Ph.D. Thesis - Physics

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4.1 The ion trap system and Hamiltonian An ion trap is an electromagnetic device for confining a charged particle in space. The charged particle that concerns us is a singly-charged atomic ion, unless otherwise noted. A trapped ion, if left undisturbed, contains two separable quantum systems: a ladder of harmonic oscillator motional states, and an internal electronic state. The key insight of trapped ion quantum computation (and simulation) is that these internal and external states can be made to interact, and even become entangled, using a set of classical control pulses, which usually take the form of laser radiation (but can also be microwaves or magnetic fields). Comparing this situation to solution-state NMR, we note that the same basic ingredients are present, but in a different form. The spin states of nuclei are replaced by the electronic states of ions, while interaction between these states is mediated not by chemical bonds, but by the coupled motion of the ions. There are two main varieties of ion trap: Paul traps and Penning traps. Paul traps use oscillating radiofrequency (rf) electric fields, possibly in combination with dc electric fields, to confine ions. In these traps, one finds that time-averaging the oscillating field leads to an effective harmonic potential. Penning traps use static magnetic fields and electric fields to confine ions. The ions execute cyclotron motion around the B-field lines, and are simultaneously confined along the axis of the B-field by static voltages applied to endcap electrodes. Both have advantages and disadvantages. Penning traps necessarily have a very large Zeeman shift due to the confining magnetic fields, whereas in Paul traps the internal states are much less sensitive to the trapping fields. However, ions in Paul traps exhibit small oscillations at the rf frequency, called micromotion, that can only be removed if the ions are positioned where the confining fields vanish. Because of the independence of the internal states from the trapping fields, but for other reasons as well, Paul traps have been the trap of choice for most quantum information experiments. In this chapter, we shall focus on them exclusively. In this section we first discuss the ion trap potential and the motion of ions confined therein. We then turn to understanding laser-ion interactions to a sufficient extent that we can study the theory of quantum simulations with trapped ions, and also review the princi- ples of laser cooling of ions, a technique which will be used extensively in the experimental work of this thesis. 4.1.1 Motional states of trapped ions All Paul traps obey the following basic principle. An electric quadrupole (or “saddle- shaped”) potential is formed in space by a set of electrodes, each of which is charged to a certain voltage. In the case of static voltages, the ion would follow the field lines out of the center of the potential. In fact, it is a well-known result, commonly known as Earnshaw’s theorem, that no static electric field configuration can trap a charged particle. However, if 84
GND RF r ENDCAP 0 z 0 ENDCAP Ring Trap Schematic RING ELECTRODE Figure 4-1: A schematic of a ring Paul trap. The ion is located a distance r0 from the nearest point on the rf electrode, and a distance z0 from the nearest point on either of the dc endcap electrodes. the potential is rotated (or, more accurately, “wobbled”) at the proper frequency, then a particle can in fact be trapped. A typical “ring” Paul trap is depicted in Fig. 4-1. It consists of one ring electrode that is oscillating at an rf voltage Vrf, along with two “endcap” electrodes that are grounded. These may have a dc voltage applied to them. Near the center of the trap, the electric potential may be written as V (x, y,z) = αx 2 + βy 2 + γz 2 . (4.1) Since the area in the trap region is free of charge (aside from that of the trapped particle, which is negligible), Laplace’s equation, ∇ 2 V = 0, must be satisfied. For our potential, this means that α + β + γ = 0. For the case of the ring trap, α = β = −2γ. To trap an ion, this potential must vary in time. Typically, an oscillating potential of the form V (t) = V0 cos (Ωt + φ) (4.2) is applied. For the remainder of the thesis, we refer to V0 as the rf amplitude and Ω as the rf frequency. In general, the solutions to this equation are solutions to the Mathieu equations, which have the general form üi = [ai + 2qi cos (Ωt)] Ω2 4 ui = 0, (4.3) where the ui are the position coordinates of the ion along each direction. The parameters qi and ai obey the following equations: 85
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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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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: Chapter 4 Theory and history of qua
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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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
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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
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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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