Ph.D. Thesis - Physics

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study should be undertaken of the effects of control errors in scalable quantum computing systems. Is it possible that the presence of control errors and the sensitivity of quantum simulations to them is superior in one system relative to others? Is it even possible that this system will be found not to be identical to the one that is best-suited for universal quantum computation? We do not yet know the answers to these questions, but they are, in our view, fascinating ones that must be pursued. Moving to the subject of analog quantum simulation, we saw in Part II how an array of trapped ions could be used for simulating physics such as Ising and Heisenberg spin models, and specifically spin frustration in 2-D antiferromagnetic spin lattices. We also saw, however, that building a scalable or even semi-scalable architecture for doing such simulations in 2-D is actually quite difficult. This problem arises for arrays of microtraps from the difficulty of maintaining both low secular frequencies and high trap depths at small ion-ion distances. This is a fairly straightforward result of the physics of coupled oscillators and the equations of motion of trapped ions. Although we have not proven that quantum simulation in lattice traps is impossible, it does appear to be quite difficult. We then looked for a different trap design for 2-D quantum simulation, and moved on to studying elliptical ion traps, calculating and measuring such properties as the motional frequencies and the structure of ion crystals in these traps. We also showed that the unavoidable micromotion is not a fundamental detriment to quantum simulation, and that interesting analog simulations, such as the search for quantum phase transitions, may be possible in such an architecture. Although other trap designs exist that create 2-D lattices of ions, elliptical traps have the advantage that they can be scaled down and microfabricated, reducing overall micromotion amplitudes and also allowing for the integration of wires that produce magnetic field gradients, which we showed can be used to provide a rich variety of simulated Hamiltonians for quantum simulation. To date, the only realization of the spin model simulation scheme has been with two ions in a linear trap. This can and should be scaled to larger numbers of ions in a linear chain, and efforts made to understand the sources of decoherence and control errors. At the same time, prototype 2-D quantum simulators should also be built. We think that elliptical traps are a good starting point. In the future, these simulations may enable us to solve very difficult problems, such as calculating the phase diagrams of 2-D antiferromagnetic lattices, that may, in turn, shed light on the causes of high-temperature superconductivity. Other condensed-matter systems, such as the Bose-Hubbard model, may be simulable in such a trap as well. Developing a good 2-D trap design is an important step toward tackling problems that classical computers cannot efficiently solve. Despite the possibility of exceeding the simulation power of classical computation even with 40 ions in a single trap, the 2-D architectures we proposed are not truly scalable to hundreds of ions or more. We mentioned a number of ideas for scaling ion trap quantum simulators to such large numbers of qubits, including moving ions between separate micro- 214
traps, linking ions using photons, and connecting ions electrically over a wire. The last of these framed the goals for the third part of this thesis. We built a system that includes a segmented surface-electrode ion trap and a moveable wire in vacuum. We calculated the theoretical coupling rates and decoherence rates, and set bounds on the acceptable experimental parameters, including the capacitance and resistance to ground of the wire. The resulting stringent requirement that the wire be highly isolated from ground at both dc and rf frequencies motivated the experimental work of Part III. Here, we studied the ways in which this wire changes the electrodynamic potentials that act on a single trapped ion. Although we have not yet observed ion-ion coupling over a wire, we have joined a recently- growing effort to exploit the fact that trapped ions are extremely sensitive detectors of electric fields. Most of the work to date has focused on measuring fluctuating fields, since they affect motional heating rates. Our work is a step toward measuring electrical properties of a macroscopic conductor from non-fluctuating fields. This experiment will progress, first into a cryogenic chamber to quell the rather high heating rates already observed, and then eventually to a point at which the wire is close enough that ion-ion coupling might be observed. Sympathetic cooling or heating of an ion in a different trap would be a good place to start. We do not yet know whether this approach will be useful to scaling up simulators, but the continued experiments will help us figure this out. If successful, the project will have bearing on other interesting approaches, such as linking a superconducting qubit (a fast processor) to a single trapped ion (a long quantum memory) over a transmission line. In sum, then, this thesis has taken steps along three quite different approaches to quantum simulation. In the course of this work several new problems were identified, which in turn motivated new questions, which we hope will form part of the efforts of researchers worldwide well into the future. We also hope that our work hastens the day when quantum simulation is reliable and commonplace, and a part of the toolbox of every researcher who can make use of it. 215
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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
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weeks. Low pressures depend on choo
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Figure 5-5: Left: Level diagram for
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Figure 5-7: (a) Schematic of the cr
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(a) Lattice Trap Schematic TOP PLAT
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mix the microspheres evenly in the
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Using the expression for ωˆr deri
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Figure 5-15: Plot of the motional c
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Figure 5-17: Dependence of the trap
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Chapter 6 Surface-electrode PCB ion
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Figure 6-1: Schematic of a linear i
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Figure 6-2: Layout of the trap elec
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Figure 6-3: Above are the cross sec
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Figure 6-6: Photograph of the trap
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Figure 6-8: CCD image of a cloud of
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Figure 6-9: Measurement results sho
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Figure 6-11: Bastille mounted in th
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Figure 6-13: A diagram of the setup
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Figure 6-14: A plot of the trapped
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allows us to upper-bound the amount
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Chapter 7 Quantum simulation in sur
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Figure 7-1: Left: Uraniborg 1, as r
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Figure 7-3: Calculated secular freq
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Figure 7-5: Left: Structure of 2-D
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Figure 7-7: Scaling of the order of
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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
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: Chapter 10 Conclusions and outlook
Page 217 and 218: Bibliography [ADM05] Paul M. Alsing
Page 219 and 220: [CGB + 94] J. I. Cirac, L. J. Garay
Page 221 and 222: [GME + 02] M. Greiner, O. Mandel, T
Page 223 and 224: [LCL + 07] D. R. Leibrandt, R. J. C
Page 225 and 226: [PC04a] D. Porras and J. I. Cirac.
Page 227 and 228: [TBZ05] L. Tian, R. Blatt, and P. Z
Page 229 and 230: Appendix A Matlab code for Ising mo
Page 231 and 232: err_result = mean(theerr) 231
Page 233 and 234: J = J0; Jt(1) = J; Bt(1) = B; for k
Page 235 and 236: A.3 Simulation with constant force
Page 237 and 238: Appendix B Mathematica code for ion
Page 239 and 240: 2 crystal_shape_6uraniborg.nb Out[6
Page 241 and 242: Appendix C How to trap ions in a cl
Page 243 and 244: Finally, we note that the chilled w
Page 245 and 246: C.4 Laser alignment and imaging As
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Ph.D. Thesis - Physics

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