational motion, along the X direction occur at, COM mode: !1=2 = 1 = 4.334 MHz Tilt mode : !2=2 = 2 = 4.074 MHz Zig-zag mode : !3=2 = 3 = 3.674 MHz. Like the simulation of the ferromagnetic Ising model, discussed in the previous chapter, the e ective spin-1/2 system is represented by the 2S1=2 jF = 1;mF = 0i and jF = 0;mF = 0i hyper ne \clock" states in each ion, depicted by j "zi and j#zi, respectively [123], and separated in frequency by HF = 12:642819 GHz (This corresponds to 5 G magnetic eld to de ne the quantization axis). To simulate Eq. (4.1), we shine Raman beams on the ions that drive the motional modes o - resonantly, as explained previously, and obtain an e ective spin-spin Ising type interaction, following the M lmer-S rensen scheme [71]. A carrier Raman transition with suitably de ned phase simulates the e ective transverse magnetic eld. 151 To simulate the frustrated spins on the corners of an equilateral triangle, we tune the M lmer-S rensen Raman beatnote ( ) near the X COM mode ( > !1). From Eq. (3.8), we see that the Ising couplings are all positive, and hence anti- ferromagnetic. The nearest neighbor and the next nearest neighbor Ising couplings are almost equal, and hence the interaction graph resembles an equilateral triangle. The far o -resonant tilt mode makes the nearest neighbor slightly stronger than the next nearest neighbor. We use a scaled M lmer-S rensen detuning ~ in this chapter, de ned as ~ 2 !21 !2Z : (4.6) With this de nition, the COM, tilt and the zigzag modes correspond to ~ = 0; 1 and 2:4 respectively. We cool all transverse x modes to near their zero point of motion and deep within the Lamb-Dicke regime, then we initialize the electronic states of each 171Yb+ ion along the y axis of the Bloch sphere through optical pumping and rotation operations [123]. Next we apply the Ising coupling along with a strong transverse eld and adiabatically ramp down the eld. An ideal triangle geometry would mean J1 = J2, which is obtained by coupling to the COM mode only. Here J1 and J2 are the nearest and the next nearest neighbor Ising couplings respectively. The far o -resonant tilt mode breaks this symmetry and makes J1 > J2. The energy spectrum in presence of unequal couplings is shown in Fig. 4.3. The nearest neighbor antiferromagnetic states j "#"i and j #"#i are lower in energy than the next nearest antiferromagnetic states, j ""#i, j "##i, j ##"i 152 Figure 4.3: Energy diagram for J1 6= J2: Weak couplings to the tilt and the zigzag vibrational modes make the Ising interactions slightly non-uniform. Here we show the energy diagram of 23 = 8 spin con gurations according to the quantum Ising Hamiltonian (Eq. (4.1)), at B = 0. The nearest neighbor AFM states (j "#"i and j#"#i are the degenerate ground states, the next nearest neighbor AFM states ( j""#i, j##"i, j"##i, and j#""i) are the rst excited states, and the FM states (j"""i and j ###i) are the third and highest excited states. The energy of each state and the di erence from the ground state energy ( E) is shown next to the levels. In our experiment, J2 0:85J1. To simulate the triangle geometry with equal couplings, we ramp the Hamiltonian in our experiment faster than the energy splitting between the ground and the rst excited states, but keep the ramping slow enough to avoid exciting the system to the FM states. 153 and j #""i by energy 1 = 2(J1 J2). The ferromagnetic states, j """i and j ###i are 2 = 2(J1 + J2) above the ground states. We mimic the triangular coupling by ramping the e ective transverse magnetic eld in Eq. (4.1) faster compared to 1, but slower compared to the splitting between the ground and the ferromagnetic states, 2 (Fig. 4.3). We nally measure the spins along the x-axis of the Bloch sphere by rotat- ing the spins from the x-basis to the z-basis and measuring the spin state (j #zi or j "zi) through standard spin-dependent uorescence techniques [123], using a charge-coupled device (CCD) imager (detection e ciency 95% per spin includ- ing initialization and rotation errors). We determine the probability of each spin con guration by repeating the above procedure 1000 times. We also measure the number of spins in state j "i by using a photomultiplier tube (PMT), which is useful for higher e ciency measurements of certain symmetric observables such as entanglement delities and witness operators. 4.5 Experimental Results In Fig. 4.4 we show the observed spin order in the quantum simulation for the ferromagnetic and the frustrated cases, for various magnetic eld end points B. The theoretical curves show both the exact ground-state populations as well as the expected population evolution from the actual applied time-dependent Hamiltonian [108], using the Trotter formula and including the contribution from phonons to lowest order in the Lamb-Dicke expansion [124]. Fig. 4.4a corresponds to nearly 154 Figure 4.4: Population of spin states with ferromagnetic and antiferro- magnetic Ising couplings: Here we show the results from quantum simulation of the ferromagnetic and the antiferromagnetic Ising models as a function of the ratio of the e ective transverse magnetic eld B to the average Ising coupling Jrms = p (2J21 + J 2 2 )=3. The green circles are the two FM states, the blue dia- monds are the two symmetric AFM states, and the red squares are the remaining four asymmetric AFM states. The dashed lines correspond to the populations in the exact ground state and the solid lines represent the expected theoretical evolution from the actual ramp, including non-adiabaticity from the initial sudden turn on of the Ising Hamiltonian. The probability of inelastic spontaneous scattering is not included in the theory curves. (a) All AFM interactions (scaled M lmer-S rensen detuning = 0:27). The FM-ordered states vanish and the six AFM states are all populated as B=Jrms ! 0. Because J2 0:8J1 for this data, a population imbal- ance also develops between symmetric and asymmetric AFM states. (b) All FM interactions (scaled M lmer-S rensen detuning = 0:27), with evolution to the two FM states as B=Jrms ! 0. 155 uniform AFM couplings and gives roughly equal probabilities for all six AFM states (3=4 of all possible spin states). The slight inequality of the Ising couplings J1 and J2 are re ected in the higher probability of the symmetric AFM states, j "#"i and j #"#i. Fig. 4.4b corresponds to FM couplings, and the two FM states are clearly predominant. In order to compare these two cases, we characterize the entanglement in the system at each point in the adiabatic evolution by measuring particular entangle- ment witness operators, with negative expectation values indicating the correspond- ing form of entanglement [122]. For the FM case, we measure the GHZ witness operator [113, 122] WGHZ (Eq. (4.4a)). For the AFM (frustrated) case, we measure the symmetric W-state witness [122], WW (Eq. (4.4b)). In both cases, as shown in Fig. 4.5, we nd that entanglement of the corresponding form develops during the adiabatic evolution. In this AFM/FM comparison, we operate with ~ 0:22 for both cases (J2 0:8J1 > 0), but for the FM case we reverse the sign of B and follow the highest excited state [39], which is formally equivalent to measuring the ground state of the sign-inverted Hamiltonian. Thus all the antiferromagnetic cou- plings are e ectively turned into ferromagnetic couplings. We may also simulate the ferromagnetic couplings by tuning the beat-note between the COM and the tilt mode ( 1 < ~ < 0). But the enhanced contribution of the tilt mode phonons due to the proximity of the mode (compared to the case of > !1, or ~ > 0) degrades the GHZ entanglement delity. In macroscopic systems, the global symmetry in the Ising Hamiltonian of Eq. 3.1 is spontaneously broken, and ground-state entanglement originating from this 156 Figure 4.5: Entanglement generation through the quantum simulation: We showed the entanglements through entanglement witness measurements, using a PMT for detection. For (a) FM and (b) AFM situations as B=Jrms is ramped down, with a negative value of the witness operator indicating entanglement. For this data, jJ2=J1j 0:85. (a) For the FM regime we measure a GHZ witness operator with = y (blue circles) and nd that entanglement occurs for B=Jrms < 1:25. The GHZ delity F (green circles), or the overlap probability with the ideal GHZ state, is also extracted from this measurement, where F > 0:5 indicates entanglement [113]. (b) For the frustrated AFM case we measure a W-state witness operator (blue circles) and nd that entanglement emerges for B=Jrms < 1:1. In both plots, the dashed lines are theoretical witness values for the exact ground states, while the solid lines theoretically describe the expected witness values given the actual ramps, not including errors due to spontaneous scattering, uctuations in control parameters, and detection errors. The error bars represent the spread over the observed witness expectations following various absolute global rotation directions, and indicate the uncertainty from parasitic e ective magnetic elds not appearing in Eq. (4.1) as well as possible drifts in the control parameters. 157 Figure 4.6: Entanglement from the frustration: The e ect of symmetry- breaking on the FM and AFM cases of the Ising model, using a PMT for detection. (a) Measured x basis populations with FM Ising model (J1; J2 < 0; B=Jrms = 0:42), labeled by the probability P (N) of N spins in state j"i. (b) Measured populations of FM Ising model with B=Jrms = 0:34, where a symmetry-breaking eld is added dur- ing the ramp, increasing linearly to Bx=Jrms = 0:87, showing the emergence of the single state j"""i. (c) Measurement of GHZ witness operator without (blue circles) and with (red squares) symmetry-breaking eld, showing a quenching of entangle- ment. In the latter case, the direction in the GHZ witness operator is coordinated with the time-dependent phase of the GHZ state. (d) Measured x basis popula- tions of AFM Ising model (J1; J2 > 0; B=Jrms = 0:36). (e) Measured populations of AFM Ising model with B=Jrms = 0:34, where a symmetry-breaking eld is added during the ramp, increasing linearly to Bx=Jrms = 1:19, showing the emergence of the three states j""#i, j"#"i, and j#""i (f) Measurement of bipartite spin-squeezing entanglement witness operator applied to the AFM case, showing that entanglement remains even after symmetry is broken. As in Fig. 4.5, the error bars in (c) and (f) represent the uncertainty from parasitic e ective magnetic elds and drifts not appearing in Eq. (4.1). 158 symmetry is expected to vanish for the non-frustrated FM case [47]. 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 during the adiabatic evolution [125]. Figure 4.6a shows the measured nal populations after adiabatic evolution to the Ising Hamiltonian (B Jrms) in the FM case without symmetry-breaking. Figure 4.6b shows the same with a symmetry-breaking eld Bx Jrms that breaks the degeneracy of the two components of the FM ground state and leaves a dominant j """i product state. Figure 4.6c shows a measurement of the corresponding GHZ witness operator, displaying a clear quenching of GHZ-type entanglement when symmetry is broken. For the frustrated AFM case, Fig. 4.6d shows the measured nal populations of the evolution of the Ising Hamiltonian, with the six AFM states dominating. But when symmetry is broken (Fig. 4.6e), the AFM system primarily evolves to the three states j ""#i, j "#"i, and j #""i, consistent with the expected W-state. The residual population in the other states is attributed to nonadiabatic evolution and a nite value of B at the end of the ramp. We characterize entanglement of the symmetry-broken frustrated AFM case by measuring the bipartite spin-squeezing witness operator [122] WSS (Eq. (4.5)). We choose this witness operator because it is less sensitive to experimental errors than the W-state witness operator WW [122]. The observation of negative values of WSS presented in Fig. 4.6f shows directly that the frustrated ground state carries entanglement even after global symmetry is 159 broken in the Ising model, and thereby establishes a link between frustration and an extra degree of entanglement. 4.6 Summary and outlook We have simulated a frustrated spin network with three spins, interacting antiferromagnetically on a triangle. The antiferromagnetic couplings induce com- petition between spin states, resulting in a massive degeneracy in the ground state. By introducing an e ective transverse magnetic eld and following the adiabatic quantum simulation protocol, we prepare an entangled state of the degenerate spin states. This entanglement is fundamentally di erent, in its origin and nature, from the case where all the couplings are ferromagnetic. Unlike the GHZ entanglement achieved in the ferromagnetic model, the W-state entanglement in the frustrated model is robust against a small biasing eld. In the next chapter we shall simulate an antiferromagnetic quantum Ising model with tunable range of interactions in a system of N > 3 spins, and observe frustration in spin ordering. It is possible to simulate an arbitrary lattice geometry in this chain of trapped ions by appropriately tailoring the interactions between each pair of spins. We shall conjecture on how to realize this in the Outlook chapter of this thesis. 160 Chapter 5 Frustrated magnetic ordering with tunable range antiferromagnetic couplings 5.1 Overview Long range antiferromagnetic interactions lead to frustration, as the individual interactions cannot be satis ed simultaneously. This may lead to massive degeneracy in the ground state and entanglement, as discussed in the previous chapter in the context of three Ising spins on a triangle. Ising models with beyond nearest neighbor interactions, the so called ANNNI models, have been studied in great detail, and a plethora of interesting phases have been theoretically uncovered [126, 127]. Here we simulate a transverse eld long range antiferromagnetism Ising model using a chain of up to N = 16 spins. The Hamiltonian is given by H = X i;j j 0) is the antiferromagnetic (AFM) Ising coupling between spins i and j, and B is an e ective transverse magnetic eld. The Ising coupling between spins i and 161 j approximately fall o with the separation between them as, Ji;j = J0 ji jj : (5.2) We observe the onset of antiferromagnetic spin ordering as the transverse eld is made weaker than the Ising couplings. We also tune the range of the inter- actions for a system of N = 10 spins, by controlling the transverse vibrational mode spectrum, and thereby control the frustration in the system. By directly measuring the spin correlations, we observe the role of frustration in the spin ordering. The Ising couplings are simulated by imparting o -resonant spin dependent forces from a mode-locked laser (at a center wavelength of 355 nm, see section 2.6.4) to the 171Yb+ ions, following the M lmer-S rensen scheme [71] as explained in the previous chapters. This chapter is organized as follows: Section 5.2- we discuss some features of the long range antiferromagnetic quan- tum Ising model, in particular the ground and a few low lying energy states. The critical energy gap between the ground and excited states in presence of the e ective transverse magnetic eld depends on the range of the interactions, which we explain in this section. Section 5.3- we describe the simulation of the AFM quantum Ising Hamiltonian Most of the results presented here are from the following manuscript in preparation. \Frus- trated magnetic ordering with variable range interactions in a trapped ion quantum simulator", R. Islam, C. Senko, W. C. Campbell, S. Korenblit, J. Smith, A. Lee, E. E. Edwards, J. C.-C. Wang, J. Freericks and C. Monroe. 162 (Eq. (5.1)), and the mechanism of tuning the range of interactions by changing the trapping parameters. We discuss the experimental protocol and some order parameters used to detect the spin order. Section 5.4- we compare the spin ordering in quantum simulation with various ranges of interactions, and present data that show suppression of the AFM or Neel order as the frustration increases in the system. 5.2 Some features of the long range antiferromagnetic quantum Ising model 5.2.1 Ground and low energy eigenstates For 6= 0 : In the absence of the transverse eld (B = 0 in Eq. (5.1)) the two fold degenerate antiferromagnetic (AFM) states j "#" :::i and j #"# :::i are the ground states for 6= 0 in Eq. (5.1). The rst excited states contain two adjacent spins ipped from the AFM states on one end, and hence has a domain wall between the second and the third spins from one of the edges. Here a domain wall consists of two neighboring spins aligned in the same direction. Due to the left-right symmetry of the couplings, and the global spin ip symmetry of the Hamiltonian, the rst excited state is four fold degenerate, namely j#" "#" :::i, j::: #"# #"i, j"# #"# :::i and j::: "#" "#i. Here we have shown the spins that are ipped from the AFM ground states in red. The energy (measured from the ground state energy) of the 163 rst excited states is, Edouble;end = 2J0 " 1 2 N 2X n=2 ( 1)n 1 n + 1 (N 1) # : (5.3) This energy ( Edouble;end) is lower than the energy of the states with two adjacent spins ipped from the ground state in the bulk of the chain, as such states have two domain walls. The states with one of the end spins ipped (from the AFM states) have an energy of Esingle;end = 2J0 " N 1X n=1 ( 1)n 1 1 n # ; (5.4) which is lower than the energy of the states with a bulk spin ipped, as they have two domain walls. Figure 5.1 shows a few excited state energies for various range of the interactions, 0 < < 3. Eqs. (5.3) and (5.4) hold for = 0 as well, but the position of the excitation becomes irrelevant in calculating the excitation energy. For = 0 : In case of a uniform antiferromagnetic Ising model ( = 0), the Hamiltonian can be written in terms of the total spin operator S = P i i ( = x; y; z) as, H = J0S 2 x BSy: (5.5) Here we have subtracted a constant term of J0N=2 from the Hamiltonian. For B = 0, states with minimum Sx belong to the ground state. For an even system size N , Sx = 0 states form the ground state manifold. The number of such states is NCN=2, which is exponential in the system size N . 164 Figure 5.1: Energy of creating spin excitations in the long range antifer- romagnetic Ising model: The ground states of the long range antiferromagnetic Ising model (Eq. (5.1) at B = 0) are the AFM states, j"#" :::i and its globally spin ipped state j #"# :::i for the exponent 6= 0 (in Eq. (5.2)). Any spin ip from this state creates domain walls, and costs some excitation energy. Here we plot the energies of creating a few types of excitations in a system of N = 10 spins, relative to the ground state energy (black line, black arrows denote one of the AFM states). The rst excited states have two adjacent spins on the end ipped (blue), which contain a single domain wall (gray dotted line) between spins 2 and 3. Only one of the four degenerate rst excited states are shown in terms of the blue arrows. It costs more energy to ip two adjacent spins in the bulk of the chain (green), as that involves creating two domain walls. The states with an end spin ipped (red) has more energy than the double spin ipped (on the end of the chain) states. States with a single bulk spin ipped (orange) have two domain walls, and hence are more energetic compared to the end spin ipped states. They are also higher in energy than the double adjacent spins ipped in the bulk states. The energies of the do- main walls created depend on the range of the interactions, . For the uniform Ising model ( = 0) the position of the excitation does not matter. 165 In presence of the transverse eld, Sx does not commute with the Hamiltonian (Eq. (5.5)), but S2 = S2x + S 2 y + S 2 z does. We probe the magnetic ordering by initializing the system in the ground state of BSy, which has a total Sy = N , and hence belongs to the S2 = N manifold, and then tune the e ective transverse eld B in Eq. (5.5). Thus the system can only access the ground states that belong to the total spin N manifold. In presence of the transverse eld B the ground state passes through a quan- tum phase transition from the paramagnetic phase (all spins polarized along the transverse eld) to the AFM phase (for 6= 0). The critical gap and the critical eld depend on the range of the interactions. 5.2.2 Frustration and the range of the interactions In Fig. 5.2a, we plot a few of the low energy excited states of Eq. (5.1) for the case of N=10 spins, and = 1:0. In our quantum simulation experiment, we initialize the spins in the ground state of a ?trivial? Hamiltonian, which is just the part of the Hamiltonian (Eq. (5.1)) that couples to the magnetic eld only. Then the Hamiltonian is ramped at a nite rate, and the dimensionless ratio of the e ective transverse eld to the Ising couplings is reduced. The system follows the ground state if the ramping is perfectly adiabatic. For a nite rate of ramping, some population is excited to excited states that have the same symmetry (w.r.t. the spin ip and spatial re ection, as discussed in section 3.2) as the ground state. The gap between the ground state and the lowest excited state with the same symmetry is 166 Figure 5.2: Dependence of the spectrum of Hamiltonian (5.1) on frustra- tion: The energy spectrum of the long range antiferromagnetic quantum Ising model (Eq. (5.1)) with Ising coupling Jij between spins i and j falling o with distance as Jij = J0=ji jj depends on the amount of frustration in the system or the range of the interaction, . (a) Few low lying energy states of the frustrated Hamiltonian (5.1) as a function of the dimensionless parameter B=J0 for = 1 or = 5. The spacing between the ground state and the second excited state reaches a bottleneck at a critical value Bc=J0. Here c is the critical energy gap of the Hamil- tonian. (b) Dependence of Bc=J0 (red dotted line) and c (black solid line) on the range of the interaction. As the interaction becomes long ranged ( increases, or decreases) the competing long range couplings make it easy to create excitations in the system, the gap is reduced, and a relatively small e ective transverse eld can break the spin ordering. Both these parameters approach zero as ! 0 or !1. Our current experiments are performed with parameters in the shaded region. The apparent kinks on the curves are due to connecting a nite number of points by straight lines in the plot. 167 known as the critical gap, c, and the value of the e ective magnetic eld (the Ising couplings are xed in our problem) at which the system passes through the critical gap is known as the critical eld, Bc. Both the critical eld Bc and the critical gap c depend on the range , as shown in Fig. 5.2b. Both the quantities decrease monotonically as the range of the interactions, and hence the amount of frustration, increases. To visualize the range of the interactions more intuitively, we de ne an e ective range as the distance between the spins at which the Ising coupling falls o to 20% of the nearest neighbor value. The choice of 20% as the cut-o is arbitrary, and motivated by our system size of N = 10 in some of the following experiments. With this de nition, = 51= , and varies between 4 and 8 sites for our experimentally accessible range of 0:76 to 1:12. As the nearest neighbor interaction always wins over the long range couplings for this chain of spins (except at = 0), the ground state shows the nearest neighbor AFM ordering, regardless of the range of the interactions. Frustration in this model brings the excited states closer to the ground state in energy. Hence to observe the e ects of frustration we ramp the e ective transverse magnetic eld B in the Hamiltonian (Eq. (5.1)) faster than the critical gap ( c) to populate the excited states. The observed spin order depends on the amount of excitations created, and hence on the frustration. The system should exhibit less ground state character for the more frustrated (longer range) couplings for the same rate of ramping, and the same nearest neighbor energy scale. 168 5.3 Experimental simulation of the model We simulate the quantum Ising model (Eq. (5.1)) by o -resonant excitations of transverse phonon modes, following the M lmer-S rensen scheme as explained in the previous chapters. The Raman beat-note detuning ( in Eq. (3.8), Fig. 5.3a) is chosen to be bigger than the COM frequency, and thus all the Ising couplings are positive, or antiferromagnetic. The e ective transverse magnetic eld is simulated by a resonant carrier transition, with phase shifted by =2 from the Raman beat-notes generating the Ising interactions. In our experiment we use global Raman beams (approximately 1 Watt each, with the 355 nm mode-locked laser) with horizontal and vertical waists of 150 m and 7 m respectively to address the ions. This produces a carrier Rabi frequency 2 600 KHz, which is more than 98% homogeneous across a chain of N = 16 spins. We set the beatnote detuning to !X + 3 , where is the single ion Lamb-Dicke parameter, and !X is the transverse COM frequency. We t a power law pro le (Eq. (5.2)) to the calculated Ising couplings, and extract the exponent . The Ising coupling, Ji;j between the spins i and j depend on the site i (in addition to the separation between the spins ji jj), due to the nite size e ects. This results in a site i dependent exponent, which varies by 10% across the chain. Figure 5.3b shows the Ising couplings J1;1+r (r = 1; 2; :::; 9) for typical experimental parameters (listed in the plot), in a system of N = 10 spins. The Ising coupling falls o as J1;1+r / 1=r1:0 ( = 1:0) in this example. The nearest neighbor Ising couplings are stronger near the end of the spin chain compared to the center (Fig. 5.3c), and they vary by 20% across the 169 Figure 5.3: Ising coupling pro le and t to a power law: a. The Ising interactions between the spins are mediated by virtual phonon excitations. We show the red and the blue sidebands of the axial (light blue for the blue sidebands, and light red for the red sidebands) and the transverse (dark blue for the blue sidebands, and dark red for the red sidebands) vibrational normal modes of N = 10 trapped ions. To simulate the Ising interactions, we shine non-co-propagating laser beams with two beatnotes (beatnote 1 and 2 in this gure), symmetrically detuned from the carrier transition. The beatnotes are detuned from the transverse COM mode by a frequency . b. Ising couplings in KHz (blue circles) between spin 1 and the others, J1;1+r (r = 1; 2; :::; 9) for a system of N = 10 spins, calculated from Eq. (3.8). The single ion sideband Rabi frequency, = 2 35 KHz and the M lmer-S rensen detuning, = 3 = 2 105 KHz. The red curve is a power law t to the Ising couplings (J1;1+r / 1=r1:0). c. Calculated nearest neighbor Ising couplings between spins i and i + 1, Ji;i+1 vs site i, according to Eq. (3.8). The interaction is the strongest at the ends of the chain, and varies by 20% across the chain. The solid black line shows the average nearest neighbor interaction. 170 chain. This is primarily due to the contribution of the ?tilt? mode, which is the next lower energy vibrational mode after the COM. The ions near the center have smaller amplitudes of motion for this mode, and hence the ?tilt? mode does not contribute (Eq. (3.8)) to the nearest neighbor couplings near the center of the chain. 5.3.1 Tuning the range of Ising interactions The range of the Ising interactions can be tuned by the following two methods: By changing the Raman beat-note frequency (or the M lmer-S rensen de- tuning ), keeping the vibrational spectrum xed. By changing the bandwidth of the vibrational spectrum, keeping the M lmer- S rensen detuning, = !COM xed. We follow the second method, as this keeps the COM phonon occupation probability approximately at the same level for various ranges. The bandwidth of the transverse vibrational modes is controlled by tuning the trap anisotropy. A higher trap anisotropy (de ned to be the ratio of the transverse COM freq to the axial COM frequency) moves the ions farther from each other, and reduce the bandwidth of the modes. On the other hand, a more isotropic trapping potential brings the ions closer together, until they cannot sustain a linear con guration and break into a zig-zag shape. We change the anisotropy of the trapping potential, and hence the bandwidth of the transverse vibrational modes, by changing the DC voltages applied on the trap electrodes, keeping the radio frequency power generating the ponderomotive potential at a constant value. Figure 5.4a shows the dependence 171 Figure 5.4: Dependence of the range of Ising interactions on the M lmer- S rensen detuning and the bandwidth of vibrational modes: a. We can tune the range of the Ising interactions by varying the Raman beatnote?s detuning from the COM mode. Here we show theoretical calculations of the dependence of the Ising exponent, (Eq. (5.2)) as a function of the detuning for normal- ized bandwidth !=!X = 0:018, (blue squares), !=!X = 0:26 (red circles), and !=!X = 0:99 (black rhombuses). For comparison, a typical COM sideband Rabi frequency is =2 = 35 KHz. The solid lines are interpolated in between the cal- culated points. b. Dependence of on the bandwidth of the transverse modes, ! (Fig. 5.3) for a xed detuning ( =2 = 105 KHz) from the COM mode. The band- width is controlled by changing the trap anisotropy. The coupling is more uniform when the normalized bandwidth is higher, and approaches a dipole interaction, i.e., = 3, in the limit of !=!X ! 0. 172 on the Ising coupling pro le exponent on the M lmer-S rensen detuning for two di erent sets of trapping parameters. A weaker trap along the axial direction leads to a larger bandwidth (red circles and line), ! (Fig. 5.3a) of the transverse vibrational modes, and thus it is easier to excite the COM modes exclusively without coupling to the other modes. This induces long range couplings, and the Ising exponent is smaller than when the bandwidth is smaller (blue squares and line). It is relatively easier to access shorter range interactions (larger ) with higher bandwidth. In the limit of very large M lmer-S rensen detuning compared to the bandwidth, ! the Ising couplings take the form of a dipolar decay, or Ji;j = J0=ji jj3, i.e., = 3. In Fig. 5.4b we set a xed M lmer-S rensen detuning =2 = 3 =2 = 105 KHz, and show the dependence of the Ising exponent on the bandwidth of the transverse modes. In the limit of !=!X ! 0 the exponent ! 3, and in the other limit of !=!X = 1, the interaction range is the longest at the speci c M lmer-S rensen detuning, for the given system size, before the mechanical instability breaks the linear con guration of the ion chain. For the current experiments, we choose four trap settings, corresponding to four di erent ranges of interactions, as shown in Table 5.1. For the range of axial frequencies used in the current experiment (between !Z = 2 0:62 MHz and 2 0.95 MHz), the Ising pro le exponent varies between 0.76 to 1.12, which corresponds to the variation of the range of interactions between = 4 to = 8 sites, the latter indicating that the range approaches the system size. 173 Settings End avg DC (Vend) Central average (Vcentral) Axial COM freq, !Z=2 (MHz) Transverse COM freq, !X=2 (MHz) Normalized mode band- width ( !=!X) Ising expo- nent, I 25 1.1 0.952 4.87 0.56 0.76 II 20 0.9 0.857 4.75 0.44 0.9 III 13 0.6 0.691 4.80 0.25 1.0 IV 10 0.4 0.62 4.80 0.19 1.12 Table 5.1: Experimental parameters used to generate long range Ising model with variable range: The bandwidth of the transverse vibrational modes ( !) is varied by changing the axial con nement of the ions to tune the range of the Ising couplings. We tabulate the average static voltages applied on the electrodes, axial and transverse COM trap frequency, the bandwidth of the modes normalized to the transverse COM frequency, and the Ising coupling exponent for experiments with N = 10 ions reported here. 5.3.2 Experimental protocol and the order parameters We initialize all the spins along the direction of the e ective transverse mag- netic eld, i.e., along the y direction of the Bloch sphere by applying a resonant Raman pulse of appropriate duration and phase (a =2 pulse about the x axis of the Bloch sphere). Then we turn on Hamiltonian 5.1 with an initial eld B0 5J0, where J0 is the average nearest neighbor Ising coupling, such that our prepared spin state approximates the ground state of the initial Hamiltonian. (The overlap be- tween the exact ground state of the initial Hamiltonian and the prepared polarized state is about 99%). The e ective magnetic eld is then ramped exponentially with a time constant of up to a nal value B of the transverse eld. We then measure the x component of each spin by rotating our measurement axes, and using state dependent uorescence signal on an intensi ed CCD imager. The experiments are 174 repeated 2000 4000 times to collect statistics. From the measured spin states, we construct the order parameters appropriate for observing the antiferromagnetic order and excitations. Various moments con- structed from a distribution of staggered magnetization, de ned asms = 1N j P odd x P even xj, would di erentiate between a paramagnetic and an antiferromagnetic state, and also quantify spin ip excitations. Here P odd(even) refers to summation over all the odd (even) sites of the lattice. In particular, we use the fourth moment, the staggered Binder cumulant, gs = h(ms hmsi)4i h(ms hmsi)2i2 : (5.6) Here h:::i denotes averaging over experimental realizations. In the paramagnetic phase the staggered magnetization, ms is distributed according to a Binomial distri- bution, which approaches a Gaussian in a very large system. The Binder cumulant is, g0s = 3 2=N in the paramagnetic phase for a system of N spins. In the AFM phase, the distribution of the staggered magnetization consists of two Knocker Delta functions, at ms = N , and ms = N . The corresponding staggered Binder cumu- lant is unity in the AFM phase. Here we scale the Binder cumulant to take the nite size e ect into account to gs g0s g g0s 1 ; (5.7) where g is the measured staggered Binder cumulant, according to Eq. (5.6). 175 We also construct two point correlation functions between the spins i and j, Ci;j from the measured spin states, according to the following formula. Ci;j = h i x j xi h i xih j xi: (5.8) The two point correlation function allows us to directly probe the spin order at each experimental realization. The Fourier transform of the correlation function, also known as the structure function S(k), provides valuable information about the spin order. The structure function is de ned as S(k) = 1 N X ji jj Ci;j cos (kji jj) (5.9) For a nite system size of N spins, we calculate the structure function at distinct points, k = 0; N ; 2 N ; :::; . Since the two point correlation function Ci;j depends on the site i itself for the nite size of the system, we use the averaged correlation of all the pairs of spins separated by ji jj sites. The structure function at k = is then S(k = ) = 1 N X ji jj ( 1)ji jjCi;j = 1; for AFM state; as Ci;j = ( 1)N for a perfect AFM order. Thus a signature of spin frustration in our experiment would be the decay of the structure function at k = . 176 5.4 Results of the quantum simulation 5.4.1 Onset of antiferromagnetic correlations in quantum simulation for N = 10 and N = 16 spins In Fig. 5.5 we plot the two point correlation between an end spin and the rest, C1;1+r (r = 1; 2; :::; N 1) for various values of the dimensionless parameter B=J0, for a range of = 4 or = 1:12 (Eq. (5.2)) in a system of N = 10 spins. For our chosen experimental parameters, J0 800Hz. For B=J0 = 5 we do not observe any appreciable correlation between the spins, consistent with a paramagnetic phase. As the ratio B=J0 is lowered, the correlation with the nearest neighbor spin starts going negative at B=J0 1:5. For lower values of B=J0, AFM correlations build up and the spins separated by even number of sites start to align themselves along the same direction, as seen by the alternating signs of the correlation coe cients with distance. The excitations created by non-adiabatic e ects reduce the correlations from the perfect AFM state value of unity, and the best (anti)correlation achieved is 60% for B=J0 0:01. Figure 5.7 shows the averaged CCD image of the N = 10 ions, all in the bright (j"zi), all in the dark (j#zi) and in the AFM states. The spins order in the two AFM states about 3% of the times (842 events out of 28,000 realizations). The spin ip symmetry is preserved in the simulation, as seen by the approximately equal number of experiments leading to either of the two AFM states (442 and 440 corresponding to spin j"i on the odd and even sites respectively). The probability of obtaining the 177 Figure 5.5: Onset of the antiferromagnetic ordering with 10 spins: As we tune the ratio of the e ective transverse magnetic eld to the antiferromagnetic Ising couplings, the spins undergo a crossover from the paramagnetic phase to the antiferromagnetic phase. a-e. Here we show the two point correlations between an edge spin, labeled as spin 1 and the others. By imaging the spin state dependent uorescence of the trapped ions with single site resolution optics on an intensi ed CCD imager (model PIMax3: 1024i, made by Princeton Instruments), we directly measure the two point correlation functions. In this example, the Ising couplings approximately fall o with distance as Ji;j = J0=ji jj1:2. The correlations do not reach unity, as expected from a perfect AFM ordering at zero temperature, primarily due to the nite speed of ramping in the quantum simulation (we ramped the external eld exponentially down from B = 5J0 with a time constant of 2 (0:4=J0)). 178 Figure 5.6: Antiferromagnetic spin ordering with 16 spins: We scale our quantum simulator up to N = 16 spins, and observe the onset of antiferromagnetic spin correlations. The Ising exponent is = 1:0 in this case. For the frustrated Ising model, the critical gap closes sharply with the system size. Thus maintaining the adiabaticity in simulation is harder compared to the ferromagnetic model. The blue circles are the two point correlations for B=J0 5, and the red squares are for B=J0 0:01. 179 AFM states is 2=210 0:2% in the paramagnetic phase. In Fig. 5.6 we compare the correlations between an end spin and others, C1;1+r (r = 1; 2; :::; 15) at B=J0 5 (blue) and B=J0 0:01 (red) for a system of N = 16 spins. At lower values of the ratio of the e ective external eld to the Ising couplings ratio, the system shows some antiferromagnetic spin ordering. The nearest neighbor correlation builds up to about 0:35, and the correlation length (de ned as the distance at which the correlation drops to 1=e of the nearest neighbor) is limited to about 5 sites. The lack of a very good ground state (AFM) order is primarily due to the vanishing critical gap c in presence of frustration when the system size grows, making our ramping of the ratio B=J0 (we ramped the e ective eld exponentially with a time constant of 450 s, keeping the Ising couplings constant) too fast to be adiabatic. 5.4.2 Frustration of the AFM order with increasing range of interac- tions We observe the spin order achieved in a quantum simulation experiment, for each of the four ranges of interactions described in table 5.1. The spins are initialized along the e ective transverse eld, B (in the y direction of the Bloch sphere), then the Hamiltonian (Eq. (5.1)) is turned on with B 5J0, where J0 is the average nearest neighbor Ising coupling. The e ective eld B is then exponentially turned down with a time constant =2 = 0:4=J0, and the spin order is detected at time t = 6 , where B=J0 0:01. Figure 5.8a shows the scaled staggered Binder cumulant, 180 Figure 5.7: CCD image of N = 10 antiferromagnetically ordered spins: a. CCD image of 10 ions in spin j "zi states. The image is obtained by averaging 440 experiments where each ion are prepared in the j "zi state by applying a carrier Raman pulse and then detected for 3 ms. b. Averaged CCD image (over 440 experiments) of the same 10 ions in j #zi state, prepared by optical pumping. c- d. Averaged CCD image of the two AFM states. We post-select the AFM states from 28,000 quantum simulation experiments with long range AFM couplings, at a ratio of the e ective transverse magnetic eld to the nearest neighbor Ising coupling B=J0 0:03. The CCD raw images of all such states with the same spin ordering (as detected by discriminating the uorescence counts from the ions) are averaged to obtain the AFM states. 440 experiments resulted in an AFM state with j"i spins on the odd sites and 442 experiments with the j "i spins on the even sites. The measurement axes are rotated before the spin state detection such as a bright ion corresponds to the state j"i (in the x basis). 181 Figure 5.8: Frustration of antiferromagnetic spin ordering with increasing range of interactions: We compare the experimental data for four di erent ranges of Ising couplings, quanti ed by the exponent, (Eq. (5.2)), which lies between 0.76 and 1.12 in our experiments. Frustration increases as the interaction range grows, and the excited states come closer to the ground state leading to a reduction in the value of the critical gap (Fig. 5.2). Thus for a given rate of ramping, the system is more excited from the ground state (non-adiabaticity) for longer range of interactions. a. The staggered Binder cumulant (Eq. (5.7)) at B=J0 0:01 vs the exponent . The order parameter goes down with increasing range, or decreasing . The error bars are from a conservative estimate of the uncertainty in detecting the spin states. b. The structure function S(k)(Eq. (5.9)) at B=J0 0:01 for various ranges of interactions. The ground state ordering is denoted by k = , which goes down with the increasing range of interactions. Characteristic error bars are conservative estimates of the uncertainty in detecting spin states. The solid lines are presented just to guide our eyes. 182 Figure 5.9: Plausible decoherence in our quantum simulation: We probe the spin order in the quantum simulation experiment as a function of the ramping speed of the Hamiltonian (Eq. (5.1)), with the Ising exponent = 1:12. The spins are polarized along the e ective external magnetic eld. The Hamiltonian is turned on with B 5J0, so that the initial spin state is an approximate ground state. Next the eld is ramped down exponentially with a time constant for a time t = 6 , keeping the Ising couplings xed, and the x components of the spins are detected. The nal value of the B=J0 0:01. The experiment should be more adiabatic for a slower ramping, and hence a longer experimental duration, consistent with our observation up to t = 6 2:5 ms. The spin order goes down for slower ramping, which indicates the presence of some form of decoherence in the system. The intensity uctuations in the Raman beams is one of the primary sources of decoherence in the system. gs (Eq. (5.7)) vs the range of the interactions, quanti ed by the Ising exponent . The staggered Binder cumulant monotonically decreases with increasing range (decreasing ) or frustration. Figure 5.8b shows the structure function, S(k) (Eq. (5.9)) vs the spatial frequency, k for various range of interactions. The ground state AFM order is shown by the spatial frequency, k = , which steadily goes down with the increasing range or frustration. 183 5.5 Discussions and conclusion In this experiment, we have qualitatively observed the e ect of frustration in the observed spin order. Our current experiment is not limited by decoherence due to spontaneous emission from the Raman beams, as the 355 nm Raman beams are far detuned from the 2S1=2 2P1=2 and 2S1=2 2P3=2 lines, and the spontaneous emission per Rabi cycle is suppressed to 10 5. The time scale for the spontaneous emission is then 100 ms, which is more than an order of magnitude slower than the experimental time scales (< 5 ms). In order to probe any decoherence e ects, we repeat the simulation experiment with various ramping speed of the e ective magnetic eld. In Fig. 5.9 we plot the antiferromagnetic order parameter gs vs the total duration for the experiment for = 1:12. Here each data point represents the spin order achieved after ramping the B eld down exponentially from B 5J0 for a total duration of t = 6 , where is the time constant with which the e ective eld is ramped down. The spin ordering into the antiferromagnetic state grows with slower ramping, as expected, for up to t 2:5 ms. Then we observe a decay in the spin order, which might indicate the presence of decoherence e ects in the system. A principal source of decoherence might be the intensity uctuations in the Raman beams, due to beam pointing instabilities, and uctuations in the optical power. 184 Chapter 6 Outlook In this thesis, I presented some proof-of-principle experiments that benchmark a quantum simulator. We can further explore the long range nature of the interac- tions in the trapped ion system to study new many body physics. From a quantum computation point of view, the simulator must be scaled to a large number of qubits, that can outperform a classical computer. While there is no road-block in principle [128], several technological challenges need to be overcome in order to build a quan- tum computer or a universal quantum simulator. Here we discuss a few directions towards that goal. 6.1 Scaling up the system - large numbers of equally spaced ions in a Paul trap To scale the ion trap system up to a size where the quantum simulator outper- forms a classical computer, a stable large chain of ions is needed. While our current trap can handle a couple of dozens of spins without any technical upgrade, new trap architectures should help us scale the system further. A shortcoming of using harmonic con ning potential in a linear ion trap is that the ions near the center of the chain are closely packed together, compared to the outermost ones. This is less preferable to a chain of uniformly spaced ion crystal 185 for the following reasons. At a given ratio of the axial to the radial con nement, a linear ion chain under- goes a structural phase transition to a zig-zag con guration. This is unwanted in quantum information processing, as the ions in the zigzag con guration experience micromotion from the driving radio-frequency eld. In a harmonic trap, this instability depends only on a few ions at the center that are very close to each other. Relatively larger spacings near the edge of the ion crystal limit the number of ions that can be trapped in a given length of the con ning zone. As we shall discuss in this section, simulation of an arbitrary fully connected spin network requires individual addressing of the ions by well focused Raman beams. A uniform chain of ions would be ideal for this, as we can image a regularly spaced grid for this purpose. Also, switching a single beam between the ions by an AOM will be easier with a uniformly spaced ions. A uniformly spaced ion crystal makes the imaging easier, such as it would allow the use of a uniformly spaced PMT array, or a regularly spaced region of interest (ROI) on a CCD. The problem of non-uniform ion spacing can be solved by adding anharmonic terms in the con ning potential, such that the ions near the center are pushed out, and the outermost ions are pushed towards the center. A potential which is atter than the harmonic potential near the center of the chain, and steeper at the edge is useful for 186 this purpose. Hence, to make a uniform chain of ions, at least a quartic potential is required to the lowest order [129]. An electrode with ve or more segments is needed to generate a quartic potential in a linear trap. 6.2 Creating an arbitrary lattice geometry It is possible to tailor the long range interactions in the trapped ion system properly to simulate a spin model on an arbitrary lattice geometry. The ultimate goal would be control every pairwise interaction in a fully connected spin network. In a system of N spins, there are NC2 = N(N 1)=2 N2=2 two body interactions, and thus we need at least N2=2 control knobs to generate an arbitrary spin network. This is feasible by addressing the ions with individual Raman beams, each of which will contain several beat-notes to drive multiple normal modes selectively. Thus, extending Eq. 2.53 to the general case [130], the Ising coupling between spins i and j becomes, Ji;j = NX n=1 i;n j;nR NX m=1 bi;mbj;m 2n !2m : (6.1) Here R is the recoil frequency, i;n is the single spin carrier Rabi frequency consistent with the optical power on the ith ion, and nth beat-note at frequency n. Thus we have N amplitude knobs i;n (i = 1; 2; :::; N for a xed beat-note index n) corresponding to each beat-note n ( n = 1; 2; :::; N) - a total of N2 controls. Thus in principle, it should be possible to solve for the N N Rabi frequency matrix i;n for any arbitrary Ji;j. 187 6.3 Other interesting spin physics In the experiments presented in this thesis, we have tuned the Raman beat- note close to the COM mode, for which all the interactions have equal sign. Tuning the Raman beat-note in between the normal modes will give more versatility in the Ising couplings, allowing us to observe new spin phases. As an example, if the beat-note is tuned in between the second (tilt) and the third mode in the order of decreasing frequency, short range ferromagnetic couplings compete with the long range antiferromagnetic couplings. The resulting spin order is predicted to undergo a rst order, or ?sharp? phase transition between a ferromagnetic and a ?kink? phase [100]. In the ferromagnetic phase all spins point in the same direction, and in the kink phase, a domain wall appears near the center of the spin chain. This prediction opens up an interesting avenue to explore, as this is an example of a truly sharp transition in a very small spin system. While the quantum Ising model with arbitrary long range interactions admits many aspects of non-trivial many body physics, an access to several other spin models, such as the xy and the Heisenberg models, will enhance the versatility of the ion trap quantum simulator. The anisotropic Heisenberg model, xxz model can also be mapped to a Bose-Hubbard Hamiltonian in the context of ultracold atoms in optical lattices [131]. The xy model can be engineered by adding another pair of Raman beams coupling to the vibrational modes along the other transverse direction (y), and o -setting the phase of the M lmer-S rensen couplings induced by the additional laser beams by =2 from the original beams. However, error terms 188 due to the cross-talk between the normal modes generating the x x and the y y interactions must be kept under control by properly shaping the mode excitations. In principle, we can add yet another beam to simulate the z z couplings, in order to simulate the anisotropic Heisenberg interactions. However, due to our choice of the 171Yb+ ?clock? hyper ne states as the spin states, the Ising couplings along the z direction may be smaller than the other two directions. This issue can be overcome by choosing one of the Zeeman states as the j "zi state. The new spin-1=2 states are now more sensitive to any magnetic eld uctuations, and hence the magnetic eld must be stabilized and the stray elds have to be shielded to maintain a long coherence time between the spin states. 189 Appendix A Quantum trajectory calculations Quantum trajectories (chapter 3) are generated by numerically integrating the Schr odinger equation, with Hamiltonian 3.1, while simultaneously executing quantum jumps to account for spontaneous emission and decoherence. The proba- bility of spontaneous emission used is consistent with the experimental parameters. The rate of dephasing (primarily due to the uctuations in the intensities of the Raman beams) is treated as a tting parameter. Spontaneous emission from ion i either localizes the spin of the ion, projecting it into 2S1=2jF = 0;mF = 0i (spin state j #zi) or 2S1=2jF = 1;mF = 0i (spin state j "zi), or it projects the ion into 2S1=2jF = 1;mF = 1i, in which case ion i is factored out of the Schr odinger evo- lution, though it is counted as spin up at the time of measurement. Decoherence (dephasing) is modeled by the quantum jump operator x; thus a jump for ion i, j i ! ixj i, introduces a phase shift between the spin states j "i and j #i (in x basis). Jump rates are taken to be xed and equal for all ions. Note that a decoherence jump rate of decoh leads to decay of the spin coherence at rate 2 decoh. To determine the entangled state of the spin ensemble after a spontaneous emission, The calculations are performed by Changsuk Noh and Prof. Howard Carmichael, Auckland University, NZ. 190 e.g. from ion i, we assume that the ground state con guration prior to emission, j "zii Y j 6=i ( i;jj "zij + i;jj #zij) + j #zii Y j 6=i ( i;jj "zij + i;jj #zij) ; is mapped, by the far detuned Raman beams, into a very small excited-state con- tribution to the overall system entangled state, j2P1=2ii Y j 6=i [( i;j + i;j)j "zij + ( i;j + i;j)j #zij)]; with 1 proportional to the amplitude of the Raman beams and inversely pro- portional to their detuning. The (unnormalized) state after the emission is j?ii Y j 6=i [( i;j + i;j)j "zij + ( i;j + i;j)j #zij)]; where j?ii is j "zii, j #zii, or the factored state 2S1=2jF = 1;mF = 1i. 191 Appendix B Detection of spin states This appendix describes the methods used to detect the spin ordering in the quantum simulation of the ferromagnetic Ising model with N = 2 to N = 9 spins, described in chapter 3. The spin states are detected by spin-dependent uorescence signals collected through f=2:1 optics by a photomultiplier tube. Spin state j "zi is resonantly excited by the 369.5 nm detection beam and uoresces from 2P1=2 states, emitting Poisson distributed photons with mean 12 in 0.8 ms. This state appears as ?bright? to PMT. The detection light is far o -resonant to spin state j #zi and this state appears ?dark? to the PMT. However, due to weak o -resonant excitation bright state leaks onto dark state, altering the photon distribution [90]. Unwanted scattered light from optics and trap electrodes also alter the photon distribution. We construct the basis function for s bright ions by convolution techniques, and include a 5% uctuation in the intensity of detection beam, which is representative of our typical experimental conditions. We then t the experimental data to these basis functions, and obtain probabilities P (s) at each time step ti in the experiment. Mean photon counts for dark (mD) and bright (mB) states are used as tting parameters so as to minimize the error residues. The best tting at time step ti is obtained for the parameters fmD;i;mB;ig. 192 These parameters uctuate at di erent time steps of the quantum simulation, pri- marily due to uctuations in the intensity of detection beam and background scat- ter, and also due to uncertainties in a multivariate tting. The tting errors are propagated to the spin state probabilities P (s) using Monte Carlo method of error analysis, as follows. We extract P (s) and compute the order parameters at time step ti with mean dark and bright state counts chosen randomly from a Gaussian distribution with mean f mD, mBg and standard deviations f mD, mBg respec- tively. Here mD and mB are averages of mD;i and mB;i respectively over di erent time steps ti. Similarly mD and mB are standard deviations of mD;i and mB;i respectively. By repeating this process 400 times we generate a histogram of each order parameter and t the histograms to a Gaussian distribution. The standard deviation of the distribution is chosen to represent the random error due to tting in that order parameter. The uncertainty in amount of uctuation of the detection beam power during the experiment is conservatively included in the error analysis by repeating the tting process for a range of uctuations. The nite width of the detection beam is taken care of by modeling the Gaussian beam having a three step intensity pro le with appropriate intensity ratios. 193 Appendix C Relevant Frequencies for 171Yb+ and 174Yb+ The following table lists the frequency lock points of various lasers used in our experiments. The 739.5 nm laser (Ti:Sapphire) is locked to a hyper ne transition of Iodine molecules. This light is sent to the Iodine saturation absorption spectroscopy set up through an EOM driven at 13.315 GHz, and the (positive) rst order side- band of this light excites the Iodine line. The 739.5 nm laser frequency lock points are about 400 MHz red detuned from the cooling transition resonance, and thus appropriate for using 400 MHz AOMs. Without changing the lock point of the 739.5 nm laser, we can address the 174Yb+ resonance by shifting the Iodine EOM drive frequency to 12.103 GHz (while locking the laser to 405.644318 THz). Laser Frequency for 171Yb+ (THz) Frequency for 174Yb+ (THz) Comments 739.5 nm 405.644318 405.645530 Iodine EOM at 13.315 GHz 935 nm 320.56922 320.57190 399 nm 751.52764 751.52680 638 nm 469.445, 469.442 469.439 scanned around these two lines Table C.1: Frequency lock points for various lasers 194 Bibliography [1] P. A. M. Dirac, The Principles of Quantum Mechanics, vol. 27 of International series of monographs on physics (Oxford, England). second ed., 1935. [2] A. Einstein, B. Podolsky, and N. Rosen, \Can quantum-mechanical description of physical reality be considered complete?," Phys. Rev., vol. 47, p. 777, 1935. [3] R. Feynman, \Simulating physics with computers," Int. J. Theor. Phys., vol. 21, pp. 467{488, 1982. [4] S. Lloyd, \Universal quantum simulators," Science, vol. 273, p. 1073, 1996. [5] T. D. Ladd, F. Jelezko, R. La amme, Y. Nakamura, C. Monroe, and J. L. O?Brien, \Quantum computers," Nature, vol. 464, p. 45, 2010. [6] D. P. DiVincenzo and P. W. 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