Ph.D. Thesis - Physics

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Contents 1 Introduction 21 1.1 Quantum information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.2 What is quantum simulation? . . . . . . . . . . . . . . . . . . . . . . . . . . 24 1.3 Models of quantum simulation . . . . . . . . . . . . . . . . . . . . . . . . . 26 1.3.1 “Digital” quantum simulation . . . . . . . . . . . . . . . . . . . . . . 26 1.3.2 “Analog” quantum simulation . . . . . . . . . . . . . . . . . . . . . . 27 1.3.3 Comparison of digital and analog quantum simulation . . . . . . . . 28 1.4 Challenges for quantum simulation . . . . . . . . . . . . . . . . . . . . . . . 29 1.4.1 Decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.4.2 Limitations to Precision . . . . . . . . . . . . . . . . . . . . . . . . . 31 1.4.3 Scalability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.5 Main results and organization of this thesis . . . . . . . . . . . . . . . . . . 37 1.6 Contributions to this work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 1.6.1 Personal contributions . . . . . . . . . . . . . . . . . . . . . . . . . . 41 1.6.2 Contributions of coworkers . . . . . . . . . . . . . . . . . . . . . . . 42 1.6.3 Publications included in this thesis . . . . . . . . . . . . . . . . . . . 42 I Digital quantum simulation with nuclear spins 45 2 Quantum simulation using nuclear magnetic resonance 47 2.1 Hamiltonian and control techniques . . . . . . . . . . . . . . . . . . . . . . . 48 2.1.1 The static Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.1.2 Effective pure states . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 2.1.3 Single-qubit operations . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.1.4 Two-qubit operations . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.1.5 Refocusing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.1.6 Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 2.1.7 Decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 2.2 Quantum simulation using NMR: prior art . . . . . . . . . . . . . . . . . . . 57 2.3 The question of precision . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 9
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 11 and 12: 5.5 Measuring interactions between
Page 13: C How to trap ions in a closed-cycl
Page 16 and 17: 5-15 Comparison of motional couplin
Page 19: List of Tables 1.1 A comparison of
Page 22 and 23: 1996, right on the heels of Peter S
Page 24 and 25: state [BW92]. Quantum communication
Page 26 and 27: PC04b]. In both cases, the idea dif
Page 28 and 29: gram of an ultracold Fermi gas with
Page 30 and 31: 1.4.1 Decoherence Decoherence is th
Page 32 and 33: that the error ǫ scales as 1/N. No
Page 34 and 35: To conclude our discussion of preci
Page 36 and 37: Proposals for scaling up ion trap q
Page 38 and 39: digits of precision in the final re
Page 40 and 41: implemented with a precision that s
Page 42 and 43: 1.6.2 Contributions of coworkers Th
Page 45: Part I Digital quantum simulation w
Page 48 and 49: precision of a quantum simulation i
Page 50 and 51: for B0 strengths on the order of 10
Page 52 and 53: Hrf(t) = −ω1 (cos (ωt + φ) X/2
Page 54 and 55: Figure 2-1: Above is a depiction of
Page 56 and 57: 2.1.7 Decoherence Decoherence, broa
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Simulation of a truncated harmonic
Page 60 and 61:
Figure 2-3: The four traces here ar
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Chapter 3 Quantum simulation of the
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superconductors exhibit the Meissne
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up the evolution a bit, it is possi
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which scales as n 4 , which is prov
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Figure 3-2: A typical sample tube f
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After loading these settings, the h
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pulses would harm the fidelity of t
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Model ∆/Hz Method ∆exp/2π·Hz
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Digital Analog Classical Quantum Sp
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Part II Two-dimensional ion arrays
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4.1 The ion trap system and Hamilto
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−2ecV qi = ζi mr2 0Ω2 4ecU ai
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Figure 4-2: Generic level structure
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coupling to the motional states by
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α = 2k 2 4Iδ I0Γ 1 + (2δ/Γ) 2
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increase the dephasing time, using
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4.2.1 The Ising and Heisenberg mode
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Figure 4-4: Schematic of the action
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problems, the others have a strong
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of the scale being tested, and trap
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104
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macroscopic ions across lattice sit
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the following quasi-static pseudopo
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and is computed, in practice, using
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Spherical octagon, feedthroughs, an
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5.3.2 Lasers and imaging Two lasers
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Figure 5-6: Top-view schematic of t
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Figure 5-8: (a) A cloud of ions (ci
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COMPRESSED AIR 00000 11111 MICROSPH
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Figure 5-13: ωˆz vs. Ω for an i
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VCoup = e2 c 8πǫ0d 3x1x2, (5.6) w
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Figure 5-16: Simulated J-coupling r
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principal method of measuring the t
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other techniques, including electro
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[CBB + 05]. Surface-electrode traps
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in trap depth, however, comes decom
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Figure 6-4: Calculated values of tr
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Figure 6-7: Photograph of the exper
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Our experimental results are presen
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Figure 6-10: Photograph of the surf
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Figure 6-12: Measured pickup from t
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The base pressure of the vacuum cha
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Figure 6-15: A plot of the trapped
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150
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the rf null in such a trap exists o
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Figure 7-2: Calculated q parameters
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Figure 7-4: Scaling of the three se
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Figure 7-6: Left: 2-D projection of
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Figure 7-8: Crystal structures for
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HTI = − i,j=i+1 Ji,jZiZj + BXi
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Figure 7-9: Calculation results for
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Figure 7-11: Calculation results fo
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Figure 7-12: The triple-Z wire conf
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Figure 7-14: The concentric rings w
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Figure 7-16: Photograph of the expa
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Figure 7-17: Photograph of the exte
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choices and heat sinking. The wire
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Figure 7-20: Left: Photograph of Ur
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Electronics The electronics for thi
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Figure 7-24: Images of ion crystals
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A major contrast between these appr
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trap depth and to keep ions in the
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188
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and calculate the relevant coupling
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espect to ground is Figure 8-2: Dia
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Ei = V 2H (H 2 − h 2 i ) ln 2H−
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Figure 8-3: Equivalent circuit of t
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in Ref. [LGA + 08] in a 75 µm trap
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nearly -3.5. We wish to undertake a
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Figure 9-1: Photograph of the vacuu
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Figure 9-3: Photograph of the 397 n
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Figure 9-6: Photograph of the fork
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Figure 9-7: Measured dc vertical co
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Figure 9-10: Example Doppler recool
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systematic change in the heating ra
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study should be undertaken of the e
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216
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[BCS57] J. Bardeen, L. N. Cooper, a
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[DLC + 09b] N. Daniilidis, T. Lee,
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[HSKH + 05] H. Häffner, F. Schmidt
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[MGW + 03] O. Mandel, M. Greiner, A
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[SCR + 06] S. Seidelin, J. Chiaveri
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[WD75] D. J. Wineland and H. G. Deh
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init1 = amps/norm(amps); init1 = in
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A.2 Simulation with a linear force
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This is the script that calls the a
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plot(t*J,expsz1) axis([min(t*J),max
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In[1]:= File: crystal_shape_6urani
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Out[18]= 0.00685854476560407 In[19]
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1. Open the safety valve on the DMX
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244
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2. Check that 422 and 1092 are roug
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