Ph.D. Thesis - Physics

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Proposals for scaling up ion trap quantum simulators Quantum control of up to eight ions in linear Paul traps has now been demonstrated [HHR + 05, LKS + 05]. The challenge in controlling that number and larger numbers has many sources. For one, the number of motional normal modes that must be effectively cooled depends linearly on the number of ions in the trap. This is a difficulty mainly for schemes that require ground state cooling. Also, the ion-ion distance, and therefore coupling rate, is not constant in a chain of more than three ions. These issues encourage one to come up with ways of networking ions that are stored in different traps, or finding single-trap designs in which quantum simulations may be implemented. We present below some of the proposals for scaling up ion trap quantum simulators. • Store a small number of ions in each of a large number of separate Paul traps, Move ions from trap to trap to allow them to interact [KMW02]. The traps for shuttling ions require microfabrication, mainly because dc control electrodes need to be close to the ions to perform shuttling. There are two broad classes of microfabricated traps, 3-D and 2-D. 3-D traps have electrodes in more than one plane, allowing for a deeper trap (all else being equal) [MHS + 04, SHO + 06, BWG + 05]. 2-D traps, also called “surface-electrode” traps, have all electrodes fabricated in a single plane. These are shallower, but simpler to fabricate [CBB + 05, SCR + 06]. The Wineland group at NIST in Boulder uses a segmented trap for their quantum operations, and already incorporates ion movement into their protocols, e.g. in [BCS + 04]. Ion movement has also been demonstrated in a T-junction [HOS + 06], and in a wider variety of planar geometries, albeit with charged microspheres rather than atomic ions [PLB + 06]. • Connect ions using photons. Some encouraging early progress has also been made in networking ions using photons. For instance, in Ref. [MMO + 07], entanglement of ions at a distance was demonstrated. This approach has the advantage that the motional states of the ions do not have a role in storing or processing quantum information, implying that the motional state of the ion can be neglected, as long as the ions are confined to a region of space much smaller than the photon wavelength (Lamb-Dicke limit). Accordingly, the cooling requirements are less stringent than those for other quantum operations, such as the Cirac-Zoller gate [CZ95]. • Connect ions using wires. The idea is that the image charges due to the oscillations of one ion are transmitted over a conductive wire to another ion. In this scheme, the ion-ion coupling arises from the motional state of the ions, even though the ions are located in different traps. This was discussed in Ref. [TBZ05] in the context of connecting ions to superconducting 36
qubits, and in Ref. [SGA + 05] in the context of connecting individual ions in Penning traps. Theoretical calculations of this situation are presented in Ref. [HW90]. To date, this approach has been explored less than ion movement or photonic coupling. However, it may allow for switchable interactions between ions, as with photonic coupling, but with a simpler experimental setup. The coupling wires could, perhaps, be integrated into the trap structure itself. • Do quantum operations on a large planar crystal of trapped ions. This has been discussed theoretically in Ref. [PC06], and may be used to implement in two dimensions the proposals of Refs. [PC04b, PC04a]. Despite the need for complex networking of trapped ions for digital quantum simulation, such simpler structures may be appropriate for certain analog simulation protocols, or for “one-way” quantum computation [RB01]. For example, a quantum simulation of 2-D spin models [PC04b] could be done using such an architecture. 1.5 Main results and organization of this thesis In this thesis, we explore experimentally and theoretically the three challenges discussed above: decoherence, precision limitations, and scalability. The thesis is divided into three parts. Part I In this part we ask the question: with what degree of precision can one (in principle, and in practice) calculate the eigenvalues of a Hamiltonian using a quantum simulator? We examine the theoretical bound on the precision, and also study the effect of control errors on the fidelity of the simulation. To this end, we report on the quantum simulation of the BCS pairing Hamiltonian using a three-qubit nuclear spin system. Although a small and non-scalable system, NMR offers the possibility to illustrate key features of quantum simulation that should be applicable also to much larger systems. Our primary question is the amount of precision that can be obtained using digital quantum simulation, both in general, and in the specific case of an NMR implementation of the digital quantum simulation presented in Ref. [WBL02]. Chapter 2 explains how NMR quantum simulation works, starting with the NMR Hamiltonian and continuing with the state of the art prior to our work. We discuss the limitations to the precision of a simple NMR quantum simulation reviewed within the chapter. In Chapter 3, we present our implementation of the quantum simulation, including the algorithm and specific parameters used, experimental apparatus, results, and analysis. We find that digital quantum simulations are inefficient with respect to the number of 37
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: 2. Initialization to a simple fiduc
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
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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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