Ph.D. Thesis - Physics

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Figure 7-14: The concentric rings wire configuration and resulting magnetic fields. The wire configuration is top left; each wire carries 1A of current in the same direction. Each other the other plots gives the magnetic field components along a given direction. In all figures, Bˆx is in blue, Bˆy is in green, and Bˆzs is in red. Note that the field gradients are quite constant across the volume occupied by the ions. Nonzero fields at the trap center must be cancelled by additional pairs of Helmholtz coils to avoid unwanted Zeeman shifts, electron alignments, and even ion crystal rotation. Figure 7-15: Scaling of the J-coupling rates for all three directions as a function of the ion height in an elliptical trap, for the three concentric square rings wire configuration. The value for each direction will only hold if the projection of the electron spin is entirely along that direction; this would give an Ising-type interaction. 170
how are single-ion operations to be applied, and what is the effect of the unequal ion-ion distance in a crystal containing more than two ions? There exist a number of ways of implementing single-ion operations. The first is simply to address the ions with a tightly-focused laser. This seems impractical, however, for the surface-electrode elliptical trap, since the laser would need to be aimed directly toward the ground electrode. Although it could be applied at an angle, it is unlikely that scatter off this electrode would not interfere with the states of neighboring ions. A much more appealing idea would be to do the addressing using rf or microwaves, tuned to the Zeeman or hyperfine transition being used. The gradient wires used for implementing the state-dependent force could be turned on to produce a small gradient that can be used for discriminating the transitions of the individual ions in frequency space. This approach has been successfully pioneered by our group with laser addressing of ions [WLG + 09], and has been demonstrated by the Wunderlich group with hyperfine qubits [JBT + 08]. With respect to the unequal ion-ion spacing, we make a few points. The first is that despite coupling rates that are not perfectly constant across all ion pairs, there is nevertheless the possibility of observing interesting physics such as phase transitions. For instance, the transition from a paramagnetic to an antiferromagnetic state is caused by competition between a static magnetic field and the ion-ion interaction. A J-coupling rate that varies in space would add to the uncertainty in measuring the transition point, but this might still be a useful measurement. The second point is that optical potentials could be used to regulate the position of ions, as was suggested in Ref. [PC04b]. Furthermore, initial trapping of ions in an elliptical trap could facilitate loading into a fully optical potential, as discussed in Ref. [SRM + 08]. In the case of an elliptical trap, additional state-dependent forces may be added by the magnetic gradient coils, in addition to any forces already present from the optical potential. Finally, we note that the same level of control required to compensate for the effects of micromotion could be used to compensate for the unequal ion-ion spacing: if one can compensate for one effect, then one can compensate for the other. A final note of caution is that the ultimate number of wires, their geometry, and their width is limited by the minimum feature size of the fabrication process used. This will in turn limit the minimum size of the trap and gradient wire structure, limiting the interaction rates achievable. 7.4 The cryostat and vacuum apparatus Given the need for reducing heating rates, and the desirability of rapid pumpdown to UHV pressures, a 4 K closed-cycle cryostat was set up to carry out the trap testing. We use an Advanced Research Systems (ARS) DE-210 system with a DMX-20 anti-vibration interface. The system is very similar to the closed-cycle system presented in Ref. [ASA + 09]. The cryostat operates by repeated expansion of high-pressure (250 psi) helium gas into 171
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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
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 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: Figure 7-13: Scaling of the J-coupl
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
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
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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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