Ph.D. Thesis - Physics

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Figure 7-24: Images of ion crystals of two and four ions in Uraniborg 2. The principal axes ˆx and ˆy of the trap are shown by depicting the corresponding ion-ion spacings dx and dy in each figure. way, and this wire forms a part of the resonant circuit that drives the trap. Therefore, some voltage differential is expected. In some papers, the rf voltage is treated as a fit parameter since it is difficult to directly measure [SCR + 06, CLBC09]. However, in this case the frequencies are not off by a constant factor, so this is not a completely satisfactory explanation. It is possible that some combination of the two effects explains the discrepancy: the dc fields at the ion location are unknown, as is the magnitude of the rf field. A more interesting question is how well the ion crystal structures match the theory for the measured frequencies, which we explore next. 7.5.3 Ion crystal structure Ion crystals consisting of between one and four ions were observed in Uraniborg 2. To obtain these images, it was necessary to readjust the imaging optics repeatedly to reduce aberrations, to set the dc voltages carefully, and to carefully coalign and focus the detection lasers. Without these conditions being met, somewhat misshapen crystals of two ions (but no more) were observed. Unfortunately, it was difficult to cool more than four ions into a crystalline state in this trap. Although ion crystals were observed, the signal was fairly weak and long exposure times of up to 5 s were required, using the HIGH resolution of the SBIG camera, in which each individual pixel is displayed (no binning). Images of crystals are plotted in Fig. 7-24. Images were analyzed as follows: the centers of each ion were determined from a Gaus- sian fit of the intensity. The magnification was calibrated to the spacing of two ions. The image of four ions was processed in the same way, using this magnification. Errors on the spacing of ions were determined from the standard deviation on the centers of the ions. For two ions, the calculated spacing is 16.5 µm, and the spacing between ions is 11±3 pixels, giving a magnification of 4.5, or 1.5 µm per pixel. For four ions, we calculate the spacing between the ions along the ˆx and ˆy directions. 182
Along ˆy, the spacing is dˆy = 28±3 µm, while along ˆx it is dˆx = 17±3 µm. These values are consistent with the theoretical values of dˆy = 29.6 µm and dˆx = 17.1µm. Again, it was not possible to crystallize larger numbers of ions, and therefore confir- mation of the crystal structure for higher numbers of ions was not done. However, we do now have some confidence that the crystal structure in elliptical traps may be accurately computed. There are a number of possible reasons for the low ion number. Among them are the stability of the lasers used, the fact that crystals were only observed at relatively low trap depths (0.3 eV), and the possibility of heating from the rf electrode rendering larger crystals unstable. Indeed, the lifetime of a single ion in the absence of cooling light was at most a few seconds, indicating that heating may indeed be a problem. 7.6 Discussion: connection to quantum simulation We have now presented the design and testing of a new method for preparing a 2-D Coulomb crystal in a Paul trap: the surface-electrode elliptical trap. The next step is to evaluate the usefulness of this trap for quantum simulation, focusing again on the challenging problem of spin frustration. We frame this discussion in terms of the requirements posed in Sec. 4.3. There, we note that a viable trap design must: 1. Provide a regular array of stationary qubits in at least two spatial dimensions. 2. Enable sufficient control over each qubit to implement the desired simulation. 3. Possess, in principle, a low enough decoherence rate to perform meaningful simulations given certain coupling rates. We address these criteria one-by-one. 7.6.1 Regular array of stationary qubits We have calculated crystal structures for the elliptical trap presented in this chapter, but it is seen from these that a truly regular array of ions is not produced. The reason for this is that the test trap was designed with more difference between the radial frequencies ωˆx and ωˆy than is strictly necessary to support a 2-D crystal with fixed ion positions. Recent calculations [BKGH08, BH09] have suggested that in a 2-D ion crystal regular lattice structures near the center of the lattice may be observed. We expect that, in our trap, as ωˆx → ωˆy, that such regular structures will also appear. The calculations of Sec. 7.1.2 support this hypothesis. 7.6.2 Sufficient controls We have outlined in this chapter two methods for implementing the state-dependent forces required for quantum simulation of spin models: optical forces and magnetic field gradients. 183
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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
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 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: Figure 7-22: Measured secular frequ
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
Page 215 and 216: traps, linking ions using photons,
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
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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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