Ph.D. Thesis - Physics

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Figure 7-11: Calculation results for a simulated Ising model for two ion-qubits with a timedependent force. Left: The Aµ = 0 (top) and Aµ = 0.1 (center) plots show significant deviations in the measured expectation value as the state-dependent force becomes large late in the simulation. Right: The error in the simulation is virtually removed by multiplying J for the Aµ = 0.1 case by the ratio of average J couplings for the Aµ = 0 and Aµ = 0.1 cases. then F = 0 for some time, and then F = 0 again; a nonlinear time dependence of F; and a nonlinear spatial gradient. With respect to the first, we note that our current simulations already treat a hard pulse. They begin with a random state (that could well be considered to be the result of evolution with F = 0), and then evolve under the Hamiltonian with F = 0. After this, F could be set again to zero and the states freely evolve until measured. In the second case, we note that any smooth and continuous force F may be approximated by a number of linear segments; for each segment then, an appropriate effective J could be applied. For the third case, we again invoke the idea that for small displacements, any force function may be approximated as a series of linear gradients, and the appropriate Jav calculated and applied. Therefore, it appears that for two ions, the effects of micromotion may be nulled for a great number of cases, provided sufficient control is available. In this section, we have focused on the effect of micromotion on the spin-spin evolution under which the internal states of ions evolve. However, this is not the only effect of micromotion. The most notable other effect is the broadening of the spectral lines through the Doppler effect, which increases the Doppler cooling limit. Therefore, even when effective controls are present to implement the correct Jav values, it is still desirable to minimize the micromotion amplitudes. One way in which this can be done is by reducing the overall dimensions of the trap; recalling the relation A = 1 2q∆x, the micromotion amplitude A may be reduced by keeping q constant and decreasing the ions’ displacement from the rf null ∆x. In contrast to methods based on 3-D linear ion traps, the surface electrode trap is amenable to microfabrication. 166
7.3 Magnetic gradient forces Now that we have seen that in many cases quantum simulations may be performed in an elliptical trap (or other trap with nonzero micromotion), we are ready to discuss the actual source of the state-dependent forces F. Optical forces have been discussed at length in Ref. [PC04b], and in the thesis of Ziliang Lin [Lin08]. We focus in this section, rather, on fields of a magnetic origin. This has been treated in Ref. [CW08], but for the case of an array of microtraps, similar to the trap design studied in Ch. 5. The weakness of interactions between ions contained in different trapping regions motivates our work here on ions in the same trap. Two questions drive this work: 1. What types of forces may be created using wires in the ground plane of the elliptical trap? 2. How does the magnitude of this force and the magnitude of the J coupling rate scale with the trap size? A state-dependent force based on magnetic fields requires a field gradient in space, giving rise to a force F = −∇(m · B), where m is the magnetic moment of the atom. For the present work, we will consider the Zeeman-split sublevels of the ground S state in a 40 Ca + or 88 Sr + -like ion. The absolute value of the magnetic moment is then m = gJµBmJ, where µB is the Bohr magneton and mJ is the magnetic quantum number for the projection on the ˆz axis of the total angular momentum J. This choice is made to facilitate straightforward estimates for the types of ions discussed in this thesis, and indeed, coherence times of several seconds have been observed for such qubits encoded in decoherence-free subspaces [HSKH + 05]. However, hyperfine levels may prove to be a better choice because of their excellent coherence times even without such encoding. The state-dependent force thus depends on the alignment of the atom’s magnetic moment in space, which follows the orientation of the local magnetic field. This force, for example along direction ˆy, is given by Fˆy = gJµB mˆx ∂Bˆx ∂y ∂Bˆy + mˆy + mˆz ∂y 7.3.1 Calculation of the gradients and interaction strengths ∂Bˆz . (7.7) ∂y The calculation of the fields and field gradients can be done by direct numerical integration of the applied surface currents. Methods used are similar to those employed by Wang et al. [WLG + 09] in their design of magnetic gradients for individual ion addressing. We assume that the wires are infinitesimally narrow; this becomes less accurate as the trap scale is decreased, and more sophisticated methods must be employed. Also, we limit the current through a given wire to 1 A, comparable to the maximal currents employed in neutral atom traps [HHHR01]. The fact that eventually, to reduce heating rates, these traps will need to 167
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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
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⎡ ⎤ a 0 0 0 ⎢ ⎥ ⎢ ρ2 =
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The same effect as number 2 above i
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where V0 is the maximum signal stre
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actions that occur between nuclei i
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Figure 2-2: The 2,3-dibromothiophen
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detail how this was done with a sim
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and thus requires invocation of the
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Renormalization Group (DMRG), with
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3.3 The bounds on precision In this
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Figure 3-1: A schematic of a generi
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Figure 3-3: The 11.7 T, 500 MHz sup
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. Figure 3-4: The above probe is a
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Figure 3-5: The CHFBr2 molecule. Al
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Figure 3-6: Frequency-domain spectr
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used to extract the answer is irrel
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Chapter 4 Theory and history of qua
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GND RF r ENDCAP 0 z 0 ENDCAP Ring T
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where Q is the charge of the trappe
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angular momentum along the quantiza
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emission from the excited state |
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selection rule ∆m = ±1 is applie
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Figure 4-3: Schematic diagram of sp
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to address an ion in state |↑〉
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the vibrational temperature is low.
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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
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Figure 5-4: The vacuum chamber for
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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
Page 153 and 154: Figure 7-1: Left: Uraniborg 1, as r
Page 155 and 156: Figure 7-3: Calculated secular freq
Page 157 and 158: Figure 7-5: Left: Structure of 2-D
Page 159 and 160: Figure 7-7: Scaling of the order of
Page 161 and 162: involved. Because ions near the cen
Page 163 and 164: Results Before presenting the numer
Page 165: Figure 7-10: Calculation results fo
Page 169 and 170: Figure 7-13: Scaling of the J-coupl
Page 171 and 172: how are single-ion operations to be
Page 173 and 174: 250 psi, which increases to around
Page 175 and 176: Figure 7-18: At the center of this
Page 177 and 178: Figure 7-19: The plastic socket tha
Page 179 and 180: Figure 7-21: The two internal lense
Page 181 and 182: Figure 7-22: Measured secular frequ
Page 183 and 184: Along ˆy, the spacing is dˆy = 28
Page 185 and 186: of O2 to 2 × 10 −12 torr. Decohe
Page 187 and 188: Part III Toward ion-ion coupling ov
Page 189 and 190: Chapter 8 Motivation for and theory
Page 191 and 192: Figure 8-1: Schematic of the experi
Page 193 and 194: are in close agreement in the case
Page 195 and 196: 8.2.3 Simulated coupling rates We n
Page 197 and 198: Johnson noise We begin our discussi
Page 199 and 200: should suffice. Note that the above
Page 201 and 202: Chapter 9 Measuring the interaction
Page 203 and 204: Figure 9-2: Left: Level diagram for
Page 205 and 206: Figure 9-5: Photograph of the micro
Page 207 and 208: A fit of the resulting curve return
Page 209 and 210: Figure 9-9: Linear fit of the ωˆy
Page 211 and 212: Figure 9-12: Heating rate as determ
Page 213 and 214: Chapter 10 Conclusions and outlook
Page 215 and 216: 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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