Ph.D. Thesis - Physics

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Figure 5-16: Simulated J-coupling rates in lattice and linear ion traps as a function of ion-ion distance d. The maximum value of d given here, 100 µm, is the point at which the Lamb-Dicke parameter is approximately 0.1. The J values for lattice and linear traps intersect at higher d values, when the ion is no longer in the Lamb-Dicke regime. values obtained in the lattice trap are much weaker than a comparable linear trap within the Lamb-Dicke regime. While some gains might be made from using the stronger field gradients of a standing wave configuration for the “pushing” laser, it is clear that the scaling of ω with 1/d is a discouraging feature of lattice traps. These scaling laws for ωex and J hold regardless of how a given lattice geometry has been “optimized,” whether for trap depth, low motional frequencies, or even ωex at some length scale. This point bears emphasizing, since recent reports [SWL09] have detailed methods of designing array trap electrodes such that the trap curvature is maximized at each site for a given set of experimental parameters. While this approach is interesting and potentially useful for some applications, it is not clear how the above scaling behavior could be circumvented. 5.6.3 Trap depth An interesting and still unanswered question is whether it is possible to modify the lattice trap design to allow for low motional frequencies even at small ion-ion spacings, with an adequate trap depth. One simple idea would be to decrease the drive voltage V (and consequently the trap depth) once the trap is loaded with ions and they have been laser- cooled to a temperature much lower than the trap depth. However, the trap depth in this case would be extremely low. To understand why this is the case, consider the following argument. Suppose that one wishes to keep ω constant a the trap scale changes. From the above formulas, we see that the trap depth is proportional to V 2 /Ω 2 , and also to qV , where q ∝ V/(r 2 0 Ω2 ). We arrive at the following relation for ω: 126
Figure 5-17: Dependence of the trap depth D on the trap scale r0 for a constant secular frequency ω. The values for r0 = 1 mm, the approximate scale of the lattice trap presented in this chapter, are D = 0.3 eV, ω = 2π×200 kHz, and Ω = 10ω. ω ∝ ecV mr 2 0 Ω. (5.10) We note that in order to hold ω constant as r0 is varied, the following scaling law must hold: r2 0 ∝ V/Ω. Whether one reduces V or increases Ω, or some combination thereof, the trap depth is reduced in proportion to either V 2 or 1/Ω2 . Taking some typical baseline experimental parameters for the lattice trap, we plot in Fig. 5-17 the trap depth as a function of the trap scale. In this particular case, we varied the drive frequency Ω, but according to the above argument, varying V would yield the same trap depth. The result given in Fig. 5-17 completes our exposition of the scaling behavior of lattice traps. For ion-ion distances for which an appreciable coupling rate might be achieved, as in our above example for J = 1 kHz at d = 50 µm and ω = 2π×200 kHz, the trap depth is reduced to the order of a mere 100 µeV. While there is no fundamental reason why a laser-cooled ion may not be trapped at such a depth, such trapping has never been reported, to our knowledge, in the literature. In fact, this value is below the Doppler limit for most ions. We again note that even a hypothetical factor of 100 improvement in the ratio of trap depth to secular frequency would result in a trap depth of only ≈ 10 meV, still a very challenging figure. 5.7 Conclusions In this chapter we have presented the design, testing, and evaluation of a macroscopic lattice ion trap. Our main experimental achievement was verifying that, for two very different systems (atomic and macroion), our theoretical model of the trap is accurate. The 127
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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
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 and 122: mix the microspheres evenly in the
Page 123 and 124: Using the expression for ωˆr deri
Page 125: Figure 5-15: Plot of the motional c
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
Page 173 and 174: 250 psi, which increases to around
Page 175 and 176: 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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