Ph.D. Thesis - Physics

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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
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Figure 3-6: Frequency-domain spectr
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used to extract the answer is irrel
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Chapter 4 Theory and history of qua
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GND RF r ENDCAP 0 z 0 ENDCAP Ring T
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where Q is the charge of the trappe
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angular momentum along the quantiza
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emission from the excited state |
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selection rule ∆m = ±1 is applie
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Figure 4-3: Schematic diagram of sp
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to address an ion in state |↑〉
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the vibrational temperature is low.
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to the design of the trap itself. M
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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
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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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Ph.D. Thesis - Physics

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