Ph.D. Thesis - Physics

More documents

Recommendations

Info

gram of an ultracold Fermi gas with respect to temperature and spin polarization [SSSK08]. It is even possible in theory to simulate quantum magnetism in a neutral atom system by tuning the optical lattice strength [DDL03]. However, such a system has not yet been realized, in part due to the very low temperatures required to observe it [BDZ08]. Analog simulations offer distinct advantages. In the case of optical lattices, one may find that the Hamiltonian governing the particles is also an interesting Hamiltonian to simulate, e.g. a Bose-Hubbard model. This is not true in every case, and sometimes other forces are added, i.e. in Ref. [JSG + 08]. Nevertheless, analog simulations provide a route to studying many-body phenomena without requiring the degree of control available in a digital quantum simulator. The advantages of analog simulation also differ from system to system. One difference between the neutral atom and ion trap implementations of the Bose-Hubbard model is the individual control over ions that ion traps offer. For instance, in the proposal of Porras and Cirac [PC04a], it is possible to adjust the trapping potentials to create site-specific interac- tions. Although still analog, such control permits a wider variety of simulated Hamiltonians than neutral atom BEC’s. The advantage, by contrast, of the optical lattice experiments is the ability to rapidly incorporate many bosons in many (O(10 5 )) sites [GME + 02], a situation that has not yet been realized in ion traps. The analog approach also has some drawbacks. For one, the ability of an experimenter to control the Hamiltonian is generally more constrained than it is for digital simulations. For instance, in the above optical lattice experiments, both the “onsite” terms, which describe repulsion between atoms at the same site, and the “hopping” terms, which describe motion of atoms from site to site, depend on the depth of the optical lattice U. Chang- ing this depth changes the ratio of onsite to hopping interaction strengths, which is what permits observation of interesting phenomena, e.g. phase transitions. However, the acces- sible regions of parameter space are also constrained by U. Another drawback of analog quantum simulation is that quantum error correction (if needed) is not possible. Digital quantum simulations, by contrast, are seen to be both more general and more robust, but also significantly more difficult to implement. 1.3.3 Comparison of digital and analog quantum simulation We would now like to compare the digital and analog approaches for both classical and quantum simulation of quantum mechanical systems. The main question is the resource requirements, in both space and time, required to perform a given simulation. In the examples below, we will consider a simulation of the dynamics of a system of n spin-1/2 systems (qubits). In this section, we also focus entirely on the dynamics of the system, neglecting for the time being the resources needed to extract the answer. We begin with classical simulation. What resources are required to simulate n qubits on a classical computer? In the digital case, the dimension of the state vector is 2 n , which is 28
Digital Analog Classical Quantum Space: 2n Time: T22n Space: n Time: Tn2 Space: 2n Space: n Time: T Time: T Table 1.1: A comparison of the resources required to implement the dynamics of digital and analog simulation of quantum systems, using both classical and quantum simulation. We consider a system of n qubits simulated for a total time T. the spatial resource requirement. In general, the time needed to simulate the system is also exponential, since an exponentiation of a 2 2n -element matrix is required to propagate the state vector forward in time. Further, the total simulation time T adds a constant factor to the total time required. For the analog case, suppose that the implementation is done using a set of voltages which specify the real and complex parts of the state vector amplitudes to the maximum possible precision. In this case, 2 n individual voltages will be required to specify the state, meaning that the spatial resource requirement is 2 n , the same as the digital case. However, the time required to perform the evolution is specified only by the total time T. Let us now consider the quantum-mechanical case. For digital quantum simulation, the space required scales as n, since each qubit in the model system may represent one qubit in the target system (and error correction adds only a polynomial number of qubits). The time required to implement the unitary evolution is proportional to n 2 , but only for Hamiltonians that can be modularly exponentiated efficiently [NC00]. The restrictions on simulable Hamiltonians are described in detail in Ref. [Llo96]. In the analog case, n model qubits again map to n target qubits, but the total simulation time is proportional only to T, as in the classical case. We summarize these results in Table 1.1, which describes the space and time resources required for implementation of the quantum dynamics, but does not include the error in the result. Although at first glance, it may seem that quantum methods are always superior to classical, and analog methods always superior to digital, this is not the case when the effects of errors are considered, or when probabilistic approaches are considered. 1.4 Challenges for quantum simulation Quantum simulation is a tantalizing prospect, but there are good reasons why we don’t already have a large-scale quantum simulator. The three main reasons can be classified broadly as decoherence, precision limitations, and scalability. We discuss all of these in this section. 29
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: Nevertheless, there are a number of
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?