Ph.D. Thesis - Physics

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−2ecV qi = ζi mr2 0Ω2 4ecU ai = ζi mr2 0Ω2 (4.4) (4.5) where ec = 1.6 × 10 −19 C is used throughout the thesis as the (positive) elementary charge, m is the mass of the ion, and r0 is defined in Fig. 4-1 as the distance from the trap center to the nearest point of the rf electrode. The constants ζi depend on the direction i; i = 1 for ˆx and ˆy, while i = −1 for ˆz. U is a voltage that may be applied to the endcaps of the ring trap. Since it is possible to trap with U = 0, a is normally zero. In the lattice trap (Ch. 5) and elliptical trap (Ch. 7), no dc voltage is required to trap the ions. In traps based on linear ion traps, however, the rf fields provide confinement along only two directions; a static voltage is required for the third. Even in this case, however, ions may be trapped if U is small compared to V . The parameters a and q determine whether the trap is stable, that is, whether an ion confined in it can remain so. The “stability regions” in a-q space have been calculated, and are given in [Gho95]. In particular, for a = 0, 0 ≤ q ≤ 0.908 results in a stable trap. The method of solution of these equations is presented in [LBMW03]. For our purposes, it will be sufficient to invoke the pseudopotential approximation. This result assumes that the motion of the ions can be decomposed into two parts: small oscillations at the drive frequency Ω, referred to as micromotion, and slower oscillations within an effective harmonic well at a lower frequency, called the secular motion. The solution has the form ui(t) = Ai cos (ωit) 1 + qi cos (Ωt) , (4.6) 2 where Ai is the amplitude of the ion’s motion along direction i. The term that oscillates at Ω represents the micromotion. It is proportional to the term of order unity, and smaller by a factor of qi 2 . Since qi is around 0.3 for many of our experiments, the amplitude of this motion is on the order of one-tenth that of the secular motion, hence the term micromotion is apt. When a = 0, the relation between the secular frequency ω and the drive frequency Ω is ωi = qi 2 √ Ω, (4.7) 2 giving rise to our oft-stated “rule of thumb” that the secular frequency is about one-tenth the drive frequency. The formula for the pseudopotential Ψ is Ψ(x, y,z) = Q2 4mΩ2 ∇V (x, y,z) 86 2 , (4.8)
where Q is the charge of the trapped particle (equal to ec for the atomic ions used in this thesis) and m is its mass. We will use this formula in all our calculations of ion trap potentials, and find that it agrees quite well with our results. Dissecting this formula a bit, we see that Ψ is proportional to ecQ/r 2 0 ; the 1/r2 0 dependence comes from the gradient of the potential squared. This parameter depends on the trap geometry, and in most of our traps is computed numerically. For ion traps that admit approximate analytical solutions, r0 has a clear physical significance; for example, in the case of the ring trap, it is the distance from the trap center to the nearest point on the rf ring electrode. For the quadrupole potential above, close to the its minimum, the pseudopotential has the form of a three-dimensional harmonic oscillator: where the ωi are the secular frequencies. Ψ(x, y, z) = 1 m 2 i ω 2 i u 2 i , (4.9) The classical solutions for a particle in a harmonic oscillator will not be repeated here. We will briefly present the quantum mechanical solution, since this is essential to the theory of quantum simulation with ion traps. The Hamiltonian of the oscillator in a given direction with frequency ω is H = ω a † a + 1 , (4.10) 2 where [a, a † ] = 1 (see Ref. [Sak85] for further details). The operators a and a † are called the annihilation and creation operators, respectively. Their effect on the state |n〉 is to remove or add one quantum of vibrational energy to the state |n〉: a |n〉 = √ n |n − 1〉 ; a † |n〉 = √ n + 1 |n + 1〉 . (4.11) The eigenbasis is the set of states {|n〉} that satisfy, in addition to the above, H |n〉 = ω n + 1 |n〉 . (4.12) 2 The ground state wave function, for which a |0〉 = 0, is written in coordinate space as where x0 = 2mω 〈0|x〉 = x0 π e− “ ” 2 x 2x0 , (4.13) is the width of the ground state wavefunction. 87
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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
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
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: GND RF r ENDCAP 0 z 0 ENDCAP Ring T
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 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
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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
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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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