Ph.D. Thesis - Physics

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Hrf(t) = −ω1 (cos (ωt + φ) X/2 + sin (ωt + φ) Y/2) , (2.13) where ω is the frequency of the applied magnetic field. For purposes of effecting rotations of the spins, we use the resonance condition ω = ω0, where again, ω0 is the Larmor frequency of the given spin. Then, in the rotating frame at frequency ω0, we can write this Hamiltonian as H rot rf = −ω1 (cos (φ) X/2 + sin (φ) Y/2) . (2.14) Because in this frame the rotations occur at a rate ω1, the total angle of rotation θ is just given by the time tp over which a resonant pulse is applied times ω1: θ = ω1tp. These rotations may be performed in either the ˆx or ˆy directions, or in both at the same time. We shall use the general form of Rˆn(θ) for describing these rotations, where e.g. Rˆx(π) is a rotation of π radians (180 ◦ ) about the x-axis, and Rˆy(−π/2) is a rotation of −π/2 radians (−90 ◦ ) about the y-axis. There are multiple ways of performing single-qubit rotations along ˆz. We present three such methods here. 1. Composite ˆx and ˆy rotations. Any rotation about ˆz can be composed of other rotations about ˆx and ˆy. The following is an example of this: 2. Compression of ˆz rotations. Rˆz(π) = Rˆx(π/2)Rˆy(π/2)Rˆx(−π/2) . (2.15) We can also use relations such as Eq. 2.15 to place all ˆz rotations at the end of the pulse sequence. An example of this is Rˆx(π/2)Rˆz(π/2) = Rˆx(π/2)Rˆy(π/2)Rˆx(−π/2)Rˆy(−π/2) = Rˆz(π/2)Rˆy(−π/2) . (2.16) Here the rotation about ˆx is replaced with one about ˆy, and the ˆz rotation is moved to the end. The advantage of this method is that all the ˆz rotations can be grouped into one single pulse, reducing the total number of single-qubit pulses that must be performed. This leads to a reduction of errors due to control errors, i.e. evolution under HI (Eq. 2.4) that occurs during the pulse duration tp. For this method to work, the ˆz rotations must commute with the free-evolution Hamiltonian, H0 + HI. Since both of these depend only on Z operators, this is clearly the case. 3. Implicit absorption of ˆz rotations. 52
The same effect as number 2 above is achieved if we regard ˆz rotations as merely a shifting of the reference frame by the angle θ in each Rˆz(θ) rotation. The effect of ˆz rotations is implemented by shifting the phases of each subsequent single-qubit pulse to account for them. Choosing between this method and number 2 is a practical matter; because it requires less modification to the pulse sequence, this method is generally preferred. 2.1.4 Two-qubit operations Now that we are armed with the ability to do arbitrary single-qubit rotations and with an interaction Hamiltonian (Eq. 2.4) we can construct two-qubit gates such as the controlled-not (CNOT). The CNOT, along with arbitrary single-qubit rotations, is sufficient to implement universal quantum computation; all the operations needed for quantum simulation are con- tained within this, and in practice less control is often required. In the computational basis ({|00〉 , |01〉,|10〉 , |11〉}), the CNOT has this matrix representation: ⎡ ⎤ 1 0 0 0 ⎢ ⎥ ⎢ CNOT12 = ⎢ 0 1 0 0 ⎥ ⎢ ⎣ 0 0 0 1 ⎥ . (2.17) ⎦ 0 0 1 0 Suppose there are two coupled spins. The basic pulse sequence rotates spin 2, the “target qubit,” from a ˆz eigenstate into the ˆx-ˆy plane of the Bloch sphere. Then, during a period of free evolution under H0 + HI, the direction of the rotation of this spin (in the rotating frame) depends on the state of spin 1, called the “control qubit.” After a time equal to 1/(2J12), the spin is rotated back into a ˆz eigenstate. This effectively flips (or doesn’t) spin 2 based on the state of spin 1. Specifically, the pulse sequence looks like: U ′ CNOT = Rˆx(π/2)UI(1/(2J12))Rˆy(π/2) = ⎡ ⎤ 1 0 0 0 ⎢ ⎥ ⎢ 0 i 0 0 ⎥ ⎢ ⎣ 0 0 0 −i ⎥ , ⎦ (2.18) 0 0 1 0 where UI(t) = exp(−iHIt/) is the unitary evolution under HI. Fig. 2-1 shows the opera- tion of the CNOT gate as described here. This does not exactly produce the CNOT presented in Eq. 2.17, but the gate can be exactly duplicated by applying appropriate single-qubit pulses to spins 1 and 2. Specifically, a Rˆz(−π/2) is required on qubit 2 and Rˆz(π/2) on qubit 1. 2.1.5 Refocusing We briefly present one more invaluable technique for NMR quantum control. In NMR, the ZZ interaction between spins is “always on,” in the sense that it’s a part of the Hamilto- 53
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 10 and 11: 2.4 Conclusions and further questio
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: ⎡ ⎤ a 0 0 0 ⎢ ⎥ ⎢ ρ2 =
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 and 102: 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
Page 111 and 112:
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
Page 117 and 118:
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
Page 195 and 196:
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
Page 205 and 206:
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
Page 225 and 226:
[PC04a] D. Porras and J. I. Cirac.
Page 227 and 228:
[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
Page 239 and 240:
2 crystal_shape_6uraniborg.nb Out[6
Page 241 and 242:
Appendix C How to trap ions in a cl
Page 243 and 244:
Finally, we note that the chilled w
Page 245 and 246:
C.4 Laser alignment and imaging As
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Ph.D. Thesis - Physics

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