Ph.D. Thesis - Physics

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of the scale being tested, and traps of other scales. These properties include the motional frequencies, which are important for calculating β and J, as well as quantities of practical importance such as the trap depth and ion position relative to the trapping electrodes. Results from a trap of one size can typically be scaled to smaller or larger sizes. Testing After theoretical calculations of a given trap are done, they must be confirmed by experiment in order to compute the coupling rates and thereby evaluate the trap design. Testing the traps consists of fabricating a trap that is representative of one of the above paradigms, mounting it in a suitable vacuum vessel, and then measuring certain properties of the trap. For both paradigms, the motional frequencies are measured. In the case of an array of ion traps, the lattice geometry is determined by the fabrication of the trap electrodes. However, for ions which form a crystal within the same trap volume, the ion crystal geometry is determined by the trapping potentials, and verifying that this crystal matches theoretical predictions is also important. Evaluation Equipped with the ion crystal structure and motional frequencies, the coupling rates may be calculated. A primary goal of this part of the thesis is to determine, for each paradigm, how the coupling rates scale with the trap size (which is typically defined as the distance from the trap center to the nearest electrode). In Ch. 5, we treat the problem of an array of Paul traps, while in Ch. 7 we study ion crystals within one example of a Paul trap that creates a 2-D array of ions: a surface-electrode elliptical ion trap. Included in the evaluation of a trap design is an estimation of the relevant decoherence rates. Decoherence of internal states depends a great deal on the choice of qubit states, the ambient fields, fluctuations in the control potentials, and other effects. Our primary concern is with motional decoherence rates. In Ch. 7, the quenching of these rates at cryogenic temperatures motivates the construction and use of a 4 K cryostat for ion trap testing. Studying internal and external decoherence is beyond the scope of this thesis, but we cite relevant results from other research efforts where appropriate. Our primary concern is to insure that the coupling rates we determine are much higher than the decoherence rates reported for similar ion traps. 4.4 Summary We have now explained how the interaction of trapped ions with laser radiation can be used to implement quantum control over both the internal and motional states of the ions. Preparation of the ions’ motional state is done effectively by first Doppler cooling the ions into a crystalline state near the Doppler limit, then further cooling them (if necessary) close 102
to the quantum-mechanical ground state using resolved sideband cooling. From this initial motional state, operations can be performed that pump the internal degrees of freedom into some fiducial initial state, then implement unitary transformations on the state vector of the ions that perform some desired quantum simulation. Since simulation of quantum spin models is a focus of this part of the thesis, we have focused on this protocol. Although in principle sufficient control exists to perform quantum simulations with high fidelity, there are a number of difficult issues when one wishes to do so in practice. In particular, we have pointed out the great challenge associated with scalable or semi- scalable trap designs for quantum simulation. We require traps that can permit a high interaction rate between trapped ions while avoiding decoherence and systematic errors. In the remainder of this part of the thesis, we describe efforts to solve these challenges using the methods described in this chapter: design, test, and evaluate two paradigms for 2-D ion arrays, based on arrays of traps and Coulomb crystals within the same trap. 103
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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
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
Page 61: detail how this was done with a sim
Page 64 and 65: and thus requires invocation of the
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
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: to the design of the trap itself. M
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 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
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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
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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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