Ph.D. Thesis - Physics

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coupling to the motional states by setting η = 0. The Hamiltonian that acts on the internal state of the ion is and the unitary transformation done on the ion-qubit is Hint = Ω σ + e −iφ + σ − e iφ , (4.17) U(t) = exp(−iHintt/) . (4.18) The rotations Rˆx(θ) and Rˆy(θ) may be performed by turning on this Hamiltonian for an appropriate period of time and setting the phase φ correctly. A rotation along ˆz may be composed of ˆx and ˆy rotations (similar to a method discussed in Sec. 2.1). We have thus shown that the interaction of a single ion with a laser may be used to perform single-qubit rotations; however, two-qubit operations are also required for the quantum simulations of interest to us. A number of references ([CZ95, MS98, LDM + 03]) detail methods for performing two-qubit gates that are sufficient for universal quantum computation. We do not go into detail on these now, since the quantum simulation schemes we study in Sec. 4.2 do not directly make use of these methods. 4.1.3 Control of the ion’s external state We now turn to laser manipulation of the external, i.e. motional states of trapped ions. The interaction Hamiltonian (Eq. 4.16) already contains this physics. It can be shown that by detuning the laser above or below the transition by an amount equal to the motional frequency ωˆz, a quantum of vibrational energy may be added to or subtracted from the ion. That is, for δl = ωˆz, ∆n = 1, and for δl = −ωˆz, ∆n = −1. These transitions are called the blue sideband (∆n = 1) and red sideband (∆n = −1). Repeated cycles of the latter process result in sideband cooling, which has been demonstrated by several ion trap groups, e.g. in Ref. [DBIW89]. Sideband cooling requires that the linewidth of the laser Γl is much less than ωˆz, for the somewhat that for Γl > ωˆz the ∆n = 1 and ∆n = −1 transitions are simultaneously addressed. The other case, Γl > ωˆz, can still be very useful for laser cooling, as we will show below. This process is called Doppler cooling. It was first observed in 1978 [WDW78], and formed the basis of the Nobel prize-winning research in laser cooling of neutral atomic ensembles. Because of the prevalence of Doppler cooling over sideband cooling in this thesis, we will focus on that from this point on. However, it is important to know that ground state cooling is possible, because many quantum information protocols rely on it to initialize the system. We will treat the problem of a single two-level atom interacting with a monochromatic radiation field (laser) that is detuned by δl from the atomic transition frequency ω0, as before. The ion is also characterized by “natural” linewidth Γ, which is the rate of spontaneous 90
emission from the excited state |↓〉. This quantity is given by Γ = e2cω 3 0 |〈↓|r |↑〉| 3πǫ0c3 . (4.19) For the types of atoms discussed in this thesis, Γ is on the order of 2π· 20 MHz. The operator ecr is the (vector) dipole operator. The principle of Doppler cooling is as follows: an atom (or ion) excited from |↑〉 to |↓〉 absorbs an amount of momentum from the laser equal to k, where k is the laser wavenumber. However, this energy is then spontaneously emitted in a random direction. Doppler cooling works by red-detuning the laser (δl < 0) so that the atom is more likely to absorb light when it is moving toward the laser than when it is moving away. This, combined with spontaneous emission, serves to cool the atom. When the atom is moving away from the laser, it is further off-resonance, and is not as affected by the beam. For Doppler cooling neutral atoms along a given direction, a total of six lasers are needed (three orthogonal pairs of counterpropagating beams). This is known as the optical molasses technique. For a trapped ion, the situation is better, because an ion is already bound in space by the trap. If the cooling laser has a component along ˆx, ˆy, and ˆz, then the back-and-forth motion of the ion allows it to be cooled in all three directions. Let us make the above description more quantitative. A result we shall need is the steady-state solution to the Optical Bloch equations that describes the average steady-state population p↓ of the excited atomic state. It reads as follows: p↓ = I/I0 1 + I/I0 + 2δv Γ , (4.20) where I0 = ω 3 0 Γ/(6c2 ) is the saturation intensity of the transition. The quantity δv is the detuning, including the Doppler shift due to the atom’s velocity: δv = δl − k ·v. The cooling force on the atom due to the laser light is calculated in the following way: the force is equal to the momentum per photon times the rate of photon emission, which is the spontaneous emission rate times the probability that the atom is in state |↓〉. Altogether, the light force is F = k Γ 2 p↓. (4.21) When the ion has been cooled to a sufficiently low velocity, one can approximate Eq. 4.21 as being linear in v: this leads to an expression for the cooling rate: with the constant α given by ˙Ecool = Fv = αv 2 , (4.22) 91
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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
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 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
Page 89: angular momentum along the quantiza
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
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
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
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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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