Ph.D. Thesis - Physics

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2.4 Conclusions and further questions . . . . . . . . . . . . . . . . . . . . . . . . 60 3 Quantum simulation of the BCS Hamiltonian 63 3.1 The BCS theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 3.2 The Wu-Byrd-Lidar proposal . . . . . . . . . . . . . . . . . . . . . . . . . . 66 3.3 The bounds on precision . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.4 The NMR system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.4.1 Sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.4.2 Magnet . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 3.4.3 Probe . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 3.4.4 RF electronics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 3.4.5 Computer control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.5 Experiment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 3.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 II Two-dimensional ion arrays for analog quantum simulation 81 4 Theory and history of quantum simulation using trapped ions 83 4.1 The ion trap system and Hamiltonian . . . . . . . . . . . . . . . . . . . . . 84 4.1.1 Motional states of trapped ions . . . . . . . . . . . . . . . . . . . . . 84 4.1.2 Control of ionic internal states . . . . . . . . . . . . . . . . . . . . . 88 4.1.3 Control of the ion’s external state . . . . . . . . . . . . . . . . . . . 90 4.1.4 State preparation and measurement . . . . . . . . . . . . . . . . . . 92 4.1.5 Decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 4.2 Quantum simulation of quantum spin models . . . . . . . . . . . . . . . . . 94 4.2.1 The Ising and Heisenberg models . . . . . . . . . . . . . . . . . . . . 96 4.2.2 Porras and Cirac’s proposal for simulating quantum spin models . . 96 4.3 Ion trap design for quantum simulation . . . . . . . . . . . . . . . . . . . . 99 4.3.1 Challenges for trap design . . . . . . . . . . . . . . . . . . . . . . . . 99 4.3.2 2-D ion arrays: prior art . . . . . . . . . . . . . . . . . . . . . . . . . 101 4.3.3 Methods of trap design, testing, and evaluation . . . . . . . . . . . . 101 4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 5 Lattice ion traps for quantum simulation 105 5.1 Proposals for quantum simulation in lattice ion traps . . . . . . . . . . . . . 106 5.2 Lattice trap design and theory . . . . . . . . . . . . . . . . . . . . . . . . . 106 5.3 Experimental setup for 88 Sr + trapping . . . . . . . . . . . . . . . . . . . . . 110 5.3.1 Vacuum chamber and electrical connections . . . . . . . . . . . . . . 110 5.3.2 Lasers and imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.4 Experimental results for 88 Sr + trapping . . . . . . . . . . . . . . . . . . . . 116 10
5.5 Measuring interactions between macroscopic ions . . . . . . . . . . . . . . . 119 5.6 Scaling laws for the simulated interactions in lattice traps . . . . . . . . . . 123 5.6.1 Motional coupling rate . . . . . . . . . . . . . . . . . . . . . . . . . . 123 5.6.2 Simulated J-coupling rate . . . . . . . . . . . . . . . . . . . . . . . . 124 5.6.3 Trap depth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126 5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 6 Surface-electrode PCB ion traps for trap development 129 6.1 Surface-electrode ion traps: history and theory . . . . . . . . . . . . . . . . 130 6.2 Design and construction of a planar PCB ion trap . . . . . . . . . . . . . . 132 6.2.1 Modeling the trap . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133 6.2.2 Constructing and mounting the trap . . . . . . . . . . . . . . . . . . 134 6.3 Buffer gas loading and micromotion compensation in a PCB ion trap . . . . 137 6.3.1 Experimental setup and ion loading . . . . . . . . . . . . . . . . . . 137 6.3.2 Measurement of stray fields . . . . . . . . . . . . . . . . . . . . . . . 139 6.3.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 6.4 The second-generation PCB ion trap . . . . . . . . . . . . . . . . . . . . . . 142 6.5 Ablation loading of planar PCB ion traps . . . . . . . . . . . . . . . . . . . 143 6.5.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 6.5.2 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 6.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 149 7 Quantum simulation in surface-electrode elliptical ion traps 151 7.1 Elliptical ion trap theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 7.1.1 Secular frequencies and trap depth . . . . . . . . . . . . . . . . . . . 154 7.1.2 Ion crystal structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 7.2 Micromotion scaling and its effect on quantum simulation . . . . . . . . . . 160 7.2.1 Scaling of the micromotion amplitude . . . . . . . . . . . . . . . . . 160 7.2.2 Effect of micromotion on quantum simulations . . . . . . . . . . . . 161 7.3 Magnetic gradient forces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 7.3.1 Calculation of the gradients and interaction strengths . . . . . . . . 167 7.3.2 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 7.4 The cryostat and vacuum apparatus . . . . . . . . . . . . . . . . . . . . . . 171 7.4.1 The cryostat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 7.4.2 Vacuum chamber and optical access . . . . . . . . . . . . . . . . . . 173 7.5 Experimental study of the elliptical trap . . . . . . . . . . . . . . . . . . . . 177 7.5.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177 7.5.2 Secular frequency measurements . . . . . . . . . . . . . . . . . . . . 180 7.5.3 Ion crystal structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 7.6 Discussion: connection to quantum simulation . . . . . . . . . . . . . . . . 183 11
Page 1: An Investigation of Precision and S
Page 5 and 6: Acknowledgments What a long, strang
Page 7: should totally start a band. I also
Page 12 and 13: 7.6.1 Regular array of stationary q
Page 15 and 16: List of Figures 2-1 The CNOT gate i
Page 17: 7-12 Magnetic fields due to a “tr
Page 21 and 22: Chapter 1 Introduction The growth o
Page 23 and 24: 1.1 Quantum information The concept
Page 25 and 26: us whether the target system obeys
Page 27 and 28: Nevertheless, there are a number of
Page 29 and 30: Digital Analog Classical Quantum Sp
Page 31 and 32: electrodes [TKK + 99]. This heating
Page 33 and 34: Digital Analog Classical Quantum Sp
Page 35 and 36: 2. Initialization to a simple fiduc
Page 37 and 38: qubits, and in Ref. [SGA + 05] in t
Page 39 and 40: the lattice architecture, we discov
Page 41 and 42: 1.6 Contributions to this work In t
Page 43: 5) Ref. [DLC + 09b] Wiring up trapp
Page 47 and 48: Chapter 2 Quantum simulation using
Page 49 and 50: µN the nuclear magneton, and B0 th
Page 51 and 52: ⎡ ⎤ a 0 0 0 ⎢ ⎥ ⎢ ρ2 =
Page 53 and 54: The same effect as number 2 above i
Page 55 and 56: where V0 is the maximum signal stre
Page 57 and 58: actions that occur between nuclei i
Page 59 and 60: 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
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Figure 6-9: Measurement results sho
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Figure 6-11: Bastille mounted in th
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Figure 6-13: A diagram of the setup
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Figure 6-14: A plot of the trapped
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allows us to upper-bound the amount
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Chapter 7 Quantum simulation in sur
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Figure 7-1: Left: Uraniborg 1, as r
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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
Page 223 and 224:
[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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