Ph.D. Thesis - Physics

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digits of precision in the final result. We also note that the inefficiency increases if error correction techniques need to be used, such that the time required scales as O(1/ǫ r ), where r ≥ 2. Errors in the final answer are also induced by systematic control errors; we find that evolution during single-qubit pulses due to the “always-on” NMR interaction Hamiltonian is the main source of these errors. The time resources needed for error correction and to compensate control errors may be compared to Table 1.2, which presents the required resources accounting only for the error in the final measurement estimates. Our results do not rule out the usefulness of digital quantum simulations. They merely point out that quantum simulation outperforms classical only for specific problems and numbers of qubits. It remains exponentially more efficient than classical simulation for quantum systems for which no good approximate classical simulation is known, as long as only fixed precision is required. Part II In the second part, we ask: how can one build a scalable two-dimensional array of trapped ions that is suitable for analog quantum simulation? We will be focused, in particular, on the goal of analog simulation of quantum spin models, particularly spin frustration. In pursuing the answer to this question, we design, test, and evaluate two types of ion trap that may be suitable for this purpose, a lattice ion trap and an elliptical surface-electrode ion trap. It is essential not only to design, model, build, and test these traps, but also to examine how analog quantum simulation might actually be performed in such traps. It will be important to estimate the expected simulated coupling rates in each type of trap, including all rf and dc fields that act on the ion. This requires us to ask, for example, how the motional frequencies and simulated coupling rates scale with the size of the trap, how the potentials that produce an effective Hamiltonian might be applied, and how rf-induced micromotion might affect the fidelity of analog quantum simulations. In Chapter 4, we begin with an introduction to quantum simulation with trapped ions, including ion trap Hamiltonians, control techniques, and proposals for analog quantum simulation. We particularly focus on the proposal of Porras and Cirac for the study of 2-D spin models [PC04b]. The goal of building a 2-D simulator of antiferromagnetic spin lattices, in which phenomena such as spin frustration could be observed, is the prime motivation for the rest of the work in this part. In Chapter 5, we examine the lattice ion trap architecture, in which ions are confined in a 2-D array of individual potential wells, enabling the trapping of ions in virtually any configuration (since sites can be loaded selectively) with a well-defined spacing between ions. Such a trap could provide the stable array of trapped ions that is required for scalable analog quantum simulation. We experimentally test certain predictions about the trapping potentials, most importantly the motional frequencies of the ions, and find that the traps agree well with our theoretical predictions. However, despite the apparent advantages of 38
the lattice architecture, we discover a serious flaw in this idea: the physics of ion traps requires an increase in the motional frequencies of the ions as the trap scale, and with it the ion-ion spacing, decreases. This means that the coupling between ions a distance d apart is actually much less than it would be if the ions were in the same trapping region, with the same d. We find that this lattice trap design is not promising for simulation of 2-D spin models. Chapter 6 introduces surface-electrode ion traps, in which all trapping electrodes reside in a single plane. The specific type of trap used in this chapter is based on printed circuit board technology, which is useful for prototyping a wide variety of ion traps, including those that could be used for analog quantum simulation. Surface-electrode traps have some particular challenges, though: they are shallower than comparable three-dimensional traps, and are more sensitive to the buildup of stray charges on the dielectrics that isolate the electrodes from one another. How to efficiently load such traps with minimal accumulation of stray charge is thus an important technical question. In this chapter, we compare and contrast the advantages of electron-gun loading in the presence of a buffer gas and laser ablation loading of these ion traps. These are compared to the photoionization technique used in Chs. 5 and 7. We find that among the three loading methods, photoionization is superior to electron impact and ablation loading for our purposes, in that it succeeds in loading shallow traps comparably to ablation loading, but without as much stray charge buildup on nearby dielectrics. However, we point out certain situations in which ablation or e-gun loading may be more useful. Most importantly, we demonstrate the PCB ion trap technology that we use for prototyping new designs for analog ion trap quantum simulators. In Chapter 7, we turn to an alternative method for realizing a 2-D array of ions for analog quantum simulation, a surface-electrode elliptical ion trap, in which ions in a single trap region align into a 2-D crystal through mutual Coulomb repulsion. Although the coupling rates are much higher than in a lattice trap with the same ion-ion spacing, this scheme suffers from two limitations: the structure of ion crystals leads to a non-uniform spacing between neighboring ions, and rf-driven micromotion is present that cannot be removed. We present calculations of the pertinent properties of the trap, including motional frequencies and ion crystal structure, and test them with experimental measurements. In an effort to reduce motional state decoherence, these traps are tested in a cryogenic system. We also demonstrate theoretically how quantum simulations may be done even in the presence of micromotion, showing that micromotion leads to a systematic shift in the simulated coupling rate between each pair of ions. Finally, we discuss the possibility of producing this force with a magnetic field gradient rather than an optical force, and discuss scaling down of the system to increase the interaction rates. Our results indicate that lattice-style ion traps suffer from poor simulated interaction rates, but that elliptical traps offer potentially much higher interaction rates, at the cost of unavoidable micromotion. However, we show that certain two-body interactions may be 39
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: qubits, and in Ref. [SGA + 05] in t
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 and 88: 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
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
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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
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
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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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Ph.D. Thesis - Physics

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