Ph.D. Thesis - Physics

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Figure 2-1: Above is a depiction of the CNOT gate in NMR. Spin 1, the control qubit, is not shown but its state alters the precession frequency of spin 2, the target qubit. At first, qubit 2 is in the |↑〉 ˆz eigenstate. A π/2 rotation about the ˆx axis places it into the ˆx-ˆy plane. Depending on the spin state of qubit 1, and in the rotating frame, qubit 2 precesses around the ˆz axis in a different direction. The solid arrow evolution is when qubit 1 is in state |↑〉, while the dotted arrow occurs when qubit 1 is in state |↓〉. After a free precession time of t = 1/(2J), a single Rˆy(−π/2) pulse completes the operation: the spin of qubit 2 has been flipped conditioned on the spin of qubit 1. This figure is due to Vandersypen, Ref. [Van01]. nian that is not directly under the experimenter’s control. We have seen above how this interaction can be used to generate the CNOT gate. However, sometimes one wishes for some of the spins to not evolve under this Hamiltonian for some period of time. This is used by us, for instance, in Sec. 3.5. A technique known as refocusing can effectively switch off the scalar coupling between one qubit and the others during a period of free evolution. This is done by applying a π-pulse to that qubit halfway through the evolution time, thereby reversing its direction of precession. Suppose one has a three qubit system, and wishes for a time t to implement HI (Eq. 2.4) on the qubits labeled a and b, while preventing qubit c from coupling to them. One would do the following pulse sequence, where the arrows indicate time-ordered pulses: U a,b I (t) = Ua,b,c I (t/2) → R ĉ x (π) → Ua,b,c I (t/2) → R ĉ x (π) . (2.19) This fairly simple technique is of great utility to us. 2.1.6 Measurement Measurement of the NMR system is done by recording the current induced in a coil of wire surrounding the spins; this coil is the same one that transmits the rf signals to the nuclei, and thus only detects magnetizations in the ˆx or ˆy directions. The measurement simultaneously records the states of the spins contained in all the many molecules in the sample, but due to the low spin polarization at room temperature, only a the small difference in spin orientations produces the signal. The induced voltage across the rf coil can be written as V (t) = V0Tr e −iHt ρe iHt (−iXk − Yk) , (2.20) 54
where V0 is the maximum signal strength and ρ is the sample density matrix. The phase of the measurement operator can be chosen so that a given spin state produces a given spectral line shape: for instance, for the above operator −iXk − Yk, a spin along −ˆy produces a positive absorptive spectral line, while one along ˆy produces a negative absorptive line. A spin along ±ˆx produces positive or negative dispersive lines. To perform this measurement in the ˆz basis, we must first rotate all spins into the ˆx-ˆy plane, since the rf coils are only sensitive to magnetizations in this direction. Thus a Rˆx(π/2) pulse is first applied to the spins being measured. The resulting free-induction decay (FID) signal oscillates at the frequency of the qubit being measured; a Fourier transform of this signal yields a frequency spectrum, from which one can determine the spin state along ˆz of each nucleus (using the chemical shift interaction, Eq. 2.4). The FID decays exponentially at a rate T ∗ 2 , due to decoherence processes which will be discussed shortly. The exponential decay in the time domain leads to a Lorentzian lineshape in the frequency domain: V (t)e −t/T ∗ 2 ↦→ F(ω) ∝ 1 1/(2T ∗ 2 )2 1 − + (ω − ω0) 2 1/(2T ∗ 2 )2 , (2.21) + (ω − ω0) 2 where the two parts represent absorptive and dispersive lineshapes. A signal may, in general, be of only one type or a mixture of the two. The full width at half-maximum (FWHM) of the NMR peaks is then given by ∆f = 1/(2πT ∗ 2 ). Although measurements in quantum information are often assumed to be strong, pro- jective measurements, those in NMR are quite weak. The constant T ∗ 2 depends on magnetic field inhomogeneities and other interactions between spins and with the environment; it is noteworthy that the decay of the FID is dominated not by the interaction of the sample with the coil, but by decoherence processes more intrinsic to the sample. This may be contrasted with the classic “strong” measurement, in which the very act of measurement induces sufficient decoherence to immediately effect wavefunction collapse! This weakness of the measurement, combined with the ensemble nature of the experiment, is also the rea- son why both ˆx and ˆy components of the magnetization can be simultaneously measured, a feat which is forbidden (for a single quantum system) by the uncertainty principle. What is recorded is the ensemble average of the magnetizations in each direction; each of the 10 18 or so molecules gives its own answer regarding its spin state, and these are added up. The result of a measurement is a frequency-domain spectrum, with peaks located at all possible locations for a given nucleus, given the various splittings due to the chemical shifts. To summarize this section so far, we have assembled all the basic concepts and techniques for performing quantum operations in solution-state NMR. In the next section we show how these principles apply to the simulation of quantum systems. First, though, we address the sources of decoherence in NMR systems. 55
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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 and 38: qubits, and in Ref. [SGA + 05] in t
Page 39 and 40: the lattice architecture, we discov
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: The same effect as number 2 above i
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
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
Page 103 and 104: 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
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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
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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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Ph.D. Thesis - Physics

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