Ph.D. Thesis - Physics

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the rf null in such a trap exists only at the center of the trap, which is the minimum of the potential in three dimensions. Thus, only one ion may be confined without excess micromotion. One of the principal questions discussed in this chapter is the effect, in theory, that this micromotion has on an example quantum simulation. The structure of large 2-D crystals, as well as the impact of rf heating and background gas collisions, was discussed in Ref. [BKGH08]. However, the question of micromotion had not previously been adequately addressed. We also construct and demonstrate a test elliptical trap, to verify a portion of the above theoretical work. Since it is crucial to reduce motional decoherence (anomalous heating) in such a trap, we construct a new closed-cycle cryostat apparatus for the testing of these traps, in order to provide an electrical noise environment in which progress toward quantum simulation can be made. We then test our predictions of the crystal structure and motional frequencies of ions in this trap. We come to the conclusion that quantum simulations with at least a few ions should be possible in this trap. The chapter is organized as follows. In Sec. 7.1, we introduce the basic model of the elliptical trap, and present calculations of the structure of ion crystals therein. In Sec. 7.2, we discuss the effects of micromotion on the fidelity of a quantum simulation, by numeri- cally simulating the quantum dynamics with a time-dependent potential resulting from the micromotion. The expected coupling rates due to laser pushing forces are also calculated. In Sec. 7.3, we study the origin of the state-dependent forces, looking at optical forces, and also at magnetic field gradients. In Sec. 7.4, we discuss the closed-cycle cryostat apparatus used for testing the elliptical traps. We put our theoretical predictions to the test in Sec. 7.5, measuring the motional frequencies and structure of crystals in the elliptical test trap. Finally, we evaluate the suitability of the elliptical trap design for quantum simulation in Sec. 7.6, and then conclude in Sec. 7.7. 7.1 Elliptical ion trap theory In this chapter, we explore the idea of using elliptical ion traps to perform quantum simulation in 2-D. Elliptical traps were proposed and demonstrated by DeVoe [DeV98] with the aim of producing a miniature linear ion trap with relatively simple fabrication and fairly favorable amounts of micromotion. Here, we study instead 2-D ion crystals in such a trap. To obtain an ordered lattice of ions in a single plane, it is desirable to have a trap with approximate cylindrical symmetry. However, in such traps (the ring trap included), the two “radial” vibrational modes are degenerate; hence, there is no preferred axis along which the ion crystal may align. This undesirable condition can be rectified by introducing an asymmetry in these two directions. In such a trap, ions will align in a 2-D array until a critical ion number is reached, at which point the crystal may minimize its energy by transitioning to a 3-D shape. This is the primary motivation for making the trap elliptical. 152
Figure 7-1: Left: Uraniborg 1, as rendered by the EAGLE scripting language. The lengths of the rf semimajor and semiminor axes are shown as dimensions A and B, while the ground semimajor and semiminor axes are shown as a and b. Four dc electrodes surround the rf electrode to provide dc compensation voltages, and the ground ellipse can be controlled as well. A gap of ≈ 0.1 mm is present between all electrodes due to fabrication constraints. Right: Uraniborg 2. The trap is identical to Uraniborg 1, except that the rf semiminor axis is different on the two sides of the ground ellipse. Our project in this section is to calculate this crystal geometry and transition point for a model trap geometry. Our model elliptical trap consists of two electrodes in a single plane. There is an elliptical rf ring electrode surrounding a grounded electrode (also elliptical). Four numbers are required to specify the semimajor and semiminor axes of the ground and rf ellipses, but in this model we will assume that the rf ellipse is similar to the ground ellipse, and thus three numbers are used to specify the trap electrodes: the dimensions of the grounded ellipse and a scaling factor that specifies the size of the outer ellipse. Calculations are presented in this section for the traps that we tested, which are named “Uraniborg.” 1 Secular frequencies for traps of a different scale may be estimated by scaling the curvature of the potentials appropriately with the trap size. There are two different elliptical traps presented here; in the first, Uraniborg 1, there is a single semiminor axis length for the rf ellipse, while in the second, Uraniborg 2, one side is made longer than the other in an effort to tilt the vertical principal axis and thereby increase the Doppler cooling efficiency. Diagrams of these traps are presented in Fig. 7-1. An appropriate rf voltage Vrf is applied to the ring electrode, while the center ellipse is grounded. These voltages result in a trap region characterized by secular frequencies ωˆn, where ωˆz = ωˆx + ωˆy. This relation should always hold for the elliptical trap, as long as the confinement is due entirely to rf fields. Below we present the calculation of the trap depth 1 After the observatory where Tycho Brahe made his measurements of planetary orbits, enabling Johannes Kepler to discover that the orbits were actually ellipses. 153
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An Investigation of Precision and S
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Acknowledgments What a long, strang
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should totally start a band. I also
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2.4 Conclusions and further questio
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7.6.1 Regular array of stationary q
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List of Figures 2-1 The CNOT gate i
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7-12 Magnetic fields due to a “tr
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Chapter 1 Introduction The growth o
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1.1 Quantum information The concept
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us whether the target system obeys
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Nevertheless, there are a number of
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Digital Analog Classical Quantum Sp
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electrodes [TKK + 99]. This heating
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Digital Analog Classical Quantum Sp
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2. Initialization to a simple fiduc
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qubits, and in Ref. [SGA + 05] in t
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the lattice architecture, we discov
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1.6 Contributions to this work In t
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5) Ref. [DLC + 09b] Wiring up trapp
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Chapter 2 Quantum simulation using
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µN the nuclear magneton, and B0 th
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⎡ ⎤ a 0 0 0 ⎢ ⎥ ⎢ ρ2 =
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The same effect as number 2 above i
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where V0 is the maximum signal stre
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actions that occur between nuclei i
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Figure 2-2: The 2,3-dibromothiophen
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detail how this was done with a sim
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and thus requires invocation of the
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Renormalization Group (DMRG), with
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3.3 The bounds on precision In this
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Figure 3-1: A schematic of a generi
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Figure 3-3: The 11.7 T, 500 MHz sup
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. Figure 3-4: The above probe is a
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Figure 3-5: The CHFBr2 molecule. Al
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Figure 3-6: Frequency-domain spectr
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used to extract the answer is irrel
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Chapter 4 Theory and history of qua
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GND RF r ENDCAP 0 z 0 ENDCAP Ring T
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where Q is the charge of the trappe
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angular momentum along the quantiza
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emission from the excited state |
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selection rule ∆m = ±1 is applie
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Figure 4-3: Schematic diagram of sp
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to address an ion in state |↑〉
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the vibrational temperature is low.
Page 101 and 102: to the design of the trap itself. M
Page 103 and 104: to the quantum-mechanical ground st
Page 105 and 106: Chapter 5 Lattice ion traps for qua
Page 107 and 108: Figure 5-1: Schematic of the lattic
Page 109 and 110: Figure 5-3: Fit to Eq. 5.2 of the C
Page 111 and 112: Figure 5-4: The vacuum chamber for
Page 113 and 114: weeks. Low pressures depend on choo
Page 115 and 116: Figure 5-5: Left: Level diagram for
Page 117 and 118: Figure 5-7: (a) Schematic of the cr
Page 119 and 120: (a) Lattice Trap Schematic TOP PLAT
Page 121 and 122: mix the microspheres evenly in the
Page 123 and 124: Using the expression for ωˆr deri
Page 125 and 126: Figure 5-15: Plot of the motional c
Page 127 and 128: Figure 5-17: Dependence of the trap
Page 129 and 130: Chapter 6 Surface-electrode PCB ion
Page 131 and 132: Figure 6-1: Schematic of a linear i
Page 133 and 134: Figure 6-2: Layout of the trap elec
Page 135 and 136: Figure 6-3: Above are the cross sec
Page 137 and 138: Figure 6-6: Photograph of the trap
Page 139 and 140: Figure 6-8: CCD image of a cloud of
Page 141 and 142: Figure 6-9: Measurement results sho
Page 143 and 144: Figure 6-11: Bastille mounted in th
Page 145 and 146: Figure 6-13: A diagram of the setup
Page 147 and 148: Figure 6-14: A plot of the trapped
Page 149 and 150: allows us to upper-bound the amount
Page 151: Chapter 7 Quantum simulation in sur
Page 155 and 156: Figure 7-3: Calculated secular freq
Page 157 and 158: Figure 7-5: Left: Structure of 2-D
Page 159 and 160: Figure 7-7: Scaling of the order of
Page 161 and 162: involved. Because ions near the cen
Page 163 and 164: Results Before presenting the numer
Page 165 and 166: Figure 7-10: Calculation results fo
Page 167 and 168: 7.3 Magnetic gradient forces Now th
Page 169 and 170: Figure 7-13: Scaling of the J-coupl
Page 171 and 172: how are single-ion operations to be
Page 173 and 174: 250 psi, which increases to around
Page 175 and 176: Figure 7-18: At the center of this
Page 177 and 178: Figure 7-19: The plastic socket tha
Page 179 and 180: Figure 7-21: The two internal lense
Page 181 and 182: Figure 7-22: Measured secular frequ
Page 183 and 184: Along ˆy, the spacing is dˆy = 28
Page 185 and 186: of O2 to 2 × 10 −12 torr. Decohe
Page 187 and 188: Part III Toward ion-ion coupling ov
Page 189 and 190: Chapter 8 Motivation for and theory
Page 191 and 192: Figure 8-1: Schematic of the experi
Page 193 and 194: are in close agreement in the case
Page 195 and 196: 8.2.3 Simulated coupling rates We n
Page 197 and 198: Johnson noise We begin our discussi
Page 199 and 200: should suffice. Note that the above
Page 201 and 202: Chapter 9 Measuring the interaction
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Figure 9-2: Left: Level diagram for
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Figure 9-5: Photograph of the micro
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A fit of the resulting curve return
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Figure 9-9: Linear fit of the ωˆy
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Figure 9-12: Heating rate as determ
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Chapter 10 Conclusions and outlook
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traps, linking ions using photons,
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Bibliography [ADM05] Paul M. Alsing
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[CGB + 94] J. I. Cirac, L. J. Garay
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[GME + 02] M. Greiner, O. Mandel, T
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[LCL + 07] D. R. Leibrandt, R. J. C
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[PC04a] D. Porras and J. I. Cirac.
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[TBZ05] L. Tian, R. Blatt, and P. Z
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Appendix A Matlab code for Ising mo
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err_result = mean(theerr) 231
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J = J0; Jt(1) = J; Bt(1) = B; for k
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A.3 Simulation with constant force
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Appendix B Mathematica code for ion
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2 crystal_shape_6uraniborg.nb Out[6
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Appendix C How to trap ions in a cl
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Finally, we note that the chilled w
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C.4 Laser alignment and imaging As
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Ph.D. Thesis - Physics

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