Ph.D. Thesis - Physics

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Figure 7-8: Crystal structures for 120 ions with ωˆx = ωˆy = 2π × 150 s −1 . Left: 2-D projection of the 3-D ion crystal that results for a vertical frequency of ωz = 2π × 500 s −1 . Right: 2-D ion crystal structure for ωˆz = 2π × 650 s −1 , greater than the critical frequency for crystal planarity (according to Eq. 7.1) of ωˆz = 2π × 608 s −1 . 7.2 Micromotion scaling and its effect on quantum simulation 7.2.1 Scaling of the micromotion amplitude We now turn to the calculation of the micromotion amplitude for the ions in the elliptical trap, as well as the effect that it has on quantum simulation. The micromotion amplitude Aµ along ˆx is given to first order by Aµ = 1 2 qˆx∆x, where qˆx is the Mathieu parameter along the direction ˆx and ∆x is the displacement of the ion from the rf null along ˆx. Similar relations apply for the other directions. Given a numerical solution for the trapping potential along with the ion geometry calculation presented in the last section, we can calculate this numerically as well. The scaling of the micromotion amplitude for two ions is straightforward. Suppose two ions align along the ˆy axis of an elliptical trap. Then, by balancing the Coulomb and trapping forces, the ion-ion distance is d = e 2 c 2πǫ0mω 2 ˆy 1/3 , (7.2) and the displacement from the rf null is just ∆y = d/2. Since the frequency ωˆy scales as 1/r0, the basic scaling law for micromotion is A ∝ r 2/3 0 , therefore it decreases as the overall size of the trap is reduced. This is one motivation for decreasing the overall size of the trap. The micromotion amplitudes of ions in larger 2-D crystals may be estimated from the ratio of displacements of ions given above in the ion crystal structure calculations. In- teractions between neighboring ions are impacted only by the relative motion of the ions 160
involved. Because ions near the center of the trap experience oscillating electric fields in opposite directions, their relative motion may be higher than ions further from the trap center, even though each individual ion experiences greater micromotion. 7.2.2 Effect of micromotion on quantum simulations The calculation of micromotion amplitudes is important, but the much more interesting (and hotly debated) question is what effect this micromotion has on the fidelity of quantum simulation. We now endeavor to answer this question. We will limit our study here to a specific Hamiltonian, the transverse Ising model, which is one example of a Hamiltonian under which spin frustration may occur. In a typical experiment, antiferromagnetism is produced by using the “radial” modes [PC04b], which in the parlance of Porras and Cirac means the modes perpendicular to the line segment that connects the two ions and perpendicular also to the plane of the trap, i.e. the vertical (ˆz) direction. In this case, the interaction is a short-range dipolar one, with the coupling constant J ∝ 1/d 3 , where d is the ion-ion distance. We are accordingly concerned with short-range, pairwise interactions. Therefore, most of the calculations study the effect of micromotion on two ions. The question we ask is this: for a given Hamiltonian, simulation time, and set of trap parameters (including micromotion amplitude), what is the fidelity of the quantum simulation as compared to the simulation in the absence of micromotion? We want to study the behavior of the system for a number of cases that we believe to be relevant for future experiments: • Constant F in space and time. • F constant in space, but adiabatically ramped up. • F varying in space, and following constant or adiabatic time dependence. The approximation that F is constant in space is good if F varies weakly across the region of space occupied by a given ion, which, in the presence of micromotion, may be up to a few hundred nanometers. If a standing-wave optical force is used, then the spatial extent of the ion trajectory is comparable to the gradient of the force, and the spatial dependence must be accounted for. Methods The Hamiltonian that we consider is the transverse Ising model. We define the ˆz, ˆx, and ˆy axes as being the principal axes of the elliptical trap in order of decreasing secular frequency, while Z, X, and Y are the Pauli operators for the ionic internal states along each of these directions. We shall simulate 161
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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
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 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: Figure 7-7: Scaling of the order of
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
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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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