Ph.D. Thesis - Physics

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espect to ground is Figure 8-2: Diagram of the model used in our calculations. V = ec ln 4πǫ0 H + h1 H − h1 + ln H + h2 H − h2 . (8.1) To obtain the coupling rate, we need to know how the induced charges in the wire due to one ion affect the electric potential seen by the other ion. We write down the potential energy of each ion due to the induced charge in the wire as Ui = ecV α ln H + hi H − hi , (8.2) where i = 1,2 indexes the ions and the geometric constant α = ln [(2H − a)/a]. Equipped with this potential energy, we can calculate the coupling constant from its second derivative. First, we switch to a more convenient coordinate system, in which the height of the ion hi is equal to its equilibrium height h0i added to its displacement from equilibrium zi. The coupling constant is given by γ ≡ ∂2 (U1 + U2) ∂z1∂z2 = 2e 2 cH 2 πǫ0αL(H 2 − h 2 1 )(H2 − h 2 2 ). (8.3) Given the fact that each ion is in a separate harmonic trap, we can write the full system Hamiltonian: H = p21 1 + 2m 2 mω2z 2 1 + p22 1 + 2m 2 mω2z 2 2 + γz1z2 , (8.4) where γz1z2 is the lowest-order interaction term between the two ions. The solution to this equation is quite well known in the classical case. For higher temperatures, this approach would be approximately correct. However, in the quantum- mechanical case the solution is a bit more difficult. Ref. [EKN68] considers both the resonant case (ω1 = ω2) exactly, as well as in the rotating wave approximation. These two approaches 192
are in close agreement in the case of small coupling constants (γ/ mω 2 ≪ 1). A more recent solution has shown that complete exchange of motional states can only occur on resonance (ω1 = ω2) and for specific initial motional states [PRDB08]. The exchange rate, also known as the coupling rate ωex, is given by ωex 2π γ = , (8.5) πωm where γ and α were defined above and ω = ω1 = ω2 (the resonant case). Incidentally, it is also equal to the classical expression! Referring to Eq. 8.3, the above formula for ωex allows us to evaluate the sensitivity of the coupling rate to the various parameters of our model system. For instance, the length of the wire and ion-wire distances enter as 1/ [L(H − h1)(H − h2)], a strong dependence, while the dependence on the wire radius a is only logarithmic (it is contained in α). Overall, we see that a shrinking of the entire system size leads to increased coupling, as the ions induce more charge when closer to the wire; also, a shorter wire length leads to a higher overall charge density on the wire. The inverse dependence on the secular frequency ω is an expected feature of coupled harmonic oscillators; we have seen similar physics at work in the lattice traps of Ch. 5. <strong>Ph</strong>ysically, tighter confinement (higher ω) reduces the effective “dipole moment” of each ion, which scales as 1/ √ ω. 8.2.2 Circuit model solution The above model leads to a physical picture of the physics of our system. In this section, we explore a different approach, which is based on a circuit model of the system, in which each component is treated as a lumped element: an inductor, capacitor, or resistor. This can have two key advantages: one is that the above physical model is based on assumptions that are not strictly true, but are good approximations. To take an example, the capacitance between the wire and ground will differ from that computed for an infinite ground plane, especially since the length of the wire is not much less than the width of the trap in our experiment. A circuit model allows one to quickly plug in more reasonable estimates of this capacitance. The second, and more compelling, reason to use a circuit model is that it makes the treatment of decoherence simpler. This is because the main decoherence sources are electrical in nature; dissipation of currents in the wire and Johnson noise heating are the two prime examples. This approach was presented in Refs. [WD75] and [HW90], and we follow it here. We begin with the equations of motion for the ions: ec m Ei = ¨zi + ω 2 i zi, (8.6) where the electric field Ei is due to the voltage in the wire induced by the other ion. We wish to write this field in terms of quantities in the above model, and the result is 193
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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
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 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: Figure 8-1: Schematic of the experi
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
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
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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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