Ph.D. Thesis - Physics

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Figure 5-13: ωˆz vs. Ω for an isolated macroion at a drive voltage of V = 255 V. The data for 0 V and 2.5 V come from the same ion. The fit was done only for data above 1500 Hz; data below this frequency begin diverging from this fit because the pseudopotential approximation no longer holds. Displacement [mm] 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 Data Fit −0.05 0.5 1 1.5 2 2.5 3 3.5 4 Frequency [kHz] Displacement [mm] 0.08 0.06 0.04 0.02 0 −0.02 Data Fit −0.04 0.5 1 1.5 2 2.5 3 3.5 4 Frequency [kHz] Figure 5-14: Ion displacement from the well center for two ions in neighboring wells, as a function of drive frequency. Due to charge asymmetry, the maximum displacements of the two ions differ by a factor of ten. The drive voltage is 350 V. The model breaks down for large displacements (high trap frequencies); fits only include data below Ω/2π = 2500 Hz. Error bars are dominated by the intrinsic error in determining the ion position from the ccd camera image. 122
Using the expression for ωˆr derived from Eq. 5.3, x1sd 2 ≈ Ω 2 mr4 1Q2 8πǫ0V 2 . (5.5) Q1 When Q1 is not equal to Q2, then the confining forces are characterized by different ωˆr and the two ions have different offsets from equilibrium. The ratio in offsets, if the masses are comparable (m1 ≈ m2), should be (Q2/Q1) 2 . We observed exactly such an asymmetry between the offsets of the two ions, where typically Q2/Q1 is between 1 and 5. There may be additional small asymmetries due to edge effects and the presence of the 4-rod trap as well as differences in charge. Fig. 5-14 shows the displacement of a pair of ions as the rf drive frequency is varied, for one experimental run. The spread in charge to mass ratios and accordingly unknown values of Q and m for each ion (as in Ref. [PLB + 06]) does not permit us to compare the observed repulsion to a theoretical model. Indeed, if these data were available, this repulsion experiment would be a very useful way to measure the screening factor s for a given trap, perhaps prior to trapping atomic ions. We conclude that ion-ion interaction in a mm-scale lattice trap is observable by the mutual Coulomb repulsion of macroions. Such an experiment could be used to measure the screening parameter s for a given trap, if knowledge of the charges and masses of the individual particles involved were available. Although we have been able to measure the ion- ion interaction of macroions and fit it to a model (in a certain region of parameter space), it will be necessary to scale the trap down further in order to observe ion-ion interactions between the atomic ions that would be used for quantum simulation. 5.6 Scaling laws for the simulated interactions in lattice traps The lattice trap discussed in this chapter provides a fairly straightforward method for real- izing a two-dimensional array of trapped ions. The first two steps enumerated in Sec. 4.3.3, design and testing of the trap, have now been reported. We turn now to evaluation of the trap design, and ask: how useful could this system be for quantum simulation of two- dimensional spin models [PC04b]? We will need to calculate both the motional coupling rate ωex, and the simulated coupling rate J. 5.6.1 Motional coupling rate We begin with the motional coupling rate, which is the rate at which two coupled ions swap motional states (provided they have the same secular frequency). Let us review the basic physics of the system formed by two trapped ions undergoing mutual Coulomb repulsion. The lowest-order term in the Taylor expansion of the Coulomb potential that contains an interaction between the ions is 123
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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
Page 72 and 73: Figure 3-3: The 11.7 T, 500 MHz sup
Page 74 and 75: . Figure 3-4: The above probe is a
Page 76 and 77: Figure 3-5: The CHFBr2 molecule. Al
Page 78 and 79: Figure 3-6: Frequency-domain spectr
Page 80 and 81: used to extract the answer is irrel
Page 83 and 84: Chapter 4 Theory and history of qua
Page 85 and 86: GND RF r ENDCAP 0 z 0 ENDCAP Ring T
Page 87 and 88: where Q is the charge of the trappe
Page 89 and 90: angular momentum along the quantiza
Page 91 and 92: emission from the excited state |
Page 93 and 94: selection rule ∆m = ±1 is applie
Page 95 and 96: Figure 4-3: Schematic diagram of sp
Page 97 and 98: to address an ion in state |↑〉
Page 99 and 100: the vibrational temperature is low.
Page 101 and 102: to the design of the trap itself. M
Page 103 and 104: to the quantum-mechanical ground st
Page 105 and 106: Chapter 5 Lattice ion traps for qua
Page 107 and 108: 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: mix the microspheres evenly in the
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 and 170: Figure 7-13: Scaling of the J-coupl
Page 171 and 172: how are single-ion operations to be
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250 psi, which increases to around
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Figure 7-18: At the center of this
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Figure 7-19: The plastic socket tha
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Figure 7-21: The two internal lense
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Figure 7-22: Measured secular frequ
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Along ˆy, the spacing is dˆy = 28
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of O2 to 2 × 10 −12 torr. Decohe
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Part III Toward ion-ion coupling ov
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Chapter 8 Motivation for and theory
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Figure 8-1: Schematic of the experi
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are in close agreement in the case
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8.2.3 Simulated coupling rates We n
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Johnson noise We begin our discussi
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should suffice. Note that the above
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Chapter 9 Measuring the interaction
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Figure 9-2: Left: Level diagram for
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Figure 9-5: Photograph of the micro
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A fit of the resulting curve return
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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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