Ph.D. Thesis - Physics

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A major contrast between these approaches is that the magnetic gradient approach is well- suited to applying global effective Hamiltonians to the system, while the optical forces may be applied either globally or pairwise between ions. Regardless of the method used, we would like to remind the reader that the simulated coupling rates in the elliptical trap are of the same order of magnitude that they would be in a “linear” ion trap (the applied force F and ion-ion spacing d being the same for both). In Sec. 5.6, we compared the interaction rates between lattice traps and traps in which ions are confined in the same potential well. Both the motional coupling rates and simulated coupling rates are therefore as high as they would be in a linear ion trap, except that they can act along multiple directions. Despite high interaction rates, the elliptical trap has certain control errors that do not occur in lattice-style traps, most especially those due to micromotion. The unequal micromotion amplitudes across the crystal mean that the effective (time-averaged) J-coupling varies across the ion crystal. If ions on the periphery of the crystal are used, such that the ion-ion distance varies from site to site, this adds an additional control error that ex- acerbates the one due to micromotion. Although it is arguably easier to apply magnetic gradients to create a global force, and although this approach removes errors due to spontaneous scattering, pairwise interactions with an optical force have the potential to correct the systematic errors. Adjusting the controls in such a manner is tractable, since the ion positions and relative micromotion amplitudes are efficient to calculate. The limited system size of about 100 ions further limits the computational resources required to do this. At the same time, doing such implementations in practice will require excellent control (with precision of O(∼ 1−10µm)) of the pushing laser position. Similar control would be required to optically perform single-qubit operations in such a trap. In summary, the question of magnetic vs. optical forces presents a tradeoff between control and simplicity. The point remains that there is no fundamental limitation to doing high-fidelity quantum operations in an elliptical trap. We note further that the results of this chapter regarding control errors due to micromotion apply generally to 2-D ion crystals in Paul traps, and are not limited to the elliptical traps that were the focus of this chapter. 7.6.3 Decoherence rates Although not the primary subject of this chapter, decoherence rates will ultimately deter- mine the feasibility of quantum simulation in elliptical traps (or other traps that generate 2-D ion arrays). Given the cryogenic temperature and relatively large (mm-scale) size of the Uraniborg traps, it is possible that the motional heating rate could be well below the interaction rate J, which may be on the order of kHz. Heating can also occur due to back- ground gas collisions, but here the cryostat provides an excellent vacuum environment, on par with or superior to the best room-temperature UHV systems. Since conventional ion- ization gauges do not function at cryogenic temperatures, other methods must be used to measure the pressure. Ref. [ASA + 09] uses ion lifetimes to upper-bound the partial pressure 184
of O2 to 2 × 10 −12 torr. Decoherence of the internal states depends on many factors, including fluctuations of the parameters (amplitude, phase, etc.) that describe the control pulses, fluctuations in ambient magnetic fields, and processes intrinsic to the atom such as spontaneous emission. The last of these can result in decoherence even for qubits that have very small spontaneous emission rates, such as hyperfine qubits, since spontaneous scattering during the Raman pulses that are used can decohere the states. Although we don’t specifically treat these decoherence processes in this thesis, we may legitimately hope that internal state coherence times of several seconds may be obtained, in the event that hyperfine states are used. 7.7 Conclusions and future work In this chapter we have explored the possibility of using surface-electrode elliptical ion traps for analog quantum simulation, particularly simulation of quantum spin models. We have presented calculations of the structure of 2-D and approximately 2-D crystals within such a trap. These predictions were confirmed experimentally for a small number of ions. We have also studied, in theory, how micromotion affects the fidelity of a quantum simulation, and have seen that studies of quantum spin phases are probably possible even in the presence of micromotion. In addition, magnetic field gradient coils embedded in the ground electrode of the elliptical trap might prove an excellent way to produce global state-dependent forces for implementing a variety of spin models. On the theoretical side, this work is a starting point. For instance, other types of quantum simulations to which ions may be well-suited have not been considered. It would be an interesting problem to study the possibility of observing Bose-Hubbard physics in this system. Furthermore, although we have calculated the effects of micromotion for small numbers of ions, the work should be extended to larger numbers. Small errors in the pairwise interactions will propagate across the system as global correlations are created, and it will be an interesting (but computationally intensive) task to examine how the global fidelity of the simulation is altered. We note, however, that such errors may be corrected for, provided adequate controls are available. Experimentally, a natural first step is attempting to form larger ion crystals in the elliptical trap. Although the source of the relatively low lifetimes of crystals is unknown, the heating rates in this system should be measured. If anomalously high heating rates are a problem, even at 4 K, materials other than copper such as silver or gold, which exhibit lower heating rates, may be used for trap fabrication. Naturally, we would someday like to see experimentally if methods based on magnetic field gradients are viable for this purpose, and whether the effects of micromotion for small ion numbers are as we predict. In all, the elliptical trap appears to be a much more promising system than arrays of microtraps for analog quantum simulations with a few tens of ions. The ability to maintain 185
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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
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Figure 5-5: Left: Level diagram for
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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
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 and 166: Figure 7-10: Calculation results fo
Page 167 and 168: 7.3 Magnetic gradient forces Now th
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: Along ˆy, the spacing is dˆy = 28
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,
Page 217 and 218: Bibliography [ADM05] Paul M. Alsing
Page 219 and 220: [CGB + 94] J. I. Cirac, L. J. Garay
Page 221 and 222: [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
Page 229 and 230: Appendix A Matlab code for Ising mo
Page 231 and 232: err_result = mean(theerr) 231
Page 233 and 234: 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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