Ph.D. Thesis - Physics

More documents

Recommendations

Info

Figure 7-12: The triple-Z wire configuration and resulting magnetic fields. The wire configuration is top left; each wire carries 1A of current in the same direction. Each other the other plots gives the magnetic field components along a given direction. In all figures, Bˆx is in blue, Bˆy is in green, and Bˆz is in red. Note that the field gradients are quite constant across the volume occupied by the ions. Nonzero fields at the trap center must be cancelled by additional pairs of Helmholtz coils to avoid unwanted Zeeman shifts, electron alignments, and even ion crystal rotation. be operated at cryogenic temperatures suggests that superconducting wires could be used, mitigating resistive heating [REL + 08]. We focus on two different wire configurations to give a flavor of the possibilities for simulated Hamiltonians. The first is a “Z” shape, the classic shape of the gradients coils used in chip-based atom traps. This shape is repeated three times, in order to further strengthen the field gradients. The second configuration is a set of three concentric square rings, that produce a different set of gradients. Results from the Z shape are given in Fig. 7-12. These are based on an elliptical trap that gives an ion height of 100 µm. We see that the gradient is of the approximate form ∂ B ∂xi = ∂Bˆz ∂x ˆz + ∂Bˆz ∂y ∂Bˆx ∂Bˆy ˆy + ∂z ˆz + ∂z ˆz. Depending on the projection of the atomic dipole, as set by the external bias fields, this yields the possibility of an Ising of Heisenberg interaction. The specific interaction induced depends upon the direction along which F acts. We now wish to calculate the actual interaction strengths and how they scale with the trap size. What do we expect? First, we recall from Ch. 5 that for constant trap depth the secular frequencies scale as 1/r0, where r0 is some characteristic length scale of the trap (usually defined as the distance from the ion to the rf electrode). At the same time, the distance between ions d is given by d 3 = e 2 c/(4πǫ0mω 2 ), as calculated by balancing the Coulomb and trapping forces. The magnetic field strengths scale as 1/r2 0 , and plugging that all into the formula for J (J ∝ F 2 /(ω4d3 )), we expect that J will scale as 1/r2 0 . 168
Figure 7-13: Scaling of the J-coupling rates for all three directions as a function of the ion height in an elliptical trap, for the three concentric square rings wire configuration. The value for each direction will only hold if the projection of the electron spin is entirely along that direction; this would give an Ising-type interaction. Numerically, this is nicely verified. We apply the above scaling law to the secular frequencies, compute the mean ion-ion distance, and calculate the J coupling as a function of h, the height of the ion above the trap surface (calculation with respect to r0 would yield the same scaling). The highest coupling rate plotted here is 700 s −1 , which will be observable if the dominant decoherence rates are significantly lower than that. By contrast, the fields due to the rings produce a field gradient at the location of the ions with the form ∂ B ∂xi ∂Bˆx ∂Bˆy ∂Bˆz = ∂x ˆx + ∂y ˆy + ∂z ˆz. We expect basically the same scaling law for the rings as for the triple-Z, due to the above arguments. However, the total interaction rates are different (and higher, overall). This is plotted in Fig. 7-15. 7.3.2 Discussion The above results are but a small sample of the interactions that may be created with magnetic field gradients. Indeed, one appealing thing about this approach is the sheer variety of forces that may be created. The source of this freedom is the fact that the wires that create the gradients are separated from the source of the trapping potentials. Thus, the tight integration of wires and trapping electrodes proposed in Ref. [CW08] is not required. However, this greater flexibility in the global potentials does not come without a cost; individually switching interactions between individual pairs of ions is not possible. The methods presented in the section will apply the same state-dependent force to every ion in a 2-D crystal in an elliptical trap. These translate into Ising or Heisenberg Hamiltonians, depending on the experimenter’s choice. There remain two major questions: 169
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 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 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 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
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
Page 141 and 142: Figure 6-9: Measurement results sho
Page 143 and 144: Figure 6-11: Bastille mounted in th
Page 145 and 146: Figure 6-13: A diagram of the setup
Page 147 and 148: Figure 6-14: A plot of the trapped
Page 149 and 150: allows us to upper-bound the amount
Page 151 and 152: Chapter 7 Quantum simulation in sur
Page 153 and 154: Figure 7-1: Left: Uraniborg 1, as r
Page 155 and 156: Figure 7-3: Calculated secular freq
Page 157 and 158: Figure 7-5: Left: Structure of 2-D
Page 159 and 160: Figure 7-7: Scaling of the order of
Page 161 and 162: involved. Because ions near the cen
Page 163 and 164: Results Before presenting the numer
Page 165 and 166: Figure 7-10: Calculation results fo
Page 167: 7.3 Magnetic gradient forces Now th
Page 171 and 172: how are single-ion operations to be
Page 173 and 174: 250 psi, which increases to around
Page 175 and 176: Figure 7-18: At the center of this
Page 177 and 178: Figure 7-19: The plastic socket tha
Page 179 and 180: Figure 7-21: The two internal lense
Page 181 and 182: Figure 7-22: Measured secular frequ
Page 183 and 184: Along ˆy, the spacing is dˆy = 28
Page 185 and 186: of O2 to 2 × 10 −12 torr. Decohe
Page 187 and 188: Part III Toward ion-ion coupling ov
Page 189 and 190: Chapter 8 Motivation for and theory
Page 191 and 192: Figure 8-1: Schematic of the experi
Page 193 and 194: are in close agreement in the case
Page 195 and 196: 8.2.3 Simulated coupling rates We n
Page 197 and 198: Johnson noise We begin our discussi
Page 199 and 200: should suffice. Note that the above
Page 201 and 202: Chapter 9 Measuring the interaction
Page 203 and 204: Figure 9-2: Left: Level diagram for
Page 205 and 206: Figure 9-5: Photograph of the micro
Page 207 and 208: A fit of the resulting curve return
Page 209 and 210: Figure 9-9: Linear fit of the ωˆy
Page 211 and 212: Figure 9-12: Heating rate as determ
Page 213 and 214: Chapter 10 Conclusions and outlook
Page 215 and 216: traps, linking ions using photons,
Page 217 and 218: Bibliography [ADM05] Paul M. Alsing
Page 219 and 220:
[CGB + 94] J. I. Cirac, L. J. Garay
Page 221 and 222:
[GME + 02] M. Greiner, O. Mandel, T
Page 223 and 224:
[LCL + 07] D. R. Leibrandt, R. J. C
Page 225 and 226:
[PC04a] D. Porras and J. I. Cirac.
Page 227 and 228:
[TBZ05] L. Tian, R. Blatt, and P. Z
Page 229 and 230:
Appendix A Matlab code for Ising mo
Page 231 and 232:
err_result = mean(theerr) 231
Page 233 and 234:
J = J0; Jt(1) = J; Bt(1) = B; for k
Page 235 and 236:
A.3 Simulation with constant force
Page 237 and 238:
Appendix B Mathematica code for ion
Page 239 and 240:
2 crystal_shape_6uraniborg.nb Out[6
Page 241 and 242:
Appendix C How to trap ions in a cl
Page 243 and 244:
Finally, we note that the chilled w
Page 245 and 246:
C.4 Laser alignment and imaging As
show all

Ph.D. Thesis - Physics

Create successful ePaper yourself

Delete template?

Save as template?