Ph.D. Thesis - Physics

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problems, the others have a strong effect upon our considerations. Estimates of decoherence rates must be calculated for each trap design, and possible control errors must also be considered. Furthermore, although we will focus on the simulation of spin frustration as the ultimate goal, we note that the traps studied in this part may be useful for other simulations as well, such as Bose-Hubbard physics. The main considerations for a trap design are interaction rates, controls, and decoherence rates. We discuss each of these below. The design must: 1. Provide a regular array of stationary qubits in at least two spatial dimensions. The array of ions desired here is an extension to two dimensions of the linear ion crystals used in most quantum information experiments. To do quantum simulations of problems that are unique to configurations of spins in two or more dimensions, such as spin frustration, we need a trap that will produce a two-dimensional array of ions. The word regular means here that the ions are stored in some configuration such that the ion-ion distance d between each pair of nearest neighbors is identical. This is important because the simulated coupling rate J depends strongly on the inter-ion distance d (varying as d −3 in the β ≪ 1 limit as discussed above). Quantum simulation is also possible in an array for which this distance is not constant, but this adds a constraint to the types of Hamiltonians that can be implemented (the J-coupling will be site-dependent). When applying a global effective Hamiltonian, this site dependence may translate into a control error, limiting the precision of the simulation, but not necessarily rendering it useless. Furthermore, such systematic control errors may be compensated if sufficient controls are available. 2. Enable sufficient control over each qubit to implement the desired simulation. A trap design that fulfills the first condition does not guarantee that quantum simulation may be effectively done. Sufficient controls are required to implement the desired effective Hamiltonian. These may include rotations of individual qubits, state- dependent forces arising from an optical or magnetic force, and global or ion-specific measurements. In addition, achieving some desired J-coupling rate requires a sufficient interaction rate between individual ions. Recall that the parameter β is the fractional transfer of the motional energy of one ion to another per secular period; although the dipolar approximation holds for β ≪ 1, the resulting J-coupling rate may prove too small to be observable within the decoherence time of the system. Therefore, in considering the controls required for quantum simulation, one must take into account the pertinent quantities β and J, and calculate each for each trap design. 3. Support a low enough decoherence rate to perform meaningful simulations given certain coupling rates. Internal state decoherence depends on the choice of qubit states, ambient field fluctu- ations, errors in the classical controls, and other factors that are not directly related 100
to the design of the trap itself. Motional state decoherence, by contrast, has a strong dependence on the size of the trap. Decoherence rates for a given trap may be estimated based on previously published measurements, and these estimated values may have a bearing on the choice of experimental conditions, such as the temperature at which the trap is operated. 4.3.2 2-D ion arrays: prior art Two-dimensional arrays of ions have been realized in both Penning [IBT + 98] and Paul [BDL + 00] traps; in both cases, an ensemble of ions was trapped and cooled within a single trapping region, and an ion crystal was formed by the mutual Coulomb repulsion of the ions. Both these approaches have certain advantages and disadvantages. The primary disad- vantage of the Penning trap approach is that the ion crystal rotates about the magnetic field axis due to the crossed E and B fields. This is inconvenient for performing ion-specific operations and measurements. While a Paul trap produces a stationary crystal of ions, each undergoes micromotion at the rf frequency, the amplitude of which increases with the distance of each ion from the center of the trap. The problem of how suitable the Paul trap approach is for quantum simulation was, up until this thesis, unaddressed. Proposals have also emerged for using arrays of individual ion traps for performing quantum operations, including quantum simulation, in two dimensions. This possibility was noted in the paper on spin model simulation of Porras and Cirac [PC04b], and was discussed in the context of universal quantum simulation in Ref. [CZ00]. A proposal was also published to use ions in microtrap arrays with gates based on microwave or radiofrequency radiation combined with magnetic field gradients to do simulation of quantum spin models [CW08]. However, an array of individual ion traps had not yet been realized prior to this thesis. Furthermore, no analysis had been published of the interaction rates for quantum simulation in such traps. 4.3.3 Methods of trap design, testing, and evaluation The main goal of this part of the thesis is to ascertain the suitability for analog quantum simulation of two trap paradigms: arrays of individual Paul traps, and 2-D Coulomb crystals within a single Paul trap region. We now describe the methods used for addressing the above challenges. The workflow consists of three steps: design, testing, and evaluation. Design At the design stage, we first conceive of a trap design that should, in principle, generate a 2-D array of trapped ions. A method of fabricating the trap must be determined, and then numerical modeling done to calculate the important properties of the trap, both for traps 101
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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
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
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: the vibrational temperature is low.
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 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
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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
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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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