Ph.D. Thesis - Physics

More documents

Recommendations

Info

PC04b]. In both cases, the idea differs from those mentioned above, in that rather than employing a sequence of discrete control pulses, they instead propose the creation of model Hamiltonians using a limited set of continuously-varying controls. Similar ideas have been set forth for the simulation of Bose-Hubbard models, also using trapped ions or neutral atoms [GME + 02, PC04a]. In these examples, the opportunity arises to observe whether the target system behaves qualitatively like the model system being studied for similar sets of parameters. 1.3 Models of quantum simulation The quantum simulations introduced above may be divided into two distinct types, or models: in the first, of which the paper of Wu et al. [WBL02] is a fine example, a quantum simulator is prepared in some initial state and then manipulated with a set of discrete pulses, a situation akin to a digital computer. In fact, this type of simulation bears other resemblances to classical digital computation, such as the fact that error correction codes may be used [CS96, Ste96, Sho96, DS96, Got97]. We refer to this type of simulation as digital quantum simulation. The second type, which the neutral atom community has recently excelled at, involves creating an effective Hamiltonian that is controlled by continuously adjusting the relevant parameters. This type is akin to classical analog computation, and we refer to it as analog quantum simulation. This approach lends itself to more qualitative questions, such as what phase (superfluid, insulator, etc.) the particles occupy within some region of parameter space. Let us dig a bit more deeply into this distinction, because both types are of interest in this work. 1.3.1 “Digital” quantum simulation Digital quantum simulations are characterized by the use of discrete quantum gates to im- plement the desired Hamiltonian. To clarify the terminology, a quantum gate is a (usually) unitary operation that is applied to some set of qubits for a finite amount of time; any gate involving more than two qubits may be decomposed into a sequence of one- and two-qubit gates. Measurement is also considered a quantum gate, and is typically assumed to be of the strong, projective variety. The weak measurements used in nuclear magnetic resonance (NMR) are a notable exception. This approach has some very appealing advantages: for one, any region of parameter space may be probed, since the interactions are entirely engineered by control pulses from the experimenter. The other very big advantage is that quantum error correction techniques may be applied, and indeed will be necessary when the system grows to a large enough size. However, quantum error correction schemes generally require a substantial increase in the required number of control pulses, increasing the likelihood that systematic control errors will reduce the accuracy of the simulation. 26
Nevertheless, there are a number of interesting problems in quantum simulation that will require the ultimate digital quantum simulator, a universal quantum computer. A quantum computer (or simulator) is universal if it employs a set of gates that can approx- imate any unitary transform to arbitrary precision. It was shown in Ref. [FKW02] that quantum computers can efficiently simulate topological field theories. Intimately related to such theories is the Jones polynomial, which is a knot invariant of knots in R 3 which is invariant under transformations of the knot. A quantum algorithm for approximately evaluating certain instances of this polynomial is presented in Ref. [AJL08], whereas no efficient classical algorithm is known to solve the same problem. Together, these results imply the possibility of using quantum computers to model some of the most fundamental aspects of nature. 1.3.2 “Analog” quantum simulation Analog quantum simulation consists of preparing a system in some initial state (which may indeed involve discrete gates), and then applying some global effective Hamiltonian, perhaps adiabatically, that permits one to mimic the exact dynamics of the target system. The key difference from digital quantum simulation is that the control variables are varied continuously. Such approach enables, for instance, the observation of the phases that the target system exhibits in various regions of parameter space, for a given Hamiltonian. In some sense, given the Ising-type interaction between spins in NMR, that system could already be considered to be an effective Hamiltonian system. However, this doesn’t give one any control over what exactly the effective Hamiltonian is; refocusing pulses that modify this interaction fall more under the blanket of circuit model simulation, in that they are discrete control pulses. The analog approach has shown much promise for trapped ion and neutral atom systems, leading to a number of stimulating papers. Two 2004 papers, Refs. [PC04b] and [PC04a], have stimulated much research on analog quantum simulators. The former describes a method for simulating quantum spin models using trapped ions; they propose the creation of effective Ising or Heisenberg interactions between ions by using state-dependent optical forces. In the latter, the quantized motional states of trapped ions are made to simulate bosons obeying a Bose-Hubbard Hamiltonian. It is predicted that in such a system, among other phenomena, Bose-Einstein condensation of phonons could be observed. Neutral atom systems have already had some remarkable experimental success with this type of quantum simulation, in that several groups have simulated fermionic and bosonic lattice models in an optical lattice. For example, the superfluid-Mott insulator phase tran- sition has been observed in ultracold 87 Rb atoms in an optical lattice [GME + 02]. Other notable papers include the simulation of the Mott insulator phase of interacting fermions [JSG + 08] and imaging of the Mott insulator shells in a Bose gas [CMB + 06]. Remarkably, a recent analog quantum simulation settled a long-standing question regarding the phase dia- 27
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: us whether the target system obeys
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 and 168:
7.3 Magnetic gradient forces Now th
Page 169 and 170:
Figure 7-13: Scaling of the J-coupl
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?