Ph.D. Thesis - Physics

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1.4.1 Decoherence Decoherence is the irreversible loss of information from a quantum system to its environ- ment. It arises from interactions between a quantum system and a “reservoir” of states too numerous to keep track of. Thus, although the full evolution of the system and reservoir together is unitary, the evolution of the system alone exhibits non-unitary dynamics. Decoherence is present in any quantum-mechanical system, but the question always is the degree to which the evolution is altered by decoherence. The main questions studied by researchers are the sources of decoherence and the methods of reducing it. Since nuclear spins and ion traps are the quantum systems used in this thesis, we discuss here the main sources of decoherence for each. The two primary types of decoherence relevant to nuclear spins are amplitude damping and dephasing. Each affects a different part of the density matrix of the quantum system. Amplitude damping affects the on-diagonal elements, or “populations,” causing population transfer from one state to another, typically from an excited state to a lower-energy (or the ground) state. However, due to the low spin polarization of a room-temperature nuclear spin system, the thermal state is a nearly equal mixture of the ground and excited spin states. In NMR, the rate at which this occurs is given by the inverse of the T1 time. Dephasing affects the off-diagonal elements, or “coherences,” effectively changing the relative phases between different terms of the quantum state, and thus destroying coherence. The rate of this process is characterized in NMR as 1/T2. The T2 time arises from a number of both macroscopic and microscopic sources. An important macroscopic source of dephasing in NMR is inhomogeneities in the static and oscillating magnetic fields used to control the nuclei. Such inhomogeneities can be corrected in one of two ways: first, by measuring and improving the field homogeneity, and second by using spin-echo techniques to correct for the inhomogeneities that remain. Still, a number of microscopic and uncontrollable processes contribute to limiting the coherence time, including the fundamental cause of the T1 time: spin transitions induced by thermal excitation of spins. Decoherence of the quantum states of trapped ions, by contrast, includes processes that affect either the internal (electronic) or external (motional) degrees of freedom. Both the electronic and motional quantum states are important for the storage and processing of quantum information. Decoherence of electronic states can be attributed to two primary sources: scattering events and control errors. For instance, the excited state lifetime in a qubit separated by optical wavelengths is fundamentally limited by spontaneous emission. For hyperfine qubits manipulated with a laser Raman transition, however, the spontaneous scattering rate defines this limit [OLJ + 95]. Some common control errors include fluctuations in the laser intensity or polarization, as well as fluctuations of ambient magnetic fields, and in practice these fluctuating controls limit the coherence times [BKRB08]. Motional states, by contrast, decohere largely due to heating caused by fluctuating potentials on the trap 30
electrodes [TKK + 99]. This heating has been observed to scale roughly as 1/d 4 , where d is the distance from the ion to the nearest electrode [DOS + 06]. Several steps have been taken to reduce these sources of decoherence in ion trap systems. Decoherence due to fluctuating magnetic fields has been suppressed by the encod- ing of qubits in a decoherence-free subspace [LOJ + 05, HSKH + 05], and also, for hyperfine qubits, by using a magnetic field-insensitive transition [LOJ + 05]. Errors due to spontaneous scattering can theoretically be eliminated by performing gates using magnetic fields and radiofrequency pulses rather than laser light [CW08, OLA + 08, JBT + 08]. Although decoherence of internal (electronic) states of the ions is of considerable importance, our efforts to develop ion trap architectures are mainly driven by the need to minimize motional state decoherence, or heating. The problem of motional decoherence has been par- tially and practically addressed by the demonstration of suppression of heating rates by several orders of magnitude through cryogenic cooling [LGA + 08, DOS + 06]. Recently, the temperature-dependence of this heating rate has been more systematically characterized [LGL + 08]. However, the sources of this noise are still not completely understood. Considerations of the effects of decoherence have a major impact on the next two limitations of quantum simulation presented. 1.4.2 Limitations to Precision In quantum simulation, one seeks to calculate some specific property of a quantum system, for instance an eigenvalue of a Hamiltonian. The precision to which a given quantity can be calculated depends on any inherent limitations in the algorithm used, as well as on factors such as imperfect controls and decoherence. Therefore, we may frame our investigations by two questions: 1. For a given quantum simulation algorithm for calculating some quantity ∆, what is, in principle, the expected error ǫ in the measurement of ∆, as a function of the space and time resources required? 2. How does ǫ change as a certain systematic or random error in the control pulses applied to the system is introduced? We begin the discussion of the first question with an example. Suppose that one wished to calculate the difference ∆ between the energy eigenvalues of two states |1〉 and |2〉. One way to do this is to prepare a superposition of the two states, then evolve under some simulated Hamiltonian for a set of discrete times tn, for N total timesteps. Measurements of a suitable operator on the system at each time will oscillate at a rate ∆. Determination of ∆ can then be done by classically Fourier-transforming the measurement results. With how much precision can ∆, in principle, be measured? Certainly, in the above scheme, the sampling rate plays a role: more points will lead to more precision. Therefore, we expect 31
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: Digital Analog Classical Quantum Sp
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
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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
Page 223 and 224:
[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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