Ph.D. Thesis - Physics

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Figure 4-2: Generic level structure for the ions used in the experimental work for this thesis. Here n is the principal quantum number; the D states have one less unit of n, as shown. The three transitions are the ones discussed in this chapter. λDopp is the Doppler cooling and detection transition. λRep is the repumper for the P 1/2 state; it is needed whenever the λDopp transition is addressed. λSB is a narrow quadrupole transition that is used for sideband cooling and state manipulation. 4.1.2 Control of ionic internal states In this section we will study the basic atomic structure of all the ions used in this work, and describe how lasers can be used to effect transitions between these levels. There are two types of atomic structure that are widely used in trapped ion quantum information processing: the first relies on a ground state connected to a metastable excited state by an optical transition (hundreds of THz), while the second connects two hyperfine levels with a transition that is in the GHz range. Here, we will focus on the former case, since it applies to both ions used in this work, 88 Sr + and 40 Ca + . We begin with a review of the atomic structure of such ions. Both are hydrogenic, meaning that there is only one valence electron. A diagram of the electronic level structure of such an ion is given in Fig. 4-2. Each transition is characterized by two numbers: the transition frequency Ωij, where i and j label the states, and the excited state lifetime τ, which is related to the spontaneous emission rate Γ by τ = 2π/γ. The term symbols follow the usual conventions for L-S (Russell-Saunders) coupling, which is valid whenever spin-orbit coupling is weak. This scheme assumes that the orbital angular momentum L and total angular momentum J are good quantum numbers, leading to an atomic state |nSLJmJ〉. The first number is the principal quantum number n. The superscript 2 is equal to 2S + 1, and is always two for such atoms since there is only one valence electron. The capital letter following it gives the orbital angular momentum, assigning each value a letter (for historical reasons): (L = 0) ↦→S, (L = 1) ↦→P, and (L = 2) ↦→D. The subscript gives the value of the total angular momentum J. J may take values ranging in integer steps from L−S to L+S. The quantum number mJ represents the projection of the total 88
angular momentum along the quantization axis, which is determined by the orientation of the external magnetic field. We now describe the interaction of the trapped ion with laser radiation, which is essential for understanding how quantum operations may be performed on trapped ions, as well as other techniques such as Doppler cooling and state measurement. We focus on electric dipole transitions, which occur between atomic levels that differ by one unit of orbital angular momentum. For simplicity, we will limit the discussion to the coupling of two discrete levels. The interaction is that of the dipole moment of the electron interacting with the oscillating laser field. The interaction is written as Hdip = − d · E = − d ·ǫE0 cos −ωlt + k · z + φ , (4.14) where d = ecz is the dipole operator, z is the displacement vector of the ion from its equilibrium position, ǫ is the polarization of the laser, E0 is the amplitude of the laser’s electric field, ωl is the frequency of the laser radiation, k is the wavevector of the radiation field, and φ is a phase that may be chosen by the experimenter. We will assume that the wavevector of the light is in the ˆz direction. We note that the dipole operator is proportional to σ + + σ − , where σ + = |↑〉 〈↓| and σ − = |↓〉 〈↑| are the raising and lowering operators for the atomic state, respectively. Making use of the relation of the creation and annihilation operators to the operators x and p, and making a rotating wave approximation, the above Eq. 4.14 takes the form HI = Ω e iη(a+a† ) + −i(ωlt+φ) σ e + h.c. , (4.15) where h.c. denotes Hermitian conjugate. The Rabi frequency Ω is the rate at which the atomic state |↑〉 is flipped to |↓〉 and vice-versa. For an electric dipole transition, Ω = |ecE0 〈↓|z |↑〉 |, that is, it depends on the applied electric field and on the expectation value of the dipole operator ecz. The Lamb-Dicke parameter η is equal to kz0, where k is the norm of the wavenumber of the radiation field and z0 is the width of the ground state wave function along ˆz: z0 = . A small η (η ≪ 1) means that the ion in its ground state 2mωˆz is well-localized in space to within the wavelength of the laser light, which implies that the light field will couple well to the harmonic oscillator mode in that direction. This condition is called the Lamb-Dicke limit, and when it is satisfied, the interaction Hamiltonian looks like HI = Ωe iφ σ + −i(ωl−ω0)t e 1 + iη ae −iωˆzt † iωˆzt + a e + h.c., (4.16) where ωˆz is the ion’s secular frequency in the ˆz direction. Eq. 4.16 actually reveals all the physics of how manipulations may be made on both the internal and external states of a single trapped ion using laser radiation. We focus first on the case in which the laser detuning δl, defined as δl ≡ ωl − ωˆz, is zero. We also neglect 89
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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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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 and 86: GND RF r ENDCAP 0 z 0 ENDCAP Ring T
Page 87: where Q is the charge of the trappe
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
Page 137 and 138: 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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