Ph.D. Thesis - Physics

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Figure 7-9: Calculation results for a simulated Ising model for two ion-qubits with a spatially and temporally constant force. Left: The trajectory of the observable 〈Z〉 ≡ 〈Z1 + Z2〉 as a function of time in the absence of micromotion is plotted on top. Below this, the trajectory with a micromotion amplitude of Aµ = 0.1 is plotted. The difference between the two is plotted on the bottom. Without micromotion, J = 10 3 s −1 . Right: The same plots as on the left, but with the correct average value Jav = 1.0305 ×10 3 s −1 used for the Aµ = 0 case. When this is done, the error drops by three orders of magnitude. but will consist of the tensor product of two arbitrary single-qubit states. Entanglement between the qubits will not form until after the application of the J-coupling, but the set of arbitrary two-qubit states states subsumes this special case. We follow the above approach, but with random, complex amplitudes for each basis state (normalized to 1). We find that for random states, the error is reduced using the same average coupling as before (Jav = 1.036 × 10 3 s −1 ). The average error during the course of each simulation, regardless of the initial state, is 1.42×10 −5 . The effects of a position-dependent force, i.e. F(x), may be considered by applying a linear gradient at the position of each ion. The maximal gradient likely to occur in an experiment is that due to an optical standing wave. We take, for the sake of argument, 400 nm to be the lower-wavelength limit of such a wave. In considering the most extreme cases, we will also assume a maximum relative micromotion amplitude of Aµ = 0.1. The reason for this is as follows. Suppose that two ions are on opposite sides of the rf null; then their displacement from the rf null is equal to d/2, and the micromotion amplitude of each is A = q q d 2∆x = 2 2 , yielding Aµ ≈ 0.08 for q = 0.3. We proceed by computing the Jav for a set of force gradients, then confirming that the error indeed vanishes. A set of gradients Fr from 0 to 1, relative to the micromotion amplitude, as used. This means that for Fr = 0, the force is constant in space, while for Fr = 1, the force falls to zero when the ion is located at the position x = A. As expected, the uncorrected error due to micromotion rises as the force is allowed to vary in space, but nevertheless these errors may be corrected by using Jav rather than J, and are quenched by three orders of magnitude. Fig. 7-10 contains plots of Jav and the uncorrected error as a function of the relative force gradient. 164
Figure 7-10: Calculation results for a simulated Ising model for two ion-qubits with a position-dependent force. Left: The time-averaged coupling Jav as a function of the relative force gradient as defined in the text. Without micromotion, J = 10 3 s −1 . Right: The average uncorrected error as a function of the relative force gradient. When using the correct average value Jav, the error drops by three orders of magnitude. The final problem we treat in this section is the calculation of the simulation error when the applied force F(t) is time-dependent. We use a linear variation in time of the force F, rising from 0 at t = 0 to F at the final time (t = 10/J). In this case, we evaluate the time- averaged force for both the Aµ = 0 and Aµ = 0.1 cases, and then multiply the Aµ = 0.1 value by the ratio between the two. This is demonstrated numerically to dramatically reduce the error. Fig. 7-11 plots these results. Discussion For a number of important cases, it has been shown that the effect of micromotion is to systematically shift the J-coupling rate under which the internal states of two ions evolve. This is an interesting result, but it raises questions about how a simulation might actually be put into practice with a 2-D array of ions. There are two cases that will be relevant: “global” forces that impact each ion equally, and “local” forces which may be applied pairwise between ions. In the latter case, the effects of micromotion may be nulled by adjusting the state- dependent force according to the site in the array being addressed. Although this requires a substantial amount of control, there is no fundamental reason why it could not work. In the former case, one will find a “spread” of Jav values across the ion crystal. Although this will reduce the fidelity of the simulation, it may still be possible to observe interesting quantum phases. We would also like to note that there are important cases that have not been addressed in this work. We believe that the examples presented here, however, shed light on the cases that are still unaddressed. Three such cases are a “hard pulse,” in which F = 0 for some time, 165
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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.
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to the design of the trap itself. M
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to the quantum-mechanical ground st
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Chapter 5 Lattice ion traps for qua
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Figure 5-1: Schematic of the lattic
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Figure 5-3: Fit to Eq. 5.2 of the C
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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 and 152: Chapter 7 Quantum simulation in sur
Page 153 and 154: Figure 7-1: Left: Uraniborg 1, as r
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: Results Before presenting the numer
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
Page 203 and 204: Figure 9-2: Left: Level diagram for
Page 205 and 206: Figure 9-5: Photograph of the micro
Page 207 and 208: A fit of the resulting curve return
Page 209 and 210: Figure 9-9: Linear fit of the ωˆy
Page 211 and 212: Figure 9-12: Heating rate as determ
Page 213 and 214: 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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