Ph.D. Thesis - Physics

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macroscopic ions across lattice sites. In Sec. 5.6, we evaluate the trap design by computing the interaction rates as a function of trap scale in lattice traps. In Sec. 5.7, we conclude on this work. 5.1 Proposals for quantum simulation in lattice ion traps Quantum information theorists have put forward several proposals for quantum operations in 2-D arrays of trapped ions. A 2000 paper from Cirac and Zoller suggests using ions in an array of microtraps to form the basis of a scalable quantum computer, in which an ion is moved from site to site within the array to interact with ions contained therein [CZ00]. Later, as we discussed in Ch. 4, the spin model proposal of Porras and Cirac suggested that an array of microtraps could be used to implement quantum simulations of interesting physics such as spin frustration [PC04b]. A subsequent proposal from Chiaverini and Lybarger [CW08] suggested using an array of microtraps to implement a 2-D quantum simulation. In their scheme, microcoils surrounding each lattice site address individual ions contained therein with microwave radiation, effecting single-qubit rotations. In addition, magnetic field gradients may be applied which serve the same purpose as the lasers in the Porras-Cirac scheme, generating state-dependent forces that translate into effective spin-spin interactions. The commonality between the two is the requirement of a 2-D array of trapped ions. The advantages of a 2-D array of microtraps are that the position of each ion is well-determined by the trap electrodes, and that, conceivably, dc compensation electrodes could be provided for each site. This is in contrast to a trap in which all ions are contained in the same potential well. In such a trap, the ion-ion distance can vary, especially near the edge of the crystal, and micromotion may pose a problem. However, in this chapter, we focus on arrays of Paul traps, or lattice ion traps. 5.2 Lattice trap design and theory The model we study in this chapter is a layered planar rf electrode geometry that creates a 2-D ion lattice. The ion trap consists of a planar electrode with a regular array of holes, held at a radiofrequency (rf) potential, mounted above a grounded planar electrode. A single ion is trapped above each hole in the rf electrode (Fig. 5-1). Ions will be preferentially loaded above the trap electrode at the intersection of the Doppler cooling and photoionization beams, allowing the user to write an arbitrary 2-D lattice structure. We view this design as an archetype of 2-D Paul trap arrays, in the sense that conclusions drawn about some properties of this trap may translate to other, similar, trap designs. Our lattice trap is an extension of the three-dimensional ring Paul trap [Gho95]. Follow- ing this reference, we first review the theory of the ring trap. The ring electrode geometry 106
Figure 5-1: Schematic of the lattice trap. An array of traps is produced by a single rf electrode with a regular array of holes, mounted above a grounded electrode. Ions will preferentially be loaded from a broad atomic beam at the intersection of the cooling and photoionization lasers. GND RF r 0 ENDCAP z 0 ENDCAP Ring Trap Schematic RING ELECTRODE Figure 5-2: Cross section of a ring trap. Ions are confined at the center of the trap. The ring is held at an alternating rf potential relative to the endcaps. Here r0 = √ 2z0 and the endcaps and ring electrode are hyperbolically shaped. is shown in Fig. 5-2. An alternating voltage of the form Vrf = V cos(Ωt) is applied to the ring electrode and the endcaps are grounded. This situation is discussed in Ch. 4, and we summarize the results here. The equations of motion for an ion in the ring trap are a set of Mathieu equations which have regions of stability depending on the dimensionless parameters a = 8QU0 mr 2 0 Ω2 and q = 2QV mr 2 0 Ω2. Q and m are charge and mass of the ion, U0 is any dc voltage applied to both endcaps of the ring trap, and the distance r0 is the distance from the trap center to the rf electrode (as shown in Fig. 5-2). When U0 = 0 and the system is in vacuum, the condition for stability is q < .908. Note that the lattice trap has two “q” parameters, qˆz and qˆr, for the vertical and radial directions, respectively. They both must meet this inequality for a stable trap to exist. We assume the trajectory of a trapped ion is well approximated by a slow secular motion superposed with a rapid oscillation, the micromotion, due to the oscillation in the potential Vrf. For U0 = 0, time-averaging the ion motion in the secular approximation (q ≪ 1) gives 107
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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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Page 57 and 58: actions that occur between nuclei i
Page 59 and 60: Figure 2-2: The 2,3-dibromothiophen
Page 61: detail how this was done with a sim
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Page 66 and 67: Renormalization Group (DMRG), with
Page 68 and 69: 3.3 The bounds on precision In this
Page 70 and 71: 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
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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
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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 |↑〉
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Page 105: Chapter 5 Lattice ion traps for qua
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
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Figure 7-5: Left: Structure of 2-D
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Figure 7-7: Scaling of the order of
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involved. Because ions near the cen
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Results Before presenting the numer
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Figure 7-10: Calculation results fo
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7.3 Magnetic gradient forces Now th
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Figure 7-13: Scaling of the J-coupl
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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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